LANDAU
LIFSHITZ
Quantum Mechanics
Non-relativistic Theory
Second edition, revised and enlarged
Course of Theoretical Physics
Volume 3
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Pergamon
D. Landau and E. M. Lifshitz
Institute of Physical Problems USSR
Academy of Sciences
Pergamon Press
QUANTUM MECHANICS
Non-relativistic Theory
L D. LANDAU and E. M. UFSHITZ
This second edition, published to meet
the consistent demand for this important and
informative book, has been considerably revised
and enlarged, the basic plan and style of the
first edition, however, being retained. The
volume gives a comprehensive treatment of
non-relativistic quantum mechanics, and an
introduction to its application to atomic and
molecular phenomena. Thetopics dealtwith
include the basic concepts, Schrodinger's
equation, angular momentum, motion in
centrally symmetric fields perturbation theory,
the quasi-classical case, spin, identity of
particles, atoms, diatomic and polyatomic
molecules, the theory of symmetry, elastic and
inelastic collisions, and motion in a magnetic
field. In this second edition, extensive changes
have been made in the sections dealing with the
theory of the addition of angular momenta and
with collision theory and a new chapter on
nuclear structure has been added. The discussion
is intended to display the physical significance
of the theory, and to be complete and
self-contained.
As with other volumes in this series, a list of
which —together with statements aboutthem by
knowledgeable and impartial reviewers — will
be found on the back cover of this jacket, this
book has been written to a level that will prove
invaluable to those carrying out undergraduate
and post-graduate study.
2217
COURSE OF THEORETICAL PHYSICS
Volume 3
QUANTUM MECHANICS
Non-relativistic Theory
Second edition, revised and enlarged
OTHER TITLES IN THE SERIES
Vol.1. MECHANICS
Vol. 2. THE CLASSICAL THEORY OF FIELDS
Vol.3. QUANTUM MECHANICS— Non-Relativistic Theory
Vol. 4. RELATIVISTIC PHYSICS
Vol. 5. STATISTICAL PHYSICS
Vol. 6. FLUID MECHANICS
Vol. 7. THEORY OF ELASTICITY
Vol. 8. ELECTRODYNAMICS OF CONTINUOUS MEDIA
Vol. 9. PHYSICAL KINETICS
QUANTUM MECHANICS
NON-RELATIVISTIC THEORY
by
L. D. LANDAU and E. M. LIFSHITZ
INSTITUTE OF PHYSICAL PROBLEMS, U.S.S.R. ACADEMY OF SCIENCES
Volume 3 of Course of Theoretical Physics
Translated from the Russian by
J. B. SYKES and J. S. BELL
Second edition, revised and enlarged
PERGAMON PRESS
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Copyright © 1958 and 1965
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First published in English 1958
2nd impression 1959
3rd impression 1962
Second {revised) edition 1965
Library of Congress Card Number 57-14444
Printed in Great Britain by J. W. Arrowsmith Ltd., Bristol.
2217/65
CONTENTS
Page
From the Preface to the first English edition xi
Preface to the second English edition xii
Notation xiii
I. THE BASIC CONCEPTS OF QUANTUM MECHANICS
§1. The uncertainty principle 1
§2. The principle of superposition 6
§3. Operators 8
§4. Addition and multiplication of operators 13
§5. The continuous spectrum 15
§6. The passage to the limiting case of classical mechanics 20
§7. The wave function and measurements 21
II. ENERGY AND MOMENTUM
§8. The Hamiltonian operator 25
§9. The differentiation of operators with respect to time 26
§10. Stationary states 27
§11. Matrices 30
§12. Transformation of matrices 35
§13. The Heisenberg representation of operators 37
§14. The density matrix 38
§15. Momentum 41
§16. Uncertainty relations 46
III. SCHRODINGER'S EQUATION
§17. Schrodinger's equation 50
§18. The fundamental properties of Schrodinger's equation 53
§19. The current density 55
§20. The variational principle 58
§21. General properties of motion in one dimension 60
§22. The potential well 63
§23. The linear oscillator 67
§24. Motion in a homogeneous field 73
§25. The transmission coefficient 75
IV. ANGULAR MOMENTUM
§26. Angular momentum 81
§27. Eigenvalues of the angular momentum 85
§28. Eigenfunctions of the angular momentum 88
§29. Matrix elements of vectors 91
§30. Parity of a state 95
§31. Addition of angular momenta 97
vi Contents
V. MOTION IN A CENTRALLY SYMMETRIC FIELD Page
§32. Motion in a centrally symmetric field 101
§33. Free motion (spherical polar co-ordinates) 104
§34. Resolution of a plane wave 111
§35. "Fall" of a particle to the centre 113
§36. Motion in a Coulomb field (spherical polar co-ordinates) 116
§37. Motion in a Coulomb field (parabolic co-ordinates) 125
VI. PERTURBATION THEORY
§38. Perturbations independent of time 129
§39. The secular equation 133
§40. Perturbations depending on time 136
§41. Transitions under a perturbation acting for a finite time 140
§42. Transitions under the action of a periodic perturbation 146
§43. Transitions in the continuous spectrum 147
§44. The uncertainty relation for energy 150
§45. Potential energy as a perturbation 153
VII. THE QUASI-CLASSICAL CASE
§46. The wave function in the quasi-classical case 158
§47. Boundary conditions in the quasi- classical case 161
§48. Bohr and Sommerfeld's quantisation rule 162
§49. Quasi-classical motion in a centrally symmetric field 167
§50. Penetration through a potential barrier 171
§51. Calculation of the quasi-classical matrix elements 177
§52. The transition probability in the quasi-classical case 181
§53. Transitions under the action of adiabatic perturbations 185
VIII. SPIN
§54. Spin 188
§55. Spinors 191
§56. Spinors of higher rank 196
§57. The wave functions of particles with arbitrary spin 198
§58. The relation between spinors and tensors 200
§59. Partial polarisation of particles 204
§60. Time reversal and Kramers' theorem 206
IX. IDENTITY OF PARTICLES
§61. The principle of indistinguishability of similar particles 209
§62. Exchange interaction 212
§63. Symmetry with respect to interchange 216
§64. Second quantisation. The case of Bose statistics 221
§65. Second quantisation. The case of Fermi statistics 227
Contents vii
X. THE ATOM
Page
§66. Atomic energy levels 231
§67. Electron states in the atom 232
§68. Hydrogen-like energy levels 236
§69. The self-consistent field 237
§70. The Thomas-Fermi equation 241
§71. Wave functions of the outer electrons near the nucleus 246
§72. Fine structure of atomic levels 247
§73. The periodic system of D. I. Mendeleev 252
§74. X-ray terms 259
§75. Multipole moments 261
§76. The Stark effect 265
§77. The Stark effect in hydrogen 269
XI. THE DIATOMIC MOLECULE
§78. Electron terms in the diatomic molecule 277
§79. The intersection of electron terms 279
§80. The relation between molecular and atomic terms 282
§81. Valency 286
§82. Vibrational and rotational structures of singlet terms in the diatomic
molecule 293
§83. Multiplet terms. Case a 299
§84. Multiplet terms. Case b 303
§85. Multiplet terms. Cases c and d 307
§86. Symmetry of molecular terms 309
§87. Matrix elements for the diatomic molecule 312
§88. A-doubling 316
§89. The interaction of atoms at large distances 319
§90. Pre-dissociation 322
XII. THE THEORY OF SYMMETRY
§91. Symmetry transformations 332
§92. Transformation groups 335
§93. Point groups 338
§94. Representations of groups 347
§95. Irreducible representations of point groups 354
§96. Irreducible representations and the classification of terms 358
§97. Selection rules for matrix elements 361
§98. Continuous groups 364
§99. Two-valued representations of finite point groups 367
viii Contents
XIII. POLYATOMIC MOLECULES
Page
§100. The classification of molecular vibrations 371
§101. Vibrational energy levels 378
§102. Stability of symmetrical configurations of the molecule 380
§103. Quantisation of the rotation of a rigid body 383
§104. The interaction between the vibrations and the rotation of the molecule 389
§105. The classification of molecular terms 394
XIV. ADDITION OF ANGULAR MOMENTA
§106. 3/-symbols 401
§107. Matrix elements of tensors 408
§108. 6/-symbols 412
§109. Matrix elements for addition of angular momenta 418
XV. MOTION IN A MAGNETIC FIELD
§110. Schrodinger's equation in a magnetic field 421
§111. Motion in a uniform magnetic field 424
§112. The Zeeman effect 427
§113. Spin in a variable magnetic field 434
§114. The current density in a magnetic field 435
XVI. NUCLEAR STRUCTURE
§115. Isotopic invariance 438
§116. Nuclear forces 442
§117. The shell model 447
§118. Non-spherical nuclei 456
§119. Isotopic shift 461
§120. Hyperfine structure of atomic levels 463
§121. Hyperfine structure of molecular levels 466
XVII. THE THEORY OF ELASTIC COLLISIONS
§122. The general theory of scattering 469
§123. An investigation of the general formula 472
§124. The unitary condition for scattering 475
§125. Born's formula 479
§126. The quasi-classical case 486
§127. Scattering at high energies 489
§128. Analytical properties of the scattering amplitude 492
§129. The dispersion relation 497
§130. The scattering of slow particles 500
Contents ix
Page
§131. Resonance scattering at low energies 505
§132. Resonance at a quasi-discrete level 511
§133. Rutherford's formula 516
§134. The system of wave functions of the continuous spectrum 519
§135. Collisions of like particles 523
§136. Resonance scattering of charged particles 526
§137. Elastic collisions between fast electrons and atoms 531
§138. Scattering with spin-orbit interaction 535
XVIII. THE THEORY OF INELASTIC COLLISIONS
§139. Elastic scattering in the presence of inelastic processes 542
§140. Inelastic scattering of slow particles 548
§141. The scattering matrix in the presence of reactions 550
§142. Breit and Wigner's formula 554
§143. Interaction in the final state in reactions 562
§144. Behaviour of cross-sections near the reaction threshold 565
§145. Inelastic collisions between fast electrons and atoms 571
§146. The effective retardation 580
§147. Inelastic collisions between heavy particles and atoms 584
§148. Scattering by molecules 587
MATHEMATICAL APPENDICES
§a. Hermite polynomials 593
§b. The Airy function 596
§c. Legendre polynomials 598
§d. The confluent hypergeometric function 600
§e. The hypergeometric function 605
§f. The calculation of integrals containing confluent hypergeometric
functions ■ 607
INDEX 611
FROM THE PREFACE TO THE FIRST ENGLISH EDITION
The present book is one of the series on Theoretical Physics, in which we
endeavour to give an up-to-date account of various departments of that science.
The complete series will contain the following nine volumes :
1. Mechanics. 2. The classical theory of fields. 3. Quantum mechanics
(non-relativistic theory). 4. Relativistic quantum theory. 5. Statistical physics.
6. Fluid mechanics. 7. Theory of elasticity. 8. Electrodynamics of continuous
media. 9. Physical kinetics.
Of these, volumes 4 and 9 remain to be written.
The scope of modern theoretical physics is very wide, and we have, of
course, made no attempt to discuss in these books all that is now included in
the subject. One of the principles which guided our choice of material was
not to deal with those topics which could not properly be expounded without
at the same time giving a detailed account of the existing experimental results.
For this reason the greater part of nuclear physics, for example, lies outside the
scope of these books. Another principle of selection was not to discuss very
complicated applications of the theory. Both these criteria are, of course,
to some extent subjective.
We have tried to deal as fully as possible with those topics that are included.
For this reason we do not, as a rule, give references to the original papers,
but simply name their authors. We give bibliographical references only to
work which contains matters not fully expounded by us, which by their com-
plexity lie "on the borderline" as regards selection or rejection. We have
tried also to indicate sources of material which might be of use for reference.
Even with these limitations, however, the bibliography given makes no pre-
tence of being exhaustive.
We attempt to discuss general topics in such a way that the physical signifi-
cance of the theory is exhibited as clearly as possible, and then to build up the
mathematical formalism. In doing so, we do not aim at "mathematical
rigour" of exposition, which in theoretical physics often amounts to self-
deception.
The present volume is devoted to non-relativistic quantum mechanics. By
"relativistic theory" we here mean, in the widest sense, the theory of all
quantum phenomena which significantly depend on the velocity of light. The
volume on this subject (volume 4) will therefore contain not only Dirac's
relativistic theory and what is now known as quantum electrodynamics, but
also the whole of the quantum theory of radiation.
Institute of Physical Problems L. D. Landau
USSR Academy of Sciences E. M. Lifshitz
August 1956
PREFACE TO THE SECOND ENGLISH EDITION
For this second edition the book has been considerably revised and en-
larged, but the general plan and style remain as before. Every chapter has
been revised. In particular, extensive changes have been made in the sections
dealing with the theory of the addition of angular momenta and with collision
theory. A new chapter on nuclear structure has been added; in accordance
with the general plan of the course, the subjects in question are discussed only
to the extent that is proper without an accompanying detailed analysis of the
experimental results.
We should like to express our thanks to all our many colleagues whose
comments have been utilised in the revision of the book. Numerous com-
ments were received from V. L. Ginzburg and Ya. A. Smorodinskii. We are
especially grateful to L. P. Pitaevskii for the great help which he has given in
checking the formulae and the problems.
Our sincere thanks are due to Dr. Sykes and Dr. Bell, who not only
translated excellently both the first and the second edition of the book, but
also made a number of useful comments and assisted in the detection of
various misprints in the first edition.
Finally, we are grateful to the Pergamon Press, which always acceded to
our requests during the production of the book.
L. D. Landau
October 1964 E. M. Lifshitz
NOTATION
Operators are denoted by a circumflex
dq element in configuration space
f nm = /" = (n\f\m) matrix elements of the quantity/ (see definition in §11)
% m = (E n — Em)!^ transition frequency
{/, g} = fg — g/ commutator of two operators
i? Hamiltonian
S, &? electric and magnetic fields
Si phase shifts of wave functions
eua antisymmetric unit tensor
■"± = "a; i ^"y
CHAPTER I
THE BASIC CONCEPTS OF QUANTUM MECHANICS
§1. The uncertainty principle
When we attempt to apply classical mechanics and electrodynamics to explain
atomic phenomena, they lead to results which are in obvious conflict with
experiment. This is very clearly seen from the contradiction obtained on
applying ordinary electrodynamics to a model of an atom in which the elec-
trons move round the nucleus in classical orbits. During such motion, as in
any accelerated motion of charges, the electrons would have to emit electro-
magnetic waves continually. By this emission, the electrons would lose their
energy, and this would eventually cause them to fall into the nucleus. Thus,
according to classical electrodynamics, the atom would be unstable, which
does not at all agree with reality.
This marked contradiction between theory and experiment indicates that
the construction of a theory applicable to atomic phenomena — that is, pheno-
mena occurring in particles of very small mass at very small distances —
demands a fundamental modification of the basic physical concepts and laws.
As a starting-point for an investigation of these modifications, it is conveni-
ent to take the experimentally observed phenomenon known as electron
diffraction.^ It is found that, when a homogeneous beam of electrons passes
through a crystal, the emergent beam exhibits a pattern of alternate maxima
and minima of intensity, wholly similar to the diffraction pattern observed
in the diffraction of electromagnetic waves. Thus, under certain conditions,
the behaviour of material particles — in this case, the electrons — displays
features belonging to wave processes.
How markedly this phenomenon contradicts the usual ideas of motion is
best seen from the following imaginary experiment, an idealisation of the
experiment of electron diffraction by a crystal. Let us imagine a screen
impermeable to electrons, in which two slits are cut. On observing the
passage of a beam of electrons^ through one of the slits, the other being
covered, we obtain, on a continuous screen placed behind the slit, some pat-
tern of intensity distribution; in the same way, by uncovering the second
slit and covering the first, we obtain another pattern. On observing the
passage of the beam through both slits, we should expect, on the basis of
ordinary classical ideas, a pattern which is a simple superposition of the other
two : each electron, moving in its path, passes through one of the slits and
f The phenomenon of electron diffraction was in fact discovered after quantum mechanics was
invented. In our discussion, however, we shall not adhere to the historical sequence of development
of the theory, but shall endeavour to construct it in such a way that the connection between the basic
principles of quantum mechanics and the experimentally observed phenomena is most clearly shown
J The beam is supposed so rarefied that the interaction of the particles in it plays no part.
1
2 The Basic Concepts of Quantum Mechanics §1
has no effect on the electrons passing through the other slit. The phenomenon
of electron diffraction shows, however, that in reality we obtain a diffraction
pattern which, owing to interference, does not at all correspond to the sum
of the patterns given by each slit separately. It is clear that this result can
in no way be reconciled with the idea that electrons move in paths.
Thus the mechanics which governs atomic phenomena — quantum mechanics
or wave mechanics — must be based on ideas of motion which are fundamentally
different from those of classical mechanics. In quantum mechanics there is
no such concept as the path of a particle. This forms the content of what is
called the uncertainty principle, one of the fundamental principles of quantum
mechanics, discovered by W. Heisenberg in 1927.f
In that it rejects the ordinary ideas of classical mechanics, the uncertainty
principle might be said to be negative in content. Of course, this principle
in itself does not suffice as a basis on which to construct a new mechanics of
particles. Such a theory must naturally be founded on some positive asser-
tions, which we shall discuss below (§2). However, in order to formulate
these assertions, we must first ascertain the statement of the problems which
confront quantum mechanics. To do so, we first examine the special nature
of the interrelation between quantum mechanics and classical mechanics. A
more general theory can usually be formulated in a logically complete manner,
independently of a less general theory which forms a limiting case of it. Thus,
relativistic mechanics can be constructed on the basis of its own fundamental
principles, without any reference to Newtonian mechanics. It is in principle
impossible, however, to formulate the basic concepts of quantum mechanics
without using classical mechanics. The fact that an electron^ has no definite
path means that it has also, in itself, no other dynamical characteristics. ||
Hence it is clear that, for a system composed only of quantum objects,
it would be entirely impossible to construct any logically independent
mechanics. The possibility of a quantitative description of the motion of an
electron requires the presence also of physical objects which obey classical
mechanics to a sufficient degree of accuracy. If an electron interacts with
such a "classical object", the state of the latter is, generally speaking, altered.
The nature and magnitude of this change depend on the state of the electron,
and therefore may serve to characterise it quantitatively.
In this connection the "classical object" is usually called apparatus, and
its interaction with the electron is spoken of as measurement. However, it
must be emphasised that we are here not discussing a process of measurement
in which the physicist-observer takes part. By measurement, in quantum
mechanics, we understand any process of interaction between classical and
f It is of interest to note that the complete mathematical formalism of quantum mechanics was
constructed by W. Heisenberg and E. Schrodinger in 1925-6, before the discovery of the uncertainty
principle, which revealed the physical content of this formalism.
X In this and the following sections we shall, for brevity, speak of "an electron", meaning in general
any object of a quantum nature, i.e. a particle or system of particles obeying quantum mechanics and
not classical mechanics.
I We refer to quantities which characterise the motion of the electron, and not to those, such as the
charge and the mass, which relate to it as a particle ; these are parameters.
§1 The uncertainty principle 3
quantum objects, occurring apart from and independently of any observer.
The importance of the concept of measurement in quantum mechanics was
elucidated by N. Bohr.
We have denned "apparatus" as a physical object which is governed, with
sufficient accuracy, by classical mechanics. Such, for instance, is a body
of large enough mass. However, it must not be supposed that apparatus is
necessarily macroscopic. Under certain conditions, the part of apparatus may
also be taken by an object which is microscopic, since the idea of "with
sufficient accuracy" depends on the actual problem proposed. Thus, the
motion of an electron in a Wilson chamber is observed by means of the
cloudy track which it leaves, and the thickness of this is large compared with
atomic dimensions; when the path is determined with such low accuracy,
the electron is an entirely classical object.
Thus quantum mechanics occupies a very unusual place among physical
theories: it contains classical mechanics as a limiting case, yet at the same
time it requires this limiting case for its own formulation.
We may now formulate the problem of quantum mechanics. A typical
problem consists in predicting the result of a subsequent measurement from
the known results of previous measurements. Moreover, we shall see later
that, in comparison with classical mechanics, quantum mechanics, generally
speaking, restricts the range of values which can be taken by various physical
quantities (for example, energy) : that is, the values which can be obtained
as a result of measuring the quantity concerned. The methods of quantum
mechanics must enable us to determine these admissible values.
The measuring process has in quantum mechanics a very important pro-
perty: it always affects the electron subjected to it, and it is in principle
impossible to make its effect arbitrarily small, for a given accuracy of measure-
ment. The more exact the measurement, the stronger the effect exerted by
it, and only in measurements of very low accuracy can the effect on the mea-
sured object be small. This property of measurements is logically related
to the fact that the dynamical characteristics of the electron appear only as a
result of the measurement itself. It is clear that, if the effect of the measuring
process on the object of it could be made arbitrarily small, this would mean
that the measured quantity has in itself a definite value independent of the
measurement.
Among the various kinds of measurement, the measurement of the co-
ordinates of the electron plays a fundamental part. Within the limits of
applicability of quantum mechanics, a measurement of the co-ordinates of an
electron can always be performed-]- with any desired accuracy.
Let us suppose that, at definite time intervals At, successive measurements of
the co-ordinates of an electron are made. The results will not in general lie
on a smooth curve. On the contrary, the more accurately the measurements
f Once again we emphasise that, in speaking of "performing a measurement", we refer to the
interaction of an electron with a classical " apparatus", which in no way presupposes the presence of
an external observer.
4 The Basic Concepts of Quantum Mechanics §1
are made, the more discontinuous and disorderly will be the variation of
their results, in accordance with the non-existence of a path of the electron.
A fairly smooth path is obtained only if the co-ordinates of the electron are
measured with a low degree of accuracy, as for instance from the condensa-
tion of vapour droplets in a Wilson chamber.
If now, leaving the accuracy of the measurements unchanged, we diminish
the intervals At between measurements, then adjacent measurements, of
course, give neighbouring values of the co-ordinates. However, the results
of a series of successive measurements, though they lie in a small region of
space, will be distributed in this region in a wholly irregular manner, lying on
no smooth curve. In particular, as A* tends to zero, the results of adjacent
measurements by no means tend to lie on one straight line.
This circumstance shows that, in quantum mechanics, there is no such
concept as the velocity of a particle in the classical sense of the word, i.e. the
limit to which the difference of the co-ordinates at two instants, divided by
the interval At between these instants, tends as At tends to zero. However,
we shall see laier that in quantum mechanics, nevertheless, a reasonable
definition of the velocity of a particle at a given instant can be constructed,
and this velocity passes into the classical velocity as we pass to classical mech-
anics. But whereas in classical mechanics a particle has definite co-ordinates
and velocity at any given instant, in quantum mechanics the situation is
entirely different. If, as a result of measurement, the electron is found to have
definite co-ordinates, then it has no definite velocity whatever. Conversely,
if the electron has a definite velocity, it cannot have a definite position in
space. For the simultaneous existence of the co-ordinates and velocity would
mean the existence of a definite path, which the electron has not. Thus, in
quantum mechanics, the co-ordinates and velocity of an electron are quantities
which cannot be simultaneously measured exactly, i.e. they cannot simultane-
ously have definite values. We may say that the co-ordinates and velocity
of the electron are quantities which do not exist simultaneously. In what
follows we shall derive the quantitative relation which determines the pos-
sibility of an inexact measurement of the co-ordinates and velocity at the same
instant.
A complete description of the state of a physical system in classical mech-
anics is effected by stating all its co-ordinates and velocities at a given instant ;
with these initial data, the equations of motion completely determine the
behaviour of the system at all subsequent instants. In quantum mechanics
such a description is in principle impossible, since the co-ordinates and the
corresponding velocities cannot exist simultaneously. Thus a description
of the state of a quantum system is effected by means of a smaller number of
quantities than in classical mechanics, i.e. it is less detailed than a classical
description.
A very important consequence follows from this regarding the nature of the
predictions made in quantum mechanics. Whereas a classical description
suffices to predict the future motion of a mechanical system with complete
§1 The uncertainty principle 5
accuracy, the less detailed description given in quantum mechanics evidently
cannot be enough to do this. This means that, even if an electron is in a state
described in the most complete manner possible in quantum mechanics, its
behaviour at subsequent instants is still in principle uncertain. Hence quan-
tum mechanics cannot make completely definite predictions concerning the
future behaviour of the electron. For a given initial state of the electron, a
subsequent measurement can give various results. The problem in
quantum mechanics consists in determining the probability of obtaining vari-
ous results on performing this measurement. It is understood, of course,
that in some cases the probability of a given result of measurement may be
equal to unity, i.e. certainty, so that the result of that measurement is unique.
All measuring processes in quantum mechanics may be divided into two
classes. In one, which contains the majority of measurements, we find those
which do not, in any state of the system, lead with certainty to a unique
result. The other class contains measurements such that for every possible
result of measurement there is a state in which the measurement leads with
certainty to that result. These latter measurements, which may be called
predictable, play an important part in quantum mechanics. The quantitative
characteristics of a state which are determined by such measurements are
what are called physical quantities in quantum mechanics. If in some state
a measurement gives with certainty a unique result, we shall say that in this
state the corresponding physical quantity has a definite value. In future we
shall always understand the expression "physical quantity" in the sense given
here.
We shall often find in what follows that by no means every set of physical
quantities in quantum mechanics can be measured simultaneously, i.e. can
all have definite values at the same time. We have already mentioned one
example, namely the velocity and co-ordinates of an electron. An important
part is played in quantum mechanics by sets of physical quantities having
the following property: these quantities can be measured simultaneously,
but if they simultaneously have definite values, no other physical quantity
(not being a function of these) can have a definite value in that state. We
shall speak of such sets of physical quantities as complete sets; in particular
cases a complete set may consist of only one quantity.
Any description of the state of an electron arises as a result of some mea-
surement. We shall now formulate the meaning of a complete description of
a state in quantum mechanics. Completely described states occur as a result
of the simultaneous measurement of a complete set of physical quanti-
ties. From the results of such a measurement we can, in particular, deter-
mine the probability of various results of any subsequent measurement,
regardless of the history of the electron prior to the first measurement.
In quantum mechanics we need concern ourselves in practice only with
completely described states, and from now on (except in §14) we shall under-
stand by the states of a quantum system just these completely described
states.
6 The Basic Concepts of Quantum Mechanics §2
§2. The principle of superposition
Passing now to an exposition of the fundamental mathematical formalism
of quantum mechanics, we shall denote by q the set of co-ordinates of a quan-
tum system, and by dq the product of the differentials of these co-ordinates.
This dq is often called an element of volume in the configuration space of the
system; for one particle, dq coincides with an element of volume dV in
ordinary space.
The basis of the mathematical formalism of quantum mechanics lies in the
fact that any state of a system can be described, at a given moment, by a
definite (in general complex) function Y(#) of the co-ordinates. The square
of the modulus of this function determines the probability distribution of the
values of the co-ordinates: |Y| 2 d# is the probability that a measurement
performed on the system will find the values of the co-ordinates to be in the
element dq of configuration space. The function Y is called the wave function
of the system (sometimes also the probability amplitude), j-
A knowledge of the wave function allows us, in principle, to calculate the
probability of the various results of any other measurement (not of the co-
ordinates) also. All these probabilities are determined by expressions bi-
linear in Y and Y*. The most general form of such an expression is
jjV{q)Wffl<Kq t <t)*qM, (2.1)
where the function (f>(q, q) depends on the nature and the result of the mea-
surement, and the integration is extended over all configuration space. The
probability YY* of various values of the co-ordinates is itself an expression
of this type. J
The state of the system, and with it the wave function, in general varies
with time. In this sense the wave function can be regarded as a function of
time also. If the wave function is known at some initial instant, then, from
the very meaning of the concept of complete description of a state, it is in
principle determined at every succeeding instant. The actual dependence
of the wave function on time is determined by equations which will be de-
rived later.
The sum of the probabilities of all possible values of the co-ordinates of
the system must, by definition, be equal to unity. It is therefore necessary
that the result of integrating |Y| 2 over all configuration space should be equal
to unity:
J>|»df = l. (2.2)
This equation is what is called the normalisation condition for wave functions.
If the integral of |Y| 2 converges, then by choosing an appropriate constant
coefficient the function Y can always be, as we say, normalised. Sometimes,
f It was first introduced into quantum mechanics by Schrodinger in 1926.
% It is obtained from (2.1) when<£(g, q') = 8(?-? ) 8(q'-q ), where 8 denotes the delta function,
defined in §5 below; q denotes the value of the co-ordinates whose probability is required.
§2 The principle of superposition 7
however, wave functions are used which are not normalised; moreover, we
shall see later that the integral of |TJ 2 may diverge, and then Y cannot be
normalised by the condition (2.2). In such cases |T| 2 does not, of course,
determine the absolute values of the probability of the co-ordinates, but the
ratio of the values of |T| 2 at two different points of configuration space deter-
mines the relative probability of the corresponding values of the co-ordinates.
Since all quantities calculated by means of the wave function, and having a
direct physical meaning, are of the form (2.1), in which T appears multiplied
by Y*, it is clear that the normalised wave function is determined only to
within a constant phase fact or of the form e ia (where a is any real number),
whose modulus is unity. This indeterminacy is in principle irremovable;
it is, however, unimportant, since it has no effect upon any physical results.
The positive content of quantum mechanics is founded on a series of
propositions concerning the properties of the wave function. These are as
follows.
Suppose that, in a state with wave function VF^), some measurement leads
with certainty to a definite result (result 1), while in a state with Y 2 (#) it
leads to result 2. Then it is assumed that every linear combination of Y x
and Y 2 , i.e. every function of the form c J T 1 +c 2 x F 2 (where c x and c % me con-
stants), gives a state in which that measurement leads to either result 1 or
result 2. Moreover, we can assert that, if we know the time dependence of
the states, which for the one case is given by the function T^, t), and for the
other by Y 2 (<7, t), then any linear combination also gives a possible dependence
of a state on time. These propositions can be immediately generalised to
any number of different states.
The above set of assertions regarding wave functions constitutes what is
called the principle of superposition of states, the chief positive principle of
quantum mechanics. In particular, it follows at once from this principle
that all equations satisfied by wave functions must be linear in Y.
Let us consider a system composed of two parts, and suppose that the state
of this system is given in such a way that each of its parts is completely
described.f Then we can say that the probabilities of the co-ordinates q x of
the first part are independent of the probabilities of the co-ordinates q % of the
second part, and therefore the probability distribution for the whole system
should be equal to the product of the probabilities of its parts. This means
that the wave function Y 12 (ft, q z ) of the system can be represented in the form
of a product of the wave functions Y^ft) and Y 2 (# 2 ) of its parts:
^i 2 (?i,? 2 )=Yi(?i)Y 2 (? 2 ). (2.3)
If the two parts do not interact, then this relation between the wave function
of the system and those of its parts will be maintained at future instants also,
f This, of course, means that the state of the whole system is completely described also. However,
we emphasise that the converse statement is by no means true: a complete description of the state
of the whole system does not in general completely determine the states of its individual parts (see
also §14).
8 The Basic Concepts of Quantum Mechanics §3
i.e. we can write
Tu(fo ?i, t) = YjCfc, /) T 2 ( ?2 , 0- (2.4)
§3. Operators
Let us consider some physical quantity / which characterises the state
of a quantum system. Strictly, we should speak in the following discussion
not of one quantity, but of a complete set of them at the same time.
However, the discussion is not essentially changed by this, and for brevity
and simplicity we shall work below in terms of only one physical quantity.
The values which a given physical quantity can take are called in quantum
mechanics its eigenvalues, and the set of these is referred to as the spectrum
of eigenvalues of the given quantity. In classical mechanics, generally speak-
ing, quantities run through a continuous series of values. In quantum mech-
anics also there are physical quantities (for instance, the co-ordinates) whose
eigenvalues occupy a continuous range ; in such cases we speak of a continuous
spectrum of eigenvalues. As well as such quantities, however, there exist in
quantum mechanics others whose eigenvalues form some discrete set; in
such cases we speak of a discrete spectrum.
We shall suppose for simplicity that the quantity / considered here has a
discrete spectrum; the case of a continuous spectrum will be discussed in §5.
The eigenvalues of the quantity / are denoted by/ n , where the suffix n takes
the values 0, 1, 2, 3 We also denote the wave function of the system, in
the state where the quantity / has the value / n , by Y n . The wave functions
Y n are called the eigenfunctions of the given physical quantity/. Each of these
functions is supposed normalised, so that
J> n |*d«z = l. (3.1)
If the system is in some arbitrary state with wave function T, a measure-
ment of the quantity / carried out on it will give as a result one of the eigen-
values f n . In accordance with the principle of superposition, we can assert
that the wave function Y must be a linear combination of those eigenfunc-
tions T TC which correspond to the values f n that can be obtained, with prob-
ability different from zero, when a measurement is made on the system and
it is in the state considered. Hence, in the general case of an arbitrary state,
the function T can be represented in the form of a series
Y=Sa w Y w , (3.2)
where the summation extends over all n, and the a n are some constant coeffi-
cients.
Thus we reach the conclusion that any wave function can be, as we say,
expanded in terms of the eigenfunctions of any physical quantity. A set of
functions in terms of which such an expansion can be made is called a complete
(or closed) set.
§3 Operators 9
The expansion (3.2) makes it possible to determine the probability of find-
ing (i.e. the probability of getting the corresponding result on measurement),
in a system in a state with wave function Y, any given value/ n of the quantity
/. For, according to what was said in the previous section, these probabili-
ties must be determined by some expressions bilinear in Y and Y*, and
therefore must be bilinear in a n and a n *. Furthermore, these expressions
must, of course, be positive. Finally, the probability of the value f n must
become unity if the system is in a state with wave function Y = Y n , and
must become zero if there is no term containing Y n in the expansion (3.2)
of the wave function Y. This means that the required probability must be
unity if all the coefficients a n except one (with the given n) are zero, that one
being unity; the probability must be zero, if the a n concerned is zero. The
only essentially positive quantity satisfying these conditions is the square of the
modulus of the coefficient a n . Thus we reach the result that the squared
modulus \a n \ % of each coefficient in the expansion (3.2) determines the prob-
ability of the corresponding value f n of the quantity / in the state with wave
function Y. The sum of the probabilities of all possible values f n must be
equal to unity; in other words, the relation
S la w | 2 = 1 (3.3)
n
must hold.
If the function Y were not normalised, then the relation (3.3) would not
hold either. The sum £ \a n \ 2 would then be given by some expression
bilinear in Y and Y*, and becoming unity when Y was normalised. Only
the integral J YY* dq is such an expression. Thus the equation
SflA* = JYY*d ? (3.4)
must hold.
On the other hand, multiplying by Y the expansion Y* = S tf n *Y n * of
the function Y* (the complex conjugate of Y), and integrating, we obtain
j YY* dq = S a n *j Y n *Y dq.
Comparing this with (3.4), we have
Sa B « tt * = Sa n *jY n *Yd ? ,
from which we derive the following formula determining the coefficients a n
in the expansion of the function Y in terms of the eigenfunctions Y n :
a n = jYY n *d 2 . (3.5)
If we substitute here from (3.2), we obtain
« n = Sa w fY TO Y n *d 9 ,
10 The Basic Concepts of Quantum Mechanics §3
from which it is evident that the eigenfunctions must satisfy the conditions
JT m V n *dq = 8 nm , (3.6)
where $ nm = 1 for n = m and S nm = for n # m. The fact that the integrals
of the products T w T n * with m ^ n vanish is called the orthogonality of the
functions T n . Thus the set of eigenfunctions T n forms a complete set of
normalised and orthogonal (or, for brevity, orthonormal) functions.
We shall now introduce the concept of the mean value f of the quantity/
in the given state. In accordance with the usual definition of mean values,
we define / as the sum of all the eigenvalues f n of the given quantity, each
multiplied by the corresponding probability |a n | 2 . Thus
/= IfnW- (3.7)
We shall write / in the form of an expression which does not contain the
coefficients a n in the expansion of the function T, but this function itself.
Since the products a n a n * appear in (3.7), it is clear that the required expres-
sion must be bilinear in T and Y*. We introduce a mathematical opera-
tor, which we denotef by /and define as follows. Let (/Y) denote the result
of the operator / acting on the function Y. We define / in such a way that
the integral of the product of (/Y) and the complex conjugate function Y*
is equal to the mean value /:
/=jV(/T)d ? . (3.8)
It is easily seen that, in the general case, the operator / is a linear J
integral operator. For, using the expression (3.5) for a n , we can rewrite the
definition (3.7) of the mean value in the form
/= \f n a n a* = J Y*(S a n f n W n ) 6q.
Comparing this with (3.8), we see that the result of the operator / acting on
the function Y has the form
(/Y) = Sa n / M Y w . (3.9)
If we substitute here the expression (3.5) for a n , we find that /is an integral
operator of the form
tfr) = JK(q,<nv(<nw, (3.io)
where the function K(q, q') (called the kernel of the operator) is
K(q, q') = S/.Y.-foT^fe). (3.11)
f By convention, we shall always denote operators by letters with circumflexes.
+ An operator is said to be linear if it has the properties
/(^i+T 2 ) =M+/Y, and/(«T) = afY,
where X F 1 and *F 2 are arbitrary functions and a is an arbitrary constant.
§3 Operators 11
Thus, for every physical quantity in quantum mechanics, there is a definite
corresponding linear operator.
It is seen from (3.9) that, if the function T is one of the eigenfunctions \F n
(so that all the a n except one are zero), then, when the operator / acts on it,
this function is simply multiplied by the corresponding eigenvalue f n :
fa =/«^n- (3.12)
(In what follows we shall always omit the parentheses in the expression
(/T), where this cannot cause any misunderstanding; the operator is taken
to act on the expression which follows it.) Thus we can say that the eigen-
functions of the given physical quantity /are the solutions of the equation
where / is a constant, and the eigenvalues are the values of this constant for
which the above equation has solutions satisfying the required conditions.
Of course, while the operator /is still defined only by the expressions (3.10)
and (3.11), which themselves contain the eigenfunctions ^I^, no further con-
clusions can be drawn from the result we have obtained. However, as we
shall see below, the form of the operators for various physical quantities can
be determined from direct physical considerations, and then the above pro-
perty of the operators enables us to find the eigenfunctions and eigenvalues
by solving the equations p¥ = f¥.
The values which can be taken by real physical quantities are obviously
real. Hence the mean value of a physical quantity must also be real, in any
state. Conversely, if the mean value of a physical quantity is real in every
state, its eigenvalues also are all real; to show this, it is sufficient to note that
the mean values coincide with the eigenvalues in the states described by the
functions T n .
From the fact that the mean values are real, we can draw some conclusions
concerning the properties of operators. Equating the expression (3.8) to its
complex conjugate, we obtain the relation
J T*(/Y) dg = j Y(f*Y*) dg, (3.13)
where /* denotes the operator which is the complex conjugate of /. j- This
relation does not hold in general for an arbitrary linear operator, so that it is
a restriction on the form of the operator /. For an arbitrary operator / we
can find what is called the transposed operator/, defined in such a way that
JY(fl>)dq=f<l>(jY)dq, (3.14)
where *F and <D are two different functions. If we take, as the function O,
the function T* which is the complex conjugate of Y, then a comparison with
(3.13) shows that we must have
/=/•• (3.15)
t By definition, if for^the operator /we h&vefifi = <f>, then the complex conjugate operator /* is
that for which we have /*^t* = <f>*.
12 The Basic Concepts of Quantum Mechanics §3
Operators satisfying this condition are said to be Hermitian.^ Thus the
operators corresponding, in the mathematical formalism of quantum
mechanics, to real physical quantities must be Hermitian.
We can formally consider complex physical quantities also, i.e. those
whose eigenvalues are complex. Let / be such a quantity. Then we can
introduce its complex conjugate quantity/*, whose eigenvalues are the com-
plex conjugates of those of/. We denote by /+ the operator corresponding
to the quantity/*. It is called the Hermitian conjugate of the operator / and,
in general, will be different from the complex conjugate operator /* : from
the condition/* = (/)* we find at once that
/ + =7*. (3.16)
from which it is clear that /+ is in general not the same as /*. For a real
physical quantity / = /+, i.e. the operator is the same as its Hermitian
conjugate (Hermitian operators are also called self -conjugate).
We shall show how the orthogonality of the eigenfunctions of an Hermitian
operator corresponding to different eigenvalues can be directly proved. Let
f n and/„, be two different eigenvalues of the quantity/, and T n , *F m the cor-
responding eigenfunctions :
f^n^fn^n, f X ¥ m =fm X r m -
Multiplying both sides of the first of these equations by T m *, and both
sides of the complex conjugate of the second by T n , and subtracting corre-
sponding terms, we find
W *fw _tF /*tF * — ( f —f W* W *
We integrate both sides of this equation over q. Since /* =/, by (3.14) the
integral on the left-hand side of the equation is zero, so that we have
(fn-f m ) j T n Y w *d^ = 0,
whence, since f n # f m , we obtain the required orthogonality property of the
functions x P n and T m .
We have spoken here of only one physical quantity /, whereas, as we said
at the beginning of this section, we should have spoken of a complete set
of physical quantities. We should then have found that to each of these
quantities/ g, ... there corresponds its operator /, g, ... . The eigenfunctions
T n then correspond to states in which all the quantities concerned have
definite values, i.e. they correspond to definite sets of eigenvalues f n , g n , ... ,
and are simultaneous solutions of the system of equations
/F=/V, iW=gY,....
f For a linear integral operator of the form (3.10), the Hermitian condition means that the kernel
of the operator must be such that K(q, q') = K*(q', q).
§4 Addition and multiplication of operators 13
§4. Addition and multiplication of operators
Let / and g be two physical quantities which can simultaneously take
definite values, and / and g their operators. The eigenvalues of the sum
f+g of these quantities are equal to the sums of the eigenvalues of/ and g.
To this new quantity f+g there will obviously correspond an operator equal
to the sum of the operators / and g. For, if Y n are the eigenfunctions com-
mon to the operators / and g } then it follows from /*F n = f^¥ ni £*F n =g n x ^n
that
(f+gWn = (fn+gnWn,
i.e. the eigenvalues of the operator f+j> are equal to the sums/ n +£ n «
If the quantities / and g cannot simultaneously take definite values, then
it is meaningless to speak of their sum in the direct sense just mentioned.
It is conventional in quantum mechanics to define the sum of the quantities
/ and g in such cases as the quantity whose mean value in an arbitrary state
is equal to the sum of the mean values /and g:
7+J=/+|. (4-1)
It is clear that, to the quantity f+g so defined, there corresponds an operator
j+£. For, by formula (3.8), we have
7+g = \^*{f+gW d? = J>*/*F dq+ JVfF dq =f+g.
The eigenvalues and eigenfunctions of the operator f+g will not, in general*
now bear any relation to those of the quantities / and g. It is evident that,
if the operators / and g are self-conjugate, the operator f+g will be so too,
so that its eigenvalues are real and are those of the new quantity f+g thus
defined.
The following theorem should be noted. Let / and g be the smallest
eigenvalues of the quantities / and g, and (f+g) that of the quantity f+g.
Then
(f+g)» >fo+go- (4.2)
The equality holds if / and g can be measured simultaneously. The proof
follows from the obvious fact that the mean value of a quantity is always
greater than or equal to its least eigenvalue. In a state in which the quantity
f+g has the value (f+g) we have f+g = (f+g) , and since, on the other
hand, f+g = f+g ^ f +g > we arrive at the inequality (4.2).
Next, let/ and g once more be quantities that can be measured simultane-
ously. Besides their sum, we can also introduce the concept of their product
as being a quantity whose eigenvalues are equal to the products of those of the
quantities / and g. It is easy to see that, to this quantity, there corresponds
an operator whose effect consists of the successive action on the function
of first one and then the other operator. Such an operator is represented
mathematically by the product of the operators / and §. For, if T n are the
14 The Basic Concepts of Quantum Mechanics §4
eigenfunctions common to the operators / and g, we have
Jg?n =JtpTJ = fgn^n = gnf^n = gnfn^n
(the symbol fg denotes an operator whose effect on a function ¥ consists of
the successive action first of the operator g on the function ¥ and then of the
operator / on the function #¥). We could equally well take the operator gf
instead of fg, the former differing from the latter in the order of its factors.
It is obvious that the result of the action of either of these operators on the
functions ¥ n will be the same. Since, however, every wave function ¥ can
be represented as a linear combination of the functions ¥ TC , it follows that
the result of the action of the operators fg and gf on an arbitrary function will
also be the same. This fact can be written in the form of the symbolic
equation fg = gf or
/g-gf=0. (4.3)
Two such operators / and g are said to commute with each other. Thus
we arrive at the important result : if two quantities/ and g can simultaneously
take definite values, then their operators commute with each other.
The converse theorem can also be proved (§11): if the operators /and g
commute, then all their eigenfunctions can be taken common to both;
physically, this means that the corresponding physical quantities can be
measured simultaneously. Thus the commutability of the operators is a
necessary and sufficient condition for the physical quantities to be simultane-
ously measurable.
A particular case of the product of operators is an operator raised to some
power. From the above discussion we can deduce that the eigenvalues of an
operator f p (where p is an integer) are equal to the pth. powers of the eigen-
values of the operator/. Any function </>(/) of an operator can be defined
as an operator whose eigenvalues are equal to the same function <£(/) of
the eigenvalues of the operator/. If the function <f>(f) can be expanded as a
Taylor series, this expresses the effect of the operator <£( / ) in terms of those
of various powers/^.
In particular, the operator / _1 is called the inverse of the operator / It is
evident that the successive action of the operators / and / _1 on any function
leaves the latter unchanged, i.e. // -1 = / _1 / = 1-
If the quantities / and g cannot simultaneously take definite values, the
concept of their product cannot be defined in the above manner. This
appears in the fact that the operator fg is not self-conjugate in this case, and
hence cannot correspond to any physical quantity. For, by the definition
of the transpose of an operator we can write
J" Yjg* dq = j¥/(i<D) dq = j (£<D)(/¥) dq.
Here the operator/ acts only on the function ¥, and the operator £ on <E>, so
that the integrand is a simple product of two functions^® and/¥. Again
§5 The continuous spectrum 15
using the definition of the transpose of an operator, we can write
Thus we obtain an integral in which the functions *F and <S> have changed
places as compared with the original one. In other words, the operator §f
is the transpose of fg, and we can write
fg=if, (4-4)
i.e. the transpose of the product fg is the product of the transposes of the
factors written in the opposite order. Taking the complex conjugate of both
sides of equation (4.4), we have
(fg) + = g + f + - (4.5)
If each of the operators / and g is Hermitian, then ( fg) + = gf It follows
from this that the operator fg is Hermitian if and only if the factors / and g
commute.
We note that, from the products)^ and gf of two non-commuting Hermitian
operators, we can form an Hermitian operator by taking the symmetrical
combination
iifg+gf)- (4.6)
Such expressions are sometimes needed ; they are called symmetrised pro-
ducts.
It is easy to see that the difference fg—gf is an anti-Hermitian operator
(i.e. one for which the transpose is equal to the complex conjugate taken with
the opposite sign). It can be made Hermitian by multiplying by i; thus
i(fg-gf)
is again an Hermitian operator.
In what follows we shall sometimes use for brevity the notation
{/£> =/£-i/> (4.7)
called the commutator of these operators. It is easily seen that
{j§A = {fM+f{g,h (4.8)
We notice that, if {/, h} = and {g, h) = 0, it does not in general follow
that / and g commute.
§5. The continuous spectrum
All the relations given in §§3 and 4, describing the properties of the eigen-
functions of a discrete spectrum, can be generalised without difficulty to the
case of a continuous spectrum of eigenvalues.
Let / be a physical quantity having a continuous spectrum. We shall
16 The Basic Concepts of Quantum Mechanics §5
denote its eigenvalues by the same letter/ simply, without suffix, in accordance
with the fact that /takes a continuous range of values. We denote by T/ the
eigenfunction corresponding to the eigenvalue /. Just as an arbitrary wave
function Y can be expanded in a series (3.2) of eigenfunctions of a quantity
having a discrete spectrum, it can also be expanded (this time as an integral)
in terms of the complete set of eigenfunctions of a quantity with a continuous
spectrum. This expansion has the form
T( ? ) = J« / T / ( 9 )d/, (5.1)
where the integration is extended over the whole range of values that can be
taken by the quantity/.
The subject of the normalisation of the eigenfunctions of a continuous
spectrum is more complex than in the case of a discrete spectrum. The
requirement that the integral of the squared modulus of the function should
be equal to unity cannot here be satisfied, as we shall see below. Instead,
we try to normalise the functions T/ in such a way that |«/| 2 d/is the prob-
ability that the physical quantity concerned, in the state described by the
wave function Y, has a value between/ and f+df. This is a direct generali-
sation of the case of a discrete spectrum, where the square \a n \ 2 determines
the probability of the eigenvalue f n . Since the sum of the probabilities of all
possible values of / must be equal to unity, we have
J>,| 2 d/=1 (5.2)
(similarly to the relation (3.3) for a discrete spectrum).
Proceeding in exactly the same way as in the derivation of formula (3.5),
and using the same arguments, we can write, firstly,
j y i n ¥*dq = j K| 2 d/
and, secondly,
j YY* dq = jj a f *¥,*¥ dfdq.
By comparing these two expressions we find the formula which determines
the expansion coefficients,
a, = JY(qy¥ f *(q)dq, (5.3)
in exact analogy to (3.5).
To derive the normalisation condition, we now substitute (5.1) in (5.3),
and obtain
a t = \a r (£¥ f W t *dq)df'.
This relation must hold for arbitrary a f , and therefore must be satisfied
identically. For this to be so, it is necessary that, first of all, the coefficient
of a r under the integral sign (i.e. the integral J" Yj'Y/* dq) should be zero for
§5 The continuous spectrum \1
all/' 7^/. For/'=/, this coefficient must become infinite (otherwise the
integral over /' would vanish). Thus the integral J Yy'Y/* dq is a function
of the difference /'■—/, which becomes zero for values of the argument
different from zero and is infinite when the argument is zero. We denote
this function by B(f'—f):
JY/r,*d<z = 8(/'-/). (5.4)
The manner in which the function S(/'— /) becomes infinite for/'— f=
is determined by the fact that we must have
J8(f'-f)a r df = a f .
It is clear that, for this to be so, we must have
j 8(f-f) d/' = l.
The function thus defined is called a delta function, and was first used in
quantum mechanics by P. A. M. Dirac. We shall write out once more the
formulae which define it. They are
S(«) = for x # 0, 8(0) = oo, (5.5)
while
f 8(x) dx = 1. (5.6)
We can take as limits of integration any numbers such that x = lies between
them. If f(x) is some function continuous at x = 0, then
js(*)/(*)d*=/(0). (5.7)
— oo *
This formula can be written in the more general form
j S(x-a)f(x)dx =/(«), (5.8)
where the range of integration includes the point x = a, and/(#) is continuous
at x = a. It is also evident that
S(-x) = 8(x), (5.9)
i.e. the delta function is even. Finally, writing
f8(a*)d*=r8(v)^ = -,
-i -i |a| |a|
we can deduce that
8(«) = (1/M) S(*), (5.10)
where a is any constant.
The formula (5.4) gives the normalisation rule for the eigenfunctions of a
continuous spectrum; it replaces the condition (3.6) for a discrete spectrum.
We see that the functions T/ and Y/» with / # /' are, as before, orthogonal.
18 The Basic Concepts of Quantum Mechanics §5
However, the integrals of the squared moduli |T/| 2 of the functions diverge
for a continuous spectrum.
The functions Yfiq) satisfy still another relation similar to (5.4). To derive
this, we substitute (5.3) in (5.1), which gives
Y(ff) = J W)tf Y/WM 4/) <¥,
whence we can at once deduce that we must have
jV f *(q'W f (q)df=S(q'-q). (5.11)
There is, of course, an analogous relation for a discrete spectrum :
HY n *(q')W n (q)=*(q'-q). (5.12)
n
Comparing the pair of formulae (5.1), (5.4) with the pair (5.3), (5.11), we
see that, on the one hand, the function x P(q) can be expanded in terms of
the functions T/(^) with expansion coefficients «/ and, on the other hand,
formula (5.3) represents an entirely analogous expansion of the function
af = a(f) in terms of the functions x F/*(g r ), while the Y(q) play the part of
expansion coefficients. The function a(f), like y ¥(q) i completely determines
the state of the system; it is sometimes called a wave function in the f repre-
sentation (while the function T(^) is called a wave function in the q representa-
tion). Just as |T(^)| 2 determines the probability for the system to have co-
ordinates lying in a given interval dq, so |«(/)| 2 determines the probability for
the values of the quantity / to lie in a given interval d/. On the one hand,
the functions Yfiq) are the eigenfunctions of the quantity /in the q representa-
tion ; on the other hand, their complex conjugates are the eigenfunctions of
the co-ordinate q in the /representation.
Let<£(/) be some function of the quantity/, such that<£ and /are related in
a one-to-one manner. Each of the functions Y^) can then be regarded as
an eigenfunction of the quantity <f>, corresponding to a value of the latter
determined by <j> = <f>(j). Here, however, the normalisation of these functions
must be changed: the eigenfunctions Y^q) of the quantity <j> must be
normalised by the condition
jT^Wd* = 8[#/')-tf/)],
whereas the functions M^, are normalised by the condition (5.4). The argu-
ment of the delta function becomes zero only for/' = /. As /' approaches/,
we have cf>(f') -<f>(f) = [d<£(/)/d/] . (/' -/). By (5.10) we can therefore writef
«/')-<£(/)] = [AM( \ IA J U'-f)- ( 5 -!3)
t In general, if<f>(x) is some one- valued function (the inverse function need not be one-valued), we
have
8[<f>(x)] = V S(x— aA
where a,- are the roots of the equation <f>(x) = 0.
§5 The continuous spectrum 19
Thus the normalisation condition for the functions T^ can be written in
the form
l
*v>Y„f d q = — -^— -scr-/).
l<W)/d/l
Comparing this with (5.4), we see that the functions T^ and ¥> are related
by
r * w =vmw? f - (5 - 14)
There are also physical quantities which in one range of values have a
discrete spectrum, and in another a continuous spectrum. For the eigen-
functions of such a quantity all the relations derived in this and the previous
sections are, of course, true. It need only be noted that the complete set
of functions is formed by combining the eigenfunctions of both spectra.
Hence the expansion of an arbitrary wave function in terms of the eigenfunc-
tions of such a quantity has the form
Yfe) = S a n r n (q)+ j a f Y f (q) d/, (5.15)
where the sum is taken over the discrete spectrum and the integral over the
whole continuous spectrum.
The co-ordinate q itself is an example of a quantity having a continuous
spectrum. It is easy to see that the operator corresponding to it is simply
multiplication by q. For, since the probability of the various values of the
co-ordinate is determined by the square |Y(?)| 2 , the mean value of the
co-ordinate is q = J q\W\ 2 dq. On the other hand, the mean value of the
co-ordinate must be determined by its operator as q^^T^qYdq. A
comparison of the two expressions shows that the operator q is simply
multiplication by q\ this may be symbolically written in the formf
§ = q- (5.16)
The eigenfunctions of this operator must be determined, according to the
usual rule, by the equation qT g<i = q Y qo , where q temporarily denotes the
actual values of the co-ordinate as distinct from the variable q. Since this
equation can be satisfied either by Y q<i = or by q = q , it is clear that the
eigenfunctions which satisfy the normalisation condition arej
% o = S(q-q ). (5.17)
t In future we shall always, for simplicity, write operators which amount to multiplication by some
quantity in the form of that quantity itself.
J The expansion coefficients for an arbitrary function Y in terms of these eigenfunctions are
\ = \nm<i-qo)dq = V(qo).
The probability that the value of the co-ordinate lies in a given interval dq is
W<, \ 2 dq = mq )\ z dq ,
as it should be.
20 The Basic Concepts of Quantum Mechanics §6
§6. The passage to the limiting case of classical mechanics
Quantum mechanics contains classical mechanics in the form of a certain
limiting case. The question arises as to how this passage to the limit is
made.
In quantum mechanics an electron is described by a wave function which
determines the various values of its co-ordinates ; of this function we so far
know only that it is the solution of a certain linear partial differential equation.
In classical mechanics, on the other hand, an electron is regarded as a material
particle, moving in a path which is completely determined by the equations
of motion. There is an interrelation, somewhat similar to that between
quantum and classical mechanics, in electrodynamics between wave optics
and geometrical optics. In wave optics, the electromagnetic waves are
described by the electric and magnetic field vectors, which satisfy a definite
system of linear differential equations, namely Maxwell's equations. In
geometrical optics, however, the propagation of light along definite paths, or
rays, is considered. Such an analogy enables us to see that the passage from
quantum mechanics to the limit of classical mechanics occurs similarly to the
passage from wave optics to geometrical optics.
Let us recall how this latter transition is made mathematically. Let u be
any of the field components in the electromagnetic wave. It can be written
in the form u = ae i( f> (with a and<£ real), where a is called the amplitude and<£
the phase of the wave. The limiting case of geometrical optics corresponds
to small wavelengths; this is expressed mathematically by saying that <f>
(called in geometrical optics the eikonal) varies by a large amount over short
distances ; this means, in particular, that it can be supposed large in absolute
value.
Similarly, we start from the hypothesis that, to the limiting case of classical
mechanics, there correspond in quantum mechanics wave functions of the
form T = ae**!*, where a is a slowly varying function and <j> takes large values.
As is well known, the path of a particle can be determined in mechanics by
means of the variational principle, according to which what is called the
action S of a mechanical system must take its least possible value (the principle
of least action, or Hamilton's principle). In geometrical optics the path of the
rays is determined by what is called Fermat's principle, according to which the
optical path length of the ray, i.e. the difference between its phases at the
beginning and end of the path, must take its least (or greatest) possible
value.
On the basis of this analogy, we can assert that the phase <j> of the wave
function, in the limiting (classical) case, must be proportional to the mech-
anical action S of the physical system considered, i.e. we must have
S = constant X <f>. The constant of proportionality is called Planck's con-
stanff and is denoted by h. It has the dimensions of action (since <f> is
t It was introduced into physics by M. Planck in 1900. The constant h, which we use everywhere
in this book, is, strictly speaking, Planck's constant divided by 2n.
§7 The wave function and measurements 21
dimensionless) and has the value
h = 1-054 xlO- 27 erg sec.
Thus, the wave function of an "almost classical" (or, as we say, quasi-
classical) physical system has the form
T = ae^/K. (6.1)
Planck's constant h plays a fundamental part in all quantum phenomena.
Its relative value (compared with other quantities of the same dimensions)
determines the "extent of quantisation" of a given physical system. The
transition from quantum mechanics to classical mechanics, corresponding to
large phase, can be formally described as a passage to the limit h ->0 (just
as the transition from wave optics to geometrical optics corresponds to a
passage to the limit of zero wavelength, A -> 0).
We have ascertained the limiting form of the wave function, but the
question still remains how it is related to classical motion in a path. In
general, the motion described by the wave function does not tend to motion
in a definite path. Its connection with classical motion is that, if at some
initial instant the wave function, and with it the probability distribution of
the co-ordinates, is given, then at subsequent instants this distribution will
change according to the laws of classical mechanics (for a more detailed dis-
cussion of this, see the end of §17).
In order to obtain motion in a definite path, we must start from a wave
function of a particular form, which is perceptibly different from zero only
in a very small region of space (what is called a wave packet); the dimensions
of this region must tend to zero with h. Then we can say that, in the quasi-
classical case, the wave packet will move in space along a classical path of a
particle.
Finally, quantum-mechanical operators must reduce, in the limit, simply
to multiplication by the corresponding physical quantity.
§7. The wave function and measurements
Let us again return to the process of measurement, whose properties have
been qualitatively discussed in §1 ; we shall show how these properties are
related to the mathematical formalism of quantum mechanics.
We consider a system consisting of two parts: a classical apparatus and
an electron (regarded as a quantum object). The process of measurement
consists in these two parts' coming into interaction with each other, as a
result of which the apparatus passes from its initial state into some other;
from this change of state we draw conclusions concerning the state of the'
electron. The states of the apparatus are distinguished by the values of some
physical quantity (or quantities) characterising it— the "readings of the ap-
paratus". We conventionally denote this quantity by g, and its eigenvalues
by g n ; these take in general, in accordance with the classical nature of the
apparatus, a continuous range of values, but we shall— merely in order to
22 The Basic Concepts of Quantum Mechanics §7
simplify the subsequent formulae — suppose the spectrum discrete. The
states of the apparatus are described by means of quasi-classical wave func-
tions, which we shall denote by <& n (£), where the suffix n corresponds to the
"reading" g n of the apparatus, and £ denotes the set of its co-ordinates. The
classical nature of the apparatus appears in the fact that, at any given instant,
we can say with certainty that it is in one of the known states <D n with some
definite value of the quantity g\ for a quantum system such an assertion
would, of course, be unjustified.
Let O (£) be the wave function of the initial state of the apparatus (before
the measurement), and T(^) some arbitrary normalised initial wave function
of the electron (q denoting its co-ordinates). These functions describe the
state of the apparatus and of the electron independently, and therefore the
initial wave function of the whole system is the product
Y(?)O (|). (7.1)
Next, the apparatus and the electron interact with each other. Applying the
equations of quantum mechanics, we can in principle follow the change of
the wave function of the system with time. After the measuring process it
may not, of course, be a product of functions of £ and q. Expanding the
wave function in terms of the eigenfunctions O n of the apparatus (which
form a complete set of functions), we obtain a sum of the form
XA n {q)® n (Z), (7-2)
where the A n (q) are some functions of q.
The classical nature of the apparatus, and the double role of classical
mechanics as both the limiting case and the foundation of quantum mechanics,
now make their appearance. As has been said above, the classical nature of
the apparatus means that, at any instant, the quantity g (the "reading of the
apparatus") has some definite value. This enables us to say that the state
of the system apparatus + electron after the measurement will in actual fact
be described, not by the entire sum (7.2), but by only the one term which
corresponds to the "reading" g n of the apparatus,
4.(«)*.tf). ( 7 - 3 )
It follows from this that A n {q) is proportional to the wave function of the
electron after the measurement. It is not the wave function itself, as is seen
from the fact that the function A n (q) is not normalised. It contains both
information concerning the properties of the resulting state of the electron
and the probability (determined by the initial state of the system) of the
occurrence of the nth "reading" of the apparatus.
Since the equations of quantum mechanics are linear, the relation between
A n {q) and the initial wave function of the electron Y(q) is in general given by
some linear integral operator :
A n (q) = JK n (q,q'W(q')dq', (7.4)
§7 The wave function and measurements 23
with a kernel K n (q, q') which characterises the measurement process con-
cerned.
We shall suppose that the measurement concerned is such that it gives a
complete description of the state of the electron. In other words (see §1),
in the resulting state the probabilities of all the quantities must be indepen-
dent of the previous state of the electron (before the measurement). Mathe-
matically, this means that the form of the functions A n (q) must be determined
by the measuring process itself, and does not depend on the initial wave
function Y(q) of the electron. Thus the A n must have the form
Mfi = " n <f>n(q), (7.5)
where the <f> n are definite functions, which we suppose normalised, and only
the constants a^ depend on T(^). In the integral relation (7.4) this corresponds
to a kernel K n {q, q') which is a product of a function of q and a function of q' :
Kn{q,q') = <i>n{qWn*(q'y, (7.6)
then the linear relation between the constants a n and the function Y(q) is
a^J^WF.-fo)^, (7.7)
where the T n (^) are certain functions depending on the process of measure-
ment.
The functions <j> n {q) are the normalised wave functions of the electron after
measurement. Thus we see how the mathematical formalism of the theory
reflects the possibility of finding by measurement a state of the electron
described by a definite wave function.
If the measurement is made on an electron with a given wave function
f (q), the constants a n have a simple physical meaning: in accordance with
the usual rules, |aj 2 is the probability that the measurement will give the
nth result. The sum of the probabilities of all results is equal to unity:
?KI 2 = 1. (7.8)
In order that equations (7.7) and (7. 8) should hold for an arbitrary normalised
function W(q\ it is necessary (cf. §3) that an arbitrary function T(g) can be
expanded in terms of the functions Y n (y). This means that the functions
^n(q) form a complete set of normalised and orthogonal functions.
If the initial wave function of the electron coincides with one of the func-
tions Y n (g), then the corresponding constant a n is evidently equal to unity,
while all the others are zero. In other words, a measurement made on an
electron in the state T n (#) gives with certainty the nth result.
All these properties of the functions Y n (q) show that they are the eigen-
functions of some physical quantity (denoted by /) which characterises the
electron, and the measurement concerned can be spoken of as a measurement
of this quantity.
It is very important to notice that the functions T TC (^) do not, in general,
24 The Basic Concepts of Quantum Mechanics §7
coincide with the functions <f> n (q) ; the latter are in general not even mutually
orthogonal, and do not form a set of eigenfunctions of any operator. This
expresses the fact that the results of measurements in quantum mechanics
cannot be reproduced. If the electron was in a state T n (^), then a measure-
ment of the quantity / carried out on it leads with certainty to the value f n .
After the measurement, however, the electron is in a state c/) n (q) different
from its initial one, and in this state the quantity / does not in general take
any definite value. Hence, on carrying out a second measurement on the
electron immediately after the first, we should obtain for /a value which did
not agree with that obtained from the first measurement.-f To predict (in the
sense of calculating probabilities) the result of the second measurement from
the known result of the first, we must take from the first measurement the
wave function <j> n {q) of the state in which it resulted, and from the second
measurement the wave function T n (^) of the state whose probability is re-
quired. This means that from the equations of quantum mechanics we deter-
mine the wave function c/> n (q, t) which, at the instant when the first measure-
ment is made, is equal to<£ n (<?) ; the probability of the mth result of the second
measurement, made at time t, is then given by the squared modulus of the
integral J^ n ( ?l t)Y m *(q) 6q.
We see that the measuring process in quantum mechanics has a "two-
faced" character: it plays different parts with respect to the past and future
of the electron. With respect to the past, it "verifies" the probabilities of the
various possible results predicted from the state brought about by the previ-
ous measurement. With respect to the future, it brings about a new state
(see also §44). Thus the very nature of the process of measurement involves
a far-reaching principle of irreversibility.
This irreversibility is of fundamental significance. We shall see later (at
the end of §18) that the basic equations of quantum mechanics are in them-
selves symmetrical with respect to a change in the sign of the time; here
quantum mechanics does not differ from classical mechanics. The irrever-
sibility of the process of measurement, however, causes the two directions
of time to be physically non-equivalent, i.e. creates a difference between the
future and the past.
t It must be remarked that there is an important exception to the statement that results of measure-
ments cannot be reproduced : the one quantity the result of whose measurement can be exactly re-
produced is the co-ordinate. Two measurements of the co-ordinates of an electron, made at a sufficiently
small interval of time, must give neighbouring values; if this were not so, it would mean that the
electron had an infinite velocity. Mathematically, this is related to the fact that the co-ordinate
commutes with the operator of the interaction energy between the electron and the apparatus, since this
energy is (in non-relativistic theory) a function of the co-ordinates only.
CHAPTER II
ENERGY AND MOMENTUM
§8. The Hamiltonian operator
The wave function Y completely determines the state of a physical system
in quantum mechanics. This means that, if this function is given at some
instant, not only are all the properties of the system at that instant described,
but its behaviour at all subsequent instants is determined (only, of course, to
the degree of completeness which is generally admissible in quantum mech-
anics). The mathematical expression of this fact is that the value of the deri-
vative <W/dt of the wave function with respect to time at any given instant
must be determined by the value of the function itself at that instant, and,
by the principle of superposition, the relation between them must be linear.
In the most general form we can write
where L is some linear operator; the factor i is introduced here for conveni-
ence.
We shall derive some properties of the operator L. Since the integral
J YY* dq is a constant independent of time, we have
Substituting here dYjdt = -iD¥, dY*{dt = it*W* and using in the first
integral the definition of the transpose of an operator, we can write
J YZ*Y* dq - J Y*&F dq = j Y*Z*Y dq - j >¥*&¥ dq
= J x F*(l*-l)Wdq = 0.
Since this equation must hold for an arbitrary function T, it follows that we
must have identically Z* — L = 0, or
The operator L is therefore Hermitian. Let us find the classical quantity to
which the operator L corresponds. To do this, we use the limiting expression
(6.1) for the wave function and write
BY _ i dS
~dt~h~dt '
the slowly varying amplitude a need not be differentiated. Comparing this
25
26 Energy and Momentum §9
equation with the definition dYjdt = —iD¥, we see that, in the limiting
case, the operator L reduces to simply multiplying by — (l/#) dSjdt. This
means that —(1//*) dSjdt is the physical quantity into which the Hermitian
operator L passes.
As is well known from mechanics, the derivative —dSjdt is just Hamilton's
function H for a mechanical system. Thus the operator hL is what corres-
ponds in quantum mechanics to Hamilton's function; this operator, which
we shall denote by i?, is called the Hamiltonian operator or, more briefly, the
Hamiltonian of the system. The relation between dY/dt and T is
ihd y ¥ldt=fi K ¥. (8.1)
If the form of the Hamiltonian is known, equation (8.1) determines the wave
functions of the physical system concerned. This fundamental equation of
quantum mechanics is called the wave equation.
§9. The differentiation of operators with respect to time
The concept of the derivative of a physical quantity with respect to time
cannot be denned in quantum mechanics in the same way as in classical mech-
anics. For the definition of the derivative in classical mechanics involves
the consideration of the values of the quantity at two neighbouring but
distinct instants of time. In quantum mechanics, however, a quantity which
at some instant has a definite value does not in general have definite values at
subsequent instants; this was discussed in detail in §1.
Hence the idea of the derivative with respect to time must be differently
defined in quantum mechanics. It is natural to define the derivative / of a
quantity / as the quantity whose mean value is equal to the derivative, with
respect to time, of the mean value /. Thus we have the definition
/=/ (9-1)
Starting from this definition, it is easy to obtain an expression for the
quantum-mechanical operator / corresponding to the quantity /. Since
f = jY*fYdq,
Here dfjdt is the operator obtained by differentiating the operator / with
respect to time; /may depend on the time as a parameter. Substituting for
BYjdt, dW^/dt their expressions according to (8.1), we obtain
/= [V*— Wdq+- f (#*Y*)/Td?-^ (wf(fr¥)dq.
J dt hj n J
Since the operator i? is Hermitian, we have
f (i?*T*)(/T) dq = JT*^/T dq:
§10 Stationary states 27
thus
/ -J**S + i* L >> d »-
Since, on the other hand, we must have, by the definition of mean values,
/ = J Y*/ Tdg-, it is seen that the expression in parentheses under the integral
is the required operator /:f
We notice that, if the operator / is independent of time, / reduces, apart
from a constant factor, to the result of commuting the operator / with the
Hamiltonian.
A very important class of physical quantities is formed by those whose
operators do not depend explicitly on time, and also commute with the
Hamiltonian, so that / = 0. Such quantities are said to be conserved.
If the operator /is identically zero, then/ =/= 0, that is, /is constant.
In other words, the mean value of the quantity remains constant in time.
We can also assert that, if in a given state the quantity /has a definite value
(i.e. the wave function is an eigenfunction of the operator /), then it will
have a definite value (the same one) at subsequent instants also.
§10. Stationary states
If the system is not in a varying external field, its Hamiltonian cannot
contain the time explicitly. This follows at once from the fact that, in the
:+ 5& + |f<>
f In classical mechanics we have for the total derivative, with respect to time, of a quantity / which
is a function of the generalised co-ordinates g, and momenta p { of the system
dt dt
Substituting, in accordance with Hamilton's equations, q t = dH/dpi and p t = —dHldq t , we obtain
d//d* = df/dt+[H,f],
where
Z-w \dq t dpi dpi dqtJ
i
is what is called the Poisson bracket for the quantities /and H (see Mechanics, §42). On comparing with
the expression (9.2), we see that, as we pass to the limit of classical mechanics, the operator i($f—fff)
reduces in the first approximation to zero, as it should, and in the second approximation (with respect
to K) to the quantity H[H,f]. This result is true also for any two quantities / and g; the operator
i(fg-gf) tends in the limit to the quantity h[f, g] , where [/, g] is the Poisson bracket
^L,\dqt dpi dpidqt)'
This follows at once from the fact that we can always formally imagine a system whose Hamiltonian
28 Energy and Momentum §10
absence of an external field (or in a constant external field) all times are
equivalent so far as the given physical system is concerned. Since, on the
other hand, any operator of course commutes with itself, we reach the con-
clusion that Hamilton's function is conserved for systems which are not in a
varying external field. As is well known, a Hamilton's function which is
conserved is called the energy. Thus we have the law of conservation of
energy in quantum mechanics. Here it signifies that, if in a given state the
energy has a definite value, this value remains constant in time.
States in which the energy has definite values are called stationary states
of a system. They are described by wave functions T n which are the eigen-
functions of the Hamiltonian operator, i.e. which satisfy the equation
i?T n = E^¥ n , where E n are the eigenvalues of the energy. Correspondingly,
the wave equation (8.1) for the function T n ,
ihdW n ldt=8Y n = E n y n
can be integrated at once with respect to time and gives
T n = erU/*>*:ntf n (q), (10.1)
where ip n is a function of the co-ordinates only. This determines the relation
between the wave functions of stationary states and the time.
We shall denote by the small letter i/j the wave functions of stationary states
without the time factor. These functions, and also the eigenvalues of the
energy, are determined by the equation
ify = E+. (10.2)
The stationary state with the smallest possible value of the energy is called
the normal or ground state of the system.
The expansion of an arbitrary wave function \F in terms of the wave func-
tions of stationary states has the form
T = S a n e-QM E «tif,Jq). (10.3)
n
The squared moduli \a n \ 2 of the expansion coefficients, as usual, determine
the probabilities of various values of the energy of the system.
The probability distribution for the co-ordinates in a stationary state is
determined by the squared modulus pFJ 2 = |</r n | 2 ; we see that it is indepen-
dent of time. The same is true of the mean values
/=jT n */Y n d ? = J^ n */0 n d 9
of any physical quantity / (whose operator does not depend explicitly on the
time), and therefore of the probabilities of its various values.
As has been said, the operator of any quantity that is conserved commutes
with the Hamiltonian. This means that any physical quantity that is con-
served can be measured simultaneously with the energy.
§10 Stationary states 29
Among the various stationary states, there may be some which correspond
to the same value of the energy, but differ in the values of some other physical
quantities. Such eigenvalues of the energy (or, as we say, energy levels of
the system), to which several different stationary states correspond, are said
to be degenerate. Physically, the possibility that degenerate levels can exist
is related to the fact that the energy does not in general form by itself a
complete set of physical quantities.
In particular, it is easy to see that, if there are two conserved physical
quantities /and g whose operators do not commute, then the energy levels of
the system are in general degenerate. For, let $ be the wave function of a
stationary state in which, besides the energy, the quantity /also has a definite
value. Then we can say that the function gifi does not coincide (apart from a
constant factor) with ip; if it did, this would mean that the quantity g also
had a definite value, which is impossible, since / and g cannot be measured
simultaneously. On the other hand, the function gip is an eigenfunction of the
Hamiltonian, corresponding to the same value E of the energy as ifs:
&m =§&+ = %&!>)•
Thus we see that the energy E corresponds to more than one eigenfunction,
i.e. the energy level is degenerate.
It is clear that any linear combination of wave functions corresponding
to the same degenerate energy level is also an eigenfunction for that value of
the energy. In other words, the choice of eigenfunctions of a degenerate
energy level is not unique. Arbitrarily selected eigenfunctions of a degener-
ate energy level are not, in general, orthogonal. By a proper choice of linear
combinations of them, however, we can always obtain a set of orthogonal
(and normalised) eigenfunctions (and this can be done in infinitely many
ways; for the number of independent coefficients in a linear transformation
of n functions is n 2 , while the number of normalisation and orthogonality
conditions for n functions is \n{n-\-\), i.e. less than w 2 ).
These statements concerning the eigenfunctions of a degenerate energy
level relate, of course, not only to eigenfunctions of the energy, but also to
those of any operator. Thus only those functions are automatically ortho-
gonal which correspond to different eigenvalues of the operator concerned ;
functions which correspond to the same degenerate eigenvalue are not in
general orthogonal.
If the Hamiltonian of the system is the sum of two (or more) parts,
H = ^+^2, one of which contains only the co-ordinates q x and the other
only the co-ordinates q 2 , then the eigenfunctions of the operator H can be
written down as products of the eigenfunctions of the operators 3 X and i? 2 ,
and the eigenvalues of the energy are equal to the sums of the eigenvalues of
these operators.
The spectrum of eigenvalues of the energy may be either discrete or
continuous. A stationary state of a discrete spectrum always corresponds to
a finite motion of the system, i.e. one in which neither the system nor any
30 Energy and Momentum §11
part of it moves off to infinity. For, with eigenfunctions of a discrete spec-
trum, the integral j |Y| 2 dq, taken over all space, is finite. This certainly
means that the squared modulus |Y| 2 decreases quite rapidly, becoming
zero at infinity. In other words, the probability of infinite values of the co-
ordinates is zero ; that is, the system executes a finite motion, and is said to
be in a bound state.
For wave functions of a continuous spectrum, the integral J" |T| 2 dq diverges.
Here the squared modulus |T| 2 of the wave function does not directly deter-
mine the probability of the various values of the co-ordinates, and must be
regarded only as a quantity proportional to this probability. The divergence
of the integral J" |T| 2 dq is always due to the fact that |T| 2 does not become
zero at infinity (or becomes zero insufficiently rapidly). Hence we can say
that the integral J" |*F| 2 dq, taken over the region of space outside any arbi-
trarily large but finite closed surface, will always diverge. This means that,
in the state considered, the system (or some part of it) is at infinity. For a
wave function which is a superposition of the wave functions of various
stationary states of a continuous spectrum, the integral J" pF| 2 dq may
converge, so that the system lies in a finite region of space. However, it can
be shown that, in the course of time, this region moves unrestrictedly, and
eventually, the system moves off to infinity.^ Thus the stationary states of a
continuous spectrum correspond to an infinite motion of the system.
§11. Matrices
We shall suppose for convenience that the system considered has a discrete
energy spectrum ; all the relations obtained below can be generalised at once
to the case of a continuous spectrum. Let *P = 1,a n , ¥ n be the expansion of
an arbitrary wave function in terms of the wave functions \F n of the stationary
states. If we substitute this expansion in the definition /= jW^f^V dq of
| This can be seen as follows. The superposition of wave functions of a continuous spectrum has
the form
T = f aEe-WEtyEiq) d£.
The squared modulus of *F can be written in the form of a double integral:
|T| 2 = J* J" a E a E '*^l^ E '- E ^iljE{q)4'E-*{q) dEdE'.
If we average this expression over some time interval T, and then let T tend to infinity, the mean values
of the oscillating factors eC/AK- 8 '-*) 1 , and therefore the whole integral, tend to zero in the limit. Thus
the mean value, with respect to time, of the probability of finding the system at any given point of
configuration space tends to zero. This is possible only if the motion takes place throughout infinite
space.
We note that, for a function *F which is a superposition of functions of a discrete spectrum, we should
have
|T| 2 = SS a n a m *e^l^ m -^t^ m * = S \a n ^ n {q)\\
nm n
i.e. the required probability remains finite on averaging over time.
§11 Matrices 31
the mean value of some quantity /, we obtain
f=maSa m f um (t)> (H.1)
n m
where f nm (t) denotes the integral
fnJt) = \V n *fV m dq. (11.2)
The set of quantities f nm (t) with all possible n and m is called the matrix^ of the
quantity/, and each of the/ nm (i) is called the matrix element corresponding
to the transition from state n to state m.%
The dependence of the matrix elements f nm (f) on time is determined (if
the operator / does not contain the time explicitly) by the dependence of the
functions T n on time. Substituting for them the expressions (10.1), we find
that
UJt) =f nm e ia, ™ t , (11.3)
where
"nm = (E n -E m )/h (11.4)
is what is called the transition frequency between the states n and m, and the
quantities
f nm = j<l>«*fK*<l (H.5)
form the matrix of the quantity / which is independent of time, and which
is commonly used. j| We note that the "frequencies" co nm satisfy the ob-
vious relation
w» m +w«i = w„|. (11.6)
The matrix elements of the derivative / are obtained by differentiating the
matrix elements of the quantity / with respect to time; this follows directly
from the fact that the mean value / is equal to /, i.e.
/= 22 a n *a m f ntn (t).
nm
From (11.3) we thus have for the matrix elements of/
fnmit) = i<*>nmfn m (t) (H-7)
or (cancelling the time factor e <a W from both sides) for the matrix elements
independent of time
(/)«m = io>nmfnm = (ijH)(E n -E m )f nm . (11,8)
f The matrix representation of physical quantities was introduced by Heisenberg in 1925, before
Schrodinger's discovery of the wave equation. "Matrix mechanics" was later developed by M.
Born, W. Heisenberg and P. Jordan.
J In some cases, when each of the suffixes n and m has to be written in the form of several letters,
we shall use the notation / m n instead of/ nm . The notation (n\f \m) is also used.
|| It must be borne in mind that, because of the indeterminacy of the phase factor in normalized
Wave functions (see §2), the matrix elements/„ m (and/ nflt (i)) also are determined only to within a factor
of the form eihm-ar). Here again this indeterminacy has no effect on any physical results.
32 Energy and Momentum §11
To simplify the notation in the formulae, we shall derive all our relations
below for the matrix elements independent of time ; exactly similar relations
hold for the matrices which depend on the time.
For the matrix elements of the complex conjugate/* of the quantity/ we
obtain, taking into account the definition of the Hermitian conjugate operator,
(/•).« = / 4>n*f + *l> m dq = j + n *f*K dq = j tJ*t n * dq -
or
(f*)nm = (fmn)*. (11.9)
For real physical quantities, which are the only ones we usually consider,
we consequently have
/nra = /m« (11.10)
{fmn* stands for (f mn )*). Such matrices, like the corresponding operators,
are said to be Hermitian.
Matrix elements with n = m are called diagonal elements. These are
independent of time, and (11.10) shows that they are real. The element f nn
is the mean value of the quantity /in the state ip n .
It is not difficult to obtain the "multiplication rule" for matrices. To do
so, we first observe that the formula
fy n = $f mn if> m (11.11)
holds. This is simply the expansion of the function fip n in terms of the func-
tions ift m , the coefficients being determined in accordance with the general
formula (3.5). Remembering this formula, let us write down the result of
the product of two operators acting on the function ip n :
fg'Pn = ftg*pn) = / S g kn *fr k = 2 g kn fy k =2 g kn f mk ^ m .
ic k fc,m
Since, on the other hand, we must have
fgtn = S {fg) m rrtm,
we arrive at the result that the matrix elements of the product fg are deter-
mined by the formula
(fg)mn = Xfmkgkn- (H-12)
This rule is the same as that used in mathematics for the multiplication of
matrices.
If the matrix is given, then so is the operator itself. In particular, if the
matrix is given, it is in principle possible to determine the eigenvalues of the
physical quantity concerned and the corresponding eigenf unctions.
We shall now consider the values of all quantities at some definite instant,
and expand an arbitrary wave function \F (at that instant) in terms of the
eigenfunctions of Hamilton's operator H, i.e. of the wave functions ip m of
the stationary states (these wave functions are independent of time) :
T=S^ m , (11.13)
m
§11 Matrices 33
where the expansion coefficients are denoted by c m . We substitute this expan-
sion in the equation /T =f¥ which determines the eigenvalues and eigen-
functions of the quantity/. We have
2 Cmifym) —f 2 C m ijj m .
m m
We multiply both sides of this equation by iff n * and integrate over q. Each
of the integrals J" i/* n *fy m d<7 on the left-hand side of the equation is the cor-
responding matrix element f nm . On the right-hand side, all the integrals
J il/ n *tp m &q with m ^ n vanish by virtue of the orthogonality of the functions
tp m , and J n *j/f n dq = 1 by virtue of their normalisation. Thus
^JnmCm == J^m
m
or
S(/„ m -/S nm V m = 0, (11.14)
m
where 8 nm = for m # n and = 1 for m = n.
Thus we have obtained a system of homogeneous algebraic equations of
the first degree (with the c m as unknowns). As is well known, such a system
has solutions different from zero only if the determinant formed by the
coefficients in the equations vanishes, i.e. only if
I fnm—finml = 0-
The roots of this equation (in which / is regarded as the unknown) are the
possible values of the quantity/. The set of values c m satisfying the equations
(11.14) when /is equal to any of these values determines the corresponding
eigenf unction.
If, in the definition (11.5) of the matrix elements of the quantity/, we take
as ift n the eigenfunctions of this quantity, then from the equation ftfi n =f n ijj n
we have
fnm = J «An*/«A«» d? = f m J >f> n *tp m d?.
By virtue of the orthogonality and normalisation of the functions j/r m , this
gives f nm = for n ^ m and f mm = f m . Thus only the diagonal matrix
elements are different from zero, and each of these is equal to the correspond-
ing eigenvalue of the quantity/. A matrix with only these elements different
from zero is said to be put in diagonal form. In particular, in the usual
representation, with the wave functions of the stationary states as the functions
iff n , the energy matrix is diagonal (and so are the matrices of all other physical
quantities having definite values in the stationary states). In general, the
matrix of a quantity /, defined with respect to the eigenfunctions of some
operator £, is said to be the matrix of fin a representation in which g is diagonal.
We shall always, except where the subject is specially mentioned, understand
in future by the matrix of a physical quantity its matrix in the usual repre-
sentation, in which the energy is diagonal. Everything that has been said
34 Energy and Momentum §11
above regarding the dependence of matrices on time refers, of course, only
to this usual representation.f
By means of the matrix representation of operators we can prove the
theorem mentioned in §4: if two operators commute with each other, they
have their entire sets of eigenfunctions in common. Let / and g be two
such operators. From fg = gf and the matrix multiplication rule (11.12),
it follows that
jfjmkSkn = 4? gmkjkn-
If we take the eigenfunctions of the operator /as the set of functions ift n with
respect to which the matrix elements are calculated, we shall have f mk =
for m # k, so that the above equation reduces to f mm g mn = g mn f nnl or
SmnKJm Jn) == U.
If all the eigenvalues f n of the quantity / are different, then for all m ^n we
nave fm—fn # 0» so that we must have g mn = 0. Thus the matrix g mn is
also diagonal, i.e. the functions ifj n are eigenfunctions of the physical quantity
g also. If, among the values / n , there are some which are equal (i.e. if there
are eigenvalues to which several different eigenfunctions correspond), then
the matrix elements g mn corresponding to each such group of functions tp n
are, in general, different from zero. However, linear combinations of the
functions ifi n which correspond to a single eigenvalue of the quantity / are
evidently also eigenfunctions of/; one can always choose these combinations
in such a way that the corresponding non-diagonal matrix elements g mn are
zero, and thus, in this case also, we obtain a set of functions which are
simultaneously the eigenfunctions of the operators / and g.
The matrix f nm can be regarded as the operator /in the energy representa-
tion. For the set of coefficients c n in the expansion (11.13) in terms of the
eigenfunctions ip n of the Hamiltonian can be considered (cf. §5) as the wave
function in the "E representation" (the variable being the suffix n which
gives the number of the stationary state). The formula
n m
for the mean value of the quantity/ then corresponds to the general expression
for the quantum-mechanical mean value of a quantity in terms of its operator
and the wave function of the state concerned.
PROBLEM
The Hamiltonian of a system, and therefore the eigenvalues E n of the energy, are functions
of some parameter A. Show that
(0H/SA)„„ = dE n ldX.
Solution. Differentiating the equation (&—E n )>]>n = with respect to A and then
f Bearing in mind the diagonality of the energy matrix, it is easy to see that equation (11.8) is the
operator relation (9.2) written in matrix form.
§12 Transformation of matrices 35
multiplying on the left by <fi n *, we obtain
On integration with respect to q, the left-hand side gives zero, since the operator i? is Hermitian
and therefore
J rf, n *(fi-E n )—dq = J ~{n-E n )*4, n *dq.
The right-hand side gives the required equation.
§12. Transformation of matrices
The matrix elements of a given physical quantity can be defined with
respect to various sets of wave functions, for example the wave functions of
stationary states described by various sets of physical quantities, or the wave
functions of stationary states of the same system in various external fields.
The problem therefore arises of the transformation of matrices from one
representation to another.
Let ip n (q) and ifs n '{q) (n = 1,2,...) be two complete sets of orthonormal
functions, related by some linear transformation :
ifjn = 2 S mn tfi m , (12.1)
which is simply an expansion of the function i// n ' in terms of the complete set
of functions ip n . This transformation may be conventionally written in the
operator form
«£«' = fyn. (12.2)
The operator S must satisfy a certain condition in order that the functions
tpn should be orthonormal if the functions ifs n are. Substituting (12.2) in
the condition
J tym*$n &q = 8 m n,
and using the definition of the transposed operator (3.14), we have
j (Sfa)£f m * dq = j $„*§•£$* dq = B mn .
If these equations hold for all m and n, we must have §*§ = 1, or
£* = §+ = £-i f (12.3)
i.e. the inverse operator is equal to the Hermitian conjugate operator.
Operators having this property are said to be unitary. Owing to this property,
the transformation iji n = S- 1 ^' inverse to (12.1) is given by
tffn = 2 S nm *xfj m '. (12.4)
36 Energy and Momentum §12
Writing the equations 3 + S = 1 and SS + = 1 in matrix form, we obtain
the following forms of the unitarity condition :
S Sim* Sin = 8 mn , (12.5)
E S m i*S n i = 8 mn . (12.6)
Let us now consider some physical quantity / and write down its matrix
elements in the "new" representation, i.e. with respect to the functions ip n '.
These are given by the integrals
j 0m'*$n' d? = J (S*iP m *)(fSiP n ) dq
= J ^m*S*f§iJj n dq
= jif>m*S-lfSifj n dq.
Hence we see that the matrix of the operator / in the new representation is
equal to the matrix of the operator
/' = S-ifS (12.7)
in the old representation.
The sums of the diagonal elements of matrices are of importance in certain
calculations in quantum mechanics. Such a sum is called the trace or spur\
of the matrix and denoted by tr/:
tr/=2/ ww . (12.8)
It may be noted first of all that the trace of a product of two matrices is
independent of the order of multiplication :
trtfc) = trfe/), (12-9)
since the rule of matrix multiplication gives
tr (fg) = 2 Yffnkgkn = S "Zg/cnfnk = tr (gf).
Similarly we can easily see that, for a product of several matrices, the trace
is unaffected by a cyclic permutation of the factors; for example,
tr(fgh) = tr(hfg) = tr(ghf). (12.10)
An important property of the trace is that it does not depend on the choice
of the set of functions with respect to which the matrix elements are defined,
since
(tr/)' = tr (S-ifS) = tr (SS^f) = tr/.
t From the German word Spur. The notation sp / is also used. The trace can be defined, of course,
only if the sum over n is convergent; we shall assume that this condition is satisfied.
§13 The Heisenberg representation of operators 37
§13. The Heisenberg representation of operators
In the mathematical formalism of quantum mechanics described here, the
operators corresponding to various physical quantities act on functions of the
co-ordinates and do not usually depend explicitly on time. The time depen-
dence of the mean values of physical quantities is due only to the time
dependence of the wave function of the state, according to the formula
f(t) = jW*(q,t)/T(q,t)dq. (13.1)
The quantum-mechanical treatment can, however, be formulated also in a
somewhat different but equivalent form, in which the time dependence is
transferred from the wave functions to the operators. Although we shall not
use this Heisenberg representation of operators in the present volume, a state-
ment of it is given here with a view to applications in the relativistic theory.
We define the operator
S = exp[-(*/£)#*], ( 13-2 )
where R is the Hamiltonian of the system. By definition, its eigenfunctions
are the same as those of the operator fi, i.e. the stationary-state wave functions
*ffn(q)y where
%(<?) = e-wn)E n t Mq) , (13>3)
Hence it follows that the expansion (10.3) of an arbitrary wave function
T in terms of the stationary-state wave functions can be written in the
operator form
V(q,t) = &¥(q,0), (13.4)
i.e. the effect of the operator £ is to convert the wave function of the system
at some initial instant into the wave function at an arbitrary instant.
According to the definition (3.16), using the fact that the operator fi is
Hermitian, we have
S+ = ex pfeM = expZ-tfA = £-1,
i.e. 3 is a unitary operator, as it should be, since formula (13.4) (with the
time t as a parameter) may be regarded as a particular case of the transforma-
tion (12.1).
Defining, as in (12.7), the time-dependent operator
, /(*) = &-W, (13.5)
we have v '
f(t) = jV*(q,0)f(t)Y(q,0)dq, (13. 6 )
and thus obtain the formula (3.8) for the mean value of the quantity / in a
form in which the time dependence is entirely transferred to the operator (for
our definition of an operator rests on formula (3.8)).
38 Energy and Momentum §14
It is evident that the matrix elements of the operator (13.5) with respect
to the stationary-state wave functions are the same as the time-dependent
matrix elements fnm(t) defined by formula (11.3).
Finally, differentiating the expression (13.5) with respect to time (assuming
that the operators / and H do not themselves involve t), we obtain
U{t) = Utif-fti), (13.7)
ot n
which is similar in form to (9.2) but has a somewhat different significance:
the expression (9.2) defines the operator / corresponding to the physical
quantity/, while the left-hand side of equation (13.7) is the time derivative
of the operator of the quantity / itself.
§14. The density matrix
Let us consider a system which is a part of some closed system. We
suppose that the closed system as a whole is in some state described by the
wave function Y(<7, x), where x denotes the set of co-ordinates of the system
considered, and q the remaining co-ordinates of the closed system. This
function in general does not fall into a product of functions of x and of q
alone, so that the system does not have its own wave function, f
Let / be some physical quantity pertaining to the system considered. Its
operator therefore acts only on the co-ordinates x, and not on q. The mean
value of this quantity in the state considered is
/= J/ Y»fo, *)/*•(&*) dfld*. (14.1)
We introduce the function p(x', x) defined by
9 {x\ x) = j Y*(2, x')W(q, x) dq, (14.2)
where the integration is extended only over the co-ordinates q ; this function
is called the density matrix of the system. From the definition (14.2) it is
evident that the function is "Hermitian":
P *(x,x') = P (x',x). (14.3)
The "diagonal elements" of the density matrix
p(x,x) = j\¥(q t x)\*6q
evidently determine the probability distribution for the co-ordinates of the
system.
f In order that W(q, x) should (at a given instant) fall into such a product, the measurement as a
result of which this state was brought about must completely describe the system considered and the
remainder of the closed system separately. In order that Wiq, x) should continue to have this form at
subsequent instants, it is necessary in addition that these parts of the closed system should not interact
(see §2). Neither of these conditions is now assumed.
§14 The density matrix 39
Using the density matrix, the mean value / can be written in the form
?= j\j P (x',x)] x >= x dx. (14.4)
Here / acts only on the variables x in the function p(x', x) ; after calculating
the result of its action, we put x' = x. We see that, if we know the density
matrix, we can calculate the mean value of any quantity characterising the
system. It follows from this that, by means of p(x', x), we can also determine
the probabilities of various values of the physical quantities in the system.
Thus we reach the conclusion that the state of a system which does not have a
wave function can be described by means of a density matrix. This does not
contain the co-ordinates q which do not belong to the system concerned,
though, of course, it depends essentially on the state of the closed system
as a whole.
The description by means of the density matrix is the most general form
of quantum-mechanical description of the system. The description by means
of the wave function, on the other hand, is a particular case of this, cor-
responding to a density matrix of the form p{x' y x) = ¥*(*')¥(*). The
following important difference exists between this particular case and the
general one.f For a state having a wave function there is always a complete
set of measuring processes such that they lead with certainty to definite
results (mathematically, this means that T is an eigenfunction of some opera-
tor). For states having only a density matrix, on the other hand, there is no
complete set of measuring processes whose result can be uniquely predicted.
Let us now suppose that the system is closed, or became so at some instant.
Then we can derive an equation giving the change in the density matrix with
time, similar to the wave equation for the Y function. The derivation can be
simplified by noticing that the required linear differential equation for
p(x', x, t) must be satisfied in the particular case where the system has a wave
function, i.e.
P (x',x,t)=W*(x',tyF(x,t).
Differentiating with respect to time and using the wave equation (8.1), we
have
dp dY(x,t) W(x',t)
ih~ = ihY*(x\ t) — K -LL+ifW(x, t) —
dt dt dt
= Y»(*\ *)#¥(*, *)-¥(*, *)#'*Y*(*', t),
where A is the Hamiltonian of the system, acting on a function of *,
and A' is the same operator acting on a function of x'. The functions
¥*(*', t) and Y(#, t) can obviously be placed behind the respective operators
tt and U\ and we thus obtain the required equation :
ih dp{x', x, t)JBt = (fi-fi'*)p(x',x, t). (14.5)
f States having a wave function are sometimes called "pure" states, as distinct from "mixed"
states, which are described by a density matrix.
40 Energy and Momentum §14
Let W n (x, t) be the wave functions of the stationary states of the system,
i.e. the eigenfunctions of its Hamiltonian. We expand the density matrix
in terms of these functions ; the expansion consists of a double series in the
functions T w (#, t) and ^¥ n (x', t), which we write in the form
9 {x\ x, t) = SS a wn Y M *(*', t)YJx, t)
= SS a mn ifj n *(x')tf; m (x)^f^ E n-E^ (i4 <6 )
tn n
For the density matrix, this expansion plays a part analogous to that of the
expansion (10.3) for wave functions. Instead of the set of coefficients a n ,
we have here the double set of coefficients a mn . These clearly have the pro-
perty of being "Hermitian", like the density matrix itself:
a nm * = a mn . (14.7)
For the mean value of some quantity/ we have, substituting (14.6) in (14.4),
/ = SS a mn j Y n *(x, t)f<¥ m {x, t) dx,
/ = SZ a mn f n Jt) = SS a mn f nm &/ME n -E m )t } (14 . 8)
or
where f mn are the matrix elements of the quantity /. This expression is
similar to formula (ll.l).f
The quantities a mn must satisfy certain inequalities. The "diagonal
elements" p(x, x) of the density matrix, which determine the probability
distribution for the co-ordinates, must obviously be positive quantities. It
therefore follows from the expression (14.6) (with x' = x) that the quadratic
form
constructed with the coefficients a nm (where the £ n are arbitrary complex
quantities) must be positive. This places certain conditions, known from the
theory of quadratic forms, on the quantities a nm . In particular, all the
"diagonal" quantities must clearly be positive:
a nn >0, (14.9)
and any three quantities a nn , a mm and a mn must satisfy the inequality
> Kn| 2 - (14.10)
*nn'*mm
To the "pure" case, where the density matrix reduces to a product of
functions, there evidently corresponds a matrix a mn of the form
We shall indicate a simple criterion which enables us to decide, from the
f The description of a system by means of the quantities a mn was introduced independently by
L. Landau and F. Bloch in 1927.
§15 Momentum 41
form of the matrix %», whether we are concerned with a "pure" or a
"mixed" state. In the pure case we have
( a )mn = ^ a mk a kn
= S a k *a m a n *a k
= a m a n * X\a k \ 2
k
or
(0 2 )«m = tf«m> (14.11)
i.e. the density matrix is equal to its own square.
§15. Momentum
Let us consider a system of particles not in an external field. Since all
positions in space of such a system as a whole are equivalent, we can say, in
particular, that the Hamiltonian of the system does not vary when the system
undergoes a parallel displacement over any distance. It is sufficient that this
condition should be fulfilled for an arbitrary small displacement.
An infinitely small parallel displacement over a distance Sr signifies a trans-
formation under which the radius vectors r a of all the particles (a being the
number of the particle) receive the same increment Sr : r a -> r -f- Sr. An
arbitrary function tfjfa, r 2 , ... ) of the co-ordinates of the particles, under
such a transformation, becomes the function
^(rj+Sr, r 2 +Sr, ... ) = ^x lt r 2 , ... )+Sr . S V a ^
= (l+8r.2Va¥(ri,r 2 ,...)
a
(V a denotes a "vector" whose components are the operators djdx a , d[dy a ,
djdz a ). The expression in parentheses, i.e.
1+Sr . 2 V«,
a
can be regarded as the operator of an infinitely small displacement, which
converts the function «/r(r 1} r 2 , ... ) into the function
^fo+Sr, r 2 +Sr, ... ).
The statement that some transformation does not change the Hamiltonian
means that, if we make this transformation on the function i?«/r, the result is
the same as if we make it only on the function if/ and then apply the operator i?.
Mathematically, this can be written as follows. Let be the operator which
effects the transformation in question. Then we have 0(fiip) = fi{Oift), whence
6ft-fi6 = o,
i.e. the Hamiltonian must commute with the operator 0.
In the case considered, the operator is the above-mentioned operator
of an infinitely small displacement. Since the unit operator (the operator
of multiplying by unity) commutes, of course, with any operator, and the
42 Energy and Momentum §15
constant factor Sr can be taken in front of 6, the condition OS—IiO =
reduces here to
(S V«)#-#(S V„) = 0. (15.1)
a a
As we know, the commutability of an operator (not containing the time
explicitly) with i? means that the physical quantity corresponding to that
operator is conserved. The quantity whose conservation for a closed system
follows from the homogeneity of space is called momentum. Thus the relation
(15.1) expresses the law of conservation of momentum in quantum mechanics ;
the operator SV« must correspond, apart from a constant factor, to the
total momentum of the system, and each term V a °f tne sum t° the momen-
tum of an individual particle.
The coefficient of proportionality between the operator p of the momentum
of a particle and the operator V can be determined by means of the passage
to the limit of classical mechanics. Putting p = c V and using the limiting
expression (6.1) for the wave function, we have
pT = (i/h) caF/MVS = c(i(hy¥S7S,
i.e. in the classical approximation the effect of the operator p reduces to
multiplication by (ijfijcS/S. The gradient S/S is, as we know from mech-
anics, the momentum p of the particle; hence we must have (ilti)c= 1,
i.e. c = — ih.
Thus the operator of the momentum of a particle is p = —ih\J, or, in
components,
p x = —ikd/dx, p v = —ihd/dy, ft z = —ihdjdz. (15.2)
It is easy to see that these operators are Hermitian, as they should be.
For, with arbitrary functions ifs(x) and <f>(x) which vanish at infinity, we have
<f>p x ip dx — —ih <f> — d# = ih ip — d# = *pp x *<f> d#,
and this is the condition that the operator should be Hermitian.
Since the result of differentiating functions with respect to two different
variables is independent of the order of differentiation, it is clear that the
operators of the three components of momentum commute with one another:
W,-M- = o. AA-M. = °» AA-A?. = o- (is.3)
This means that all three components of the momentum of a particle can
simultaneously have definite values.
Let us find the eigenf unctions and eigenvalues of the momentum operators.
They are determined by the equations
-ihdiMdx=p 3 xl>, -ihd/dy =p v f, -ihfy/dz = prf. (15.4)
The solution of the first of these equations is
§15 Momentum 43
where /is independent of x. This solution remains finite for all values of x,
for any real value of p x . Thus the eigenvalues of the component p x of the
momentum form a continuous spectrum extending from — oo to +00; the
same is true, of course, of the components p y and p z .
The three equations (15.4) have, in particular, common solutions, which
correspond to states with definite values of all three momentum components
forming the vector p. These solutions are of the form
= C^/»>P- r , (15.5)
where C is a constant. If all three components of the momentum are given
simultaneously, we see that this completely determines the wave function
of the particle. In other words, the quantities ^/ty, p z form one of the poss-
ible complete sets of physical quantities.
Let us determine the normalisation coefficient in (15.5). According to the
rule (5.4) for normalising the eigenfunctions of a continuous spectrum, we
must have
J* <Mp* dV = S(p'- p) (15.6)
(where dV = dxdydz), the integration being extended over all space; S(p'— p)
is the three-dimensional delta function, defined similarly to the one-dimen-
sional function (the delta function of one variable). f The integration can
be immediately effected by means of the formulaj
(1/2tt) J e l « x dx = 8(a). (15.7)
We have
f «£ p/ p * dV = C 2 I e<*/»XP'-P).r dV
= C 2 (2^) 3 S(p'-p).
Hence we see that we must have C^ilirKf = 1. Thus the normalised func-
tion «/r is
^ p = (277-^)- 3 / 2 ^/»)P- r . (15.8)
t The three-dimensional function S(r) can, in particular, be represented as a product of delta
functions of the Cartesian components of the vector r: S(r) = 8(*)8(y)S(sr).
I The conventional meaning of this formula is that the function on the left-hand side has the
property of the delta function expressed by the equation
J/(*)8(*)d*=/(0).
This follows from the Fourier integral formula
f(x') = (1/2tt) \\f{x)e* x ~ x '* dada,
if we put *' = 0.
Separating the real part, we can also write formula (15.7) in the form
00
(1/277-) J cosouc d;v = 8(a). (15.7a)
44 Energy and Momentum §15
The expansion of an arbitrary wave function ^(r) of a particle in terms
of the eigenfunctions «/r p of its momentum operator is simply the expansion
as a Fourier integral:
0(r) = j a(p)Mr) d 3 /> = (2^)- 3 /2 J a(p)e^l^- r d*p (15.9)
(where d 3 /> = dp x dp y dp z ). The expansion coefficients a(p) are, according
to formula (5.3),
«(P) = J 0(r)0P*(r) dF = (2tt^)- 3 / 2 J 0(r)<r<W dF. (15.10)
The function a(p) can be regarded (see §5) as the wave function of the
particle in the "p representation"; \a(p)\ 2 d 3 p is the probability that the
momentum has a value in the interval d 3 p. The formulae (15.9) and (15.10)
give the relation between the wave functions in the two representations.
Just as the operator p corresponds to the momentum, determining its
eigenfunctions in the "r representation", we can introduce the idea of the
operator f of the radius vector of the particle in the "p representation". It
must be defined so that the mean value of the co-ordinates can be represented
in the form
r= J a*(p)*a(p) &p. (15.11)
On the other hand, this mean value is determined from the wave function
0(r) by
r = jif,*rjdV.
Writing «/r(r) in the form (15.9) we have (integrating by parts)f
r0(r) = (Zirh)- 3 ' 2 f ra(p)^/« p - r d^
= (2tt/*)- 3 / 2 f *M/«P- r [d a (p)/dp] &p.
Using this expression and (15.10), we find
f=j 0*r^ dV = (2ttH)- s / 2 j j 0*(r)#*[dtf(p)/dpy*"/A)p.r d 3 pdV
= J iha*(p)[da(p)/dp] d*p.
Comparing with (15.11), we see that the radius vector operator in the "p
representation" is
r = ihd/dp. (15.12)
t The derivative with respect to the vector p is understood as the vector whose components are the
derivatives with respect to p z , p y , p t .
§15 Momentum 45
The momentum operator in this representation reduces simply to multipli-
cation by p.
PROBLEMS
Problem 1. Express the operator T a of a parallel displacement over a finite distance a
in terms of the momentum operator.
Solution. By the definition of the operator T a we must have
T*l>{r) = flr+a).
Expanding the function ^(r+a) in a Taylor series, we have
0(r+a) = 0(r)+a . &£(r)/dr+ ... ,
or, introducing the operator p = —ih\J,
«A(r+a) = £l+ia -P+~(^ -p) 2 + ••• ]tf(r).
The expression in brackets is the operator
This is the required operator of the finite displacement.
Problem 2. Find the law of transformation of the wave function under a Galilean trans-
formation.
Solution. We shall carry out the transformation on a wave function of free motion of
a particle (a plane wave). Since any function T can be expanded in terms of plane waves,
this gives also the transformation law for an arbitrary wave function.
The wave functions of free motion in frames of reference K and K' (where K' moves
relative to K with velocity V) are
T = constant xe ( */*> <!»•*-**>,
Y' = constant x ««'»"»»'•*'-*'*>,
with r = r'+\t, and the momenta and energies in the two systems are related byt
p = p' + wV,
E = E' + V.p'+^mV 2 .
Substituting these expressions for r, p and £ in T, we obtain
Y =Y' exp\-mV.(r' + iVt) 1
= T'expUmV.i(r+r')l. (1)
In this form the formula involves no quantities characterising the free motion of the particle,
and gives the required general law of transformation of the wave function of an arbitrary
state of the particle. For a system of particles a sum over the particles appears in the ex-
ponent in (1).
t See Mechanics,
46 Energy and Momentum §16
§16. Uncertainty relations
Let us derive the rules for commutation between momentum and co-
ordinate operators. Since the result of successively differentiating with
respect to one of the variables x, y, z and multiplying by another of them
does not depend on the order of these operations, we have
Pxy~ypx = 0, p x z-zp x = 0, (16.1)
and similarly for P v , ft z .
To derive the commutation rule for p x and x, we write
(ftxX—xfixJtp = —ih d(xifi)ldx+ihx difi/dx
= —ihifj.
We see that the result of the action of the operator $xX—xp x reduces to
multiplication by — ih; the same is true, of course, of the commutation of p y
with y and p s with z. Thus we havef
p x x—xp x = —ih, p v y—yp y = —ih, p z z—zp g = —ih. (16.2)
All the relations (16.1) and (16.2) can be written in the form
fiiX k —x k fii = —ihS ik (i,k = x,y,z). (16.3)
Before going on to examine the physical significance of these relations and
their consequences, we shall derive two formulae which will be useful later.
Let /(r) be some function of the co-ordinates. Then
p/(r)-/(r)P = -«»V/. (16.4)
For
(p/-/p)<A = -«[V(#)-/V0] - -«tyV/.
A similar relation holds for the commutation of r with a "function" /(p) of
the momentum operator:
/(P)r-r/(p) = -ihdfldp. (16.5)
It can be derived in the same way as (16.4) if we calculate in the p representa-
tion, using the expression (15.12) for the co-ordinate operators.
The relations (16.1) and (16.2) show that the co-ordinate of a particle along
one of the axes can have a definite value at the same time as the components
of the momentum along the other two axes ; the co-ordinate and momentum
component along the same axis, however, cannot exist simultaneously. In
particular, the particle cannot be at a definite point in space and at the same
time have a definite momentum p.
Let us suppose that the particle is in some finite region of space, whose
dimensions along the three axes are (of the order of magnitude of) A#, Ay, Az.
t These relations, discovered in matrix form by Heisenberg in 1925, formed the genesis of modern
quantum mechanics.
§16 Uncertainty relations 47
Also, let the mean value of the momentum of the particle be p . Mathe-
matically, this means that the wave function has the form ijs = M(r)e< f /A > p «* r >
where u(t) is a function which differs considerably from zero only in the
region of space concerned. We expand the function ip in terms of the eigen-
functions of the momentum operator (i.e. as a Fourier integral). The co-
efficients a(p) in this expansion are determined by the integrals (15.10) of
functions of the form u(r)e< i/n ^ p '~ p >- r . If this integral is to differ consider-
ably from zero, the periods of the oscillatory factor g(*/»)(Po-p)- r must not be
small in comparison with the dimensions Ax, Ay, Az of the region in which
the function u(r) is different from zero. This means that a(p) will be con-
siderably different from zero only for values of p such that (llh)(p 0x —p x )Ax <;
1, etc. Since |«(p)| 2 determines the probability of the various values of the
momentum, the ranges of values oi p x , p y , p z m which a(p) differs from zero
are just those in which the components of the momentum of the particle may
be found, in the state considered. Denoting these ranges by Ap x , Ap v , Ap et
we thus have
ApJ^x ~ h, Ap y Ay ~ h, Ap z Az ~ h. (16.6)
These relations, known as the uncertainty relations, were obtained by
Heisenberg.
We see that, the greater the accuracy with which the co-ordinate of the
particle is known (i.e. the less Ax), the greater the uncertainty Ap x in the
component of the momentum along the same axis, and vice versa. In parti-
cular, if the particle is at some completely definite point in space (Ax =
Ay = Az = 0), then Ap x = Ap y = Ap z = co. This means that all values
of the momentum are equally probable. Conversely, if the particle has a
completely definite momentum p, then all positions of it in space are equally
probable (this is seen directly from the wave function (15.8), whose squared
modulus is quite independent of the co-ordinates).
As an example, let us consider a particle in a state described by the wave
function
ift = constant x (MUM**-*** I™ (16.7)
(for simplicity, we consider a one-dimensional case, with the wave function
depending on only one co-ordinate). The probabilities of the various values
of the co-ordinates are
|0| 2 = constant x e~ aX *l n ,
i.e. are distributed about the origin of co-ordinates (the mean value x = 0)
according to a Gaussian law, with a standard deviation \Z[(AxY] = -\/(^/2a)
(Ax denotes the difference x— x)-\. If the expansion coefficients a(p x ) of
f As is well known, the Gaussian distribution for the probability w(x) of the values of some quantity
x has the form
w(x) = [2 7 r(Aj ( ;)2]-l/2 e -(A a; )V2(Ax) 2
48 Energy and Momentum §16
this function are calculated as a Fourier integral according to the formula
a(p x ) = (2irA)-i/2 j ^( x )e-a/h)P x x dX)
we obtain an expression of the form
a(p x ) = constant x e-fcv-pJ'/M**.
The distribution of probabilities of values of the momentum is \a\ 2 =
constant X erl'i-'x)*/**, an( j consequently is also of Gaussian form, with a
standard deviation
(where &p x =p x —p )- The product of the standard deviations of co-
ordinate and momentum is thus
V[(A/0 2 (A*) 2 ]=p, (16.8)
in agreement with the relation (16.6).f
Finally, we shall derive another useful relation. Let / and g be two
physical quantities whose operators obey the commutation rule
M-gf=-ihc, (16.9)
f It can be shown that this value of the product of the standard deviations is the least possible.
To do this, we give the following formal derivation (H. Weyl). Let the state of the particle be de-
scribed by the function *ji(x); for simplicity, we suppose the mean values of co-ordinate and momen-
tum in this state to be zero. We consider the obvious inequality
I
dx
dx >0,
where a is an arbitrary real constant (the equality sign holds for a function of the form (16.7), and for
no other). On calculating this integral, noticing that
!
we obtain
* 2 |0| 2 d* = (A*) 2 ,
Kx— f+xif,*— ) dx = f *— dx = - f M 2 dx = - 1,
dx dx/ J dx J
r d0* diL r d 2 <A 1 r „ 1
a 2 (A*) 2 -a+(l//* 2 )(A/g 2 > 0.
If this quadratic (in a) trinomial is positive for all a, the condition
4(A*) 2 (1//* 2 )(A^) 2 >1
V[(A*) 2 (A/>x) 2 ]>P (16.8a)
must be fulfilled.
§16 Uncertainty relations 49
where 6 is the operator of some physical quantity c. On the right-hand side
of the equation the factor h is introduced in accordance with the fact that in
the classical limit (i.e. as h -» 0) all operators of physical quantities reduce
to multiplication by these quantities and commute with one another. Thus,
in the "quasi-classical" case, we can, to a first approximation, regard the right-
hand side of equation (16.9) as being zero. In the next approximation, the
operator c can be replaced by the operator of simple multiplication by the
quantity c. We then have
f£-£f= -fa-
This equation is exactly analogous to the relation pxX— x Px = — ih, the only
difference being that, instead of the constant h, we havef the quantity he.
We can therefore conclude, by analogy with the relation AxAp x ~ h, that
in the quasi-classical case there is an uncertainty relation
A/<V ~ he (16.10)
for the quantities / and g.
In particular, if one of these quantities is the energy (/=i?) and the
operator (§) of the other does not depend explicitly on the time, then by
(9.2) c = g, and the uncertainty relation in the quasi-classical case is
&EAg~hg. (16.11)
f The classical quantity c is the Poisson bracket of the quantities / and g; see the footnote in §9.
CHAPTER III
SCHRODINGER'S equation
§17. Schrodinger's equation
Let us now turn to determining the form of the Hamiltonian — a problem of
the greatest importance, since its solution determines the form of the wave
equation.
We shall begin by considering one free particle, i.e. a particle which is not
in any external field. Because of the complete homogeneity of space for such
a particle, its Hamiltonian cannot explicitly contain the co-ordinates, and
must be expressible in terms of the momentum operator only. Moreover,
for a free particle both the energy and the momentum are conserved, and
hence both these quantities can exist simultaneously. Since the value of the
momentum vector completely determines the state of the particle, the eigen-
values of the energy E must be expressible in the form of functions of the
value of the momentum in the same state. Here E is a function only of the
absolute value of the momentum, and not of its direction ; this follows from
the complete isotropy of space relative to the free particle, i.e. the equivalence
of all directions in space. The actual form of the function E(p) is completely
determined by the requirements of what is called Galileo's relativity principle,
which must hold in non-relativistic quantum mechanics just as much as in
classical (non-relativistic) mechanics. As is found in mechanics,! this re-
quirement leads to a quadratic dependence of the energy on the momentum :
E = p 2 j2m, where the constant m is called the mass of the particle.
If the relation E = p 2 j2m holds for every eigenvalue of the energy and
momentum, the same relation must hold for their operators also:
# = (l/2«)tf.«+/,»+M (17.1)
Substituting here from (15.2), we obtain the Hamiltonian of a freely moving
particle in the form
#=-(#/2m)A, (17.2)
where A = d 2 / dx 2 + d 2 j dy 2 + d 2 jdz 2 is the Laplacian operator.
If we have a system of non-interacting particles, its Hamiltonian is equal
to the sum of the Hamiltonians of the separate particles :
#=-P*S(l/m a )A (17.3)
a
(the suffix a is the number of the particle; A a is the Laplacian operator in
f See Mechanics, §4.
50
§17 Schrddinger's equation 51
which the differentiation is with respect to the co-ordinates of the ath
particle).
The form of the Hamiltonian for a system of particles which interact with
one another cannot be derived from the general principles of quantum mech-
anics alone. It is found that it has in fact a form similar to that of Hamilton's
function in classical mechanics: it is obtained by adding to the Hamiltonian
of the non-interacting particles a certain function U(r ly r 2 , ... ) of their co-
ordinates :
#=-**■ S A^m a +Z/(r 1 ,r lf ...). (17.4)
a
The first term can be regarded as the operator of the kinetic energy and the
second as that of the potential energy. The latter reduces to simple multipli-
cation by the function U, and it follows from the passage to the limiting case
of classical mechanics that this function must coincide with the one which
gives the potential energy in classical mechanics. In particular, the Hamil-
tonian for a single particle in an external field is
ti =p 2 /2m+ U(x,y, z) = -(h*/2m) A + U(x,y, *), (17.5)
where U(x, y, z) is the potential energy of the particle in the external field.
The eigenvalues of the kinetic energy operator are positive; this follows
at once from the fact that this operator is equal to the sum of the squares
of the operators of the momentum components with positive coefficients.
Hence the mean value of the kinetic energy in any state is also positive.
Substituting the expressions (17.2) to (17.5) in the general equation (8.1),
we obtain the wave equations for the corresponding systems. We shall write
out here the wave equation for a particle in an external field:
ih BV/dt = -(h*/2m)A x ¥+ U{x,y, zj¥. (17.6)
The equation (10.2), which determines the stationary states, takes the form
(h*/2m)A*fi +[E- U(x,y, z)]f = 0. (17.7)
The equations (17.6) and (17.7) were obtained by Schrodinger in 1926 and
are called Schrddinger's equations, with and without the time respectively.
For a free particle, Schrodinger's equation (17.7) has the form
(h*/2m)A<P+E+ = 0. (17.8)
This equation has solutions finite in all space for any positive (or zero) value
of the energy E. These solutions can be taken to be the common eigenfunc-
tions (15.5) of the operators of the three momentum components. The com-
plete wave functions of the stationary states will then have the form
¥ = constant x^/^W/fllP-r (g _ pZj2 m ). (17.9)
Each such function describes a state in which the particle has a definite
energy E and momentum p. This is a plane wave propagated in the direction
3
52 Schrodinger's Equation §17
of p and having an angular frequency E\h and wavelength 2irh\p (the latter
is called the de Broglie wavelength of the particle).-)-
The energy spectrum of a freely moving particle is thus found to be con-
tinuous, extending from zero to + 00. Each of these eigenvalues (except
E = 0) is degenerate, and the degeneracy is infinite. For there corresponds
to every value of E, different from zero, an infinite number of eigenfunctions
(17.9), differing in the direction of the vector p, which has a constant absolute
magnitude.
Let us enquire how the passage to the limit of classical mechanics occurs
in Schrodinger's equation, considering for simplicity only a single particle
in an external field. Substituting in Schrodinger's equation (17.6) the limit-
ing expression (6.1) for the wave function, T = fle< i/a > s , we obtain, on per-
forming the differentiation,
dS da a ih . ih n h?
a ih— +— (VS) 2 aAS \/S . \/a Aa+C/a = 0.
dt Bt 2m 2m m 2m
In this equation there are purely real and purely imaginary terms (we recall
that S and a are real); equating each separately to zero, we obtain two
equations
dS 1 * a a
— +-(\/S)*+U-— Aa = 0,
dt 2m 2ma
da a . 1
_ + _AS+— S/S . Vfl = 0.
dt 2m m
Neglecting the term containing h 2 in the first of these equations, we obtain
-^+i-(V5) 2 +t/ = 0, (17.10)
dt 2m
that is, the familiar classical Hamilton- Jacobi equation for the action S of a
particle, as it should be. We see, incidentally, that, as h -> 0, classical mech-
anics is valid as far as quantities of the first (and not only the zero) order in
h inclusive.
The second equation obtained above, on multiplication by 2a, can be re-
written in the form
™ +di J a *El)=0. (17.11)
dt \ m J
This equation has an obvious physical meaning: a 2 is the probability density
for finding the particle at some point in space (|T*| 2 = a 2 ); VSj/i = p/p
is the classical velocity v of the particle. Hence equation (17.11) is simply
the equation of continuity, which shows that the probability density "moves"
according to the laws of classical mechanics with the classical velocity v at
every point.
f The idea of a wave related to a particle was first introduced by L. db Broglie in 1924.
§18 The fundamental properties of Schrodinger's equation 5 3
§18. The fundamental properties of Schrodinger's equation
The conditions which must be satisfied by solutions of Schrodinger's
equation are very general in character. First of all, the wave function must
be single-valued and continuous in all space. The requirement of continuity
is maintained even in cases where the field U(x, y, z) itself has a surface of
discontinuity. At such a surface both the wave function and its derivatives
must remain continuous. Concerning the continuity of the derivatives, how-
ever, it must be added that this does not hold if there is some surface beyond
which the potential energy U becomes infinite. A particle clearly cannot
penetrate at all into a region of space where U = oo, i.e. we must have
if/ = everywhere in this region. The continuity of if* means that ifi vanishes
at the boundary of this region; the derivatives of i(j, however, in general are
discontinuous in this case.
If the field U(x, j, z) nowhere becomes infinite, then the wave function
also must be finite in all space. The same condition must hold in cases where
U becomes infinite at some point but does so only as l/r s with s < 2 (see
also §35).
Let U min be the least value of the function U(x, y, z). Since the Hamil-
tonian of a particle is the sum of two terms, the operators of the kinetic
energy (T ) and of the potential energy, the mean value E of the energy in any
state is equal to the sum T+ V. But all the eigenvalues of the operator t
(which is the Hamiltonian of a free particle) are positive; hence the mean
value T > 0. Recalling also the obvious inequality V > £/ min , we find that
& > ^min- Since this inequality holds for any state, it is clear that it is valid
for all the eigenvalues of the energy:
E n>U min . (18.1)
Let us consider a particle moving in an external field which vanishes at
infinity ; we define the function U(x, y y z), in the usual way, so that it vanishes
at infinity. It is easy to see that the spectrum of negative eigenvalues of the
energy will then be discrete, i.e. all states with E < in a field which vanishes
at infinity are bound states. For, in the stationary states of a continuous
spectrum, which correspond to infinite motion, the particle reaches infinity
(see §10); however, at sufficiently large distances the field may be neglected,
the motion of the particle may be regarded as free, and the energy of a freely
moving particle can only be positive.
The positive eigenvalues, on the other hand, form a continuous spectrum
and correspond to an infinite motion; for E > 0, Schrodinger's equation
in general has no solutions (in the field concerned) for which the integral
/ \if>\ 2 dV converges. f
Attention must be drawn to the fact that, in quantum mechanics, a particle
in a finite motion may be found in those regions of space where E < U;
t However it must be mentioned that, for some particular mathematical forms of the function
U(*, y, z) (which have no physical significance), a discrete set of values may be absent from the
otherwise continuous spectrum.
54 Schrddinger's Equation §18
the probability |«/r| 2 of rinding the particle tends rapidly to zero as the distance
into such a region increases, yet it differs from zero at all finite distances.
Here there is a fundamental difference from classical mechanics, in which
a particle cannot penetrate into a region where U > E. In classical mechanics
the impossibility of penetrating into this region is related to the fact that,
for E < U, the kinetic energy would be negative, that is, the velocity would
be imaginary, which is meaningless. In quantum mechanics, the eigen-
values of the kinetic energy are likewise positive; nevertheless, we do not
reach a contradiction here, since, if by a process of measurement a particle
is localised at some definite point of space, the state of the particle is changed,
as a result of this process, in such a way that it ceases in general to have any
definite kinetic energy.
If U(x,y, z) > in all space (and U -» at infinity), then, by the inequality
(18.1), we have E n > 0. Since, on the other hand, for E > the spectrum
must be continuous, we conclude that, in this case, the discrete spectrum
is absent altogether, i.e. only an infinite motion of the particle is possible.
Let us suppose that, at some point (which we take as origin), U tends to
— oo in the manner
U « -ocr- s (a > 0). (18.2)
We consider a wave function finite in some small region (of radius r ) about
the origin, and equal to zero outside this region. The uncertainty in the
values of the co-ordinates of a particle in such a wave packet is of the order
of r ; hence the uncertainty in the value of the momentum is ~ hjr Q . The
mean value of the kinetic energy in this state is of the order of h 2 jmr 2 , and
the mean value of the potential energy is ~ — a/r *. Let us first suppose
that s > 2. Then the sum
h 2 lmr 2 — a/r s
takes arbitrarily large negative values for sufficiently small r . If, however,
the mean energy can take such values, this always means that the energy has
negative eigenvalues which are arbitrarily large in absolute value. The mo-
tion of the particle in a very small region of space near the origin corresponds
to the energy levels with large \E\. The "normal" state corresponds to a
particle at the origin itself, i.e. the particle "falls" to the point r = 0.
If, however, s < 2, the energy cannot take arbitrarily large negative values.
The discrete spectrum begins at some finite negative value. In this case the
particle does not ' 'fall" to the centre. It should be mentioned that, in classical
mechanics, the "fall" of a particle to the centre would be possible in principle
in any attractive field (i.e. for any positive s). The case 5 = 2 will be specially
considered in §35.
Next, let us investigate how the nature of the energy spectrum depends on
the behaviour of the field at large distances. We suppose that, as r -* oo,
the potential energy, which is negative, tends to zero according to the power
law (18.2) (r is now large in this formula), and consider a wave packet "filling"
a spherical shell of large radius r and thickness Ar <^ r . Then the order
§19 The current density 55
of magnitude of the kinetic energy is again £ 2 /m(Ar) 2 , and of the potential
energy, — <x/r *. We increase r , at the same time increasing Ar, in such a
way that Ar increases proportionally to r . If s < 2, then the sum h 2 jm(Ar) 2 —
<x/r * becomes negative for sufficiently large r . Hence it follows that there
are stationary states of negative energy, in which the particle may be found,
with a fair probability, at large distances from the origin. This, however,
means that there are levels of arbitrarily small negative energy (it must be
recalled that the wave functions rapidly tend to zero in the region of space
where U > E). Thus, in this case, the discrete spectrum contains an infinite
number of levels, which become denser and denser towards the level E = 0.
If the field diminishes as — \\r* at infinity, with s > 2, then there are not
levels of arbitrarily small negative energy. The discrete spectrum terminates
at a level with a non-zero absolute value, so that the total number of levels is
finite.
Schrodinger's equation (without the time) is real, as are the conditions
imposed on its solution. Hence its solutions ifs can always be taken as real.f
The eigenfunctions of non-degenerate values of the energy are automatically
real, apart from the unimportant phase factor. For ifj* satisfies the same
equation as tfi, and therefore must also be an eigenfunction for the same value
of the energy; hence, if this value is not degenerate, i/t and «/»* must be essen-
tially the same, i.e. they can differ only by a constant factor (of modulus unity).
The wave functions corresponding to the same degenerate energy level need
not be real, however, but by a suitable choice of linear combinations of them
we can always obtain a set of real functions.
The complete wave functions Y are determined by an equation in whose
coefficients i appears. This equation, however, retains the same form if we
replace t in it by — t and at the same time take the complex conjugate. %
Hence we can always choose the functions Y in such a way that Y and Y*
differ only by the sign of the time, a result which we know already from
formulae (10.1) and (10.3).
As is well known, the equations of classical mechanics are unchanged by
time reversal, i.e. when the sign of the time is reversed. In quantum mechanics,
the symmetry with respect to the two directions of time is expressed, as we
see, in the invariance of the wave equation when the sign of t is changed and
Y is simultaneously replaced by Y*. However, it must be recalled that this
symmetry here relates only to the equation, and not to the concept of
measurement itself, which plays a fundamental part in quantum mechanics
(as we have explained in detail in §7).
§19. The current density
In classical mechanics, the velocity of a particle is equal to its momentum
divided by its mass. We shall show that the same relation holds in quantum
mechanics, as we should expect.
t These assertions are not valid for systems in a magnetic field (see Chapter XV).
| It is assumed that the potential energy U does not depend explicitly on the time: the system
is either closed or in a constant (non-magnetic) field.
56 Schrodinger *s Equation §19
According to the general formula (9.2) for the differentiation of operators
with respect to time, we have for the velocity operator v = f
v = (Hh){8t-tfl).
Using the expression (17.5) for fi and formula (16.5), we obtain
v=p/m. (19.1)
Similar relations will clearly hold between the eigenvalues of the velocity
and momentum, and between their mean values in any state.
The velocity, like the momentum of a particle, cannot have a definite value
simultaneously with the co-ordinates. But the velocity multiplied by an
infinitely short time interval dt gives the displacement of the particle in the
time dt. Hence the fact that the velocity cannot exist at the same time as
the co-ordinates means that, if the particle is at a definite point in space at
some instant, it has no definite position at an infinitely close subsequent
instant.
We may notice a useful formula for the operator / of the derivative, with
respect to time, of some quantity /(r) which is a function of the radius vector
of the particle. Bearing in mind that/ commutes with U(r) t we find
/= (i/k)(8f-ffi) = (il2mh)(&f-m.
Using (16.4), we can write
p 2 / = P • (/P-^ V/) = p/ • P-ihP . V/,
/P 2 = (P/+*W/).P = p/.p+^V/.p.
Substituting in the formula for/, we obtain the required expression:
/= (l/2m)(£ . V/+V/-P). (19-2)
Next, let us find the acceleration operator. We have
v = (P)(#v - v#) = (i/mh)(fip - p#) = (ilmh)(Up -pU)
(all the terms in i? except U(r) commute with p).
Using formula (16.4), we find
mfi = -\/U. (19.3)
This operator equation is exactly the same in form as the equation of motion
(Newton's equation) in classical mechanics.
The integral / |T| 2 dV, taken over some finite volume V, is the probability
of finding the particle in this volume. Let us calculate the derivative of this
probability with respect to time. We have
j /. f / gvp** d x ¥\ i r
§19 The current density 57
Substituting here
fi = 8* = -(#/2m) A + U(x,y, z)
and using the identity
T FA X F*- X F*AT = div (TV^-T* VY),
we obtain
v
where i denotes the vector
— JVpdF=- fdividF,
i = (^/2m)(YV x F*-T*V 1 F). (19.4)
The integral of div i can be transformed by Gauss's theorem into an integral
over the closed surface S which boundsf the volume V:
d
It
JV|»dF=- fi.df. (19.5)
It is seen from this that the vector i may be called the probability current density
vector. The integral of this vector over a surface is the probability that the
particle will cross the surface during unit time. The vector i and the prob-
ability density |T| 2 satisfy the equation
apF|2/dH-divi = 0, (19.6)
which is analogous to the classical equation of continuity.
Introducing the momentum operator, we can write the vector i in the form
i = (l/2/n)CFp*r*+*F*pF). (19 j)
It is useful to show how the orthogonality of the wave functions of
states with different energies follows immediately from Schrodinger's
equation. Let tf> m and t// n be two such functions; they satisfy the equations
-(h*/2m)A,l, m +Uf m = E m f mi
-(*V2») A£»+ Ety n * = BJS.
We multiply the first of these by n * and the second by i/, m and subtract
corresponding terms ; this gives
(£ m -2? n )«M«* = (£ 2 /2m)(«A ro A^ B *-&*A«
= (k*l2m) div ty m V«A»*-^„* 7W.
If we now integrate both sides of this equation over all space, the right-hand
t The surface element df is denned as a vector equal in magnitude to the area d/ of the element
and directed along the outward normal.
58 Schrodinger's Equation §20
side, on transformation by Gauss's theorem, reduces to zero, and we obtain
whence, by the hypothesis E m # E n , there follows the required orthogonality
relation
§20. The variational principle
Schrodinger's equation, in the general form Zfy = Etff, can be obtained
from the variational principle
8 J" «/»*(#-£)«£ d ? = 0. (20.1)
Since <p is complex, we can vary iff and $* independently. Varying «/r*, we
have
\ Stfj*(fi-E)ip dq = 0,
whence, because Sip* is arbitrary, we obtain the required equation Biff = Eif/.
The variation of if* gives nothing different. For, varying iff and using the
fact that the operator i? is Hermitian, we have
f i}j*(fi-E)8iJj dq = J 80(#*-£)«/<* dq = 0,
from which we obtain the complex conjugate equation i?*«/r* = Eifs*.
The variational principle (20.1) requires an unconditional extremum of
the integral. It can be stated in a different form by regarding J? as a Lagran-
gian multiplier in a problem with the conditional extremum requirement
sJ^ifyd^O, (20.2)
the condition being
j ifjif,* dq = 1. (20.3)
The least value of the integral in (20.2) (with the condition (20.3)) is the first
eigenvalue of the energy, i.e. the energy E of the normal state. The func-
tion ift which gives this minimum is accordingly the wave function «/»o of the
normal state, f The wave functions ifs n (n > 0) of the other stationary states
correspond only to an extremum, and not to a true minimum of the integral.
In order to obtain, from the condition that the integral in (20.2) is a mini-
mum, the wave function «/a and the energy E\ of the state next to the normal
one, we must restrict our choice to those functions ifs which satisfy not only the
f In the rest of this section we shall suppose the wave functions tf> to be real; they can always be
so chosen (if there is no magnetic field).
§20 The variational principle 59
normalisation condition (20.3) but also the condition of orthogonality with
the wave function $ of the normal state: j i[n[/ dq = 0. In general, if the
wave functions i(f Qy X , ... , if} n _ x of the first n states (arranged in order of in-
creasing energy) are known, the wave function of the next state gives a mini-
mum of the integral in (20.2) with the additional conditions
J>d$ = l, J# ro d ? = (m = 0,1, 2,... ,fi-l). (20.4)
We shall give here some general theorems which can be proved from the
variational principle, f
The wave function i/f of the normal state does not become zero (or, as we
say, has no nodes) for any finite values of the co-ordinates. £ In other words,
it has the same sign in all space. Hence, it follows that the wave functions
i// n (n > 0) of the other stationary states, being orthogonal to tjt 0t must have
nodes (if tfj n is also of constant sign, the integral j ift if/ n dq cannot vanish).
Next, from the fact that i/i has no nodes, it follows that the normal energy
level cannot be degenerate. For, suppose the contrary to be true, and let
«Ao> to be two different eigenfunctions corresponding to the level E Q . Any
linear combination Ci// -\-c't/j f will also be an eigenf unction ; but by choosing
the appropriate constants c, c\ we can always make this function vanish at
any given point in space, i.e. we can obtain an eigenfunction with nodes.
If the motion takes place in a bounded region of space, we must have
ifj = at the boundary of this region (see §18). To determine the energy
levels, it is necessary to find, from the variational principle, the minimum
value of the integral in (20.2) with this boundary condition. The theorem that
the wave function of the normal state has no nodes means in this case that
«/r does not vanish anywhere inside this region.
We notice that, as the dimensions of the region containing the motion
increase, all the energy levels E n decrease; this follows immediately from
the fact that an extension of the region increases the range of functions which
can make the integral a minimum, and consequently the least value of the
integral can only diminish.
The expression
J* 0#0 dq = f [- S (£ 2 /2m a )«AA a 0+ Ut/fi] dq
for the states of the discrete spectrum of a particle may be transformed into
another expression which is more convenient in practice. In the first term
of the ^integrand we write
4>Aa*l> = div a (0Va«A)-(Va«A) 2 .
t The proof of theorems concerning the zeros of eigenfunctions (see also §21) is given by M. A.
Lavrent'ev and L. A. Lyusternik, The Calculus of Variations (Kurs variatsionnogo ischisleniya),
2nd edition, chapter IX, Moscow 1950; R. Courant and D. Hilbert, Methods of Mathematical
Physics, volume I, chapter VI, Interscience, New York 1953.
% This theorem and its consequences are not in general valid for the wave functions of systems
consisting of several identical particles (see footnote at the end of §63).
60 Schrodinger's Equation §21
The integral of div a (^Va^) over all space is transformed into an integral
over an infinitely distant closed surface, and since the wave functions of the
states of a discrete spectrum tend to zero sufficiently rapidly at infinity, this
integral vanishes. Thus
j 0f dq = j [S (h*l2ma)( Va<£) 2 + £ty 2 ] dq. (20.5)
§21. General properties of motion in one dimension
If the potential energy of a particle depends on only one co-ordinate (x),
then the wave function can be sought as the product of a function of y and z
and a function of x only. The former of these is determined by Schro-
dinger's equation for free motion, and the second by the one-dimensional
Schrodinger's equation
-l+-[E-U(xM = Q. (21.1)
dx z ft 2
Similar one-dimensional equations are evidently obtained for the problem of
motion in a field whose potential energy is U(x, y, z) = Ui(x) + Uz{y) + U^{z),
i.e. can be written as a sum of functions each of which depends on only one
of the co-ordinates. In §§22-24 we shall discuss a number of actual examples
of such "one-dimensional" motion. Here we shall obtain some general
properties of the motion.
We shall show first of all that, in a one-dimensional problem, none of the
energy levels of a discrete spectrum is degenerate. To prove this, suppose
the contrary to be true, and let ift x and ip 2 be two different eigenfunctions
corresponding to the same value of the energy. Since both of these satisfy
the same equation (21.1), we have
or *lti"*l't~- l l t i" , l l i s= (th e prime denotes differentiation with respect to x).
Integrating this relation, we find
0i'02-"~0i02' — constant. (21.2)
Since ifs x = tfj 2 = at infinity, the constant must be zero, and so
or ipx'tyi = 0«7^2- Integrating again, we obtain ^ = constant x ^ 2 » i.e. the
two functions are essentially identical.
The following theorem (called the oscillation theorem) may be stated for the
wave functions iftjx) of a discrete spectrum. The function *jt n (x) correspond-
ing to the (n+l)th eigenvalue E n (the eigenvalues being arranged in order of
magnitude), vanishes n times (for finitef values of x).
t If the particle can be found only on a limited segment of the x-axis, we must consider the zeros
of *l> n (x) within that segment.
§21 General properties of motion in one dimension 61
We shall suppose that the function U(x) tends to finite limiting values as
x -> ± oo (though it need not be a monotonic function). We take the limiting
value U(+ oo) as the zero of energy (i.e. we put U( + oo) = 0), and we denote
U(—co) by U , supposing that U > 0. The discrete spectrum lies in the
range of energy values for which the particle cannot move off to infinity; for
this to be so, the energy must be less than both limiting values C/(±oo),
i.e. it must be negative:
E < 0, (21.3)
and we must, of course, have in any case E > t/ min , i.e. the function U(x)
must have at least one minimum with U mia < 0.
Let us now consider the range of positive energy values less than U :
0<E<U . (21.4)
In this range the spectrum will be continuous, and the motion of the particle
in the corresponding stationary states will be infinite, the particle moving off
towards x = + oo. It is easy to see that none of the eigenvalues of the energy
in this part of the spectrum is degenerate either. To show this, it is sufficient
to notice that the proof given above (for the discrete spectrum) still holds if
the functions if/ v if> 2 are zero at only one infinity (in the present case they tend
to zero as x -> — oo).
For sufficiently large positive values of x, we can neglect U(x) in Schro-
dinger's equation (21.1):
0" +(2»#W = 0.
This equation has real solutions in the form of a stationary plane wave
if* = a cos(fcc+8), (21.5)
where a and 8 are constants, and the wave number k = pjh = ^(ImE)^.
This formula determines the asymptotic form (for x ->+ oo) of the wave
functions of the non-degenerate energy levels in the range (21.4) of the
continuous spectrum. For large negative values of x, Schrodinger's equation
is
«£" -(2m/£ 2 )(£/ o -£)0 = 0.
The solution which does not become infinite as x -> — oo is
if> = be* x t where k = V[2m{U -E)]/h. (21.6)
This is the asymptotic form of the wave function as x -> — oo. Thus the
wave function decreases exponentially in the region where E < U.
Finally, for
E > U Q (21.7)
the spectrum will be continuous, and the motion will be infinite in both
directions. In this part of the spectrum all the levels are doubly degenerate.
62 Schrodinger's Equation §21
This follows from the fact that the corresponding wave functions are deter-
mined by the second-order equation (21.1), and both of the two independent
solutions of this equation satisfy the necessary conditions at infinity (whereas,
for instance, in the previous case one of the solutions became infinite as
x -> — oo, and therefore had to be rejected). The asymptotic form of the
wave function as x -> + oo is
if, = ai e ikx +a 2 e- ikx , (21.8)
and similarly for x -> — oo. The term e ikx corresponds to a particle moving
to the right, and e~ ikx corresponds to one moving to the left.
Let us suppose that the function U(x) is even [U(—x) = U(x)]. Then
Schrodinger's equation (21.1) is unchanged when the sign of the co-
ordinate is reversed. It follows that, if tf/(x) is some solution of this equation,
then iff(-x) is also a solution, and coincides with ip(x) apart from a constant
factor: if*(—x) = ap(x). Changing the sign of x again, we obtain if/(x) =
c 2 ifj(x), whence c = ±1. Thus, for a potential energy which is symmetrical
(relative to x = 0), the wave functions of the stationary states must be either
even [ip( — x) = ifi(x)] or odd [ifi — {x) = — ^(*)].f In particular, the wave
function of the ground state is even, since it cannot have a node, while an
odd function always vanishes for x — [^(0) = —$(0) = 0].
To normalise the wave functions of one- dimensional motion (in a continu-
ous spectrum), there is a simple method of determining the normalisation
coefficient directly from the asymptotic expression for the wave function for
large values of \x\.
Let us consider the wave function of a motion infinite in one direction,
i.e. of a stationary state in the range (21.4) of the continuous spectrum.
The normalisation integral diverges as x -> oo (as x -> — oo, the function
decreases exponentially, so that the integral rapidly converges). Hence, to
determine the normalisation constant, we can replace «/r by its asymptotic
value (for large positive x), and perform the integration, taking as the lower
limit any finite value of x, say zero ; this amounts to neglecting a finite quantity
in comparison with an infinite one. We shall show that the wave function
normalised by the delta function of p (the momentum of the particle at
infinity) must have the asymptotic form (21.5) with a = ^(IJrrh), i.e.
iftp « V( 2 M) cos(kx+8) = \/(l/27rA)[e« feB ^>+e-'**<-»>]. (21.9)
Since we do not intend to verify the orthogonality of the functions corre-
sponding to different^), on substituting the functions (21.9) in the normali-
sation integral J $p*typ dx we shall suppose the momenta p to be arbitrarily
close; we can therefore put 8 = 8' (in general 8 is a function of p). Next, we
t In this discussion it is assumed that the stationary state is not degenerate, i.e. the motion is not
infinite in both directions. Otherwise, when the sign of x is changed, two wave functions belonging
to the energy level concerned may be transformed into each other. In this case, however, although
the wave functions of the stationary states need not be even or odd, they can always be made so (by
choosing appropriate linear combinations ofrthe original functions).
§22 The potential well 63
retain in the integrand only those terms which diverge for p = p'\ in other
words, we omit terms containing the factor e±*(fc+fc')£. Thus we obtain
f i/j p *iP P ' dx = {Ijlrrh) ( j e«*'-*>* dx+ j «-«*'-*>* dA
or
f $f$ v > dx = (1/2tt£) f «<*'-*)* d*.
This integral, however, is identical with the normalisation integral for the
wave function of free motion
tfj p = (27rA)-i/2e**« (21.10)
which is normalised by the delta function of momentum (cf. (15.8)).
The change to normalisation by the delta function of energy is effected, in
accordance with (5.14), by multiplying i/j p by s/{dpjdE) = lj-y/v, where v is
the velocity. Thus for free motion we have
4s E = {2irhv)-Vty* x . (21.11)
The probability current density in this wave is
©|0*|2 = 1/2tt*. (21.12)
Dividing the function (21.9) by \/v and using equation (21.12), we can
formulate the following rule for the normalisation of the wave function for a
motion infinite in one direction by the delta function of energy: having
represented the asymptotic expression for the wave function in the form of
a sum of two plane waves travelling in opposite directions, we must choose
the normalisation coefficient in such a way that the probability current
density in the wave travelling towards (or away from) the origin is I/IttH.
Similarly, we can obtain an analogous rule for normalising the wave func-
tions of a motion infinite in both directions. The wave function will be
normalised by the delta function of energy if the sum of the probability cur-
rents in the waves travelling towards the origin from x = + oo and x = — oo
is H2ttH.
§22. The potential well
As a simple example of one-dimensional motion, let us consider motion in
a square potential well, i.e. in a field where U(x) has the form shown in Fig. 1 :
U(x) = for < x < a, U(x) = U f or x < and x > a. It is evident
a priori that for E < U the spectrum will be discrete, while for E > U
we have a continuous spectrum of doubly degenerate levels.
In the region < x < a we have SchrSdinger's equation
^'+(2mlh z )Ei/ t = (22.1)
(the prime denotes differentiation with respect to x) y while in the region
64 Schrddinger's Equation §22
uix)
a
Fig. 1
outside the well
f +(2»/#)(£- UM = 0. (22.2)
For # = and a; = a the solutions of these equations must be continuous
together with their derivatives, while for x = ± oo the solution of equation
(22.2) must remain finite (for the discrete spectrum when E < U , it must
vanish).
For E < U , the solution of equation (22.2) which vanishes at infinity is
tfi = constant xe***, where k = V[(2mJh 2 )(U -E)]; (22.3)
the signs — and + in the exponent refer to the regions x > a and * <
respectively. The probability \ift\ 2 of finding the particle decreases exponen-
tially in the region where E < U(x). Instead of the continuity of t// and tfj'
at the edge of the potential well, it is convenient to require the continuity of
iff and of its logarithmic derivative ip'/ift. Taking account of (22.3), we obtain
the boundary condition in the form
f/0 = =F*. (22.4)
We shall not pause here to determine the energy levels in a well of arbitrary
depth U (see Problem 2), and shall analyse fully only the limiting case of
infinitely high walls (U -> oo).
For U = oo, the motion takes place only between the points x = and
x = a and, as was pointed out in §18, the boundary condition at these points
must be
# = 0. (22.5)
(It is easy to see that this condition is also obtained from the general condition
(22.4). For, when U -> oo, we have also k -> oo and hence tf/'ji// -+ oo;
since ip' cannot become infinite, it follows that ift = 0.) We seek a solution
of equation (22.1) inside the well in the form
ift = c sin(foc+8), where k = </(2mEI&). (22.6)
The condition ^ = for x = gives S = 0, and then the same condition for
§22 The potential well 65
x = a gives sin ka = 0, whence ka — rnr, n being a positive integer,! or
E n = (7T 2 ^ 2 /2ma 2 )» 2 , n = 1,2,3,.... (22.7)
This determines the energy levels of a particle in a potential well. The
normalised wave functions of the stationary states are
*l>n = vWO sin(7rnxla). (22.8)
From these results we can immediately write down the energy levels for a
particle in a rectangular "potential box", i.e. for three-dimensional motion
in a field whose potential energy U = QforO < x < a y <y <b,0 < z <c
and U = oo outside this region. In fact, these levels are given by the sums
ir 2 h 2 /n, 2 Wo 2 « 3 2 \
*w, = T-( ^-+1F + -T- ) (*» W2 ' n * = 1,2,3,...), (22.9)
128 2m \ a 2 6 2 c 2 /
and the corresponding wave functions by the products
/ 8 Trn x 7rw 2 rm 3
0«i",«« = H~ sin — ^^sin— ^ysin — z. (22.10)
1 s ' V abc a b c
It may be noted that the energy Eq of the ground state is, by (22.7) or
(22.9), of the order of h 2 fml 2 , where / is the linear dimension of the region
in which the particle moves. This result is in accordance with the uncertainty
relation ; when the uncertainty in the co-ordinate is ~ /, the uncertainty in
the momentum, and therefore the order of magnitude of the momentum
itself, is ~ hjl. The corresponding energy is ~ (hjl) 2 lm.
PROBLEMS
Problem 1. Determine the probability distribution for various values of the momentum
for the normal state of a particle in an infinitely deep square potential well.
Solution. The coefficients a(p) in the expansion of the function tp t (22.8) in terms of the
eigenfunctions (21.10) of the momentum are
a(p) = tf/j*^ dx = sinf -* )«-&/*>»>* dx.
J -\/{iTah) J \a /
Calculating the integral and squaring its modulus, we obtain the required probability distri-
bution :
47rh z a pa
|a(/>)| 2 = cos 2 —.
' ^ ' (p 2 a 2 -7r 2 ^ 2 ) 2 2H
Problem 2. Determine the energy levels for the potential well shown in Fig. 2.
Solution. The spectrum of energy values E < U u which we shall consider, is discrete.
In the region * < the wave function is
= aefi't where k x = ^/[{ImJh^U^-E)},
f For « = 0we should have = identically.
66
Schrodinger's Equation
u(x)
§22
Us
<A
Fig. 2
while in the region x > a
if, = c 2 e-"* x , where /c 2 = V[(2m/h 2 )(U 2 -E)].
Inside the well (0 < x < a) we look for >p in the form
iff = c sin(foc+8), where k=^(2mE/h 2 ).
The condition of the continuity of ifi'/*fi at the edges of the well gives the equations
k cot 3 = k x = V[(2™lh 2 )U 1 -k 2 ], k cot(ka+S) = -k 2 = -^/[(2m/h 2 )U 2 -k 2 ] y
or
sin 8 = kh/ViZmUJ, sin(&*+8) = -kh/V(2mU 2 ).
Eliminating S, we obtain the transcendental equation
ka = rnr- sirr^kh/ V(2m C/j)] - sivr 1 [khl^{2m U 2 )] (1)
(where n = 1, 2, 3, ... , and the values of the inverse sine are taken between and Jtt), whose
roots determine the energy levels E = k 2 h z /2m. For each n there is in general one root;
the values of n number the levels in order of increasing energy.
Since the argument of the inverse sine cannot exceed unity, it is clear that the values of k
can lie only in the range from to V(2mU 1 /h 2 ). The left-hand side of equation (1) increases
monotonically with k, and the right-hand side decreases monotonically. Hence it is neces-
sary, for a root of equation (1) to exist, that for k = V(2mU 1 /h i ) the right-hand side should
be less than the left-hand side. In particular, the inequality
ay/ilmU^fh >frr-sm-i</(UJUJ,
(2)
which is obtained for « = 1, is the condition that at least one energy level exists in the well.
We see that for given and unequal U u U 2 there are always widths a of the well which are so
small that there is no discrete energy level. For t/j = U 2 , the condition (2) is evidently always
satisfied.
For Ui = U 2 = U (a symmetrical well), equation (1) reduces to
sm- 1 [hk/V(2mU )] = \{mr-hd). (3)
Introducing the variable £ = \ka, we obtain for odd n the equation
COS £ = ±yi, where y = {t\\a)^{2\mVJ) y (4)
and those roots of this equation must be taken for which tan g > 0. For even n we obtain
the equation
sin $ = ±yg t (5)
and we must take those roots for which tan £ < 0. The roots of these two equations deter-
mine the energy levels E = 2£ 2 h 2 /ma 2 . The number of levels is finite when y ^ 0.
§23 The linear oscillator 67
In particular, for a shallow well in which U < h 2 /ma 2 , we have y > 1 and equation (5)
has no root. Equation (4) has one root (with the upper sign on the right-hand side),
s = 1/y— l/2y 3 . Thus the well contains only one energy level,
E £ U -(ma*l2HV)Uo*,
which is near the top of the well.
Problem 3. Determine the pressure exerted on the walls of a rectangular "potential
box" by a particle inside it.
Solution. The force on the the wall perpendicular to the #-axis is the mean value of the
derivative -dH/da of the Hamilton's function of the particle with respect to the length of
the box in the direction of the *-axis. The pressure is obtained by dividing this force by the
area be of the wall. According to the formula derived in §11, Problem, the required mean
value is found by differentiating the eigenvalue (22.9) of the energy. The result is
pU) _ -nZffin^jtnaZbc.
§23. The linear oscillator
Let us consider a particle executing small oscillations in one dimension
(what is called a linear oscillator). The potential energy of such a particle
is well known to be %mco 2 x 2 , where co is, in classical mechanics, the character-
istic (angular) frequency of the oscillations. Accordingly, the Hamiltonian
of the oscillator is
& =iP 2 /m+ \mco 2 x 2 . (23.1)
Since the potential energy becomes infinite for x = ±00, while its least
value (at x = 0) is zero, it is clear from general principles that the energy
spectrum of the oscillator is discrete and the energy values are positive.
Let us determine the energy levels of the oscillator, using the matrix
methodf. We shall start from the "equations of motion" in the form (19.3) ;
in this case they give
x+co 2 x = 0. (23.2)
In matrix form, this equation reads
(x)mn+<*>*x mn = 0.
For the matrix elements of the acceleration we have, according to (11.8),
(*)«m = lco mn(x)mn = — ^mn^mn' Hence we obtain
(w WB 2 -w 2 )* mM = 0.
Hence it is evident that all the matrix elements x mn vanish except those for
which to mn = at or co mn = - co. We number all the stationary states so that
the frequencies ± co correspond to transitions n ->w+ 1, i.e. co n ,r^i = ± co.
Then the only non-zero matrix elements are x n , n±1 .
t This was done by Heisenberg in 1925, before Schrodinger's discovery of the wave equation.
68 Schrodinger's Equation §23
We shall suppose that the wave functions ift n are taken real. Since x is a real
quantity, all the matrix elements x mn are real. The Hermitian condition
(11.10) now shows that the matrix x mn is symmetrical:
To calculate the matrix elements of the co-ordinate which are different
from zero, we use the commutation rule
&x— x& = —thjm,
written in the matrix form
(xx) mn -(xx) mn = -(ihlm)8 mn .
By the matrix multiplication rule (11.12) we hence have for m = n
i 2 ((*inlXnlXln—Xnlo»lnXm) = 2* 2 a>nlXnl 2 = —th/m.
I •
In this sum, only the terms with / = n ± 1 are different from zero, so that
we have
(Xn + l,nY-{n,Xn-lY = h/lmw. (23.3)
From this equation we deduce that the quantities (#n+i,n)* form an arith-
metic progression, which is unbounded above, but is certainly bounded
below, since it can contain only positive terms. Since we have as yet fixed
only the relative positions of the numbers n of the states, but not their abso-
lute values, we can arbitrarily choose the value of n corresponding to the first
(normal) state of the oscillator, and put this value equal to zero. Accordingly
x _! must be regarded as being zero identically, and the application of equa-
tions (23.3) with n = 0, 1, ... successively leads to the result
(*n,n-i) 2 = nh\2moi.
Thus we finally obtain the following expression for the matrix elements of
the co-ordinate which are different from zero:f
*«,«-! = *-l« = vW2»c). ( 23 - 4 )
The matrix of the operator tt is diagonal, and the matrix elements H nn
are the required eigenvalues E n of the energy of the oscillator. To calculate
them, we write
= \m[ S ioi nl x nl iw ln x ln +oi 2 S x n ix ln ]
=|wS(a> 2 +co n? 2 )^ n 2 .
In the sum over /, only the terms with /= n±\ are different from zero;
t We choose the indeterminate phases o„ (see the third footnote to §11) so as to obtain the plus
sign in front of the radical in all the matrix elements (23.4). Such a choice is always possible for a
matrix in which only those elements are different from zero which correspond to transitions between
states with adjacent numbers.
§23 The linear oscillator 69
substituting (23.4), we obtain
E n = (n+$)ha>, n = 0,1,2 (23.5)
Thus the energy levels of the oscillator lie at equal intervals of hot from
one another. The energy of the normal state (« = 0) is \ha\ we call atten-
tion to the fact that it is not zero.
The result (23.5) can also be obtained by solving Schrodinger's equation.
For an oscillator, this has the form
dV 2w
~^-^<E-lmo^x^ = 0. (23.6)
Here it is convenient to introduce, instead of the co-ordinate x, the dimension-
less variable g by the relation
£ = ^{moijh)x. (23.7)
Then we have the equation
r+[(2Elko)-£ 2 W = 0; (23.8)
here the prime denotes differentiation with respect to £.
For large £, we can neglect lEjhco in comparison with £ 2 ; the equation
«/," = ^tf, has the asymptotic integrals = c ±«' (for differentiation of this
function gives ifi" = ^j, on neglecting terms of order less than that of the
term retained). Since the wave function fi must remain finite as g ->±oo,
the index must be taken with the minus sign. It is therefore natural to make
in equation (23.8) the substitution
^ = «" f8/2 *(£)• (23.9)
For the function x{£) we obtain the equation (with the notation {lEjhoi) — 1 =
2n; since we already know that E > 0, we have n > — £)
x"-2&'+2»x = 0, (23.10)
where the function x must be finite for all finite £, and for $ -»± oo must not
tend to infinity more rapidly than every finite power of $ (in order that the
function tf> should tend to zero).
Such solutions of equation (23.10) exist only for positive integral (and
zero) values of n (see §a of the Mathematical Appendices); this gives the
eigenvalues (23.5) for the energy, which we know already. The solutions of
equation (23.10) corresponding to various integral values of n are x = con-
stant xH n (£), where H n (g) are what are called Hermite polynomials; these
are polynomials of the nth degree in £, defined by the formula
H n (€) = (-l)V>(<r**)/d£ n . (23.11)
70 Schrodinger's Equation §23
Determining the constants so that the functions ijs n satisfy the normalisation
condition
oo
j ifr n 2 (x) dx = 1,
—GO
we obtain (see (a. 7))
-r) on/, // e-m^y^H n (xV[mcolh]). (23.12)
Thus the wave function of the normal state is
O («) = (mwlTrh) 1 ^-™"**/™. (23.13)
It has no zeros for finite x, which is as it should be.
By calculating the integrals J ift n *p m € d£, we can determine the matrix ele-
ments of the co-ordinate; this calculation leads, of course, to the same
values (23.4).
Finally, we shall show how the wave functions ip n may be calculated by the
matrix method. We notice that, in the matrices of the operators £±icox,
the only elements different from zero are
(x— tW)„_ 1>n = —{x+io}x) nn _ x = —{^(liohn/m). (23.14)
Using the general formula- (11.11), and taking into account the fact that
ift-i = 0, we conclude that
(£—ia}x)tfi = 0.
After substituting the expression £ = —{(hjmjdldx, we obtain the equation
difijdx = —(mcolh)xif/ Qt
whose normalised solution is (23.13). And, since
(£+ia)x)if/ n _ 1 = (x+iu)x) n3t ^_ x tfj n = i^(2coknlm)ift n ,
we obtain the recurrence formula
ty n = ■y/(ml2o)hn)[—(hlm) d/dx+coxtyn^
i / a \ i d
V(2«)\ dg J V(2«) d£
when this is applied n times to the function (23.13), we obtain the expression
(23.12) for the normalised functions ip n -
PROBLEMS
Problem 1. Determine the probability distribution of the various values of the momentum
for an oscillator.
Solution. Instead of expanding the wave function of the stationary state in terms of the
eigenfunctions of momentum, it is simpler in the case of the oscillator to start directly from
§23 The linear oscillator 7 1
Schrodinger's equation in the "p representation". Substituting in (23.1) the co-ordinate
operator £ = ihdjdp (15.12), we obtain the Hamiltonian in the p representation,
GER'i
d 2 a(/>) 2
dp 2 rnuPW'
The corresponding Schrodinger's equation Ba(p) — Ea(p) for the wave function a(p) in
the p representation is
( E -ty p)= °-
This equation is of exactly the same form as (23.6); hence its solutions can be written down
at once by analogy with (23.12) (replacing xVitno/h) in this formula by p/V(mtoft)). Thus
we find the required probability distribution to be
2 n n\y{Trma>n)
Problem 2. Determine the lower limit of the possible values of the energy of an oscillator,
using the uncertainty relation (16.8a).
Solution. We have for the mean value of the energy of the oscillator
or, using the relation (16.8a),
E > (Ap) 2 /2m+ma) 2 h 2 /8(Ap)\
On determining the minimum value of this expression (regarded as a function of (Ap) 2 ), we
find the lower limit of the mean values of the energy, and therefore that of all possible values :
E>ihco.
Problem 3. Determine the energy levels for a particle moving in a field of potential enerev
(Fig. 3)
U(x) = A(e- %ax — 2e~ ax )
(P. M. Morse). '
Solution. The spectrum of positive eigenvalues of the energy is continuous (and the levels
are not degenerate), while the spectrum of negative eigenvalues is discrete.
Schrodinger's equation reads
We introduce a new variable
2V(2mA)
£ = e -ax
Cf.k
72
Schrodinger's Equation
§23
(taking values from to oo) and the notation (we consider the discrete spectrum, so that
E<0)
s = ^(—2mE)laLh, n = V(2mA)/*h-(s+$). (1)
Schrodinger's equation then takes the form
r v ,+ (-<
n+s+$ s 2
■>
0.
As £ -> oo, the function $ behaves asymptotically as e ± if, while as £ -v it is proportional to
£±*. From considerations of finiteness we must choose the solution which behaves as e~it
as £ -> oo and as £ s as £ -» 0. We make the substitution
= e -t'H s z»($)
and obtain for w the equation
£w"+(2*+ 1-|)«>'+««; = 0,
(2)
which has to be solved with the conditions that w is finite as £ -> 0, while as £ -> oo, w tends
to infinity not more rapidly than every finite power of £. Equation (2) is the equation for a
confluent hypergeometric function (see §d of the Mathematical Appendices) :
to = F(-n,2s+l y g).
A solution satisfying the required conditions is obtained for non-negative integral n (when
the function F reduces to a polynomial). According to the definitions (1), we thus obtain
for the energy levels the values
-E n = A^\
a.h
.(»+*)
]'■
V(2mA)
where n takes positive integral values from zero to the greatest value for which \/(2mA)[a.h >
n-\- J (so that the parameter s is positive in accordance with its definition). Thus the discrete
spectrum contains only a limited number of levels. If V(2mA)JctH < %, there is no discrete
spectrum at all.
Problem 4. The same as Problem 3, but with U = — L7 /cosh 2 ax (Fig. 4).
Solution. The spectrum of positive eigenvalues of the energy is continuous, while that of
negative values is discrete ; we shall consider the latter. Schrodinger's equation is
dV 2m/
_L + _(£+-
cbc 2 h 2 \ cosb^aa:.
Up \
sh 2 aa: /
0.
obtaining
§24 Motion in a homogeneous field 73
We put £ = tanh auc and use the notation
e = V(-2w£)//wc, 2mC/ /a 2 ^2 = *(s+l),
5[ ( '- p ?]*[* t "-ii>- 1
This is the equation of the associated Legendre polynomials; it can be brought to hyper-
geometric form by making the substitution «/- = (1— £ 2 ) e/2 w(£) and temporarily changing
the variable to « = J(l— £):
M(l-ttK' + (e+l)(l-2z/)«;'-( e -s)(e + s+i)a> = 0.
The solution finite for £ = 1 (i.e. for * = oo) is
If tfi remains finite for £ = — 1 (i.e. for x = — oo), we must have e—s = —n, where
n = 0, 1, 2, ...; then F is a polynomial of degree n, which is finite for f = —1.
Thus the energy levels are determined by s — € = «, or
/* 2 <x 2 r .... //. 8mU n
WC)
8m
There is a finite number of levels, determined by the condition e > 0, i.e. « < $.
§24. Motion in a homogeneous field
Let us consider the motion of a particle in a homogeneous external field.
We take the direction of the field as the axis of x; let F be the force acting
on the particle in this field. In an electric field of intensity E, this force is
F = eE, where e is the charge on the particle.
The potential energy of the particle in the homogeneous field is of the
form U = —Fx+ constant; choosing the constant so that U = for x = 0,
we have U = —Fx. Schrodinger's equation for this problem is
d 2 if,ldx 2 +(2m/h 2 )(E+Fx)if, = 0. (24.1)
Since U tends to + oo as x -> — oo, and vice versa, it is clear that the energy
levels form a continuous spectrum occupying the whole range of energy
values £from -co to +oo. None of these eigenvalues is degenerate, and
they correspond to motion which is finite towards x = — oo and infinite to-
wards X = +00.
Instead of the co-ordinate x, we introduce the dimensionless variable
f = (x+ElF)(2mF/&)V*. (24.2)
Equation (24.1) then takes the form
*"+# = 0. (24.3)
74 Schrodinger's Equation §24
This equation does not contain the energy parameter. Hence, if we obtain a
solution of it which satisfies the necessary conditions of finiteness, we at
once have the eigenfunction for arbitrary values of the energy.
The solution of equation (24.3) which is finite for all x has the form (see
§b of the Mathematical Appendices)
m = A®(-£) t (24.4)
where
<D(£) = cos(£tt 3 +«!) du
\/tt J
v
is called the Airy function, while A is a normalisation factor which we shall
determine below.
As | -> — oo, the function ifj(g) tends exponentially to zero. The asymp-
totic expression which determines «/r(|) for large negative values of £ is (see
(b.4))
' A(a ~2J^ eXp[ " fI ^ |3/2] ' (24 " 5)
For large positive values of £, the asymptotic expression for ?/r(|) is (see
(b.5))t
</,(£) = AiJT^> sin(f^/ 2 +|7r). (24.6)
Using the general rule (5.4) for the normalisation of eigenfunctions of a
continuous spectrum, let us reduce the function (24.4) to the form normalised
by the delta function of energy, for which
jW(f ) d* = 8(E'-E). (24.7)
In §21 we gave a simple method of determining the normalisation coefficient
by means of the asymptotic expression for the wave functions. Following
this method, we represent the function (24.6) as the sum of two travelling
waves :
0(0 a^^- 1 /4exp(f[f^/ 2 -|7r])+|^- 1 / 4 exp(-t[f^/ 2 -i7r]).
The probability current density v |^| 2 , calculated from each of these two terms,
must be \\2rnh:
«/[2(E+Fx)ltn](AI2pi*)* = A\2hFyi*l4m*l* = 1/2^,
whence we find
(2m) 1 /»
A = — — - . (24.8)
^1/2^1/6^2/3
f It may be noted, by way of anticipation, that the asymptotic expressions (24.5) and (24.6) cor-
respond to the quasi-classical expressions (47.1) and (47.4a) for the wave function.
§25 The transmission coefficient 75
PROBLEM
Determine the wave functions in thep representation for a particle in a homogeneous field.
Solution. The Hamiltonian operator in the p representation is
fl = p*/2m-ihF d/dp,
so that Schrodinger's equation for the wave function a(p) has the form
da fp % \
-ihF—+[- — E)a = Q.
dp \2m J
Solving this equation, we find the required functions
a E (p) = (277-^F)-i/ 2 ^'/ft^^P-P 3 /6m).
These functions are normalised by the condition
ja E *(p)a E {p)dp = S(E'-E).
§25. The transmission coefficient
Let us consider the motion of particles in a field of the type shown in
Fig. 5 : U(x) increases monotonically from one constant limit ( U = as
x -> — oo) to another (U = U Q as x -> +oo). According to classical mech-
anics, a particle of energy E < U moving in such a field from left to right,
on reaching such a "potential wall", is "reflected" from it, and begins to move
in the opposite direction ; if, however, E > U , the particle continues to
move in its original direction, though with diminished velocity. In quantum
mechanics, a new phenomenon appears : even for E > U Q , the particle may
be "reflected" from the potential wall. The probability of reflection must in
principle be calculated as follows.
Fig. 5
Let the particle be moving from left to right. For large positive values of
x, the wave function must describe a particle which has passed "above the
wall" and is moving in the positive direction of x, i.e. it must have the asymp-
totic form
for * -> oo, « Ae ik * x , where k 2 = (l/h)y/[2m(E— U Q )] (25.1)
and A is a constant. To find the solution of Schrodinger's equation which
satisfies this boundary condition, we calculate the asymptotic expression for
76 Schrodinger's Equation §25
x -> — oo ; it is a linear combination of the two solutions of the equation of
free motion, i.e. it has the form
for x -» - oo, tjt « e ile * x +Be- ik * x t where ^ = ^/(2mE)/h. (25.2)
The first term corresponds to a particle incident on the "wall" (we suppose
if> normalised so that the coefficient of this term is unity) ; the second term
represents a particle reflected from the "wall". The probability current
density in the incident wave is k ly in the reflected wave k^B] 2 , and in the
transmitted wave £ 2 I^I 2 - We define the transmission coefficient D of the par-
ticle as the ratio of the probability current density in the transmitted wave
to that in the incident wave :
D = (*A)M». (25.3)
Similarly we can define the reflection coefficient R as the ratio of the density
in the reflected wave to that in the incident wave. Evidently R = 1— D:
R = \B\ 2 = l-(k 2 lk 1 )\A\ 2 (25.4)
(this relation between A and B is automatically satisfied).
If the particle moves from left to right with energy E < U , then k 2 is
purely imaginary, and the wave function decreases exponentially as # -» + oo.
The reflected current is equal to the incident one, i.e. we have "total reflec-
tion" of the particle from the potential wall. We emphasise, however, that
in this case the probability of finding the particle in the region where E < U
is still different from zero, though it diminishes rapidly as x increases.
In the general case of an arbitrary stationary state (with energy E > C/o),
the asymptotic form of the wave function is given, both for x ->— oo and for
x -> + oo, by a sum of waves propagated in each direction:
di = A\e iklX +B\er ilc ^ x for x -> — oo,
(25.5)
*fi = A2,e ik * x +B2fir ik * x for x -> +oo.
Since these expressions are asymptotic forms of the same solution of a linear
differential equation, there must be a linear relation between the coefficients
Ai, B± and A2, B%. Let A% = a.Ai+f$B\, where a, j8 are constants (in general
complex) which depend on the specific form of the field U(x). The corres-
ponding relation for B2 can then be written down from the fact that Schro-
dinger's equation is real. This shows that, if ip is a solution of a given
Schrodinger's equation, the complex conjugate function iff* is also a solution.
The asymptotic forms
«/»* = A 1 *e~ ik ^ x + B±*e iklX for *-^— 00,
tjj# = A 2 * e -ik2x + B 2 *e ik * x for *->+oo
differ from (25.5) only in the nomenclature of the constant coefficients; we
therefore have B 2 * = aBi*+fiAi* or B 2 = ol*Bi+P*Ai. Thus the coefficients
§25 The transmission coefficient 77
in (25.5) are related by equations of the form
A 2 = o^i+jSBi, j? 2 = jS*^i+a*#i. (25.6)
The condition of constant probability current along the jc-axis leads to
the relation
k^A^-^) = k*{\A 2 \*-\B 2 \%
Expressing A 2 , B 2 in terms of Ai, B\ by (25.6), we find
|a| 2 -|j3| 2 = hlk 2 . (25.7)
Using the relation (25.6), we can show, in particular, that the reflection
coefficients are equal (for a given energy E > Uq) for particles moving in the
positive and negative directions of the x-axis ; the former case corresponds to
putting B 2 = in (25.5), and the latter case to A\ = 0. The corresponding
reflection coefficients are
Ri = |#i/^i| 2 = |£*/a*|2,
R 2 = \A 2 jB 2 \* = |)3/a*|2,
whence it is clear that R± = R 2 .
PROBLEMS
Problem 1. Determine the reflection coefficient of a particle from a rectangular potential
wall (Fig. 6) ; the energy of the particle E > U .
U(x)
Fig. 6
Solution. Throughout the region x > 0, the wave function has the form (25.1), while in
the region x < its form is (25.2). The constants A and B are determined from the condi-
tion that i/i and dift/dx are continuous at * = :
1+B=A, k x {\-B) =M,
whence
A = 2k x i{k x +kt\ B = {k x -k 2 )l{k x +k 2 ).
The reflection coefficient f is (25.4)
R = / v-^y = / p ± -p 2 \*
\k x +kj Kpx+pi)'
For E = U (k 3 = 0), R becomes unity, while for E -> oo it tends to zero as (C7 /4J?) 2 .
f In the limiting case of classical mechanics, the reflection coefficient must become zero. The
expression obtained here, however, does not contain the quantum constant at all. This apparent
contradiction is explained as follows. The classical limiting case is that in which the de Broglie
wavelength of the particle A ~ hip is small in comparison with the characteristic dimensions of the
problem, i.e. the distances over which the field U(x) changes noticeably. In the schematic example
considered, however, this distance is zero (at the point x = 0), so that the passage to the limit cannot
be effected.
78 Schrddinger' s Equation §25
Problem 2. Determine the transmission coefficient for a rectangular potential barrier (Fig. 7).
U„
y(x)
Fig. 7
Solution. Let E be greater than U , and suppose that the incident particle is moving from
left to right. Then we have for the wave function in the different regions expressions of the
form
for * < 0, iff = e ik i x +Ae- ik i x ,
for0<x<a,ip= Be ik * x +B'e- ik i x ,
for * > a, ifi = Ce ik i x
(on the side x > a there can be only the transmitted wave, propagated in the positive direc-
tion of x). The constants A, B, B' and C are determined from the conditions of continuity
of tfi and dtfi/dx at the points x = and a. The transmission coefficient is determined as
D = ^|C| 2 /^i = |C| 2 . On calculating this, we obtain
D =
lf\i f\a
For E < U , &a is a purely imaginary quantity; the corresponding expression for D is
obtained by replacing k% by ik 2 , where Hk z = \/[2m(U — E)]:
D =
lantity ;
= V[2m(
t/Jj /c 2
(&! 2 +/c 2 2 ) 2 sinh«fl#c 8 +4ife 1 a f2 a '
Problem 3. Determine the reflection coefficient for a potential wall defined by the formula
U(x) = J7 /(l +e~ ax ) (Fig. 5); the energy of the particle is E > U .
Solution. Schrodinger's equation is
d 2 if/ 2m/
d 2 «/r 2m/ U \
—-+—[E °—U =
d* 2 h 2 \ l+e-^J
We have to find a solution which, as x -> +oo, has the form
iff = constant x e ik * x .
We introduce a new variable
£ = — e~ aX
(which takes values from -co to 0), and seek a solution of the form
= £-**,/««;(£),
where w(£) tends to a constant as £ -> (i.e., as x ->■ co). For w(£) we find an equation of
hypergeometric type:
£(l-£) w "+(l-2ikjK)(l-0w'^k 2 *-k 1 *)wl«? = 0,
§25
The transmission coefficient
79
which has as its solution the hypergeometric function
w = F(i[^ 1 -^ 2 ]/a,-^ 1 +A 2 ]/a,-2^ 2 /a+l, $)
(we omit a constant factor). As £ ->■ 0, this function tends to 1, i.e. it satisfies the condition
imposed.
The asymptotic form of the function ^ as £ -*■ — oo (i.e. x -» — oo) isf
if, « |-^,/«[C 1 (-|) i (V* ; i)/»+C 2 (-^ fc i+ A: i >/ a ] = (-l)-**./°[C 1 e < *i*+C a e-«*i a! ],
where
r(-2* 1 /a)r(-2tfe 2 /a+ 1)
C 1 =
c, =
r(_z-(^ 1 +^)/a)r(-^ 1 +A 2 )/a+l)
T(2ife 1 /a)r(-2«fe 2 /a+l)
r(^ 1 -^ 2 )/a)r(t(^-^ 2 )/a+ 1)*
The required reflection coefficient is R = IC2/CJ 2 ; on calculating it by means of the well-
known formulae
we have
T(x+1) =xT(x), r(*)r(l-*) =7r/sin7r^,
r = / sinhfrrfo-fej/tt] \ 2
Vsinh[7r(^ 1 +^ 2 )/a] /
For E = U (& 2 = 0), i? becomes unity, while for E -> co it tends to zero as
7rU n \ 2 2m
/7rU \ z 2m
e -i*\/(2mE)/ah,
In the limiting case of classical mechanics, R becomes zero, as it should.
Problem 4. Determine the transmission coefficient for a potential barrier denned by the
formula
U(x) = U Q lcosh 2 ctx
(Fig. 8) ; the energy of the particle is E < U Q .
U(x)
Fig. 8
Solution. The Schrodinger's equation is the same as that obtained in the solution of
Problem 4, §23 ; it is necessary merely to alter the sign of Uo and to regard the energy E
now as positive. A similar calculation gives the solution
4> = (l~P)- ik ' 2 "F[-iklx-s, -ik/x+s+1, -ik/a+ !,&!-£)],
(1)
f See formula (e.6), in each of whose two terms we must take only the first term of the expansion
i.e. replace the hypergeometric functions of 1/a by unity.
80 Schrddinger' s Equation §25
where
$ = tanh cox, k = y/{lmE)\%,
SmU -
-i-»A<-m
This solution satisfies the condition that, as x -*■ oo (i.e. as $ -> 1, (1 — f) *» 2e~ x ), the wave
function should include only the transmitted wave (~e* fcz ). The asymptotic form of the
wave function as x -> — oo (£ -*■ ~ 1) is found by transforming the hypergeometric function
with the aid of formula (e.7) :
rmm - ik) rr - ik)va - ik)
j, „ e -ik x v ' v L +e ikx 1 L± i . (2)
T(-s)T(l+s) T(-ik-s)r(-ik+s+l) K J
Talcing the squared modulus of the ratio of coefficients in this function, we obtain the follow-
ing expression for the transmission coefficient D = 1 — R :
sinh 2 (7r£/a)
(if 8mt/o/£ 2 <x 2 <l), or
D =
sinh 2 (7rfc/a) + cos 2 [£tt V( 1 - 8m U /h 2 oi. 2 )]
sinh 2 (7r£/a)
sinh 2 (7r£/a)+cosh 2 |>- V(8m£y# 2 a 2 - 1)]
(if 8tnUo/h 2 ofi > 1). The first of these formulae holds also for the case Uo < 0, i.e. when
the particle is passing over a potential well instead of a potential barrier. It is interesting to
note that in that case D = 1 if 1 +8m\Uo\lfr 2 a? = (2« + l) 2 ; thus, for certain values of the
depth | Uq\ of the well, particles passing over it are not reflected. This is evident from equation
(2), where the term in e~ ikx vanishes for positive integral s.
CHAPTER IV
ANGULAR MOMENTUM
§26. Angular momentum
In §15, to derive the law of conservation of momentum, we have made use
of the homogeneity of space relative to a closed system of particles. Besides
its homogeneity, space has also the property of isotropy: all directions in it
are equivalent. Hence the Hamiltonian of a closed system cannot change
when the system rotates as a whole through an arbitrary angle about an
arbitrary axis. It is sufficient, as in §15, to require the fulfilment of this con-
dition for an infinitely small rotation.
Let 8<p be the vector of an infinitely small rotation, equal in magnitude
to the angle 8<f> of the rotation and directed along the axis about which the
rotation takes place. The changes Sr a (in the radius vectors r a of the par-
ticles) in such a rotation are well known to be
8r = 8<p x r a .
An arbitrary function j/rfo, r 2 , ... ) is thereby transformed into the function
^1+8^,^+8^,...) = «A(r 1 ,r 2 ,...)+S8r . V a ^
= #*i» r 2 , ... )+ S 8<p x r a . V a ^
= (l+8<p.Sr a xV a )0(r 1 ,r 2 ,...).
The expression
l+8<p.Sr a xV«
can be regarded as the "operator of an infinitely small rotation". The fact
that an infinitely small rotation does not alter the Hamiltonian of the system
is expressed (see §15) by the commutability of the "rotation operator" with
the operator i?. Since the operator of multiplication by unity commutes
with any operator, while 8<p is a constant vector, this condition reduces to the
relation
(2 r a x V«)#-#(2 r a x V a ) = 0, (26.1)
which expresses a certain law of conservation.
The quantity whose conservation for a closed system follows from the
property of isotropy of space is the angular momentum of the system. Thus
the operator Sr a x V a must correspond exactly, apart from a constant factor,
to the total angular momentum of the system, and each of the terms r a x V a
of this sum corresponds to the angular momentum of an individual particle.
81
82 Angular Momentum §26
The coefficient of proportionality must be put equal to — ih\ this follows
immediately, because then the expression for the angular momentum operator
of a particle is —ihrx V = rxp and corresponds exactly to the familiar
classical expression rxp. Henceforward we shall always use the angular
momentum measured in units of h. The angular momentum operator of a
particle, so defined, will be denoted by 1, and that of the whole system by L.
Thus we have for the angular momentum component operators of a particle
the expressions
ht x = yp z —zpy, ht y = zp x —xp z , hl z = xp y —yp x . (26.2)
For a system which is in an external field, the angular momentum is in
general not conserved. However, it may still be conserved if the field has a
certain symmetry. Thus, if the system is in a centrally symmetric field, all
directions in space at the centre are equivalent, and hence the angular momen-
tum about this centre will be conserved. Similarly, in an axially symmetric
field, the component of angular momentum along the axis of symmetry is
conserved. All these conservation laws holding in classical mechanics are
valid in quantum mechanics also.
In a system where angular momentum is not conserved, it does not have
definite values in the stationary states. In such cases the mean value of the
angular momentum in a given stationary state is sometimes of interest. It is
easily seen that, in any non-degenerate stationary state, the mean value of the
angular momentum is zero. For, when the sign of the time is changed, the
energy does not alter, and, since only one stationary state corresponds to a
given energy level, it follows that when t is changed into — t the state of the
system must remain the same. This means that the mean values of all
quantities, and in particular that of the angular momentum, must remain
unchanged. But when the sign of the time is changed, so is that of the angular
momentum, and we have L = — L, whence it follows that L = 0. The same
result can be obtained by starting from the mathematical definition of the
mean value L as being the integral of i/i*Lift. The wave functions of non-
degenerate states are real (see the end of §18). Hence the expression
L = — ih J 0(Sr o x V o )0d#
is purely imaginary, and since L must, of course, be real, it is evident that
L=0.
Let us derive the rules for commutation of the angular momentum operators
with those of co-ordinates and linear momenta. By means of the relations
(16.2) we easily find
{! x ,x} = 0, {l x ,y} = tar, {t x ,z} = —iy,
tfv»3'}= » &»*}=**» {/„*}= —tar, \ (26.3)
{/„*} = 0, {t z ,x} = iy, {t z ,y} = -ix.
§26 Angular momentum 83
For instance,
Ly-yL = (m)(yh-4v)y-y(yfa-4v)(W)
= -(zlh){$v,y} = iz-
All the relations (26.3) can be written in tensor form as follows:
{h,x k } =te m x u (26.4)
where e iU is the antisymmetric unit tensor of rank three,f and summation is
implied over those suffixes which appear twice (called dummy suffixes).
It is easily seen that a similar commutation rule holds for the angular
momentum and linear momentum operators :
{4 h) = i*ikipi- (26.5)
By means of these formulae, it is easy to find the rules for commutation of
the operators ! x , t y , l z with one another. We have
HUv-tJx) = L(zfix- x Pz)-(zpx-x$ g )L
= (lxZ-zL)Px—x(t x } z —$J x )
= —iyfrx+ixpy = iht z .
{l v Jz}=iL {tzJx}=il v , {/«,/,}=«/„ (26.6)
Thus
or
{4,4} =**«,/,. (26.7)
Exactly the same relations hold for the operators L x ,L y ,L z of the total
angular momentum of the system. For, since the angular momentum oper-
ators of different individual particles commute, we have, for instance,
5'«»|''«— S/asS/ay = 2>{hvhz—hzhv) = »S t ax .
Thus
{t vt l z }=iL x , {L z ,l x } =it y , {L x ,L y }=iL z . (26.8)
The relations (26.8) show that the three components of the angular momen-
tum cannot simultaneously have definite values (except in the case where all
three components simultaneously vanish: see below). In this respect the
angular momentum is fundamentally different from the linear momentum,
whose three components can simultaneously have definite values.
f The antisymmetric unit tensor of rank three, e m (also called the unit axial tensor), is denned as a
tensor antisymmetric in all three suffixes, with e nz = 1 . It is evident that, of its 27 components, only 6
are not zero, namely those in which the suffixes i, k, I form some permutation of 1, 2, 3. Such a com-
ponent is +1 if the permutation i, k, I is obtained from 1, 2, 3 by an even number of transpositions
ot pairs of figures, and is - 1 if the number of transpositions is odd. Clearly e m e ikm = 2Sim
emenci = 6. The components of the vector C = AxB which is the vector product of the two vectors
A and B can be written by means of the tensor em in the form
c, =
84 Angular Momentum §26
From the operators L x , L yy L z we can form the operator L x 2 +L y 2 +L*,
which can be regarded as the operator of the square of the modulus of the
angular momentum vector, and which we denote by L 2 :
L* = Lf+Lf+L*. (26.9)
This operator commutes with each of the operators L x ,L yi L e :
{L\t x }=0, {L*,Z„}=0, {L*,4}=0. (26.10)
Using (26.8), we have
{l x \l z } =l x {l x ,l z }+{L x ,t z }l x
= l^Lgljy + LyLgc),
{Ly t Lg} = l^Lgljy + LyL;,)),
{L,\L.} =o.
Adding these equations, we have {L 2 , L z ) = 0. Physically, the relations
(26.10) mean that the square of the angular momentum, i.e. its modulus, can
have a definite value at the same time as one of its components.
Instead of the operators L x , L y it is often more convenient to use the
complex combinations
L+ = Lx+iLy, ■£- = L x —iLy. (26.11)
It is easily verified by direct calculation using (26.8) that the following
commutation rules hold :
{L + , I-} = 2t z , {L z , L + } =L + ,
{l z ,l-}=-l- t
and it is also not difficult to see that
L2 = t + t-+t z *-l z
= L-L + +L z *+l*. (26.13)
Finally, we shall give some frequently used expressions for the angular
momentum operator of a single particle in spherical polar co-ordinates.
Defining the latter by means of the usual relations
* = r sin 8 cos <f>, y = r sin 6 sin <f>, z = r cos 0,
we have after a simple calculation
U = -A (26.14)
/. = e±**[ + — +* cot 9— ). (26.15)
§27 Eigenvalues of the angular momentum
85
Substitution in (26.13) gives the squared angular momentum operator of the
particle :
12 r 1 ^ 1 d f d \i
It should be noticed that this is, apart from a factor, the angular part of the
Laplacian operator.
§27. Eigenvalues of the angular momentum
In order to determine the eigenvalues of the component, in some direction,
of the angular momentum of a particle, it is convenient to use the expression
for its operator in spherical polar co-ordinates, taking the direction in question
as the polar axis. According to formula (26.14), the equation ty = 1$ can
be written in the form
-i ty/ty = Irf. (27.1)
Its solution is
*!> = /(r,0)A*
where/(r, 0) is an arbitrary function of r and 0. If the function $ is to be single-
valued, it must be periodic in <f>, with period 2tt. Hence we findf
h = m, where m = 0, ±1,± 2, ... . (27.2)
Thus the eigenvalues l z are the positive and negative integers, including
zero. The factor depending on <j>, which characterises the eigenfunctions of
the operator t„ is denoted by
Q n (t) = (27r)-W»* (27.3)
These functions are normalised so that
j ® m *(<t>)® m <<!>) d0 = h mm '. (27.4)
The eigenvalues of the ^-component of the total angular momentum of the
system are evidently also equal to the positive and negative integers:
L s = M, where M = 0,±1, ±2, ... (27.5)
(this follows at once from the fact that the operator L z is equal to the sum of
the commuting operators 4 for the individual particles).
Since the direction of the ^-axis is in no way distinctive, it is clear that the
same result is obtained for L x , i y and in general for the component of the
angular momentum in any direction: they can all take integral values only.
At first sight this result may appear paradoxical, particularly if we apply
it to two directi ons infinitely close to each other. In fact, however, it must
i f J h V US l° mary no r tation . fo , r the eigenvalues of the angular momentum component is m, which
also denotes the mass of a particle, but this should not lead to any confusion.
86 Angular Momentum §27
be remembered that the only common eigenfunction of the operators
L x , L y , L z corresponds to the simultaneous values
L x = L y = L z = u;
in this case the angular momentum vector is zero, and consequently so is its
projection upon any direction. If even one of the eigenvalues L x ,L y ,L s is
not zero, the operators L x ,L y ,L e have no common eigenfunctions. In other
words, there is no state in which two or three of the angular momentum
components in different directions simultaneously have definite values differ-
ent from zero, so that we can say only that one of them is integral.
The stationary states of a system which differ only in the value of L z have
the same energy; this follows from general considerations, based on the
fact that the direction of the sr-axis is in no way distinctive. Thus the
energy levels of a system whose angular momentum is conserved (and is not
zero) are always degenerate, f
Let us now look for the eigenvalues of the square L 2 of the angular momen-
tum. We shall show how these values may be found, starting from the
commutation conditions (26.8) only. We denote by \Jj m the wave functions
of the stationary states belonging to one degenerate energy level and distin-
guished by the value of L z = M. Besides the energy, the square L 2 of the
angular momentum also has a (single) definite value in these states. J
First of all, we note that since the difference
L 2 -4 2 = £* 2 + V
is equal to the operator of the essentially positive physical quantity L x 2 +L y 2 ,
it follows that, for a given value of the squared angular momentum L 2 and
any possible eigenvalue of the quantity L s , the inequality L 2 ^ L s 2 , or
- VL 2 < L z < + VL 2 , (27.6)
must hold. Thus the possible values of L z (for a given L 2 ) are bounded by
certain upper and lower limits; we denote by L the integer corresponding
to the greatest value of \L Z \.
Next, by applying the operator L Z L ± to the eigenfunction j/tm of the
operator L z and using the commutation rule {L z , £ ± } = ±L ± (26.12), we
obtain
LzLJjM = {M± l)£ ± 0a*.
t This is a particular case of the general theorem, mentioned in §10, which states that the levels
are degenerate when two or more conserved quantities exist whose operators do not commute. Here
the components of the angular momentum are such quantities.
t Here it is supposed that there is no additional degeneracy leading to the same value of the energy
for different values of the squared angular momentum. This is true for a discrete spectrum (except
for the case of what is called accidental degeneracy in a Coulomb field; see §36) and in general untrue
for the energy levels of a continuous spectrum. However, even when such additional degeneracy is
present, we can always choose the eigenfunctions so that they correspond to states with definite values
of L 2 , and then we can choose from these the states with the same values of E and L 2 . This is mathe-
matically expressed by the fact that the matrices of commuting operators can always be simultaneously
brought into diagonal form. In what follows we shall, in such cases, speak, for the sake of brevity, as
if there were no additional degeneracy, bearing in mind that the results obtained do not in fact depend
on this assumption, by what we have just said.
§27 Eigenvalues of the angular momentum 87
Hence we see that the function Lj, M is (apart from a normalisation constant)
the eigenfunction corresponding to the value M± 1 of the quantity L z - we
can write '
fa+i = constant xL + fa, ^m-i = constant x Z_0 M . (27.7)
If we put M = L in the first of these equations, we must have identically
£-4l = 0, (27.8)
since there is by definition no state M > L. Applying the operator L-
to this equation and using the relation (26.13), we obtain
LXrf L = (Ifi-lf-L^ = o.
Since, however, the fa are common eigenfunctions of the operators L 2 and
L z , we have
so that the equation found above gives
L2=Z(£+1). (279)
Formula (27.9) determines the required eigenvalues of the square of the
angular momentum; the number L takes all positive integral values, including
zero. For a given value of Z, the component L z = M of the angular momen-
tum can take the values
M = L,L-1,...,-L, (27.10)
i.e. 2L + 1 different values in all. The energy level corresponding to the
angular momentumf L thus has (2L + l)-fold degeneracy. A state with
angular momentum L = (when all three components are also zero) is not
degenerate; we notice that the wave function of such a state is spherically
symmetric. This follows from the mere fact that, when acted on by the
angular momentum operator, it becomes zero, i.e. it is unchanged as a result
of any infinitely small rotation.
For the angular momentum of a single particle we write formula (27.9) in
the form
I 2 ='('+!)> (27.11)
i.e. we denote the angular momentum of an individual particle by the small
letter /. J
Let us calculate the matrix elements of the quantities L x and L y in a
representation in which L z and V, as well as the energy, are diagonal
(M. Born, W. Heisenberg and P. Jordan 1926). First of all, we note that,
since the operators L x and L y commute with the operator 8, their matrices
are diagonal with respect to the energy, i.e. all matrix elements for transitions
t We shall often, for the sake of brevity, and in accordance with custom, speak of the "angular
momentum L of a system, understanding by this a momentum whose greatest possible component
88 Angular Momentum §28
between states of different energy (and different angular momentum L)
are zero. Thus it is sufficient to consider the matrix elements for transitions
within a group of states with different values of M, corresponding to a single
degenerate energy level.
It is seen from formulae (27.7) that, in the matrices of the operators
L+ and L-, only those elements are different from zero which correspond
to transitions M+ 1 -> M and M— 1 -> M respectively. Taking this into
account, we find the diagonal matrix elements on both sides of the equation
(26.13), obtainingf
L(L+1) = (L+) m , M -i(L-)m-i,m+M*-M.
Noticing that, since the operators L x and L y are Hermitian,
(L-)m-1,M = (L+)*m,m-i>
we can rewrite this equation in the form
\(L + )m,m-i\ 2 = L(L+1)-M(M-1)
= (L-M+1)(L+M),
whencej
(L+)m,m-i = (L-)m-i,m
= V[(L+M)(L-M+1)]. (27.12)
Hence we have for the non-zero matrix elements of the quantities L x and L y
themselves
(L x )m,m-i = (L x ) M -i,m = * V[(L+M)(L-M+ 1)],
{L v )m,m-x = -(L y ) M -i,M = -VV[(L+M)(L-M+1)].
In the corresponding formulae for the angular momentum of a particle, we
must write /, m instead of L, M.
§28. Eigenfunctions of the angular momentum
The wave function of a particle is not completely determined when the
values of l 2 and l z are prescribed. This is seen from the fact that the expres-
sions for the operators of these quantities in spherical polar co-ordinates
contain only the angles 6 and <f>, so that their eigenfunctions can contain an
arbitrary factor depending on r. We shall here consider only the angular
part of the wave function which characterises the eigenfunctions of the
f In the symbols for the matrix elements, we omit for brevity all suffixes with respect to which
they are diagonal (including L).
% The choice of sign in this formula corresponds to the choice of the phase factors in the eigenfunc-
tions of the angular momentum.
§ 28 Eigenfunctions of the angular momentum 89
angular momentum, and denote this by Y^fi, with the normalisation
condition
J|rj»do«i,
where do = sin 6 ddd<f> is an element of solid angle.
We shall see that the problem of determining the common eigenfunctions
of the operators i 2 and l s admits of separation of the variables B and <f> and
these functions can be sought in the form
Ylm = «Ufl©,„(0), (28.1)
where O m (<f>) are the eigenfunctions of the operator / 3 , which are given by
formula (27.3). Since the functions O m are already normalised by the condi-
tion (27.4), the 0, m must be normalised by the condition
V
J|0, w | 2 sin0d0 = l. ( 28.2)
o
The functions Y lm with different / or m are automatically orthogonal:
/ / Y Vm * Yim sin ddd<f> = M..', (28.3)
o o
as being the eigenfunctions of angular momentum operators corresponding
to different eigenvalues. The functions OJfl separately are themselves
orthogonal (see (27.4)), as being the eigenfunctions of the operator l e cor-
responding to different eigenvalues m of this operator. The functions (0)
are not themselves eigenfunctions of any of the angular momentum operators •
they are mutually orthogonal for different /, but not for different m.
The most direct method of calculating the required functions is by directly
solving the problem of finding the eigenfunctions of the operator I 2 written
in spherical polar co-ordinates (formula (26.16)). The equation f 2 <£ = IV is
Substituting in this equation the form (28.1) for 0, we obtain for the function
&i m the equation
1 d /. d0 Zw \ m 2
5n7SV Sm '^rJ-^ - +/ ( /+1 ) - - °- (28.4)
This equation is well known in the theory of spherical harmonics. It has
solutions satisfying the conditions of finiteness and single-valuedness for
positive mtegral values of / > | m |, in agreement with the eigenvalues of the
angular momentum obtained above by the matrix method. The correspond-
ing solutions are what are called associated Legendre polynomials P?(cos 6)
90 Angular Momentum §28
(see §c of the Mathematical Appendices). Using the normalisation condition
(28.2), we findf
€U0) = {-l) m iWmi+W-^)W+m)\]Pi m (cos9). (28.5)
Here it is supposed that m ^ 0. For negative m, we use the definition
©i.-i»i=(-1)"®«i«i- ( 28 - 6 )
In other words, @i m for m < is given by (28.5) with \m\ instead of m and
the factor (— l) m omitted. For m = 0, the associated Legendre polynomials
are called simply Legendre polynomials Pi(cos 6) ; we have
0*o = i l V[«2/+ l)]P,(oo8 0). (28.7)
Thus the eigenfunctions of the angular momentum are mathematically
just spherical harmonic functions normalised in a particular way. From (27.3)
and (28.6) it is seen that functions differing in the sign of m are related by the
equation
(-l) l - m Y l ,- m = Y lm *. (28.8)
We shall make some remarks concerning the eigenfunctions of the angular
momentum. For / = (so that m = also) this function reduces to a con-
stant. In other words, the wave functions of the states of a particle with
zero angular momentum depend only on r, i.e. they have complete spherical
symmetry.
For a given m, the values of / starting from \m\ denumerate the successive
eigenvalues of the quantity / in order of increasing magnitude. Hence, from
the general theory of the zeros of eigenfunctions ( §21), we can deduce that the
function @ lm becomes zero for l—\m\ different values of the angle 6; in
other words, it has as nodal lines /— \m\ "lines of latitude" on the sphere. If
the complete angular functions are taken with the real factors cos m<j> or
sin m<f> instead of J «±*»#, they have as further nodal lines \m\ "lines of longi-
tude"; the total number of nodal lines is thus /.
Finally, we shall show how the functions S lm may be calculated by the
matrix method. This is done similarly to the calculation of the wave func-
tions of an oscillator in §23. We start from the equation (27.8) :
t+Y a = 0.
Using the expression (26.15) for the operator /+ and substituting
Yu = (27r)-V 7 *0 w (0), we obtain for ®u the equation
d0„/d0-Zcot0 0„ =0,
t The choice of the phase factor is not, of course, determined by the normalisation condition.
The definition (28.5) used in this book is the most natural from the viewpoint of the theory of addition
of angular momenta (see Chapter XIV). It differs by a factor i l from the one usually adopted.
X Each such function corresponds to a state in which l z does not have a definite value, but can have
the values ±m with equal probability.
§29 Matrix elements of vectors 91
whence ®u = constant X sin'0. Determining the constant from the normali-
sation condition, we find
% = (-*M£(2/+1)!]2-'(1/Z!) sin'0. (28.9)
Next, using (27.12), we write
*-^,w+l = ('-)m,m+l^im
= V[(l-m)(l+m+l)]Y ln .
A repeated application of this formula gives
V[(l-m)\l(l+m)\]Y lm = [(20!]- 1 /2(/_)^F„.
The right-hand side of this equation is easily calculated by means of the
expression (26.15) for the operator L. We have
L[f(ey™<t>] = e^m-m & { n i-me d(/sin^)/d(cos d).
A repeated application of this formula gives
{!_)l-m e -04>® n = fdrrul, ^-mg d *-m( sin * q 0^)/ d ( cos Qf-m^
Finally, using these relations and the expression (28.9) for ZZ , we obtain
the formula
which is the same as (28.5).
§29. Matrix elements of vectors
Let us again consider a closed system of particles ;f let/ be any scalar
physical quantity characterising the system, and /the operator corresponding
to this quantity. Every scalar is invariant with respect to rotation of the
co-ordinate system. Hence the scalar operator / does not vary when acted
on by a rotation operator, i.e. it commutes with a rotation operator. We know,
however, that the operator of an infinitely small rotation is the same, apart
from a constant factor, as the angular momentum operator, so that
{/, L.) = {/, Lj = {/ 4} = (29.1)
(and also {/, L 2 } = 0).
From the commutability of / with the angular momentum operator it
follows that, in a representation where L 2 and L z are diagonal, the matrix of
the quantity / will also be diagonal with respect to the suffixes LM. We
shall conventionally denote by n all the remaining suffixes which define the
t AH the results in this section are valid also for a particle in a centrally symmetric field (and in
general whenever the total angular momentum of the system is conserved).
92 Angular Momentum §29
state of the system, and we shall show that the matrix elements f^L are
independent of the suffix M. To do this, we use the commutability of /with
/£+-£+/= 0.
Let us write down the matrix element of this equation corresponding to the
transition «, L, M -> «', L, M— 1. Taking into account the fact that the
matrix of the quantity L + has only elements with n,L,M-+ n, L, M—l, we
obtain
nLM n'LM nLM n L,M-l
J n'LM{^+)n'L,M-l— \L>+)nL,M-lJ n'L,M-l = «,
and since the matrix elements of the quantity L+ are independent of the suffix
n, we find
nLM nL,M-l
J n'LM =Jn'L,M-l,
whence it follows that all the quantities ffilh for different M (the other
suffixes being the same) are equal.
Thus the matrix elements of the quantity / that are different from zero
will be
nLM nL •
J n'LM =Jn'L, (29.2)
where ffi L denotes quantities depending on the values of the suffixes «, «', L.
If we apply this result to the Hamiltonian itself, we obtain our previous
result that the energy of the stationary states is independent of M, i.e. that
the energy levels have (2L+l)-fold degeneracy.
Next, let A be some vector physical quantity characterising a closed
system. When the system of co-ordinates is rotated (and, in particular, when
the operator of an infinitely small rotation, i.e. the angular momentum opera-
tor, is applied), the components of a vector are transformed into linear
functions of one another. Hence, as a result of the commutation of the
operators L t with the operators A t> we must again obtain components of the
same vector, A%. The exact form can be found by noticing that, in the
particular case where A is the radius vector of the particle, the formulae
(26.3) must be obtained. Thus we find the commutation rules
{l x ,A x } = 0, {l x ,A y } = iA z , {L X ,A Z }= -iA y , ... (29.3)
(the remaining rules are obtained by cyclic permutation of the suffixes
x, y, z), or
{l u A k } = ie m A x . (29.4)
These relations enable us to obtain several results concerning the form
of the matrices of the components of the vector A (M. Born, W. Heisenberg
and P. Jordan 1926). First of all, it is possible to derive selection rules which
determine the transitions for which the matrix elements can be different
from zero. We shall not go through the fairly lengthy calculations here,
however, since it will appear later (§107) that these rules are actually a direct
§29 Matrix elements of vectors 93
consequence of the general transformation properties of vector quantities
and can be derived from the latter with hardly any calculation at all. Here
we shall merely give the rules, without proof.
The matrix elements of all the components of a vector can be different
from zero only for transitions in which the angular momentum L changes by
not more than one unit :
L-+L or L±l. (29.5)
There is a further selection rule which forbids transitions between any two
states with L = 0. This rule is an obvious consequence of the complete
spherical symmetry of states with angular momentum zero.
The selection rules for the angular momentum component M are different
for the different components of a vector: the matrix elements can be different
from zero for transitions where M changes as follows:
for A+ = A x +iA y , M-+M-1, n
for A- = A x -iA y , M-+M+1, (29.6)
for A z , M^M. J
Moreover, it is possible to determine a general form for the matrix elements
of a vector as functions of the number M. These important and frequently
used formulae are given here, also without proof, since they are actually
a particular case of more general relations derived in §107 for any tensor
quantities.
The non-zero matrix elements of the quantity A z are given by the formulae
M
(A^ n ^ M = A^ L
Wn'LM V[L(L+1)(2L+1) fn'L>
W^ 1 . M = y £(2£ _ 1)(2£+1) ^ .,- 1 . | (29.7)
yL 2 —M 2
L(2L-1)(2L+1) nL
Here the A^ L . are quantities! independent of M and related by the "Her-
mitian" condition
a I>l> = (^£?>. (29.8)
which follows directly from the fact that the operator A z is Hermitian.
t These quantities are sometimes called reduced matrix elements. The appearance in formulae
(29.7) and (29.9) of denominators which depend on L is in accordance with the general notation used
in §107. The convenience of these denominators is shown, in particular, by the simple form of equation
(29.12) for the matrix elements of the scalar product of two vectors.
It may be noted for reference that for the vector L itself the reduced matrix elements are
L\ = VW+l)(2L+l)l L^ 1 = Ll_ x = 0,
since the matrix of L z is diagonal with respect to L and the diagonal elements are equal to M. The
same values are, of course, obtained from a comparison of formulae (29.9) and (27.12).
94 Angular Momentum §29
The matrix elements of the quantities A- and A + are also determined by
the A^\j. The non-zero matrix elements of A- are
(A \nL ,M-1 _ 12 /v ' An L
Wwlm -J L(L+l)(2L+ir n ' L '
(A \nL,M-l _ ± '2 L/jnL
1 -V,l-i,m ./ L{2 L-1)(2L+1) n '' L ~ v
(A )n ,L-l,M-l - _ ^l^Zl^l^l A n,L-l
(29.9)
(L + M-1)(L + M)
L(2L-1)(2L+1)
The matrix elements of A+ need not be written out separately: since A x
and A y are real we have, using (11.9),
(4)ZI&M> = U-)nLM M ']*- (29.10)
It is useful to notice a formula which expresses the matrix elements of a
scalar A . B (where A, B are two vector physical quantities) in terms of the
coefficients A^, B n J- L , in formulae (29.7)-(29.9); in applications such
products are usually involved. The calculation is conveniently carried out
by writing the operator A . B in the form
A . B = i(A + B-+A~B+)+A z B z . (29.11)
It is evident a priori that the matrix of A . B (like that of any scalar) is diagonal
with respect to L and M. A calculation by means of formulae (29.7)-(29.9)
gives the result
(A.B)»f* f = i ^- r 2^-i" B »"'f. (29.12)
n",L"
where L" takes the values L, L ± 1.
PROBLEMS
Problem 1. Determine the matrix elements (with respect to the eigenfunctions of the
angular momentum) of a unit vector n along the radius vector of the particle.
Solution. The matrix elements of a polar vector for an individual particle are non-zero
only for transitions / -*■ l± 1 (see §30). Their dependence on the quantum number m is given
by the general formulae (29.7)-(29.9), so that it is sufficient to calculate the coefficient
n \-i = ( n l X )* (corresponding to A % _ t in those formulae). To do this, we can find, for
example, the matrix element of n z = cos 6 as follows:
7 1 n
(^I 1)(2 / + i) "J" 1 = (C ° S d) "~'° = / 0; - 1 ' * cos 6 • ® 10 sin 6 dd >
with the functions ©jo given by (28.7); the integral is calculated from formula (107.15)
with h = \,h = 1—1. The result is
Problem 2. Average the tensor nitik—^ik (where n is a unit vector along the radius vector
of a particle) over a state where the magnitude but not the direction of the vector 1 is given
(i.e. l z is indeterminate).
§30 Parity of a state 95
Solution.^ The required mean value is an operator which can be expressed in terms of
the operator 1 alone. We seek it in the form
nink ~ %$ik — a[hh + tJi — §Sifc/(/+ 1)] ;
this is the most general symmetrical tensor of rank two with zero trace that can be formed
from the components of I. ^ To determine the constant a we multiply this equation on the left
by U and on the right by h (summing over i and k). Since ^the vector n is perpendicular to
the vector M = rxp, we have mU = 0. The product UlihU = (l 2 ) 2 is replaced by its
eigenvalue l\l+\) z , and the product UUUh is transformed by means of the commutation
relations (26.7) as follows:
hhhh = hhhh—iemhhh
= fi 2 ) 2 -¥ e mk(tih-hli)
= (l 2 ) 2 +ieikieikmUm
= (12)2 _ J2
= / 2 (/+l) 2 -/(/+l)
(using the fact that eme m ki = 2Sj m ). After a simple reduction we obtain the result
a= -l/(2/-l)(2/+3).
§30. Parity of a state
Besides the parallel displacement of the co-ordinate system (used in §15)
and the rotation of it (used in §26), there is another transformation which
leaves unaltered the Hamiltonian of a closed system, f This is what is called
the inversion transformation, which consists in simultaneously changing
the sign of all the co-ordinates. In classical mechanics, the invariance of
Hamilton's function with respect to inversion does not lead to a conser-
vation law, but the situation is different in quantum mechanics.
Let us denote by / the inversion operator; its effect on a function is to
change the sign of all the co-ordinates. The invariance of ff with respect to
inversion means that
fil-lfi = 0. (30.1)
The operator / also commutes with the angular momentum operators :
{/, L x } = {/, L y ) = {/, L z } = o, {/, L*} = o (30.2)
(on inversion, both the co-ordinates themselves and the operators of differ-
entiation with respect to them change sign, so that the angular momentum
operators remain unchanged).
It is easy to find the eigenvalues J of the inversion operator, which are deter-
mined by the equation
U = fy.
f The same is true of a system in a centrally symmetric field.
96 Angular Momentum §30
To do this, we notice that a double application of the operator / amounts to
identity : no co-ordinate is altered. In other words, we have
Piff = Pift — fa i.e. P = 1, whence
/ = ±1. (30.3)
Thus the eigenfunctions of the inversion operator are either unchanged or
change in sign when acted upon by this operator. In the first case, the wave
function (and the corresponding state) is said to be even, and in the second
it is said to be odd.
The equation (30.1) thus expresses the "law of conservation of parity";
if the state of a closed system has a given parity (i.e. if it is even, or odd),
then this parity is conserved.
The physical meaning of equations (30.2) is that the system can have defin-
ite values of L and M and, at the same time, a definite parity of its state. We
can also say that all states differing only in the value of M have the same
parity. This can be shown by starting from the relation
L4-1U = o
and proceeding in exactly the same way as in obtaining the result (29.2).
When the inversion transformation is applied to scalar quantities, either
they do not change at all (true scalars) or they change sign (what are called
pseudoscalars).-\ If a physical quantity / is a true scalar, its operator com-
mutes with /:
7/-// = 0. (30.4)
It follows from this that, if the matrix of I is diagonal, then the matrix of
/ is diagonal with respect to the suffix which shows the parity of the state, i.e.
only the matrix elements for transitions u -> u and g ->• g are not zero (the
suffixes g and u denote even and odd states respectively).
For the operator of a pseudoscalar quantity we have // = — //, or
//+// = 0; (30.5)
/ anticommutes with /. The matrix element of this equation for a transition
and since I gg =1 we have f gg = (we omit all suffixes apart from that
showing the parity). Similarly we find that f uu — 0. Thus, in the matrix
of a pseudoscalar quantity, only those elements can be different from zero
which are non-diagonal with respect to the parity suffix (transitions with
change of parity).
\ An example of a pseudoscalar is the scalar product of an axial and a polar vector.
§31 Addition of angular momenta 97
Similar results are obtained for vector quantities. The operators of a
polar vectorf anticommute with /, and in their matrices (in a representation
where 1 is diagonal) only the elements for transitions with change of parity
are not zero. The operators of an axial vector, however, commute with / ,
and their matrices have non-zero elements only for transitions without change
of parity.
It is useful to point out another method of obtaining these results. For
example, the matrix element of a scalar / for a transition between states of
opposite parity is the integral f ug = $ tft u *ftfi g dq, where the function \\t g is
even and tj/ u is odd. When all the co-ordinates change sign, the integrand
does so if / is a true scalar; on the other hand, the integral taken over all
space cannot change when the variables of integration are re-named. Hence
it follows that/ u? = -/ ua , i.e. / w s 0.
Let us determine the parity of the state of a single particle with angular
momentum /. The inversion transformation (x -> — x, y -> —y, z -> —z)
is, in spherical polar co-ordinates, the transformation
r^r, 6-+TT-9, ^->tt+0. (30.6)
The dependence of the wave function of the particle on the angle is given by
the eigenfunction Y Jm of the angular momentum, which, apart from a constant
which is here unimportant, has the form P, m (cos &)e im *. When <j> is replaced
by ir+<f>, the factor e im * is multiplied by ( — l) m , and when 6 is replaced by
77—0, P, m (cos0) becomes P^-cos 0) = (-l) l - m P J wl (cos 0). Thus the
whole function is multiplied by ( — l) 1 (independent of m, in agreement with
what was said above), i.e. the parity of a state with a given value of / is
/ = (-1) 1 . (30.7)
We see that all states with even I are even, and all those with odd / are odd.
A vector physical quantity relating to an individual particle can have non-
zero matrix elements only for transitions with / -> / or /±1 (§29). Remem-
bering this, and comparing formula (30.7) with what was said above regarding
the change of parity in the matrix elements of vectors, we reach the result
that the matrix elements of a polar vector are non-zero only for transitions
with / -> 7±1, and those of an axial vector for transitions with / -> /.
§31. Addition of angular momenta
Let us consider a system composed of two parts whose interaction is weak.
If the interaction is entirely neglected, then for each part the law of conserva-
tion of angular momentum holds. The angular momentum L of the whole
system can be regarded as the sum of the angular momenta Li and L2 of its
parts. In the next approximation, when the weak interaction is taken into
f Ordinary (polar) vectors change sign under the inversion transformation, whilst axial vectors
(for instance, the vector product of two polar vectors) are unchanged by this transformation.
98 Angular Momentum §31
account, Li and L2 are not exactly conserved, but the numbers L\ and L 2
which determine their squares remain "good" quantum numbers suitable for
an approximate description of the state of the system. Regarding the angular
momenta in a classical manner, we can say that in this approximation Li and
L2 rotate round the direction of L while remaining unchanged in magnitude.
For such systems the question arises regarding the "law of addition" of
angular momenta : what are the possible values of L for given values of L\
and L 2 1 The law of addition for the components of angular momentum is
evident: since L z = L\ z + L 2z , it follows that M = M\ + M 2 . There is no
such simple relation for the operators of the squared angular momenta, how-
ever, and to derive their "law of addition" we reason as follows.
If we take the quantities L-, 2 , L 2 2 , L Xz , L 2z as a complete set of physi-
cal quantities,f every state will be determined by the values of the numbers
L x , L 2 , M x , M 2 . For given L x and L 2 , the numbers M x and M 2 take (21^+1)
and (2L 2 +1) different values respectively, so that there are altogether
(2L 1 +1)(2L 2 +1) different states with the same L x and L 2 . We denote the
wave functions of the states for this representation by <f> L LiMM .
Instead of the above four quantities, we can take the four quantities
Li 2 , L 2 2 , L 2 , L z as a complete set. Then every state is characterised by
the values of the numbers L x , L 2 , L, M (we denote the corresponding wave
functions by ^l^lm)- For given L x and L 2 , there must of course be
(2L 1 +1)(2L 2 +1) different states as before, i.e. for given L x and L 2 the pair
of numbers L and M must take (2L 1 +1)(2L 2 +1) pairs of values. These
values can be determined as follows.
To each value of L, there correspond 2L + 1 different possible values of M,
from — L to +L. The greatest possible value of M in the states ^ (for given
L x and L 2 ) is M = L x +L 2 , which is obtained when M x = L x and M 2 = L 2 .
Hence the greatest possible value of M in the states iff is Lj+1,2, and this is
therefore the greatest possible value of L also. Next, there are two states <f>
with M = L ± +L 2 — 1, namely those where M x = L X ,M 2 — L 2 —\ and
M x = L x — 1, Af 2 = L 2 . Consequently, there must also be two states ifj with
this value of M ; one of them is the state with L = L x +L 2 (and M = L— 1),
and the other is clearly that with L = L^+L^ — 1 (and M = L). For the
value M = L x +L 2 —2 there are three different states <j>, with the following
pairs of values of M lf M 2 : (L ly L 2 — 2), (L x — 1, L 2 — 1), and (L X —2,L 2 ).
This means that, besides the values L = L x +L 2 , L = L x +L 2 —1, the value
L = L x +L 2 — 2 can occur.
The argument can be continued in this way so long as a decrease of M by
1 increases by 1 the number of states with a given M. It is easily seen that
this is so until M reaches the value \L± — L 2 \. When M decreases further, the
number of states no longer increases, remaining equal to 2L2+ 1 (if L 2 ^ L{).
Thus \L\ — L 2 \ is the least possible value of L, and we arrive at the result
f Together with such other quantities as form a complete set when combined with these four.
These other quantities play no part in the subsequent discussion, and for brevity we shall ignore
them entirely, and conventionally call the above four quantities a complete set.
§31 Addition of angular momenta 99
that, for given L\ and L% the number L can take the values
L = Li+Z^+La-l, ..., I^-L^, (31.1)
that is 2L 2 +1 different values altogether (supposing that L 2 < jy. It is
easy to verify that we do in fact obtain (2L 1 +1)(2L 2 +1) different values of
the pair of numbers M, L. Here it is important to note that, if we ignore
the 2L+\ values of M for a given L, then only one state will correspond to
each of the possible values (31.1) of L.
This result can be illustrated by means of what is called the vector model.
If we take two vectors L l5 L 2 of lengths L x and L 2 , then the values of L are
represented by the integral lengths of the vectors L which are obtained by
vector addition of L x and Lg; the greatest value of L is L x -\-L 2 , which is
obtained when L 2 and L 2 are parallel, and the least value is \L L —L 2 \, when
Li and L 2 are antiparallel.
The addition rule for angular momenta which we have obtained also makes
it possible, of course, to add any number (more than two) of angular momenta
by successive applications of this rule.
In states with definite values of the angular momenta Z^, L 2 and of the
total angular momentum L, the scalar products L x . L 2 , L . L x and L . L 2 also
have definite values. These values are easily found. To calculate L x . L 2 ,
we write L = I^+Lg or, squaring and transposing,
Replacing the operators on the right-hand side of this equation by their
eigenvalues, we obtain the eigenvalue of the operator on the left-hand side:
L x . L 2 = J{X(L+1)-L 1 (Z 1 + !)-£,(£,+ 1)}. (31.2)
Similarly we find
L . L x = ^^+l)+L 1 (L l +l)-L^L t +l)}. (31.3)
Let us now determine the "addition rule for parities". As we know, the
wave function T of a system consisting of two independent parts is the pro-
duct of the wave functions W ± and T 2 of these parts. Hence it is clear that, if
the latter are of the same parity (i.e. both change sign, or both do not
change sign, when the sign of all the co-ordinates is reversed), then the
wave function of the whole system is even. On the other hand, if T x and
T 2 are of opposite parity, then the function Y is odd.
This rule can, of course, be generalised at once to the case of a system
composed of any number n of non-interacting parts. If these parts are in
states with definite parities determined by the corresponding eigenvalues
Ii = ± 1 of the operator /, then the parity I of the state of the whole system
is given by the product
/=7 1 7 2 .../ n . (31.4)
100 Angular Momentum §31
In particular, if we are concerned with a system of particles in a centrally
symmetric field (the mutual interaction of the particles being supposed weak),
then I t = ( — l) 1 *, where l t is the angular momentum of the *'th particle
(see (30.7)), so that the parity of the state of the whole system is given by
7 = (-l)Wwtf„. (31.5)
We emphasise that the exponent here contains the algebraic sum of the
angular momenta l if and this is not in general the same as their "vector sum",
i.e. the angular momentum L of the system.
If a closed system disintegrates (under the action of internal forces), the
total angular momentum and parity must be conserved. This circumstance
may render it impossible for a system to disintegrate, even if this is energetic-
ally possible.
For instance, let us consider an atom in an even state with angular momen-
tum L = 0, which is able, so far as energy considerations go, to disintegrate
into a free electron and an ion in an odd state with the same angular momen-
tum L = 0. It is easy to see that in fact no such disintegration can occur
(it is, as we say, forbidden). For, by virtue of the law of conservation of angu-
lar momentum, the free electron would also have to have zero angular momen-
tum, and therefore be in an even state (I = ( — 1)° = +1); the state of the
system ion + electron would then be odd, however, whereas the original state
of the atom was even.
CHAPTER V
MOTION IN A CENTRALLY SYMMETRIC FIELD
§32. Motion in a centrally symmetric field
The problem of the motion of two interacting particles can be reduced in
quantum mechanics to that of one particle, as can be done in classical mech-
anics. The Hamiltonian of the two particles (of masses mi, mz) interacting in
accordance with the law U(r) (where r is the distance between the particles)
is of the form
# = -— Ax-— A 2 + U(r), (32.1)
2m x 2m %
where Ai arid A 2 are the Laplacian operators with respect to the co-ordinates
of the particles. Instead of the radius vectors r x and r 2 of the particles, we
introduce new variables R and r:
r = r 2— *i> R = ( m i r i+™jr 2 )l(m x +m 2 ); (32.2)
r is the vector of the distance between the particles, and R the radius vector
of their centre of mass. A simple calculation gives
8 = - 0/ , An— -A + I7(r), (32.3)
2(m 1 +m 2 ) 2m v J
where A R and A are the Laplacian operators with respect to the components
of the vectors R and r respectively, m x +m 2 is thetotal mass of the system, and
m = m 1 ni2l(m 1 +m i ) is what is called the reduced mass. Thus the Hamiltonian
falls into the sum of two independent parts. Hence we can look for 0(r lf r 2 )
in the form of a product ^(R)^r(r), where the function ^(R) describes the mo-
tion of the centre of mass (as a free particle of mass m^m^), and «/r(r) describes
the relative motion of the particles (as a particle of mass m moving in the cen-
trally symmetric field U(r)).
Schrodinger's equation for the motion of a particle in a centrally sym-
metric field is
A0+(2m//* 2 )[£- U(r)W = 0. (32.4)
Using the familiar expression for the Laplacian operator in spherical polar
co-ordinates, we can write this equation in the form
1 0/ &K lr 1 d/ M\ 1 Sty! 2m
(32.5)
101
102 Motion in a Centrally Symmetric Field §32
If we introduce here the operator i 2 (26.16) of the squared angular momentum
we obtainf
h 2 r 18/ dib\ I 2 "I
£L - 7a;raVj +DW *-*- (326)
The angular momentum is conserved during motion in a centrally sym-
metric field. We shall consider stationary states in which l 2 and l z have
definite values. In other words, we shall seek the common eigenfunctions
of the operators i?, i 2 and l z .
The requirement that iff is an eigenfunction of the operators i 2 and l z
determines its angular dependence. We thus seek solutions of equation
(32.6) in the form
^ = R(r)Y lm (9,<f>), (32.7)
where the functions Y lm (6, <f>) are defined by the formulae of §28.
Since l 2 Y lm = l(l+l)Y lm , we obtain for the radial function R(r) the equa-
tion
1 d / dR\ 1(1+1) 2m
We note that this equation does not contain the value of l z = m at all, in
accordance with the (2/+l)-fold degeneracy of the levels, with which we are
already familiar.
Let us investigate the radial part of the wave functions. By the substitu-
tion
R(r) = x (r)/r (32.9)
equation (32.8) is brought to the form
d 2 x r2m 1(1+1)1
_ + [_ (£ _ C0 _l_l], =0 . ( 3 2 ,0)
If the potential energy U(r) is everywhere finite, the wave function if/ must
also be finite in all space, including the origin, and consequently so must its
radial part R(r). Hence it follows that x { r ) must vanish for r = :
X(0)=0. (32.11)
t If we introduce the operator of the radial component p r of the linear momentum, in the form
1 2 /d 1\
prf = -ih-—(r$) = -ih( —+- U,
r or \or rj
the Hamiltonian can be written in the form
H = (l/2m)(/ r 2 +M 2 /r 2 )+E/(r),
which is the same in form as the classical Hamilton's function in spherical polar co-ordinates.
§32 Motion in a centrally symmetric field 103
This condition actually holds also (see §35) for a field which becomes infinite
as r -> 0.
The normalisation condition for the radial function R(r) is determined by
the integral J \R\ 2 r 2 dr, and therefore that for the function x(r) is determined
by the integral J" |x| 2 dr.
Equation (32.10) is formally identical with Schrodinger's equation for
one-dimensional motion in a field of potential energy
h 2 Z(Z+1)
£/,(r) = £/(,)+ J_ J-, (32.12)
2m r l
which is the sum of the energy U(r) and a term
M(/+l)/2mr 2 = hH*l2mr\
which may be called the centrifugal energy. Thus the problem of motion in a
centrally symmetric field reduces to that of one-dimensional motion in a
region bounded on one side (the boundary condition for r = 0).
In one-dimensional motion in a region bounded on one side, the energy
levels are not degenerate ( §21). Hence we can say that, if the energy is given,
the solution of equation (32.10), i.e. the radial part of the wave function, is
completely determined. Bearing in mind also that the angular part of the
wave function is completely determined by the values of / and m, we reach
the conclusion that, for motion in a centrally symmetric field, the wave func-
tion is completely determined by the values of E, I and m. In other words,
the energy, the squared angular momentum and the ^-component of the
angular momentum together form a complete set of physical quantities for
such a motion.
The reduction of the problem of motion in a centrally symmetric field to a
one-dimensional problem enables us to apply the oscillation theorem (see
§21). We arrange the eigenvalues of the energy (discrete spectrum) for a
given / in order of increasing magnitude, and give them numbers n r , the lowest
level being given the number % = 0. Then n r determines the number of
nodes of the radial part of the wave function for finite values of r (excluding
the point r = 0). The number nr is called the radial quantum number. The
number / for motion in a centrally symmetric field is sometimes called the
azimuthal quantum number, and m the magnetic quantum number.
There is an accepted notation for states with various values of the angular
momentum / of the particle: they are denoted by Latin letters, as follows:
Z = 01234567...
s p d f g h i k ...
The normal state of a particle moving in a centrally symmetric field is
always the s state; for, if / # 0, the angular part of the wave function in-
variably has nodes, whereas the wave function of the normal state can have
no nodes. We can also say that the least possible eigenvalue of the energy,
104 Motion in a Centrally Symmetric Field §33
for a given /, increases with /. This follows from the fact that the presence
of an angular momentum involves the addition of the essentially positive
term /* 2 /(/+l)/2mr 2 , which increases with /, to the Hamiltonian.
Let us determine the form of the radial function near the origin. Here
we shall suppose that
lim U(r)r 2 = 0.
r-*0
We seek R(r) in the form of a power series in r, retaining only the first term
of the series for small r; in other words, we seek R(r) in the form R = con-
stant Xr 1 . Substituting this in the equation
d(r 2 d#/dr)/dr-/(/+l)i? = 0,
which is obtained from (32.8) by multiplying by r 2 and taking the limit as
r -> 0, we find
s(s+l) =1(1+1).
Hence
s = I or s = — (/+1).
The solution with s = — (/+1) does not satisfy the necessary conditions;
it becomes infinite for r = (we recall that / ^ 0). Thus the solution with
s = I remains, i.e. near the origin the wave functions of states with a given /
are proportional to r l :
R t £ constant xr*. (32.13)
The probability of a particle's being at a distance between r and r+dr from
the centre is determined by the value of r*\R\ 2 and is thus proportional to
r 2(i+i) w e see t j lat - lt b ecomes 2ero at ^ or igi n tne more rapidly, the
greater the value of /.
§33. Free motion (spherical polar co-ordinates)
The wave function of a freely moving particle
t/tp = constant xc (t '/ A) P- r
describes a stationary state in which the particle has a definite momentum p
(and energy E = p 2 j2m). Let us now consider stationary states of a free
particle in which it has a definite value, not only of the energy, but also of the
absolute value and component of the angular momentum. Instead of the
energy, it is convenient to introduce the wave number
k=p/h = V(2mE)/h. (33.1)
The wave function of a state with angular momentum / and projection
§33 Free motion {spherical polar co-ordinates) 105
thereof m has the form
lAwm = RM Y lm (d f <j>), (33.2)
where the radial function is determined by the equation
2 r Z(/+l)n
*«"+-*h'+ L* 1 """^]^ = ° ( 33 - 3 >
(equation (32.8) with U(r) = 0). The wave functions ^ Wm satisfy the condi-
tions of normalisation and orthogonality:
/ kffm+fium W = Sll'KmW-k).
The orthogonality for different /, /' and m, m' is ensured by the angular func-
tions. The radial functions must be normalised by the condition
oo
jrW^Rudr-Sik'-k). (33.4)
o
If we normalise the wave functions, not on the "k scale", but on the "energy
scale", i.e. by the condition
oo
j rtR^REt dr = 8(E'-E),
o
then, by the general formula (5.14), we have
Rbi = Ra VCdkJdE) = (l/h) V(m/k)R kl . (33.5)
For / = 0, equation (33.3) can be written
d 2 (r**o)
dr 2
•khRn = 0;
its solution finite for r = and normalised by the condition (33 4) is fcf
(21.9)) v
72 sin kr
.
77 r
(33.6)
To solve equation (33.3) with / ^ 0, we make the substitution
R u = i*Xki- (33.7)
For Xfci we have the equation
X»"+2(/+lW/r+**«=0.
If we differentiate this equation with respect to r, we obtain
W" + ^," + [^-^> tt '=0.
106 Motion in a Centrally Symmetric Field §33
By the substitution xui = r Xk,i+i it becomes
2(1+2)
Xfc,l+i"H Xk,l+i'+k 2 x k ,i+ 1 = 0,
r
which is in fact the equation satisfied by xn,i+i- Tnus the successive func-
tions Xki are related by
Xm+i = Xki/r, (33.8)
and hence
,'1 dV
Xkl
~ {;&)*»*
wner e Xko = Rko is determined by formula (33.6) (this expression can, of
course, be multiplied by an arbitrary constant).
Thus we finally have the following expression for the radial functions in
the free motion of a particle :
/ *s7 I 2 rl Z 1 d V sin ^
(the factor k~ l is introduced for normalisation purposes — see below — and
the factor ( — l) 1 for convenience).!
To obtain an asymptotic expression for the radial function (33.9) at large
distances, we notice that the term which decreases least rapidly as r -» oo is
obtained by differentiating sin kr I times :
12 1 d*
Rki ~ (— 1) / — 77- — 7 sin kr.
V 77 k l r dr l
d • , / dV
— smkr =ksm(kr-%7r),..., I J sinkr = # sin (Ar- A/77),
t The functions R u can be expressed in terms of Bessel functions of half-integral order, in the form
Ru = V(k/r)J l+1/2 (kr). (33.9a)
The first few functions R kl are :
\2 sinkr
Rkn = l-k
kr
2 rsin&r coskr"
y2 rsmkr cos&r"]
7t L (kr) z k~r~J
(l r/ 3 1\ 3cosArT
#* 2 = -k\ ( hinkr .
V77 L\(kr)* kr) (kr) 2 J
§33 Free motion {spherical polar co-ordinates) 107
we have the following asymptotic expression :
V2 sin(kr— Utt)
— . (33.10)
it r
The normalisation of the functions R kl can be effected by means of their
asymptotic expressions, as was explained in §21. Comparing the asymptotic
formula (33.10) with the normalised function R kQ (33.6), we see that the func-
tions R kl , with the coefficient used in (33.9), are in fact normalised as they
should be.
Near the origin (r small) we have, retaining only the term containing the
lowest power of r,
1 dVsin&r /l d\*^ #2n+i r 2n
/l dysin&r /ldVy #Jn+i r 2n
\r drj r ~ \r drj ^ (2«+l)I
/l dV^+V 2 '
= (-m — )
\rdr/(2/+l)I
2. 4... 21
= (_ l)^+i
(2/+1)!
= (-l)^+V1.3...(2/+l).
Thus the functions i? fci near the origin have the formf
in agreement with the general result (32.13).
In some problems it is necessary to consider wave functions which do not
satisfy the usual conditions of finiteness, but correspond to a flux of particles
from the origin. The wave function which describes such a flux of particles
with angular momentum / = is obtained by taking, instead of the "station-
ary spherical wave" (33.6), a solution in the form of an "outgoing spherical
wave",
# ft0 + = Ae ikr Jr. (33.12)
This function becomes infinite at the origin.
Similarly, a flux of particles incident on the centre (with angular momen-
tum / = 0) is described by a wave function in the form of an "ingoing
spherical wave",
R k0 ~ = Ae~ ik rjr. (33.13)
f The symbol !! denotes the product of all integers of the same parity up to and including the num-
ber in question.
108 Motion in a Centrally Symmetric Field §33
In the general case of an angular momentum / which is not zero, we obtain
a solution of equation (33.3) in the formf
, r 1 /l d \ l e ±ikr
^ =( _^_(._)_. (33 , +)
The asymptotic expression for these functions is
R kl ± « Ae ±{ < kr - ln ^Jr. (33.15)
Near the origin, it has the form
(2/-1)!!
22„± « A — H- 1 . (33.16)
We normalise these functions so that they correspond to the emission (or
absorption) of one particle per unit time. To do so, we notice that, at large
distances, the spherical wave can be regarded as plane in any small interval,
and the probability current density in it is i = viffif,*, where v = kh/m is the
velocity of a particle. The normalisation is determined by the condition
§idf =1, where the integration is carried out over a spherical surface of
large radius r, i.e. j ir % do = 1, where do is an element of solid angle. If the
angular functions are normalised as before, the coefficient A in the radial
function must be put equal to
A =1/Vv = V(mjkh). (33.17)
An asymptotic expression similar to (33.10) holds, not only for the radial
part of the wave function of free motion, but also for motion (with positive
energy) in any field which falls off sufficiently rapidly with distance.:}: At
large distances we can neglect both the field and the centrifugal energy in
Schrodinger's equation, and there remains the approximate equation
1 d*(rR kl )
r dr z
The general solution of this equation is
R*
*J IT
2 sin(Ar— ^/tt+Si)
unct:
rd.2),
(33.18)
t These functions can be expressed in terms of Hankel functions :
Rm ± = ±iA V(^/2r)H?4W), (33.14a)
of the first and second kinds for the signs + and — respectively. The asymptotic expansion of the
functions i?*j± for large r is
e ±i(kr-ln/2) p /(/ + 1) (/-l)/(/+l)(/+2)
r 1 *(f+i) (f-i)*(*+i)(/+2) -j
1 l!2^r 2\(2ikrY T "j
r L l!2^r 2\(2ikrf
% As we shall show in §123, the field must decrease more rapidly than 1/r.
§33 Free motion {spherical polar co-ordinates) 109
where S t is a constant, called the phase shift, and the common factor is chosen
in accordance with the normalisation of the wave function on the "k scale".t
The constant phase shift S z is determined by the boundary condition (Ria
is finite as r -» 0) ; to do this, the exact Schrodinger's equation must be
solved, and 8 Z cannot be calculated in a general form. The phase shifts S,
are, of course, functions of both / and k, and are an important property of the
eigenfunctions of the continuous spectrum.
PROBLEMS
Problem 1 . Determine the energy levels for the motion of a particle with angular momen-
tum / = in a centrally symmetric potential well:
U(r)[= — U for r < a, U(r) = for r > a.
Solution. For / = the wave functions depend only on r. Inside the well, Schrodinger's
equation has the form
~ TiWH*** = 0, * = - V[2m(U - |tf|)].
r dr a n
The solution finite for r = is
For r > a, we have the equation
sinkr
1 d 2 1
" 77^)- *V - °» «=r V(2m\E\).
r dr z n
The solution vanishing at infinity is
ifi = A'erKr/r.
The condition of the continuity of the logarithmic derivative of rtf> at r = a gives
k cot ka = -k = - V[(2mU /h 2 )-k 2 ], (1)
or
sinka = ±V(h*l2ma 2 U )ka. (2)
This equation determines in implicit form the required energy levels (we must take those
roots of the equation for which cot ka < 0, as follows from (1)). The first of these levels
(with / = 0) is at the same time the deepest of all energy levels whatsoever, i.e. it corresponds
to the normal state of the particle.
If the depth U of the potential well is small enough, there are no levels of negative energy,
and the particle cannot "stay" in the well. This is easily seen from equation (2), by means of
the following graphical construction. The roots of an equation of the form ± sin * = ax
are given by the points of intersection of the line y = oue with the curves y = ±sin x, and
we must take only those points of intersection for which cot x < ; the corresponding parts
of the curve y = sin x are shown in Fig. 9 by a continuous line. We see that, if a is
sufficiently large (U small), there are no such points of intersection. The first such point
appears when the line y = atx occupies the position Oa, i.e. for a = 2/w, and is at * = Jtt.
t The term — \Itt in the argument of the sine is added so that Si = when the field is absent.
Since the sign of the wave function as a whole is not significant, the phase shifts Si are determined to
within nir (not 2nir). Their values may therefore always be chosen in the range between and n.
110 Motion in a Centrally Symmetric Field §33
Fig. 9
Putting a = h/V(2ma 2 U ), x = ka, we hence obtain for the minimum well depth to give a
single negative level
^o.min = **h*l8ma*. (3)
This quantity is the greater, the smaller the well radius a. The position of the first level
Z?i at the point where it first appears is determined from ka = \tt and is E x = 0, as we should
expect. As the well depth increases further, the normal level E x descends.
Problem 2. Determine the order of the energy levels with various values of the angular
momentum / in a very deep potential well (Uo > % % \mcF) (W. Elsasser 1933).
Solution. The condition at the boundary of the well requires that -> as Uo -> oo
(see §22). Writing the radial wave function within the well in the form (33.9a), we thus have
the equation
Jl+l/2(ka) = 0,
whose roots give the position of the levels above the bottom of the well (Uo — \E\ = H 2 k 2 l2m)
for various values of I. The order of the levels from the ground state is found to be
1*, lp, Id, 2s, If, 2p, \g, 2d, \h, 3s, 2/, ... .
The numbers preceding the letters give the sequence of levels for each /.f
Problem 3. Determine the order of appearance of levels with various I as the depth
Uo of the well increases.
Solution. When it first appears, each new level has energy E = 0. The corresponding
wave function in the region outside the well, which vanishes asr-> oo, is
Ri = constant x r~ (l +U
(the solution of equation (33.3) with k = 0). From the continuity of i? ? and Ri' at the boundary
of the well it follows, in particular, that the derivative (r l+x R{)' is continuous, and so we have
the following condition for the wave function within the well:
(ri+iRtf = for r = a.
This is equivalent to the condition for the function Ri-± to vanish and, from (33.9a), we
obtain the equation
Ji-i/2(aV(2mU )lh) = 0;
for I = the function 7?_i/ 2 must be replaced by the cosine. This gives the following order
of appearance of new levels as Uo increases :
1*, lp, Id, 2s, If, 2p, \g, 2d, 3s, \h, 2f, ... .
It may be noted that differences from the order of levels in a deep well occur only for compar-
atively high levels.
t This notation is customary for particle levels in the nucleus (see Chapter XVI).
% According to (33.7) and (33.8) we have (r~ l Ri)' ~ r-'Ri +1 . Since the equation (33.3) is unaltered
when / is replaced by — /— 1, we also have (r 1+1 R-i-i)' ~ r'+ii?_ ( . Finally, since the functions i?_j
and Ri-i satisfy the same equation, we obtain (r l+1 Ri)' ~ r l+1 Ri_i, the formula used in the text.
§34 Resolution of a plane wave 111
Problem 4. Determine the energy levels of a three-dimensional oscillator (a particle in a
field U — $fi(o*r 2 ), their degrees of degeneracy, and the possible values of the orbital angular
momentum in the corresponding stationary states.
Solution. Schrodinger's equation for a particle in a field U = $na> 2 (x*+y t +z*) allows
separation of the variables, leading to three equations like that of a linear oscillator. The
energy levels are therefore
E n = foofa+wj+ns+f) = ha>{n+f).
The degree of degeneracy of the nth level is equal to the number of ways in which n can be
divided into the sum of three positive integral (or zero) numbers ;f this is
K»+i)(»+2).
The wave functions of the stationary states are
«£ WlW ,n 3 = constant x e-^V2H n ^)H n J < pLy)H n3 {a.z), (1)
where a = \/(ma>lfi) and m is the mass of the particle. When the sign of the co-ordinate
is changed, the polynomial H n is multiplied by ( — 1)". The parity of the function (1) is
therefore ( — l)»i+»s+»8 = ( — 1)". Taking linear combinations of these functions with a
given sum ni + m + nz = n, we can form the functions
4>nim = constant x e-* 2 r 2 l*r n ®i m {d)e ±im <l>, (2)
where m = 0,1, ..., / and I takes the values 0, 2, ..., n for even n and 1, 3, ..., « for odd n.
This is evident from a comparison of the parities ( — 1)" of the functions (1) and ( — I) 1 of the
functions (2), which must be the same. This determines the possible values of the orbital
angular momentum corresponding to the energy levels considered.
The order of levels of the three-dimensional oscillator is, therefore, with the same notation
as in Problems 2 and 3,
(Is), (l/>), (Id, 2.), (1/, 2p), (\g, 2d, 3s), ... ,
where the parentheses enclose sets of degenerate states.
§34. Resolution of a plane wave
Let us consider a free particle moving with a given momentum p = kk
in the positive direction of the ^-axis. The wave function of such a particle
is of the form
iff = constant xe ikz .
Let us expand this function in terms of the wave functions rji klm oi free motion
with various angular momenta. Since, in the state considered, the energy
has the definite value k 2 h 2 ]2m, it is clear that only functions with this k will
appear in the required expansion. Moreover, since the function e ikz has
axial symmetry about the ^-axis, its expansion can contain only functions
independent of the angle <f>, i.e. functions with m = 0. Thus we must have
t In other words, this is the number of ways in which n similar balls can be distributed among
three urns.
112 Motion in a Centrally Symmetric Field §34
where the ai are constants. Substituting the expressions (27.3), (28.7) and
(33.9) for the functions <I>, 0, R, we obtain
v^ /r\ l /l dVsinAr
e , te= gc,p,(-*)(-)(--)_ <»-»»•>.
where the C, are other constants. These constants are conveniently deter-
mined by comparing the coefficients of (r cos 6) n in the expansions of the two
sides of the equation in powers of r. On the right-hand side of the equation
this term occurs only in the nth summand ; for / > n, the expansion of the
radial function begins at a higher power of r, while for / < n the polynomial
P i (cos 6) contains only lower powers of cos 0. The term in cos * 6 in P, (cos 6)
has the coefficient (2/)!/2*(/!) 2 (see formula (c.2)). Using also formula
(33.11), we find the desired term of the expansion of the right-hand side of the
equation to be
2'(Z!) 2 (2/+1)H
On the left-hand side of the equation the corresponding term (in the expansion
of e ikr cos d ) is
(ikr cos ey III
Equating these two quantities, we find C, = (— *y(2/+l). Thus we finally
obtain the required expansion:
<- -g^^+DIM^Q (--) — . (34.1)
At large distances this relation takes the asymptotic form
e ikz ~ _ y ;/( 2 /+ l)P,(cos 6) sm(kr-ilir). (34.2)
kr r—'
We normalise the wave function e ik * to give a probability current density
of unity, i.e. so that it corresponds to a flux of particles (parallel to the #-axis)
with one particle passing through unit area in unit time. It is easy to see that
this function is
xfj = v -i/2 e tkz = v(m/&#)e** z> (34.3)
where v is the velocity of the particles. Multiplying both sides of equation
(34.1) by y/{mjkh) and introducing on the right-hand side the normalised
functions ^ ffl ± = R kl ±(r)Y lm (d,<f>), we obtain
1 <» 1
—em = 2 VK2/+l)]-(^ ro +-^ -).
§35 "Fall" of a particle to the centre 1 13
The squared modulus of the coefficient of ip kW ~ (or <p kt0 +) in this expansion
determines, according to the usual rules, the probability that a particle in a
current converging to (or diverging from) the centre has an angular momen-
tum / (about the origin). Since the wave function «rV** corresponds to a
current of particles of unit density, this "probability" has the dimensions of
length squared; it can be conveniently interpreted as the magnitude of the
"cross-section" (in the *y-plane) on which the particle must fall if its angular
momentum is /. Denoting this quantity by a h we have
ex, = ir(2l+ 1W = *W2ir)«(2/+ 1), (34.4)
where A is the de Broglie wavelength of the particle.
For large values of /, the sum of the cross-sections over a range A/ of I
(such that 1 <^ A/ < /) is
2 it ih 2
<t, £ — 2/AZ = 2tt — A/.
Al R P
On substituting the classical expression for the angular momentum, hi = pp
(where p is what is called the impact parameter), this expression becomes
27TpAp t
in agreement with the classical result. This is no accident; we shall see
below that, for large values of /, the motion is quasi-classical (see Chapter
VII).
§35. "Fall" of a particle to the centre
To reveal certain properties of quantum-mechanical motion it is useful to
examine a case which, it is true, has no direct physical meaning: the motion
of a particle in a field where the potential energy becomes infinite at some
point (the origin) according to the law U(r) « -£/r 2 , > 0; the form of the
field at large distances from the origin is here immaterial. We have seen in
§18 that this is a case intermediate between those where there are ordinary
stationary states and those where a "fall" of the particle to the origin takes
place.
Near the origin, Schrodinger's equation in the present case is
R"+2R'lr+yRlr* = 0, (35.1)
where R(r) is the radial part of the wave function, and we have introduced
the constant
Y «2«0/*»-Z(Z+l) (35.2)
and have omitted all terms of lower orders in 1/r; the value of the energy E
is supposed finite, and so the corresponding term in the equation is omitted
also.
114 Motion in a Centrally Symmetric Field §35
Let us seek R in the form R ~r s ; we then obtain for s the quadratic
equation
*(*+l)+y =0,
which has the two roots
*i = -B- V(i-y), *, = -*-V(i-y). (35.3)
For further investigations it is convenient to proceed as follows. We draw
a small region of radius r round the origin, and replace the function — yjr 2
in this region by the constant — y/r 2 . After determining the wave functions
in this "cut off" field, we then examine the result of passing to the limit
'o->0.
Let us first suppose that y < \. Then s x and s 2 are real negative quantities,
and s x > s 2 . For r > r Qy the general solution of Schrodinger's equation
has the form (always restricting ourselves to small r),
R = Ar\+Br\ (35.4)
A and B being constants. For r < r , the solution of the equation
R"+2R'/r+yRlr * =
which is finite at the origin has the form
sin&r
R = C , k = Vylr . (35.5)
r
For r = r , the function R and its derivative R' must be continuous. It is
convenient to write one of the conditions as a condition of continuity of the
logarithmic derivative of rR. This gives the equation
^i+lK Sl +£(*2+l)V' t 7
= k cot kr .
i4r *»+ 1 +-Br ».+ 1
or
= Vy cot \/y.
Ar **+Br *>
On solving for the ratio B/A, this equation gives an expression of the form
B/A = constant xr s ^-^. (35.6)
Passing now to the limit r -> 0, we find that BjA -*• (recalling that
h > h)- Thus, of the two solutions of Schrodinger's equation (35.1) which
diverge at the origin, we must choose that which becomes infinite less rapidly:
R = A/rW. (35.7)
§ 3 *> "Fall" of a particle to the centre 115
Next, let y > £. Then s x and s 2 are complex:
*i = -i+*V(y-£), * 2 = V-
Repeating the above analysis, we again arrive at equation (35.6), which, on
substituting the values of ^ and s 2 , gives
B/A = constant xr^^H. (35.8)
On passing to the limit r -» 0, this expression does not tend to any definite
limit, so that a direct passage to the limit is not possible. Using (35.8), the
general form of the real solution can be written
R = constant xrW cos( V(y-£) log (r/r )+ constant). (35.9)
This function has a number of zeros which increases without limit as r
decreases. Since, on the one hand, the expression (35.9) is valid for the
wave function (when r is sufficiently small) with any finite value of the energy
E of the particle, and, on the other hand, the wave function of the normal
state can have no zeros, we can infer that the "normal state" of a particle in
the field considered corresponds to the energy E = - oo. In every state of a
discrete spectrum, however, the particle is mainly in a region of space where
E > U. Hence, for E -> - oo, the particle is in an infinitely small region
round the origin, i.e. the particle "falls" to the centre.
The "critical" field U cr for which the "fall" of a particle to the centre
becomes possible corresponds to the value y = J. The smallest value of the
coefficient of -1/r 2 is obtained for / = 0, i.e.
Ucr = -h 2 /8mr 2 . (35.10)
It is seen from formula (35.3) (for s x ) that the permissible solution of Schro-
dinger's equation (near the point where U ~ 1/r 2 ) diverges, as r -> 0, not
more rapidly than 1/yV. If the field becomes infinite, as r -> 0, more slowly
than 1/r 2 , we can neglect U{r), in Schrodinger's equation near the
origin, in comparison with the other terms, and we obtain the same solutions
as for free motion, i.e. if, ~ r l (see §33). Finally, if the field becomes infinite
more rapidly than 1/r 2 (as -1/r 8 with s > 2), the wave function near the
origin is proportional to r**-i (see §49, Problem). In all these cases the
product rip tends to zero at r = 0.
Next, let us investigate the properties of the solutions of Schrodinger's
equation in a field which diminishes at large distances according to the law
U « -£/r 2 , and has any form at small distances. We first suppose that
y <h It is easy to see that in this case only a finite number of negative
energy levels can exist.f For with energy E = Schrodinger's equation
at large distances has the form (35.1), with the general solution (35.4). The
function (35.4), however, has no zeros (for r # 0); hence all zeros of the
required radial wave function lie at finite distances from the origin, and their
t It is assumed that for small r the field is such that the particle does not "fall".
5
116 Motion in a Centrally Symmetric Field §36
number is always finite. In other words, the ordinal number of the level
E = which terminates the discrete spectrum is finite.
If y > I, on the other hand, the discrete spectrum contains an infinite
number of negative energy levels. For the wave function of the state with
E = has, at large distances, the form (35.9), with an infinite number of
zeros, so that its ordinal number is always infinite.
Finally, let the field be U = -jS/r 2 in all space. Then, for y > £, the
particle "falls", but if y < £ there are no negative energy levels. For the
wave function of the state with E = is of the form (35.7) in all space ; it has
no zeros at finite distances, i.e. it corresponds to the lowest energy level (for
the given /).
§36. Motion in a Coulomb field (spherical polar co-ordinates)
A very important case of motion in a centrally symmetric field is that of
motion in a Coulomb field
U = ±«/r
(where a is a positive constant). We shall first consider a Coulomb attraction,
and shall therefore write U = —<xjr. It is evident from general considera-
tions that the spectrum of negative eigenvalues of the energy will be discrete
(with an infinite number of levels), while that of the positive eigenvalues will
be continuous.
Equation (32.8) for the radial functions has the form
d 2 R 2dR 1(1+1) nt 2m/
— —- 1 —
2m/ a\
.#+_(£+-)# =0. (36.1)
H 2 \ rj
dr 2 r dr
If we are concerned with the relative motion of two attracting particles, m
must be taken as the reduced mass.
In calculations connected with the Coulomb field it is convenient to use,
instead of the ordinary units, special units for the measurement of all quanti-
ties, which we shall call Coulomb units. As the units of measurement of
mass, length and time, we take respectively
m, h 2 /mx, ft? I ma. 2 .
All the remaining units are derived from these ; thus the unit of energy is
mx 2 /ft 2 .
From now on, in this section and the following one, we shall always (unless
explicitly stated otherwise) use these units.f
t Ifm = 911xl0 _28 gis the mass of the electron, anda = e 2 (where e is the charge on the electron),
the Coulomb units are the same as what are called atomic units. The atomic unit of length is
ft 2 \me 2 = 0-529 x lO" 8 cm
(what is called the Bohr radius). The atomic unit of energy is
me^lh 2 = 4-36 XIO" 11 erg = 27-21 electron-volts.
The atomic unit of charge is e = 4-80 X 10 -10 esu. We formally obtain the formulae in atomic units
by putting e = m = H = 1 . For a = Ze 2 the Coulomb and atomic units are not the same.
§36 Motion in a Coulomb field {spherical polar co-ordinates) 117
Let us rewrite equation (36.1) in the new units:
d 2 R 2 dR 7(7+1) / 1\
Instead of the parameter E and the variable r, we introduce the new quantities
n = l/V(-2£), P = 2r/n. (36.3)
For negative E (which we shall first consider), n is a real positive number.
The equation (36.2), on making the substitutions (36.3), becomes
2 r n 7(7+1) n
#"+-#'+ -£+— - L \R=o (36.4)
P L p p 2 J
(the primes denote differentiation with respect to p).
For small p, the solution which satisfies the necessary conditions of finite-
ness is proportional to p l (see (32.13)). To calculate the asymptotic be-
haviour of R for large p, we omit from (36.4) the terms in 1/p and 1/p 2 and
obtain the equation
R" = IR,
whence R = e ±iP . The solution in which we are interested, which vanishes
at infinity, consequently behaves as e~ ip for large p.
It is therefore natural to make the substitution
R = P l e- P l*w{p\ (36.5)
when equation (36.4) becomes
pw"+(2l+2-p)w'+(n-l-l)w = 0. (36.6)
The solution of this equation must diverge at infinity not more rapidly than
every finite power of p, while for p = it must be finite. The solution which
satisfies the latter condition is the confluent hypergeometric function
to = F(-n+l+l, 27+2, p) (36.7)
(see §d of the Mathematical Appendices).f A solution which satisfies the
condition at infinity is obtained only for negative integral (or zero) values of
— n+7+1, when the function (36.7) reduces to a polynomial of degree
n—l—1. Otherwise it diverges at infinity as eP (see (d.14)).
Thus we reach the conclusion that the number n must be a positive integer,
and for a given 7 we must have
n>l+l. (36.8)
Recalling the definition (36.3) of the parameter n, we find
E = -l/2n\ n = l,2,.... (36.9)
t The second solution of equation (36.6) diverges as p -2 ' -1 as p -> 0.
118 Motion in a Centrally Symmetric Field §36
This solves the problem of determining the energy levels of the discrete
spectrum in a Coulomb field. We see that there are an infinite number of
levels between the normal level Ei = — \ and zero. The distances between
successive levels diminish as n increases; the levels become more crowded
as we approach the value E = 0, where the discrete spectrum closes up into
the continuous spectrum. In ordinary units, formula (36.9) isf
E = -ma?l2h*n 2 . (36.10)
The integer n is called the principal quantum number. The radial quantum
number defined in §32 is
n r = n—l—l.
For a given value of the principal quantum number, / can take the values
Z = 0,l,...,w-1, (36.11)
i.e. n different values in all. Only n appears in the expression (36.9) for the
energy. Hence all states with different / but the same n have the same energy.
Thus each eigenvalue is degenerate, not only with respect to the magnetic
quantum number m (as in any motion in a centrally symmetric field) but
also with respect to the number /. This latter degeneracy (called accidental) is
a specific property of the Coulomb field. To each value of / there correspond,
as we know, 2/+1 different values of m. Hence the degree of degeneracy
of the nth. energy level is
S X (2/+1)=« 2 . (36.12)
1=0
The wave functions of the stationary states are determined by formulae
(36.5) and (36.7). The confluent hypergeometric functions with both
parameters integral are the same, apart from a factor, as what are called the
generalised Laguerre polynomials (see §d of the Mathematical Appendices).
Hence
_ 7 /or 2I+l
R n i = constant xp l e-<>' 2 L n +i (p).
The radial functions must be normalised by the condition
j R n i 2 r 2 dr = 1.
t Formula (36.10) was first derived by N. Bohr in 1913, before the discovery of quantum mechanics.
In quantum mechanics it was derived by W. Pauli in 1926 using the matrix method, and a few months
later by E. Schrodinger using the wave equation.
§36 Motion in a Coulomb field {spherical polar co-ordinates) 119
Their final form isf
2 / («+/)
/ (n+l)\
V (n-/-1)! ( 2 ^- r/ ^(-«+/+l, 2/+ 2, 2r/n) ;
«'+ 2 (2/+l)! V(n-/-l)I
(36.13)
the normalisation integral is calculated by (f.6)4
Near the origin, R ni has the form
At large distances,
R nl « (-l)n-l-l r n-l e -r/n ,35 15)
The wave function R 10 of the normal state decreases exponentially at distances
of the order r ~ 1, i.e. r ~ h^jnix in ordinary units.
The mean values of the various powers of r are calculated from the formula
00
r k = J* ^+2^2 dr<
The general formula for r k can be obtained by means of formula (f.7). Here
we shall give the first few values of r* (for positive and negative k) :
r =i[3n»-/(/+l)], r* =4»*[5»*+l-3/(/+l)] f
^ =r =l/»2, r "=i = l/„3(/ + J).
(36.16)
t We give the first few functions R nl explicitly:
#20=(1/V2>-^(1-^),
R 21 =(l/2V6)e-r/2 r ,
R 30 = (2/3 v3)e-'/3A--r+-A
fl 31 = (8/27V6)e-^/3/l— X
R 32 = (4/81 V30)e-r/3 r 2.
X The normalisation integral can also be calculated by substituting the expression (d.13) for the
Laguerre polynomials and integrating by parts (similarly to the calculation of the integral (c.ll) for
the Legendre polynomials).
120 Motion in a Centrally Symmetric Field §36
The spectrum of positive eigenvalues of the energy is continuous and
extends from zero to infinity. Each of these eigenvalues is infinitely degener-
ate ; to each value of E there corresponds an infinite number of states, with
/ taking all integral values from to oo (and with all possible values of m for
the given /).
The number n and the variable p, defined by the formulae (36.3), are now
purely imaginary :
n = _;/v(2E) = -ijk, P = 2ikr (36.17)
(we have introduced the wave number k = y/(2E) in place of the energy).
The radial eigenfunctions of the continuous spectrum are of the form
R» = —^—(2krye-i k 'F(ilk+l+l, 21+2, 2ikr), (36.18)
where the C k are normalisation factors. They can be represented as a
complex integral (see §d) :
1 r / 2ikr\- i l k - l -' i ,
R M = C^krfe^— i> et(l —J t*~ 2 df,
(36.19)
which is taken along the contourf shown in Fig. 10. The substitution
,, i-zikr
Fig. 10
| = 2ikr(t + £) converts this integral to the more symmetrical form
(-2kr)~ 1 - 1 r
R kl = c k - — &> <* l * r *(t+l) i / k -*- 1 (t-t)-*' k -i- 1 dt; (36.20)
the path of integration passes in the positive direction round the points
t = ± \. It is seen at once from this representation that the functions R kl
are real.
The asymptotic expansion (d.14) of the confluent hypergeometric function
enables us to obtain immediately a similar expansion for the wave functions
R kt . The two terms in (d.14) give two complex conjugate expressions in the
| Instead of this contour we could use any closed loop passing round the singular points £ = and
£ = 2ikr in the positive direction. For integral /, the function V(g) = ^ Jn - l ^—2ikr) n ^ (see §d)
returns to its initial value on passing round such a contour.
§36 Motion in a Coulomb field {spherical polar co-ordinates) 121
function R hh and as a result we obtain
e -"/2k t e -Hkr-irQ+i)/ 2+(l Ik) log 2kr]
Rh = C k — — re G(/+l+i/*, i/*-Z, -2ftr)
&r ( r(Z+l— */&)
(36.21)
If we normalise the wave functions on the "k scale" (i.e. by the condition
(33.4)), the normalisation coefficient is
C h = V(2/7r)fo"/ 2 *|r(Z+l-*7A)|. (36.22)
For the asymptotic expression for R kl when r is large (the first term of the
expansion (36.21)) is then of the form
/2 1 1
R ki ~ / - -sin(Ar+- log 2kr— |Ztt+Sj),
V irr k
(36.23)
8,=argr(/+l-*/*),
in agreement with the general form (33.18) of the normalised wave functions
of the continuous spectrum in a centrally symmetric field. The expression
(36.23) differs from (33.18) by the presence of a logarithmic term in the argu-
ment of the sine ; however, since log r increases only slowly compared with r
itself, the presence of this term is immaterial in calculating a normalisation
integral which diverges at infinity.
The modulus of the gamma function which appears in the expression
(36.22) for the normalisation factor can be expressed in terms of elementary
functions. Using the familiar properties of gamma functions:
we have
and also
r(*+l) =zT(z), r(s)r(l-*) =7r/sin7rs,
r(/+i+f/*) = (/+»/*) ... (i+»/*)(i/ft)r(i/*),
ry+i-i/k) = (i-ilk) ... (i-i/k)r(i-i/k),
|r(/+i-t/ft)| = [r(/+i-^)r(/+i+^)] 1/2
-J\i\J(?+i)***l
Thus
2Vk
V(l-g-2"/*)
for / = the product is replaced by unity
tlJi***)'' (3624)
122 Motion in a Centrally Symmetric Field §36
The radial functions R El normalised on the "energy scale" are obtained
from the functions R kl by dividing by \/k: R El = k~ i R Jel (see (33.5)). By
passing to the limit as E -» (i.e. k -> 0), we can correctly obtain from R m
the normalised radial function R ol for the particular case of zero energy, f
The limit of the series F(ijk+l+l, 2/ +2, likr) as k -> is
r% try \«
1 — + — ... = (2/+l)!(2r)-*-i/2/ 2l+1 ( V[8r]),
(2/+ 2)1! (2/+2)(2/+3)2!
where J21+1 is the Bessel function of order 21+ 1. The coefficient C& (36.24)
behaves as 2k~ l+1/2 as k -> 0. We hence easily obtain
R Ql = V(2/r)J 2l+1 (V[8r]). (36.25)
The asymptotic form of this function for large r isj
R Ql « (2/7rV)i/4 S in(v[8r]-/7r-j7r). (36.26)
In a repulsive Coulomb field (U = a/r) there is only a continuous spectrum
of positive eigenvalues of the energy. Schrodinger's equation in this field
can be formally obtained from the equation for an attractive field by changing
the sign of r. Hence the wave functions of the stationary states are found
immediately from (36.18) by the same alteration. The normalisation co-
efficient is again determined from the asymptotic expression, and as a result
we obtain
R H = —j^—(2kr)¥^F(ilk+l+l, 21+2, -2ikr) t
2
-ke-"l* k \T(l+ l+i/k)\
= v(* 2 */*-i) \ J v V + *v (36 ' 27)
The asymptotic expression for this function for large r is
Rm ~ /--sinf &-— -log2&r— J/w+8,J,
B t =ar g r(/+l +*y&).
(36.28)
PROBLEMS
Problem 1. Determine the probability distribution of various values of the momentum
in the ground state of the hydrogen atom.
t It is found that in fact the function RBi normalised on the energy scale remains finite as E -> 0,
while Rki -> as k -*■ 0.
% It may be noted that this function corresponds to the quasi-classical approximation (§49) applied
to motion in the region (/+i) 2 <^ r <^ k~ 2 .
§36 Motion in a Coulomb field {spherical polar co-ordinates) 123
Solution.! The wave function of the ground state is «A = R10Y00 = (l/\Z"-)«~ r . The
wave function of this state in the p representation is then given by the integral
a(p) = (2tt)-3/2 f 0(r)<r*p.r dV
(see (15.10)). The integral is calculated by changing to spherical polar co-ordinates with the
polar axis along p; the result is$
«(p) =
77 (1+^ 2 ) 2
and the probability density in p-space is |a(p)| 2 .
Problem 2. Determine the mean potential of the field created by the nucleus and the
electron in the ground state of the hydrogen atom.
Solution. The mean potential <f> e created by an "electron cloud" at an arbitrary point r
is most simply found as the spherically symmetric solution of Poisson's equation with charge
density p = — \ifi\ 2 :
1 d 2
—ri r h) = 4e ~ 2r -
r dr 1
Integrating this equation, and choosing the constants so that <f> e (0) is finite and <£ e (oo) = (),
and adding the potential of the field of the nucleus, we obtain
t = -+M r ) = (-+iV 2r -
For r^lwe have <f> ~ 1/r (the field of the nucleus), and for r > 1 the potential 4> ^ e~ 2r
(the nucleus is screened by the electron).
Problem 3. Determine the energy levels of a particle moving in a centrally symmetric
field with potential energy U = A/r 2 —B/r (Fig. 11).
Solution. The spectrum of positive energy levels is continuous, while that of negative
levels is discrete; we shall consider the latter. Schrodinger's equation for the radial func-
tion is
d 2 i? 2dR 2m/ h* 1 A B\
+ +— ( E /(/+ 1) +— )R = 0. (1)
dr 2 r dr h*\ 2rn V r* r J l '
We introduce the new variable
p =2V(-2mE)r/h,
and the notation
2mAJh a +l(l+l) =s(s+l), (2)
BV(m/-2E)Jh =n. (3)
f In Problems 1 and 2, atomic units are used.
% The wave functions in the p representation for excited states in a Coulomb field are given by
H. A. Bethe and E. E. Salpeter, Handbuch der Physik 35, 88, Springer, Berlin 1957. These functions
were analysed by V. A. Fok and applied by him to calculate a number of complicated sums (Izvestiya
Akademii Nauk SSSR, Seriya fizicheskaya No. 2, 169, 1935).
124
Motion in a Centrally Symmetric Field
u(r)
§36
Fig. 11
Then equation (1) takes the form
2
P
2 /In s(s+ 1)\
#"+-*'+( --+— ^— - )R = 0,
P \ 4 p p 2 /
which is formally identical with (36.4). Hence we can at once conclude that the solution
satisfying the necessary conditions is
R = p» e -Pl*F(-n+s+l, 2s+2, p),
where n—s — 1 = p must be a positive integer (or zero), and s must be taken as the positive
root of equation (2). From the definition (3) we consequently obtain the energy levels
■E n =
2B 2 m
h*
-RM-1+ V{(2/+l) 2 4-8m,4//*2}]-2.
Problem 4. The same as Problem 3, but with U = A/^+Br 2 (Fig. 12).
Solution. There is only a discrete spectrum. Schrodinger's equation is
d 2 R 2dR 2m r k 2 l(l+l) A
1 1 — I E Br 2
dr 2 r dr h 2 L 2mr 2 r 2
]*-
0.
Introducing the variable
| = V(2mB)r i /H
Fig. 12
§37 Motion in a Coulomb field {parabolic co-ordinates) 125
and the notation
/(/ + 1) + 2m,4//* 2 = 2s(2s+l),
V(2m/B)Elh = 4(n+s)+3,
we obtain the equation
#r+-ir+[n+H-|-tf-*(*+i)/fl* = o.
The solution required behaves asymptotically as e~$£ when £ -> co, while for small f it is
proportional to £*, where s must be taken as the positive quantity
s = i[-l+ V{(2/+l) 2 +8m^ 2 }].
Hence we seek a solution in the form
R = e-UH'zo,
obtaining for w the equation
£w"+(2s+ — nw'+nw = 0,
whence
w
= F(-n,2s+-^y
where n must be a non-negative integer. We consequently find as the energy levels the infinite
set of equidistant values
E n =^V(5/2m)[4n+2+V{(2/+l) 2 +8m^ 2 }],n =0,1,2,....
§37. Motion in a Coulomb field (parabolic co-ordinates)
The separation of the variables in Schrodinger's equation written in
spherical polar co-ordinates is always possible for motion in any centrally
symmetric field. In the case of a Coulomb field, the separation of the variables
is also possible in what are called parabolic co-ordinates. The solution of the
problem of motion in a Coulomb field in terms of parabolic co-ordinates is
useful in investigating a number of problems where a certain direction in
space is distinctive; for example, for an atom in an external electric field
(see §77).
The parabolic co-ordinates £, 77, (f> are defined by the formulae
x = V(£})W, y = V(£})sin<£, * =l(£—q),
r = v(* 2 +:y 2 +2 2 ) = &£+!?),
or conversely
|=r+*, r, =r-z, <j>=tBir 1 {ylxy, (37.2)
f and rj take values from to 00, and <j> from to 2tt. The surfaces £ =
126 Motion in a Centrally Symmetric Field §37
constant and rj = constant are paraboloids of revolution about the z-sotis,
with focus at the origin. This system of co-ordinates is orthogonal. The
element of length is given by the expression
(dZ)* = ^(d^+^Cd^+^d^, (37.3)
and the element of volume is
dV = £(£+77)d£Vty. (37.4)
From (37.3) we have the Laplacian operator
4 rd / d\ d / d\~] 1 8*
A= sAii(^K(%)]w (375)
Schrodinger's equation for a particle in an attractive Coulomb field with
U = -1/r = -2/(| + i7) is
4 rd/ddt\ d / 8ifr\-i 1 2Ni / 2 \
irfii(y + 4wJ + i^ + r w - °- (376)
Let us seek the eigenfunctions «/r in the form
=mm-ny im \ (37.7)
where m is the magnetic quantum number. Substituting this expression in
equation (37.6) multiplied by £(£+*?)» and separating the variables £ and rj,
we obtain for/i and/2 the equations
d/ d/i\
^-(^)+[^-Jm^+)8 2 ]/ 2 =0,
(37.8)
diy \ d?7
where the separation parameters j8i, /?2 are related by
ft+Ai = 1. (37.9)
Let us consider the discrete energy spectrum (E < 0). We introduce in
place of E, £, 77 the quantities
« = 1IV(-2E), Px = €V(-2E) = £ln, p 2 = ,/», (37.10)
whereupon we obtain the equation for /1
"P7+-— + -!+-( — — +«i )-— /1 = 0, (37.11)
d/>i 2 Pi d Pl L Pi \ 2 / 4 Pl 2 J
§37 Motion in a Coulomb field {parabolic co-ordinates) 127
and a similar equation for/2, with the notation
"1 = -MM + l)+"ft, n 2 = -£(|m| + l)+njS 2 . (37.12)
Similarly to the calculation for equation (36.4), we find thatyj behaves as
e'tP 1 for large p x and as p ± iw for small p x . Accordingly, we seek a solution
of equation (37.11) in the form
A(Pi) = r^w^^,
and similarly for/2, obtaining for zoi the equation
■Pi«'i"+(M + 1 — Pi) w i+ n i w i =°-
This is again the equation for a confluent hypergeometric function. The
solution satisfying the conditions of finiteness is
wi =F(—n 1 , |w| + l,p x ),
where n\ must be a non-negative integer.
Thus each stationary state of the discrete spectrum is determined in para-
bolic co-ordinates by three integers : the parabolic quantum numbers «i and
«2, and the magnetic quantum number m. For n, the principal quantum
number, we have from (37.9) and (37.12)
n =w 1 +« 2 +|m| + l. (37.13)
For the energy levels, of course, we obtain our previous result (36.9).
For given «, the number \m\ can take n different values from to n — \.
For fixed n and \m\ the number n x takes n—\m\ values, from to n— \m\ — 1.
Taking into account also that for given \m\ we can choose the functions with
m = ±N|, we find that for a given n there are altogether
n-l
2 2 («-ot)+(k-0) = w 2
m—l
different states, in agreement with the result obtained in §36.
The wave functions ifin^m of the discrete spectrum must be normalised
by the condition
00 00 2W
j I^J 2 dV = J JJJ l-An^l 2 ^) d^d, = 1. (37.14)
000
The normalised functions are
V2 / £\ /*?\ e im *
*.%- - *A»(jJA-kh^ (37 ' 15)
where
1 /(/>+|m|)!
fpm(p) = — / ^ (-A M + 1. /»)^V""». (37.16)
128 Motion in a Centrally Symmetric Field §37
The wave functions in parabolic co-ordinates, unlike those in spherical
polar co-ordinates, are not symmetrical about the plane z = 0. For n± > w 2
the probability of finding the particle in the direction z > is greater than
that for z < 0, and vice versa for n\ < n%.
To the continuous spectrum (E > 0) there corresponds a continuous spec-
trum of real values of the parameters ft, ft in equations (37.8) (connected as
before, of course, by the relation (37.9)). We shall not pause to write out here
the corresponding wave functions, since it is not usually necessary to employ
them. Equations (37.8), regarded as equations for the "eigenvalues" of the
quantities ft, ft, have also (for E > 0) a spectrum of complex values of ft
and ft. The corresponding wave functions are written out in §133, where
we shall use them to solve a problem of scattering in a Coulomb field.
In classical motion of a particle in a Coulomb field there is a conservation
law peculiar to this type of field :f
A = pxl — x\r = constant. (37.17)
In quantum mechanics the three components of this vector cannot simul-
taneously have definite values, since the operators^, A y , A z do not commute.
Any one of these operators, A z say, commutes (like the ^-component of any
vector; see (29.3)) with l z , but does not commute with the conserved square
of the angular momentum, I 2 . The existence of another conserved quantity
which does not commute with the others leads (see §10) to an additional
degeneracy of the levels, and this is the accidental degeneracy peculiar to a
Coulomb field.
The description of motion in a Coulomb field by means of the wave
functions ijt nlm in spherical polar co-ordinates corresponds to states in which
not only the energy but also the square of the angular momentum and its
sr-component have definite values. The wave functions ifj ni n,n in parabolic
co-ordinates, on the other hand, describe stationary states in which l z and A z
have definite values. It can be shown that the value of A z is given in terms of
the quantum numbers n\, n%, n by
A z = (nz-n{)\n. (37.18)
t See Mechanics, §15.
CHAPTER VI
PERTURBATION THEORY
§38. Perturbations independent of time
The exact solution of Schrodinger's equation can be found only in a com-
paratively small number of the simplest cases. The majority of problems
in quantum mechanics lead to equations which are too complex to be solved
exactly. Often, however, quantities of different orders of magnitude appear
in the conditions of the problem ; among them there may be small quantities
such that, when they are neglected, the problem is so much simplified that its
exact solution becomes possible. In such cases, the first step in solving the
physical problem concerned is to solve exactly the simplified problem, and
the second step is to calculate approximately the errors due to the small terms
that have been neglected in the simplified problem. There is a general
method of calculating these errors; it is called perturbation theory.
Let us suppose that the Hamiltonian of a given physical system is of the
form
where V is a small correction {ox perturbation) to the unperturbed operator i? .
In §§38, 39 we shall consider perturbations V which do not depend explicitly
on time (the same is assumed regarding i? also). The conditions which are
necessary for it to be permissible to regard the operator V as "small" com-
pared with the operator fi will be derived below.
The problem of perturbation theory for a discrete spectrum can be formu-
lated as follows. It is assumed that the eigenfunctions «/r n (0) and eigenvalues
E n w of the discrete spectrum of the unperturbed operator ff are known, i.e.
the exact solutions of the equation
#«# > = &°ty<» (38.1)
are known. It is desired to find approximate solutions of the equation
ify= (#0+^)0=^, (38.2)
i.e. approximate expressions for the eigenfunctions ift n and eigenvalues E n of
the perturbed operator i?.
In this section we shall assume that no eigenvalue of the operator i? is
degenerate. Moreover, to simplify our results, we shall suppose that there
is only a discrete spectrum of eigenvalues ; all the formulae can be at once
generalised to the case where there is a continuous spectrum.
The calculations are conveniently performed in matrix form throughout
129
130 Perturbation Theory §38
To do this, we expand the required function $ in terms of the functions
&* (0) :
t = Zc m i/, m «». (38.3)
Substituting this expansion in (38.2) we obtain
multiplying both sides of this equation by i/; k W* and integrating, we find
{E-E^)c k = S V km c m . (38.4)
Here we have introduced the matrix V km of the perturbation operator V,
defined with respect to the unperturbed functions <^ m (0) :
V km =j^*^ m (o) dq , (38>5)
We shall seek the values of the coefficients c m and the energy E in the form
of series
E = #»+jEa> +jE » + ... , Cm = cj»+c m u+cj»+
where the quantities E™ and cJU are of the same order of smallness as the
perturbation P, the quantities £< 2 > and c m < 2 > are of the second order of small-
ness (if V is of the first order), and so on.
Let us determine the corrections to the nth. eigenvalue and eigenfunction,
putting accordingly * B <°> = 1, c m <°> = for m # n. To find the first approxi-
mation, we substitute in equation (38.4) E = E n ^+E n ( x \ c k = c k W+cJ*\
and retain only terms of the first order. The equation with k = n gives
E n {1) = V nn =j +®*fy® dq. (38.6)
Thus the first-order correction to the eigenvalue £M°> is equal to the mean
value of the perturbation in the state tp n (0) .
The equation (38.4) with k ^ n gives
'* (1) = V^KE^-E^) for k^n,
while c n <u remains arbitrary; it must be chosen so that the function tp n =
*Pn 0) + l f'n (1) is normalised up to and including terms of the first order. For
this we can put c TC (1 > = 0. For the functions
m ^n ■L'm
(the prime means that the term with m = n is omitted from the sum) are
orthogonal to n <«>, and hence the integral of l^^+^l 2 differs from unity
only by a quantity of the second order of smallness.
§38 Perturbations independent of time 131
Formula (38.7) determines the correction to the wave functions in the first
approximation. Incidentally, we see from this formula the condition for the
applicability of the above method of perturbation theory. This condition is
that the inequality
\V mn \ <\E®-EJ>\ (38.8)
must hold, i.e. the matrix elements of the operator ^must be small compared
with the corresponding differences between the unperturbed energy levels.
Next, let us determine the correction to the eigenvalue £ n <°> in the second
approximation. To do this, we substitute in (38.4) E = E n <°)+E n M+E (2)
c k = £fc (0) +£ft (1) +c& (2) , and examine the terms of the second order of small-
ness. The equation with k = n gives
E ®>C <°> = S' V r CD
whence
E,
« (2) =2
' \V I 2
mn\
E <®—E <°>
(38.9)
(we have substituted cj" = V m J(E n W-E m ^) t and used the fact that, since
the operator V is Hermitian, V mn = V nm *).
We notice that the correction in the second approximation to the energy
of the normal state is always negative; for, since £ n <°> then corresponds to
the lowest value of the energy, all the terms in the sum (38.9) are negative.
The further approximations can be calculated in an exactly similar manner.
The results obtained can be generalised at once to the case where the
operator H has also a continuous spectrum (but the perturbation is applied,
as before, to a state of the discrete spectrum). To do so, we need only add to
the sums over the discrete spectrum the corresponding integrals over the
continuous spectrum. We shall distinguish the various states of the continu-
ous spectrum by the suffix v, which takes a continuous range of values; by v
we conventionally understand an assembly of values of quantities sufficient
for a complete description of the state (if the states of the continuous spec-
trum are degenerate, which is almost always the case, the value of the energy
alone does not suffice to determine the state).f Then, for instance, we must
write instead of (38.7)
and similarly for the other formulae.
It is useful to note also the formula for the perturbed value of the matrix
element of a physical quantity/, calculated as far as terms of the first order
by using the functions n = ^<°>+0 n tt> f with ^a) g i ven by (38.7). The
t Here the wave functions «£„(») must be normalised by delta functions of the quantities v.
132 Perturbation Theory §38
following expression is easily obtained:
^' *W*m (0) ^' V km f nk W
f =f (o)+ > u > (38 U)
k *-"n. ^k k ^m ^k
In the first sum k ^ n, while in the second k ^ m.
PROBLEMS
Problem 1. Determine the correction ^„ (2) in the second approximation to the eigen-
functions.
Solution. The coefficients Ck (2) (k ^ n) are calculated from equations (38.4) with k ^ n,
written out up to terms of the second order, and the coefficient c„< 2 > is chosen so that the
function t(i n = ^n (0) +^n (1) +^» (2) is normalised up to terms of the second order. As a result
we find
*•* =22 j£S^-*-2 ^---**.-2'fe^
m k ul nk w nm m '* w nm m n w nm
where we have introduced the frequencies
Problem 2. Determine the correction in the third approximation to the eigenvalues of the
energy.
Solution. Writing out the terms of the third order of smallness in equation (38.4) with
k — n, we obtain
„ (3) _ V"V' ^nmYmkVkn NT^' \Vn,
Problem 3. Determine the energy levels of an anharmonic linear oscillator whose Hamil-
tonian is
Solution. The matrix elements of x 3 and ** can be obtained directly according to the
rule of matrix multiplication, using the expression (23.4) for the matrix elements of x. We
find for the matrix elements of x 3 that are not zero
(^)n- 3 ,« = («■)..-. = (hlmcof/W[Mn-l)(n-2)],
(*■)-!.. = (^)«,«-i = (W 3/ V(M8).
The diagonal elements in this matrix vanish, so that the correction in the first approximation
due to the term ax 3 in the Hamiltonian (regarded as a perturbation of the harmonic oscillator)
is zero. The correction in the second approximation due to this term is of the same order as
that in the first approximation due to the term fa*. The diagonal matrix elements of a: 4 are
(**)«,« = (h/mojy . |(2» 2 +2 W +1).
Using the general formulae (38.6) and (38.9), we find the following approximate expression
for the energy levels of the anharmonic oscillator:
15 a 2 / h \ V 11\ 3 / h \ 2
E n =ha>{n+l)--—{ — ) (»■+„+_ )+-p(_) < w2+w +*)-
4 hat \mco/ \ 30/ 2 \mco/
§39 The secular equation 133
§39. The secular equation
Let us now turn to the case where the unperturbed operator # has de-
generate eigenvalues. We denote by </r n <°>, ^ n ,<o) f ... the eigenfunctions be-
longing to the same eigenvalue £ n <°> of the energy. The choice of these func-
tions is, as we know, not unique; instead of them we can choose any s (where
s is the degree of degeneracy of the level £ n <°>) independent linear combina-
tions of these functions. The choice ceases to be arbitrary, however, if we
subject the wave functions to the requirement that the change in them' under
the action of the small applied perturbation should be small.
At present we shall understand by if, n «>\ f n S<>\ ... some arbitrarily selected
unperturbed eigenfunctions. The correct functions in the zeroth approxima-
tion are linear combinations of the form %<°ty n <°> + c n > «»ip n ' «»+.... The co-
efficients in these combinations are determined, together with the corrections
in the first approximation to the eigenvalues, as follows.
We write out equations (38.4) with £=»,»',..., and substitute in them,
in the first approximation, E = £ n «» + £(i) ; for the quantities c h it suffices
to take the zero-order values c n = c n <°\ c n . = cj°\ ... ; c m = for m # n,
n', ... . We then obtain
or
S(F nn ,-#»S wn 0^<°>=0, (39.1)
where n, n' take all values denumerating states belonging to the given un-
perturbed eigenvalue £ TC <°>. This system of homogeneous linear equations
for the quantities c„< 0) has solutions which are not all zero if the determinant
of the coefficients of the unknowns vanishes. Thus we obtain the equation
|F nn ,-2?a>S nn ,|=(). (39.2)
This equation is of the rth degree in E™ and has, in general, s different real
roots. These roots are the required corrections to the eigenvalues in the first
approximation. Equation (39.2) is called the secular equation.} We notice
that the sum of its roots is equal to the sum. of the diagonal matrix elements
V nn > Vnn; »• ( tni s being the coefficient of [2?fl)]*-i in the equation).
Substituting in turn the roots of equation (39.2) in the system (39.1) and
solving, we find the coefficients <: n < > and so determine the eigenfunctions
in the zeroth approximation.
We notice that, as a result of the perturbation, an originally degenerate
energy level ceases in general to be degenerate (the roots of equation (39.2)
are in general distinct); the perturbation removes the degeneracy, as we say.
The removal of the degeneracy may be either total or partial (in the latter
case, after the perturbation has been applied, there remains a degeneracy of
degree less than the original one).
t The name is taken from celestial mechanics.
134 Perturbation Theory §39
It may happen that all the matrix elements for transitions between the
states «, n\ ... with a single energy are zero. The correction to the energy
then vanishes in the first approximation. Let us calculate the correction in
the second approximation for this case. In equation (38.4) with k = n we
put on the left-hand side E = E n w +E®\ and write c n w in place of c n . Only
the terms with m ^ w, »', ... on the right-hand side are different from zero,
and since c m w = we have
EPcf* = S V nm c^\ (39.3)
m
The equation (38.4) with k = m ^ n, ri , ... , on the other hand, gives, correct
to terms of the first order,
n
whence
c 0> = V Vmn ' c .<o>
n
Substituting in (39.3), we obtain
£(2) c (0) = V c ,(0)VM!!!l
n Z-, n Z-, e <°>— £• <°> '
This system of equations for the c n (0) now replaces the system (39.1); the
condition that these equations are compatible is
xr-^ V V >
^ nm mn -&% nn .
£i F. (o)—
E (°)— E <°>
= 0. (39.4)
Thus here also the corrections to the energy are calculated as the roots of a
secular equation, in which, instead of the matrix elements V nn >, we now have
the sums '
2* nm* mn'
E (°)— E (o)
PROBLEMS
Problem 1. Determine the corrections to the eigenvalue in the first approximation and
the correct functions in the zeroth approximation, for a doubly degenerate level.
Solution. Equation (39.2) here has the form
V u -E»> V 12
V» V,,-E^
=
(the suffixes 1 and 2 correspond to two arbitrarily chosen unperturbed eigenfunctions i/^ *
§39 The secular equation 135
and 0a (o) of the doubly degenerate level in question). Solving, we find
£ (1) = *[(F U + v 22 )±v{(v 11 - v 22 y+4\ F 12 p}]. (i)
Solving also equations (39.1) with these values of £<D, we obtain for the coefficients in the
correct normalised function in the zeroth approximation, i/r«» = ci< >«Ai<°) + c2< ><A2<°> the
values '
w= f v 12 r . F,i-F« 1U
b r
c a <°>
fl± ^^ I)*
|L V{(^i-^ 2 ) 2 +4|F 12 |2}J/ ,
(2)
- =fc l^LTlT ^^ I)*
(2| V 12 \ I V{(V X1 - V 22 )*+4\ F 12 p}Jj '
Problem 2. Derive the formulae for the correction to the eigenfunctions in the first
approximation and to the eigenvalues in the second approximation.
Solution. We shall suppose that the correct functions in the zeroth approximation are
chosen as the functions *„(»>. The matrix V nn , defined with respect to these is clearly diagonal
with respect to the suffixes n, n' (belonging to the same group of functions of a degenerate
Zfx i*^ ■ di f go ? al elements V-> Vww are equal to the corresponding corrections
*V \ i4n»* ', ... in the first approximation.
Let us consider a perturbation of the eigenfunction if> n M, so that in the zeroth approxima-
tion E = E n (0) , c„(°) = 1, c »(°) = for m # n. In the first approximation E = E n ^ + V nn
c » - 1 + c » (1, » Cr» = c m M. We write out from the system (38.4) the equation with k ^ n n'
retaining in it terms of the first order: ' ''"'
W-E^p = V kn c n «» = V kn ,
whence
<* (1) = VjJVZjn-EP) for h ± n, n\ ... . (1)
Next we write out the equation with k = n\ retaining in it terms of the second order:
E n <»c n 'to = V n > n > c„'<«-f E' V > c «
(the terms with m = n, n', ... are omitted in the sum over m). Substituting E <*> = V and
the expression (1) for c m W, we obtain for n' # n " ""
1 xrV V > V
(X) ~ X ' n m r mn
V — V • > ^-i F (0)_ F (0) ' (2)
(In this approximation the coefficient c»U> is zero.) Formulae (1) and (2) determine the
correction 0.W = Sc^tyJ") to the eigenfunctions in the first approximation.
Finally, writing out the second-order terms in equation (38.4) with k = n, we obtain for
the second-order corrections to the energy the formula
V V
' nm v mn
2) = y _Vnm_
r E « (0) -
-£<<»' (3)
which is formally identical with (38.9).
Problem 3. At the initial instant * = 0, a system is in a state &<») which belongs to a
doubly degenerate level. Determine the probability that, at a subsequent instant t, the
system will be in the state &«» with the same energy; the transition occurs under the action
of a constant perturbation.
Solution. We form the correct functions in the zeroth approximation,
136 Perturbation Theory §40
where c lt c t ; c^, c t ' are two pairs of coefficients determined by formulae (2) of Problem 1
(for brevity, we omit the index (°) on all quantities).
Conversely,
«Ai = ; — ■
The functions ifi and ifi' belong to states with perturbed energies E+EW and E+EW, where
EW and EW are the two values of the correction (1) in Problem 1. On introducing the time
factors we pass to the time-dependent wave functions :
g-dfldEt
T x = ; —[c^e-QlMEt-Citft'e-QlME <]
tjtg — ^l ^2
(at time t = 0, Tj = fa). Finally, again expressing tft, tfi' in terms of fa, fa, we obtain Yj
as a linear combination of fa and fa, with coefficients depending on time. The squared modu-
lus of the coefficient of fa determines the required transition probability w 12 . Calculation
gives
/ Q
\ e -(i/h)E Kl 't_ e -(i/h)E Kl ' <|2 j
or, substituting formulae (1) and (2) from Problem 1,
2|r M |»
MV 12 \'+(V 11 -V 22 f
l-cos^-VliV^-V^f+^V^ty^ . (1)
We see that the probability varies periodically with time, with frequency (EW—EW)jH.
For times t which are small compared with the period in question, the expression in the
braces, and therefore W12, is proportional to t 2 : W12 = \Vi2\ 2 t 2 /H z . This formula can be very
simply obtained by the method given in the next section (using equation (40.4)).
§40. Perturbations depending on time
Let us now go on to study perturbations depending explicitly on time. We
cannot speak in this case of corrections to the eigenvalues, since, when the
Hamiltonian is time-dependent (as will be the perturbed operator & = $ Q +
+ !?"(£)), the energy is not conserved, so that there are no stationary states.
The problem here consists in approximately calculating the wave functions
from those of the stationary states of the unperturbed system.
To do this, we shall apply a method analogous to the well-known method
of varying the constants to solve linear differential equations (P. A. M. Dirac
1926). Let Y& (0) be the wave functions (including the time factor) of the
stationary states of the unperturbed system. Then an arbitrary solution of the
unperturbed wave equation can be written in the form of a sum:
T = 2flfc¥V°). We shall now seek the solution of the perturbed equation
ih dW/dt = (# + P)¥ (40.1)
in the form of a sum
T = S<z fc (0T fc <o>, (40.2)
where the expansion coefficients are functions of time. Substituting (40.2)
§40 Perturbations depending on time 137
in (40.1), and recalling that the functions Y^ satisfy the equation
we obtain
k "t k
Multiplying both sides of this equation on the left by T OT (0) * and integrating,
we have
where
are the matrix elements of the perturbation, including the time factor (and
it must be borne in mind that, when V depends explicitly on time, the quanti-
ties V mk also are functions of time).
As the unperturbed wave function we take the wave function of the nth
stationary state, for which the corresponding values of the coefficients in
(40.2) are a n (0) = 1, a k w = for k # n. To find the first approximation,
we seek a h in the form a k = a k w +a k a) , substituting a k = a k {0) on the
right-hand side of equation (40.3), which already contains the small quantities
V mk . This gives
ihda k V>ldt = V kn (t)- (40.4)
In order to show the unperturbed function to which the correction is being
calculated, we introduce a second suffix in the coefficients a k , writing
Y n = Sa fcn (*)V.
Accordingly, we write the result of integrating equation (40.4) in the form
a*» (1) = -(«/*) J* V kn (t) & = -(ilh) j V kn e*»*nt dt, (40.5)
where we have introduced the frequencies (o kn = (E k w —E n {0) )lh. This
determines the wave functions in the first approximation.
We can similarly determine the subsequent approximations (in practice,
however, the first approximation is adequate in the majority of cases).
Let us now consider in more detail the case of a perturbation which is
periodic with respect to time, of the form
■p z= p e -i<ot + @ e ic*t i (40.6)
138 Perturbation Theory §40
where P and are operators independent of time. Since V is Hermitian,
we must have V nm = V mn *, or
J? p-ioit 1/7 oittit _ K 1 * p i(oiir; # p -io>t
from which
which determines the relation between the operators and P. Using this
relation, we have
= Ftoe* a *- ai *+F ia *e 1 l a> *»+ ai *. (40.8)
Substituting in (40.5) and integrating, we obtain the following expression for
the expansion coefficients of the wave functions :
Fto/to**-"* F nl *e*- M kn+<»V
K(<x) kn —aj) H(aj kn +co)
These expressions are applicable if none of the denominators vanishes,! i.e.
if for all k (and the given n)
EjP-E® * ±hco. (40.10)
In a number of applications it is useful to have expressions for the matrix
elements of an arbitrary quantity /, defined with respect to the perturbed
wave functions. In the first approximation we have
JnmSJ) = Jnm \f)ijnm \r)i
where
fnm ( °Kt) = jT n W*/T w (0)d ? =/ B J¥M
/™ (1) (0 = j W 0) */ V+^« (1) */^ 0) ] dq.
Substituting here T n < 1) = S a &n cl,, *V 0) » with « fcTC (1) determined by formula
(40.9), it is easy to obtain the required expression
fnnP(t) = -*"*"* V [\ f<*® F *» + f*»® F ** 1 e -^< +
^ lL^(a) fcm — cu) ^(eo^+a^J
+ r /A* A^V-i^a (40 . n)
L^(co ftTO +co) n{oi kn — u))j )
f More precisely, if none is so small that the quantities aje n W are no longer small compared with
unity.
§40 Perturbations depending on time 139
This formula is applicable if none of its terms becomes large, i.e. if none of
the frequencies o)^ n , co^ m is too close to co. For <o = we return to formula
(38.11).
In all the formulae given here, it is understood that there is only a discrete
spectrum of unperturbed energy levels. However, these formulae can be
immediately generalised to the case where there is also a continuous spectrum
(as before, we are concerned with the perturbation of states of the discrete
spectrum) ; this is done by simply adding to the sums over the levels of the
discrete spectrum the corresponding integrals over the continuous spectrum.
Here it is necessary for the denominators a) kn ±eo in formulae (40.9), (40.11)
to be non-zero when the energy E k w takes all values, not only of the discrete
but also of the continuous spectrum. If, as usually happens, the continuous
spectrum lies above all the levels of the discrete spectrum, then, for instance,
the condition (40.10) must be supplemented by the condition
EnJQ-E® > ha>, (40.12)
where J? min (0) is the energy of the lowest level of the continuous spectrum.
PROBLEM
Determine the change in the nth and mth solutions of Schrodinger's equation in the
presence of a periodic perturbation (of the form (40.6)), of frequency a» such that £' m < >— EJ'1
= h(a>+e), where e is a small quantity.
Solution. The method developed in the text is here inapplicable, since the coefficient
aj 1 ' in (40.9) becomes large. We start afresh from the exact equations (40.3), with V mlc (t)
given by (40.8). It is evident that the most important effect is due to those terms, in the
sums on the right-hand side of equations (40.3), in which the time dependence is determined
by the small frequency to mn — to. Omitting all other terms, we obtain a system of two equa-
tions :
ihdajdt = F mn e*<° mn -<»)ta n = F mn e™a ni
ihdajdt = F mn *e-^a m .
We make the substitution
a e iet = h
and obtain the equations
ihd m = F mn b nt th(b n —t€b n ) = F mn *a m .
Eliminating a m , we have
h n -kb n +\F mn \*b n l¥ = 0.
We can take as two independent solutions of these equations
a n = AeW, a m = ~Ah^ x e^\F m * (1)
and
a n = Be-W, a m = Bkx^-W/F^*, (2)
140 Perturbation Theory §41
where A and B are constants (which have to be determined from the normalisation condition),
and we have used the notation
«1 = -h+Vih 2 +\Fmn\ 2 m a 2 =h+V[k 2 +\Fmn\Wl
Thus, under the action of the perturbation, the functions Y„(°>, T m <°) become a n ¥ n <°> +
+a m Y TO <°), with a n and a m given by (1) and (2).
Let the system be in the state T m <°> at the initial instant (t — 0). The state of the system
at subsequent instants is given by a linear combination of the two functions which we have
obtained, which becomes Y ro <°) for t — 0:
I*" win 1
■ — e-^/2sinV[ie 2 +|F mn |2/^ 2 ]« T n «». (3)
The squared modulus of the coefficient of Y n (°) is
^•\r mr>
/* 2 e 2 +4|^„
^l-cosV[ € 2 +4|^ wn | 2 /^]0. (4)
This gives the probability of finding the system in the state VF n (°> at time t. We see that it is a
periodic function with period 2irh(e 2 h 2 +4\F mn \ z )~ i , and varies from to 4l.F mB |V(J* 2 e 2 4-
+4| J F ran | 2 ).
For e = (exact resonance) the probability (4) becomes
*(l-cos2|F mn |^).
It varies periodically, with period nH/\F mn \, between and 1; in other words, the system
makes periodic transitions from the state T m <°) to the state Y,,* ).
§41. Transitions under a perturbation acting for a finite time
Let us suppose that the perturbation V(t) acts only during some finite
interval of time (or that V(t) diminishes sufficiently rapidly as t -> ± oo).
Let the system be in the nth stationary state (of a discrete spectrum) before
the perturbation begins to act (or in the limit as t -> — oo). At any subsequent
instant the state of the system will be determined by the function T =
2 a k1 ^¥ k w , where, in the first approximation,
«*n = a k v> = -- f V^e^Jdt for k*n,
a nn = l+a Bn <« = 1-- I V n
(41.1)
d*;
the limits of integration in (40.5) are taken so that, as t -> — oo, all the tf &n (1>
tend to zero. After the perturbation has ceased to act (or in the limit t ->
+ oo), the coefficients a hn take constant values a kn {co), and the system is in
§41 Transitions under a perturbation acting for a finite time 141
the state with wave function
Y = S« ftn (a>W 0) ,
which again satisfies the unperturbed wave equation, but is different from
the original function ¥ n < 0) . According to the general rule, the squared
modulus of the coefficient a kn (co) determines the probability for the system
to have an energy E k <°\ i.e. to be in the Mi stationary state.
Thus, under the action of the perturbation, the system may pass from its
initial stationary state to any other. The probability of a transition from
the original (wth) to the Ath stationary state is
h*\J
V*^°>*f dt
(41.2)
Let us now consider a perturbation which, once having begun, continues
to act for an indefinite time (always, of course, remaining small). In other
words, V(t ) tends to zero as t -» — oo and to a finite non-zero limit as t ->
+ oo. Formula (41.2) cannot be applied directly here, since the integral in
it diverges. This divergence, however, is physically unimportant and can
easily be removed. To do this, we integrate by parts :
-— IJ^*--[^T+J— — *•
The value of the first term vanishes at the lower limit, while at the upper
limit it is formally identical with the expansion coefficients in formula (38.7);
the presence of an additional periodic factor e %0iknt is merely due to the fact
that the a kn are the expansion coefficients of the complete wave function T,
while the c kn in §38 are the expansion coefficients of the time-independent
function *Jj. Hence it is clear that its limit as t -*■ oo gives simply the change
in the original wave function *F n (0 > under the action of the "constant part"
F( + oo) of the perturbation, and consequently has no relation to transitions
into other states. The probability of a transition is given by the squared
modulus of the second term and is
w n k = — **"*■' dt
W I J dt
/
(41.3)
The derivation is also valid when the transition is from a state of the discrete
spectrum to a state of the continuous spectrum. The only difference is that
here we have the probability of the transition from a given (nth) state to states
in a range of values of v (see §38) from v to v + dv, so that, for example, formula
(41.2) must be written
°°
1 I /* 2
dw nv = — Vv n e i( °vnt dt dv. (41.4)
h 2 \ J
142 Perturbation Theory §41
If the perturbation V(t) varies little during time intervals of the order of
the period l/co fcn the value of the integral in (41.2) or (41.3) will be very
small. In the limit when the applied perturbation varies arbitrarily slowly, the
probability of any transition with change of energy (i.e. with a non-zero
frequency ca kn ) tends to zero. Thus, when the applied perturbation changes
sufficiently slowly {adiabatically), a system in any non-degenerate stationary
state will remain in that state (see also §53).
In the opposite limiting case of a very rapid, "instantaneous" application of
the perturbation, the derivatives dVjcn/dt become infinite at the "instant of
application". In the integral of (8V kn ldt)e i ' ^ t , we can take outside the
integral the comparatively slowly varying factor e* w *»* and use its value at
this instant. The integral is then found at once, and we obtain
v>nk=\V kn \ 2 /h*a> kn *. (41.5)
The transition probabilities in instantaneous perturbations can also be
found in cases where the perturbation is not small. Let the system be in a
state described by one of the eigenfunctions ifj n (Q) of the original Hamiltonian
Ho. If the change in the Hamiltonian occurs instantaneously (i.e. in a time
short compared with the periods l/to to of transitions from the given state n
to other states), then the wave function of the system is "unable" to vary and
remains the same as before the perturbation. It will no longer, however, be
an eigenfunction of the new Hamiltonian H of the system, i.e. the state
j/f w <°> will not be a stationary state. The probabilities w nk for transitions of
the system into the new stationary states are determined, according to the
general rules of quantum mechanics, by the coefficients in the expansion of the
function ift n (°> in terms of the eigenfunctions *p n of the Hamiltonian i?:
™nk = | \ W m H* <*/. (41.6)
We shall show how this general formula becomes (41.5) if the change
V = A— i? in the Hamiltonian is small. We multiply the equations
by ifjjc* and i/%<°)* respectively, integrate with respect to q and subtract.
Using also the self-conjugacy of the operator ]2, we obtain
(E k -E n ™) j <A**<An (0) d? = j 0**Jty» (O) 6q.
If the perturbation V is small, in the first approximation we can replace
Ejc by the adjoining unperturbed level E^, and the wave function ifj^ (on
the right-hand side of the equation) by the corresponding function i/jjc^.
This gives
f «A*W 0) dq = [ W<»*fyn {0) dq,
J natjcn J
and formula (41.6) becomes (41.5).
§41 Transitions under a perturbation acting for a finite time 143
PROBLEMS
Problem 1. A uniform electric field is suddenly applied to a charged oscillator in the
ground state. Determine the probabilities of transitions of the oscillator to excited states
under the action of this perturbation.
Solution. The potential energy of the oscillator in the uniform field (which exerts a
force F on it) is
U(x) = \moi 2 x 2 —Fx
= %maj 2 (x—xo) 2 + constant
(where xo = F/mw 2 ), i.e. has still the pure oscillator form but with the equilibrium position
shifted. Hence the wave functions of the stationary states of the perturbed oscillator are
<lik(x— xo), where >/ijc(x) are the oscillator functions (23.12); the initial wave function is
tfio(x) (23.13). Using these functions and the expression (23.11) for the Hermite polynomials,
we find
00 00
f (-1)* r d *
«£o (0 ty* dx = — — e-£o<>/2 e -Uo e -^+2Ko d£,
J y/(2*irk\) J d£*
-00 v \ / _ K)
with the notation |o = xo-\Z(tnu)jh). On integrating k times by parts, the integral on the
right becomes
00
—00
Thus the transition probability (41.6) is
£ 2*
As a function of the number k it represents a Poisson distribution for which the mean value
of k is
k = #„ z = F*l2mha>*.
Perturbation theory is applicable when F is small, so that k <^ 1. Then the excitation
probabilities are small, and decrease rapidly with increasing k. The largest is woi = k.
In the opposite case of large F (k ^> 1), excitation of the oscillator occurs with very high
probability: the probability that the oscillator will remain in the normal state is woo = e~*.
Problem 2. The nucleus of an atom in the normal state receives an impulse which gives
it a velocity v; the duration t of the impulse is assumed short in comparison both with the
electron periods and with ajv, where a is the dimension of the atom. Determine the probability
of excitation of the atom under the influence of such a "jolt" (A. B. Migdal 1939).
Solution. We use a frame of reference K' moving with the nucleus after the impact.
By virtue of the condition r <^ afv, the nucleus may be regarded as practically stationary
during the impact, so that the co-ordinates of the electrons in K' and in the original frame K
immediately after the perturbation are the same. The initial wave function in K' is
*i>o = ^0 exp( — tq . 2 r fl ), q = mvjh,
where ^o is the wave function of the normal state with the nucleus at rest, and the summation
144 Perturbation Theory §41
in the exponent is over all Z electrons in the atom; see §15, Problem 2. The required prob-
ability of transition to the Ath excited state is now given, according to (41.6), by
WOK
J fa* exp(-iq . 2 r a )^ dFi ... dV z \2 .
In particular, if qa <^ 1 , then by expanding the exponential factor in the integrand and noting
that the integral of <pk*<jio is zero because the functions ifio and ipk are orthogonal, we obtain
»0*
| f^»(q.Sro)0odFi...dKz| 2 .
J a
Problem 3. Determine the total probability of excitation and ionisation of an atom of
hydrogen which receives a sudden "jolt" (see Problem 2).
Solution. The required probability can be calculated as the difference
1 -WOO = 1 - 1 J Wer*** d j/|2 f
where woo is the probability that the atom will remain in the ground state (^o = TT~ U2 e~ r l a
being the wave function of the ground state of the hydrogen atom, with a the Bohr radius).
Calculation of the integral gives
l-«oo= l-l/(l+& 2 « 2 ) 4 .
In the limiting case qa <^ 1 this probability tends to zero as q 2 a 2 , while for qa ^> 1 it tends to
unity as 1 — (2/qa) s .
Problem 4. Determine the probability that an electron will leave the K-shell of an atom
with large atomic number Z when the nucleus undergoes /3-decay. The velocity of the
^-particle is assumed large in comparison with that of the iC-electron (A. B. Migdal and
E. L. Feinberg 1941).
Solution.! In the conditions stated the time taken by the ^-particle to pass through the
iC-shell is small compared with the period of revolution of the electron, so that the change in
the nuclear charge can be regarded as instantaneous. The perturbation is here represented
by the change V = 1/r in the field of the nucleus when the change in its charge is small
(1 compared with Z). According to (41.5) the transition probability for one of the two iC-shell
electrons with energy Eo = —\Z 2 (here and below we use the fact that the state of the K-
electrons is hydrogen-like; see §74) to a state of the continuous spectrum with energy
E = $k 2 in the range dE = k dk is
dw = 2 d&.
(£2 + Z2)2
In the range which determines the matrix element Voic, the important part is that of short
distances (~l/Z) from the nucleus, in which the hydrogen-like expression can again be used
for the wave function of a state of the continuous spectrum. The final state of the electron
must have angular momentum I = (the same as that of the initial state). By means of the
functions Rio and i?&o (normalised on the k scale) derived in §36 and formula (f.3) in the
Mathematical Appendices we find J
1 \ 4 Vk ( 1 + tkjZ)W*( 1 - ikjZ)-iz/k
\r/ok
Ok y/(l-er*»V*) 1+&2/Z2
t In Problems 4 and 5, atomic units are used.
| In the calculation it is convenient to use Coulomb units and then return to atomic units in the
final result.
§41 Transitions under a perturbation acting for a finite time 145
and, since
|(l+w)</«| 2 = exp[-(2/a)tan- 1 a],
we obtain finally
dw = f(k/Z)k dk,
with
1
/(°0 = : TT ex P[~(4/a) tan" 1 a].
The limiting values of the function /(a) are e~4 for a <^ 1 and v.\2-n for a > 1.
The total probability of ionisation of the X-shell is obtained by integration of dw over all
energies of the emergent electron. A numerical evaluation gives w = 0-652T 2 .
Problem 5. Determine the probability of emergence of an electron from the i^-shell
of an atom with large Z in oc-decay of the nucleus. The velocity of the oc-particle is small
compared with that of the ^-electron, but the time which it takes to leave the nucleus is small
in comparison with the time of revolution of the electron.!
Solution. After the emergence of the oc-particle, the perturbation acting on the electron
is adiabatic. The required effect is therefore determined essentially by the interval of time
close to the "instant of application' ' of the perturbation which destroys the adiabaticity, when
the a-particle, leaving the nucleus and moving freely, is still at a distance small compared with
the radius of the iC-orbit. The perturbation V which causes the ionisation of the atom is
here represented by the deviation of the combined field of the nucleus and the a-particle
from the purely Coulomb field Zjr. The dipole moment of two particles with atomic weights
4 and A—\, and charges 2 and Z— 2, at a distance vt apart (where v is the relative velocity of
the nucleus and the a-particle), is{
2(4 - 4) -(Z- 2)4 2(A-2Z)
vt = vt.
A A
Hence the dipole term in the field of the nucleus and the a-particle is||
2(A-2Z) z
V = vt—,
A f3'
where the 2-axis is in the direction of the velocity v. The matrix element of this perturbation
reduces to that of z: taking the matrix element of the equation of motion of the electron
z = —Zz/r 3 , we obtain
(zlr*)o k = (E-E )*z iclZ.
The required transition probability for one of the two electrons in the K-shell is, by (41.2),
dw = 2| J Voke^Eo-Eu dt\ dk
o
$(A-2Z) 2 v2
A 2 Z*
Fo*| 2 dk;
t This problem was first discussed by A. B. Migdal (1941).
} The necessity of simultaneously considering the motion of the a-particle and that of the nucleus
in this problem has been pointed out by J. S. Levinger (1953).
|| If the difference A— 2Z is small, it may be necessary to take account of the next (quadrupole)
term also.
146 Perturbation Theory §42
to calculate the integral, we include in the integrand an additional damping factor e~ M with
A > 0, and then make A -> in the result. To calculate the matrix element of z — r cos 6,
we note that, since the orbital angular momentum in the initial state is I = 0, cos has a non-
zero matrix element only for the transition to a state with 1=1, and
|(cos0)oi| 2 = (COS0) OO = |
and
M 2 = i\r 0k \ 2 .
Calculating rok by means of the radial functions Roo and Rki, we find
211(A-2ZW
dzv = f(k/Z)k dk,
3A 2 Z%l+k*IZ 2 )S
the function / being as in Problem 4.
§42. Transitions under the action of a periodic perturbation
The results are different for the probability of transitions to the states of
the continuous spectrum under the action of a periodic perturbation. Let
us suppose that, at some initial instant t = 0, the system is in the wth station-
ary state of the discrete spectrum. We shall assume that the frequency co of
the periodic perturbation is such that
hco>E min -E^\ (42.1)
where E min is the value of the energy where the continuous spectrum begins.
It is evident from the results of §40 that the chief part will be played by
states with energies E v very close to the resonance energy E n w +hco, i.e. those
for which the difference co vn —co is small.f For this reason it is sufficient to
consider, in the matrix elements (40.8) of the perturbation, only the first
term (with the frequency co vn — co close to zero). Substituting this term in
(40.5) and integrating, we obtain
" vn = -t K„(t) dt = -F- _. (42.2)
hj % n -w)
The lower limit of integration is chosen so that a vn = for t =0, in accord-
ance with the initial condition imposed.
Hence we find for the squared modulus of a vn
KJ 2 = l*UI 2 • 4 sin*[Kc^-a,)*]//*V,,„-") 2 . (42.3)
It is easy to see that, for large t, this function can be regarded as propor-
tional to t. To show this, we notice that
sin 2 a£
lim = 8(a). (42.4)
f->oo Trta?
For when a # this limit is zero, while for a = we have (sin 2 otf)/ta 2 = t,
so that the limit is infinite; finally, integrating over a from — oo to +a>,
f We recall that the suffix v refers to the continuous spectrum (see the end of §38).
§™ Transitions in the continuous spectrum 147
we have (with the substitution at = £)
1 r sin 2 af 1 j* sin 2 £
1 r sm 2 xt 1 f sin 2 £
- — —da =- d$ = 1.
77-J to 2 77 J £ 2
Thus the function on the left-hand side of equation (42.4) in fact satisfies all
the conditions which define the delta function. Accordingly, we can write
for large t
kJ 2 = ( i/£VJ 2 ^s(K«-K),
or, substituting hco vn = E v -E n ™ an d using the fact that 8(ax) = (l/a)8(«),
\a vn \* = (2*lh)\FJWB v -E n m-ha>)t.
The expression |a m | 2 dv is the probability of a transition from the original
state to one in the interval from v to v+dv (cf. §5). We see that, for large t,
it is proportional to the time interval elapsed since t = 0. The probability
dzv nv of the transition per unit time isf
d«v = (2iT/h)\FJ*8(E,,-EM-hco) dv. (42.5)
As we should expect, it is zero except for transitions to states with energy
E v = E n W+hco. If the energy levels of the continuous spectrum are not
degenerate, so that v can be taken as the value of the energy alone, then the
whole "interval" of states dv reduces to a single state with energy E = E n w +
+ hm, and the probability of a transition to this state is
VnE = {27rlh)\F En \\ (42.6)
§43. Transitions in the continuous spectrum
One of the most important applications of perturbation theory is to calculate
the probability of a transition in the continuous spectrum under the action
of a constant (time-independent) perturbation. We have already mentioned
that the states of the continuous spectrum are almost always degenerate.
Having chosen in some manner the set of unperturbed wave functions cor-
responding to some given energy level, we can put the problem as follows.
It is known that, at the initial instant, the system is in one of these states;
it is required to determine the probability of the transition to another state
with the same energy. If we denote the initial state by the suffix v , then
for transitions to states between v and v+dv we have at once from (42.5)
(putting co = and changing the notation)
<K.„ = ( 27r ^)l V w} 2 KE-E v ) dv. (43.1)
This expression is, as we should expect, zero except for E V =E V : under
.*!•«? iS i CaSy t0 ™ rify th&t l °? tBking acCOunt of the second term in < 40 - 8 >' which we have omitted,
additional expressions are obtained which, on being divided by t, tend to zero as *-» +00.
148 Perturbation Theory §43
the action of a constant perturbation, transitions occur only between states
with the same energy. It must be noticed that, for transitions from states
of the continuous spectrum, the quantity dw VtV cannot be regarded directly
as the transition probability; it is not even of the right dimensions (1/time).
Formula (43.1) represents the number of transitions per unit time, and its
dimensions depend on the chosen method of normalisation of the wave
functions of the continuous spectrum.
Let us calculate the perturbed wave function which at the initial instant
coincides with the initial unperturbed function $ Vt w . According to formula
(42.2) (putting co = and changing the notation) we have
AilME v -E )t_\
a a) = V
**HM w It'll
"• E vr E v
The perturbed wave function has the form
T„ o =T v W+J^T/)dv,
or
r r \- e (iimE Vo -E v )t -,
1 J Ev ~ Ev J (43.2)
where the integration is extended over the whole continuous spectrum.f
Let us ascertain the limiting form of this function for large t. To do so,
we separate from dv the differential dE v of the energy (writing dv = dE v dr,
where dr is the product of the differentials of the remaining quantities which
determine a state in the continuous spectrum), and formally regard E v as a
complex variable. The integral
f
\— e {i/ME Vo -E v )t
r -*' m E..-E.. "*
in (43.2) is taken along the real axis. We slightly displace the path of inte-
gration into the lower half-plane; this can be done without changing the
value of the integral, since the integrand has no singularities on the real axis.
The integral can then be divided into two parts
r dE r Vw <A V (0)
J vv ^ v E v -E v J E v -E v
(these integrals have no meaning when the integration is along the real axis,
since they diverge at the point E v = E v ). Since im E v < on the contour
of integration, the second of the above integrals tends to zero as t tends J:o
t If there is also a discrete spectrum, then we must add to the integral in this formula (and subse-
quent ones) the appropriate sum over the states of the discrete spectrum.
§43 Transitions in the continuous spectrum 149
infinity (because of the factor exp[fr-iim(£ w )*] in the integrand). In the first
integral, we can again make the path of integration the real axis, but pass
round the point E v = E v ^ below. This method of integration can be conven-
iently represented in another form by adding to the constant E„ in the
denominator of the integrand a small positive imaginary quantity tS. The
pole of the integrand is thereby shifted into the upper half-plane, and the
integration can be taken simply along the real axis (which now passes below
the pole), after which 8 is allowed to tend to zero.
Thus we obtain for the wave function the expression
T ». = [fv t m + j E "° E +i8 W 0) dJje-UmE^ S ^o+ . (43.3)
The time factor shows that this function belongs, as it should, to the same
energy E Vt as the initial unperturbed function. In other words, the function
K =K {0) + » — ^ (0)d »
satisfies Schrodinger's equation (# + P)^ = E v $ n . For this reason it
is natural that the expression above should correspond exactly to formula
(38.7).f
The calculations given above correspond to the first approximation of
perturbation theory. It is not difficult to calculate the second approximation
as well. To do this, we must derive the formula for the next approximation
to Y Vo ; this is easily effected by using the method of §38 (now that we know
the method of dealing with the "divergent" integrals). A simple calculation
gives the formula
with the same method of indentation in the integrals.
Comparing this expression with formula (43.3), we can write down the
corresponding formula for the probability (or, more precisely, the number)
of transitions, by direct analogy with (43.1):
dzo = —
n
r V VV .V V . V , 2
Kv+ F -p dv ' 8 ^.~^) d". (43.5)
It may happen that the matrix element V vv% for the transition considered
>u f Sta iS ng fr u m l he latter formuIa > the wa y ^ which the integral must be taken can be found from
!»L C ^f? 0!? " *? "J"** *? ^Pression for ^. at large distances must contain only an outgoing
(and not an ingoing) wave (see §134). " i s"«'s
150 Perturbation Theory §44
vanishes. The effect is then zero in the first approximation, and we have for
the number of transitions
^ J i\i^ d ^- E ^- (43 ' 6)
In applications of this formula, the point E v , = E Va is not usually a singularity
of the integrand; the manner of integrating with respect to E v , is therefore
unimportant in general, and the integral can be taken along the real axis.
The states v for which V vv , and V v . Vv are not zero are usually called
intermediate states for the transition v -> v. It may happen that the transi-
tion v -> v can take place not through one but only through several successive
intermediate states. Formula (43.6) can be at once generalised to such cases.
Thus, if two intermediate states are needed, we have
2tt\ r r VwnVv.'v.Vv^, 7/ 2
dw = — I I dv dv
"•" h\)](E v -E v .){E-E v .)
8(E-E Vo )dv.
(43.7)
§44. The uncertainty relation for energy
Let us consider a system composed of two weakly interacting parts. We
suppose that it is known that at some instant these parts have definite values
of the energy, which we denote by E and e respectively. Let the energy be
measured again after some time interval At; the values E', e obtained are
in general different from E, e. It is easy to determine the order of magnitude
of the most probable value of the difference E' + e —E— e which is found as a
result of the measurement.
According to formula (42.3) with <o = 0, the probability of a transition of
the system (after time t), under the action of a time-independent perturbation,
from a state with energy E to one with energy E' is proportional to
sm*[(E'-E)tl2h]l(E'-E)\
Hence we see that the most probable value of the difference E'—E is of the
order of hjt.
Applying this result to the case we are considering (the perturbation being
the interaction between the parts of the system), we obtain the relation
\E+ € -E'-€'\At~h. (44.1)
Thus the smaller the time interval A*, the greater the energy change that is
observed. It is important to notice that its order of magnitude HjAt is inde-
pendent of the amount of the perturbation. The energy change determined
by the relation (44.1) will be observed, however weak the interaction between
the two parts of the system. This result is peculiar to quantum theory and has
a deep physical significance. It shows that, in quantum mechanics, the law
a™" The uncertainty relation for energy 151
of conservation of energy can be verified by means of two measurements only
to an accuracy of the order of h/At, where At is the time interval between the
measurements.
The relation (44. 1) is often called the uncertainty relation for energy. How-
ever, it must be emphasised that its significance is entirely different from
that of the uncertainty relation ApAx ~ h for the co-ordinate and momen-
tum. In the latter, Ap and Ax are the uncertainties in the values of the
momentum and co-ordinate at the same instant; they show that these two
quantities can never have entirely definite values simultaneously. The
energies E, e, on the other hand, can be measured to any degree of accuracy
at any instant. The quantity (E+e)-(E'+e f ) in (44.1) is the difference
between two exactly measured values of the energy E+e at two different
instants, and not the uncertainty in the value of the energy at a given instant.
If we regard E as the energy of some system and e as that of a "measuring
apparatus", we can say that the energy of interaction between them can be
taken into account only to within h/At. Let us denote by AE, As, ... the
errors in the measurements of the corresponding quantities. In the favour-
able case when € , e' are known exactly (Ae = Ae' = 0), we have
A(E-E')~hlAt. (44.2)
From this relation we can derive important consequences concerning the
measurement of momentum. The process of measuring the momentum of a
particle (for definiteness, we shall speak of an electron) consists in a collision
of the electron with some other ("measuring") particle, whose momenta
before and after the collision can be regarded as known exactly.f If we apply
to this collision the law of conservation of momentum, we obtain three equa-
tions (the three components of a single vector equation) in six unknowns
(the components of the momentum of the electron before and after the col-
lision). The number of equations can be increased by bringing about a
series of further collisions between the electron and "measuring" particles,
and applying to each collision the law of conservation of momentum. This)
however, increases the number of unknowns also (the momenta of the electron
between collisions), and it is easy to see that, whatever the number of col-
lisions, the number of unknowns will always be three more than the number
of equations. Hence, in order to measure the momentum of the electron,
it is necessary to bring in the law of conservation of energy at each collision,'
as well as that of momentum. The former, however, can be applied, as we
have seen, only to an accuracy of the order of kjAt, where At is the time be-
tween the beginning and end of the process in question.
To simplify the subsequent discussion, it is convenient to consider an
imaginary idealised experiment in which the "measuring particle" is a
perfectly reflecting plane mirror; only one momentum component is then
of importance, namely that perpendicular to the plane of the mirror. To
tailed 1 thC PrCSent analysis h is ° f n0 im POrtance how the energy of the "measuring" particle is ascer-
152 Perturbation Theory §44
determine the momentum P of the particle, the laws of conservation of
momentum and energy give the equations
p' + P'-p-P = 0, (44.3)
\ € ' + E'-e-E\ ~ h/At, (44.4)
where P, E are the momentum and energy of the particle, and p, e those of
the mirror ; the unprimed and primed quantities refer to the instants before
and after the collision respectively. The quantities p, p\ e, e' relating to the
"measuring particle" can be regarded as known exactly, i.e. the errors in
them are zero. Then we have for the errors in the remaining quantities,
from the above equations :
AP = AP', AE'-AE ~ hi At.
But AE = (BE/dP)AP = vAP, where v is the velocity of the electron (before
the collision), and similarly AE' = v'AP' = v'AP. Hence we obtain
(v' x -v x )AP x ~hlAt. (44.5)
We have here added the suffix x to the velocity and momentum, in order to
emphasise that this relation holds for each of their components separately.
This is the required relation. It shows that the measurement of the
momentum of the electron (with a given degree of accuracy AP) necessarily
involves a change in its velocity (i.e. in the momentum itself). This change
is the greater, the shorter the duration of the measuring process. The change
in velocity can be made arbitrarily small only as At -» oo, but measurements
of momentum occupying a long time can be significant only for a free particle.
The non-repeatability of a measurement of momentum after short intervals
of time, and the "two-faced" nature of measurement in quantum mechanics —
the necessity of a distinction between the measured value of a quantity and
the value resulting from the process of measurement — are here exhibited with
particular clarity, f
The conclusion reached at the beginning of this section, which was based
on perturbation theory, can also be derived from another standpoint by con-
sidering the decay of a system under the action of some perturbation. Let
E be some energy level of the system, calculated without any allowance for
the possibility of its decay. We denote by t the lifetime of this state of the
system, i.e. the reciprocal of the probability of decay per unit time. Then
we find by the same method that
\E -E-e\~hlr, (44.6)
where E, e are the energies of the two parts into which the system decays.
The sum £+e, however, gives us an estimate of the energy of the system
before it decays. Hence the above relation shows that the energy of a system,
t The relation (44.5) and the elucidation of the physical significance of the uncertainty relation for
energy are due to N. Bohr (1928).
§45 Potential energy as a perturbation
153
in some "quasi-stationary" state, which is free to decay can be determined
only to within a quantity of the order of hjr. This quantity is usually called
the width Y of the level. Thus
T ~ hJT. (447)
§45. Potential energy as a perturbation
The case where the total potential energy of the particle in an external
field can be regarded as a perturbation merits special consideration. The
unperturbed Schrodinger's equation is then the equation of free motion of
the particle:
A0<o) + #ty(o> = 0> k = V(2mElh*) =pjh, (45.1)
and has solutions which represent plane waves. The energy spectrum of
free motion is continuous, so that we are concerned with an unusual case of
perturbation theory in a continuous spectrum. The solution of the problem
is here more conveniently obtained directly, without having recourse to
general formulae.
The equation for the correction 0<*> to the wave function in the first ap-
proximation is
A^i)+AV (1) = {2mV\W)^\ (45.2)
where U is the potential energy. The solution of this equation, as we know
from electrodynamics, can be written in the form of retarded potentials, i.e.
in the formf
^\x, y, z) = -(m/lnfr) J ^)U{x' t y\ z')e"<r ^v'\r % (45.3)
where
dV = dx'dy'dz', r 2 = (*-*') 2 +(v-/) 2 +(*-*') 2 .
Let us find what conditions must be satisfied by the field Urn order that
it may be regarded as a perturbation. The condition of applicability of per-
turbation theory is contained in the requirement that 0&> <^ ^<»). Let a be
the order of magnitude of the dimensions of the region of space in which the
field is noticeably different from zero. We shall first suppose that the energy
of the particle is so small that ka is at most of the order of unity. Then the
factor e^r i n the integrand of (45.3) is unimportant in an order-of-magnitude
estimate, and the integral is of the order of ifjM\U\a 2 , so that
iP ~ m\ U\a*i]J®lh 2 ,
t This is a particular integral of equation (45.2), to which we may add any solution of the same
equauon with zero on the right-hand side, i.e. the unperturbed equation (45.1).
154 Perturbation Theory §45
and we have the condition
\U\ < h 2 lma 2 (for ka < 1). (45.4)
We notice that the expression on the right has a simple physical meaning;
it is the order of magnitude of the kinetic energy which the particle would
have if enclosed in a volume of linear dimensions a (since, by the uncertainty
relation, its momentum would be of the order of hja).
Let us consider, in particular, a potential well so shallow that the condition
(45.4) holds for it. It is easy to see that in such a well there are no negative
energy levels (R. Peierls 1929); this has been shown, for the particular case
of a spherically symmetric well, in §33, Problem. For, when E = 0, the
unperturbed wave function reduces to a constant, which can be arbitrarily
taken as unity: j/» (0) = 1. Since «/» (1) <4 (o) , it is clear that the wave function
tf, = 1 +if/<U for motion in the well nowhere vanishes ; the eigenf unction,
being without nodes, belongs to the normal state, so that E = remains the
least possible value of the energy of the particle. Thus, if the well is suffi-
ciently shallow, only an infinite motion of the particle is possible : the particle
cannot be "captured" by the well. Attention must be paid to the fact that
this result is peculiar to quantum theory; in classical mechanics a particle can
execute a finite motion in any potential well.
It must be emphasised that all that has been said refers only to a three-
dimensional well. In a one- or two-dimensional well (i.e. one in which the
field is a function of only one or two co-ordinates), there are always negative
energy levels (see the Problems at the end of this section). This is related to
the fact that, in the one- and two-dimensional cases, the perturbation theory
under consideration is inapplicable for an energy E which is zero (or very
small).f
For large energies, when ka > 1, the factor e ikr in the integrand plays an
important part, and markedly reduces the value of the integral. The solution
(45.3) in this case can be transformed; the alternative form, however, is more
conveniently derived by returning to equation (45.2). We take as #-axis the
direction of the unperturbed motion; the unperturbed wave function then
has the form ^ (0) = e ikx (the constant factor is arbitrarily taken as unity).
Let us seek a solution of the equation
AifW+ktyV = {2m\}i % )TJe ikx
in the form ip {1) = e ikx f; in view of the assumed large value of k, it is suffi-
cient to retain in A<£ (1) only those terms in which the factor e ikx is differen-
tiated one or more times. We then obtain for / the equation
2ik 8/1 dx = 2mUlh 2 ,
t In the two-dimensional case ^ is expressed (as is known from the theory of the two-dimensional
wave equation) as an integral similar to (45.3), in which, instead of e ikr dx'dy'dz'jr, we hwtiirH^Kkr)
dx'dy', where H Q W is the Hankel function of the first kind, of zero order, andr 2 = (x— x') + (y—y ) .
As k -> 0, the Hankel function, and therefore the whole integral, tend logarithmically to infinity.
Similarly, in the one-dimensional case, we have, in the integrand, 2me ikr dx' IK where r = \x— x'\,
and as k -> i/*' 1 ) tends to infinity as Ijk.
§45 Potential energy as a perturbation 155
whence
0(D = e ikxy = -(i m /^k)e ikx ! Udx. (45.5)
An estimation of this integral gives |0&>| ~ mUajfPk, so that the condition
of applicability of perturbation theory in this case is
\U\ ^ (h 2 jma z )ka = hvja {ka > 1), (45.6)
where v = khjm is the velocity of the particle. It is to be observed that this
condition is weaker than (45.4). Hence, if the field can be regarded as a
perturbation at small energies of the particle, it can always be so regarded at
large energies, whereas the converse is not necessarily true.f
The applicability of the perturbation theory developed here to a Coulomb
field requires special consideration. In a field where U = <x/r, it is impossible
to separate a finite region of space outside which U is considerably less than
inside it. The required condition can be obtained by writing in (45.6) a
variable distance r instead of the parameter a ; this leads to the inequality
a//fo <^ 1. (45.7)
Thus, for large energies of the particle, a Coulomb field can be regarded as a
perturbation.:}:
Finally, we shall derive a formula which approximately determines the
wave function of a particle whose energy E everywhere considerably exceeds
the potential energy U (no other conditions being imposed). In the first
approximation, the wave function depends on the co-ordinates in the same
way as for free motion (whose direction is taken as the x-axis). Accordingly,
let us look for ifs in the form if* = e ikx F, where F is a function of the co-
ordinates which varies slowly in comparison with the factor e ikx (but we
cannot in general say that it is close to unity). Substituting in Schrodinger's
equation, we obtain for F the equation
2ik dF/8x = (2m/# 2 ) UF, (45 . 8)
whence
if, = e ikx F = constant xe ikx e^/ hv) ^ dx . (45.9)
This is the required expression. It should, however, be borne in mind that
this formula is not valid at large distances. In equation (45.8) a term AF
has been omitted which contains second derivatives of F. The derivative
d 2 F/8x 2 , together with the first derivative dFjdx, tends to zero at large
distances, but the derivatives with respect to the transverse co-ordinates y
and z do not tend to zero, and can be neglected only if x <^ ka 2 .
f In the one-dimensional case the condition for perturbation theory to be applicable is given by
the inequality (45.6) for all ka. The derivation of the condition (45.4) given above for the three-
dimensional case is not valid in the one-dimensional case, owing to the divergence of the resulting
function ift^ (see the preceding footnote).
% It must be borne in mind that the integral (45.5) with a field U = aijr diverges (logarithmically)
when xl\/(y z +z 2 ) is large. Hence the wave function in a Coulomb field, obtained by means of pertur-
bation theory, is inapplicable within a narrow cone about the ar-axis.
156 Perturbation Theory §45
PROBLEMS
Problem 1. Determine the energy level in a one-dimensional potential well whose depth
is small. It is assumed that the condition (45.4) is satisfied.
Solution. We make the hypothesis, which will be confirmed by the result, that the
energy level \E\ <^ \U\. Then, on the right-hand side of Schrodinger's equation
dY/d* 8 = (2tn]h*)[U{x)-Ey,,
we can neglect E in the region of the well, and regard ip as a constant, which without loss of
generality can be taken as unity:
dV/d* 2 = 2m U/h 2 .
We integrate this equation with respect to x between two points ±* x such that a <^ x x <^ l//c,
where a is the width of the well and k = \/(2m\E\lh 2 ). Since the integral of U(x) converges,
the integration on the right can be extended to the whole range from — oo to + oo :
d,-*/™- (1)
1 —00
At large distances from the well, the wave function is of the form <\i = e± KX . Substituting
this in (1), we find
-2k = (2mjk 2 ) j U
dx
\E\ = (m\2W)
h
dx
We see that, in accordance with the hypothesis, the energy of the level is a small quantity of a
higher order (the second) than the depth of the well.
Problem 2. Determine the energy level in a two-dimensional potential well U(r) (where
00
r is the polar co-ordinate in the plane) of small depth; it is assumed that the integral J rU dr
converges. °
Solution. Proceeding as in the previous problem, we have in the region of the well the
equation
1 d / d«/r\ 2m
-— (r— ) =—U.
r dr\ dr J h 2
Integrating this with respect to r from to r t (where a <^ r x <^ l//c), we find
[d</ri 2m }
o
At large distances from the well, the equation of free motion in two dimensions is
1 d / di£\ 2m
r dr\ dry h 2
and has a solution (vanishing at infinity) ifr = constant x Ko(Kr), where Ko is the Hankel
function of the first kind, of zero order and imaginary argument; for small values of the
argument, the leading term in Ko(*cr) is proportional to log kt. Bearing this in mind, we
equate the logarithmic derivatives of ^ for r ~ a inside the well (the right-hand side of (1))
§45 Potential energy as a perturbation 157
and outside it, obtaining
1 2m r
=77- U(r)rdr,
a log K.a n z a J
log
whence
h* ( h 2
\E\ -exp( [ jlC/jrdrl-A.
ma 2 [ m \_j j )
We see that the energy of the level is exponentially small compared with the depth of this
well.
CHAPTER VII
THE QUASI-CLASSICAL CASE
§46. The wave function in the quasi-classical case
If the de Broglie wavelengths of particles are small in comparison with the
characteristic dimensions which determine the conditions of a given problem,
then the properties of the system are close to being classical, j- just as wave
optics passes into geometrical optics as the wavelength tends to zero.
Let us now investigate more closely the properties of "quasi-classical"
systems. To do this, we make in Schrodinger's equation
V 2m «
the substitution
= ««/»)* (46.1)
For the function a we obtain the equation
y — (V a cr)2- V l —H a a = E- U. (46.2)
t— 1 2m a *-* 2m„
a a a a
Since the system is supposed almost classical in its properties, we seek a in
the form of a series :
a = ff +(A/«> 1 +(A/«) 8 a 8 + ... , (46.3)
expanded in powers of h.
We begin by considering the simplest case, that of one-dimensional motion
of a single particle. Equation (46.2) then reduces to
a'*l2m-iha"l2m = E-U{x), (46.4)
where the prime denotes differentiation with respect to the co-ordinate x.
In the first approximation we write a = a and omit from the equation the
term containing h :
(T ' 2 /2m = E- U(x).
f We may point out, in particular, that the states of the discrete spectrum with large values of the
quantum number n are quasi-classical. For the number n (the ordinal number of the state) determines
the number of nodes of the eigenfunction (see §21). The distance between adjoining nodes, however,
is of the same order of magnitude as the de Broglie wavelength. For large n this distance is small,
so that the wavelength is small in comparison with the dimensions of the region of motion.
158
§46 The wave junction in the quasi-classical case 159
Hence we find
c = ± j \/{Zm[E-U{x)]} dx.
The integrand is simply the classical momentum p(x) of the particle, expres-
sed as a function of the co-ordinate. Defining the function p{x) with the -f
sign in front of the radical, we have
a =±jpdx, p = ^/[2m{E-U)], (46.5)
as we should expect from the limiting expression (6.1) for the wave function.-)-
The approximation made in equation (46.4) is legitimate only if the second
term on the left-hand side is small compared with the first, i.e. we must have
h\<j"la' 2 \ 4 1 or
\d(h/a')ldx\ < 1.
In the first approximation we have, according to (46.5), a' =p, so that the
condition obtained can be written
|d(A/27r)/d*| <^ 1, (46.6)
where X(x) = 2irhlp(x) is the de Broglie wavelength of the particle, expressed,
as a function of x by means of the classical function p(x). Thus we have
obtained a quantitative "quasi-classical" condition: the wavelength of the:
particle must vary only slightly over distances of the order of itself. The
formulae here derived are not applicable in regions of space where this condi-
tion is not satisfied.
The condition (46.6) can be written in another form by noticing that
dft d ntdU m\F\
dx dx p dx p
where F = —dUjdx is the classical force acting on the particle in the external
field. In terms of this force we find
mh\F\/p* <4 1. (46.7)
It is seen from this that the quasi-classical approximation becomes inapplic-
able if the momentum of the particle is too small. In particular, it is clearly
inapplicable near turning points, i.e. near points where the particle, according
to classical mechanics, would stop and begin to move in the opposite direction.
These points are given by the equation p(x) = 0, i.e. E = U(x). As p -> 0,
the de Broglie wavelength tends to infinity, and hence cannot possibly be
supposed small.
t As is well known, J p dx is the time-independent part of the action. The total mechanical action
5 of a particle is S = —Et± J p dx. The term — Et is absent from o , since we are considering a time-
independent wave function tfi.
160 The Quasi-Classical Case §46
Let us now calculate the next term in the expansion (46.3). The first-order
terms in % in equation (46.4) give
whence
a/ = -a "/2a ' = -p'\lp.
Integrating, we find
o 1 = -ilogp, (46.8)
omitting the constant of integration.
Substituting this expression in (46.1) and (46.3), we find the wave function
in the form
tjj = c 1 p- 1 ^ i ^JP dx +C 2 p- 1 ^e^ i / n) JP dx . (46.9)
The subsequent terms in the expansion (46.3) lead to the appearance, in the
coefficients of the exponentials, of terms in the first and higher powers of h\
it is not usually necessary to calculate these terms.
The presence of the factor \jy/p in the wave function has a simple inter-
pretation. The probability of finding the particle at a point with co-ordinate
between x and x+dx is given by the square |«/f| 2 , i.e. is essentially propor-
tional to ljp. This is exactly what we should expect for a "quasi-classical"
particle, since, in classical motion, the time spent by a particle in the segment
dx is inversely proportional to the velocity (or momentum) of the particle.
In the "classically inaccessible" parts of space, where E < U(x), the func-
tion p(x) is purely imaginary, so that the exponents are real. The wave func-
tion in these regions can be written in the form
C ' C '
V\P\ V\P\ ' ' }
PROBLEM
Determine the wave function in the quasi-classical approximation up to terms of the
order of H in the coefficient of the exponent.
Solution. The terms of order h 2 in equation (46.4) give
<V<V+ W 2 + W = o,
whence (substituting (46.5) and (46.8) for a and a ± )
G2 ' =p"l4p2-3p'*l8p3.
Integrating (by parts in the first term) and introducing the force F — pp'/tn, we obtain
o- 2 = lmF/p*+$m* J* {F*lp) dx.
§47 Boundary conditions in the quasi-classical case 161
The wave function in this approximation is of the form
= £tlKp = «tf/M».+» 1 (l—^[(j a )
constant r , „ r
= [\-limhFlp z -\ihmH (F 2 /p 5 ) dx]eWMb d *.
VP J
§47. Boundary conditions in the quasi-classical case
Let x — a be a turning point, so that U(a) = E, and let U > E for all
x > a, so that the region to the right of the turning point is classically
inaccessible. The wave function must be damped in this region. Sufficiently
far from the turning point, it has the form
C / 1 I r |\
tb = expf ad* ) for x > a, (47.1)
corresponding to the first term in (46.10). To the left of the turning point,
the wave function must be represented by a real combination (46.9) of two
quasi-classical solutions of Schrodinger's equation:
* = ^KiJH + :S exp H.M for x<a - (47 - 2)
a a
To determine the coefficients in this combination we must follow the
variation in the wave function for positive x—a (where (47.1) holds) to
negative x — a. In doing so, however, it would be necessary to pass through
a region near the turning point where the quasi-classical approximation is
invalid, and the exact solution of Schrodinger's equation must be con-
sidered.f
This can be avoided if we formally regard f asa function of a complex
variable x and go from positive to negative x— a along a path which is always
sufficiently far from the point a, so that the quasi-classical condition is formally
satisfied.
Let us first examine the variation of the wave function (47.1) on passing
round the point a from right to left along a semicircle of large radius in the upper
half-plane of the complex variable x. For clarity, we may rewrite (47.1) in
the form
X
ftx) = iC[2m(U- E)]-U* exp ( - ^ f y/[U(x) - E] dx\ , (47.3 )
t Near the turning point, E—U^.Fo(x—a) (where Fq = [— dU/dx] x =a); the corresponding
exact solution of Schrodinger's equation is given by the formulae derived in §24.
162 The Quasi-Classical Case §48
where the function p(x) has been written explicitly. It is evident a priori
that, as a result of passing along the path indicated above, the function (47.3)
must become the second term in (47.2), since along the whole of the path
this term predominates over the first term, which decreases exponentially into
the upper half -plane; and in fact, when x varies along this path, the phase
of the difference U(x) — E, like that of x— a, increases by it. Consequently
the function (47.3) becomes the second term in (47.2) with coefficient
C*2 = \Ce~ in,A . Similarly, on passing from right to left along a semicircle in
the lower half-plane, the function (47.3) becomes the first term in (47.2) with
coefficient C\ = \Ce inl *.
Thus the wave function (47.1) for x > a corresponds for x < a to the
functionf
ib = cosf- p dx+hr )
V'P VU /
a
a
= sinf- p dx+hr), for x < a. (47.4)
Vp \*J /
X
The functions (47.1) and (47.4) are approximate expressions to the right and
left of the turning point for the same exact solution of Schrodinger's equa-
tion (H. A. Kramers 1926).
If the classically accessible region is bounded (at x = a) by an infinitely
high "potential wall", the boundary condition for the wave function at x = a
is ifj = (see §18). The quasi-classical approximation is then valid up to
the wall itself, and the wave function is
1 C
ib = sin- I p dx for x < a,
hj
C
Vp HJ x (47.5)
a
ib = for x > a.
§48. Bohr and Sommerfeld's quantisation rule
The results which we have obtained enable us to derive the condition which
determines the quantum energy levels in the quasi-classical case. To do this
we consider a finite one-dimensional motion of a particle in a potential well :
t If the region U < E lay to the right of a turning point x = b, (47.4) would be replaced by
b
c
ib = cos
VP
(i j p *,+*,) = -£ sing j p d* +i „). (47.4a)
§48 Bohr and Sommerf eld's quantisation rule 163
the classically accessible region b ^ x ^ a is bounded by two turning points.f
The boundary condition at x = b gives (in the region right of this point) the
wave function (47.4a) :
= — sirX- J p dx+fr\. (48.1)
b
Applying formula (47.4) to the region left of the point x = a, we obtain the
same function in the form
^ = S sin G/' d * +i 4
VP
X
If these two expressions are the same throughout the region, the sum of their
phases (which is a constant) must be an integral multiple of it:
a
hi
pdX+ilT =(»+1)tT,
withC = (-l)"C". Hence
ipdx = 2irh(n+i), (48.2)
a
where §pdx=2$pdxis the integral taken over the whole period of the
quasi-classical motion of the particle. This is the condition which determines
the stationary states of the particle in the quasi-classical case. It corresponds
to Bohr and Sommerfeld's quantisation rule in the old quantum theory.
It is easy to see that the integer n is equal to the number of zeros of the
wave function, and hence it is the ordinal number of the stationary state.
For the phase of the wave function (48.1) increases from ^n at x — b to
(« + f)7r at x = a y so that the sine vanishes n times in this range (outside
the range b ^ x < a, the wave function decreases monotonically and has
no zeros at a finite distance) .% We recall, incidentally, that the quasi-classical
f In classical mechanics, a particle in such a field would execute a periodic motion with period (time
taken in moving from * = b to x = a and back)
a u
T = 2 f dxjv = 1m J dxjp,
b b
where v is the velocity of the particle.
J Strictly speaking, the zeros should be counted by means of the exact form of the wave
function near the turning points. If this is done, the result given in the text is confirmed.
164 The Quasi-Classical Case §48
approximation, and therefore the quantisation rule (48.2), are applicable
only when n is large, f
In normalising these wave functions, the integration of J^fJ 2 can be restricted
to the range b ^ x ^ a, since outside this range «/r decreases exponentially.
Since the argument of the sine in (48.1) is a rapidly varying function, we can
with sufficient accuracy replace the squared sine by its mean value \. This
gives
b rK '
~ 7rC 2 l2m<o = 1,
where w = InJT is the frequency of the classical periodic motion. Thus
the normalised quasi-classical function is
* = f^ sin [ll pdx+iir } (483)
b
It must be recalled that the frequency co is in general different for different
levels, being a function of energy.
The relation (48.2) can also be interpreted in another manner. The
integral j> p dx is the area enclosed by the closed classical phase trajectory
of the particle (i.e. the curve in the px-plane, which is the phase space of the
particle). Dividing this area into cells, each of area 2ttH, we have n cells
altogether ; n, however, is the number of states with energies not exceeding
the given value (corresponding to the phase trajectory considered). Thus
we can say that, in the quasi-classical case, there corresponds to each quantum
state a cell in phase space of area 2mh. In other words, the number of states
belonging to the volume element ApAx of phase space is
ApAx/27rk. (48.4)
If we introduce, instead of the momentum, the wave number k =p/h, this
number can be written
AIiAx/2tt.
It is, as we should expect, the same as the familiar expression for the number
of proper vibrations of a wave field.J
Starting from the quantisation rule (48.2), we can ascertain the general
nature of the distribution of levels in the energy spectrum. Let AE be the
f In some cases the exact expression for the energy levels E(n) (as a function of the quantum
number «), obtained from the exact Schrodinger's equation, is such that it retains its form as n — ► co ;
examples are the energy levels in a Coulomb field, and those of a harmonic oscillator. In these cases,
of course, Bohr's quantisation rule, although really applicable only for large n, gives for the function
E(n) an expression which is the exact one.
J See, for example, The Classical Theory of Fields, §52.
§48 Bohr and Sommerfeld's quantisation rule 165
distance between two neighbouring levels, i.e. levels whose quantum numbers
n differ by unity. Since AE is small (for large n) compared with the energy
itself of the levels, we can write, from (48.2),
AE i (dp/dE) dx = 2-nh.
But dEjdp = v, so that
i (dp/dE) dx = i dxjv = T.
Hence we have
AE = IttH/T = hco. (48.5)
Thus the distance between two neighbouring levels is hco. The frequencies
co may be regarded as approximately the same for several adjacent levels (the
difference in whose numbers n is small compared with n itself). Hence we
reach the conclusion that, in any small range of a quasi-classical part of the
spectrum, the levels are equidistant, at intervals of hco. This result could
have been foreseen, since, in the quasi-classical case, the frequencies cor-
responding to transitions between different energy levels must be integral
multiples of the classical frequency co.
It is of interest to investigate what the matrix elements of any physical
quantity / become in the limit of classical mechanics. To do this, we start
from the fact that the mean value / in any quantum state must become, in
the limit, simply the classical value of the quantity, provided that the state
itself gives, in the limit, a motion of the particle in a definite path. A wave
packet (see §6) corresponds to such a state ; it is obtained by superposition of
a number of stationary states with nearly the same energy. The wave func-
tion of such a state is of the form
n
where the coefficients a n are noticeably different from zero only in some
range Aw of values of the quantum number n such that 1 < An <^ n\ the
numbers n are supposed large, because the stationary states are quasi-classical.
The mean value of/ is, by definition,
/ = j T*/T d* = 22 a m *a n f mn #»™\
or, replacing the summation over n and m by a summation over n and the
difference m —n = s,
where we have put co mn = sco in accordance with (48.5).
166 The Quasi-Classical Case §48
The matrix elements f nm calculated by means of the quasi-classical wave
functions decrease rapidly in magnitude as the difference m—n increases,
though at the same time they vary only slowly with n itself (m—n being fixed).
Hence we can write approximately
/ = SS a*a n f/<»« = 2 K|* S//*—,
where we have introduced the notation f s =f- +s -, n being some mean value
of the quantum number in the range An. But 2 \a n \ 2 = 1 ; hence
dost
The sum obtained is in the form of an ordinary Fourier series. Since /
must, in the limit, coincide with the classical quantity f(t), we arrive at the
result that the matrix elements f mn in the limit become the components f m _ n
in the expansion of the classical function f(t) as a Fourier series.
Similarly, the matrix elements for transitions between states of the con-
tinuous spectrum become the components in the expansion of/(/) as a Fourier
integral. Here the wave functions of the stationary states must be normalised
by (l//z) times the delta function of energy.
All the above results can be generalised immediately to systems with
several degrees of freedom, executing a finite motion for which the problem
in classical mechanics allows a complete separation of the variables in the
Hamilton-Jacobi method (called a conditionally periodic motionf). After
separation of the variables for each degree of freedom, the problem reduces to
a one-dimensional problem, and the corresponding quantisation conditions
are
j>pi dqi = 2Trh(ni + yi), (48.6)
where the integral is taken over the period of variation of the generalised
co-ordinate qi, and yi is a number of the order of unity which depends on
the nature of the boundary conditions for the degree of freedom considered .J
In the general case of an arbitrary (not conditionally periodic) motion in
several dimensions, there are no quantum numbers nt. The concept of
cells in phase space is, however, always valid in the quasi-classical approxima-
tion. This is clear from the relationship noted above with the number of
proper vibrations of the wave field in a given volume of space. In the general
t See Mechanics, §50.
X For example, in motion in a centrally symmetric field we have
j>p r dr = 2Trh(n r +%), j>p d d6 = 2iTh(l—m+\) > j>p^ d<f> = 2-nhm,
where n r = n— I— 1 is the radial quantum number. The last of the three equations simply expresses
the fact that ptj> is the z-component of the angular momentum, equal to Hm.
§49 Quasi-classical motion in a centrally symmetric field 167
case of a system with s degrees of freedom, there are
AIV = A 2 i ... A? 5 A£i ... ^p s |(2^^hy (48.7)
quantum states in a volume element in phase space.
PROBLEMS
Problem 1 . Determine (approximately) the number of discrete energy levels of a particle
moving in an arbitrary (not central) field C7(r) which satisfies the quasi-classical condition.
Solution. The number of states belonging to a volume of phase space which corresponds
to momenta in the range ^ p < £max and particle co-ordinates in the volume element dV
is f Trpmax 3 dVI(2irh) 3 . For given r the particle can have (in its classical motion) a momentum
satisfying the condition E = p 2 /2m + U(r) «£ 0. Substituting />max = V[ — 2m£7(r)], we
obtain the total number of states of the discrete spectrum:
V (-Ufl*&V,
3tt2 #*
where the integration is over the region of space in which U < 0. This integral diverges
(i.e. the number of states is infinite) if U decreases at infinity as r~ s with s< 2, in accordance
with the results of §18.
Problem 2. The same as Problem 1, but for a quasi-classical centrally symmetric field
U(r) (V. L. Pokrovskii).
Solution. In a centrally symmetric field the number of states is not the same as the
number of energy levels, on account of the degeneracy of the latter with respect to the
direction of the angular momentum. The required number can be found by noting that the
number of levels with a given value of the angular momentum M is the same as the number
of (non-degenerate) levels for a one-dimensional motion in a field with potential energy
Uett = U(r)+M 2 l2tnr' 2 . The maximum possible value of the momentum p r for given r and
energies E =s£ is /v.max = V(— 2mU e ti). The number of states (i.e. the required number of
levels) is therefore
f^#r = V(2m)r /(_ u _^!\ dr
J Ink 2tt% J V V 2mry
The required total number of discrete levels is obtained from this by integration with respect
to M/H (which replaces in the quasi-classical case the summation with respect to I), and is
(m/4£2) j(-U)rdr.
§49. Quasi-classical motion in a centrally symmetric field
In motion in a centrally symmetric field the wave function of a particle
falls, as we know, into an angular and a radial part. Let us first consider the
former.
The dependence of the angular wave function on the angle <f> (determined
by the quantum number m) is so simple that the question of finding approxi-
mate formulae for it does not arise. The dependence on the polar angle 9 is,
according to the general rule, quasi-classical if the corresponding quantum
number / is large (this condition will be more precisely formulated below).
168 The Quasi-Classical Case §49
We shall here confine ourselves to deriving the quasi-classical expression
for the angular function for the case (the most important one in applications)
of states whose magnetic quantum number is zero (m = 0). This function
is, apart from a constant factor, the Legendre polynomial P, (cos0) (see
(28.7)), and satisfies the differential equation
d 2 Pj/d0 2 +cot d dP t ldd+l(l+ l)Pt = 0. (49.1)
The substitution
Pi(cos0) = x (0)/Vsin0 (49.2)
reduces this to
x"+[('+£) 2 +i cosec 2 0] x = 0, (49.3)
which does not contain the first derivative and is similar in appearance to
the one-dimensional Schrodinger's equation.
In equation (49.3), the part of the de Broglie wavelength is played by
A = 2tt [(/+$)«+£ cosec^J-i/a.
The requirement that the derivative d(A/27r)/da: is small (the condition (46.6))
gives the inequalities
61 > 1, (77-0)/ > 1, (49.4)
which are the conditions that the angular part of the wave function is quasi-
classical. For large / these conditions hold for almost all values of 0, exclud-
ing only a range of angles very close to or n.
When the conditions (49.4) are satisfied, we can neglect the second term
in the brackets in (49.3) compared with the first:
x"+{i+lfx = o.
The solution of this equation is
x = V sin e ^(cos 0) = A sin[(/+i)0+ a], (49.5)
where A and a are constants.
For angles <^ 1, we can put in equation (49.1) cot 6 ^ 1/0; replacing
also 1(1+1) by the approximation (/+|) 2 , we obtain the equation
d 2 P, 1 dPj
-+ i +(/+^) 2 P I = 0,
d0 2 d0 V '
which has as solution the Bessel function of zero order :
P,(cos0) =M(l+i)0]> 6<1- (49.6)
The constant factor is put equal to unity, since we must have P z = 1 for
0=0. The approximate expression (49.6) for P, is valid for all angles
<^ 1. In particular, it can be applied for angles in the range 1// <^ <^ 1,
§49 Quasi-classical motion in a centrally symmetric field 169
where it must agree with the expression (49.5), which holds for all dp 1//.
For 61 > 1 the Bessel function can be replaced by its asymptotic expression
for large values of the argument, and we obtain
2 8in[(Z+l)0+lir]
/-
N ttI
VO
(we can neglect \ in the coefficient compared with /). On comparison with
(49.5), we find that A = V(2/w/), a = \n. Thus we obtain finally the
following expression for P,(cos 0), applicable in the quasi-classical case:f
Pl (cos9) S /- U 7! ; fl . (49.7)
V ttI ysinfl
The normalised wave function © l0 is obtained, according to (28.7), by
multiplying by V"\/(l+\) = i l y/l'.
\2 sin[(Z4-i)0+ Jtt]
V it ysin0
Let us now turn to the radial part of the wave function. It has been
shown in §32 that the function x{ r ) = rR(r) satisfies an equation identical
with the one-dimensional Schrodinger's equation, with the potential energy
h 2 1(1+1)
U l (r) = U(r)+-±-^.
2m r 2
Hence we can apply the results obtained in the previous sections, if the
potential energy is understood to be the function C/,(r).
The case / = is the simplest. The centrifugal energy vanishes and, if
the field U(r) satisfies the necessary condition (46.6), the radial wave
function will be quasi-classical in all space. For r = we must have x = 0,
and hence the quasi-classical function %(r) is determined by formulae (47.5).
If / # 0, the centrifugal energy also must satisfy the condition (46.6). In
the region of small r, where the centrifugal energy is of the same order as
the total energy, the wavelength A = Qmhjp ~ rjl, and the condition (46.6)
gives / > 1 . Thus, if / is small, the quasi-classical condition is violated by the
centrifugal energy in the region of small r. It is easily seen that we obtain
the correct value of the phase of the quasi-classical wave function %[r) by
calculating it from the formulae for one-dimensional motion, replacing the
coefficient /(/+1) in the potential energy U^r) by (/+|) 2 4
Ui(r)=U(r)+—±-^-. (49.9)
2m r 2
t Attention is drawn to the fact that, as a result of replacing /(/+ 1) by (/+i) 2 » we have obtained an
expression which is multiplied by (—1)' when 6 is replaced by it— 0; this is as it should be for the
function Pi(cos 6).
% For example, in the simple case of free motion (U = 0) the phase of the function calculated from
formula (47.4) with Ui from (49.9) will be kr— JwZ for large r, as it should be.
170 The Quasi-Classical Case §49
The question of the applicability of the quasi-classical approximation to a
Coulomb field U = ±a/r requires special consideration. The most import-
ant part of the whole region of the motion is that corresponding to distances
r for which \U\ ~ \E\, i.e. r ~ cc/|£|. The condition for quasi-classical
motion in this region amounts to the requirement that the wavelength
A ~ h/ y/(2m\E\) is small compared with the dimensions <x.j\E\ of the
region; this gives
\E\ ^ ma 2 //* 2 , (49.10)
i.e. the absolute value of the energy must be small compared with the energy
of the particle in the first Bohr orbit. This condition can also be written in
the form
d/Hv > 1, (49.11)
where v ~ <s/(\E\jm) is the velocity of the particle. It should be noticed
that this condition is the opposite of the condition (45.7) for the applicability
of perturbation theory to a Coulomb field.
The region of small distances ( | U{r) \ > E) is without interest in a repulsive
Coulomb field, since for U > E the quasi-classical wave functions diminish
exponentially. In an attractive field, however, when / is small it is possible
for the particle to penetrate into the region where \U\ p E, so that we have
to consider the limits of applicability of the quasi-classical approximation in
this case. We use the general condition (46.7), putting there
F = -dU/dr = - a/r 2 , p ~ y/(2m\ U\) ~ y/(mijr).
As a result, we find that the region of applicability of the quasi-classical
approximation is restricted to distances such that
r>/* 2 /ma, (49.12)
i.e. distances large in comparison with the "radius" of the first Bohr orbit.
PROBLEM
Determine the behaviour of the wave function near the origin, if the field becomes infinite
as ± oc/r s , with s > 2, when r -» 0.
Solution. For sufficiently small r, the wavelength A ~ hl\/(m\U\) ~ Hr*l 2 j\/(maL),
so that dA/dr ~ hr*' 2 - 1 / V0»°0 <€ 1 1 thus the quasi-classical condition is satisfied. In an
attractive field Ui -> — oo when r -> 0. The region near the origin is in this case classically
accessible, and the radial wave function x ~ 1/Vp, whence
if, ~ y */4-l #
In a repulsive field, the region of small r is classically inaccessible. In this case the wave
function tends exponentially to zero as r -> 0. Omitting the coefficient of the exponential
function, we have
T
■HH]'
2 v / (2*«a)
exp| — -| | p dr\ |,or ^r~exp{ r -{s/2-i)
(s—2)h
§50
Penetration through a potential barrier
171
§50. Penetration through a potential barrier
Let us consider the motion of a particle in a field of the type shown in
Fig. 13, characterised by the presence of a potential barrier, i.e. a region in
which the potential energy U(x) exceeds the total energy E of the particle.
Fig. 13
In classical mechanics, a potential barrier is "impenetrable" to a particle;
in quantum mechanics, however, a particle can pass "through the barrier":
the probability of this is not zero (see also §25, Problem 2). If the field U(x)
satisfies the quasi-classical conditions, the transmission coefficient for the bar-
rier can be calculated in a general form. We may remark that, in particular,
these conditions give the result that the barrier must be "wide", and hence
the transmission coefficient is small in the quasi-classical case.
In order not to interrupt the subsequent calculations, we shall first solve
the following problem. Let the quasi-classical wave function in the region
to the right of the turning point x = b (where U(x) < E) have the form of a
travelling wave :
C ri } 1
ip = expl - I p 6.x— \vn I.
y/p LHJ J
(50.1)
We require to find the wave function of this state in the regionf x < b. We
shall seek it in the form
C" rl) f n
ib = exp - p dx\ ,
V V\P\ Ul J U
(50.2)
which increases as we go into the region x < b. The exponentially decreasing
term is neglected in comparison with the increasing one. To determine the
coefficient C we proceed as follows. We notice that, according to formulae
| In the problem of penetration through a potential barrier, we are concerned with a motion infinite
in both directions; the corresponding levels are doubly degenerate (see §21), and hence the wave
functions need not be real.
172 The Quasi-Classical Case §50
(47.1) and (47.4a), there is a correspondence between the functions
X
1
^ = 2VpL eXP |^ J P d *-i £7r )+ ex p{-^ J P <**+!**}] for x > b,
(50.3)
On the other hand, between two different exact solutions X and 2 of the one-
dimensional Schrodinger's equation we have the relation (21.2)
tits'— 020i' = constant.
We apply this relation with *fj x the solution given by formulae (50.1), (50.2),
and 2 the solution (50.3). To the left of the point x = b, we have
fcfc'-fck' = -ffl&IW = C7A,
while to the right we have
Mt-Mi = &W0i)' = -icjh.
Equating these two expressions, we obtain C" = — iC. Thus the required
quasi-classical wave function is of the form
iC
for * < b, 1I1 = exp
VlPl
fl/Hl-
c ■■- (50 - 4)
for * > 6, = expl- I p dx—liiA.
b
Let us now go on to calculate the coefficient for the penetration of the
barrier by a particle. Let the particle be incident on the barrier from left
to right. Since the probability of penetrating the barrier is small in the quasi-
classical case, we can with sufficient accuracy write the wave function in
region I (Fig. 13), in front of the barrier, the same as if the barrier were
completely impenetrable, i.e. in the form (47.4):
xf> = — cosl- \p d^+^rj, (50.5)
§50 Penetration through a potential barrier 173
where we have introduced the velocity v = p\m\ see below regarding the
choice of the normalisation coefficient. If this is written as the sum of two
complex expressions,
a a
the first term (which becomes a plane wave ifi ~ e {i/n)px as x -> -oo)
represents a particle incident on the barrier, and the second a particle reflected
from the barrier. The normalisation chosen corresponds to a unit probability
current density in the incident wave.
On the other side of the turning point x = a (in region II, inside the
barrier), the wave function (50.5) corresponds, according to the results of
§47, to the function
x
♦-^[-SHl (50 - 6)
a
Writing this in the form
6 x
i r ii r i ii f n
«£ = exp -- pM+t\ PM , (50.7)
V VN L h\ J I h\ J U
a J>
and applying formula (50.4), we find the wave function in region III:
b x
if, = exp| — I | pdxW- p&x+liir . (50.8)
\A> L W J \ nj J
v a b
The current density in region III, calculated by means of this function, is
6
D=exp [~il \ pdx W' (50 ' 9)
Since the current density in the wave incident on the barrier is taken as unity,
D is in fact the required transmission coefficient for the barrier. We emphasise
that this formula is applicable only when the exponent is large, so that D
itself is small.
It has been assumed in the foregoing that the field U(x) satisfies the quasi-
classical condition over the whole extent of the barrier (excluding only the
immediate neighbourhood of the turning points). In practice, however, we
often have to deal with barriers where the potential energy curve on one side
drops so steeply that the quasi-classical approximation is inapplicable. The
174
The Quasi-Classical Case
§50
exponential factor in D remains the same in this case as in formula (50.9),
but the coefficient of the exponential (equal to unity in (50.9)) is different.
To calculate it we must, essentially, calculate the exact wave function in the
"non-quasi-classical" region and determine the quasi-classical wave function
inside the barrier in accordance with this. In formula (50.6) a coefficient
P # 1 then appears, and in (50.9) a coefficient £ 2 .
PROBLEMS
Problem 1. Determine the transmission coefficient for the potential barrier shown in
Fig. 14: U(x) = for * < 0, U(x) = U —Fx for x > 0; only the exponential factor need
be calculated.
UM
Fig. 14
Solution. A simple calculation gives the result
V(2m)
D
>exp
ZhF
{U -E)*/j.
Problem 2. Determine the probability that a particle (with zero angular momentum) will
emerge from a centrally symmetric potential well with U(r) = — U a for r < r„ U(r) = air
for r > r (Fig. 15). W
Fig. 15
Solution. The centrally symmetric problem reduces to a one-dimensional one, so that
the formulae obtained above can be applied directly. We have
a/E
■H-ii 7Kr*)] d j
§50 Penetration through a potential barrier
Evaluating the integral, we finally obtain
175
W r*->
M-m-fi-M'-m
In the limiting case r -> 0, this formula becomes
W ~ e -{™/nW&m/E) _ e -2ira/nv m
These formulae are applicable when the exponent is large, i.e. when ajhv > 1. This condi-
tion agrees, as it should, with the condition (49.11) for quasi-classical motion in a Coulomb
field.
Problem 3. The field U(x) consists of two symmetrical potential wells (I and II in Fig.
16), separated by a barrier. If the barrier were impenetrable to a particle, there would be energy
levels corresponding to the motion of the particle in one or other well, the same for both
wells. The fact that a passage through the barrier is possible results in a splitting of each of
these levels into two neighbouring ones, corresponding to states in which the particle moves
simultaneously in both wells. Determine the magnitude of the splitting (the field U(x) is
supposed quasi-classical).
Solution. Let 2?,, be some level for the motion of the particle in one well (I, say), and
ifi (x) the corresponding wave function (so normalised that the integral of ip * over well I is
unity). When the small probability of penetration through the barrier is taken into account,
the level splits into levels E t and E t with wave functions which are symmetric and anti-
symmetric combinations of ^ o 0*0 and 'I'oi—x):
M*) = ( W2)[«Ao(*)+0o(-*)]» M*) = (WQM*)-M-*)]- (!)
The quasi-classical function tfi (x) diminishes exponentially outside the well, and in particular
VW
Fig. 16
in the direction of negative x. Hence, within well I, t/> ( _ x) is vanishingly small in compari-
son with <Ao(*)> and vice versa in well II. The functions (1) are so normalised that their
squares integrated over wells I and II are unity.
Schrodinger's equations are
,A "+(2m/£ 2 )(£o- Wo = 0, ^'+(2tnj^){E 1 -U)4, 1 = 0;
we multiply the former by <l>i and the latter by ^ , subtract corresponding terms, and integrate
176 The Quasi-Classical Case §50
over * from to oo. Bearing in mind that, for * = 0, ^ = \Z2^ and <Ai' = 0, and that
OO 00
J 0o0i <** S — j 0o 2 dx = 1/ V2,
we find
£i-#o = -(^/^)0o(O)<Ao'(O).
Similarly, we find for E 2 —E the same expression with the sign changed. Thus
E 2 -E x = (2/z»0 o (O)0 o '(O).
By means of formula (47.1), with the coefficient C from (48.3), we find that
a
ya> r 1 /* n mz;,,
— exp|_--j,„d,J, vm—jfm,
o
where t> = V[2((7 -£ )/m]. Thus
a
a>/* r 1 r -]
E i -E 1 = —exp^-- J |/>| d*J.
—a
Problem 4. Determine the exact value of the transmission coefficient D for the passage
of a particle through a parabolic potential barrier U(x) = — £fe* 2 (supposing that D is mo*
small) (E. Kemble 1935).t
Solution. Whatever the values of k and E, the motion is quasi-classical at sufficiently
large distances |*|, with
p = V[2m(E+%kx 2 )] ~ x^(mk)+E^(m/k)fx,
and the asymptotic form of the solutions of Schrodinger's equation is
= constant xe^ 2 / 2 ^' 6 - 1 / 2 ,
where we have introduced the notation
$ = ximk/h 2 ) 1 ^, e = (EJhW(mlk).
We are interested in the solution which, as x -> + oo, contains only a wave which has
passed the barrier, i.e. is propagated from left to right. We put
as X -> OO, i/j = Be^ 2 / 2 ^- 1 ? 2 , (1)
as * -> — 00, ifs = e-i^ a /2|||-ie-i/2 +^£ 8 /2|£|«-i/2 > (2)
In the expression (2), the first term represents the incident wave, and the second the reflected
wave (the direction of propagation of a wave is that in which its phase increases). The
relation between A and B can be found by using the fact that in this case the asymptotic
expression for tft is valid in the whole of a sufficiently distant region of the plane of the complex
variable £ . Let us follow the variation of the function (1) as we go round a semicircle of large
radius in the upper half-plane of £ (cf. §47). Over the whole of this path the term in e*^ 2 is
the dominant part of the solution, and hence the function (1) must be converted, by traversing
t The solution of this problem can also be applied to penetration sufficiently near the top of any
barrier U(x) whose dependence on x near the maximum is quadratic.
§51 Calculation of the quasi-classical matrix elements \11
this path, into the second term of the function (2). Hence we find
A = B^)"- 1 / 2 = —iBe-™.
On the other hand, the condition that the number of particles should be conserved is
\A\*+\B\* = 1.
From these two relations we find the required transmission coefficient D = \B\*:
D = l/(l+e- 2ffe ).
This formula holds for any E. When E is large and negative, it gives JD £ e~ mM , in accord-
ance with formula (50.9). For E > 0, the quantity R = 1 —D = 1/(1 +e 27te ) is the coefficient
of reflection "above the barrier".
§51. Calculation of the quasi-classical matrix elements
A direct calculation of the matrix elements of any physical quantity/ with
respect to the quasi-classical wave functions presents great difficulty. We may
suppose that the energies of the states between which the matrix element is
calculated are not close to each other, so that the element does not reduce to
the Fourier component of the quantity/ ( §48). The difficulties arise because,
owing to the fact that the wave functions are exponential (with a large imagin-
ary exponent), the integrand oscillates rapidly, and this makes it very
troublesome to obtain even an approximate estimate of the integral.
We shall consider a one-dimensional case (motion in a field U(x)), and sup-
pose for simplicity that the operator of the physical quantity is merely a func-
tion/^) of the co-ordinate. Let ^ and «/» 2 be the wave functions correspond-
ing to some values E x and E 2 of the energy of the particle (with E 2 > E x ,
Fig. 17) ; we shall suppose that ^ and if/ 2 are taken real. We have to calculate
the integral
/l2 = J 0l/& d *«
(51.1)
\U(x)
Fig. 17
178 The Quasi-Classical Case §51
The wave function ifj x in the regions on both sides of the turning point
* = a lt but not in its immediate neighbourhood, is of the form (47.1), (47.4a):
for x < a lt ifj x
C\
-exp[-|j A d,|],
W\Pi\ L A|
(51.2)
for * > fll , if, x = — -L cosf - pj d*— Jfl-V
a 1
and similarly for j/t 2 (replacing the suffix 1 by 2).
However, the calculation of the integral (51.1) by substituting in it these
asymptotic expressions for the wave functions would not give the correct
result. The reason is, as we shall see below, that this integral is an exponen-
tially small quantity, whereas the integrand is not itself small. Hence even a
relatively small change in the integrand will in general change the order of
magnitude of the integral. This difficulty can be circumvented as follows.
We represent the function ip 2 as a sum if> 2 = ip 2 + +i/if, expressing the cosine
(in the region x > a 2 ) as the sum of two exponentials. According to formulae
(50.4), we have
f or*<* 2) ^ = _=g. exp Q|J fodj( |],
x
(51.3)
the function fa- is the complex conjugate of tfs 2 + : i/t 2 ~ = (iff 2 + )*.
The integral (51.1) is also divided into the sum of two complex conjugate
integrals /i 2 =/i2 + 4-/i2~, which we shall proceed to calculate. First of all,
we note that the integral
00
fl2 + = / 0l/^2 + d*
—oo
converges. For, although the function «/r 2 + tends exponentially to infinity in
the region x < a 2 , the function ifj x , in the region x < a lt tends exponentially
to zero still more rapidly (since we have \p x \ > \p 2 \ everywhere in the region
x < a 2 ).
We shall regard the co-ordinate # as a complex variable, and displace the
path of integration off the real axis into the upper half-plane. When x
receives a positive imaginary increment, an increasing term appears in the
function tp x (in the region * > a^, but the function i/j 2 + decreases still more
§51 Calculation of the quasi-classical matrix elements 179
rapidly, since we have p 2 > p x everywhere in the region x > a x . Hence the
integrand decreases.
The displaced path of integration does not pass through the points * = a lt
a 2 on the real axis, near which the quasi-classical approximation is inapplic-
able. Hence we can use for i]i x and ip 2 + , over the whole path, the functions
which are their asymptotic expressions in the upper half-plane. These are
«Ai = expf- [ ^{ImCU-Ei)} &x\
2[2m(?7-£' 1 )]i/4 F UJ J
(51.4)
&j + =- — expf — [ ^{2m(U-E 2 )}dx\
where the roots are taken so as to be positive on the real axis for x < a 2 .
In the integral
— ZCjC2
/ia + = , ,L ? f expl"- fv{2m(C/-£' 1 )} dx— (v{2 m (U-E 2 )} dxl x
(51.5)
4V(2m)
f(x) dx
[(U-Ei)(U-E)]V*
we desire to displace the path of integration in such a way that the exponential
factor is diminished as much as possible. The exponent has an extreme value
only where U(x) = co (for E ± # E 2 , its derivative with respect to x vanishes
at no other point). Hence the displacement of the contour of integration into
the upper half-plane is restricted only by the necessity of passing round the
singular points of the function U(x) ; according to the general theory of linear
differential equations, these coincide with the singular points of the wave
function ifi(x). The actual choice of the contour depends on the actual form
of the field U(x). Thus, if the function U(x) has only one singular point
x = x in the upper half-plane, the integration can be effected along the type
of path shown in Fig. 18. The immediate neighbourhood of the singular
point plays the important part in the integral, so that the matrix element
/12 = 2 re/i2 + required is practically proportional to an exponentially small
expression of the form
/12 - expj- -imf" J ^[2m(E 2 - U)] dx- f V[2m(£i- U)] dal! (51.6)
(L. Landau 1932).f
t In deriving formulae (51.5) and (51.6), we have replaced the wave functions by their asymptotic
expressions, since, in the integral taken along the contour shown in Fig. 18, the order of magnitude
of the integral is determined by that of the integrand; hence a relatively small change in the latter
does not have any great effect on the value of the integral.
180
The Quasi-Classical Case
§51
Fig. 18
The lower limits of the integrals may be any points in the classically
accessible regions ; their particular values evidently do not affect the imaginary
parts of the integrals. If the function U(x) has several singular points in the
upper half-plane, xo in (51.6) must be taken as that for which the exponent is
smallest in absolute value.f
The quasi-classical matrix elements for motion in a centrally symmetric
field must be calculated by the same method. However, we must now replace
U(r) by the effective potential energy (the sum of the potential energy and
the centrifugal energy), which will be different for states with different /.
In view of further applications of the method in question, we shall write the
effective potential energies in the two states in a general form, as Ui(r) and
Uzir). Then the exponent in the exponential factor in the integrand in (51.5)
has an extreme value not only at the points where U\{r) or U2(r) becomes
infinite, but also at those where
U^-U^r) = E 2 -E v
(51.7)
Hence, in the formula
r r
fe~ exd--im[ ( ^[2m(E 2 -U 2 )]dr- JV[2m(£i- U{)] drJJ (51.8)
the possible values of r include not only the singular points of U x {r) and
C/ 2 ( r )» but also the roots of equation (51.7).
The centrally symmetric case differs also in that the integration over r in
(51.1) is taken from (and not from -co) to oo:
/12 = J Xifx* dr.
o
Here two cases must be distinguished. If the integrand is an even function
t We assume that the quantity f(x) itself has no singular points.
§52 The transition probability in the quasi-classical case 181
of r, the integration can be formally extended to the whole range from -oo
to oo, so that there is no difference from the previous case. This may occur
if U x {r) and U 2 (r) are even functions of r [U(-r) = U(r)]. Then the wave
functions Xl (r) and Xi (r) are either even or odd functionsf (see §21), and,
if the function /(r) is also even or odd, the product x ±fx% may be even.
If, on the other hand, the integrand is not even (as always happens if U(r)
is not even), the start of the path of integration cannot be moved away from
the point r = 0, and this point must be included among the possible values of
r in (51.8).
PROBLEMS
field*?/ ^Ue-a? 10 " 1 ** thC q uasi - classica l matrix elements (exponential factor only) in a
• ^ L YP°£; U(X) becomes infinite °nly for x -> -oo. Accordingly, we put * oo
in (51.6). We can extend the integration to + oo.
Each of the integrals diverges at the lower limit. Hence we first calculate them from -x
to oo, and then pass to the limit x -> oo. We find
/l2 r ^ / e^mfaMva-Vi)
where vi = V(2fii/«), c* = V(2E 2 lm) are the velocities of the particle at infinity (* -> oo)
where the motion is free. "
Problem 2 The same as Problem 1, but in a Coulomb field U = et/r, for transitions be-
tween states with / = 0.
Solution The only singular point of the function U(r) is r = 0. The corresponding
integral has been calculated in §50, Problem 2. As a result we have by formula (51.8)
f 12 ~ exp — ( —
Lh\v 2 vjj
§52. The transition probability in the quasi-classical case
Penetration through a potential barrier is an example of a process which
is entirely impossible in classical mechanics. Another example is "reflection
above the barrier" of a particle whose energy exceeds the height of the barrier.
In the quasi-classical case the probability of such processes is exponentially
small. The relevant exponent can be determined as follows.
Considering a transition of any system from one state to another, we
solve the corresponding classical equations of motion and find the "path" of
the transition; this, however, is complex, in accordance with the fact that the
process cannot occur in classical mechanics. In particular, it is found that
in general, the "transition point" q at which the formal transition of the
system from one state to the other occurs is complex; the position of this
point is determined by the classical conservation laws. We next calculate
the action Si(?i, q ) + S 2 (q , ? 2 ) for the motion of the system in the first
state from some initial position qi to the "transition point" q , and then in the
ftl to bS^fa ^rTSLS??^ iS CVen ( ° r ° dd) when ' is even (or odd >' - is seen
w ~
182 The Quasi-Classical Case §52
second state from qo to the final position q<i. The required probability of the
process is then given by the formulaf
exp im [Si (qi,q ) + S 2 (qo,qz)] • (52.1)
If the position of the "transition point" is not unique, it must be chosen
so that the exponent in (52.1) has the smallest absolute value (which must yet,
of course, be sufficiently large for formula (52.1) to be valid). $
Formula (52.1) is in accordance with the rule derived in §51 for calculating
the quasi-classical matrix elements. It should be emphasised, however, that
it would not be correct to use the square of the matrix element in calculating
the coefficient before the exponential in the probability of such transitions.
If formula (52.1) is applied to "reflection above the barrier" of a particle
(in the one-dimensional case), qo must be taken as the complex co-ordinate
xo of the "turning point" at which the particle reverses its direction of
motion, i.e. the complex root of the equation U(x) = E. We shall show how
the reflection coefficient can then be calculated more precisely, including
the coefficient of the exponential.
We must again (as in §50) establish the relation between the wave functions
far to the right of the barrier (the transmitted wave) and far to the left (the
incident and reflected waves). This is easily done by a method similar to
that used in §47, regarding as a function of the complex variable x.
We write the transmitted wave in the form
X
Xi
where x± is any point on the real axis, and follow its variation on passing along
a path C in the upper half -plane which encloses (at a sufficient distance) the
turning point x (Fig. 19) ; the whole of the latter part of this path must
lie so far to the left that the error in the approximate (quasi-classical) wave
function of the incident wave is less than the required small quantity iff-.
Passage round the point x causes a change in the sign of the root \/[E- C/(^)],
and after the return to the real axis the function if/+ therefore becomes
«/r_, a wave propagated to the left (i.e. the reflected wave). || Since the ampli-
tudes of the incident and transmitted waves may be regarded as equal, the
t If the transition point is real but lies in the classically inaccessible region, formula (52.1) corres-
ponds (in the simple case of one-dimensional motion) to formula (50.9) for the probability of penetra-
tion through the potential barrier. An example of the application of formula (52.1) to a problem with
several degrees of freedom (the stripping of a deuteron in the field of a nucleus) is given by E. M.
Lifshitz, Zhurnal eksperimental'noi i teoreticheskoi fiziki 8, 930, 1938.
% If the potential energy of the system has itself singular points, these also must be considered as
possible values of go.
|| A passage along a path below the point *o (simply going along the real axis, for example) converts
the function tp+ into the incident wave.
§52 The transition probability in the quasi-classical case 183
Fig. 19
required reflection coefficient R is simply the ratio of the squared moduli
of «/r_ and ift+ :
R =
= expf im pdxj. (52.2)
Having derived this formula, we can deform the path of integration in the
exponent in any manner; if we convert it into the path C shown in Fig. 19,
the integral jp dx reduces to twice the integral from x± to x , givingf
*0
R = expf im p dxf . (52.3)
Xi
Since p is real everywhere on the real axis, the choice of x\ does not affect
the imaginary part of the integral in the exponent.
As already mentioned, among the possible values of xo we must select the
one for which the exponent in (52.3) is smallest in absolute magnitude (and
this value must be large compared with unity).J It is also implied that, if
the potential energy U(x) itself has singularities in the upper half-plane,
the integral
im- I
hj
p dx
has larger values for such points; otherwise the exponent would be deter-
mined by one of these points, but the coefficient of the exponential would not
be unity as in (52.3). This condition is certainly not satisfied with increasing
t This formula with the coefficient of the exponential (equal to unity) was first obtained by V. L.
Pokrovskh, S. K. Sawinykh and F. R. Ulinich (1958).
t Of course, only points *o are considered for which
x
im I p dx > 0,
i.e. points lying in the upper half-plane.
184 The Quasi-Classical Case §52
energy E if U{x) becomes infinite anywhere in the upper half-plane: ulti-
mately the point xq at which U = E becomes so close to the point #«, where
U = oo that the two points give comparable contributions to the reflection
coefficient (the integral
im- \ p dx ~ 1),
and formula (52.3) becomes invalid. In the limit where E is so large that this
integral is small compared with unity, perturbation theory becomes applicable
(see Problem l).f
PROBLEMS
Problem 1. Determine the coefficient of reflection above the barrier for particle energies
such that perturbation theory is applicable.
Solution. Formula (43.1) is used, the initial and final wave functions being plane waves
propagated in opposite directions and normalised respectively by unit current density and the
delta function of momentum, with dv = dp' and p' the momentum after reflection. Carrying
out the integration with respect to p' (taking account of the delta function), we obtain
m? I f
Wp*\ J
(1)
—00
This formula is valid if the conditions for perturbation theory to be applicable are satisfied :
Ua/Hv <^ 1, where a is the width of the barrier (see the third footnote to §45), and also
pa/H £ 1. The latter condition ensures that the function R(p) is not exponential; otherwise
the question of the validity of formula (1) would require further investigation.
Problem 2. Determine the coefficient of reflection above the barrier for a quasi-classical
barrier when the function U(x) has a discontinuity of slope.
Solution. If the function U(x) has a singularity for real x, the reflection coefficient is
determined mainly by the field near that point, and perturbation theory can be formally
applied to calculate it, without having to be valid for all x ; the fulfilment of the quasi-classical
condition is sufficient. We then have formula (1) of Problem 1 , the only difference being that
the momentum of the incident particle must be replaced by the value ofp(x) at the singular
point.
In this case we take the point of discontinuous slope as * = 0, and thus have near this point
U = — F\x for * > 0, U = —F 2 x for x < 0,
with different Fi and F%. The integration with respect to x is effected by including in the
integrand a damping factor e ±Xx and then letting A -> 0. The result is
m 2 H 2
where po = p(0).
t An intermediate case is discussed by V. L. Pokrovskii and I. M. Khalatnikov, Soviet Physics
JETP 13, 1207, 1961.
§53 Transitions under the action of adiabatic perturbations 185
§53. Transitions under the action of adiabatic perturbations
It has already been mentioned in §41 that, in the limit of a perturbation
which varies arbitrarily slowly with time, the probability of a transition of
a system from one state to another tends to zero. Let us now consider this
problem quantitatively, by calculating the transition probability under the
action of a slowly varying (adiabatic) perturbation.
Let the Hamiltonian of the system be a slowly varying function of time,
tending to definite limits as t -> ± oo, and let ifi n (q, t) and E n {t) be the eigen-
f unctions and the eigenvalues of the energy (depending on time as a para-
meter) obtained by solving Schrodinger's equation i?(*)0» = E n ty n ; on
account of the adiabatic variation of i? with time, the time variation of E n
and ift n with time will also be slow. The problem is to determine the proba-
bility «?i2 of finding the system in a certain state 02 as t -> + oo, if it was in
the state 0i as t -> — oo.
The slow variation of the perturbation means that the duration of the
"transition process" is very long, and therefore the change in the action during
this time (given by the integral — J E(t) dt) is large. In this sense the problem
is quasi-classical, and the required probability is mainly determined by the
values t o of t for which
Ei(t ) = E 2 (t ) (53.1)
and which correspond, as it were, to the "instant of transition" in classical
mechanics (cf. §52); in reality, of course, such a transition is classically
impossible, as is shown by the fact that the roots of equation (53.1) are
complex. It is therefore necessary to examine the properties of the solutions
of Schrodinger's equation for complex values of the parameter t in the
neighbourhood of the point t = t o at which the two eigenvalues of the energy
become equal.
As we shall see, the eigenfunctions 0i, 02 vary rapidly with t near this
point. To determine this dependence, we first define linear combinations
01, 02 of 0i, 02 which satisfy the conditions
j fa* dq = j 02 2 dq = 0, J* fafa dq = 1. (53.2)
This can always be achieved by suitable choice of the complex coefficients
(which are functions of t). The functions 0i, 02 have no singularity at t = to.
We now seek the eigenfunctions as linear combinations
= «i0i+a 2 02. (53.3)
Here it must borne in mind that, when the "time" t is complex, the operator
fi(t) (of the form (17.4)) is still equal to its transpose (i? = i?), but is no
longer Hermitian (i? # i?*), since the potential energy U(t) ^ U*(t).
We substitute (53.3) in Schrodinger's equation, multiply on the left by
186 The Quasi-Classical Case §53
fa or fa, and integrate with respect to q. With the notation
H tk (t) = jfififadq, (53.4)
and using the fact that H±2 = H21 owing to the above-mentioned property
of the Hamiltonian, we obtain the equations
H n a 1 +H 12 a 2 = Ea 2 ,
(53.5)
Hi 2 a 1 + H 22 a 2 = Ea-y.
The condition for these equations to have non-zero solutions is (#12 — E) 2 =
H11H22, and the roots of this give the energy eigenvalues
E = H 12 ±V(HiiH 22 ). (53.6)
Then (53.5) gives
a 2 / ai = ±V(HnlH 22 ). (53.7)
It is seen from (53.6) that, for a coincidence at the point t = Jo of the two
eigenvalues, either Hn or H22 must vanish at that point ; let Hn vanish there.
At a regular point, a function in general vanishes as t — to, and therefore
E(t) - E(t ) = ± constant x y/(t- 1 ), (53.8)
i.e. E(t) has a branch point at t = to. We also have #2 ~ V(* — *o)> an< ^ so
there is at the point t = to only one eigenfunction, fa.
We now see that the problem is formally completely analogous to the
problem of reflection above the barrier discussed in §52. We have a wave
function *F(£) which is "quasi-classical" with respect to time, instead of the
function quasi-classical with respect to the co-ordinate in §52, and wish to
find the term of the form C2fae~ iE £ ln in the wave function for t -> + 00,
if the wave function \F(f) = fae~ iE t ln as t -> — 00. This is analogous to the
problem of determining the reflected wave for x -> — 00 from the transmitted
wave for x -> + 00. The required transition probability w±2 = |c2| 2 - The
action S = — J E(t) dt is given by the time integral of a function having
complex branch points (just as the function p(x) in the integral j p dx had
complex branch points). The problem under consideration is therefore
dealt with by means of a contour in the plane of the complex variable t
from large negative to large positive values, just as in §52 for the plane of the
variable x, and we shall not repeat the derivation here.
We shall suppose that E2 > E\ on the real axis. Then the contour must lie
in the upper half-plane of the complex variable t (where the ratio
g-iEfihje-iEjih increases). The resulting formula (analogous to (52.2)) is
«12 = exp ( - - im J E(t) dt\ , (53.9)
c
where the integration is along the contour shown in Fig. 19 (from left to
right).
§53 Transitions under the action of adiabatic perturbations 187
On the left-hand branch of this contour E = Ei, and on the right-hand
branch E = E 2 . We can therefore write (53.9) in the form
to
wi2 = exp ( - 2 im j a>2i(t) dtj , (53.10)
where o> 2 i = {E^-E^fh, and t± is any point on the real axis of t\ t must be
taken as that root of equation (53.1) lying in the upper half-plane for which
the exponent in (53.10) is smallest in absolute value.f In addition, besides
the direct transition from state 1 to state 2, there may be possible paths
through various intermediate states; the probabilities of these are given by
analogous formulae. For example, for a transition 1 -> 3 -> 2 the integral
in (53.10) is replaced by a sum of integrals:
f ( 31 ) t (23)
J <*Ki(t)dt+ J oi2s,{t)dt y
where the upper limits are the "points of intersection" of the terms #i(*),
E 3 (t) and E 3 (t), E 2 (t) respectively. This result is obtained by means of a
contour which encloses both these complex points. J
t The possible values of t must include points at which E(t) becomes infinite; for such points the
coefficient of the exponential in (53.9) will not be unity.
t The intermediate states of a continuous spectrum require a special discussion.
CHAPTER VIII
SPIN
§54. Spin
Let us consider a system, such as an atomic nucleus, which executes some
motion as a whole. We shall suppose that the internal energy of the nucleus
has a definite value. The internal state of the nucleus, however, is in general
not completely determined by this value; for the "internal" angular mo-
mentum L of the nucleus (i.e. the angular momentum of the particles in
their motion within the nucleus) may still have various directions in space.
The number of different possible orientations of this angular momentum
is, as we know, 2L+1. Thus, in considering the motion of the nucleus (in
a given internal state) as a whole, we must examine, as well as its co-ordinates,
another discrete variable: the projection of its internal angular momentum
on some chosen direction in space.
Consequently, we see that the formalism of quantum mechanics allows us,
in considering the motion of any particle, to introduce, besides its co-ordinates,
another variable quantity specific to any given particle, which can take a
limited number of discrete values. We have no reason to suppose, a priori,
that this variable is absent when the particle is elementary. In other words,
we must in general suppose that, in quantum mechanics, some "intrinsic"
angular momentum must be ascribed to an elementary particle, regardless of
its motion in space. This property of elementary particles is peculiar to
quantum theory (it disappears in the limit h -> ; see the second footnote to
this section), and hence is essentially incapable of a classical interpretation.
In particular, it would be wholly meaningless to imagine the "intrinsic"
angular momentum of an elementary particle as being the result of its rotation
about "its own axis".
The intrinsic angular momentum of a particle is called its spin, as distinct
from the angular momentum due to the motion of the particle in space,
called the orbital angular momentum.^ The particle concerned may be either
elementary, or composite but behaving in some respect as an elementary
particle (e.g. an atomic nucleus). The spin of a particle (measured, like the
orbital angular momentum, in units of h) will be denoted by s.
In the preceding chapters we have always supposed that the three co-
ordinates of a particle form a complete set of quantities, so that, if they are
given, its state is completely determined. We now see that this is in general
t The physical idea that an electron has an intrinsic angular momentum was put forward by
G. Uhlenbeck and S. Goudsmit in 1925. Spin was introduced into quantum mechanics in 1927 by
W. Pauli.
188
§54 Spin 189
not true : for a complete description of the state of a particle, not only its co-
ordinates, but also the direction of the spin vector, must be specified. Hence
the wave function of a particle must be a function of four variables : the three
co-ordinates, and the spin variable which gives the value of the projection of
the spin on a selected direction in space, and takes a limited number of
discrete values. These values can be added as a suffix to the wave functions.
Thus the wave function of a particle which has a non-zero spin is in fact not
one function, but a set of several different functions of the co-ordinates,
differing in their spin suffixes.
The quantum-mechanical operator corresponding to the spin of the particle,
on being applied to the wave function, acts on the spin variable. In other
words, it in some way linearly transforms the functions differing only in
the spin suffix into one another. The form of this operator will be established
later. However, it is easy to see from very general considerations that the
operators § x , i y , s z satisfy the same commutation conditions as the operators
of the orbital angular momentum.
The angular momentum operator is essentially the same as that of an
infinitely small rotation. In deriving, in §26, the expression for the orbital
angular momentum operator, we considered the result of applying the rotation
operator to a function of the co-ordinates. In the case of the spin, this
derivation becomes invalid, since the spin operator acts on the spin variable,
and not on the co-ordinates. Hence, to obtain the required commutation
relations, we must consider the operation of an infinitely small rotation in a
general form, as a rotation of the system of co-ordinates. If we successively
perform infinitely small rotations about the *-axis and the jy-axis, and then
about the same axes in the reverse order, it is easy to see by direct calculation
that the difference between the results of these two operations is equivalent
to an infinitely small rotation about the #-axis (through an angle equal to the
product of the angles of rotation about the x and j-axes). We shall not pause
here to carry out these simple calculations, as a result of which we again
obtain the usual commutation relations between the operators of the com-
ponents of angular momentum ; these must therefore hold for the spin oper-
ators also:
{$ v J e }=is x , {s z> s x }=is y , {f x ,$ y }=i$z, (54.1)
together with all the physical consequences resulting from them.
The commutation relations (54.1) enable us to determine the possible
values of the absolute magnitude and components of the spin. All the results
derived in §27 (formulae (27.7)-(27.9)) were based only on the commutation
relations, and hence are fully applicable here also; we need only replace L
in these formulae by s. It follows from formula (27.7) that the eigen-
values of the sr-component of the spin form a sequence of numbers differing
by unity. However, we cannot now assert that these values must be integral,
as we could for the component L z of the orbital angular momentum (the
derivation given at the beginning of §27 is invalid here, since it was based
190 Spin §54
on the expression (26.14) for the operator l z , which holds only for the orbital
angular momentum).
Moreover, we find that the sequence of eigenvalues s z is limited above and
below by values equal in absolute magnitude and opposite in sign, which we
denote by ±s. The difference 2s between the greatest and least values of s e
must be an integer or zero. Consequently s can take the values 0, |, 1, f , ... .
Thus the eigenvalues of the square of the spin are
s 2 = s{s+\), (54.2)
where s can be either an integer (including zero) or half an integer. For given
s, the component s g of the spin can take the values s, s— 1, ... , — s, i.e. 2s+ 1
values in all. From what was said above, we conclude that the state of a
particle whose spin is s must be described by a wave function which is a set
of 2s + 1 functions of the co-ordinates, f
Experiment shows that the majority of the elementary particles (electrons,
positrons, protons, neutrons, jit-mesons and all hyperons (A, 2, H)) have a spin
of |. There are also elementary particles, the 7r-mesons and the .K-mesons,
whose spin is zero.
The total angular momentum of a particle is composed of its orbital angular
momentum 1 and its spin s. Their operators act on functions of different
variables, and therefore, of course, commute. The eigenvalues of the total
angular momentum
j = 1+s (54.3)
are determined by the same "vector model" rule as the sum of the orbital
angular momenta of two different particles (§31). That is, for given
values of / and s, the total angular momentum can take the values l+s,
l+s— 1, ... , \l— s\. Thus, for an electron (spin |) with non-zero orbital angular
momentum /, the total angular momentum can be j = /±|; for / = the
angular momentum j has, of course, only the one value/ = \.
The operator of the total angular momentum J of a system of particles is
equal to the sum of the operators of the angular momentum j of each particle,
so that its values are again determined by the vector model rules. The angular
momentum J can be put in the form
J = L+ S, L = 2 l a , S = 2 s a , (54.4)
a
where S may be called the total spin and L the total orbital angular momentum
of the system. We notice that, if the total spin of the system is half-integral
(or integral), the same is true of the total angular momentum, since the orbital
angular momentum is always integral. In particular, if the system consists
of an even number of similar particles, its total spin is always integral, and
therefore so is the total angular momentum.
f Since s is fixed for each kind of particle, the spin angular momentum Hs becomes zer o in the limit
of classical mechanics (H-> 0). This consideration does not apply to the orbital angular momentum,
since I can take any value. The transition to classical mechanics is represented by H ten ding to zero
and / simultaneously tending to infinity, in such a way that the product hi remains finite .
§55 Spinors 191
The operators of the total angular momentum j of a particle (or J, of a
system of particles) satisfy the same commutation rules as the operators of
the orbital angular momentum or the spin, since these rules are general com-
mutation rules holding for any angular momentum. The formulae (27.13)
for the matrix elements of angular momentum, which follow from the com-
mutation rules, are also valid for any angular momentum, provided that the
matrix elements are defined with respect to the eigenstates of this angular
momentum. Formulae (29.7) — (29.10) for the matrix elements of arbitrary
vector quantities also remain valid (with appropriate change of notation).
PROBLEM
A particle with spin J is in a state with a definite value s z = %. Determine the probabilities
of the possible values of the component of the spin along an axis z' at an angle 9 to the s'-axis.
Solution. The mean spin vector s is evidently along the ar-axis and has magnitude J.
Taking the component along the s'-axis, we find that the mean value of the spin in that
direction is sT' = i cos 0. We also have s7' = i(w+— w~), where io± are the probabilities
of the values Sz- = ±£. Since w++w- — 1, we find w+ = cos 2 J0, w- = sin 2 £0.
§55. Spinors
Let if/(x, y, z; a) be the wave function of a particle with spin a; a denotes
the ^-component of the spin, and takes values from — s to +s. We shall call
the functions ip(a) with various values of a the "components" of the wave
function. We impose on the choice of these "components" the condition
that the integral J |«/r(a) | 2 dV determines the probability that the ^-component
of the spin of the particle is equal to a. The probability that the particle is in
s
an element of volume dV in space is dV 21 |j^(cr) | 2 . If the particle is in a state
cr = -s
with a definite a-value o- , only the component ifj(o) with a = a is not zero,
i.e. the wave function is of the form
il*(x,y,z; a) = i/^.jy,*^.
In this chapter we shall not be interested in the dependence of the wave func-
tion on the co-ordinates. For example, in speaking of the behaviour of the
function ifi(a) when the system of co-ordinates is rotated, we can suppose that
the particle is at the origin, so that its co-ordinates remain unchanged by
such a rotation, and the results obtained will characterise the behaviour of the
function ^r(o-) with regard to the spin variable a.
Let us effect an infinitely small rotation through an angle 8(f> about the
sr-axis. The operator of such a rotation can be expressed in terms of the
angular momentum operator (in this case the spin operator), in the form
1 +i8(f> . s s . Hence, as a result of the rotation, the functions ^r(a) become
ifj(cr) + Sifj(<r), where 8ip(cr) =i8<f> . s z t[i(a). But f s ^(a) = o^r(cr), so that
8if/(a) = iaift(a)8(f>. By a rotation through a finite angle <f> the functions ift{a)
are therefore transformed into
■f(a) =«^(<t). (55.1)
192 Spin §55
In particular, by a rotation through an angle 2tt, they are multiplied by a
factor e 27ria , which is the same for all a and is ( - l) 2s (2a is always of the same
parity as 2s). Thus we see that, when the system of co-ordinates is com-
pletely rotated about an axis, the wave functions of a particle of integral spin
return to their original values, while those of a particle of half-integral spin
change sign.
The variable a differs from the ordinary variables (the co-ordinates) by
being discrete. The most general form of a linear operator acting on func-
tions of a discrete variable a is
(/fl(cr) = S/^aO, (55.2)
where the f^ are constants. The parentheses round fift emphasise that the
spin argument a which follows does not relate to the original function «/» but
to that which results from it when the operator / is applied. It is easy to see that
the quantities/^' are the same as the matrix elements of the operator/ defined
in the usual manner. For the "eigenfunction" of the operator s z corresponding
to the value s z = ct is ip(a) = S^. For this function we have
Ao* = g/aA'a,, =/a V
The right-hand side of this equation can be rewritten in the form ]}fa f a, 8 aa'
and then
Aw. = S/a'aA*" (55.3)
This equation, however, agrees with the usual definition of the matrix of the
operator / with respect to the eigenfunctions of the operator s z .
Thus the operators acting on functions of a can be represented in the form
of (2*+l)-rowed matrices. In particular, we have for the operators of the
spin components themselves
(4Mff)= 2(^0(a')» (55.4)
and similarlyf for s y > K- According to what has been said above, the matrices
s x , s y , s g are identical with the matrices L x , L y , L z obtained in §27, where the
letters L and M need only be replaced by s and a. Thus the non- vanishing
matrix elements of the spin operators are
i s x) a ,a-l = (**)<r-l,a = i\/[(*+ °)(s- (7+ 1)], \
( s v)o,o-i = -(s y )<r-i, a = -¥V[(s+o)(s-a+l)] t (55.5)
In the important case of a spin of \{s = £, a = ± £), these matrices have
t Attention is drawn to the fact that the sequence of suffixes in the matrix elements on the right-hand
side of equation (55.4) is the reverse of the usual sequence (in (55.3)).
§55 Spinors 193
two rows, and are of the form
<*•>-€ 3- w "€ "J- w=i C -D- (556)
These are called (without the factor ^) Pauli matrices. We may also give the
matrices of the complex combinations s ± = s x ±is y :
rO In r0 °T
(s+) =[o J w =[i 0} (557)
By direct multiplication of the Pauli matrices, it is easy to verify that the
relations
2s J z = is x , 2s J x = is v , 2s J v = iS t (55.8)
hold. Combining these with the usual commutation rules (54.1) we find that
sJv+sJ x =0, 4vK4=0, $J g +§J y =Q, (55.9)
i.e. the Pauli matrices anticommute with one another.
By means of these relations, we can easily verify the following useful formulae :
§ 2 = |, (55.10)
(s.a)(s.b) = i(a.b)+ii§.(axb), (55.11)
where a and b are any vectors.f
It may be noted that any expression quadratic in the components of s thus
reduces to terms independent of s and terms linear in s. Hence it follows that
any function of the operator s (of spin \) reduces to a linear function (see
Problem 1).
Let us consider more closely the "spin" properties of wave functions.
When the spin is zero, the wave function has only one component, ^(0). When
the spin operators act upon it, the result is zero :
S X lp = Sylfj = S z *ji = 0.
Since the spin operators are related to the rotation operators, this means that
the wave function of a particle with spin zero is invariant under rotation of
the co-ordinate system, i.e. it is a scalar.
The wave functions of particles with spin \ have two components, i}j{\) and
^( — £). For convenience in later generalisations, we shall call these compo-
nents i/j 1 and i/j 2 respectively (with upper indices 1 and 2). In any rotation of
the co-ordinate system, tp 1 and ip 2 undergo a linear transformation :
0v = a^+j^ 2 , V 2 ' = rl> x +W- (55.12)
t The terms on the right which are independent of s must, of course, be understood as constants
multiplying the unit two-by-two matrix.
194 Spin §55
The coefhcientsf a, jS, y, 8 are in general complex and functions of the angles
of rotation. They are connected by a relation which we derive by considering
the bilinear form
0^-0^1, (55.13)
where (ifi 1 , xjj % ) and (<£\ cf> 2 ) are two wave functions transformed according to
(55.12). A simple calculation gives
0i'02'-02'0i' = ( a 8- i 8y)(^ 2 -^i) )
i.e. the quantity (55.13) is transformed into itself when the co-ordinate system
is rotated. If, however, there is only one function which is transformed into
itself, it can be regarded as corresponding to zero spin, and therefore must be
a scalar, i.e. must remain unchanged when the co-ordinate system is rotated
in any manner. Hence we have
aS-j8y = 1. (55.14)
This is the required relation.
The linear transformations (55.12) which leave the bilinear form (55.13)
invariant are called binary transformations. A quantity having two compo-
nents which undergoes a binary transformation when the co-ordinate system
is rotated is called a spinor. Thus the wave function of a particle with spin \
is a spinor.
It is possible to put the algebra of spinors in a form analogous to that
of tensor algebra. This is done by introducing a vector space of two dimen-
sions, in which the metric is defined by an antisymmetrical "metric tensor":
The vectors in this space are spinors. Besides the contravariant components
0\ ^ 2 of the spinor, we may introduce the covariant components in accordance
with the usual formulae of tensor algebra :
fa = 5«v^,
so that
fa = <A 2 , fa = -V- (55.16)
The binary transformations for the covariant components of a spinor are
obviously of the form
fa' = Zfa-yfa, fa' = -j3<Ai+a«A 2 . (55.17)
t Called the Cayley-Klein parameters.
§55 Spinors 195
The converse transformation from covariant to contravariant components
can be written in the form
«A A = S^ty M , (55.18)
where the contravariant "metric tensor" g^ has the components
which are the same as the components g^.
The invariant combination (55.13) can be written as a "scalar product"
+Hx = Wi+tfVi = W-0V = 5VA A ^; (55.20)
here, and in what follows, summation is implied over repeated {dummy)
indices, as in tensor algebra. We may note the following rule which has to
be borne in mind in spinor algebra. We have
tf Va = Wi+tfVi = -M 2 -^ 1 = -M A -
Thus
*Va = ~^ A - (55.21)
Hence it is evident that the scalar product of any spinor with itself is zero :
^ = 0. (55.22)
The expression
|^l|2 + |02|2 = ^l*^^
which gives the probability of finding the particle at a given point in space,
must clearly be a scalar. Comparing it with the scalar (55.20), we see that
the components j/t 1 *, 2 * of the wave function which is the complex conjugate
of j/t 1 , ifs 2 are transformed as covariant components of a spinor, i.e. as ip 2 ,—^ 1
respectively :
0i*' = 80 1 *— yifi 2 *, if> 2 *' = — j80 1# +a^ 2 *.
On the other hand, by taking the complex conjugate equations to (55.12)
0i*' = a*^ 1 *-^*^ 2 *, 2 *' = y*0!*+8*0 2 *
and comparing them with the above, we find that the coefficients a, jS, y, S
are related also by
a =5*, j8 = -y*. (55.23)
By virtue of the relations (55.14), (55.23), the four complex quantities
a, jS, y, S actually contain only three independent real parameters, correspond-
ing to the three angles which define a rotation of a three-dimensional system
of co-ordinates.
196 Spin §56
The fact that 1 *, iff 2 * are transformed as if/ 2 , —if* 1 is closely related to the
symmetry with respect to a change in the sign of the time. As was remarked
in §18, in quantum mechanics a change in the sign of the time corresponds
to a replacement of the wave function by its complex conjugate. When the
sign of the time is changed, however, so is that of the angular momentum.
Hence the functions which are the complex conjugates of the components
1 , ifj 2 corresponding to projections of the spin a = \ and a = —\ must be
equivalent in their properties to the components corresponding respectively
to projections of the spin a = — \ and a = \.
PROBLEMS
Problem 1. Reduce an arbitrary function of the scalar a+2b . s (where s is the operator
of spin i) to another linear function of s.
Solution. To determine the coefficients in the required formula f(a + 2b . s) =
oc+2/?b . s/b, we note that, when the sr-axis is taken in the direction of b, the eigenvalues of
the operator a + 2b . s are a ± b, and the corresponding eigenvalues of the operator f(a +2b . s)
are/(a±6). Hence we find a = h[f(a + b)+f(a-b)], j3 = £[/( a +&)_/( a _£)].
Problem 2. Determine the values of the scalar product si . S2 of spins (£) of two particles
in states in which the total spin of the system, S = Si +S2, has definite values (0 or 1).
Solution. From the general formula (31.2), which is valid for the addition of any two
angular momenta, we find si . S2 = i for S = 1, Si . S2 = —J for S = 0.
Problem 3. Which powers of the operator s of an arbitrary spin s are independent?
Solution. The operator
(S z -s)(s z -s+l) ... (s z + s),
formed from the differences between S z and all possible eigenvalues s z , gives zero when it is
applied to any wave^ function, and is therefore itself zero. Hence it follows that (S z ) 28+1 is
expressed in terms of lower powers of the operator S z , so that only its powers from 1 to 2s are
independent.
§56. Spinors of higher rank
Analogously to the transition from vectors to tensors in ordinary tensor
algebra, we can introduce the idea of spinors of higher rank. Thus, a quantity
ip^ t having four components which are transformed as the products j/t\^*
of the components of two spinors of rank one, is called a spinor of rank two.
Besides the contravariant components «/rV we can consider the covariant
components j/t^ and the mixed components 0^ which are transformed as
the products ifix<f>n and ipx^ respectively. The transition from one set of
components to another is effected by means of a "metric tensor" gx „, in
accordance with the usual formulae
Thus ip 12 = —ip! 1 = —iff 21 , 0ii = *Pi 2 = ip 22 , and so on. Spinors of any
rank are similarly defined. The quantities g^ themselves form an anti-
symmetrical spinor of rank two. It is easy to see that the values of its com-
ponents remain unchanged under binary transformations.
§56 Spinors of higher rank 197
It is easily verified that the product g\ v g^ v is, as it should be, a unit spinor
of rank two, i.e. a spinor with components 8\ = S| = 1, Sf = S2 = 0. Thus
ibF* = V- (56.1)
As in ordinary tensor algebra, there are two fundamental operations in
spinor algebra: multiplication, and contraction with respect to a pair of in-
dices. The multiplication of two spinors gives a spinor of higher rank ; thus,
from two spinors of ranks two and three, iff^ and ip v P a , we can form a spinor
of rank five, i/fXfi l P vpa ' Contraction with respect to a pair of indices (i.e. sum-
mation of the components over corresponding values of one covariant and
one contravariant index) decreases the rank of a spinor by two. Thus, a
contraction of the spinor ^x^ 9 " with respect to the indices ju and v gives the
spinor ^A^ po °f rank three; the contraction of the spinor ^y* gives the scalar
tp\*. Here there is a rule similar to that expressed by formula (55.21): if we
interchange the upper and lower indices with respect to which the contraction
is effected, the sign is changed (i.e. ^ A = — if**\). Hence, in particular, it
follows that, if a spinor is symmetrical with respect to any two of its indices,
the result of a contraction with respect to these indices is zero. Thus, for a
symmetrical spinor ijj X/l of rank two, we have ift\^ = 0.
A spinor of rank n symmetrical with respect to all its indices is called a
symmetrical spinor of rank n. From an asymmetrical spinor we can construct a
symmetrical one by the process of symmetrisation, i.e. summation of the compo-
nents obtained by all possible interchanges of the indices. From what has
been said above, it is impossible to construct (by contraction) a spinor of lower
rank from the components of a symmetrical spinor.
Only a spinor of rank two can be antisymmetrical with respect to all its
indices. For, since each index can take only two values, at least two out of
three or more indices must have the same value, and therefore the compo-
nents of the spinor are zero identically. Any antisymmetrical spinor of rank
two is a scalar multiple of the unit spinor g^. We may notice here the fol-
lowing relation :
gAv&v+g^X+g^ = (56.2)
(where ip\ is any spinor), which follows from the above; this rule is simply a
consequence of the fact that the expression on the left is (as we may easily
verify) an antisymmetrical spinor of rank three.
The spinor which is the product of a spinor j/^ with itself, on contraction
with respect to one pair of indices, becomes antisymmetrical with respect to
the other pair:
Hence, from what was said above, this spinor must be a scalar multiple of
the spinor gx^. Defining the scalar factor so that contraction with respect to
the second pair of indices gives the correct result, we find
hA v = -W*^Tsv ( 56 - 3 )
198 Spin §57
The components of the spinor ^ M * which is the complex conjugate of
f/fXfi... are transformed as the components of the contravariant spinor tp^f 1 —,
and conversely. The sum of the squared moduli of the components of any
spinor is consequently invariant.
§57. The wave functions of particles with arbitrary spin
Having developed a formal algebra for spinors of any rank, we can now
turn to our immediate problem, to study the properties of wave functions of
particles with arbitrary spin.
This subject is conveniently approached by considering an assembly of
particles with spin |. The greatest possible value of the ^-component of the
total spin is |w, which is obtained when s g = \ for every particle (i.e. all the
spins are directed the same way, along the #-axis). In this case we can
evidently say that the total spin S of the system is also \n.
All the components of the wave function i/t(a lt cr 2 , ... , a n ) of the system
of particles are then zero, except for «/r(|, \, ... , §). If we write the wave
function as a product of n spinors if/ty ... , each of which refers to one of the
particles, only the component with A, p, ... = 1 in each spinor is not zero.
Thus only the product j/r 1 ^ 1 ... is not zero. The set of all these products,
however, is a spinor of rank n which is symmetrical with respect to all its
indices. If we transform the co-ordinate system (so that the spins are not
directed along the #-axis), we obtain a spinor of rank n, general in form except
that it is symmetrical as before.
The "spin" properties of wave functions, being essentially their properties
with respect to rotations of the co-ordinate system, are evidently identical for
a particle with spin 5 and for a system of n = 2s particles each with spin \
directed so that the total spin of the system is s. Hence we conclude that
the wave function of a particle with spin s is a symmetrical spinor of rank
n = 2s.
It is easy to see that the number of independent components of a sym-
metrical spinor of rank 2s is equal to 2s +1, as it should be. For all those
components are the same whose indices include 2s ones and twos; so are
all those with 2s— 1 ones and 1 two, and so on up to ones and 2s twos.
Mathematically we can say that the symmetrical spinors provide a classifica-
tion of the possible types of transformation of quantities when the co-ordinate
system is rotated. If there are 2s +1 different quantities which are transformed
linearly into one another (and which cannot be reduced in number by any
choice of linear combinations of them), then we can assert that their law of
transformation is equivalent to that of the components of a symmetrical spinor
of rank 2s."f" Any set of any number of functions which are transformed linearly
into one another when the co-ordinate system is rotated can be reduced (by
an appropriate linear transformation) to one or more symmetrical spinors.
f In other words, the symmetrical spinors form what are called irreducible representations of the
rotation group (see §98).
§57 The wave functions of particles vdth arbitrary spin 199
Thus an arbitrary spinor ^A/xv... °f rank n can De reduced to symmetrical
spinors of ranks », n — 2, n — 4, ... . In practice, such a reduction can be made
as follows. By symmetrising the spinor j/^... with respect to all its indices,
we form a symmetrical spinor of the same rank n. Next, by contracting the
original spinor j/^v... with respect to various pairs of indices, we obtain
spinors of rank n— 2, of the form «/f A Av , which, in turn, we symmetrise, so
that symmetrical spinors of rank n—2 are obtained. By symmetrising the
spinors obtained by contracting j/^... w i tn respect to two pairs of indices,
we obtain symmetrical spinors of rank n— 4, and so on.
We have still to establish the relation between the components of a sym-
metrical spinor of rank 2s and the 2s +1 functions ^(cx), where a — s, s— 1, ... ,
—s. The component
.11. ..1 22. ..2
s+a s—cr
in whose indices 1 occurs s+a times and 2 s— a times, corresponds to a value
a of the projection of the spin on the sr-axis. For, if we again consider a system
of n = 2s particles with spin ^, instead of one particle with spin s, the product
tfjty 1 . . . x 2 p 2 - • • corresponds to the above component; this product belongs to a
s+a
state in which s+a particles have a projection of the spin equal to £, and
s — a a projection of — $, so that the total projection is ?(s + a) — %(s — a) = a.
Finally, the proportionality coefficient between the above component of
the spinor and ifj(o) is chosen so that the equation
J s |*(a)|»- jy**-|« ( 57 -!)
holds; this sum is a scalar, as it should be, since it determines the probability
of finding the particle at a given point in space. In the sum on the right-hand
side, the components with (s + a) indices 1 occur
(2*)!
(,+ ff )!(,-a)l
times. Hence it is clear that the relation between the functions ip(a) and the
components of the spinor is given by the formula
0(
^ = /r (2 * )! V 1 - 1 22 - 2 (57 2)
G) V U+a)!(*-<?)U s+ff *- a '
The relation (57.2) ensures the fulfilment not only of the condition (57.1),
but also, as we easily see, of the more general condition
«A v ~£w. = 5 (- 1) 8 -^)^- *)> ( 57 - 3 )
where ifj^'" and <f>Xa... are two different spinors of the same rank, while
200 Spin §58
iff(<j),(f>(o) are functions derived from these spinors by formula (57.2); the
factor ( — l) s -° is due to the fact that, when all the indices of the spinor com-
ponents are raised, the sign changes as many times as there are twos among
the indices.
Formulae (55.4) determine the result of the action of the spin operator
on the wave functions ip(a). It is not difficult to find how these operators act
on a wave function written in the form of a spinor of rank 2s. For a spin £,
the functions i/f(£), «/r( — |) are the same as the components ^r 1 , 2 of the spinor.
According to (55.4) and (55.6), the result of the spin operators' acting on them
will be
(40) x =# 2 , (W = -^ 2 , (W=#\
(40) 2 = W, {Syrfsf = W, (U) 2 = -W- 7A)
To pass to the general case of arbitrary spin, we again consider a system
of 2s particles with spin |, and write its wave function as a product of 2s
spinors. The spin operator of the system is the sum of the spin operators
of each particle, acting only on the corresponding spinor, the result of this
action being given by formulae (57.4). Next, returning to arbitrary symmetri-
cal spinors, i.e. to the wave functions of a particle with spin s, we obtain
, g ..11. ..22... .. N .11... 82... ... . 11 ... 22
s+a 8-a ' s+a-1 s-a+1 ' T 8+a+l s-a-X
, g ..11. .22... ... ..11... 22... . ..11... 22
8+a 8-a v ' 8+a-l s-a+1 v ' r 8 +o+l s-a-1
,a ,.H •••22 ... .11 ...22...
(**#) = # •
s+o s-a s+a 8-a
(57.5)
We notice that, by starting from these formulae and the relations (57.2),
we could derive the expressions (55.5) for the matrix elements of the spin
operator acting on the functions ift(o).
§58. The relation between spinors and tensors
Hitherto we have spoken of spinors as wave functions of the intrinsic angular
momentum of elementary particles. Formally, however, there is no difference
between the spin of a single particle and the total angular momentum of any
system regarded as a whole, neglecting its internal structure. It is therefore
evident that the transformation properties of spinors apply equally to the
behaviour, with respect to rotations in space, of the wave functions ipj m of any
particle or system of particles with total angular momentum /, independent of
whether orbital or spin angular momentum is concerned. There must therefore
be some definite relation between the laws of transformation for the eigen-
functions ^ m under rotations of the co-ordinate system and those for the
components of a symmetrical spinor of rank 2/.
§58 The relation between spinors and tensors 201
In establishing this relation we must, however, make a clear distinction
between two aspects of the dependence of the wave functions on the component
m (for a given value of j). The wave function may be regarded as the probability
amplitude for various values of m, or may be considered for a given value of m.
These two aspects have already been discussed at the beginning of §55,
where we dealt with the "eigenf unction" S^ of the operator s z which corres-
ponds to s z = <to. The mathematical difference between them is especially
clear for a particle of spin s = |. In this case the spin function is, with respect
to the variable a, a contravariant spinor of rank 1, i.e. must be written in spinor
notation as 8°"^ . With respect to cto it is therefore a covariant spinor.
This is evidently a general result: the eigenfunctions 0y m can be put in
correspondence with the components of a covariant symmetrical spinor of
rank 2/ by means of formulae analogous to (57.2) :f
0/m = — — -011... 22...- (58.1)
V (i + tn)\(i — m)\
The eigenfunctions of integral angular momentum j are spherical harmonics,
and formula (58.1) relates them to the components of a covariant spinor of even
rank.
The case; = 1 is of particular importance. The three spherical harmonics
Y lm are
J^ee^ = TiJ^n x± in v)t
3 .. /3
Fi ± i = T i '
where n is a unit vector along the radius vector. Comparing with (58.1), we
see that the components of a spinor of rank two can be brought into corres-
pondence with the components of some vector by the formulae J
i i *
012 = a z , 0n= -A a x+ia y ), 022 = — -riflx—fay), ( 58 - 2 )
V 2 v 2 v 2
t This result can also be regarded somewhat differently. If the wave function ^ of a particle in
state with angular momentum j is expanded in terms of the eigenfunctions tftj m :
ifj = S a m ifij m>
m
then the coefficients a m are the probability amplitudes for various values of m. In this sense they
correspond to the "components" t/)(m) of a spin wave function, and this gives their law of transformation.
On the other hand, the value of at a given point in space cannot depend on the choice of the co-
ordinate system, i.e. the sum S a m iffj m must be a scalar. Comparing with the scalar (57.3), we see that
a m must transform as (— \) i ~ m 'jsj ,-m-
% Here the functions ifsj m and the components of the vector are related by
0io = ia z , 0n= —(ax+iay), 01 -1 = — r(%-«fy)- (58.2a)
V 2 V 2
202 Spin §58
or
V = - -Z&> <£ U = —z{a x -ia y \ <p = _ Jt ax+iay y (58 3)
Conversely
a z = iV20i2, «* = -i-(022_0ii) 5 % = ( 0ii + 022). (58>4)
v^ V 2
It is easily verified that with these definitions we have
</v£ A /<= a.b, (58.5)
where a and b are vectors corresponding to the symmetrical spinors i/t^ and
(f)^. It is also not difficult to see that the spinor ift/<f>t iv + 0/0 A " corresponds
to the vector <y/2a x b.
The relations (58.2) or (58.3) are a particular case of a general rule: any sym-
metrical spinor of even rank 2j, where j is integral, can be correlated with a
symmetrical tensor of half the rank (j) which gives zero on contraction with
respect to any pair of indices; we call this an irreducible tensor.
This follows from the fact that the numbers of independent components
of the spinor and of the tensor are the same (2/+ 1), as may easily be seen.f
The relation between the components of the spinor and of the tensor can be
found by means of formulae (58.2) — (58.4), if we consider a spinor of the rank
concerned as the product of several spinors of rank two, and the tensor as a
product of vectors.
Finally, let us determine the relation between the angles of rotation of the
co-ordinate system and the coefficients <x, /?, y, 8 of the binary transformation.
This is done by noticing that, on the one hand, the cosines of the angles
between the original and final axes of co-ordinates are the coefficients in the
formulae for the transformation of the components of a vector:
<*'t =g*afi* (58.6)
and, on the other hand, this same transformation can be performed by means
of a binary transformation, using formulae (58.4), (55.12). Thus, for instance,
we have
a' z =*V 2 «A 12 ' =^V 2 [ a y^ 1 +W 22 +(a3+ i 8y)0 12 ]
= (-ay+)88)a e +i(ay+iS8)fl,+(a8+j8y)fl.
We can similarly determine the remaining oi ik , and thus obtain the following
scheme of transformation coefficients :
(«-ik) =
l (a 2_£2_ y 2 + S 2) ^(_ a 2_ / J2 + y 2 + S 2) (-a/J+yg)"
^ai-jgi+yi-S") Ka^+yHS 2 ) -/(ajS+yS)
(-ay+jSS) *(«y+j88) ocS+Py
(58.7)
t Mathematically we can say that the 2/+ 1 components of an irreducible tensor of rank j (an
integer), the 2/+1 spherical harmonics Yj m , and the 2/+1 components of a symmetrical spinor of
rank 2/ give the same irreducible representation of the rotation group.
§58
The relation between spinors and tensors
203
The inverse expressions for the coefficients a, jS, y, 8 in terms of the angles
of rotation of the co-ordinate system can be found by using the Eulerian
angles to define the rotation. The matrix of the coefficients a, /?, y, S for a rota-
tion through an angle <f> about the sr-axis (denoted by co(<f>)), according to formula
(55.1) with cr = + J, is of the form
The matrix Q(0) expressing a rotation through an angle 6 about the x-axis
is easily calculated from formulae (58.7), in which oc X x = 1> <*-yy — <*zz — cos #>
V-yz = ~ a zy = sin "> 'X-xy = Kyx = a #z = &-zx = :
[cosA0 *sinA0~|
• • i/i I '
ism*v cos*0J
(58.9)
A rotation specified by the Eulerian angles <f>, 6, ifs (Fig. 20 ; OAT is the line of
intersection of the xy and x'y' planes) is carried out in three stages : a rotation
*>K
through an angle <f> about the #-axis, one through an angle 6 about the new
position of the x-axis, and finally one through an angle tfj about the final direc-
tion of the #-axis. Accordingly, the matrix of the complete transformation
is equal to the product co(i[t)Cl(d)co(<f>). By direct multiplication of the matrices,
we finally obtain
Ly 8J \j
cos^.e^ + *)/ 2
sinA0.e^-* )/2
i sin \Q
COSifl
g-£(0-0)/2'
g -t(^+0)/2
]
(58.10)
204 Spin §59
In particular, a rotation through an angle it about the jy-axis corresponds
to Eulerian angles = 7r, 0-0 = tt, so that a = S = 0, j8 = 1, y = — 1. This
means that the spinor components are transformed by such a rotation according
to0!' = 08,02' = -01, or
0i' = 0!, 02' = 2 . (58.il)
PROBLEMS
Problem 1. Rewrite the definition (57.4) of the operator of spin J in terms of the spinor
components of the vector s.
Solution. By means of formulae (58.3), which give the relation between the vector s
and the spinor S X|X , the definition (57.4) can be written as
i
2\/2
Problem 2. Derive formulae which determine the effect of the spin operator on a vector
wave function of a particle with spin 1 .
Solution. The relation between the components of the vector function l|> and the com-
ponents of the spinor ^ is given by formulae (58.3), and from (57.5) we have
40+ — -0+, S z tfi- = 0_, s z i/j z =
or
403 = -*"02/, S z lf> v = tlfl z , 402 = 0.
The remaining formulae are derived from these by cyclic permutation of the suffixes x, y, z.
They can be written together as
40* = —iemvph
The complex vector %\> can be put in the form l|> = e ia (u+iv), where u and v are real
vectors, which can be taken to be mutually perpendicular if the common phase a is suitably
chosen. The two vectors u and v determine a plane which has the property that the spin
component perpendicular to it can take only the values ± 1 .
§59. Partial polarisation of particles
By a suitable choice of the direction of the #-axis, we can always cause one
component (e.g. 2 ) of a given spinor A , the wave function of a particle with
spin ^, to vanish. This is evident from the fact that a direction in space is
determined by two quantities (angles), i.e. the number of disposable parameters
is just equal to the number of quantities (the real and imaginary parts of the
complex 2 ) which it is desired to make zero.
Physically this means that, if a particle with spin \ (for definiteness, we shall
speak of an electron) is in a state described by a spin wave function, then there
is a direction in space in which the component of the particle spin has the
definite value a = \. We can say that in such a state the electron is completely
polarised.
There are also, however, states of an electron which may be said to be
partially polarised. Such states are not described by wave functions but only
by density matrices, i.e. they are mixed states (with respect to spin) (see §14).
§59 Partial polarisation of particles 205
The spin density matrix of an electron is a spinor p A <" of rank two normalised
by the condition
p/ = P1 i+ P2 2 = 1, (59.1)
and satisfying the "Hermitian" condition
(pa")* = p/- (59.2)
For a pure (i.e. completely polarised) spin state of the electron the spinor
p A -" reduces to a product of components of the wave function «/r A :
Px p = (0A)#^. (59.3)
The "diagonal" components pi 1 and p2 2 of the density matrix determine the
probabilities of the values + 1 and — f of thcsr-component of the electron spin.
The mean value of this component is therefore
^ = KP1 1 — /»2 2 )i
or, using (59.1),
Pi 1 = H*> P2 2 = i-Tz- (59.4)
In a pure state the mean value of the quantities s ± = s x ± is y is calculated as
sL = ifjtes-ifj*- = ift 2 *ifj l
(see (55.7)). Accordingly we have in a mixed state
Pi 2 = n, P2 1 = r-. (59.5)
Thus we see that all the components of the spin density matrix of the electron
are expressed in terms of the mean values of components of its spin vector. In
other words, the real vector s entirely determines the polarisation properties of
a particle with spin J. In the limit of complete polarisation one of the com-
ponents of this vector (with an appropriate choice of the directions of the
axes) is ^ and the other two are zero. In the opposite case of an unpolarised
state all three components are zero. In the general case of an arbitrary partial
polarisation and any choice of the co-ordinate system we have ^ p ^ 1,
where
p = 2(^2 + ^2 + sl 2)l/2
is a quantity which may be called the degree of polarisation of the electron.
For a particle of arbitrary spin s, the density matrix is a spinor p\ /l ... f " T -"
of rank 4s, symmetrical in the first 2s and the last 2s indices and satisfying the
conditions
Pa,../*- = 1, (59.6)
(PA/*...^-)* = P^...^-. (59.7)
To calculate the number of independent components of the density matrix,
we note that, among the possible sets of values of the indices A, /*, ... (or />, a, . . .)
206 Spin §60
there are only 2s +1 which are essentially different. Using also the fact that
the components of the spinor p X/l ... p(T — are related by (59.6), we find that the
number of different components is (2s+l) 2 -l = 4s(s+l). Although these
components are complex, the relation (59.7) shows that this does not increase
the total number of independent quantities describing the state of partial
polarisation of the particle, which is therefore 4s(s+l).f For comparison, it
may be remarked that the state of complete polarisation of the particle is de-
scribed by only 4s quantities (the 2s + 1 complex components of the wave function
j/r'M"-, related by one normalisation condition and containing one common phase
which is unimportant in the description of the state).
Like any spinor of rank 4s, the spinor ^..Z '— is equivalent to a set of
irreducible tensors of ranks 4s, 4s -2, ... , 0. In the present case there is only
one tensor of each rank, since, on account of the symmetry properties of the
spinor ^Xfi... p(T "'i eacn contraction of it can be carried out in only one way:
with respect to any one of the indices A, fi, ... , and one of p, a, ... . In addition,
the scalar (tensor of rank 0) does not appear, reducing to unity by virtue of the
condition (59.6).
§60. Time reversal and Kramers' theorem
The symmetry of motion with respect to a change in the sign of the time is
expressed in quantum mechanics by the fact that, if «/r is the wave function of
a stationary state of the system, the "time-reversed" wave function (which we
denote by «/i rev ) describes a possible state with the same energy. At the end of
§18 it has been pointed out that */r rev is the same as the complex conjugate
function </r*. In this simple form the statement applies to wave functions where
the spin of particles is neglected. When spin is present, a refinement is necessary.
Let us take the wave function of a particle of spin s in the form of the contra-
variant spinor A / t — (of rank 2s). On taking the complex conjugate function
^,A/*...# we obtain a set of quantities which are transformed as components of a
covariant spinor. Hence the operation of time reversal corresponds to a change
from the wave function ifi^-- to a new wave function whose covariant com-
ponents are given by
.rev
0a,... = «A^- # . (60.1)
For a given set of values of the indices A, /x, ... , the components of covariant
and contravariant spinors correspond to values of the angular-momentum
component which differ in sign. In terms of the functions «/r So ., therefore, time
reversal corresponds to a change from ifj S(T to «/f S ,_ ff , as it should, since a change
in the sign of the time changes the direction of the angular momentum. The
exact relation is given by (60.1):
t When these quantities are given, so are the mean values of the components of the vector s and all
their powers and products 2, 3, ..., 2s at a time, which do not reduce to lower powers (see §55, Problem
3).
§60 Time reversal and Kramers' theorem 207
In other words, the operation of time reversal requires the change
^->^,-.(-l) s -*- ( 60 - 2 )
Let us consider an arbitrary system of interacting particles. The orbital and
spin angular momenta of such a system are not in general separately conserved
when relativistic interactions are taken into account. Only the total angular
momentum J is conserved. If there is no external field, each energy level of the
system has (27+l)-fold degeneracy. When an external field is applied, the
degeneracy is removed. The question arises whether the degeneracy can be
removed completely, i.e. so that the system has only simple levels. This is
closely related to the symmetry with respect to time reversal.
In classical electrodynamics the equations are invariant with respect to a
change in the sign of the time, if the electric field is left unchanged and the sign
of the magnetic field is reversed.! This fundamental property of motion must
be preserved in quantum mechanics. Hence, not only in a closed system but in
any external electric field (there being no magnetic field), there is symmetry
with respect to time reversal.
The wave functions of the system are spinors xp^—, whose rank n is twice
the sum of the spins s a of all the particles (n = 2 2 s a ); this sum may not be
equal to the total spin S of the system.
According to what was said above, we can assert that, in any electric field,
the wave function and its time reversal must correspond to states with the same
energy. If a level is non-degenerate, it is necessary that these states should be
identical, i.e. the corresponding wave functions must be the same apart from a
constant factor (both, of course, being expressed as similar (covariant or contra-
variant) spinors).
We write ^ v = C0 A/l ... or, by (60.1),
^...* = ClflX[i f (60.3)
where C is a constant.
Taking the complex conjugate of both sides of this equation, we obtain
^... = C *0 V *.
We lower the indices on the left-hand side of the equation and correspond-
ingly raise them on the right. This means that we multiply both sides of the
equation by gaXgpn ••• an ^ sum over the indices A, /*, ... ; on the right-hand side
we must use the fact that
BcSJBfy. •••=(- l)V a ^--
As a result we have
<Aam~. = C*(-i) n <A v "*.
t See, for example, The Classical Theory of Fields, §17, and the end of §110 below.
208 Spin §60
Substituting 0V-* from (60.3), we find
^.. =(-i) n cc*^.
This equation must be satisfied identically, i.e. we must have (— 1) W CC* = 1.
Since, however, \C\ 2 is always positive, it is clear that this is possible only
for even n (i.e. for integral values of the sum 2 s a ). For odd n (half-integral
values of S s a ) the condition (60.3) cannot be fulfilled.f
Thus we reach the result that an electric field can completely remove the
degeneracy only for a system with an integral value of the sum of the spins of
the particles. For a system with a half-integral value of this sum, in an
arbitrary electric field, all the levels must be doubly degenerate, and complex
conjugate spinors correspond to two different states with the same energy!
(H. A. Kramers 1930).
One further, mathematical, comment may be made. A relation of the form
(60.3) with a real constant C is mathematically the condition that the components
of the spinor may be put in correspondence with a set of real quantities, and
may be called the condition for the spinor to be "real".|| The impossibility of
fulfilling the condition (60.3) for odd n signifies that no real quantity can
correspond to a spinor of odd rank. For even n, on the other hand, the condition
(60.3) can be satisfied, and C can be real. In particular, a real vector can
correspond to a symmetrical spinor of rank two if the condition (60.3) is
satisfied with C = 1 :
<A A "* = K
(as is easily seen by means of (58.2) and (58.3)). The condition (60.3) with
C = 1 is in fact the condition for a symmetrical spinor of any even rank to be
"real".
t When the sum S s„ is integral (or half-integral), all possible values of the total spin S of the
system are also integral (or half-integral).
X If the electric field possesses a high (cubic) symmetry, fourfold degeneracy may occur (see
§99, including the Problem).
|| It is meaningless to call the spinor real in the literal sense, since complex conjugate spinors have
different laws of transformation.
CHAPTER IX
IDENTITY OF PARTICLES
§61. The principle of indistinguishability of similar particles
In classical mechanics, identical particles (electrons, say) do not lose their
"individuality", despite the identity of their physical properties. For we
can imagine the particles at some instant to be "numbered", and follow the
subsequent motion of each of these in its path ; then at any instant the particles
can be identified.
In quantum mechanics the situation is entirely different, as follows at once
from the uncertainty principle. We have already mentioned several times
that, by virtue of the uncertainty principle, the concept of the path of an
electron ceases to have any meaning. If the position of an electron is exactly
known at a given instant, its co-ordinates have no definite values even at an
infinitely close subsequent instant. Hence, by localising and numbering the
electrons at some instant, we make no progress towards identifying them at
subsequent instants ; if we localise one of the electrons, at some other instant,
at some point in space, we cannot say which of the electrons has arrived at
this point.
Thus, in quantum mechanics, there is in principle no possibility of separ-
ately following each of a number of similar particles and thereby distinguish-
ing them. We may say that, in quantum mechanics, identical particles
entirely lose their "individuality". The identity of the particles with respect
to their physical properties is here very far-reaching: it results in the complete
indistinguishability of the particles.
This principle of the indistinguishability of similar particles, as it is called,
plays a fundamental part in the quantum-mechanical investigation of systems
composed of identical particles. Let us start by considering a system of only
two particles. Because of the identity of the particles, the states of the system
obtained from each other by merely interchanging the two particles must be
completely equivalent physically. This means that, as a result of this inter-
change, the wave function of the system can change only by an unimportant
phase factor. Let ip(g lf | 2 ) be the wave function of the system, £ x and £ 2 con-
ventionally denoting the three co-ordinates and the spin projection for each
particle. Then we must have
where a is some real constant. By repeating the interchange, we return to
the original state, while the function is multiplied by e 2toc . Hence it follows
that e %i(X = 1, or e ia = ±1. Thus
*&.&)= ±*(&,*i).
209
210 Identity of Particles §61
We thus reach the result that there are only two possibilities: the wave
function is either symmetrical (i.e. it is unchanged when the particles are inter-
changed) or antisymmetrical (i.e. it changes sign when this interchange is
made). It is obvious that the wave functions of all the states of a given system
must have the same symmetry; otherwise, the wave function of a state which
was a superposition of states of different symmetry would be neither sym-
metrical nor antisymmetrical.
This result can be immediately generalised to systems consisting of any
number of identical particles. For it is clear from the identity of the particles
that, if any pair of them has the property of being described by, say, sym-
metrical wave functions, any other pair of such particles has the same pro-
perty. Hence the wave function of identical particles must either be un-
changed when any pair of particles are interchanged (and hence when the
particles are permuted in any manner), or change sign when any pair are
interchanged. In the first case we speak of a symmetrical wave function, and in
the second case of an antisymmetrical one.
The property of being described by symmetrical or antisymmetrical wave
functions depends on the nature of the particles. Particles described by
antisymmetrical functions are said to obey Fermi-Dirac statistics (or to be
fermions), while those which are described by symmetrical functions are
said to obey Bose-Einstein statistics (or to be bosons).^
Relativistic quantum mechanics shows that the statistics obeyed by particles
is uniquely related to their spin : particles with half- integral spin are fermions,
and those with integral spin are bosons.
The statistics of complex particles is determined by the parity of the
number of elementary fermions entering into their composition. For an
interchange of two identical complex particles is equivalent to the simul-
taneous interchange of several pairs of identical elementary particles. The
interchange of bosons does not change the wave function, while the inter-
change of fermions changes its sign. Hence complex particles containing
an odd number of elementary fermions obey Fermi statistics, while those
containing an even number obey Bose statistics. This result is, of course,
in agreement with the above rule, since a complex particle has an integral
or a half-integral spin according as the number of particles with half-integral
spin entering into its composition is even or odd.
Thus atomic nuclei of odd atomic weight (i.e. containing an odd number of
neutrons and protons) obey Fermi statistics, and those of even atomic weight
obey Bose statistics. For atoms, which contain both nuclei and electrons, the
statistics is evidently determined by the parity of the difference between the
atomic weight and the atomic number.
t This terminology refers to the statistics which describes a perfect gas composed of particles with
antisymmetrical and symmetrical wave functions respectively. In actual fact we are concerned here
not only with a different statistics, but essentially with a different mechanics. Fermi statistics was
proposed by E. Fermi for electrons in 1926, and its relation to quantum mechanics was elucidated by
P. A. M. Dirac (1926). Bose statistics was proposed by S. N. Bose for light quanta, and generalised
by A. Einstein (1924).
§61
The principle of indistinguishability of similar particles
211
Let us consider a system composed of N identical particles, whose mutual
interaction can be neglected. Let «/» 1} i/r 2 , ... be the wave functions of the vari-
ous stationary states which each of the particles separately may occupy, f
The state of the system as a whole can be denned by giving the numbers of
the states which the individual particles occupy. The question arises how
the wave function Y of the whole system should be constructed from the
functions if/ ly ift 2 , ... .
Let Px,p 2 , ... ,p N be the numbers of the states occupied by the individual
particles (some of these numbers may be the same). For a system of bosons,
the wave function T(£i, £ 2 , ..., $n) is given by a sum of products of the form
1^(6)*, (*•>•" **<**>'
with all possible permutations of the different suffixes p lt p 2 , ... ; this sum
clearly possesses the required symmetry property. Thus, for example, for
a system of two particles
T&.&) = [&> (*iW, (&)+*„ ($M>p &)]/V2;
(61.1)
we suppose that^ # p 2 . The factor \\y/2 is introduced for normalisation
purposes; all the functions iff lt 2 , ... are orthogonal and are supposed normal-
ised.
For a system of fermions, the wave function *F is an antisymmetrical
combination of these products. It can be written in the form of a deter-
minant
iM6) M&)
Y =
VNl
l^tfi)
M«
iM&) *..(&)
^ (for)
i
<Ap 2 (^)
(61.2)
Here an interchange of two particles corresponds to an interchange of two
columns of the determinant, as a result of which the latter, as is well known,
changes sign. For a system composed of two particles we have
y = [M^M^-MtoMfiWa.
(61.3)
The following important result is a consequence of the expression
(61.2). If among the numbers p lf p 2 ,... any two are the same, two rows of the
determinant are the same, and it therefore vanishes identically. It will be
different from zero only when all the numbers p lt p 2 , ... are different. Thus,
in a system consisting of identical fermions, no two (or more) particles can
be in the same state at the same time. This is called PaulV s principle (1925).
t If there is a strong interaction between the particles we cannot, of course, speak of such states.
212 Identity of Particles §62
§62. Exchange interaction
The fact that Schrodinger's equation does not take account of the spin
of particles does not invalidate this equation or the results obtained by means
of it. This is because the electrical interaction of the particles does not
depend on their spins.f Mathematically, this means that the Hamiltonian
of a system of electrically interacting particles (in the absence of a magnetic
field) does not contain the spin operators, and hence, when it is applied to the
wave function, it has no effect on the spin variables. Hence Schrodinger's
equation is actually satisfied by each component of the wave function; in
other words, the wave function ^(r^ a x \ r 2 , <r 2 ; ...) of the system of particles
can be written in the form of a product
x(°i> ff a»— W( r i> r »— )
of a function <j> of the co-ordinates of the particles only and a function x of the
spins. We call the former a co-ordinate or orbital wave function, and the latter
a spin wave function. Schrodinger's equation essentially determines only
the co-ordinate function <f>, the function x remaining arbitrary. In any
instance where we are not interested in the actual spin of the particles, we
can therefore use Schrodinger's equation and regard as the wave function
the co-ordinate function alone, as we have done hitherto.
However, despite the fact that the electrical interaction of the particles
is independent of their spin, there is a peculiar dependence of the energy
of the system on its total spin, arising ultimately from the principle of
indistinguishability of similar particles.
Let us consider a system consisting of only two identical particles. By
solving Schrodinger's equation we find a series of energy levels, to each of
which there corresponds a definite symmetrical or antisymmetrical co-
ordinate wave function </>(ri, r 2 ). For, by virtue of the identity of the particles,
the Hamiltonian (and therefore the Schrodinger's equation) of the system
is invariant with respect to interchange of the particles. If the energy levels
are not degenerate, the function <f>(ri, r 2 ) can change only by a constant
factor when the co-ordinates ri and r 2 are interchanged; repeating this
interchange, we see that this factor can only be J ± 1.
Let us first suppose that the particles have zero spin. The spin factor for
such particles is absent altogether, and the wave function reduces to the
co-ordinate function ^(r 1? r 2 ), which must be symmetrical (since particles with
zero spin obey Bose statistics). Thus not all the energy levels obtained by a
formal solution of Schrodinger's equation can actually exist; those to which
antisymmetrical functions <j> correspond are not possible for the system under
consideration.
t This is true only so long as we consider the non-relativistic approximation. When relativistic
effects are taken into account, the interaction of charged particles does depend on their spin.
J When there is degeneracy we can always choose linear combinations of the functions belonging
to a given level, such that this condition is again satisfied.
§62 Exchange interaction 213
The interchange of two similar particles is equivalent to the operation of
inversion of the co-ordinate system (the origin being taken to bisect the line
joining the two particles). On the other hand, the result of inversion is to
multiply the wave function <j> by ( — 1)*, where / is the orbital angular momen-
tum of the relative motion of the two particles (see §30). By comparing
these considerations with those given above, we conclude that a system of two
identical particles with zero spin can have only an even orbital angular mo-
mentum.
Next, let us suppose that the system consists of two particles with spin £
(say, electrons). Then the complete wave function of the system (i.e. the
product of the function ^(r 1} r 2 ) and the spin function x(<*v cr 2 )) must certainly
be antisymmetrical with respect to an interchange of the two electrons.
Hence, if the co-ordinate function is symmetrical, the spin function must
be antisymmetrical, and -vice versa. We shall write the spin function in spinor
form, i.e. as a spinor x V of rank two, each of whose indices corresponds to
the spin of one of the electrons. A symmetrical spinor (^V = x^) corre-
sponds to a function symmetrical with respect to the spins of the two particles,
and an antisymmetrical spinor (x^ = — x^) to an antisymmetrical func-
tion. We know, however, that a symmetrical spinor of rank two describes a
system with total spin unity, while an antisymmetrical spinor reduces to a
scalar, corresponding to zero spin.
Thus we reach the following conclusion. The energy levels to which there
correspond symmetrical solutions ^(r 1? r 2 ) of Schrodinger's equation can
actually occur when the total spin of the system is zero, i.e. when the spins
of the two electrons are "antiparallel", giving a sum of zero. The values of
the energy belonging to antisymmetrical functions (frfa, r 2 ), on the other hand,
require a value of unity for the total spin, i.e. the spins of the two electrons
must be "parallel".
In other words, the possible values of the energy of a system of electrons
depend on their total spin. For this reason we can speak of a peculiar inter-
action of the particles which results in this dependence. This is called
exchange interaction. It is a purely quantum effect, which entirely vanishes
(like the spin itself) in the passage to the limit of classical mechanics.
The following situation is characteristic of the case of a system of two
electrons which we have discussed. To each energy level there corresponds
one definite value of the total spin, or 1. This one-to-one correspondence be-
tween the spin values and the energy levels is preserved, as we shall see below
(§63), in systems containing any number of electrons. It does not hold,
however, for systems composed of particles whose spin exceeds \.
Let us consider a system of two particles, each with arbitrary spin s. Its
spin wave function is a spinor of rank 4s:
Aji... po...
*~28 IT"'
half (2s) of whose indices correspond to the spin of one particle, and the other
214 Identity of Particles §62
half to that of the other particle. The spinor is symmetrical with respect to
the indices in each group. An interchange of the two particles corresponds
to an interchange of all the indices A, [i, ... of the first group with the indices
p, a, ... of the second group. In order to obtain the spin function of a state of
the system with total spin S, we must contract this spinor with respect to
2s— S pairs of indices (each pair containing one index from X, /u, ... and one
from p, a, ...), and symmetrise it with respect to the remainder; as a result
we obtain a symmetrical spinor of rank 2S. However, the contraction of a
spinor with respect to a pair of indices means, as we know, the construction
of a combination antisymmetrical with respect to these indices. Hence,
when the particles are interchanged, the spin wave function is multiplied
by(-l)»-«
On the other hand, the complete wave function of a system of two particles
must be multiplied by ( — l) 2s when they are interchanged (i.e. by +1 for
integral s and by — 1 for half-integral s). Hence it follows that the symmetry
of the co-ordinate wave function with respect to an interchange of the particles
is given by the factor ( — l) s , which depends only on S. Thus we reach the
result that the co-ordinate wave function of a system of two identical particles
is symmetrical when the total spin is even, and antisymmetrical when it is
odd.
Recalling what was said above concerning the relation between interchange
of the particles and inversion of the co-ordinate system, we conclude also
that, when the spin S is even (odd), the system can have only an even (odd)
orbital angular momentum.
We see that here also a certain dependence is revealed between the possible
values of the energy of the system and the total spin, but this dependence is
not necessarily one-to-one. The energy levels to which there correspond
symmetrical (antisymmetrical) co-ordinate wave functions can occur for any
even (odd) value of S.
Let us calculate how many different states of the system there are with even
and odd S. The quantity S takes 2s +1 values: 2s, 2s— 1, ..., 0. For any
given S there are 2S+1 states differing in the value of the s-component of the
spin ((2s+ 1) 2 different states altogether). Let s be integral. Then, among
the 2s +1 values of S, s+1 are even and s odd. The total number of states
with even S is equal to the sum
s iUF +1 > =(2s+1)(s+1):
the remaining s(2s+l) states have odd S. Similarly, we find that, when s is
half-integral, there are s(2s+l) states with even values of S and (s+l)(2s+l)
with odd values.
PROBLEMS
Problem 1. Determine the exchange splitting of the energy levels of a system of two
electrons, regarding the interaction of the electrons as a perturbation.
Solution. Let the particles be (when their interaction is neglected) in states with orbital
§62 Exchange interaction 215
wave functions <i>i(r) and fa(r). The states of the system with total spin S = and S = 1
correspond to symmetrised and antisymmetrised products respectively:
1
<f> = —dM r i)M r 2)±M r 2)M r i)l
The mean value of the operator of the interaction U(rz— ri) of the particles in these states
is A ± J, where
A = J j UlMriTlU^WdVidVz,
J= \ \ C/0i(ri)^i*(r 2 )^2(r2)^2*(ri) dFi dV 2 ,
the latter being called the exchange integral. Omitting the additive constant A, which is not
an exchange term, we therefore find the level shifts Aj£ = J, Aj?i = —J (where the suffix
indicates the value of S). These quantities can be represented as the eigenvalues of the spin
"exchange operator" t
Pexch= -17(1+48!. s 2 ); (1)
the eigenvalues of the product si . S2 are derived in §55, Problem 2.
If the electrons belong to different atoms, for example, the exchange integral decreases
exponentially with increasing distance R between the atoms. It is clear from the form of the
integrand that this integral is determined by the "overlap" of the wave functions of the states
<f>(ri) and ^2(1*2) ; using the asymptotic law of decrease of the wave functions of states of a
discrete spectrum (cf. (21.6)), we find that
J „ r-lKi+icm, K1 = V(2m|£i|)/£, K2 = ^/(2m\E 2 \)jh,
where Ei and E2 are the energy levels of the electron in the two atoms.
Problem 2. The same as Problem 1, but for a system of three electrons.
Solution. Using formula (1), Problem 1, we can write the operator of pairwise exchange
interaction in a system of three electrons as
^exch = — 2 J a& (^ + 2s a .S 6 ), (1)
where the summation is over pairs of particles 12, 13 and 23. The matrix elements of the
operators s a . s& between states with different values of the pair of numbers o a> at, are given
by formulae (55.6) as
11 1 — i i 1
(Sa.S&) u = 4, (Sa.Sftji,-! = — 4, ( s a« s fc)-i,i = 2 .
We first determine the energy corresponding to the greatest possible value of the total-spin
component Ms = 01+02 + 03, viz. Ms = 3/2. This gives the energy of the state with total
spin »S = 3/2. On calculating the corresponding diagonal matrix element of the operator
(1), we find
A-E3/2 = — (7i2 + /l3 + /23)-
Next we take states with Ms = i. This value can occur in three ways, depending on which
of the numbers 01, 02, 031s —$ (the other two being $). Thus for these states we should
have a secular equation of the third degree. The calculation can, however, be simplified
immediately by noting that one of the roots of this equation must correspond to the energy
already found for the state with 5 = 3/2, and the secular equation must therefore have the
factor AE—AE3/2. In this way the calculation of the free term in the cubic equation can
be avoided. J
t First used by Dirac.
} This device is particularly useful in similar calculations for systems with a larger number of
particles .
216 Identity of Particles §63
The leading terms of the equation are found to be
(A£)3 + (7i2 + /l3 + /23)(A£ 2 )+[/i27l3 + A2/23 + A3^3-
-9(7i2 2 + 7l3 2 + ^3 2 )/4]AE+ ... = 0.
Dividing by A£ '+ J12 + J13+ J23, we find the two energy levels corresponding to states with
spins S = i:
AEl/ 2 = ±[(/l2 2 + /l3 2 + ^23 2 )-/l2^13-il2i23-Jl3i23] 1/2 .
Thus there are three energy levels, in accordance with the calculation in §63, Problem.
Problem 3. In which states can the Be 8 nucleus decay into two oe-particles?
Solution. Since the a-particle has no spin, a system of two a-particles can only have an
even orbital angular momentum (equal to the total angular momentum), and its states are
even. The decay in question is therefore possible only from even states of the Be 8 nucleus
with even total angular momentum.
§63. Symmetry with respect to interchange
By considering a system composed of only two particles, we have been able
to show that its co-ordinate wave functions <£(r x , r 2 ) for the stationary states
must be either symmetrical or antisymmetrical. In the general case of a sys-
tem of an arbitrary number of particles, the solutions of Schrodinger's
equation (the co-ordinate wave functions) need not necessarily be either sym-
metrical or antisymmetrical with respect to the interchange of any pair of
particles, as the complete wave functions (which include the spin factor)
must be. This is because an interchange of only the co-ordinates of two par-
ticles does not correspond to a physical interchange of them. The physical
identity of the particles here leads only to the fact that the Hamiltonian of the
system is invariant with respect to the interchange of the particles, and hence,
if some function is a solution of Schrodinger's equation, the functions ob-
tained from it by various interchanges of the variables will also be solutions.
Let us first of all make some remarks regarding interchanges in general.
In a system of N particles, N\ different permutations in all are possible. If
we imagine all the particles to be numbered, each permutation can be
represented by a definite sequence of the numbers 1, 2, 3, ... . Every such
sequence can be obtained from the natural sequence 1, 2, 3, ... by successive
interchanges of pairs of particles. The permutation is called even or odd,
according as it is brought about by an even or odd number of such inter-
changes. We denote by P the operators of permutations of N particles, and
introduce a quantity S P which is +1 if P is an even permutation and —1
if it is odd. If <f> is a function symmetrical with respect to all the particles,
we have
while, if <f> is antisymmetrical with respect to all the particles, then
P<f> = bp4>.
§63 Symmetry zvith respect to interchange 217
From an arbitrary function ^(r^ r 2 , ... , r^), we can form a symmetrical
function by the operation of symmetrisation, which can be written
<f> sym = constant xS^, (63.1)
where the summation extends over all possible permutations. The formation
of an antisymmetrical function (an operation sometimes called alternation)
can be written as
0^ = constant x S 8 P P<f>. (63.2)
Let us return to considering the behaviour, with respect to permutations,
of the wave functions ^ of a system of identical particles, j- The fact that the
Hamiltonian i? of the system is symmetrical with respect to all the particles
means, mathematically, that jff commutes with all the permutation operators
P. These operators, however, do not commute with one another, and so
they cannot be simultaneously brought into diagonal form. This means that
the wave functions^ cannot be so chosen that each of them is either symmetri-
cal or antisymmetrical with respect to all interchanges separately.];
Let us try to determine the possible types of symmetry of the functions
<f>{r x , r 2 , ... , r N ) of N variables (or of sets of several such functions) with
respect to permutations of the variables. The symmetry must be such that
it "cannot be increased", i.e. such that any additional operation of symmetri-
sation or alternation, on being applied to these functions, would reduce them
either to linear combinations of themselves or to zero identically.
We already know two operations which give functions with the greatest
possible symmetry: symmetrisation with respect to all the variables, and
alternation with respect to all the variables. These operations can be general-
ised as follows.
We divide the set of all the N variables r 1} r 2 , ... , r^ (or, what is the same
thing, the suffixes 1, 2, 3, ... , N) into several sets, containing N ly N 2 , ... ele-
ments (variables); N 1 +N 2 + ... = N. This division can be conveniently
shown by a diagram (known as a Young diagram) in which each of the num-
bers N lt N 2 , ... is represented by a line of several cells (thus, Fig. 21 gives a
diagram of the divisions 6+4+4+3+3 + 1+1 and 7+5+5+3 + 1 + 1 for
N = 22); one of the numbers 1, 2, 3, ... is to be placed in each square. If
we place the lines in order of decreasing length (as in Fig. 21), the diagram
contains not only successive horizontal rows, but also vertical columns.
Let us symmetrise an arbitrary function ^(r x , r 2 , ... , r N ) with respect to the
variables in each row. The alternation operation can then be performed only
with respect to the variables in different rows; alternation with respect to a
pair of variables in the same row clearly gives zero identically.
t From the mathematical point of view, the problem is to find irreducible representations of the
permutation group. A detailed account of the mathematical theory of permutation (or symmetry)
groups is given by H. Weyl, The Theory of Groups and Quantum Mechanics, Methuen, London 1931 ;
D. E. Rutherford, Substitutional Analysis, University Press, Edinburgh 1948; F. D. Murnaghan,
The Theory of Group Representations, Johns Hopkins Press, Baltimore 1938.
J Except for a system of only two particles, where there is a single interchange operator, which
can be brought into diagonal form simultaneously with ff.
218
Identity of Particles
§63
Fig. 21
Having chosen one variable from each row, we can, without loss of gener-
ality, regard them as being in the first cells in each row (after symmetrisation,
the order of the variables among the cells in each row is immaterial) ; let us
alternate with respect to these variables. Having then deleted the first column,
we alternate with respect to variables chosen one from each row in the thus
"curtailed" diagram; these variables can again be regarded as being in the
first cells of the "curtailed" rows. Continuing this process, we finally have
the function first symmetrised with respect to the variables in each row and
then alternated with respect to the variables in each column. After alternation,
of course, the function in general ceases to be symmetrical with respect to
the variables in each row. The symmetry is preserved only with respect to
the variables in the cells of the first row which project beyond the other rows.
Having distributed the N variables in various ways among the rows of a
Young diagram (the distribution among the cells in each row is immaterial),
we thus obtain a series of functions, which are transformed linearly into one
another when the variables are permuted in any manner, f However, it must
be emphasised that not all these functions are linearly independent ; the number
of independent functions is in general less than the number of possible distri-
butions of the variables among the rows of the diagram. We shall not pause
here, however, to discuss this more closely. %
Thus any Young diagram determines some type of symmetry of functions
with respect to permutations. By constructing all the possible Young dia-
grams (for a given N), we find all possible types of symmetry. This amounts
to dividing the number N in all possible ways into a sum of smaller terms,
including the number N itself; thus for N = 4 the possible partitions are
4,3 + 1,2+2,2+1 + 1,1 + 1 + 1 + 1.
To each energy level of the system we can make correspond a Young dia-
gram which determines the permutational symmetry of the appropriate
solutions of Schrodinger's equation; in general, several different functions
correspond to each value of the energy, and these are transformed linearly into
f It would be possible to perform the symmetrisation and alternation in the reverse order: to alter-
nate with respect to the variables in each column, and then to symmetrise with respect to those in the
rows. This, however, would give effectively the same thing, since the functions obtained by the two
methods are linear combinations of one another.
J For particles with spin i the number of independent functions (i.e. the dimension of the irreducible
representation) is derived in the Problem below.
§63 Symmetry with respect to interchange 219
each other by permutations.! However, it must be emphasised that this does
not signify any additional physical degeneracy of the energy levels. All these
different co-ordinate wave functions, multiplied by the spin functions, enter
into a single definite combination — the complete wave function — which
satisfies (according to the number of particles) the condition of symmetry or
antisymmetry.
Among the various types of symmetry there are always (for any given N)
two to each of which only one function corresponds. One of these cor-
responds to a function symmetrical with respect to all the variables, and the
other to one which is similarly antisymmetrical ; in the first case, the Young
diagram consists of a single row of N cells, and in the second case of a single
column.
Let us now consider the spin wave functions x(cr l5 tr 2 , ... , a N ). Their kinds
of symmetry with respect to permutations of the particles are given by the
same Young diagrams, with the components of the spins of the particles
taking the part of variables. There arises the question of what diagram must
correspond to the spin function for a given diagram of the co-ordinate func-
tion. Let us first suppose that the spin of the particles is integral. Then the
complete wave function if/ must be symmetrical with respect to all the particles.
For this to be so, the symmetry of the spin and co-ordinate functions must be
given by the same Young diagram, and the complete wave function ijj is
expressed as definite bilinear combinations of the two ; we shall not here pause
to examine more closely the problem of constructing these combinations.
Next, suppose the spin of the particles to be half-integral. Then the com-
plete wave function must be antisymmetrical with respect to all the particles.
It can be shown that, for this to be so, the Young diagrams for the co-ordinate
and spin functions must be obtained from each other by interchanging rows
and columns (as in the two diagrams shown in Fig. 21).
Let us consider in more detaii the important case of particles with spin \
(electrons, for instance). Each of the spin variables ct 1? o- 2 , ... here takes only
the two values ±|. Since a function antisymmetrical with respect to any
two variables vanishes when these variables take the same value, it is clear
that the function x can be alternated only with respect to pairs of variables ;
if we alternate with respect to even three variables, two of them must always
take the same value, so that we have zero identically.
Thus, for a system of electrons, the Young diagrams for the spin functions
can contain columns of only one or two cells (i.e. only one or two rows) ; in
the Young diagrams for the co-ordinate functions, the same is true of the
number of columns. The number of possible types of permutational sym-
metry for a system of N electrons is therefore equal to the number of possible
partitions of the number N into a sum of ones and twos. When N is even,
this number is %N+1 (partitions with 0, 1, ... , %N twos), while if N is odd
t The existence of this "permutational degeneracy" is related to the fact that the permutation oper-
ators commuting with the Hamiltonian do not in general commute with one another (see the middle
of §10).
220 Identity of Particles §63
it is f(iV+l) (partitions with 0, 1, ... , \{N— 1) twos). Thus, for instance,
Fig. 22 shows the possible Young diagrams (co-ordinate and spin) for N = 4.
It is easy to see that each of these types of symmetry (i.e. each of the
Young diagrams) corresponds to a definite total spin S of the system of
electrons. We shall consider the spin functions in spinor form, i.e. as spinors
^A/x... f ran k JV, whose indices (each of which corresponds to the spin of
an individual particle) will be the variables that are arranged in the cells of
the Young diagrams. Let us examine the Young diagram consisting of two
rows with N x and N % cells {N x +N% = N, and N x > iV 2 ). In each of the first
N 2 columns there are two cells, and the spinor must be antisymmetrical with
respect to the corresponding pairs of indices. With respect to the indices in
the last n = N 1 —N. 3 ells in the first row, however, it must be symmetrical.
As we know, such a spinor of rank N reduces to a symmetrical spinor of rank
n, to which there corresponds a total spin S = \n. Returning to the Young
diagrams for the co-ordinate functions, we can say that the diagram with n
rows each of one cell corresponds to a total spin S = \n. For even N, the
total spin can take integral values from to |iV, while for odd N it can take
half-integral values from | to $N, as it should.
We emphasise that this one-to-one correspondence between the Young
diagrams and the total spin holds only for systems of particles with spin | ;
we have seen this, for a system of two particles, in the previous section. f
PROBLEM
Determine the number of energy levels with different values of the total spin S, for a system
of N particles of spin J.
Solution. A given value of the projection of the total spin of the system, M s = 2 a, can
be obtained in
f(M s ) = N\l(W+M s )\(iN-M s )\
ways; this is the number of combinations of N elements £N+M S at a time, since we put
a = £ for %N+M S particles and a = —\ for the remainder. To each energy level with a
given S, there correspond 2S+1 states with values M s = S, S—l, ... , —S of the projected
spin. Hence it is easy to see that the number of different energy levels with a given value of
S is
n(S) =f(S)-f(S+l) =Nl (2S+ l)l(iN+ S+1)\QN-S)\
f In the third footnote to §20, we have remarked that, for a system of several identical particles, we
cannot assert that the wave function of the stationary state of lowest energy is without nodes. We shall
now amplify this statement and elucidate its origin.
The wave function (that is, the co-ordinate function), if it has no nodes, must certainly be symmetri-
cal with respect to all the particles; for, if it were antisymmetrical with respect to the interchange of
any pair of particles 1, 2, it would vanish for r x = r 2 . If, however, the system consists of three or
more electrons, no completely symmetrical co-ordinate wave function is possible : the Young diagram
of the co-ordinate function cannot have rows with more than two cells.
Thus, although the solution of Schrodinger's equation which corresponds to the lowest eigenvalue
is without nodes (by the theorem of the variational calculus), this solution may be physically inadmis-
sible; the smallest eigenvalue of Schrodinger's equation will not then correspond to the normal state
of the system, and the wave function of this state will in general have nodes.
For particles with a half-integral spin s, this situation occurs in systems with more than 2s +1
particles. For systems of bosons, a completely symmetrical co-ordinate wave function is always
possible.
§64
Second quantisation. The case of Bose statistics 221
J
S=2
S = 1
Fig. 22
SrQ
The total number of different energy levels is
n = Sn(5)=/(0)=AT!/[(|iV)!p
s
for even N, and
»=/(i)=M/(*tf +*)!(*#-*)'
for odd JV.
§64. Second quantisation. The case of Bose statistics
In the quantum-mechanical investigation of systems consisting of a very
large number of identical particles, interacting in any manner, there is a
useful method of considering the problem, known as second quantisation.
This method is necessary in relativistic theory also, where we have to deal
with systems in which the number of particles is itself variable^ Let us
first consider systems of bosons.
We denote by 0i(£), ^2(£)> ••• some complete set of orthogonal and normal-
ised wave functions. These may, for instance, correspond to the stationary
states of a single particle in some external field. We emphasise that the choice
of this field is arbitrary ; it need not be the same as the actual field acting on
the particles in the physical system considered. As in §61, | denotes the
assembly of the co-ordinates and spin projection a of a particle.
Let us consider, in a purely formal manner, a system of N non-interacting
particles, in the field selected. Then every particle is in one of the
states «/f 1} ?/f 2 , ... . Let N t be the number of particles in the state ^; it may,
of course, be zero. If the numbers N±, N%, ... are given (clearly 2 Nt = N),
the state of the system as a whole is determined; we shall indicate these
numbers by suffixes to the wave function M^n,... °f tne system. Let us
seek to construct a mathematical formalism in which the occupation numbers
N lt N 2 , ... of the states (and not the co-ordinates of the particles) play the
part of independent variables.
f The method of second quantisation was developed by P. A. M. Dirac (1927) for particles obeying
Bose statistics, and later extended to Fermi particles by E. Wigner and P. Jordan (1928).
222 Identity of Particles §64
The function ^¥ NiNz _ is a symmetrised (the particles obeying Bose statistics)
sum of products of the functions fa. Let us write it in the form
Yur v... = VWNJ ...INI) S fa (Qfa (&) ... fa (fr). (64.1)
i. «s 1 2 N -
Here p ly p 2 , ... , ^v are the ordinal numbers of the states in which the indivi-
dual particles are, and the sum is taken over all permutations of those suffixes
Pi>p2> ••• >Pn which are different. The numbers A^ show how many of the
suffixes p x , p 2 , ... , p N have the value i. The total number of terms in the sum
(64.1) is evidently
Nl/NjlNJi....
The constant factor in (64.1) is chosen so that the function is normalised;
by virtue of the orthogonality of the functions fa, on integrating! tne square
IT/v^... ! 2 with respect to | x , £ 2 , ... , g N all the terms vanish except the squared
modulus of each term in the sum.
Next, let / (1) a be the operator of some physical quantity pertaining to the
<zth particle, i.e. acting only on functions of £ a . We introduce the operator
^ (1) = S/(« a , (64.2)
which is symmetrical with respect to all the particles (the summation being
over all particles), and determine its matrix elements with respect to the wave
functions (64.1). First of all, it is easy to see that the matrix elements will
be different from zero only for transitions which leave the numbers N ± , N 2 , ...
unchanged (diagonal elements) and for transitions where one of these numbers
is increased, and another decreased, by unity. For, since each of the operators
/<!>„ acts only on one function in the product fajiijfa^) ... fa^),
its matrix elements can be different from zero only for transitions whereby
the state of a single particle is changed ; this, however, means that the number
of particles in one state is diminished by unity, while the number in another
state is correspondingly increased. The calculation of these matrix elements
is in principle very simple ; it is easier to do it oneself than to follow an account
of it. Hence we shall give only the result of this calculation. The non-
diagonal elements are
Fll) N^N k =f a) ikV(NiN k ). (64.3)
We shall indicate only those suffixes with respect to which the matrix element
is non-diagonal, omitting the remainder for brevity. Here/ (1) <fc is the matrix
element
f w nc=jfa*(t)f w fa(t)dt. (64.4)
f By integration over £ we conventionally understand integration over the co-ordinates and sum-
mation over a.
§64 Second quantisation. The case of Bose statistics 223
It must be borne in mind that the operators / (1) differ only in the naming of
the variables on which they act, and hence the integrals / (1) i& are indepen-
dent of a. The diagonal matrix elements of F (1) are the mean values of the
quantity FV> in the states Yj^jy,.... Calculation gives
F» = ?/««#<• (64.5)
i
We now introduce the operators d iy which play a leading part in the method
of second quantisation; they act, not on functions of the co-ordinates, but on
the variables N lt N it ... , and are defined as follows. When acting on the
function Y# # ., the operator d t decreases the suffix N t by unity, and at the
same time it multiplies the wave function by ^N t :
d^ Nl N2 ... Ni ... = V% n 2 .... *hu. (64.6)
We can say that the operator a\ diminishes by one the number of particles in
the ith state; it is called an annihilation operator. It can be represented in
the form of a matrix whose only non-zero element is
(a,VV* = VNi. (64.7)
The operator &f which is the Hermitian conjugate of d i is, by definition
(see §3), represented by a matrix with an element
i.e.
ta + #-i =VNi ' (64 ' 8)
This means that, when acting on the function x i r NlNt ..., it increases the suffix
N t by unity:
d^¥ Nl n 2 ...N t ... = VW+Wn, n 2 ...,N i+1 (64.9)
In other words, the operator di + increases by one the number of particles in
the tth state, and is therefore called a creation operator.
The product of operators d^d*, acting on the wave function, evidently
multiplies it by a constant simply, leaving unchanged all the variables
N lt N 2 , ... : the operator d t diminishes N t by unity, and d t + then restores it
to its original value. Direct multiplication of the matrices (64.7) and (64.8)
shows that afdi is represented, as we should expect, by a diagonal matrix
whose diagonal elements are N t . We can write
dt+dt = N t . (64.10)
Similarly, we find that
<W=iVVr-l. (64.11)
224 Identity of Particles §64
Hence the commutation rule for the operators d t and d t + is
d i d i +—d i + d i = 1. (64.12)
The operators d t and d k (or d t and d k + ) with i and k different act on different
variables (N t and N k ), and of course commute:
didic—dicdi = 0, didk + -dk + di = (i ^ A). (64.13)
From the above properties of the operators d t , df it is easy to see that the
operator
/» = S/ (1) a^ + 4 (64.14)
is the same as the operator (64.2). For all the matrix elements calculated from
(64.7), (64.8) are the same as the elements (64.3), (64.5). This is a very
important result. In formula (64. 14), the quantities/ Q-\ k are simply numbers.
Thus we have been able to express an ordinary operator (of the form (64.2)),
acting on functions of the co-ordinates, in the form of an operator acting on
functions of new variables, the occupation numbersf N t .
The result which we have obtained is easily generalised to operators of other
forms. Let
jft2) = S/(2) a6j (6415)
a>o
where / (2) a6 is the operator of a physical quantity pertaining to two particles
at once, and hence acts on functions of g a and g b . Similar calculations show
that this operator can be expressed in terms of the operators d t , d t + by
where
(f ( X = If ^(^*tfi)M(6)ao d&d*,.
The matrices calculated for (64.15) and (64.16) are the same. The generalisa-
tion of these formulae to operators of any other form symmetrical with
respect to all the particles (of the form / (3) = S/ 3 > a6c etc.) is obvious.
Finally, it remains to express, in terms of the operators d iy the Hamiltonian
& of the physical system of N identical interacting particles that is actually
being considered. The operator i? is, of course, symmetrical with respect
to all the particles. In the non-relativistic approximation, J it is independent
f Formula (64.14) bears a similarity to the expression (11.1)
f = ^>fika*a k
for the mean value of a quantity/, expressed in terms of the coefficients a { in the expansion of the wave
function of a given state in terms of the wave functions of the stationary states. This is the reason for
calling this method the second quantisation method.
% In the absence of a magnetic field.
§64 Second quantisation. The case of Bose statistics 225
of the spins of the particles, and can be represented in a general form as
follows :
i? = S^)+S^(W)+ s £/< 3 >(r a ,r 6 ,r c )+.... (64.17)
a a>b a>b>c
Here # (1) is the part of the Hamiltonian which depends on the co-ordinates
of the ath particle only :
#(D a = -(#72m)Aa+tf (1) (»-«), ( 64 - 18 )
where C/ (1) (r a ) is the potential energy of a single particle in the external field.
The remaining terms in (64.17) correspond to the mutual interaction energy
of the particles; for convenience, the terms depending on the co-ordinates of
two, three, etc. particles have been separated.
This representation of the Hamiltonian enables us to apply formulae (64.14),
(64.16) and their analogues directly. Thus
# = JflW^A+^+1 J^Xfl i +a?d 1 A+ ... • (64.19)
This gives the required expression for the Hamiltonian in the form of an
operator acting on functions of the occupation numbers.
For a system of non-interacting particles, only the first term in the expres-
sion (64.19) remains:
i,k
If the functions ^ are taken to be the eigenfunctions of the Hamiltonian
#(*> of an individual particle, the matrix H (X \ k is diagonal, and its diagonal
elements are the eigenvalues e i of the energy of the particle. Thus
ft = S efdi+di',
i
replacing the operator a^ by its eigenvalues (64.10), we have for the energy
levels of the system the expression
E = 2 €i N it
a trivial result which could have been foreseen.
The formalism which we have developed can be put in a somewhat more
compact form by introducing the operatorsf
t Attention is drawn to the analogy between these expressions and the expansion
T = S aWt
of the wave function in terms of the eigenfunctions of an operator (cf . the third footnote to this sec-
tion).
226 Identity of Particles §64
where the variables £ are regarded as parameters. By what has been said
above concerning the operators d it df, it is clear that the operator *F de-
creases the total number of particles in the system by one, while ¥+ increases
it by one.
It is easy to see that the operator ¥ + (f ) creates a particle at the point | .
For the result of the action of the operator of is to create a particle in a state
with wave function «/^(£). Hence it follows that the result of the action of the
operator ^(lo) is to create a particle in a state with wave function
S <Ai*(£)^(£o)> or ( bv the general formula (5.12)) with wave functionf S(£— | ),
which corresponds to a particle with definite values of the co-ordinates (and
spin).
The commutation rules for ¥ and 4^+ are obtained at once from those for
d it d t + . It is evident that
*(*)¥(*' )-¥(*')*(0 = 0; (64.21)
we also have
^)T+(f)-^+(n^) = 2&GrW(f),
or
n^ + (n-^ + (n^) = w-n (64.22)
The expression (64.14) for the operator (64.2) can be written, using these
new operators, in the form
fr>=f¥+{f)fw*(QiX, (64.23)
where it is understood that the operator / (1) acts on functions of the parameters
| in ^"(l)- For, substituting (64.20), we have
/» =^J4>i*(i)f w Ui)^.d^d k = s/ (1 W< + 4,
which is the same as (64.14). Similarly, we have instead of (64.16)
pm = |J j ^+(|)^+(^')/ (2 >^(r)^(0 d*d£'. (64.24)
In particular, to a physical quantity /(£) that is simply a function of £,
there corresponds the operator (64.23), which in this case can be written in
the form
f §(£ — 1 ) conventionally denotes the product
8(x-x )8(y-y Q )8(z-z )8 a
§65
Second quantisation. The case of Fermi statistics
227
Hence it is clear that ^ + (£)^(£) d| is the operator of the number of particles
in the range d|.
When expressed by means of the operators ^F, ^ + , the operator H takes
the form
ti = j {(^ 2 /2m)VT+(0V^(l)+ U^)^ + (0^(i)} <*£+
+
ij-J xr+(^^+(f)t/< 2 ^,f )^(f )"¥($) d^df +
(64.25)
We have used here the expression (64.18) for i? (1) , and have integrated by
parts (with respect to the co-ordinates) the term containing the Laplacian.
We can clarify the formula (64.25) by noticing the following point. Sup-
pose that we have a system of particles, each of which is described (at a given
instant) by the same wave function «/»(£), which we suppose normalised so that
J \tjj\ 2 d£ = N. Then it is immediately evident that, if we replace the operator
ty in the expression (64.25) by the function ifj, this expression becomes the
mean energy of the system in the state considered. This gives the following
rule for deriving the Hamiltonian in the second quantisation formalism. The
expression for the mean energy is written in terms of the wave function of an
individual particle (normalised as stated above), and this function is then
replaced by the operator T, the Hermitian conjugate operators ^ + being
written to the left of the operators T.
If the system consists of bosons of different kinds, operators a, d + or
ty, y i' + must be defined in the second quantisation method for each kind of
particle. Operators pertaining to particles of different kinds of course
commute.
§65. Second quantisation. The case of Fermi statistics
The basic theory of the method of second quantisation remains wholly
unchanged for systems of identical fermions, but the actual formulae for the
matrix elements of quantities and for the operators di are naturally different.
The wave function Tjvat s ... now has the form (61.2):
Vnn
1 2
VNl
Mfor)
0,_(*l) 0. (& - *»-(**)
(65.1)
Because of the antisymmetry of this function, the question of its sign arises
first of all. This question did not arise in the case of Bose statistics, since,
228 Identity of Particles §65
because of the symmetry of the wave function, its sign, once chosen, was
preserved under all permutations of the particles. In order to make definite
the sign of the function (65.1), we shall agree to choose it as follows. We
number successively, once and for all, all the states fa. We then complete
the rows of the determinant (65.1) so that always
Px<p2<P 3 <-<pN, (65.2)
whilst in the successive columns we have functions of the different variables
in the order f lf £ 2 , ... , g N . No two of the numbers p^p 2 , ... can be equal,
since otherwise the determinant would vanish. In other words, the occupa-
tion numbers N t can take only the values and 1.
Let us again consider an operator of the form (64.2), F a) = S/ (1) tt . As
in §64, its matrix elements will be non-zero only for transitions where all the
occupation numbers remain unchanged and for those where one occupation
number (A^) is diminished by unity (becoming zero instead of one) and an-
other (N k ) is increased by unity (becoming one instead of zero). We easily
find that, for i < k,
(F ll) Y<°* =/ t V-l)** w -*- 1 \ (65.3)
V*;
where by t , l t we signify N ( = 0, Nt = 1 and the symbol 2(&, /) denotes
the sum of the occupation numbers of all states from the Mi to the /thrj-
S (*,/)= S N».
For the diagonal elements we obtain our previous formula (64.5) :
FW = ?/w,AT,. (65.4)
In order to represent the operator ^ (1) in the form (64.14), the operators
d t must be defined as matrices whose elements are
(«*)£ = (^^(-l)* 1 -*-". (65.5)
On multiplying these matrices, we find, for k > i,
(ai+a^hh = («, + ) 1 A(a ft ) * * = (-1) I(1 « *'- 1) (- l)3i.*-iHS(*fi.*-i)
°i 1 fc °i°k Vfc
or
(aSa^ = (- l)**w. *-». (65.6)
If/ = k, the matrix of ^ + a i is diagonal, and its elements are unity for N t = 1,
and zero for N t = ; this can be written
_^^___ di+di = N t . (65.7)
t For i > k the exponent in (65.3) becomes S (k + 1, i — 1). The sum must be taken as zero
when i = k + 1.
§65 Second quantisation. The case of Fermi statistics 229
On substituting these expressions in (64.14), we in fact obtain (65.3), (65.4).
Multiplying a/*", d k in the opposite order, we have
(a^+yi^ = (a k yi^(ai+yiH == (_l)i(i.^-i)+i+i(*fi.A-i)+^a.*-i)+i,
or
(ajflt+ffi* = -(- l) I(i+1 ' & - 1) . (65.8)
Comparing (65.8) with (65.6), we see that these quantities have opposite signs,
i.e. we can write
di + d k +d k di + = (i # k).
For the diagonal matrix a i a/ f , we find
&.&* = i-N t . (65.9)
Adding this to (65.7), we obtain
didi + +di + di = 1.
Both the above equations can be written in the form
didif+dk+di = S ik . (65.10)
On carrying out similar calculations, we find for the products d i d k the re-
lations
did k +d k di = 0, (65.11)
and in particular d^ = 0.
Thus we see that the operators d i and d k (or d k + ) for i # k anticommute,
whereas in the case of Bose statistics they commuted with one another. This
difference is perfectly natural. In the case of Bose statistics, the operators
di and d k were completely independent; each of the operators d t acted only
on a single variable N it and the result of this action did not depend on the
values of the other occupation numbers. In the case of Fermi statistics,
however, the result of the action of the operator d t depends not only on the
number N t itself, but also on the occupation numbers of all the preceding
states, as we see from the definition (65.5). Hence the action of the various
operators d it d k cannot be considered independent.
The properties of the operators d if d t + having been thus defined, all the
remaining formulae (64.14)-(64.19) remain valid. The formulae (64.23)-
(64.25), which express the operators of physical quantities in terms of the
operators x F(f), 1 F + (£) defined by (64.20), also hold good. The commutation
rules (64.21), (64.22), however, are now obviously replaced by
230 Identity of Particles §65
If the system consists of particles of different kinds, second quantisation
operators must be defined for each kind of particle (as already mentioned at
the end of §64). Operators belonging to bosons and fermions commute;
those belonging to different fermions may formally be regarded as either
commutative or anticommutative within the limits of non-relativistic theory.
On either assumption the results obtained by means of the second quantisa-
tion method are the same.
However, with a view to later applications in the relativistic theory, which
allows different particles to be transformed into one another, we should
assume that the creation and annihilation operators for different fermions
anticommute. This becomes evident if we regard as "different" particles
two different "internal" states of a single complex particle.
CHAPTER X
THE ATOM
§66. Atomic energy levels
In the non-relativistic approximation, the stationary states of the atom are
determined by Schrodinger's equation for the system of electrons, which
move in the Coulomb field of the nucleus and interact electrically with one
another; the spin operators of the electrons do not appear in this equation.
As we know, for a system of particles in a centrally symmetric external field
the total orbital angular momentum L and the parity of the state are conserved.
Hence each stationary state of the atom will be characterised by a definite
value of the orbital angular momentum L and by its parity. Moreover, the
co-ordinate wave functions of the stationary states of a system of identical
particles have a certain permutational symmetry. We have seen in §63 that,
for a system of electrons, a definite value of the total spin of the system cor-
responds to each type of permutational symmetry (i.e. to each Young dia-
gram). Hence every stationary state of the atom is characterised also by the
total spin S of the electrons.
The converse, however, is of course not true; if L, S and the parity are
given, the energy of the state is not uniquely determined.
The energy level having given values of S and L is degenerate to a degree
equal to the number of different possible directions in space of the vectors S
and L. The degree of the degeneracy from the directions of L and S is re-
spectively 2L + 1 and 2S + 1 . Consequently, the total degree of the degener-
acy of a level with given L and S is equal to the product (2L + 1)(2S+1).
In fact, however, there is always some relativistic electromagnetic inter-
action of the electrons, which depends on their spins. It has the result that
the energy of the atom depends not only on the absolute values of the orbital
angular momentum and spin vectors, but also on their relative positions.
Strictly speaking, when the relativistic terms in the Hamiltonian operator are
taken into account, it no longer commutes with the operators L and §, i.e.
the orbital angular momentum and the spin are not separately conserved.
Only the total angular momentum J = L+S is conserved. The conservation
of the total angular momentum is an exact law which follows at once from the
isotropy of space relative to a closed system. For this reason the energy levels
must be characterised by the values / of the total angular momentum.
However, if the relativistic effects are comparatively small (as happens in
many cases), they can be allowed for as a perturbation. Under the action of
this perturbation, a level with given L and S, having (2L+l)(2S+l)-fold
degeneracy, is "split" into a number of distinct (though close) levels, which
differ in the value of the total angular momentum /. These levels are
231
232 The Atom §67
determined (in the first approximation) by the appropriate secular equation
(§39), while their wave functions (in the zeroth approximation) are definite
linear combinations of the wave functions of the initial degenerate level with
the given L and S. In this approximation we can therefore, as before, regard
the absolute values of the orbital angular momentum and spin (but not their
directions) as being conserved, and characterise the levels by the values of
L and S also.
Thus, as a result of the relativistic effects, a level with given values of L
and S is split into a number of levels with different values of /. This splitting
is called the fine structure (or the multiplet splitting) of the level. As we know,
J takes values from L+S to \L— S\ ; hence a level with given L and S is split
into 25+1 (if L > S) or 2L + 1 (if L < S) distinct levels. Each of these is
still degenerate with respect to the directions of the vector J; the degree of
this degeneracy is 2/ + 1. It is easily verified that the sum of the numbers
2/ + 1 for all possible values of /is equal to (2L + l)(2S+l), as it should be.
There is a generally accepted notation to denote the atomic energy levels
(or, as they are called, the spectral terms of the atoms), similar to that used
for the states of individual particles with definite values of the angular
momentum (§32): states with different values of the total orbital angular
momentum L are denoted by capital Latin letters, as follows :
L = 0123456789 10...
SPDFGHIKLMN ...
Above and to the left of this letter is placed the number 2S+1, called the
multiplicity of the term (though it must be borne in mind that this number
gives the number of fine-structure components of the level only when
L ^ S). Below and to the right of the letter is placed the value of the total
angular momentum J. Thus the symbols 2 P 1/2 , 2 P 3/2 denote levels with
L = 1, S = h J = \ and f .
§67. Electron states in the atom
An atom with more than one electron is a complex system of mutually
interacting electrons moving in the field of the nucleus. For such a system
we can, strictly speaking, consider only states of the system as a whole.
Nevertheless, it is found that we can, with fair accuracy, introduce the idea
of the states of each individual electron in the atom, as being the stationary
states of the motion of each electron in some effective centrally symmetric
field due to the nucleus and to all the other electrons. These fields are in
general different for different electrons in the atom, and they must all be
defined simultaneously, since each of them depends on the states of all the
other electrons. Such a field is said to be self-consistent (see §69).
Since the self-consistent field is centrally symmetric, each state of the elec-
tron is characterised by a definite value of its orbital angular momentum /.
The states of an individual electron with a given / are numbered (in order
§67 Electron states in the atom 233
of increasing energy) by the principal quantum number n, which takes the
values » = /+l,/+2, ... ; this choice of the order of numbering is made in
accordance with what is usual for the hydrogen atom. However, it must be
noticed that the sequence of levels of increasing energy for various / in com-
plex atoms is in general different from that found in the hydrogen atom. In
the latter, the energy is independent of /, so that the states with larger values
of n always have higher energies. In complex atoms, on the other hand, the
level with n= 5,1=0, for example, is found to lie below that with n = 4,
1=2 (this is discussed in more detail in §73).
The states of individual electrons with different values of n and / are
customarily denoted by a figure which gives the value of the principal
quantum number, followed by a letter which gives the value of /:f thus \d
denotes the state with n = 4, / = 2. A complete description of the atom
demands that, besides the values of the total L, S and/, the states of all the
electrons should also be enumerated. Thus the symbol 1* 2p 3 P denotes a
state of the helium atom in which L = 1,S = 1,J = and the two electrons
are in the Is and 2p states. If several electrons are in states with the same /
and w, this is usually shown for brevity by means of an index: thus 3p 2
denotes two electrons in the 3p state. The distribution of the electrons in the
atom among states with different / and n is called the electron configuration.
For given values of n and /, the electron can have different values of the
projections of the orbital angular momentum (m) and of the spin (a) on the
#-axis. For a given /, the number m takes 2/+1 values; the number a is
restricted to only two values, ±$. Hence there are altogether 2(2/+ 1)
different states with the same n and /; these states are said to be equivalent.
According to Pauli's principle there can be only one electron in each such
state. Thus at most 2(2/+ 1) electrons in an atom can simultaneously have
the same n and /. An assembly of electrons occupying all the states with the
given n and / is called a closed shell of the type concerned.
The difference in energy between atomic levels having different L and S
but the same electron configuration:}: is due to the electrostatic interaction
of the electrons. These energy differences are usually small, and several
times less than the distances between the levels of different configurations.
The following empirical principle (Hund's rule) is known concerning the
relative position of levels with the same configuration but different L and S:
The term with the greatest possible value of S (for the given electron con-
figuration) and the greatest possible value of L (for this S) has the lowest energy. ||
f Another terminology often used is that in which electrons with principal quantum numbers
n = 1, 2, 3, ... are said to belong to the K, L, M, . . . shells (see §74).
J We here ignore the fine structure of each multiplet level.
|| The requirement that S should be as large as possible can be explained as follows. Let us consider,
for example, a system of two electrons. Here we can have S = or S = 1 ; the spin 1 corresponds to
an antisymmetrical co-ordinate wave function <f>(r lt r 2 ). For r x = r 2 , this function vanishes; in other
words, in the state with 5=1 the probability of finding the two electrons close together is small.
This means that their electrostatic repulsion is comparatively small, and hence the energy is less.
Similarly, for a system of several electrons, the "most antisymmetrical" co-ordinate wave function
corresponds to the greatest spin.
234 The Atom §67
We shall show how the possible atomic terms can be found for a given elec-
tron configuration. If the electrons are not equivalent, the possible values
of L and S are determined immediately from the rule for the addition of
angular momenta. Thus, for instance, with the configurations np, n'p (n, ri
being different) the total angular momentum L can take the values 2, 1,0,
and the total spin S = 0, 1 ; combining these, we obtain the terms ^S,
If we are concerned with equivalent electrons, however, restrictions im-
posed by Pauli's principle make their appearance. Let us consider, for
example, a configuration of three equivalent p electrons. For 1=1 (the p
state), the projection m of the orbital angular momentum can take the values
m = 1, 0, — 1, so that there are six possible states, with the following values
of m and a:
(«)l,i (b)0,i (c)-l,i
(a') 1,-1 (b') 0,-1 (0-l,-i.
The three electrons can be one in each of any three of these states. As a
result we obtain states of the atom with the following values of the projections
M L = 2m, M s = £<r of the total orbital angular momentum and spin:
{a+a'+b)2,\ ( a +a'+c)l,i (a+b+c) 0, f
(a+b+b')l,i (a+b+c') 0,$
(a+b'+c)0,\
(a'+b+c)0,l
The states with M L or M s negative need not be written out, since they give
nothing different. The presence of a state with M L = 2, M s = \ shows that
there must be a 2 D term, and to this term there must correspond one state
(1, £) and one (0, \). Next, there remains one state with (1, Q, so that there
must be a 2 P term; one of the states (0, £) corresponds to this. Finally, there
remain the states (0, f ) and (0, £), corresponding to a 4 S term. Thus, for a
configuration of three equivalent p electrons, the only possibilities are one
term of each of the types 2 Z), 2 P, *S.
Table 1 gives the possible terms for various configurations of equivalent
p and d electrons. The figures below the letters of the terms show the num-
ber of terms of the type concerned that exist for the given configuration, if
this number is more than one. For the configuration with the greatest
possible number of equivalent electrons (s 2 ,p 6 , d 10 , ...), the term is always X S.
Like terms always correspond to configurations which differ in that one of
them has as many electrons as the other lacks to form a closed shell. This is
an evident result of the fact that the absence of an electron from the shell can
be regarded as a "hole", whose state is defined by the same quantum numbers
as the state of the missing electron.
§67
Electron states in the atom
Table 1
Possible terms for configurations of equivalent electrons
235
p,p s
p*,p*
P 3
*P
2 PZ>
3 P
4 5
d,d 3
d\d a
d 3 ,d 7
1 SDG
*PDFGH
S PF
*PF
d^d*
iSDFGI
2 2 2
3 PDFGH
2 2
5 D
d 5
*SPDFGHI
3 2 2
*PDFG
6 S
When Hund's rule is applied to determine the ground term of an atom
from a known electron configuration, only the unfilled shell need be con-
sidered, since the moments of electrons in closed shells cancel out. For
example, let there be four d electrons outside the closed shells in an atom.
The magnetic quantum number of the d electron can take five values:
0, ±1, ±2. Hence all four electrons can have the same spin component
<j = |, and the maximum possible total spin is S = 2. We must then assign
to the electrons different values of m so as to give the maximum value of
M L = 2w, namely 2, 1, 0, — 1, M L = 2. This means that the maximum
value of L for S = 2 is also 2, and the term is 5 D.
PROBLEM
Find the orbital wave functions of the possible states of a system of three equivalent p
electrons.
Solution. In the states 4 S the spins a of all the electrons are the same, and the values of
m are therefore different. The wave function is given by a determinant of the form (61.2)
composed of the functions ^o, 01, <A-i (where the suffix shows the value of w).
For the 2 Z) term we consider the state with the maximum possible value Ml = 2. Two
of the components m will be 1 and the other —1. Let electrons 2 and 3 have a = + \ and
electron 1 have a = —\ (corresponding to total spin S = J). The orbital wave function
having the required symmetry is
= —-0i(l)[0o(2)^i(3)-0o(3)0i(2)] f
V 2
the argument of each function being the number of the electron to which it refers.
For the 2 P term we consider the state with Ml = 1 and the same values of the electron
spin components as previously. This state can be obtained with two different sets of values
of m, so that the orbital wave function is given by the linear combination
j/j = aift-ui + bifjioo,
0-m = «Ai(l)[«A-i(2)0i(3)-«A-i(3)0i(2)],
0ioo = 0o(l)[0i(2)0o(3)-0i(3)0 o (2)].
236 The Atom §68
To determine the coefficients, we use the relation
Z+0 = (/+(!) + /+( 2 >+ /+(3))0 = 0,
which must be satisfied by the wave function with Ml = M (see (27.8)). Using the matrix
elements (27.12), we find that
/+«Al = 0, l+ifj-i = -\/2i/j 0y t+ipo = \Z2ifjx,
and so
L+if>=^/2(a + b)ifj 011 = 0.
Hence a + b = 0, and using also the normalisation condition, we have a = — b = \.
The wave functions of states with Ml < L are obtained from those found above by apply-
ing to them the operator £-.
§68. Hydrogen-like energy levels
The only atom for which Schrodinger's equation can be exactly solved
is the simplest of all atoms, that of hydrogen. The energy levels of the hydro-
gen atom, and of the ions He+, Li++, ... which each have only one electron,
are given by Bohr's formula (36.10)
mZ 2 e* 1
E = . _. (68.1)
2h%l + mJM) n 2 V '
Here Ze is the charge on the nucleus, M its mass, and m the mass of the elec-
tron. We notice that the dependence on the mass of the nucleus is only very
slight.
The formula (68.1) does not take account of any relativistic effects. In
this approximation there is an additional {accidental) degeneracy, peculiar
to the hydrogen atom, of which we have already spoken in §36; for a given
principal quantum number n, the energy is independent of the orbital angular
momentum /.
Other atoms have states whose properties recall those of hydrogen. We
refer to highly excited states, in which one of the electrons has a large principal
quantum number, and so is mostly at large distances from the nucleus. The
motion of such an electron can be regarded, to a certain approximation, as
motion in the Coulomb field of the rest of the atom, whose effective charge is
unity. The values of the energy levels thus obtained are, however, too in-
exact; it is necessary to apply to them a correction to take account of the devia-
tion of the field from the pure Coulomb field at small distances. The nature
of this correction is easily ascertained from the following considerations.
Since the states with large quantum numbers are quasi-classical, the energy
levels can be determined from Bohr and Sommerfeld's quantisation rule
(48.6). The deviation from the Coulomb field at distances from the nucleus
small compared with the "orbit radius" can be formally allowed for by an
alteration in the boundary condition imposed on the wave function at
t = 0. This brings about a change in the constant y in the quantisation
condition for radial motion. Since this condition is otherwise unchanged,
§69 The self-consistent field 237
we can conclude that we obtain for the energy levels an expression which
differs from that for hydrogen in that the radial, that is, the principal, quantum
number n is replaced by n + A/, where A* is some constant (known as
Rydberg's correction) :
m< £ 1
E = . (68.2)
1W- («+A,) 2
Rydberg's correction is (by definition) independent of n, but it is of course
a function of the azimuthal quantum number / of the excited electron (which
we add as a suffix to A), and of the angular momenta L and S of the whole
atom. For given L and S, A* decreases rapidly as / increases. The greater /,
the less time the electron spends near the nucleus, and hence the energy levels
must approach more and more closely those of hydrogen as / increases.f
§69. The self-consistent field
Schrodinger's equation for atoms containing more than one electron can-
not be directly solved in practice, even by numerical methods. Approximate
methods of calculating the energies and wave functions of the stationary states
of the atoms are therefore important. The most important of these methods
is what is called the self-consistent field method. The idea of this method con-
sists in regarding each electron in the atom as being in motion in the "self-
consistent field" due to the nucleus together with all the other electrons.
As an example, let us consider the helium atom, restricting ourselves to
those terms in which both the electrons are in s states (with or without the
same n) ; the states of the whole atom will then be S states also. Let ^(ri)
and »/f 2 (r 2 ) be the wave functions of the electrons; in the s states they are
functions only of the distances r v r 2 of the electrons from the nuclei. The
wave function ^(r l5 r 2 ) of the atom as a whole is a symmetrised
4> = H^H^+H^H^) (69-1)
or antisymmetrised
= Ur^UH)-Mr*)Uri) (69 2)
product of the two functions, according as we are concerned with states of
t As an illustration, we may give the experimental values of Rydberg's correction for the highly
excited states of the helium atom. The total spin of this atom can have the values S = and 1, while
the total orbital angular momentum L is, in the states considered, the same as the angular momentum
/ of the excited electron (the other electron being in the state Is). Rydberg's corrections are
for S = 0: A = -0-140, Ai = +0-012, A 2 = -00022;
for S = 1 : A = -0-296, Ai = -0068, A 2 = -00029.
238 The Atom §69
total spinf S =. or S = 1 . We shall consider the second of these. The
functions i/f x and i{j 2 can then be regarded as orthogonal. J
Let us try to determine the function of the form (69.2) which is the best
approximation to the true wave function of the atom. To do so, it is natural
to start from the variational principle, allowing only functions of the form
(69.2) to be considered; this method was proposed by V. A. Fok (1930).
As we know, Schrodinger's equation can be obtained from the variational
principle
I i/t*lJipdV 1 dV 2 = minimum,
with the additional condition
(the integration is extended over the co-ordinates of both electrons in the
helium atom). The variation gives the equation
jj S<A*(#-£)«£ dV l( iV 2 = 0, (69.3)
and hence, with an arbitrary variation of the wave function ip, we obtain the
usual Schrodinger's equation. In the self-consistent field method, the
expression (69.2) for ip is substituted in (69.3), and the variation is effected
with respect to the functions ^ and ifj 2 separately. In other words, we seek
an extremum of the integral with respect to functions iff of the form (69.2) ;
as a result we obtain, of course, an inexact eigenvalue of the energy and an
inexact wave function, but the best of the functions that can be represented
in this form.
The Hamiltonian for the helium atom is of the form||
8 =#H-#,+ 1/ru, #x = -*Ai-2/r lf (69.4)
where r 12 is the distance between the electrons. Substituting (69.2) in (69.3),
carrying out the variation, and equating to zero the coefficients of 8^ and 8ip 2
in the integrand, we easily obtain the following equations:
[lA+2/r+^-^ 22 -G 22 (r)]e/- 1 (r)+[H 12 +G 12 (r)]^ 2 (r) = 0,
[*A+2/r+E-H 11 -G u (r)]^(r)+[ff 11 +G 11 (r)M 1 (r) =0,
where *
GaM = J «Aa(r 2 )<A b (r 2 ) dV 2 /r 12 ,
H ab = j 0.[-4A-2/r]& dV (a, b = 1, 2). (69.6)
f The states of the helium atom with S = are usually called parahelium states, and those with
S = 1 orthohelium states.
J The wave functions tfi lt ip 2 , •■• of the various states of the electron which are obtained by the self-
consistent field method are not in general orthogonal, since they are solutions of different equations,
not of the same equation. In (69.2), however, without altering the function tp of the whole atom, we
can replace tp 2 by tp 2 ' = 2 + constant X0 X ; by an appropriate choice of the constant, we can always
ensure that ipj and tfi 2 ' are orthogonal.
II In this section (including the Problems) we use atomic units (see the first footnote to §36).
§69 The self-consistent field 239
These are the final equations resulting from the self-consistent field method;
they can, of course, be solved only numerically, f
The equations are similarly derived in more complex cases. The wave
function of the atom to be substituted in the integral in the variational
principle is in the form of a linear combination of products of the wave func-
tions of the individual electrons. This combination must be so chosen that,
firstly, its permutational symmetry corresponds to the total spin S of the
state of the atom considered and, secondly, it corresponds to the given value
of the total orbital angular momentum L of the atom.
By using, in the variational principle, the wave function having the neces-
sary permutational symmetry, we automatically take account of the exchange
interaction of the electrons in the atom. Simpler equations (though leading
to less accurate results) are obtained if we neglect the exchange interaction
and also the dependence on L of the energy of the atom for a given electron
configuration (D. R. Hartree 1928). As an example, let us again consider the
helium atom; we can then write the equations for the wave functions of the
electrons immediately in the form of ordinary Schrodinger's equations :
[I Aa+E a - V a (r a )]Ur a ) =0 (a = 1, 2), (69.7)
where V a is the potential energy of one electron moving in the field of the
nucleus and in that of the distributed charge of the other electron:
V 1 (r 1 ) = -2M- j (l!r 12 W(r 2 ) dV z , (69.8)
and similarly for V % . In order to find the energy E of the whole atom, we
must notice that, in the sum E x +E 2 , the electrostatic interaction between the
two electrons is counted twice, since it appears in the potential energy Fifa)
of the first electron and in that — V 2 (r 2 ) — of the second. Hence E is obtained
from the sum E x -\-E^ by subtracting once the mean energy of this interaction;
that is,
E = E x +E 2 - Jj (l/r 12 W(r x ),A 2 2 (r 2 ) dV.dV,. (69.9)
To refine the results obtained by this simplified method, the exchange
interaction and the dependence of the energy on L can afterwards be taken
into account as perturbations.
PROBLEMS
Problem 1. Determine approximately the energy of the ground level of the helium atom
and helium-like ions (a nucleus of charge Z and two electrons), regarding the interaction
between the electrons as a perturbation.
Solution. In the ground state of the ion, both electrons are in s states. The unperturbed
t A comparison of the energy levels of light atoms, calculated by the self-consistent field method,
with spectroscopic data enables us to estimate the accuracy of the method at about 5 per cent.
240 The Atom §69
value of the energy is twice the ground level of a hydrogen-like ion (because of the two
electrons) :
EP» = 2{-\Z*)=-Z*.
The correction in the first approximation is given by the mean value of the electron inter-
action energy in a state with wave function
Z3
77
(the product of two hydrogen functions with / = 0). The integral
=/H
EW= | 102 dPWa
ri2
is most simply calculated as
oo r 2
EP> = 2 J d V 2 . P2— fpid V h d Fi = 4 7 rr 1 2dr 1 ,
dV 2 = 477r 2 2 dr2,
the energy of the charge distribution p 2 = I fa I 2 in the field of the spherically symmetric
distribution pi = | fa |2; the integrand with dF 2 is the energy of the charge pn(r 2 ) in the field
of the sphere n <r 2 , and the factor 2 takes account of the contribution from configurations
in which n > r%. Thus we find E^ = 5Z/8, and finally
E=Ef® +#!>= -Z2+|Z.
For the helium atom (Z = 2) this gives —J? = 11/4 = 2-75; the actual value of the ground-
state energy of this atom is — E = 2-90 atomic units = 78-9 eV.
Problem 2. The same as Problem 1, but using the variational principle, approximating
the wave function by a product of two hydrogen functions with some effective nuclear charge.
Solution. We calculate the integral
IS
ifJlfdVidVz, #=-&Ai+A 2 ) +—
n r 2 r 12
with the function tji given by (1), Problem 1 , but with Zett instead of Z. The integral of fa/m
is calculated in Problem 1 ; the integral of if> Ai "A can be reduced to that of ^ 2 /n, since, by
Schrodinger's equation,
(-*Ai )<Al = -iZen^L
The result is
jjifsff l / J dV 1 dV 2 =Z efl 2-2ZZe fl + fZ efl .
This expression as a function of Zett has a minimum at Zett = Z—&. The corresponding
value of the energy is
For the helium atom this gives — E = 2-85.
It may be noted that the wave function (1) with the above value of Zett is in fact the best
not only of all functions of the form (1) but of all functions which depend only on the sum
r\ + r 2 .
§70 The Thomas-Fermi equation 241
§70. The Thomas-Fermi equation
Numerical calculations of the charge distribution and field in the atom by
the self-consistent field method are extremely cumbersome, especially for
complex atoms. For these, however, there is another approximate method,
whose value lies in its simplicity; its results are admittedly much less accurate
than those of the self-consistent field method.
The basis of this method (E. Fermi, and L. Thomas, 1927) is the fact that,
in complex atoms with a large number of electrons, the majority of the elec-
trons have comparatively large principal quantum numbers. In these condi-
tions the quasi-classical approximation is applicable. Hence we can apply
the concept of "cells in phase space" (§48) to the states of the individual
electrons.
The volume of phase space corresponding to electrons which have momenta
less than p and are in the volume element dV of physical space is iirp z dV.
The number of "cells", i.e. possible states, corresponding to this volume isf
4ttP 3 dVjZilrtf, and in these states there cannot at any one time be more than
4it/> 3 p z
2-J—dV = ^—dV
3(2t7) 3 3?t 2
electrons (two electrons, with opposite spins, in each "cell"). In the normal
state of the atom, the electrons in each volume element dV must occupy (in
phase space) the cells corresponding to momenta from zero up to some maxi-
mum value p . Then the kinetic energy of the electrons will have its smallest
possible value at every point. If we write the number of electrons in the
volume dV as ndV (where n is the number density of electrons), we can say
that the maximum value Po of the momenta of the electrons at every point
is related to n by
P*jZtt* = n.
The greatest value of the kinetic energy of an electron at a point where the
electron density is n is therefore
W=\{Wnyi*. (70.1)
Next, let ${r) be the electrostatic potential, which we suppose zero at
infinity. The total energy of the electron is ip 2 —<f>. It is evident that the
total energy of each electron must be negative, since otherwise the electron
moves off to infinity. We denote the maximum value of the total energy of
the electron at each point by — <£ , where <£ is a positive constant ; if this quan-
tity were not constant, the electrons would move from points with smaller
<j> to those with greater <f> . Thus we can write
W = <t>~K (70.2)
t In this section we use atomic units.
242 The Atom §70
Equating the expressions (70.1) and (70.2), we obtain
n = [2(^ )] 3 / 2 /37r 2 , (70.3)
a relation between the electron density and the potential at every point in the
atom.
For <j>=(j> Q the density n vanishes ; n must clearly be put equal to zero also
in the whole of the region where <f> < <j> , and where the relation (70.2) would
give a negative maximum kinetic energy. Thus the equation <f> = <f> deter-
mines the boundary of the atom. There is, however, no field outside a
centrally symmetric system of charges whose total charge is zero. Hence we
must have <f> = at the boundary of a neutral atom. It follows from this that,
for a neutral atom, the constant <£ must be put equal to zero. On the other
hand, <f> is not zero for an ion.
Below we shall consider a neutral atom, putting accordingly <j> Q = 0.
According to Poisson's electrostatic equation, we have A<f> = 4irn; substitut-
ing (70.3) in this, we obtain the fundamental equation of the Thomas-Fermi
method :
A«£ = (8V2/377-)^/ 2 . (70.4)
The field distribution in the normal state of the atom is determined by the
centrally symmetric solution of this equation that satisfies the following
boundary conditions: for r -> the field must become the Coulomb field of
the nucleus, i.e. (f>r -> Z, while for r -> oo we must have^r -» 0. Introducing
here, in place of the variable r, a new variable x according to the definitions
r = xbZ-y*, b = Kf^) 2/3 = 0-885, (70.5)
and, in place of (f>, a new unknown function x by f
Z /rZ l l*\ ZW x(x)
<f>( r )=-x(-ir)=-T- — • < 70 - 6 )
r \ b J b x
we obtain the equation
x i>2 tf x jdx 2 = x s/2 , (70.7)
with the boundary conditions x = 1 for ^ = and x = for x = oo. This
equation contains no parameters, and thus defines a universal function x(x).
Table 2 gives values of this function obtained by numerical integration of
equation (70.7). The function x(x) decreases monotonically, and vanishes
only at infinity 4 In other words, the atom has no boundaries in the Thomas-
Fermi model, and formally extends to infinity.
t In ordinary units,
<f>(r) = {Ze\r)x{rZV*me*\W&5W-).
% The equation (70.7) has the exact solution \( x ) — 144x~ 3 , which vanishes at infinity but does not
satisfy the boundary condition at x = 0. It could be used as an asymptotic expression for the function
X(x) for large x. However, this expression gives fairly exact values only for very large x, whilst the
Thomas-Fermi equation becomes inapplicable at large distances (see below).
§70
The Thomas-Fermi equation
Table 2
Values of the function x(x)
243
X
X(x)
X
X(x)
X
X(x)
0-00
1-000
1-4
0-333
6
0-0594
0-02
0-972
1-6
0-298
7
0-0461
0-04
0-947
1-8
0-268
8
0-0366
0-06
0-924
2-0
0-243
9
0-0296
0-08
0-902
2-2
0-221
10
0-0243
0-10
0-882
2-4
0-202
11
0-0202
0-2
0-793
2-6
0-185
12
0-0171
0-3
0-721
2-8
0-170
13
0-0145
0-4
0-660
3-0
0-157
14
0-0125
0-5
0-607
3-2
0-145
15
0-0108
0-6
0-561
3-4
0-134
20
0-0058
0-7
0-521
3-6
0-125
25
0-0035
0-8
0-485
3-8
0-116
30
0-0023
0-9
0-453
4-0
0-108
40
0-0011
1-0
0-424
4-5
0-0919
50
0-00063
1-2
0-374
5-0
0-0788
60
0-00039
The value of the derivative x '(x) for x = is x'(°) = —1-59. Hence, as
x -> 0, the function x(%) is of the form x = 1 — 159*, and accordingly the
potential <f>(r) is
<f>(r) ^ Z/r-l-80Z 4 / 3 .
(70.8)
The first term is the potential of the field of the nucleus, while the second
( — \%0me z Z m jh 2 ' in ordinary units) is the potential at the origin due to the
electrons.
Substituting (70.6) in (70.3), we find for the electron density an expression
of the form
n = Z*f{rZ™\b\ f(x) = (32/9tt%/*)3/2.
(70.9)
We see that, in the Thomas- Fermi model, the charge density distribution in
different atoms is similar, with Z~ 1/3 as the characteristic length (in ordinary
units h 2 /me 2 Z 1/3 , i.e. the Bohr radius divided by Z 1/3 ). If we measure
distances in atomic units, the distances at which the electron density has its
maximum value are the same for all Z. Hence we can say that the majority
of the electrons in an atom of atomic number Z are at distances from the
nucleus of the order of Z~ 1/3 . A numerical calculation shows that half the
total electron charge in an atom lies inside a sphere of radius 1-33 Z~ 1/3 .
Similar considerations show that the mean velocity of the electrons in the
9
244 The Atom §70
atom (taken, as an order of magnitude, as the square root of the energy) is of
the order of Z 2/3 .
The Thomas- Fermi equation becomes inapplicable both at very small and
at very large distances from the nucleus. Its range of applicability for small r
is restricted by the inequality (49.12); at smaller distances the quasi-classical
approximation becomes invalid in the Coulomb field of the nucleus. Putting
in (49.12) a = Z, We find 1/Z as the lower limit of distance. The quasi-
classical approximation becomes invalid for large r also in a complex atom.
In fact, it is easy to see that, for r ~ 1, the de Broglie wavelength of the elec-
tron becomes of the same order of magnitude as the distance itself, so that the
quasi-classical condition is undoubtedly violated. This can be seen by esti-
mating the values of the terms in equations (70.2) and (70.4); indeed, the
result is obvious without calculation, since equation (70.4) does not involve
Z. Thus the applicability of the Thomas-Fermi equation is limited to dis-
tances large compared with \\Z and small compared with unity. In complex
atoms, however, the majority of the electrons in fact lie in this region.
This means that the "outer boundary" of the atom in the Thomas-Fermi
model is at r~ 1, i.e. the dimensions of the atom do not depend on Z. The
energy of the outer electrons, i.e. the ionisation potential of the atom, is
likewise independent of Z.f
By means of the Thomas-Fermi method we can calculate the total ionisa-
tion energy E y i.e. the energy needed to remove all the electrons from the
neutral atom. To do this, we must calculate the electrostatic energy of the
Thomas-Fermi distribution for the charges in the atom; the required total
energy is half this electrostatic energy, since the mean kinetic energy in a
system of particles interacting in accordance with Coulomb's law is (by the virial
theorem) minus half the mean potential energy. The dependence of E on Z
can be determined a priori from simple considerations: the electrostatic
energy of Z electrons at a mean distance Z -1/3 from a nucleus of charge Z,
and moving in its field, is proportional to Z Z\Z~ VZ = Z 7/3 . A numerical
calculation gives the result E = 20-8Z 7/3 eV. The dependence on Z is in
good agreement with the experimental data, though the empirical value of
the coefficient is close to 16.
We have already mentioned that positive (non-zero) values of the constant
cf> correspond to ionised atoms. If we define the function x by <f> —<f> = Z%\r y
we obtain the same equation (70.7) for x as previously. We must now, how-
ever, consider only solutions which vanish not at infinity as for the neutral
atom, but for finite values x of x. Such solutions exist for any x . At the
point x = x 0f the charge density vanishes together with x, but the potential
remains finite. The value of x is related to the degree of ionisation in the
following manner. The total charge inside a sphere of radius r is, by Gauss's
t This model does not, of course, show the periodic dependence of the dimensions and ionisation
potential of the atom on Z, which appears in the periodic system of the elements. Moreover, experi-
mental data indicate the existence of a slight but steady increase in dimensions and decrease in the
ionisation potential as Z increases.
§70
The Thomas-Fermi equation
245
theorem, —r 2 d(f>jdr = Z[x(x)— xx'(x)]. The total charge z on the ion is
obtained by putting x = x in this ; since x( x o) = 0» we have
z = -Zx x'(x ).
(70.10)
The thick line in Fig. 23 shows the curve of x(x) for a neutral atom; below
it are two curves for ions of different degrees of ionisation. The quantity
z\Z is shown graphically by the length of the segment intercepted on the
axis of ordinates by the tangent to the curve at x = x .
Fig. 23
Equation (70.7) also has solutions which are nowhere zero; these diverge
at infinity. They can be regarded as corresponding to negative values of the
constant (f> . Fig. 23 also shows two such curves of x(x) ; they lie above the
curve for the neutral atom. At the point x = x lt where
X(*i)-*i*'(*i) = 0,
(70.11)
the total charge inside the sphere x < x x is zero (graphically, this point is
evidently the one where the tangent to the curve passes through the origin).
If we cut off the curve at this point, we can say that it defines x(x) for a neutral
atom at whose boundary the charge density remains non-zero. Physically,
this corresponds to a "compressed" atom confined to some given finite
volume.f
The Thomas- Fermi equation does not take account of the exchange inter-
action between electrons. The effects which this involves are of the next
order of magnitude with respect to Z~ 2/3 . Hence an allowance for the ex-
change interaction in the Thomas-Fermi method requires a simultaneous
consideration of both these effects and others of the same order of magnitude. J
t This approach may be useful in studying the equation of state of highly compressed matter.
t This has been done by A. S. Kompaneets and E. S. Pavlovskii (Soviet Physics JETP 4 328
1957) and by D. Kirzhnits (ibid. 5, 64, 1338, 1957).
246 The Atom §71
PROBLEM
Find the relation between the energy of the electrostatic interaction between electrons
and that of their interaction with the nucleus in a neutral atom, using the Thomas- Fermi
model.
Solution. The potential 4> e of the field due to the electrons is found by subtracting the
potential Zjr of the nucleus front the total potential 4>. The energy of the interaction between
the electrons is therefore
= $z(-?-dV-$[<l>ndV
-**Jr ,r - L T-J :
(3tt2)2/3
(where <f> has been expressed in terms of n by means of (70.3)). The energy U e n of the
interaction between the electrons and the nucleus and their kinetic energy T are therefore
U en =-ZJ^dV,
(3tt2)2/3
= 3- — » 5 /3dF.
10
■/■
Comparing these expressions with the previous equation, we find
Uee = — $Uen e-**
According to the virial theoremf, for a system of particles interacting according to Coulomb's
law we have IT = — U = —Uen — Uee. Thus finally
§71. Wave functions of the outer electrons near the nucleus
We have seen, on the basis of the Thomas- Fermi model, that the outer
electrons in complex atoms (Z large) are mainly at distances r ~ 1 from the
nucleus. $ A number of properties of atoms, however, depend significantly
on the electron density near the nucleus; such properties will be considered
in §§72 and 120. To determine the order of magnitude of this density we
may examine the variation of the wave function ?/»(r) of the electron in the
atom when r varies from large (r ~ 1) to small distances.
t See Mechanics, §10.
j In this section we use atomic units.
§72 Fine structure of atomic levels 247
In the region r ~ 1, the field of the nucleus is screened by the remaining
electrons, so that the potential energy U{r) ~ 1/r ~ 1. The energy of the
electron level in this field E ~ 1. At distances of the order of the Bohr
radius in the field of a charge Z, r ~ 1/Z, the field of the nucleus may be
regarded as unscreened, and U = — Zjr. In the transitional region, 1/Z < r
< 1, the potential energy j t7 j is large compared with the electron energy E,
and the condition
d 1
1(1)
dr\pj
pj dr ,/\U\
holds (where/> is the momentum), so that the motion of the electron is quasi-
classical. The spherically symmetrical quasi-classical wave function is
1 1 1
U( r )| f or _ <^1, (71.1)
the order of magnitude of the coefficient (~ 1) being determined by the con-
dition «/r ~ 1 for "joining" to the wave function for r ~ 1.
Applying the expression (71.1) in order of magnitude for r ~ 1/Z (sub-
stituting U = — Z/r), we obtain the required value of the wave function
near the nucleus :f
4l\\Z)~y/Z. (71.2)
In accordance with the general properties of wave functions in a central
field (§32), when the distance decreases further ift(r) either remains constant
in order of magnitude (for an s electron) or begins to decrease (for 7^0).
The probability of finding the electron in the region r < 1/Z is
w~\iP\W~l/Z2. (71.3)
The formulae (71.2) and (71.3) of course determine only the systematic
variation with increasing Z, and do not take into account non-systematic
variations from one element to the next.
§72. Fine structure of atomic levels
A detailed study of relativistic interactions will be given in Volume 4,
but some properties of such interactions may be mentioned here. It is found
that the relativistic terms in the Hamiltonian of an atom fall into two classes.
One of these contains terms linear with respect to the spin operators of the
electrons, while the other includes quadratic terms. The former correspond
to the interaction between the orbital motion of the electrons and their spin
(this interaction is called spin-orbit interaction), while the latter correspond
to the interaction between the spins of the electrons (spin-spin interaction).
t To determine the coefficient in this formula (when the wave function is known in the region
r ~ 1), we should have to use the expression (36.25) in the range r ;S 1/Z.
248 The Atom §72
Both interactions are of the same order (the second) with respect to vjc y the
ratio of the velocity of the electrons to that of light ; in practice, the spin-
orbit interaction considerably exceeds the spin-spin interaction in heavy
atoms. This is because the spin-orbit interaction increases rapidly with
the atomic number, whereas the spin-spin interaction is essentially inde-
pendent of Z (see below).
The spin-orbit interaction operator is of the form
^ = SA s .s 8 (72.1)
(the summation being over all the electrons in the atom), where s a are the
spin operators of the electrons, and A a are some "orbital" operators, i.e. opera-
tors acting on functions of the co-ordinates. In the self- consistent field
approximation the operators A a are proportional to the operators \ a of the
orbital angular momentum of the electrons, and V s i can then be written in
the form
V s i = 2a a l a . s a . (72.2)
The coefficients in the sum are given in terms of the potential energy U(r)
of the electron in the self-consistent field by
h 2 dU(r a )
2m 2 c 2 r a dr a (72.3)
Since U < and | U(r)\ decreases away from the nucleus, all the <x a > 0.
Regarding the interaction (72.1) as a perturbation, we should, in order to
calculate the energy, average it with respect to the unperturbed state. The
main contribution to the energy is given by distances close to the nucleus,
of the order of the Bohr radius ( ~ h 2 /Zme 2 ) for a nucleus with charge Ze.
In this region the field of the nucleus is almost unscreened and the potential
energy is
£/(r)~Ze 2 /r~Z 2 »*e 4 //* 2 ,
so that
a.~h 2 Ulm 2 c 2 r 2
~Z\e 2 \hcfm<A\h 2 .
The mean value of a is obtained by multiplying by the probability w of
finding the electron near the nucleus. According to (71.3), w~Z~ 2 , so that
we have finally that the energy of the spin-orbit interaction of the electron is
given by
/ Ze 2 \ 2 me 4
\ he J h 2
i.e. differs from the fundamental energy of the outer electrons in the atom
(~me*/h 2 ) only by the factor (Ze 2 /hc) 2 . This factor increases rapidly with
the atomic number, and reaches values of the order of unity in heavy atoms.
§72 Fine structure of atomic levels 249
The actual averaging of the operator (72.1) is done in two steps. First of
all, we average over electron states with given absolute values L and S of the
total orbital angular momentum and spin, but not with given directions of
these. After this averaging V' sl is still, of course, an operator, which we de-
note by f Psl' From considerations of symmetry it is evident that the mean
values of s a must be "directed" along S, which is the only spin "vector"
characterising the atom as a whole (it must be recalled that, in the zeroth ap-
proximation, the wave functions are products of a spin part and a co-ordinate
part). Similarly, the mean values of l a must be "directed" along L. Thus
the operator V~ 8L is of the form
V 8 L=A%l h (72.4)
where A is a constant characterising a given (unsplit) term, i.e. depending
on S and L but not on the total angular momentum / of the atom.
To calculate the energy of the splitting of a degenerate level (with given
S and L), we must solve the secular equation formed from the matrix elements
of the operator (72.4). In this case, however, we already know the correct
functions in the zeroth approximation, in which the matrix of V$l is diagonal.
These are the wave functions of states with definite values of the total
angular momentum J. The averaging with respect to such a state involves
replacing the operator S.L by its eigenvalues, which, according to the general
formula (31.2), are
L-S =IU(J+1)-L(L+1)-S(S+1)].
Since the values of L and S are the same for all the components of a multiplet,
and we are interested only in their relative position, we can write the energy
of the multiplet splitting in the form
44/C7+1). (72.5)
The intervals between adjacent components (with numbers/ and /— 1) are
consequently
AEjj-! = AJ. (72.6)
This formula gives what is called Lande's interval rule (1923).
The constant A can be either positive or negative. For A > the lowest
component of the multiplet level is the one with the smallest possible /, i.e.
J = \L— S\ ; such multiplets are said to be normal. If A < 0, on the other
hand, the lowest level of the multiplet is that with J = L+S; these multi-
plets are said to be inverted.
t This averaging signifies essentially the construction of a matrix with elements
(nM'iM's Wn\ nMiMs)
with all possible Mi, M'l and Ms, M's and diagonal with respect to all the other quantum numbers
(the assembly of which we denote by n).
250 The Atom §72
It is easy to determine the sign of A for the normal states of atoms if the
electron configuration is such that there is only one shell not completely
filled. If this shell is not more than half filled, then according to Hund's
rule (§67) all n electrons in it have parallel spins, so that the total spin has
the greatest possible value, S = \n. Substituting in (72.2) s a = S/n and
taking cc a (which is the same for all electrons in a given shell) outside the
sum we obtain
fitt = («/2S)8.L,
i.e. A = a/2»S> 0. If the shell is more than half full, we first add and sub-
tract in (72.2) the same sum taken over the unoccupied places or "holes" in
the incomplete shell. Since, for a completely filled shell, we should have
V s i = 0, the operator frsi is thereby represented as a sum
V'si = -2 <x a l a .s a ,
taken only over the "holes", the total spin and orbital angular momentum
of the atom being S = — 2 s a , L = — 2 l a . By the same method as pre-
viously we therefore find A — — <x./2S, i.e. A < 0.
From the above we have a simple rule which gives the value of / in the
normal state of an atom with one incompletely filled shell. If this shell
contains not more than half the greatest possible number of electrons for
that shell, then/ = \ L-S\; if the shell is more than half full, J = L + S.
As already mentioned, the spin-spin interaction, unlike the spin-orbit
interaction, is essentially independent of Z. This is evident from the fact
that it is a direct interaction between electrons and does not involve the field
of the nucleus.
For the averaged spin-spin interaction operator we should obtain, analog-
ously to formula (72.4), an expression quadratic in S. The expressions S 2
and (S.L) 2 are quadratic in §. The former has eigenvalues independent
of J, and therefore does not give any splitting of the term. Hence it can be
omitted, and we can write
P SS =:B(S.L)\ (72.7)
where B is a constant. The eigenvalues of this operator contain terms inde-
pendent of J, terms proportional to J(J+l), and finally a term proportional
to J\J+ 1) 2 . The first of these do not give any splitting and hence are without
interest; the second can be included in the expression (72.5), which simply
means a change in the constant A. Finally, the last term gives an energy
iB/ 2 C/+l) 2 . (72.8)
The scheme for the construction of the atomic levels discussed in §§66-67
§72 Fine structure of atomic levels 251
is based on the supposition that the orbital angular momenta of the electrons
combine to give the total orbital angular momentum L of the atom, and their
spins to give the total spin S. As has already been mentioned, this supposi-
tion is legitimate only when the relativistic effects are small; more exactly,
the intervals in the fine structure must be small compared with the differences
between levels with different L and S. This approximation is called the
Russell- Saunders case, and we speak also of LS coupling.
In practice, however, this approximation has a limited range of applica-
bility. The levels of the light atoms are arranged in accordance with the LS
model, but as the atomic number increases the relativistic interactions in the
atom become stronger, and the Russell-Saunders approximation becomes
inapplicable.f It must also be noticed that this approximation is, in parti-
cular, inapplicable to highly excited levels, in which the atom contains an
electron which is in a state with large «, and which is therefore mainly at large
distances from the nucleus (see §68). The electrostatic interaction of this
electron with the motion of the other electrons is comparatively weak, but
the relativistic interaction in the rest of the atom is not diminished.
In the opposite limiting case the relativistic interaction is large compared
with the electrostatic (or, more precisely, compared with that part of it which
governs the dependence of the energy on L and S). In this case we cannot
speak of the orbital angular momentum and spin separately, since they are
not conserved. The individual electrons are characterised by their total
angular momenta/, which combine to give the total angular momentum/ of
the atom. This scheme of arrangement of the atomic levels is called jj coupl-
ing. In practice, this coupling is not found in the pure state, but various types
of coupling intermediate between LS and jj are observed among the levels of
very heavy atoms. J
A peculiar type of coupling is observed in certain highly excited states.
Here the rest of the atom may be in a Russell-Saunders state, i.e. may be
characterised by the values of L and S, while its coupling with the highly
excited electron is of the;)" type ; this is again due to the weakness of the elec-
trostatic interaction for this electron.
The fine structure of the energy levels of the hydrogen atom has certain
characteristic properties. It will be calculated exactly in Volume 4, but here
we shall only mention that, for a given principal quantum number n, the energy
depends only on the total angular momentum j of the electron. Thus the
degeneracy of the levels is not completely removed; to a level with given
n and/ there correspond two states with orbital angular momenta I = j ± |
(unless/ has the value « — -J, which is the greatest possible for a given n).
t Nevertheless, it must be mentioned that, although the quantitative formulae which describe
this type of coupling become inapplicable, the method of classifying levels according to this scheme
may itself remain meaningful for heavier atoms, especially for the lowest states (including the normal
state).
% For further details regarding types of coupling and the quantitative aspect of the problem, see,
for instance, E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, Cambridge University
Press 1935
252 The Atom §73
Thus the level with n = 3 is split into three levels, of which the states
S\I2, pi/2 correspond to one, ^3/2 and ^3/2 to another, and ^5/2 to the third.
§73. The periodic system of D. I. Mendeleev
The elucidation of the nature of the periodic variation of properties, ob-
served in the series of elements when they are placed in order of increasing
atomic number, requires an examination of the peculiarities in the successive
completion of the electron shells of atoms. The theory of the periodic system
is due to N. Bohr (1922).
When we pass from one atom to the next, the charge is increased by unity
and one electron is added to the envelope. At first sight we might expect
the binding energy of each of the successively added electrons to vary
monotonically as the atomic number increases. The actual variation, how-
ever, is entirely different.
In the normal state of the hydrogen atom there is only one electron, in
the Is state. In the atom of the next element, helium, another Is electron is
added; the binding energy of the Is electrons in the helium atom is, however,
considerably greater than in the hydrogen atom. This is a natural conse-
quence of the difference between the field in which the electron moves
in the hydrogen atom and the field encountered by an electron added to the
He + ion. At large distances these fields are approximately the same, but
near the nucleus with charge Z = 2 the field of the He + ion is stronger than
that of the hydrogen nucleus with Z = 1. In the lithium atom (Z = 3), the
third electron enters the 2s state, since no more than two electrons can be
in Is states at the same time. For a given Z the 2s level lies above the Is
level; as the nuclear charge increases, both levels become lower. In the
transition from Z = 2 to Z = 3, however, the former effect is predominant,
and so the binding energy of the third electron in the lithium atom is con-
siderably less than those of the electrons in the helium atom. Next, in the
atoms from Be (Z = 4) to Ne (Z = 10), first one more 2s electron and then
six 2p electrons are successively added. The binding energies of these
electrons increase on the average, owing to the increasing nuclear charge.
The next electron added, on going to the sodium atom (Z = 11), enters
the 3s state, and the binding energy again diminishes markedly, since the
effect of going to a higher shell predominates over that of the increase of the
nuclear charge.
This picture of the filling up of the electron envelope is characteristic of the
whole sequence of elements. All the electron states can be divided into
successively occupied groups such that, as the states of each group are occu-
pied in a series of elements, the binding energy increases on the average, but
when the states of the next group begin to be occupied the binding energy
decreases noticeably. Fig. 24 shows those ionisation potentials of elements
that are known from spectroscopic data; they give the binding energies of
the electrons added as we pass from each element to the next.
§73
The periodic system of D. I. Mendeleev
253
254 The Atom §73
The different states are distributed as follows into successively occupied
groups :
1* 2 electrons \
2s, Ip 8
3s , Zp 8
4s , 3 d, \p 18
Is, 4d t Sp 18
(73.1)
6s, 4/, Sd, dp 32
7*,6<f,5/,... /
The first group is occupied in H and He; the occupation of the second and
third groups corresponds to the first two (short) periods of the periodic
system, containing 8 elements each. Next follow two long periods of 18 ele-
ments each, and a long period containing the rare-earth elements and 32
elements in all. The final group of states is not completely occupied in the
natural (and artificial transuranic) elements.
To understand the variation of the properties of the elements as the states
of each group are occupied, the following property of d and / states, which
distinguishes them from s and/) states, is important. The curves of the effec-
tive potential energy of the centrally symmetric field (composed of the electro-
static field and the centrifugal field) for an electron in a heavy atom have a
rapid and almost vertical drop to a deep minimum near the origin; they then
begin to rise, and approach zero asymptotically. For s and p states, the rising
parts of these curves are very close together. This means that the electron
is at approximately the same distance from the nucleus in these states. The
curves for the d states, and particularly for the / states, on the other hand,
pass considerably further to the left; the "classically accessible" region
which they delimit ends considerably closer in than that for the 5 and^ states
with the same total electron energy. In other words, an electron in the d and/
states is mainly much closer to the nucleus than in the s and p states.
Many properties of atoms (including the chemical properties of elements;
see §81) depend principally on the outer regions of the electron envelopes. The
above characteristic of the d and /states is very important in this connection.
Thus, for instance, when the 4/ states are being filled (in the rare-earth ele-
ments ; see below), the added electrons are located considerably closer to the
nucleus than those in the states previously occupied. As a result, these
electrons have practically no effect on the chemical properties, and all the
rare-earth elements are chemically very similar.
The elements containing complete d and / shells (or not containing these
shells at ail) are called elements of the principal groups; those in which the
filling up of these states is actually in progress are called elements of the inter-
mediate groups. These groups of elements are conveniently considered separ-
ately.
§73
The periodic system of D. I. Mendeleev
255
Let us begin with the elements of the principal groups,
helium have the following normal states:
Hydrogen and
iH:b 2 5 1/2
2 He:ls 21 »So
(the number with the chemical symbol always signifies the atomic number).
The electron configurations of the remaining elements of the principal groups
are shown in Table 3.
In each atom, the shells shown on the right of the table in the same line
and above are completely filled. The electron configuration in the shells
that are being filled is shown at the top, while the principal quantum number
of the electrons in these states is shown by the figure on the left of the table
in the same line. The normal states of the whole atom are shown at the bot-
tom. Thus, the aluminium atom has the electron configuration Is 2 2s 2 2p* 3s 2
The values of L and S in the normal state of the atom can be determined
(the electron configuration being known) by means of Hund's rule (§67), and
the value of/ is determined by the rule given in §72.
Table 3
Electron configurations of the atoms of elements in the principal groups
5
s*
s*p
s*p*
5 4 i>»
*V
s*p s
s*p*
n = 2
3Li
4 Be
5 B
eC
7N
8 o
9F
ioNe
U*
3
iiNa
12Mg
13AI
i 4 Si
15P
ioS
17C1
iaAr
2s* 2p*
4
ioK
2oCa
3s 2 3p*
4
29C11
3oZn
3iGa
32Ge
33AS
34Se
3 5 Br
3eKr
3d 10
5
37 Rb
ssSr
4s a Ap*
5
47Ag
48Cd
49I11
soSn
5iSb
52Te
53I
54Xe
4d 10
6
55CS
56Ba
5s* 5p*
6
79AU
soHg
81TI
8 2 Pb
83Bi
84P0
85At
8eRn
4/ 1 * 5d l °
7
87Fr
ssRa
6s % 6p*
*s 1/t
'So
■Pi/»
'Po
4 S./«
'P*
*Pt/t
l S
The atoms of the inert gases (He, Ne, Ar, Kr, Xe, Rn) occupy a special
position in the table: the filling up of one of the groups of states listed in
(73.1) is completed in each of them. Their electron configurations have
unusual stability (their ionisation potentials are the greatest in their respective
series). This causes the chemical inertness of these elements.
We see that the occupation of different states occurs very regularly in the
series of elements of the principal groups : first the s states and then the p
states are occupied for each principal quantum number n. The electron
configurations of the ions of these elements are also regular (until electrons
from the d and /shells are removed in the ionisation): each ion has the con-
figuration corresponding to the preceding atom. Thus, the Mg+ ion has
the configuration of the sodium atom, and the Mg++ ion that of neon.
256
The Atom
§73
Let us now turn to the elements of the intermediate groups. The filling
up of the 3d, 4d, and Sd shells takes place in groups of elements called
respectively the iron group, the palladium group and the platinum group.
Table 4 gives those electron configurations and terms of the atoms in these
groups that are known from experimental spectroscopic data. As is seen from
this table, the d shells are filled up with considerably less regularity than the
s and p shells in the atoms of elements of the principal groups. Here a
Table 4
Electron configurations of the atoms of elements in the iron, palladium and
platinum groups
Iron group
2lSc
22 Ti
23V
24Cr
25Mn
26Fe
27C0
28Ni
Ar envelope +
3d As 2
2 D 3/2
3d 2 4s 2
*F 2
3d* 4s 2
*F 3/2
3d 6 4s
'S 3
3d* 4s*
<s 5/2
3d 9 4s 2
8 #4
3d 7 4s 2
*F tlt
3d 6 4s 2
*F t
Palladium group
39Y
4oZr
4lNb
42M0
43TC
44RU
4sRh
4ePd
Kr envelope +
4d5s 2
2 D 3/2
4<f 2 5s 2
S F 2
4d*5s
"A/2
4d 5 5s
'S 3
4rfs 5s 2
6 5 6 /2
4J' 5s
5 F 6
4d*5s
*F 9/2
4d 10
'S
Platinum group
57La
Xe envelope +
5d6s s
*D 3/2
7iLu
7 2 Hf
73Ta
74W
7sRe
7eOs
nlr
7 8 Pt
Xe envelope 1
+4/"+ )
5 d 6s 2
2 D 3/2
5d* 6s 2
5d* 6s 2
4 F 3 /2
5d l 6s 2
6 D
5d 5 6s 2
e s 6/2
5d« 6s 2
5 £>4
5J7 6s 2
5d°6s
*D 3
characteristic feature is the "competition" between the s and d states. It is
seen in the fact that, instead of a regular sequence of configurations of the
type dP s 2 with increasing/), configurations of the type d& +1 s or dv+ 2 are often
found. Thus, in the iron group, the chromium atom has the configuration
3d 5 4s, and not 3d 4 4s 2 ; after nickel with 8 d electrons, there follows at once
the copper atom with a completely filled d shell (and hence we place this
§73 The periodic system of D. I. Mendeleev 257
element in the principal groups). This lack of regularity is observed in the
terms of ions also: the electron configurations of the ions do not usually
agree with those of the preceding atoms. For instance, the V+ ion has the
configuration 3d* (and not 3d 2 4-s 2 like titanium) ; the Fe+ ion has 3d 6 4s
(instead of 3d 5 4s 2 as in manganese). We may remark that all ions found
naturally in crystals and solutions contain only d (not s or p) electrons in their
incomplete shells. Thus iron is found in crystals or solutions only as the
ions Fe++ and Fe +++ , whose configurations are 3d 6 and 3d 5 respectively.
A similar situation occurs in the filling up of the 4/ shell ; this takes place
in the series of elements known as the rare earths (Table 5).f The filling up
of the 4/ shell also occurs in a slightly irregular manner characterised by the
"competition" between 4/, Sd and 6s states.
The last group of intermediate elements begins with actinium. In this
group the 6d and 5/ shells are filled, similarly to what happens in the group
of rare- earth elements (Table 6).
To conclude this section, let us examine an interesting application of the
Thomas-Fermi method. We have seen that the electrons in the p shell first
appear in the fifth element (boron), the d electrons for Z = 21 (scandium),
and the/electrons for Z = 58 (cerium). These values of Z can be predicted
by the Thomas-Fermi method, as follows.
An electron with orbital angular momentum / in a complex atom moves
with an "effective potential energy" J of
The first term is the potential energy in an electric field described by the
Thomas-Fermi potential <f>(r). The second term is the centrifugal energy,
in which we put (/+i) 2 instead of /(/+1), since the motion is quasi-classical.
Since the total energy of the electron in the atom is negative, it is clear that,
if (for given values of Z and /) U t {r) > for all r, there can be no electrons
in the atom concerned with the given value of the angular momentum /. If
we consider any definite value of /and vary Z, it is found that in fact U^r) >
everywhere when Z is sufficiently small. As Z is increased, a value is reached
for which the curve of U t (r) touches the axis of abscissae, while for larger Z
there is a region where C/ Z (r) < 0. Thus the value of Z at which electrons
with the given / appear in the atom is determined by the condition that the
curve of £/j(r) touches the axis of abscissae, i.e. by the equations
Ufc) = -t+W+Wlr 2 = 0, U t \r) = _^(r)-(/+£) 2 /r* = .
Substituting here the expression (70.6) for the potential, we obtain the
equations
Z*l\{x)lx = (4/3^/3(/+i) 2 /^,
Z*l*[x x '{x)- X {x)\lx = -2(4/37r) 2 / 3 (/+i) 2 /* 2 .
f In books on chemistry, lutetium is also usually placed with the rare-earth elements. This, how-
ever, is incorrect, since the 4/ shell is complete in lutetium; it must therefore be placed in the platinum
group, as in Table 4.
% As in §70, we use atomic units.
258
The Atom
§73
s\s
<3
iv.
ft
f
o
fel
w
X
Q
&3
^
^
i>
^
i>
i>
*S>
§
CO
i.
&3
i->
«
u
vO
IM
OS
>-l
S>
•*
C3
•O
\o
k* 1
to
^
+
X
2
<s
9 g
K
.O
■*»»
s
ft
o
SJ
«
fel
6
o
to
o
>0 <N
4
■» Co
3
Til
03
CM
CO
^ r°
<M
«0
eg
«o
if £
X!
o
o>
<o
vO
00
SO
+
V
a
o
c
5 1
Dividing each side of the second equation by the corresponding side of the
first, we find for x the equation
X '(x)lx(x) = -1/*,
§74 X-ray terms 259
and we then calculate Z from the first of equations (73.2). A numerical
calculation gives
Z = 0- 155(2/+ 1) 3 .
This formula determines the value of Z for which electrons with a given /
first appear in the atom; the error is about 10 per cent.
Very accurate values are obtained by taking the coefficient as 017 instead
of 0155:
Z =0-17(2/+ 1) 3 . (73.3)
For / = 1, 2, 3 this formula gives respectively, after rounding to the nearest
integer, the correct values 5, 21, 58. For / = 4, formula (73.3) gives Z = 124;
this means that^ electrons should first appear only in the 124th element.
§74. X-ray terms
The binding energy of the inner electrons in the atom is so large that, if
such an electron makes a transition into an outer unfilled shell (or is removed
from the atom), the excited atom (or ion) is mechanically unstable with
respect to ionisation, which is accompanied by the reconstruction of the
electron envelope and the formation of a stable ion. However, because of the
comparatively weak interaction between the electrons in the atom, the prob-
ability of such a transition is comparatively small, so that the lifetime t of
the excited state is long. Hence the "width" k/r of the level (see §44) is so
small that it is reasonable to regard the energies of an atom with an excited
inner electron as discrete energy levels of "quasi-stationary" states of the
atom. These levels are called X-ray terms.-f
The X-ray terms are primarily classified according to the shell from which
the electron is removed, or in which, as we say, a "hole" is formed. Where
the electron goes has almost no effect on the energy of the atom, and hence
is unimportant.
The total angular momentum of the set of electrons occupying any shell is
zero. When one electron has been removed, the shell acquires some angular
momentum j. For the («, /) shell, the angular momentum j can obviously
take the values /db£. Thus we obtain levels which might be denoted by
ls 1/2 , 2s 1/2 , 2p 1/2 , 2p 3/2 , ..., where the value of j is added as a suffix to the
letter giving the position of the "hole". It is usual, however, to employ
special symbols as follows :
ls 1/2 2s 1/2 2p 1/2 2p3/z 3s 1/2 3p 1/2 3p 3/2 3d 3/2 3d 5/2 ...
K Li Lh Liu Mi M u M iu M w My ...
The levels with « = 4, 5, 6 are similarly denoted by the letters N, O, P.
Levels with the same n (denoted by the same capital letter) lie close together
f The name is due to the fact that transitions between these levels cause the emission of X-rays
by the atom.
260 The Atom §74
and at a distance from levels with a different n. The reason for this is that,
owing to the relative nearness of the inner electrons to the nucleus, they are
in the almost unscreened field of the nucleus, and hence their states are
"hydrogen-like"; the energy is, to a first approximation, ~Z 2 /2n 2 (in atomic
units), i.e. depends only on n. If relativistic effects are taken into account,
terms with different j are separated (cf. the discussion in §72 of the fine
structure of the hydrogen levels), such as, for example, L\ and Lu from Lm,
and Mi and M\\ from Mm and Miy. These pairs of levels are said to be
regular (or relativistic) doublets. The separation of terms with different / and
the same 7 (for instance L\ and Lu, M\ and Mu) is due to the deviation of the
field in which the inner electrons move from the Coulomb field of the nucleus,
i.e. to the taking into account of the interaction of the electron with other
electrons. These are said to be irregular (or screening) doublets. The main
correction term to the "hydrogen- like" energy of the electron results from
the potential due to the remaining electrons in the region near the nucleus,
and is proportional to Z 4/3 (see (70.8)). However, since this correction does
not depend on either n or /, it does not affect the level spacings. The principal
correction terms in the level differences are therefore due to the interaction
of one electron with those adjoining it. Since the distances between the inner
electrons are r ~ 1/Z (the Bohr radius in the field of a charge Z), the energy
of this interaction is ~ \jr ~ Z. Taking this correction into account, we can
write the energy of an X-ray term, to the same accuracy, as —(Z—8) 2 j2n 2 ,
where 8 = 8{n,l) is a quantity small compared with Z, and may be regarded
as the magnitude of the screening of the nuclear charge.
Terms with two and three "holes" may exist in the electron shells together
with the X-ray terms with one "hole". Since the spin-orbit interaction is
strong for the inner electrons, the holes are subject toji)' coupling.
The width of an X-ray term is determined by the total probability of all
possible processes by rearrangement of the electron envelope of the atom so
as to fill the "hole" in question. In the heavy atoms, transitions of the hole
from a given shell to a higher one (i.e. electron transitions in the opposite
direction) are the most important, and are accompanied by the emission of
X-ray quanta. The probability of these "radiative" transitions, and therefore
the corresponding part of the level width, increase very rapidly with the
atomic number (as Z 4 ) but decrease towards higher levels for a given Z.f
For lighter atoms (and higher levels) an important or even predominant
part is played by radiationless transitions, in which the energy liberated when
a hole is filled by an electron from above goes to remove another inner
electron from the atom (called the Auger effect). As a result of this process
the atom is in a state with two holes. The probabilities of these processes and
the corresponding contribution to the level width are independent of the
atomic number to a first approximation with respect to 1/Z (see the Problem).:):
t See Volume 4.
X As an example it may be mentioned that the Auger width of the K level is about 1 eV, and reaches
values of the order of 10 eV for higher levels.
§75 Multipole moments 261
PROBLEM
Find the limiting law of dependence of the Auger width of X-ray terms on atomic number
when the latter is sufficiently large.
Solution. The Auger transition probability is proportional to the square of a matrix
element of the form
M = ( Uw^VfafcdVidVz,
where ipi, <p2 and if/^, tp'a are the initial and final wave functions of the two electrons involved
in the transition, and V = e 2 /ri2 is their interaction energy. When Z is sufficiently large, the
wave functions of the inner electrons may be regarded as hydrogen-like and the screening of
the field of the nucleus by other electrons may be neglected (the wave function of the ionisa-
tion electron is also hydrogen-like in the region within the atom which is of importance in the
integral M). If we carry out the calculations and all quantities are expressed in Coulomb
units (with the constant a = Ze 2 ; see §36), then the only quantity in the integral M which
depends on Z is V = \jZriz, so that M ~ \\Z. The transition probability, and therefore the
Auger width AE of the level, are proportional to \jZ 2 . On returning to ordinary units (the
Coulomb unit of energy being Z 2 me i jh 2 ), we find that AE is independent of Z.
§75. Multipole moments
In classical theory, the electrical properties of a system of particles are
described by its multipole moments of various orders, expressed in terms of
the charges and co-ordinates of the particles. In the quantum theory, the
definitions of these quantities are the same in form, but they must now be
regarded as operators.
The first multipole moment is the dipole moment, defined as the vector
d= Ser,
where the summation is over all the particles, and the suffix which numbers
the particles is omitted for brevity. The matrix of this operator, like that of
any polar vector (see §30), has non-zero elements only for transitions between
states of different parity. The diagonal elements are therefore always zero.
In other words, the mean values of the dipole moment of any system of par-
ticles (e.g. an atom) in stationary states are zero.f
The same is evidently true of all 2*-pole moments with odd /. The com-
ponents of such a moment are polynomials of odd degree / in the co-ordinates,
which, like the components of a polar vector, change sign on inversion of
the co-ordinates. The same parity selection rule therefore applies.
The quadrupole moment of a system is defined as the symmetrical tensor
Q tk ='Ze(3xiX k -8i k r 2 ), (75.1)
t To avoid misunderstanding it should be emphasised that this refers to a closed system of particles
or to a system of particles in a centrally symmetric external electric field. For example, if the nuclei
are regarded as "fixed", the above statement is valid for the electrons in an atom, but not for those in
a molecule.
It is also assumed that there is no additional ("accidental") degeneracy of the energy level other than
that with respect to directions of the total angular momentum. If this is not so, wave functions of
stationary states can be constructed which do not have any definite parity, and the corresponding
diagonal elements of the dipole moment need not vanish.
262 The Atom §75
the sum of whose diagonal terms is zero. The determination of the values of
these quantities in a particular state of a system (an atom, say) requires an
averaging of the operator (75.1) over the corresponding wave function. This
averaging is conveniently carried out in two stages (cf. §72).
Let Q ile denote the quadrupole moment operator averaged over the electron
states with a given value of the total angular momentum / (but not of its com-
ponent Mj). The only vector which pertains to the atom as a whole is the
"vector" J. The only symmetrical tensor operator with zero trace is therefore
of the form
IQ
Q« = UtJic+Mi-mvc)- (75.2)
The constant Q is defined so that Q zz = Q in the state with Mj = J; this
quantity is usually called simply the quadrupole moment.
The operators J\ must be regarded as the known matrices of the angular
momentum with a given value of J but non- diagonal with respect to Mj\ the
operator j 2 can, of course, be simply replaced by its eigenvalue /(/+ 1).
For / = (so that Mj = also) all the elements of these matrices are
zero, i.e. the operators vanish identically. This occurs also when / = \.
This is easily seen by direct multiplication of the Pauli matrices (55.6),
wh ch are the matrices of the components of any angular momentum equal
to£.
This is no accident, but is a particular case of the general rule that the tensor
of a 2*-pole moment (with even /) is non-zero only for states of the system
with total angular momentum / > |/.f
PROBLEMS
Problem 1. Find the relation between the operators of the quadrupole moment of an
atom in states corresponding to various components of the fine structure of a level (i.e. states
with different values of J but given values of L and S).
Solution. In states with given values of L and S, the operator of the quadrupole moment,
a purely "orbital" quantity, depends only on the operator L, and so is given by the same
formula (75.2) with J replaced by L and with a different constant Q. The operator (75.2)
is obtained by a further averaging over the state with a given value of/:
SQj . .
Qik = 27(27- 1) W* + -foft-&fl/+ ^
3Ql
2L(L-1)
[til k +t k ti-%L(L+l)8t k ]. (1)
t This rule is a consequence of the general properties of symmetry with respect to rotations. The
tensor of the 2*-pole moment £)(') is an irreducible tensor of rank / (see The Classical Theory of Fields,
§41); its transformation properties correspond to those of a symmetrical spinor of rank 21 (§58).
The wave functions t/ij correspond to a spinor of rank 2J. The matrix elements of the operator
Z)W are non-zero if, in the integrals which determine them, the integrand (tfu* DW xfij) contains a
scalar part. This part is obtained by contraction with respect to all pairs of spinor indices, with each
pair belonging to different factors (tfu*, ipj and DM); otherwise the result is zero. It is clear that
such a contraction is possible only if the sum of the numbers of indices in tfu and tfu* is not less than
the number of indices in £>('), i.e. if 4/ ^ 21.
§75 Multipole moments 263
It is required to find the relation between the coefficients Qj and Ql> To do so, we multiply
equation (1) on the left by ji and on the right by /*, sum over i and k, and take the eigen-
values of the diagonal operators. We have
where, by (31.3),
2JX =/(/+l)+L(L+l)-S(S+l).
The product ji £* Li j* can be transformed by means of the formulae
{LiLic} = ieacitu {Jiti} = ieumLm,
as in §29, Problem 2; the result is
jit k ij k = (m*-j.L.
Similarly
MW* = (J 2 ) 2 ,
AW*/*=J 2 (J 2 -i)-
Thus we obtain from (1) the relation
3J.L(2J.L- 1 )-2/(7 + l)L(£ + l)
(7+l)(2/+3)L(2i-l)
In particular, for S = \ this formula gives
Qj=Ql for/ = L+J,
(L-l)(2L+3)
Qj= Ql- for 7 = L-*.
* * L(2L+1)
(3)
Problem 2. Express the quadrupole moment of an electron (charge —e) with orbital
angular momentum / in terms of the mean square of its distance from the centre.
Solution. We have to average the expression
Q zz = - er\Z cos20- 1) = -er\ln z *-\)
over a state with given angular momentum I and component m — I. The mean value of the
angle factor is found immediately from the formula derived in §29, Problem 2 (where l z
must be replaced by /) ; the result is
- 2/
Qi = er* . (4)
* 2/+3
The sign of this quantity is opposite to that of the electron charge (— e), as it should be:
a particle moving with an angular momentum in the jsr-direction is mainly near the plane
z = 0, and hence cos 2 < \.
For an electron with a given value of / = / ± £, formulae (3) give
& = " 2 (2/-l)/(2/+2). (5)
Problem 3. Determine the quadrupole moment of an atom (in the ground state) in which
all v electrons in excess of closed shells are in equivalent states with orbital angular momen-
tum /.
264 The Atom §75
Solution. Since the total quadrupole moment of completed shells is zero, the quad-
ruple moment operator of the atom is given by the sum
Qa = (2/-1X2/+3) 2 |> +/ ' ? '-^+ D«4
taken over the v outer electrons (here we have used formula (4)).
Let us first suppose that v ^ 21+1, i.e. at most half the places in the shell are occupied.
Then, by Hund's rule (§67), the spins of all the v electrons are parallel (so that S = £v).
This means that the spin wave function of the atom is symmetrical, and the co-ordinate
wave function therefore antisymmetrical, with respect to these electrons. Thus the electrons
must all have different values of m, so that the greatest possible value of Ml (and the value
of L, which is the same) is
I
L = (M L ) m&x = SJ m = \v{2l-v+\).
The required Ql is the eigenvalue Q zz for Ml = L. We therefore have
6er 2
(2/-l)(2/+3)
6er 2 ^-v
(2/-1Y2/+3) ^ L 3V }b
whence, on calculating the sum,
2/(2/- 2v+\) _
Ql = — -er\
(2/-l)(2/+3)
The final change from Ql to Qj is effected by means of formula (2).
The case of an atom whose outer shell is more than half filled is reduced to the previous
one by considering "holes" instead of electrons: the result is therefore given by the same
formula (6) with the opposite sign (the "hole" charge being +e), v being now taken not as
the number of electrons but as the number of unoccupied places in the shell.
Problem 4. Determine the quadrupole splitting of the levels in an axially symmetric
external electric field.f
Solution. Owing to the axial symmetry of the field (taking the axis of symmetry to be
the 2-axis) we have d^/dx 2 = d^/dy 2 , where <f> is the field potential, and A<f> = from the
electrostatic equation; hence d^/dz 2 = -28m dx 2 . The Hamiltonian of the quadrupole
moment in the external field isf
#=£
3 a2 + 1>
whence in this case
8=AJx+L 2 )-2AJ z 2
= A{p-ZJ z %
"2/(2/- 1) dx*.
Replacing the operators by their eigenvalues, we obtain the displacement of the levels:
AE = A[J(J+l)-3Mj2].
t A similar problem for an arbitrary field is discussed in §103, Problem 5.
t See The Classical Theory of Fields, §42.
§76 The Stark effect 265
§76. The Stark effect
If an atom is placed in an external electric field, its energy levels are altered ;
this phenomenon is known as the Stark effect.
In an atom placed in a homogeneous external electric field, we have a
system of electrons in an axially symmetric field (the field of the nucleus
together with the external field). The total angular momentum of the atom
is therefore, strictly speaking, no longer conserved; only the projection Mj
of the total angular momentum J on the direction of the field is conserved.
The states with different values of Mj have different energies, i.e. the electric
field removes the degeneracy with respect to directions of the angular
momentum. The removal is, however, incomplete: the states differing only
in the sign of Mj are degenerate as before. For an atom in a homogeneous
external electric field is symmetrical with respect to reflection in any plane
passing through the axis of symmetry (i.e. the axis passing through the nucleus
in the direction of the field; we shall take this as the sr-axis). Hence the states
obtained from one another by such a reflection must have the same energy.
On reflection in a plane passing through some axis, however, the angular
momentum about this axis changes sign (the direction of a positive revolution
about the axis becomes that of a negative one).
We shall suppose that the electric field is so weak that the additional energy
due to it is small compared with the distances between neighbouring energy
levels of the atom, including the fine-structure intervals. Then, in order to
calculate the displacement of the levels in the electric field, we can use the
perturbation theory developed in §§38 and 39. Here the perturbation
operator is the energy of the system of electrons in the homogeneous field <f,
and this is
V= -d. «?= -£d 2 , (76.1)
where d is the dipole moment of the system. In the zeroth approximation,
the energy levels are degenerate (with respect to directions of the total angular
momentum); in the present case, however, this degeneracy is unimportant,
and in applying perturbation theory we can proceed as if we were dealing
with non- degenerate levels. This follows from the fact that, in the matrix
of the quantity d z (as in that of the ^-component of any vector), only the
elements for transitions without change of Mj are not zero (see §29), and hence
states with different values of Mj behave independently when perturbation
theory is applied.
The displacement of the energy levels is determined, in the first approxi-
mation, by the diagonal matrix elements of the perturbation. But all the
diagonal matrix elements of the dipole moment vanish identically (§75).
Thus the splitting of the levels in an electric field is a second-order effect
and is proportional to the square of the field.f It must be calculated according
f The hydrogen atom forms an exception; here the Stark effect is linear in the field (see the next
section). The atoms of other elements, when in highly excited states (and therefore hydrogen-like;
see §68), behave like hydrogen in sufficiently strong fields.
266 The Atom §76
to the general rules of perturbation theory, but the dependence of the dis-
placement of the levels on the quantum number Mj can be established from
more general considerations.
Being quadratic in the field, the displacement AE n of the level E n must be
of the form
AE n = - £a a (n) <W*, (76.2)
where a^") is a symmetrical tensor of rank two; taking the s-axis in the
direction of the field, we obtain
AE n = -\^n)^. (7 6 .3)
The tensor a tt <»> is specific to the (unsplit) level concerned, and depends
also on the number Mj. The values of a tt <»> for various values of Mj can
be regarded as eigenvalues of the operator
a« (B) = KnSvc+pniJiJk+JicJi-iSap). (76.4)
This is the most general symmetrical tensor of rank 2 depending on the
vector j (cf. §75). From (76.3) and (76.4) we have
AE n = -K 2 K+2£„[M./2-i/C/ + 1)]}. (76.5)
It may be noted that, on summation over all values of Mj t the second term
in the braces vanishes, so that the first term is the displacement of the
"centre of gravity" of the split level. Moreover, according to (76.5) a level
with J =\ remains unsplit, in accordance with Kramers' theorem (§60).
The tensor a«f<») which appears in the above formulae is also the polaris-
ability of the atom in the external electric field. According to a general
formula,
(8Hld\) nn = dE n ld\
(see §11, Problem). Taking the parameters A to be the components of the
vector € and putting fi = i? -^.d, we find for the mean value of the dipole
moment of the atom
dt = a^Hf*. (76.6)
If the atom is in a non-uniform external field (which varies only slightly
over the dimensions of the atom), there can also exist a splitting effect linear
in the field, due to the quadrupole moment of the atom. The operator of the
quadrupole interaction between the system and the field has the form which
corresponds to the classical expression-)- for the quadrupole energy:
^ =i ^:^' (76 - 7)
f See The Classical Theory of Fields, §42.
76§ The Stark effect 267
where <f> is the potential of the electric field (the derivatives being understood
to be taken at the position of the atom).
We can, in particular, apply this formula to a neutral atom in the field of
an electron which is at a distance large compared with atomic dimensions.
Then the field of the electron at the position of the atom satisfies the con-
dition of quasi-uniformity, and in the first approximation of perturbation
theory we find that the energy of interaction between the electron and the
atom is proportional to 1/r 3 (since j> ~ 1/r).
This result applies, however, only to states with given values of the com-
ponent Mj of the total angular momentum. On averaging over all directions
of the angular momentum (i.e. over all values of Mj\ the interaction pro-
portional to 1/r 3 vanishes, since Qu = 0. Furthermore, this interaction does
not exist if the angular momentum J of the atom is or |, since then Qm = ;
see §75.
The next term in powers of 1/r, which is never zero, is the interaction which
appears in the second-order perturbation theory with respect to the dipole
operator (76.1). Since the field of the electron & ~ 1/r 2 , the energy U of this
interaction is proportional to 1/r 4 . If the atom is in the normal state, this
energy (like any second-order correction to the energy of the ground state ;
see §38) is negative, i.e. there is a force of attraction between the atom and
the electron.
This attraction is the reason why some atoms are able to form negative
ions by the attachment of an electron.! This property is not shared by all
atoms, however, the reason being that, in a field which decreases at large
distances as 1/r 4 (or 1/r 3 ), the number of levels (corresponding to a finite
motion of the electron) is always finite, and in some cases there may be no
such levels.
PROBLEMS
Problem 1. Determine the Stark splitting of the different components of a multiplet
level as a function of /.
Solution. The problem is conveniently solved by changing the order in which the per-
turbations are applied ; we first consider the Stark splitting of the level in the absence of fine
structure, and then bring in the spin-orbit interaction. Since the spin of the atom does not
interact with the external electric field, the Stark splitting of a level with orbital angular
momentum L is given by a formula of the same form (76.2), with a tensor a.ac which is ex-
pressed in terms of the operator L in the same way as oeifc in (76.4) is expressed in terms of J:
«4k = a8ik+b(LiLjc+LicLi—%8ikL 2 ),
the suffixes n being everywhere omitted. When the spin-orbit interaction is included, the
states of the atom must be described by the total angular momentum J. The averaging of
the operator oloc over states with a given value of the angular momentum J (but not of its
component Mj) is formally identical with the averaging carried out in §75, Problem 1. We
hus return to formulae (76.4), (76.5), with constants a,0 which are given in terms of the
f For example, the halogen atoms attach an electron with a binding energy of the order of 2 to 4 eV,
and the hydrogen atom does so with a binding energy 0*7 eV.
268 The Atom §76
constants a, b by
3J.L[2J.L-l]-2/(7+l)L(L+l)
a = a, p = b — .
7(/+l)(2/-l)(2/+3)
This determines the splitting as a function of J (but not of L and S, of course; these are
characteristics of the unsplit term on which the constants a and b also depend).
Problem 2. Determine the splitting of a doublet level (spin S = £) in an arbitrary
(not weak) electric field.
Solution. If the splitting is not small in comparison with the interval between the
components of the doublet, the perturbation from the electric field and the spin-orbit inter-
action must be taken into account simultaneously, i.e. the perturbation operator is the sum
P = AS.L-$<?2{a+2b[L z 2-±L(L+l)]}
(cf. (72.4) and Problem 1). Omitting the constant terms which do not affect the splitting,
we can write this operator in the form
V = \A[S + L-+ S-L + +2£ z l z -b^L z z]
(see (29.11)). For each given value of Mj the eigenvalues of this operator are determined by
the roots of the secular equation formed from its matrix elements with respect to the states :
(2)M L = Mj+l M s = -%.
From formulae (27.12) we find
Vn = \A(Mj-\)-b£\Mj-\)\
V22 = -lA(Mj+i)-b£2(Mj+i)2,
V 12 = iAV[(L+Mj+$)(L-Mj+$].
Thus (see §39, Problem 1) the level displacement is
AE= -bg2M J 2±V[lA2(L+i)2+b<?2(b£ , 2+A)Mj2], (1)
where all terms which are the same for all components of the split doublet are omitted.
This formula (with both signs of the root) applies to all levels with \Mj\ < L— J. For
Mj = L + \ {or Mj = — (L + i)) there is no state 2 (or 1); for this level the displacement is
given simply by the matrix element Vn (or F 22 ), i.e., with the same choice of the additive
constant as in (1),
AE = {\A+b£*){L+\)-b£Z{L+®z. (2)
This is the same as the result obtained from formula (1) with only one sign of the root.
Problem 3. Determine the quadrupole splitting of levels in an axially symmetric electric
field.
Solution. In a field symmetrical about the s-axis we have 2 «£/&e 2 = 8 2 <f>/dy 2 = a,
8^<f>ldz 2 = —2a, the remaining second derivatives being zero. The quadrupole energy
operator (76.7) is
a - - - Qo<* „
-iQxx+Qyy-2Q zz )= (J 2 -3/ 2 2).
6 2/(2/- 1)
Replacing the operators by their eigenvalues, we obtain for the displacement of the levels
AE = a [J(J+ 1) - 3Mj2].
§77 The Stark effect in hydrogen 269
§77. The Stark effect in hydrogen
The levels of the hydrogen atom, unlike those of other atoms, undergo
a splitting proportional to the field (the linear Stark effect) in a uniform
electric field. This is due to the occurrence of an accidental degeneracy in
the hydrogen terms, whereby states with different / (for a given principal
quantum number n) have the same energy. The matrix elements of the dipole
moment for transitions between these states are not zero, and hence the secular
equation gives a non-zero displacement of the levels, even in the first approxi-
mation.
For purposes of calculation-}- it is convenient to choose the unperturbed
wave functions so that the perturbation matrix is diagonal with respect to
each group of mutually degenerate states. It is found that this is achieved
by quantising the hydrogen atom in parabolic co-ordinates. The wave func-
tions ift nintm of the stationary states of the hydrogen atom in parabolic co-
ordinates are given by formulae (37.15) and (37.16).
The perturbation operator]: (the energy of an electron in the field &) is
Sz = \ &($—-n)> the field is directed along the positive s-axis, and the force
on the electron along the negative sr-axis. We are interested in the matrix
elements for transitions n^m -> n^n^m', for which the energy (i.e. the prin-
cipal quantum number n) is unaltered. It is easy to see that, of these, only
the diagonal matrix elements
CO0O27T
J ^..J 1 '* dV = K jjj (e~V 2 )\Kn,m\ 2 d^drj
= K fjfn, m 2 (Pl)fn 2 m 2 (P2)(Pl 2 -P2 2 ) <WP2 (77-1)
are non-zero (we have made the substitution £ = np v t] = »p 2 ). The matrix
concerned is evidently diagonal with respect to the number m, while its
diagonality with respect to the numbers n 1 , n 2 follows from the orthogonality
of the functions f niTn for different n x and the same n (see below). The integra-
tions over p x and p 2 in (77.1) are separable; the integrals obtained are calcu-
lated in §f of the Mathematical Appendices (integral (f.6)). After a simple
calculation, we find for the corrections to the energy levels in the first approxi-
mation ||
£(« = | £n{n x -n 2 \ (77.2)
or, in absolute units,
f In the following calculations we do not take account of the fine structure of the hydrogen levels.
Hence the field must be, though not strong (for perturbation theory to be applicable), yet such that
the Stark splitting is large in comparison with the fine structure.
% In this section we use atomic units.
|| This result was derived by K. Schwarzschild and P. Epstein (1916), using the old quantum
theory, and by W. Pauli and E. Schrodinger (1926) using quantum mechanics.
270 The Atom §77
The two extreme components of the split level correspond to Wj = n— 1,
«2 = and % = 0, w 2 = n— 1. The distance between these two extreme
levels is, by (77.2),
Z£n{n-\\
i.e. the total splitting of the level by the Stark effect is approximately pro-
portional to n 2 . It is natural that the splitting should increase with the prin-
cipal quantum number: the further the electrons are from the nucleus, the
greater the dipole moment of the atom.
The presence of the linear effect means that, in the unperturbed state, the
atom has a dipole moment whose mean value is
d~ e = -fw^-^). (77.3)
This is in accordance with the fact that, in a state determined by parabolic
quantum numbers, the distribution of the charges in the atom is not sym-
metrical about the plane z = (see §37). Thus, for % > n^, the electron is
predominantly on the side of positive z, and hence the atom has a dipole
moment opposite to the external field (the charge on the electron being
negative).
In the previous section we have shown that a uniform electric field cannot
entirely remove degeneracy: there always remains a twofold degeneracy of
states differing in the sign of the projection of the angular momentum on the
direction of the field (in this case, states whose projected angular momenta
are + m). However, we see from formula (77.2) that even this removal of
the degeneracy does not occur in the linear Stark effect in hydrogen: the
displacement of the levels (for given n and n± - n 2 ) is independent of m and n 2 .
A further removal of the degeneracy occurs in the second approximation;
the calculation of this effect is the more interesting in that the linear Stark
effect is altogether absent in states with n\ = w 2 -
To calculate the quadratic effect, it is not convenient to use ordinary
perturbation theory, since it would be necessary to deal with infinite sums
of complicated form. Instead we use the following slightly modified method.
Schrodinger's equation for the hydrogen atom in a uniform electric
field is of the form
(£A+£+i/r-«f*)«A = o.
Like the equation with «f = 0, it allows separation of the variables in
parabolic co-ordinates. The same substitution (37.7) as was used in §37
gives the two equations
d / d/„ \ / m 2 \
d^Vd^J + (^ _i ~ +i<f V /2 = ~^ 2y
(77.4)
&+& = 1,
§77 The Stark effect in hydrogen 271
which differ from (37.8) by the presence of the terms in S. We shall regard
the energy E in these equations as a parameter which has a definite value,
and the quantities jS 1} j3 2 as eigenvalues of corresponding operators; it is easy
to see that these operators are self-conjugate. These quantities are deter-
mined, by solving the equations, as functions of E and S, and then the condi-
tion j8i+j82 = 1 gives the required relation between E and S, i.e. the energy
as a function of the external field.
For an approximate solution of equations (77.4), we regard the terms con-
taining the field ^ as a small perturbation. In the zeroth approximation
( g = 0), the equations have the familiar solutions
A = Ve/n^e),
(77.5)
/■ = V e fn t mM,
where the functions f nitn are the same as in (37.16), and instead of the energy
we have introduced the parameter
€ = V(-2E). ( 776 >
The corresponding values of &, jS 2 (from the equations (37.12), in which n
must be replaced by 1/e) are
&(« = K+iM+Jh j8 2 «» = ( W2 +i|m|+£)e. (77.7)
The functions j x with different % for a given e are orthogonal, as are the eigen-
functions of any self-conjugate operator; we have already used this fact above
in discussing the linear effect. In (77.5) these functions are normalised by
the conditions
o o
The corrections to & and jS 2 in the first approximation are determined by
the diagonal matrix elements of the perturbation:
oo oo
Calculation gives
&W = i^(6n 1 2 +6n 1 |m|+m 2 +6n 1 +3|m|+2)/€ 2 .
The expression for j8 2 (1) is obtained by replacing n x by n^ and changing the
sign.
272 The Atom §77
In the second approximation we have, by the general formulae of perturba-
tion theory,
<f 2 ^
Kaw
The integrals appearing in the matrix elements (£ 2 )„ „ . are calculated in S f
of the Mathematical Appendices. The only non-zero elements are
(^ 2 )« 1 ,« 1 -i = (P ) ni -i,n = -2(2« 1 +M)V[h 1 (« 1 +M)]/ 6 2,
(?)n it n r 2 = {¥\-2,n x = VM^- 1)K+ |m|)K+|m|- 1)]{ € *.
The differences occurring in the denominators are
As a result of the calculations we have
ft(2> = - < r 2 (|m|+2 % +l)[4m 2 +17(2|m| % +2« 1 H|M+2n 1 )+18]/1665;
the expression for j8 2 < 2 > is obtained by replacing « x by n^. Combining the
expressions obtained and substituting in the relation ft+/? 2 = 1, we have
the equation
en- < r 2 «[17n 2 +51K~« 2 )2-9m 2 +19]/16 € 5+3 ( f w ( Wi _ M2 )/ € 2 = x
Solving by successive approximations, we have in the second approximation
for the energy E = — \e % the expression
1 «f 2
E = ~2^ 2 " +l ^" (ni ~ M2) ~l6" w4[17w2 ~ 3(Wl ~ W2)2 ~ 9m2+19] ' (77 * 8)
The second term is the already familiar linear Stark effect, and the third is
the required quadratic effect (G. Wentzel, I. Waller and P. Epstein 1926).
We notice that this quantity is always negative, i.e. the terms are always
displaced downwards by the quadratic effect. The mean value of the dipole
moment is obtained by differentiating (77.8) with respect to the field; in the
states with n\ = n^ it is
d z =£M 4 (17» 2 -9m 2 +19)«f. (77.9)
Thus the polarisability of the hydrogen atom in the normal state (»=1, m=0)
is 9/2 (in absolute units 9(^ a /we 2 ) 3 /2).
The absolute value of the energy of the hydrogen terms falls rapidly as the
principal quantum number n increases, while the Stark splitting is increased.
Hence it is of interest to examine the Stark effect for highly excited levels in
§77 The Stark effect in hydrogen 273
fields so strong that the splitting they cause is comparable with the energy
of the level itself, and perturbation theory is inapplicable.! This can be done
by using the fact that states with large values of n are quasi-classical.
By the substitution
A=xJVe, h=xM (77-10)
the equations (77.4) are brought into the form
/ ft w 2 -l a \
(77.11)
drf
Each of these equations, however, is the same in form as the one-dimensional
Schrodinger's equation, the part of the total energy of the particle being
taken by \E, and that of the potential energy by the functions
ft m 2 -l
ft w 2 -l
(77.12)
2tj Sr, 2
respectively.
Figs. 25 and 26 respectively show the approximate form of these functions
(for m > 1). By Bohr and Sommerfeld's quantisation rule (48.2) we write
i,
j y/{2&E-Ujm d£ = K+iK
ix (77.13)
VWhE-UtivWdrt =(« 2 +|K
where %, w 2 are integers. J These equations determine implicitly the depen-
dence of the parameters & and jS 2 on E. Together with the equation ft + 2 = 1 ,
they therefore give the energies of the levels when displaced by the electric
field. The integrals in equations (77.13) can be reduced to elliptic integrals;
these equations can be solved only numerically.
The applicability of perturbation theory to high levels requires the perturbation to be small only
in comparison with the energy of the level itself (the binding energy of the electron), and not with the
intervals between the levels. For in the quasi-classical case (which corresponds to highly excited
states) the perturbation can be regarded as small if the force due to it is small in comparison with those
acting on the particle in the unperturbed system; and this condition is equivalent to the one given
above.
% A detailed investigation shows that a more exact result is obtained by writing m z instead of m'— 1
in the expressions for U x and U 2 . The integers n lt Wj are then equal to the parabolic quantum numbers.
274
The Atom
§77
Fig. 25
The Stark effect in strong fields is complicated by another phenomenon,
the ionisation of the atom by the electric field. The potential energy £z of
an electron in the external field takes arbitrarily large negative values as
z -» — co. Added to the potential energy of the electron within the atom,
it has the effect that the region of possible motion for the electron (whose
total energy E is negative) includes, besides the region inside the atom, the
region of large distances from the nucleus in the direction of the anode.
These two regions are separated by a potential barrier, whose width dimin-
ishes as the field increases. We know, however, that in quantum mechanics
there is always a certain non-zero probability that a particle will penetrate a
potential barrier. In the case we are considering, the emergence of the elec-
tron from the region within the atom, through the barrier, is simply the
ionisation of the atom. In weak fields the probability of this ionisation is
vanishingly small. It increases exponentially with the field, however, and
becomes considerable in fairly strong fields.-f
Fig. 26
t This phenomenon may serve to illustrate how a small perturbation may alter the nature of the
energy spectrum. Even a weak field $ is sufficient to create a potential barrier and produce a region,
far from the nucleus, which is in principle accessible to the electron. As a result, the motion of the
electron becomes, strictly speaking, infinite, and hence the energy spectrum becomes continuous
instead of discrete. Nevertheless, the formal solution obtained by the methods of perturbation theory
has a physical significance: it gives the energy levels of states which are not quite but "almost" station-
ary. An atom that is in such a state at some initial instant remains in it for a long period of time.
However, the series given by perturbation theory for the Stark splitting of the levels cannot be
convergent in the strict sense, but is merely an asymptotic series : after a certain point in the series
(which becomes later as the perturbation is reduced in magnitude) the terms increase, not decrease.
§77 The Stark effect in hydrogen 275
PROBLEM
Determine the probability (per unit time) of the ionisation of a hydrogen atom (in the
ground state) in a strong electric field.
Solution. In parabolic co-ordinates there is a potential barrier "along the 17 co-ordinate"
(Fig. 26); the "extraction" of the electron from the atom in the direction z -> — 00
corresponds to its passage into the region of large 17. To determine the ionisation probability,
it is necessary to investigate the form of the wave function for large 17 (and small £ ; we shall
see below that small values of £ are the important ones in the integral which determines the
total probability current for the emerging electron). The wave function of the electron in the
normal state (in the absence of the field) is
= «-tf+W2/yV. (1)
When the field is present, the dependence of iff on £ in the region in which we are interested
can be regarded as being the same as in (1), while to determine its dependence on 77 we have
the equation
d\ r 11 1
v + [- i+ ^ + v +K, ] x = 0, (2)
where \ =■ y/-r\i\> (equation (77.11) with E = — £, m = 0, /3 2 = £). Let 17 be some value of 17
("within" the barrier) such that 1 <^ 7? <C l/<^. For 17 <; 179, the wave function is quasi-
classical. Since, on the other hand, equation (2) has the form of the one-dimensional
Schrodinger's equation, we can use formulae (50.4). Using the exact expression (1) for
the wave function at the point 17 = 170, we obtain in the region outside the barrier the expres-
sion
1 x l\Po\ }
X = — -€ r -* ( * f "«V 1 7o / ex p[* \pdr)— f«r],
where
We shall be interested only in the square |x| 2 - Hence the imaginary part of the exponent is
unimportant. Denoting by -q x the root of the equation p(rj) = 0, we have
| x |2 = lV^exp[-2f|/> |d,-, ].
71 P i
In the coefficient of the exponential we put
in the exponent we must keep also the next term of the expansion :
Vi Vi
\x\ 2 = er€ exp[~- f y/ll-g-q) d v + f « "|,
where ^ ^ \\S. Effecting the integration and neglecting 17 <^ compared with 1 wherever
possible, we obtain
|x| 2 = e- 2 /^ . (3)
TtS \Z(<a7)—l)
276 The Atom §77
The total probability current through a plane perpendicular to the ar-axis (i.e. the required
ionisation probability w) is
00
w = \*fi\H z 2TTr dr.
o
For large rj (and small £) we can put
Substituting also for the velocity of the electron
v. s V[2(-l+*^)] = VW-1),
we have
00
W=j\x\ 2 7rV(<?ri-l)d$
o
4 °?
^ o
that is, finally,
or, in ordinary units,
w = (4m3<?9/«f/z7)exp {-2m^\ZSh%
CHAPTER XI
THE DIATOMIC MOLECULE
§78. Electron terms in the diatomic molecule
In the theory of molecules an important part is played by the fact that the
masses of atomic nuclei are very large compared with those of the electrons.
Because of this difference in mass, the rates of motion of the nuclei in the
molecule are small in comparison with the velocities of the electrons. This
makes it possible to regard the motion of the electrons as being about fixed
nuclei placed at given distances from one another. On determining the
energy levels U n for such a system, we find what are called the electron terms
for the molecule. Unlike those for atoms, where the energy levels were
certain numbers, the electron terms here are not numbers but functions of
parameters, the distances between the nuclei in the molecule. The energy
U n includes also the electrostatic energy of the mutual interaction of the nu-
clei, so that U n is essentially the total energy of the molecule for a given
arrangement of the fixed nuclei.
We shall begin the study of molecules by taking the simplest type, the
diatomic molecules, which permit the most complete theoretical investigation.
One of the chief principles in the classification of the atomic terms was
the classification according to the values of the total orbital angular momen-
tum L. In molecules, however, there is no law of conservation of the total
orbital angular momentum of the electrons, since the electric field of several
nuclei is not centrally symmetric.
In diatomic molecules, however, the field has axial symmetry about an
axis passing through the two nuclei. Hence the projection of the orbital
angular momentum on this axis is here conserved, and we can classify the
electron terms of the molecules according to the values of this projection.
The absolute value of the projected orbital angular momentum along the
axis of the molecule is customarily denoted by the letter A; it takes the values
0, 1, 2, ... . The terms with different values of A are denoted by the capital
Greek letters corresponding to the Latin letters for the atomic terms with
various L. Thus, for A = 0, 1, 2 we speak of S, II and A terms respectively;
higher values of A usually need not be considered.
Next, each electron state of the molecule is characterised by the total spin
S of all the electrons in the molecule. If S is not zero, there is degeneracy
of degree 2S+1 with respect to the directions of the total spin. f The number
2S+1 is, as in atoms, called the multiplicity of the term, and is written as an
index before the letter for the term; thus 3 II denotes a term with A = 1
S=l.
f We here neglect the fine structure due to relativistic interactions (see §§83 and 84 below).
277
278 The Diatomic Molecule §78
Besides rotations through any angle about the axis, the symmetry of the
molecule allows also a reflection in any plane passing through the axis. If
we effect such a reflection, the energy of the molecule is obviously unchanged.
The state obtained from the reflection is, however, not completely identical
with the initial state. For, on reflection in a plane passing through the axis
of the molecule, the sign of the angular momentum about this axis is changed.
Thus we conclude that all electron terms with non-zero values of A are doubly
degenerate: to each value of the energy, there correspond two states which
differ in the direction of the projection of the orbital angular momentum on
the axis of the molecule. In the case where A = the state of the molecule
is not changed at all on reflection, so that the S terms are not degenerate.
The wave function of a S term can only be multiplied by a constant as a
result of the reflection. Since a double reflection in the same plane is an iden-
tity transformation, this constant is ±1. Thus we must distinguish S terms
whose wave functions are unaltered on reflection and those whose wave
functions change sign. The former are denoted by S+, and the latter by S~.
If the molecule consists of two similar atoms, a new symmetry appears,
and with it an additional characteristic of the electron terms. A diatomic
molecule with identical nuclei has a centre of symmetry at the point bisecting
the line joining the nuclei.f (We shall take this point as the origin.) Hence
the Hamiltonian is invariant with respect to a simultaneous change of sign
of the co-ordinates of all the electrons in the molecule (the co-ordinates of the
nuclei remaining unchanged). Since the operator of this transformation J
also commutes with the orbital angular momentum operator, we have the
possibility of classifying terms with a given value of A according to their
parity: the wave functions of even (g) states are unchanged when the co-
ordinates of the electrons change sign, while those of odd (u) states change sign.
The suffixes u, g indicating the parity are customarily written with the letter
for the term: Ii u , Rg, and so on.
Finally, we shall mention an empirical rule, according to which the normal
electron state in the great majority of chemically stable diatomic molecules
is completely symmetrical: the electron wave function is invariant with
respect to all symmetry transformations in the molecule. As we shall show
in §81, the total spin S is zero too, in the great majority of cases, in the
normal state. In other words, the ground term of the molecule is X S+, and
it is 1 E+ i? if the molecule consists of two similar atoms. ||
PROBLEM
Effect the separation of variables in Schrodinger's equation for the electron terms of the
ion H2 + , using elliptic co-ordinates.
f It has also a plane of symmetry perpendicularly bisecting the axis of the molecule. This element
of symmetry need not be considered separately, however, since the existence of such a plane follows
automatically from the existence of a centre of symmetry and of an axis of symmetry.
% Not to be confused with that of inversion of the co-ordinates of all the particles in the molecule
|| Exceptions to these rules are formed by the molecules 2 (whose normal term is 3 S~ g ) and NO
(normal term 2 II).
§79 The intersection of electron terms 279
Solution. Schrodinger's equation for an electron in the field of two protons at rest
is (using atomic units)
^+2(e+— +— y =o-
\ r\ r 2 /
n r Zi
The elliptic co-ordinates £, 17 are defined by
£ = (n + r 2 )IR, v = (rz-r^R; 1<£<oo, -1<i?<1,
and the third co-ordinate <£ is the angle of rotation about an axis passing through the two
nuclei at a distance R apart.f The Laplacian operator in these co-ordinates is
4 rd 8 8 8n
A = — (^ 2 — 1) 1 (1 — t? 2 ) — +
4 r8 8 8 8~[ 1 8*
\ =
R\{
Putting
|J2(|2_1)(1_^2) 0^2
= X(g)Y(riy**,
we obtain for X and Y the equations
dr dX~] / A2 \
dr dY~] ( A2 \
i 1 -' !2) ^] + (-^ v ^-w) y = '
where A is the separation parameter.
Each electron term is described by three quantum numbers : A, and two numbers ng , tin
which determine the number of zeros of the functions X(£) and Y(-q). Since all these numbers
are related to functions of different variables, there is in general nothing to prevent the inter-
section of terms E(R) having different values of any two quantum numbers, including the
pair ng , «tj with the same A, even though such terms have the same symmetry (see the footnote
to §79).
§79. The intersection of electron terms
The electron terms in a diatomic molecule are functions of a single para-
meter, the distance r between the nuclei. They can be represented graphically
by plotting the energy as a function of r. It is of considerable interest to
examine the intersection of the curves representing the different terms.
Let U x {r), U 2 (r) be two different electron terms. If they intersect at some
point, then the functions U 1 and U 2 will have neighbouring values near this
point. To decide whether such an intersection can occur, it is convenient to
put the problem as follows. Let us consider a point r where the functions
U x (r), U%(r) have very close but not equal values (which we denote by E v E 2 ),
and examine whether or not we can make U x and U 2 equal by displacing the
point a short distance 8r. The energies E x and E 2 are eigenvalues of the
Hamiltonian ff of the system of electrons in the field of the nuclei, which
are at a distance r from each other. If we add to the distance r an increment
f See Mechanics, §48.
280 The Diatomic Molecule §79
8r, the Hamiltonian becomes fi Q +1/, where V = 8r . dffjdr is a small cor-
rection ; the values of the functions U lt U 2 at the point r + 8r can be regarded
as eigenvalues of the new Hamiltonian. This point of view enables us to
determine the values of the terms U x (r), U 2 (r) at the point r +8r by means of
perturbation theory, V being regarded as a perturbation to the operator
The ordinary method of perturbation theory is here inapplicable, however,
since the eigenvalues E v E 2 of the energy in the unperturbed problem are very
close to each other, and their difference is in general small compared with the
magnitude of the perturbation; the condition (38.8) is not fulfilled. Since,
in the limit as the difference E 2 —E x tends to zero, we have the case of degener-
ate eigenvalues, it is natural to attempt to apply to the case of close eigenvalues
a method similar to that developed in §39.
Let tf/ v «/r 2 be the eigenfunctions of the unperturbed operator 3 which
correspond to the energies E x , E 2 . As an initial zero-order approximation we
take, instead of fa and fa themselves, linear combinations of them of the form
<A = c x fa+c 2 fa. C 79 - 1 )
Substituting this expression in the perturbed equation
&+?)</,= Efa (79.2)
we obtain
c x {E x +V-E)fa+c z {E 2 +t-E)fa = 0.
Multiplying this equation on the left by fa* and 02* in turn, and integrating,
we have two algebraic equations :
Ci(E 1 + V 11 -E)+c 2 V 12 = 0,
CiV n +c t (E t + V 2t -E) = 0,
where V ik = j" «^ # ty*fj k dq. Since the operator V is Hermitian, the quantities
V n and F 22 are real, while V x% = V 2X *. The compatibility condition for these
equations is
E 1 +V n -E V u '
V 21 E 2 +V 22 -E
whence we obtain after some calculation
= 0,
E = K£i+£ 2 + Vu+ V tt )±V\WB 1 -E*+ V n - V 22 ?+\ V 12 \*]. (79.4)
This formula gives the required eigenvalues of the energy in the first approxi-
mation.
If the energy values of the two terms become equal at the point r + 8r
(i.e. the terms intersect), this means that the two values of E given by formula
§79 The intersection of electron terms 281
(79.4) are the same. For this to happen, the expression under the radical in
(79.4) must vanish. Since it is the sum of two squares, we obtain, as the
condition for there to be points of intersection of the terms, the equations
E 1 -E 2 +V 11 -V 22 =0 1 F 12 =0. (79.5)
However, we have at our disposal only one arbitrary parameter giving the
perturbation V, namely the magnitude Sr of the displacement. Hence the
two equations (79.5) cannot in general be simultaneously satisfied (we sup-
pose that the functions tf/ lt «/r 2 are chosen to be real, so that V 12 also is real).
It may happen, however, that the matrix element V 12 vanishes identically ;
there then remains only one equation (79.5), which can be satisfied by a suit-
able choice of 8r. This happens in all cases where the two terms considered
are of different symmetry. By symmetry we here understand all possible forms
of symmetry: with respect to rotations about an axis, reflections in planes,
inversion, and also with respect to interchanges of electrons. In the diatomic
molecule this means that we may be dealing with terms of different A, differ-
ent parity or multiplicity, or (for S terms) £+ and 2~ terms.
To prove this statement it is essential that the operator V (like the Hamil-
tonian itself) commutes with all the symmetry operators for the molecule:
the operator of the angular momentum about an axis, the reflection and in-
version operators, and the operators of interchanges of electrons. It has been
shown in §§29 and 30 that, for a scalar quantity whose operator commutes
with the angular momentum and inversion operators, only the matrix elements
for transitions between states of the same angular momentum and parity are
non-zero. This proof remains valid, in essentially the same form, for the
general case of an arbitrary symmetry operator. We shall not pause to repeat
it here, especially since in §97 we shall give another general proof, based on
group theory.
Thus we reach the result that, in a diatomic molecule, only terms of differ-
ent symmetry can intersect, while the intersection of terms of like symmetry is
impossible (E. Wigner and J. von Neumann 1929). If, as a result of some
approximate calculation, we obtain two intersecting terms of the same
symmetry, they are found to move apart on calculating the next approxi-
mation, as shown by the continuous lines in Fig. 27.f
We emphasise that this result not only is true for the diatomic molecule,
but is a general theorem of quantum mechanics ; it holds for any case where
the Hamiltonian contains some parameter and its eigenvalues are consequently
functions of that parameter.
In a polyatomic molecule, the electron terms are functions of not one but
several parameters, the distances between the various nuclei. Let s be the
number of independent distances between the nuclei ; in a molecule of iV( > 2)
atoms, this number is s = 3N— 6 for an arbitrary arrangement of the nuclei.
t There is a curious exception to this rule in the case where the problem of determining the electron
terms allows a complete separation of the variables (see §78, Problem).
282
The Diatomic Molecule
m
U(r)
Fig. 27
Each term U n {r ly ... , r s ) is, from the geometrical point of view, a surface in
a space of s + 1 dimensions, and we can speak of the intersections of these
surfaces in manifolds of varying numbers of dimensions, from (intersection
in a point) to s— 1. The derivation given above is wholly valid, except that
the perturbation V is here determined not by one but by s parameters, the
displacements 8r x , ... , 8r s . Even with two parameters, the two equations
(79.5) can in general be satisfied. Thus we conclude that, in polyatomic
molecules, any two terms may intersect. If the terms are of like symmetry,
the intersection is given by the two conditions (79.5), from which it follows
that the number of dimensions of the manifold in which the intersection
occurs is s—2. If the terms are of different symmetry, on the other hand,
there remains only one condition, and the intersection takes place in a mani-
fold of s—1 dimensions.
Thus for s = 2 the terms are represented by surfaces in a three-dimensional
system of co-ordinates. The intersection of these surfaces occurs in lines
(s — 1 = 1) when the symmetry of the terms is different, and in points
(s — 2 = 0) when it is the same. It is easy to ascertain the form of the surfaces
near the point of intersection in the latter case. The value of the energy near
the points of intersection of the terms is given by formula (79.4). In this
expression the matrix elements V u , V^, V 12 are linear functions of the dis-
placements 8r lt 8r 2 , and hence are linear functions of the distances r lt r 2
themselves. Such an equation determines an elliptic cone, as we know from
analytical geometry. Thus, near the points of intersection, the terms are
represented by the surface of an arbitrarily situated double elliptic cone
(Fig. 28).
§80. The relation between molecular and atomic terms
As we increase the distance between the nuclei in a diatomic molecule, we
have in the limit two isolated atoms (or ions). The question thus arises of
the correspondence between the electron terms of the molecule and the states
of the atoms obtained by moving them apart. This relation is not one-to-one ;
if we bring together two atoms in given states, we may obtain a molecule in
various electron states.
§80
The relation between molecular and atomic terms
u
283
Fig. 28
Let us first suppose that the molecule consists of two different atoms. Let
the isolated atoms be in states with orbital angular momenta L^, L 2 and spins
S lt S 2 , and let Z^ ^ L 2 . The projections of the angular momenta on the
line joining the nuclei take the values M 1 = —L v —L x +1, ... , L^ and
M 2 = — L 2 , — L 2 +l, ... , L 2 . The absolute value of the sum Mj+ik/a deter-
mines the angular momentum A obtained on bringing the atoms together.
On combining all possible values of M x and M 2 , we find the following values
for the numbers of times that we obtain the various values of A = jiW^+Mjl :
A = L x +L 2
L x +L 2 -l
twice
four times
L x —L 2
Lx-£ 2 -l
2(2L 2 +1) times
2(2L 2 +1) times
2(2L 2 +1) times
2L 2 +1 times.
Remembering that all terms with A ^ are doubly degenerate, while
those with A = are not degenerate, we find that there will be
1 term with A = L x -\-L 2y
2 terms with A = L 1 -\-L 2 — 1,
2L 2 +1 terms with A = L x — L 2
2L 2 +1 terms with A = L 1 —L 2 —l,
2L 2 +1 terms with A = 0;
(80.1)
284 The Diatomic Molecule §80
in all, (2L 2 +1)(L 1 + 1) terms with values of A from to L x -\-L 2 .
The spins S lt S 2 of the two atoms combine to form the total spin of the
molecule in accordance with the general rule for the addition of angular
momenta, giving the following possible values of S:
S = S 1 +S 2 , S^Sz-l, ..., | Si-Sal . (80.2)
On combining each of these values with each value of A in (80.1), we obtain
the complete list of all possible terms in the molecule formed.
For 2 terms there is also the question of sign. This is easily resolved by
noticing that the wave functions of the molecule can be written, as r -» oo,
in the form of products (or sums of products) of the wave functions of the
two atoms. An angular momentum A = can be obtained either by adding
two non-zero angular momenta of the atoms such that M x = — M 2 , or from
M 1 = M 2 = 0. We denote the wave functions of the first and second atoms
b y «A (1) m 1 > ^ (2 W For M = \M X \ = \M 2 \ # 0, we form the symmetrised and
antisymmetrised products
A reflection in a vertical plane (i.e. one passing through the axis of the mole-
cule) changes the sign of the projection of the angular momentum on the
axis, so that ifj a) M , «A (2) m are changed into ^ a) - M > <A (2) -m respectively, and vice
versa. The function \Js + is thereby unchanged, while ifs~ changes sign; the
former therefore corresponds to a 2+ term and the latter to a 2~ term. Thus,
for each value of M, we obtain one 2+ and one 2~ term. Since M can take
L% different values (M = 1, ... , L 2 ), we have in all L 2 2+ terms and L 2 2~
terms.
If, on the other hand, M 1 = M 2 = 0, the wave function of the molecule
is of the form j/f = ^f (1) ^ (2) - In order to ascertain the behaviour of the func-
tion ?/r (1) on reflection in a vertical plane, we take a co-ordinate system with
its origin at the centre of the first atom, and the #-axis along the axis of the
molecule, and we notice that a reflection in the vertical xs-plane is equivalent
to an inversion with respect to the origin, followed by a rotation through 180°
about the ^-axis. On inversion, the function (1) o is multiplied by I lt where
I x = Jb 1 is the parity of the given state of the first atom. Next, the result
of applying to the wave function the operation of an infinitely small rotation
(and therefore that of any finite rotation) is entirely determined by the total
orbital angular momentum of the atom. Hence it is sufficient to consider
the particular case of an atom having one electron, with orbital angular
momentum / (and a .sr-component of the angular momentum m = 0) ; on
putting L in place of / in the result, we obtain the required solution for any
atom. The angular part of the wave function of an electron with m = is, apart
from a constant coefficient, P/(cos 6) (see (28.7)). A rotation through 180°
about the j-axis is the transformation x -> — x, y -> j, z -> —z or, in
§80 The relation between molecular and atomic terms 285
spherical polar co-ordinates, r -+r, 6 -+n— 0, <£->7r — (f>. Then cos 6 ->
— cos 6, and the function Pj(cos 6) is multiplied by (— 1)*.
Thus we conclude that, as a result of reflection in a vertical plane, the func-
tion «^ (1) is multiplied by ( — V) Ll I x . Similarly, i/j (2) is multiplied by ( — 1) L 2 / 2 ,
so that the wave function «/r = «A (1) «A (2) is multiplied by ( — l) Ll+L2 7 1 7 2 . The
term is 2+ or 2~ according as this factor is +1 or — 1.
Summarising the results obtained, we find that, of the total number
2L 2 +1 of S terms (each of the appropriate multiplicity), L 2 +l terms are
2+ andL 2 are 2", if ( - 1) L > +Lt I 1 I 2 = + 1 , and vice versa if ( - l) Ll +L *IJ 2 = - 1 .
Let us now turn to a molecule consisting of similar atoms. The rules for
the addition of the spins and orbital angular momenta of the atoms to form
the total S and A for the molecule remain the same here as for a molecule
composed of different atoms. The difference is that the terms may be even
or odd. Here we must distinguish two cases, according as the combined
atoms are in the same or different states.
If the atoms are in different states, f the total number of possible terms is
doubled in comparison with the number when the atoms are different. For a
reflection with respect to the origin (this being the point bisecting the axis of
the molecule) results in an interchange of the states of the two atoms. Sym-
metrising or antisymmetrising the wave function of the molecule with respect
to an interchange of the states of the atoms, we obtain two terms (with the
same A and S), of which one is even and the other odd. Thus we have al-
together the same number of even and odd terms.
If, on the other hand, both atoms are in the same state, the total number of
states is the same as for a molecule with different atoms. An investigation
which we shall not give here on account of its length J leads to the following
results for the parity of these states. Let N g , N u be the numbers of even and
odd terms with given values of A and S. Then
if A is odd, N g = N u ;
if A is even and S is even (S = 0, 2, 4, ... ), N g = N u + 1 ;
if A is even and S is odd (S = 1, 3, 5, ... ), N u = N g +1.
Finally, we must distinguish, among the 2 terms, between 2+ and 2 - . Here,
if S is even, N g + = iW+l = L+l ;
if S is odd, N u + = N g ~+1 = L+l,
where L x = L 2 = L. All the 2+ terms are of parity ( — 1) & ', and all 2 -
terms are of parity ( — l) s+1 .
Besides the problem that we have examined of the relation between the
molecular terms and those of the atoms obtained as r -> oo, we may also
propose the question of the relation between the molecular terms and those
of the "composite atom" obtained as r -> 0, i.e. when both nuclei are brought
to a single point (for example, between the terms of the H 2 molecule and those
of the He atom). The following rules can be deduced without difficulty.
t In particular, we may be discussing the combination of a neutral and an ionised atom.
% It can be found in the original paper by E. Wigner and E. Witmer, Zeitschrift fur Physik 51,
859, 1928.
286 The Diatomic Molecule §81
From a term of the "composite" atom having spin S, orbital angular momen-
tum L and parity /, we can obtain, on "moving the constituent atoms apart",
molecular terms with spin S and angular momentum about the axis A =
0, 1, ... , L, with one term for each of these values of A. The parity of the
molecular term is the same as the parity / of the atomic term (g for / = +1
and u for / = — 1). The molecular term with A = is a 2+ term if
(-1) L I= + 1, and a 2- term if (-1)^7= -1.
PROBLEMS
Problem 1. Determine the possible terms for the molecules H 2 , N 2 , 2 , Cl 2 which can
be obtained by combining atoms in the normal state.
Solution. According to the rules given above, we find the following possible terms:
H 2 molecule (atoms in the 2 S state) :
*2+ 3 2+ •
N 2 molecule (atoms in the *S state) :
1 2+
32+ 5 2+ 7 2+ •
Cl 2 molecule (atoms in the 2 P state) :
2 X E+ X T.- iTT iTT
iA„ 2«2+ tt , 3 S - fl> 3n „
3 n„,
3 A U
O a molecule (atoms in the 3 P state) :
2*2+,, *2- M , m g , in.,
iA ff , 232\, 32-,, m w
3 n„
3 A,
2 5 S+„,
*2- tt , *n g , ^n M , *\.
The figures in front of the symbols indicate the number of terms of the type concerned, if
this number exceeds unity.
Problem 2. The same as Problem 1, but for the molecules HC1, CO.
Solution. When unlike atoms are combined, the parity of their states is important
also. From formula (31.5) we find that the normal states of the H, O and C atoms are even,
while that of the CI atom is odd (see Table 3 for the electron configurations of these atoms).
From the rules given above, we have
HC1 molecule (atoms in the 2 S g and 2 P U states) :
1,3 S + 1.3TJ;
CO molecule (both atoms in the *P g state) :
2 1.3,5 S + 1,3.5 S - 2L3.5II, WA.
§81. Valency
The property of atoms of combining with one another to form molecules
is described by means of the concept of valency. To each atom we ascribe
a definite valency, and when atoms combine their valencies must be mutually
satisfied, i.e. to each valency bond of an atom there must correspond a
valency bond of another atom. For example, in the methane molecule
CH4, the four valency bonds of the quadrivalent carbon atom are satisfied
by the four univalent hydrogen atoms. In going on to give a physical inter-
pretation of valency, we shall begin with the simplest example, the com-
bination of two hydrogen atoms to form the molecule H 2 .
§81
Valency
287
Let us consider two hydrogen atoms in the ground state ( 2 S). When they
approach, the resulting system may be in the molecular state 1 S+ S or 3 E+„.
The singlet term corresponds to an antisymmetrical spin wave function, and
the triplet term to a symmetrical function. The co-ordinate wave function,
on the other hand, is symmetrical for the *E term and antisymmetrical for the
3 E term. It is evident that the ground term of the H 2 molecule can only be
the X E term. |For an antisymmetrical wave function (f>(r ly r 2 ) (where r x and r 2
are the radius vectors of the two electrons) always has nodes (since it vanishes
for r x = r 2 ), and hence cannot belong to the lowest state of the system.
A numerical calculation shows that the electron term X S in fact has a deep
minimum corresponding to the formation of a stable H 2 molecule. In the
3 E state, the energy U(r) decreases monotonically as the distance between
the nuclei increases, corresponding to the mutual repulsion of the two hydro-
gen atomsf (Fig. 29).
\U{r)
Fig. 29
Thus, in the ground state, the total spin of the hydrogen molecule is zero,
5 = 0. It is found that the molecules of practically all chemically stable
compounds of elements of the principal groups have this property. Among
inorganic molecules, exceptions are formed by the diatomic molecules 2
(ground state 3 E) and NO (ground state 2 II) and the triatomic molecules
N0 2 , C10 2 (total spin S = \). Elements of the intermediate groups have
special properties which we shall discuss below, after studying the valency
properties of the elements of the principal groups.
The property of atoms of combining with one another is thus related to
their spin (W. Heitler and H. London 1927). The combination occurs in
t Here we ignore the van der Waals attraction forces between the atoms (see §89). The existence
of these forces causes a minimum (at a greater distance) on the U(r) curve for the 3 S term also. This
minimum, however, is very shallow in comparison with that on the X E curve, and would not be per-
ceptible on the scale of Fig. 29.
288 The Diatomic Molecule §81
such a way that the spins of the atoms compensate one another. As a quanti-
tative characteristic of the mutual combining powers of atoms, it is convenient
to use an integer, twice the spin of the atom. This is equal to the chemical
valency of the atom. Here it must be borne in mind that the same atom may
have different valencies according to the state it is in.
Let us examine, from this point of view, the elements of the principal
groups in the periodic system. The elements of the first group (the first
column in Table 3, the group of alkali metals) have a spin S = I in the normal
state, and accordingly their valencies are unity. An excited state with a higher
spin can be attained only by exciting an electron from a completed shell.
Accordingly, these states are so high that the excited atom cannot form a stable
molecule, f
The atoms of elements in the second group (the second column in Table 3,
the group of alkaline-earth metals) have a spin S = in the normal state.
Hence these atoms cannot enter into chemical compounds in the normal state.
However, comparatively close to the ground state there is an excited state
having a configuration sp instead of s 2 in the incomplete shell, and a total
spin S = 1. The valency of an atom in this state is 2, and this is the principal
valency of the elements in the second group.
The elements of the third group have an electron configuration s 2 p in the
normal state, with a spin S — \. However, by exciting an electron from the
completed s-shell, an excited state is obtained having a configuration sp 2 and a
spin S = 3/2, and this state lies close to the normal one. Accordingly, the
elements of this group are both univalent and tervalent. The first two ele-
ments in the group (boron, aluminium) behave only as tervalent elements.
The tendency to exhibit a valency 1 increases with the atomic number, and
thallium behaves equally as a univalent and as a tervalent element (for example,
in the compounds T1C1 and T1C1 3 ). This is due to the fact that, in the first
few elements, the binding energy in the tervalent compounds is greater than
for the univalent compounds, and this difference exceeds the excitation energy
of the atom.
In the elements of the fourth group, the ground state has the configuration
s 2 p 2 with a spin of 1, and the adjacent excited state has a configuration sp 3
with a spin 2. The valencies 2 and 4 correspond to these states. As in the
third group, the first two elements (carbon, silicon) exhibit mainly the higher
valency (though the compound CO, for example, forms an exception), and
the tendency to exhibit the lower valency increases with the atomic number.
In the atoms of the elements of the fifth group, the ground state has the
configuration s 2 p 3 with a spin S =3/2, so that the corresponding valency
is three. An excited state of higher spin can be obtained only by the transi-
tion of one of the electrons into the shell with the next higher value of the
principal quantum number. The nearest such state has the configuration
sph' and a spin S = 5/2 (by s' we conventionally denote here an s state of an
t See the end of this section for the elements copper, silver and gold.
§81 Valency 289
electron with a principal quantum number one greater than in the state s).
Although the excitation energy of this state is comparatively high, the
excited atom can still form a stable compound. Accordingly, the elements of
the fifth group behave as both tervalent and quinquevalent elements (thus,
nitrogen is tervalent in NH 3 and quinquevalent in HN0 3 ).
In the sixth group of elements, the spin is 1 in the ground state (configura-
tion s^p 9 ), so that the atom is bivalent. The excitation of one of the p electrons
leads to a state s*p 3 s' of spin 2, while the excitation of an s electron in addition
gives a state sph'p' of spin 3. In both excited states the atom can enter into
stable molecules, and accordingly exhibits valencies of 4 and 6. The first
element of the sixth group (oxygen) shows only the valency 2, while the sub-
sequent elements show higher valencies also (thus, sulphur in H 2 S, S0 2 , S0 3
is respectively bivalent, quadrivalent and sexivalent).
In the seventh group (the halogen group), the atoms are univalent in the
ground state (configuration s 2 ^ 5 , spin S = . $). They can, however, enter into
stable compounds when they are in excited states having configurations
s 2 p*s', s 2 p 3 s'p', sp s s'p' 2 with spins 3/2, 5/2, 7/2 and valencies 3, 5, 7 respec-
tively. The first element in the group (fluorine) is always univalent, but the
subsequent elements also exhibit the higher valencies (thus, chlorine in HC1,
HC10 2 , HC10 3 , HC10 4 is respectively univalent, tervalent, quinquevalent
and septivalent).
Finally, the atoms of the elements in the group of inert gases have com-
pletely filled shells in their ground states (so that the spin S = 0), and their
excitation energies are high. Accordingly, the valency is zero, and these
elements are chemically inactive.f
The following general remark should be made concerning all these discus-
sions. The assertion that an atom enters into a molecule with a valency per-
taining to an excited state does not mean that, on moving the atoms apart to
large distances, we necessarily obtain an excited atom. It means only that the
distribution of the electron density in the molecule is such that, near the
nucleus of the atom in question, it is close to that in the isolated and excited
atom; but the limit to which the electron distribution tends as the distance
between the nuclei is increased may correspond to non-excited atoms.
When atoms combine to form a molecule, the completed electron shells in
the atoms are not much changed. The distribution of the electron density
in the incomplete shells, on the other hand, may be considerably altered. In
the most clearly denned cases of what is called heteropolar binding, all the
valency electrons pass over from their own atoms to other atoms, so that we
t The elements xenon and radon (and less easily krypton) nevertheless form stable compounds
with fluorine. These valencies may be due to a transfer of electrons from the outermost complete
shell to the incomplete / or d states, whose energies are comparatively near.
There is also an attraction which occurs in the interaction of an inert gas atom with an excited atom
of the same element. This is due to the doubling in the number of possible states obtained on bringing
together two atoms, if these atoms are of the same element but in different states (see §80). The transi-
tion of the excitation from one atom to the other here replaces the exchange interaction which brings
about the ordinary valency. The molecule He2 is an example of such a molecule. The same type of
bond occurs in molecular ions composed of two similar atoms (for instance, H2 + ).
290 The Diatomic Molecule §81
may say that the molecule consists of ions with charges equal (in units of e)
to the valency. The elements of the first group are electropositive : in hetero-
polar compounds they lose electrons, forming positive ions. As we pass to
the subsequent groups the electropositive character of the elements becomes
gradually less marked and changes into electronegative character, which is
present to the greatest extent in the elements of the seventh group. Regard-
ing heteropolarity the same remark should be made as was made above con-
cerning excited atoms in the molecule. If a molecule is heteropolar, this does
not mean that, on moving the atoms apart, we necessarily obtain two ions.
Thus, from the molecule KC1 we should in fact obtain the ions K+ and CI - ,
but the molecule NaCl gives in the limit the neutral atoms Na and CI (since
the affinity of chlorine for an electron is greater than the ionisation potential
of potassium but less than that of sodium).
In the opposite limiting case of what is called homopolar binding, the atoms
in the molecule remain neutral on the average. Homopolar molecules, un-
like heteropolar ones, have no appreciable dipole moment. The difference
between the heteropolar and homopolar types is purely quantitative, and any
intermediate case may occur.
Let us now turn to the elements of the intermediate groups. Those of
the palladium and platinum groups are very similar to the elements of the
principal groups as regards their valency properties. The only difference is
that, owing to the comparatively deep position of the d electrons inside the
atom, they interact only slightly with the other atoms in the molecule. As a
result, "unsaturated" compounds, whose molecules have non-zero spin
(though in practice not exceeding £), are often found among the compounds
of these elements. Each of the elements can exhibit various valencies, and
these may differ by unity, and not only by two as with the elements of the
principal groups (where the change in valency is due to the excitation of some
electron whose spin is compensated, so that the spins of two electrons are
simultaneously released).
The elements of the rare-earth group are characterised by the presence of an
incomplete / shell. The / electrons lie much deeper than the d electrons,
and therefore take no part in the valency. Thus the valency of the rare-
earth elements is determined only by the s and p electrons in the incomplete
shells.f However, it must be borne in mind that, when the atom is excited,
/ electrons may pass into s and p states, thereby increasing the valency by
one. Hence the rare-earth elements too exhibit valencies differing by unity
(in practice they are all tervalent and quadrivalent).
The elements of the actinium group occupy a unique position. Actinium
and thorium have no / electrons, and their valencies involve d electrons.
In their chemical properties they are therefore analogous to elements of the
palladium and platinum groups, not to the rare earths. The uranium atom in
t The d electrons which are found in the incomplete shells of the atoms of some rare-earth elements
are unimportant, since these atoms in practice always form compounds in excited states where there
are no d electrons.
§81 Valency 291
the normal state contains / electrons, but in its compounds it too has no /
electrons. Finally, the atoms of the elements neptunium, plutonium,
americium and curium contain/ electrons in compounds also, but the electrons
which participate in their valencies are again s and d electrons. In this sense
they are homologues of uranium. The maximum possible number of "un-
paired" s and d electrons is one and five respectively, and so the maximum
valency of elements in the actinium group is six, whereas the maximum
valency of the rare-earth elements (with 5 and p electrons participating in
the valency) is 1 + 3 = 4.
The elements of the iron group occupy, as regards their valency properties,
a position intermediate between the rare-earth elements and those of the palla-
dium and platinum groups. In their atoms, the d electrons lie comparatively
deep, and in many compounds take no part in the valency bonds. In these
compounds, therefore, the elements of the iron group behave like rare-earth
elements. Such compounds include those of ionic type (for instance FeCl 2 ,
FeCl 3 ), in which the metal atom enters as a simple cation. Like the rare-
earth elements, the elements of the iron group can show very various valencies
in these compounds.
Another type of compound of the iron-group elements is formed by what
are called complex compounds. These are characterised by the fact that the
atom of the intermediate element enters into the molecule not as a simple
ion, but as part of a complex ion (for instance the ion Mn0 4 ~~ in KMn0 4 , or
the ion Fe(CN) 6 4_ in K 4 Fe(CN) 6 ). In these complex ions, the atoms are
closer together than in simple ionic compounds, and in them the d electrons
take part in the valency bond. Accordingly, the elements of the iron group
behave in complex compounds like those of the palladium and platinum groups.
Finally, it must be mentioned that the elements copper, silver and gold,
which in §73 we placed among the principal groups, behave as intermediate
elements in some of their compounds. These elements can exhibit valencies
of more than one, on account of a transition of an electron from a d shell to a
p shell of nearly the same energy (for example, from 3d to 4p in copper). In
such compounds the atoms have an incomplete d shell, and hence behave as
intermediate elements : copper like the elements of the iron group, and silver
and gold like those of the palladium and platinum groups.
PROBLEM
Determine the electron terms of the molecular ion H2 + obtained when a hydrogen atom in
the normal state combines with an H + ion, for distances R between the nuclei large compared
with the Bohr radius.
Solution. This problem is analogous in form to §50, Problem 3: instead of two
one-dimensional potential wells we have here two three-dimensional wells (round the two
nuclei) with axial symmetry about the line joining the nuclei. The levelf E0 = —i (the ground
level of the hydrogen atom) is split into two levels U g (R) and U U (R) (the terms 2 S ff + and
2 E U +), corresponding to the electron wave functions
1
<A<7,uO*y>#) = — -[M x >y> z )±M- x >y> z )l
v 2
t Here we are using'atomic units.
292 The Diatomic Molecule §81
which are symmetrical and antisymmetrical about the plane x = which bisects the line
joining the nuclei (which are at (±iR, 0, 0)). Here *f>o(x, y, z) is the wave function of the
electron in one of the potential wells. Exactly as in §50, Problem 3, we find
ndfo
00 dydz, (1)
8x
where the integration is over the plane x = 0.
The function ^o (corresponding to motion around nucleus 1, say, at x = %R) is sought
in the form
h = -^e-r*, (2)
V 77-
where a is a slowly varying function (for a hydrogen atom, a = 1). The function ^o must
satisfy Schrodinger's equation
/ 1 1 1 \
£A<A+(-£-- + - + - <£ = o,
\ R n r 2 /
(3)
where n, r% are the distances of the electron from nuclei 1 and 2. In this equation the total
energy of the electron is Eo — l/R, since Eq itself includes the energy 1/R of the Coulomb
repulsion of the nuclei.
Since the function tfio decreases rapidly away from the #-axis, only the region where y
and z are small compared with R is important in the integral (1). For y, z <^ R, substitution
of (2) in (3) gives
da a a
— + - = 0;
dx \R+x R
here we have neglected the second derivatives of the slowly varying function a and put
?2 S iR+x. The solution of this equation which becomes unity as x -*■ %R (i.e. in the
neighbourhood of nucleus 1) is
2R
exp
R+2x
a->}
Formula (1) now gives
Ug, u -Eo=+— I e-^i . 2ttt 1 dr ±
- S
ire J
R/2
= +(2/e)Re-R
At sufficiently large distances this expression decreases exponentially and becomes less
than the effect in the second approximation with respect to the dipole interaction of the H
atom and the H + ion. Since the polarisability of the hydrogen atom in the normal state is
9/2 (see (77.9)), and the field of the H"' ion is $ = 1/R 2 , the corresponding interaction energy
is —9/4R 4 , and when this is taken into account we have
U g , u (R) - Eo = + -Re-* - — . (4)
e 4/c 4
The second term becomes comparable with the first when R = 10-8. It may also be noted
that the term U u has a minimum of — 5 -8 x 10~ 5 atomic unit ( — 1 -6 x 10~ 3 eV) when R = 12-6.f
t This minimum, which is due to van der Waals forces, is very shallow compared with that of the
term U g (R) which corresponds to the normal state of the stable ion H2 + : the latter minimum is
-0-60 atomic unit (-16-3 eV), at R = 2-0.
§82 Vibrational and rotational structures of singlet terms 293
§82. Vibrational and rotational structures of singlet terms in the
diatomic molecule
As has been pointed out at the beginning of this chapter, the great differ-
ence in the masses of the nuclei and the electrons makes it possible to divide
the problem of determining the energy levels of a molecule into two parts.
We first determine the energy levels of the system of electrons, for nuclei at
rest, as functions of the distance between the nuclei (the electron terms).
We can then consider the motion of the nuclei for a given electron state ; this
amounts to regarding the nuclei as particles interacting with one another in
accordance with the law U n (r), where U n is the corresponding electron term.
The motion of the molecule is composed of its translational displacement as a
whole, together with the motion of the nuclei about their centre of mass. The
translational motion is, of course, without interest, and we can regard the
centre of mass as fixed.
For convenience of discussion, let us first consider the electron terms in
which the total spin S of the molecule is zero (the singlet terms). The
problem of the relative motion of two particles (the nuclei) which interact
according to the law U(r) reduces, as we know, to that of the motion of a
single particle of mass M (the reduced mass of the two particles) in a centrally
symmetric field U(r). By U(r) we mean the energy of the electron term
considered. The problem of motion in a centrally symmetric field U(r),
however, reduces in turn to that of a one-dimensional motion in a field where
the effective energy is equal to the sum of U(r) and the centrifugal energy.
We denote by K the total angular momentum of the molecule, composed
of the orbital angular momentum L of the electrons and the angular momen-
tum of the nuclei. Then the operator of the centrifugal energy of the nuclei
B(r)(K-L)\
where we have introduced the notation
B(r) = H 2 /2Mr 2 (82.1)
for a convenient simplification of the formulae in the theory of diatomic
molecules. Averaging this quantity (for a given r), we obtain the centrifugal
energy as a function of r, which must appear in the effective potential energy
U K (r). Thus
U K (r) = t/(r)+5(r)(K-L)*,
where the line denotes the average mentioned. Expanding the square and
recalling that the square K 2 of a conserved total angular momentum has the
definite value K(K+1) (where K is integral), we can rewrite this expression
in the form
U K (r) = U(r)+B(r)K(K+l)+B(r)(l}-2L . K); (82.2)
we omit the line over the quantity K, since it is conserved.
294 The Diatomic Molecule §82
In a state with a definite value of L z = A, the mean values of the other two
components of the orbital angular momentum are zero, L x = L y = 0; this
follows at once from the fact that, in a representation in which L z is diagonal,
the diagonal matrix elements of the operators L x and L are zero (see §27).
Hence the mean value of the vector L is plirected along the sr-axis and we can
write
L = nA,
where n is a unit vector along the axis of the molecule. Next, in classical
mechanics the angular momentum of a system of two particles (the nuclei) is
directed perpendicular to the line joining them ; in quantum mechanics the
same is true for the angular momentum operator. Hence we can write
(R— L) . n = 0, and R . n = L . n. Hence, for the eigenvalues,
K.n=L.n=A. (82.3)
Thus the projection of the total angular momentum K on the axis of the mole-
cule is also A. Hence it follows that, in a state with a given value of A, the
quantum number K can take only values from A upwards :
K > A. (82.4)
Finally, substituting in (82.2) L . K = An . K = A 2 , we obtain
U K (r) = U(r)+B(r)K(K+l)+B(r)(V-2A*). (82.5)
The last term on the right-hand side is some function of r, depending only
on the electron state, and not on the quantum number K. This function
can be included in the energy U(r), and (82.5) then takes the form
U K (r) = U(r)+B(r)K(K+l). (82.6)
On solving the one-dimensional Schrodinger's equation with this potential
energy, we obtain a series of energy levels. We arbitrarily number these
levels (for each given K) in order of increasing energy, using a number
v = 0, 1, 2, ... ; v = corresponds to the lowest level. Thus the motion
of the nuclei causes a splitting of each electron term into a series of levels
characterised by the values of the two quantum numbers K and v.
The number of these levels (for a given electron term) may be either finite
or infinite. If the electron state is such that, as r -> oo, the molecule becomes
two isolated neutral atoms, then as r -> oo the potential energy U(r) (and
therefore U K (r)) tends to a constant limiting value U(oo) (the sum of the
energies of the two isolated atoms) more rapidly than 1/r tends to zero (see
§89). The number of levels in such a field is finite (see §18), though in actual
molecules it is very large. The levels are so distributed that, for any given
value of K, there is a definite number of levels (with different values of v),
while the number of levels with the same K diminishes as K increases, until
a value of K is reached for which there are no levels at all.
§82 Vibrational and rotational structures of singlet terms 295
If, on the other hand, as r -» oo the molecule disintegrates into two ions,
at large distances U(r) — U(oo) becomes the energy of the attraction of the
ions according to Coulomb's law (^ 1/r). In such a field there is an infinite
number of levels, which become closer and closer as we approach the limiting
value E/(oo). We may remark that, for the majority of molecules, the previous
case is found in the normal state ; only a comparatively small number of mole-
cules become pairs of ions when their nuclei are moved apart.
The dependence of the energy levels on the quantum numbers cannot be
completely calculated in a general form. Such a calculation is possible only
for low excited levels which lie not too far above the ground level.f Small
values of the quantum numbers K and v correspond to these levels. It is
with such levels that we are in fact most often concerned in the study of
molecular spectra, and hence they are of particular interest.
The motion of the nuclei in slightly excited states can be regarded as small
vibrations about the equilibrium position. Accordingly we can expand U(r)
in a series of powers of £ = r— r e , where r e is the value of r for which U(r)
has a minimum. Since U'(r e ) — 0, we have as far as terms of the second
order
U(r) = t/ 8 +|Mov^,
where U e = U(r e ), and a> e is the frequency of the vibrations. J
In the second term in (82.6) — the centrifugal energy — it is sufficient to
put r = r e , since it already contains the small quantity K(K+1). Thus we
have
U K (r) = U.+BJC{K+l)+tM<**?, (82.7)
where B e = W>\2Mr? = h % \71 is what is called the rotational constant
(I ='"Mr e 2 is the moment of inertia of the molecule).
The first two terms in (82.7) are constants, while the third corresponds
to a one-dimensional harmonic oscillator. Hence we can at once write down
the required energy levels :
E = U e +B e K(K+l)+hco e (v+l). (82.8)
Thus, in the approximation considered, the energy levels are composed of
three independent parts :
E = E el +E r +E v . (82.9)
Here E a = U e is the electron energy (including the energy of the Coulomb
interaction of the nuclei for r = r e ),
E r =B e K{K+\) (82.10)
t We refer always to levels belonging to the same electron term.
J We here use the notation customary in the theory of diatomic molecules.
296 The Diatomic Molecule §82
is the rotational energy from the rotation of the molecule, f and
E v = h<* e {v+\) (82.11)
is the energy of the vibrations of the nuclei within the molecule. The number
v denumerates, by definition, the levels with a given K in order of increasing
energy; it is called the vibrational quantum number.
For a given form of the potential energy curve U(r), the frequency <o e is
inversely proportional to ^JM. Hence the intervals AE V between the
vibrational levels are proportional to 1[\/M. The intervals AE r between
the rotational levels contain in the denominator the moment of inertia /, and
are therefore proportional to \jM. The intervals AE el between the electron
levels, however, are independent of M, like the levels themselves. Since mjM
(m being the electron mass) is a small parameter in the theory of diatomic
molecules, we see that
AE el > AE V > AE r .
Thus the distribution of the energy levels of the molecule is rather unusual.
The vibrational motion of the nuclei splits the electron terms into levels lying
comparatively close together. These levels, in turn, exhibit a fine splitting
due to the rotational motion of the molecule. J
In subsequent approximations, the separation of the energy into indepen-
dent vibrational and rotational parts is impossible; rotational- vibrational
terms appear, which contain both K and v. On calculating the successive
approximations, we should obtain the levels E as an expansion in powers of the
quantum numbers K and v.
We shall calculate here the next approximation after (82.8). To do this,
we must continue the expansion of U(r) in powers of £ up to terms of the
fourth order (cf. the problem of an anharmonic oscillator in §38). Similarly,
the expansion of the centrifugal energy is extended as far as the terms in | 2 .
We then obtain
U K {r) = U e +iM^e+(h 2 l2Mr e ^)K(K+l)~
-a£*+b?-(h 2 IMr*)K(K+ l)£+(3k*l2Mr t *)K(K+ 1)£ 2 . (82.12)
Let us now calculate the correction to the eigenvalues (82.8), using
perturbation theory and regarding the last four terms in (82.12) as the per-
turbation operator. Here it is sufficient, for the terms in | 2 and £ 4 , to take the
first approximation of perturbation theory, but for those in f and £ 3 we must
t A rotating system of two rigidly connected particles is often called a rotator. Formula (82.10)
gives the quantum-mechanical energy levels for a rotator. The wave functions of the stationary states
of a rotator evidently correspond to the case A = and are ordinary spherical harmonic functions
(see the Problem at the end of this section).
J As an example, we give the values of U e , hu>e and B e (in electron-volts) for a few molecules :
H 2 N 2 2
-U e 4-7 7-5 5-2
Hcoe 0-54 0-29 0-20
10 3 x5 e 7-6 0-25 0-18
§82
Vibrational and rotational structures of singlet terms
297
calculate the second approximation, since the diagonal matrix elements of |
and | 3 vanish identically. All the matrix elements needed for the calculation
are derived in §23 and in §38, Problem 3. As a result, we obtain an expres-
sion which is usually written in the form
E - E tl +ha>Jiv+\)-x-hco 6 {v+}tf+B v K{K+ \)-D € K%K+ 1) 2 , (82.13)
where
B v = B e — a. e (v+%) = B —cc e v.
(82.14)
The constants x ey B e , <x e , D e are related to the constants appearing in (82.12)
by
B e = /* 2 /27, D = AB e z lh*<* e \
6B*Y ah I 2 \ 3 /h Yf 5 a% b\
(82.15)
The terms independent of v and K are included in E el .
PROBLEM
Determine the angular part of the wave function for a diatomic molecule with zero spin
(F. Reiche 1926).
Solution. The required functions are just the eigenfunctions of the total angular momen-
tum K of the molecule. The operator of the total angular momentum is the sum
R =rxp+ Sr a xp ,
where p is the linear momentum of the relative motion of the nuclei, r the radius vector be-
tween them, r„ and p tt the radius vectors and linear momenta of the electrons (relative to the
centre of mass of the molecule). Introducing the polar angle 6 and the azimuthal angle <f> of
the axis of the molecule relative to a fixed system of co-ordinates *, y, z, we have for the
components of the operator & expressions similar to (26.14), (26.15), so that
&
= e x H \-i
cot0 — )+L+,
— + i cot e— )+£_,
dd d<f>J
(i)
it.
■ d ' t
■i \-L z ,
where L . = Lx + ilj.
. / 8 8 \
are the operators of the angular momenta of the electrons; the primes on djdd and djd<j>
signify that the differentiation is to be performed for constant x a , ya, Za-
Besides the fixed system of co-ordinates x, y, z, we introduce a moving system £, t), £, with
the same origin, the £-axis directed along the axis of the molecule, and the £-axis lying in the
298 The Diatomic Molecule §82
«y-plane. The co-ordinates £ a , ya, U of the electrons in this system are related to the co-
ordinates X a , ya, Za by
€ a = —x a sin <f>+y a coscf),
-q a = — x a cos 6 cos<f>— y a cos0 sin^+.s'a sin0,
t a = x a sin 6 cos <f>+y a sin 9 sin (f>+z a cos 6.
Using these formulae, we can transform the derivatives:
8 8 d
= sin0 [-cosd , etc.,
fea fya %a
d 8 Z\ 8 d 8$ a 86 8 Va 86 8tJ
8 \? / d 8 \
where 9/90 and djd<f> (unprimed) denote differentiation for constant £ a , 17a, U- As a result,
we have for the operators of the components of the total angular momentum relative to the
fixed system the expressions
K+ = e«P [ — \-i cot0 — )+ L,,
\86 8<f>J sin 6 ^
£ z = -i8j8<j>,
where
is the operator of the angular momentum of the electrons about the axis of the molecule.
Let
*PnAKM K = <f>nAK(€ a > Va, £a"> r)p n AK(r)@ A KM K (9)^>M K (<f>) (3)
be the wave function of a state with definite values of the absolute value K and ^-component
Mk of the total angular momentum of the molecule, a nd a definite value A of the ^-component
of the electron angular momentum ; n denotes the ass embly of the remaining quantum num-
bers which determine the state of the molecule. <f>nAK is the electron wave function, depend-
ing on r as a parameter, p n AK is the "radial part" of the nuclear wave function, &AKMK is the
required function of the angle 0, and the dependence of ifi on the angle <f> is obvious :
1
*>**&) = —^rr eiMR<l> '
v( 27r )
When the operators J2" z , £j act on the function (3), we can replace them by their eigenvalues
§83 Multiplet terms 299
Mr, A, so that
/d \ e x< f
g = e m M K cot0 ) + A,
\dd / sin 9
^ ,/ 3 \ e"^
it- = e-^( M x cot0 )+ A.
\ 36 J sin9
The subsequent argument exactly follows that at the end of §28. When the operator
&+ acts on the function «A„Akk (with Mr = K), the result is zero; hence we have the
equation
d A \
Kcot6+ )®AKK = 0,
G
whose solution ist
(2*1+1)
2*k+i(K+A)\(K-A)\
the function is normalised by the condition
®AKK=(-i) K I (2 ^ +1)! x(l-cos^- A )/2(l + cos^+ A )/2; (4)
AKK K ' V 2^+UK+A)\(K-A)\
je AKK 2 sin ede = 1,
and the normalisation integral reduces to Euler's beta function.
The remaining functions are then calculated from the formula
- — — -Qakm =R- k - Mk &akk,
V (K+M K )\
and as a result we obtain
/ (2K+1)\(K+M K )\ (I - cos 8)( a -Mk) 12
G\ KT71l/r = t-.i\K / 1 _ 1 x X
VAKM K \ ) ,J ( K+fi)] ( K _ A) l( K _ MK )\ (l + cOsdy A +M«)V
KQ k K-Mk -i
) (l-cos^- A (l+cos^+ A .
d cos 9 J J
For A = these functions become ordinary spherical harmonic functions, as they should:
®0KM = constant x P^*(cos 6),
and are the wave functions of a rotator (eigenfunctions of the free angular momentum K).
§83. Multiplet terms. Case a
Let us now turn to the question of the classification of molecular levels with
non-zero spin S. In the zero-order approximation, when relativistic effects
are entirely neglected, the energy of the molecule, like that of any system of
particles, is independent of the direction of the spin (the spin is "free"), and
this results in a (2S+l)-fold degeneracy of the levels. When relativistic
f The choice of the phase factor accords with the definition of the eigenfunctions of the free angular
momentum (§28), which are obtained for A = 0.
300 The Diatomic Molecule §83
effects are taken into account, however, the degenerate levels are split, and
the energy consequently becomes a function of the projection of the spin on
the axis of the molecule. We shall refer to relativistic interactions in mole-
cules as the spin-axis interaction. The chief part in this is played (as in the
case of atoms) by the interaction of the spins with the orbital motion of the
electrons.^
The nature and classification of molecular levels depend markedly on the
relative parts played by the interaction of the spin with the orbital motion,
on the one hand, and the rotation of the molecule, on the other. The part
played by the latter is characterised by the distances between adjacent rota-
tional levels. Accordingly, we have to consider two limiting cases. In one,
the energy of the spin-axis interaction is large compared with the energy
differences between the rotational levels, while in the other it is small. The
first case is usually called case (or coupling type) a, following Hund, and the
second is called case b.
Case a is the one most often found. An exception is formed by the 2
terms, where case b chiefly occurs, since the effect of the spin-axis interaction
is very small for these terms! (see below). For other terms, case b is some-
times found in the lightest molecules, since the spin-axis interaction is here
comparatively weak, while the distances between the rotational levels are
large (the moment of inertia being small).
Of course, cases intermediate between a and b are also possible. It must
also be borne in mind that the same electron state may pass continuously
from case a to case b as the rotational quantum number changes. This is due
to the fact that the distances between adjacent rotational levels increase with
the rotational quantum number, and hence, when this is large, the distances
may become large compared with the energy of the spin-axis coupling (case
b), even if case a is found for the lower rotational levels.
In case a, the classification of the levels is in principle little different from
that of the terms with zero spin. We first consider the electron terms for
nuclei at rest, i.e. we neglect rotation entirely; besides the projection A of the
orbital angular momentum of the electrons, we must now take into account the
projection of the total spin on the axis of the molecule. This projection is
denoted by|| S; it takes the values S, 5-1, ... , -S. We arbitrarily regard
2 as positive when the projection of the spin is in the same direction as
that of the orbital angular momentum about the axis (we recall that A denotes
the absolute value of the latter). The quantities A and S combine to give
the total angular momentum of the electrons about the axis of the molecule :
Q=A+S; (83.1)
f Besides the spin-orbit and spin-spin interactions there is also an interaction of the spin and orbital
motion of the electrons with the rotation of the molecule. This part of the interaction is very small,
however, and it is of possible interest only for terms with spin S = J (see §84).
X A special case is the normal electron term of the molecule 2 (the term S S). For this we have a
type of coupling intermediate between a and b (see §84, Problem 3).
|| Not to be confused with the symbol for terms with A == 0.
§83 Multiplet terms 301
this takes the values A+S, A+S-l, ... , A-S. Thus the electron term
with orbital angular momentum A is split into 25+1 terms with different
values of ft; this splitting, as with atomic terms, is called the^ine structure
or multiplet splitting of the electron levels. The value of H is usually indicated
as a suffix to the symbol for the term: thus, for A = 1, S = \ we obtain the
terms 2 IIi/2, 2 n 3/2 .
When the motion of the nuclei is taken into account, vibrational and rota-
tional structures appear in each of these terms. The various rotational levels
are characterised by the values of the quantum number /, which gives the
total angular momentum of the molecule, including the orbital and spin
angular momenta of the electrons and the angular momentum of the rotation
of the nuclei.f This number takes all integral values from \€i\ upwards:
7>|0|, ( 83 - 2 )
which is an obvious generalisation of (82.4).
Let us now derive quantitative formulae to determine the molecular levels
in case a. First of all, we consider the fine structure of an electron term.
In discussing the fine structure of atomic terms in §72, we used formula
(72.4), according to which the mean value of the spin-orbit interaction is
proportional to the projection of the total spin of the atom on the orbital
angular momentum vector. Similarly, the spin-axis interaction in a diatomic
molecule (averaged over electron states for a given distance r between the
nuclei) is proportional to the projection S of the total spin of the molecule on
its axis, so that we can write the split electron term in the form
C/(r)+^(r)S,
where U(r) is the energy of the original (unsplit) term, and A(r) is some func-
tion of r; this function depends on the original term (and in particular on A),
but not on S. Since one usually uses the quantum number Q. and not 2,
it is more convenient to put AQ. in place of AZ; these expressions differ by
AA, which can be included in U(r). Thus we have for an electron term the
expression /M „
U(r)+A(r)Cl. (83.3)
We may notice that the components of the split term are equidistant from
one another: the distance between adjacent components (with values of Q.
differing by unity) is A(r), independent of £1.
It is easy to see from general considerations that the value of A for S terms
is zero. To show this, we perform the operation of changing the sign of the
time. The energy must then remain unchanged, but the state of the mole-
cule changes in that the direction of the orbital and spin angular momenta
about the axis is reversed. In the energy ^4(r)S, the sign of S is changed, and
if the energy remains unchanged A(r) must change sign. If A # 0, we can
f The notation K is, as usual, reserved for the total angular momentum of the molecule without
allowance for its spin. In case a there is no quantum number K, since the angular momentum K is
not even approximately conserved.
302 The Diatomic Molecule §83
draw no conclusions regarding the value of A(r), since this depends on the
orbital angular momentum, which itself changes sign. If A = 0, however,
we can say that A(r) is certainly unchanged, and consequently it must vanish
identically. Thus, for the S terms, the spin-orbit interaction causes no split-
ting in the first approximation; splitting (proportional to S 2 ) would occur
only on taking account of this interaction in the second approximation or the
spin-spin interaction in the first approximation, and would be relatively
small. This is the reason for the fact, already mentioned, that case b usually
occurs for 21 terms.
When the multiplet splitting has been determined, we can take account of
the rotation of the molecule as a perturbation, just as in the derivation given
at the beginning of §82. The angular momentum of the rotation of the nuclei
is obtained from the total angular momentum by subtracting the orbital
angular momentum and spin of the electrons. Hence the operator of the
centrifugal energy now has the form
B(r)(J-L-S)\
Averaging this quantity with respect to the electron state and adding to
(83.3), we obtain the required effective potential energy Uj(r):
Uj(r) = U(r)+A(r)Q+B(r)(]-L-Sf
= tf(r)+^(r)Q+£(r)[J 2 -2J.(L+S)+L 2 +2L. S+S 2 ].
The eigenvalue of J 2 is /(/ + 1). Next, by the same argument as in §82, we
have
L = nA, S = nS, (83.4)
and also (J— L— S) . n = 0, whence we have for the eigenvalues
J.n =(L+S).n =A+S = O. (83.5)
Substituting these values, we find
Uj(r) = U(r)+A(r)£l+B(r)[J(J+l)-2Q. 2 +V+2L.S+S 2 ].
The averaging with respect to the electron state is effected by means of the
wave functions of the zero-orderf approximation. In this approximation,
however, the magnitude of the spin is conserved, and hence S 2 = S^S+l).
The wave function is the product of the spin and co-ordinate functions ; hence
the averaging of the angular momenta L and S takes place independently, and
we obtain
L.S =An.S =AS.
f That is, the zero-order approximation with respect to both the effect of the rotation of the mole-
cule and the spin-axis interaction.
§84 Multiplet terms 303
Finally, the mean value of the squared orbital angular momentum L 2 is
independent of the spin, and is some function of r characterising the given
(unsplit) electron term. All the terms which are functions of r but indepen-
dent of J and S can be included in U(r), while the term proportional to S
(or, what is the same thing, to Q.) can be included in the expression A{r)Cl.
Thus we have for the effective potential energy the formula
Uj(r) = U(r)+A(r)£l+B(r)[J(J+l)-2&]. (83.6)
The energy levels of the molecule can be obtained from this by the same
method as in §82 when using the formula (82.6). Expanding U(r) and A(r)
in series of powers of £, and retaining the terms up to and including the
second order in the expansion of U(r), but only the terms of zero order in the
second and third terms, we obtain the energy levels in the form
E = U 6 +A 6 a+h<* e {v+\)+B e [](J+\)-2Wl (83.7)
where A e = A(r e ) and B e are constants characterising the given (unsplit)
electron term. On continuing the expansion to higher terms, we obtain a
series of terms in higher powers of the quantum numbers, but we shall not
pause to write these out here.
§84. Multiplet terms. Case b
Let us now turn to case b. Here the effect of the rotation of the molecule
predominates over the multiplet splitting. Hence we must first consider the
effect of rotation, neglecting the spin-axis interaction, and then the latter
must be taken into account as a perturbation.
In a molecule with "free" spin, not only the total angular momentum J
but also the sum K of the orbital angular momentum of the electrons and the
angular momentum of the nuclei are conserved; the latter is related to J by
J=K+S. (84.1)
The quantum number K distinguishes different states of a rotating molecule
with free spin that are obtained from a given electron term. The effective
potential energy Uj^r) in a state with a given value of K is evidently deter-
mined by the same formula (82.6) as for terms with S = 0:
U K (r) = U(r)+B(r)K(K+l), (84.2)
where K takes the values A, A+l, ... .
When the spin-axis interaction is included, there is a splitting of each
term into 2S+1 terms in general (or 2K+1 if K < S), which differ in the
value of the total angular momentum! /. According to the general rule for
the addition of angular momenta, the number / takes (for a given K) values
f In case b, the projection n.S of the spin on the axis of the molecule does not have definite
values, so that there is no quantum number S (or Q).
304 The Diatomic Molecule §84
from K+S to \K-S\:
\K-S\<J<K+S. (84.3)
To calculate the energy of the splitting (in the first approximation of per-
turbation theory), we must determine the mean value of the operator of the
spin-axis interaction energy for the state in the zero-order approximation
(with respect to this interaction). In the case considered, this means averag-
ing with respect to both the electron state and the rotation of the molecule
(for a given r). The result of the first averaging is, as we know, an operator
of the form A(r)n . S, which is proportional to the projection n . S of the spin
operator on the axis of the molecule. Next we average this operator with
respect to the rotation of the molecule, taking the direction of the spin vector
to be arbitrary; then n . § = h . S. The mean value n is a vector which,
from considerations of symmetry, must have the same direction as the
"vector" R, the only vector which characterises the rotation of the molecule.
Thus we can write
n = constant x £.
The coefficient of proportionality is easily determined by multiplying both
sides of this equation by R; noting that the eigenvalues of n . K and K 2 are
respectively A (see (82.3)) and K(K+ 1), we find the constant to be AIKIK+ 1).
Thus
n.S =AR.S/K(K+l).
Finally, the eigenvalue of the product K . S, according to the general formula
(31.2), is
K. S = W(J+V-K(K+1)-S(S+1)]. (84.4)
As a result, we arrive at the following expression for the required mean
value of the energy of the spin-axis interaction :
Ar)A[J(J+ 1)-S(S+ \)-K{K+ 1)]I2K(K+ 1)
= Ar)A[(J-S)(J+S+l)]l2K(K+l)-lA(r)A.
This expression must be added to the energy (84.2). The term \A(r)A, being
independent of K and /, can be included in U(r), so that we have finally for
the effective potential energy the expression
U K (r) = U(r)+B(r)K(K+l)+A(r)A(J-S)(J+S+l)l2K(K+l). (84.5)
An expansion in powers of £ = r— r e gives, in the usual manner, an expres-
sion for the energy levels of the molecule in case b :
E = U e +h<* e {v+\)+B e K{K+\)+A e A{J-S){J+S+\)}K{K+\). (84.6)
§84 Multiplet terms 305
As has been pointed out in the previous section, the spin-orbit interaction
for 2 terms does not give a multiplet splitting in the first approximation, and
to determine the fine structure we must take into account the spin-spin
interaction, whose operator is quadratic with respect to the spins of the elec-
trons. We are at present interested not in this operator itself, but in the result
of averaging it with respect to the electron state of the molecule, as was done
for the operator of the spin-orbit interaction. It is evident from considera-
tions of symmetry that tie required averaged operator must be proportional
to the squared projection of the total spin of the molecule on the axis, i.e.
it can be written in the form
a(r)(S.n) 2 , (84.7)
where a(r) is again some function of the distance r, characterising the given
electron state. Symmetry allows also a term proportional to S 2 , but this is
immaterial since the absolute value of the spin is just a constant. We shall
not pause here to derive the lengthy general formula for the splitting due to
the operator (84.7) ; in Problem 1 of this section we give the derivation of the
formula for triplet 21 terms.
The doublet S terms form a special case. According to Kramers' theorem
(§60), the double degeneracy in a system of particles with total spin S = \
certainly persists, even when the internal relativistic interactions in the system
are fully allowed for. Hence the 2 I! terms remain unsplit, even when we
take account of both the spin-orbit and the spin-spin interaction, and in any
approximation.
The splitting is obtained here only by taking into account the relativistic
interaction of the spin with the rotation of the molecule; this effect is very
small. The averaged operator of this interaction must evidently be of the
form y& . S, and its eigenvalues are determined by the formula (84.4), in
which we must put S = $, / = K±%. As a result, we obtain for the 2 S
terms the formula
E = U e +h<* e (v+$+B e K(K+l)±b>(K+$; (84.8)
a constant — £y is included in U e .
PROBLEMS
Problem 1. Determine the multiplet splitting of a 3 2 term in case b (H. A. Kramers
1929).
Solution. The required splitting is determined by the operator (84.7), which must be
averaged with respect to the rotation of the molecule. We write it in the form <x e ninicSiSk,
where oe e = oc(ro). Since the vector S is conserved, only the products mmc need be averaged.
According to the formula derived in §29, Problem 2, we have
KiKjc + KjcKi
ntnjc = h ... ;
(2a:-i)(2*:+3)
here we have not written out the terms proportional to 8a whose contribution to the energy
is independent of J and therefore does not cause any splitting of the type under consideration.
306 The Diatomic Molecule §84
Thus the splitting is given by the operator
(2K-l)(2K+3)
Since S commutes with &,
SiSicRtK* = SiKiSjcK* = (S . K)2,
where the eigenvalue S . K is given by (84.4). We also have
StSjcKjcKi = SiSjcKiKjc+iSiSjceMiKi
= (S . E)*+U£ i S t -£ k S t )ie ai &i
= (S . K) 2 +^en c ien, ;m i§ m Ki
= (S.K)2 + S.K.
The values/ = K, K± 1 correspond to the three components E K of the triplet 3 S (S = 1).
For the intervals between these components we find
K+\ K
Ek+i - E K = — g< £#_! ~ E K = —a.
'2K+1' e 2K-l
Problem 2. Determine the energy of a doublet term (with A#0) for cases intermediate
between a and 6 (E. Hill and J. H. van Vleck 1928).
Solution. Since the rotational energy and the energy of the spin-axis interaction are
supposed of the same order of magnitude, they must be considered together in perturbation
theory, so that the perturbation operator is of the formf
As wave functions in the zero-order approximation it is convenient to use those of states in
which the angular momenta .Kand / have definite values (i.e. those of case b). Since S — £
for a doublet term, the quantum number K, for a given /, can take the values K = /±J.
To construct the secular equation, we must calculate the matrix elements VZIk v (« denoting
the assembly of quantum numbers defining the electron term), where K,K' take the above
values. The matrix of the operator K 2 is diagonal ; the diagonal elements are K(K+ 1). The
matrix elements of n . S are calculated from the general formula (109.5), in which we must
putji = S, 72 = K; the matrix elements of n are given by (87.1). Calculation gives the
secular equation
B e (J+m+%)-AAI(2J+ 1)-£W AvUm) 2 -A 2 ]/(2/+ 1)
^V[C/+i) 2 -A 2 ]/(2/+l) B e (J+W-l)+AAI(2J+l)-EV
Solving this equation and adding E^ to the unperturbed energy, we have
E = u e +h^v+±)+B e j{j+\)±^[B*u+W-ABA+lA 2 Y,
a constant \B e is included in U e . The inequality A e ^> B e J corresponds to case a, and the
opposite one to case b.
Problem 3. Determine the intervals between the components of a triplet level 3 Sina
case intermediate between a and b.
= o.
t The averaging with respect to vibrations must be done before that with respect to rotation.
Hence, restricting ourselves to the first terms of the expansions in £, we have replaced the functions
B(r ) and A(r) by the values B e and A e , and the unperturbed energy levels are 2?(°> = U e + Hco e (v + J).
§85 Multiplet terms 307
Solution. As in Problem 2, the rotational energy and the energy of the spin-spin inter-
action are considered together in the perturbation theory. The perturbation operator is of
the form
t = 5 e & 2 +a 6 (n.S) 2 .
As wave functions in the zero-order approximation we use those of case b. The matrix elements
(n . S)f . (we omit all suffixes with respect to which the matrix is diagonal) are again calcu-
lated from (109.5) and (87.1), this time with A = 0, S = 1. The non-zero elements are of
the form
(n.S)^ = VK7+i)/(2/+i)L ( n - s )/ +1 = VL//(2J+i)].
For a given /, the number K can take the values K = J, J±\. For the matrix elements V& ,
we find
V J j = B e J(J+\)+* e , V£l = B e {J-\)J+* e {J+\)l{2J+\),
V jtl = V i-l = a eV[/(/+l)]/(2/+l).
We see that there are no transitions between states with K = J and those with K = J±.\.
Hence one of the levels is simply E x = Vj. The other two (E 2 , E a ) are obtained by solving
the quadratic secular equation formed from the matrix elements V J j~\, V J j%\, F/+i- Since
we are here interested only in the relative position of the components of the triplet, we
subtract the constant a e from all three energies E lt E 2 , E 3 . As a result we obtain
*i =*./(/+!),
E 2S = J B 6 (7 2 +/+l)-K±V[B e 2 (2/+l) 2 -a e J5 e +K 2 ].
In case b (a small), by considering three levels with the same K and different J (J = K,
K±l), we again obtain the formulae of Problem 1.
§85. Multiplet terms. Cases c and d
Besides cases of a and b coupling and those intermediate between them,
there are also other types of coupling. These originate as follows. The
occurrence of the quantum number A is due ultimately to the electric interac-
tion of the two atoms in the molecule, which results in the axial symmetry
of the problem of determining the electron terms (this interaction in the
molecule is called the coupling between the orbital angular momentum and
the axis). The distances between terms with different values of A give a
measure of the magnitude of this interaction. Previously we have tacitly
supposed this interaction so strong that these distances are large both com-
pared with the intervals in the multiplet splitting and compared with those
in the rotational structure of the terms. There are, however, opposite cases
where the interaction of the orbital angular momentum with the axis is com-
parable with or even small compared with the other effects ; in such cases, of
course, we cannot in any approximation speak of a conservation of the pro-
jection of the orbital angular momentum on the axis, so that the number A
is no longer meaningful.
308 The Diatomic Molecule §85
If the coupling of the orbital angular momentum with the axis is small
in comparison with the spin-orbit coupling, we say that we have case c. It
is found in molecules which contain an atom of a rare-earth element. These
atoms are characterised by the presence of / electrons with uncompensated
angular momenta ; their interaction with the axis of the molecule is weakened
by the deep position of the / electrons in the atom. Cases intermediate be-
tween the a and c types of coupling are found in molecules consisting of
heavy atoms.
If the coupling of the orbital angular momentum with the axis is small
compared with the intervals in the rotational structure, we say that we have
case d. This case is found for high rotational levels (with large J) in some elec-
tron terms of the lightest molecules (H 2 , He 2 ). These terms are characterised
by the presence in the molecule of a highly excited electron, whose interaction
with the remaining electrons (or, as we say, with the "core" of the mole-
cule) is so weak that its orbital angular momentum is not quantised along the
axis of the molecule (whereas the "core" has a definite angular momen-
tum Acore about the axis).
As the distance r between the nuclei increases, the interaction between the
atoms is diminished, and finally becomes small compared with the spin-orbit
interaction within the atoms. Hence, if we consider the electron terms for
fairly large r, we shall have case c. This must be borne in mind when
ascertaining the relation between the electron terms of the molecule and the
states of the atoms obtained as r -> oo. In §80 we have already discussed this
relation, neglecting the spin-orbit interaction. When the fine structure of
the terms is included, there arises also the question of the relation between
the values J x and / 2 of the total angular momenta of the isolated atoms and
the values of the quantum number Q, for the molecule. We shall give the
results here, without reiterating arguments which are entirely similar to those
of §80.
If the molecule consists of different atoms, the possible values off |0|
obtained on combining atoms with angular momenta J lt J 2 (J x ^ J 2 ) are
given by the same table (80.1), in which we must put^i, J 2 m place of L ly L 2 ,
and jO| in place of A. The only difference is that, for half-integral Jx+J z ,
the smallest value of \C1 \ is not zero as shown in the table, but \. For integral
7i+/ 2 , on the other hand, there are 2/ 2 +l terms with Q. = 0, for which (as
for S terms when the fine structure is neglected) we have to decide the
question of sign. If J x and/ 2 are each half-integral, the number 2/ 2 +l is
even, and there are equal numbers of terms, which we shall denote by +
and 0~. If J x and/ 2 are both integral, however, then/ 2 +l terms are 0+ and
J 2 are 0" (if (-lf^IJ, = 1) or vice versa (if {-\) J ^ J *IJ 2 = -1).
If the molecule consists of similar atoms in different states, the resulting
molecular states are the same as in the case of different atoms, the only
f In adding the two total angular momenta J lt J t of the atoms to form the resultant angular momen-
tum Q, the sign of Q is clearly immaterial.
§86 Symmetry of molecular terms 309
difference being that the total number of terms is doubled, with each term
appearing once as an even and once as an odd term.
Finally, if the molecule consists of similar atoms in the same state (with
angular momenta J x = J 2 = J), the total number of states is the same as in
the case of different atoms, while their distribution in parity is such that,
if / is integral and Q, is even, N g = N u +1;
if J is integral and Q. is odd, N g = N u ;
if J is half-integral and Q is even, N u = N g ;
if J is half-integral and Q. is odd, N u = N g +1.
All the 0+ terms are even and all the 0~ terms odd.
As the nuclei approach, a coupling of type c usually passes into one of
type a\. Here the following interesting circumstance may arise. As already
mentioned, the term with A = belongs to case b, and as regards the classi-
fication of case a this means that multiplet levels with different values of Q,
(and the same A = 0) have the same energy; but such levels can occur on
the approach of atoms which are in different fine-structure states.
Thus it may happen that the same molecular term corresponds to different
pairs of atomic fine-structure states. A similar situation may occur for terms
with O = which, on the approach of the nuclei, become a molecular term
with A # (and therefore S = —A). Such levels are doubly degenerate,
since in case a the same energy corresponds to the terms 0+ and 0~
(which may arise from different pairs of atomic states). J
§86. Symmetry of molecular terms
In §78 we have already examined some symmetry properties of the terms
of a diatomic molecule. These properties characterised the behaviour of the
wave functions in transformations which leave the co-ordinates of the nuclei
unaltered. Thus the symmetry of the molecule with respect to reflection in
a plane passing through its axis brings about the difference between 2+ and
S~ terms ; the symmetry with respect to a change in sign of the co-ordinates ||
of all the electrons (for molecules composed of like atoms) gives rise to the
classification of terms into even and odd. These symmetry properties char-
acterise the electron terms, and are the same for all rotational levels belonging
to the same electron term.
The states of the molecule, like those of any system of particles (see §30),
are characterised by their behaviour with respect to inversion, i.e. a simul-
taneous change in sign of the co-ordinates of all the electrons and the nuclei.
For this reason, all the terms for the molecule can be divided into positive
(whose wave functions are unaltered when the sign of the co-ordinates of the
f The correspondence between the classification of terms of types a and c cannot be derived in a
general form. Its derivation necessitates a consideration of the actual potential energy curves, taking
into account the rule that levels of like symmetry cannot intersect (§79).
t We here neglect what is called A-doubling (see §88).
|| The origin is supposed to be taken on the axis of the molecule, and half-way between the two
nuclei.
310 The Diatomic Molecule §86
electrons and nuclei is reversed) and negative (whose wave functions change
sign on inversion).f
For A # 0, each term is doubly degenerate, on account of the two possible
directions of the angular momentum about the axis of the molecule. As a
result of inversion, the angular momentum itself does not change sign, but
the direction of the axis of the molecule is reversed (since the atoms change
places), and hence the direction of the angular momentum A relative to the
axis of the molecule is reversed. Hence two wave functions belonging to the
same energy level are transformed into each other, and from them we can
always form a linear combination that is invariant with respect to inversion
and one that changes sign under this transformation. Thus we obtain for
each term two states, of which one is positive and the other negative. In
practice, every term with A ^ is split, however (see §88), and so these two
states correspond to different values of the energy.
The S terms require special consideration to determine their sign. First
of all, it is clear that the spin bears no relation to the sign of the term : the
inversion operation changes only the co-ordinates of the particles, leaving
the spin part of the wave function unaltered. Hence all the components of
the multiplet structure of any given term have the same sign. In other words,
the sign of the term depends only on K, and not on J.\
The wave function of the molecule is the product of the electron and
nuclear wave functions. It has been shown in §82 that, in a S state, the motion
of the nuclei is equivalent to that of a single particle, of orbital angular
momentum K, in a centrally symmetric field U(r). Hence we can say that,
when the sign of the co-ordinates is changed, the nuclear wave function is
multiplied by (-1) K (see (30.7)).
The electron wave function characterises the electron term, and to ascertain
its behaviour under inversion we must consider it in a system of co-ordinates
rigidly connected to the nuclei and rotating with them. Let x, y, z be a sys-
tem of co-ordinates fixed in space, and £, rj, £ a. rotating system of co-ordinates
in which the molecule is fixed. The direction of the axes of |, rj, £ is defined
so that the £-axis coincides with the axis of the molecule from (say) nucleus
1 to nucleus 2, and the relative position of the positive directions of the axes
of £, rj, £ is the same as in the system x, y, z (i.e. if the system x, y, z is left-
handed, the system £, rj, £ is so too). As a result of the inversion operation,
the direction of the axes of x, y, z is reversed, and the system changes from
left-handed to right-handed. The system £, rj, £ must also become right-
handed, but the £-axis, being rigidly connected to the nuclei, retains its
former direction. Hence the direction of either one of the axes of £, rj must
f We retain the customary terminology. It is unfortunate, however, since in the case of an atom
the behaviour of the terms with respect to the operation of inversion is referred to as parity and not
sign.
The sign of which we are here speaking must not be confused with the + and — which are added
as indices to S terms.
J We recall that case b usually holds for S terms, and so it is necessary to use the quantum numbers
K and /.
§86 Symmetry of molecular terms 311
be reversed. Thus the operation of inversion in the fixed system of co-
ordinates is equivalent in the moving system to a reflection in a plane passing
through the axis of the molecule. Under such a reflection, however, the
electron wave function of a 2+ term is unaltered, while that of a 2~" term
changes sign.
Thus the sign of the rotational components of a 2+ term is determined by
the factor ( — 1)^; all the levels with even K are positive, while those with
odd K are negative. For a 1r term, the sign of the rotational levels is deter-
mined by the factor ( — 1) K+1 ; all levels with even K are negative, while those
with odd K are positive.
If the molecule consists of similar atoms, f its Hamiltonian is also invariant
with respect to an interchange of the co-ordinates of the two nuclei. A term
is said to be symmetric with respect to the nuclei if its wave function is un-
altered when they are interchanged, and antisymmetric if its wave function
changes sign. The symmetry with respect to the nuclei is closely related to
the parity and sign of the term. An interchange of the co-ordinates of the
nuclei is equivalent to a change in sign of the co-ordinates of all the particles
(electrons and nuclei), followed by a change in sign of the co-ordinates of the
electrons only. Hence it follows that, if the term is even and positive (or
odd and negative), it is symmetric with respect to the nuclei. If, on the other
hand, it is even and negative (or odd and positive), then it is antisymmetric
with respect to the nuclei.
At the end of §62 we have established a general theorem that the co-ordinate
wave function of a system of two identical particles is symmetrical when the
total spin of the system is even, and antisymmetrical when it is odd. If we
apply this result to the two nuclei of a molecule composed of similar atoms,
we find that the symmetry of a term is related to the parity of the total spin
/ obtained by adding the spins i of the two nuclei. The term is symmetric
when i" is even, and antisymmetric when I is odd. % In particular, if the nuclei
have no spin (i = 0), / is zero also ; hence the molecule has no antisymmetric
terms. We see that the nuclear spin has an important indirect influence on
the molecular terms, although its direct influence (the hyperfine structure of
the terms) is quite unimportant.
When the spin of the levels is taken into account, an additional degeneracy
of the levels results. Again in §62, we have calculated the number of states
with even and odd values of / that are obtained on adding two spins i. Thus,
when i is half-integral, the number of states with even / is i(2«+l), and with
odd / is (j+1)(2*"+1). From what was said above, we conclude that the
ratio of the degrees g 8 , g a of the degeneracy || of symmetric and antisymmetric
f The two atoms must be not only of the same element, but also of the same isotope.
% Recalling the relation between the parity, sign and symmetry of terms, we conclude that, when the
total spin I of the nuclei is even, the positive levels are even and the negative levels odd, and vice
versa when J is odd.
|| The degree of degeneracy of a level is often referred to in this connection as its statistical weight.
Formulae (86.1), (86.2) determine the ratio of the nuclear statistical weights of symmetric and anti-
symmetric levels.
312 The Diatomic Molecule §87
terms for terms with half-integral i is
gslga=il(i+V- (86.1)
For integral /, we similarly find that this ratio is
SJg a =(i+m (86.2)
We have seen that the sign of the rotational components of a S + term is
determined by the number (—1) K . Hence, for example, the rotational com-
ponents of a 2+0 term for even K are positive, and therefore symmetric, while
for odd K they are negative and consequently antisymmetric. Bearing in
mind the results obtained above, we conclude that the nuclear statistical
weights of the rotational components of a S + ff level with successive values of
K take alternate values, in the ratios (86.1) or (86.2). A wholly similar situa-
tion is found for E+ M , 1r g and S~ M levels. In particular, for i = the
statistical weights of levels with even K for S + M and T,~ g terms, and of levels
with odd K for T, + g and S~ M terms, are zero. In other words, in the electron
states 2+ M , Tr g there are no rotational states with even K, and in 2+ , Hr u
states there are none with odd K.
Because of the extremely weak interaction of the nuclear spins with the
electrons, the probability of a change in / is very small, even in collisions of
molecules. Hence molecules differing in the parity of /, and accordingly
having only symmetric or only antisymmetric terms, behave almost as differ-
ent forms of matter. Such, for instance, are orthohydrogen and parahydro-
gen; in the molecule of the former, the spins i = \ of the two nuclei are
parallel (/ = 1), while in that of the latter they are antiparallel (/ = 0).
§87. Matrix elements for the diatomic molecule
In calculating the matrices of various quantities in the diatomic molecule,
let us begin with the matrix elements for transitions between states with
zero spin.
Let n be a unit vector along the axis of the molecule. The vector n, re-
garded as an operator, commutes with the operator of the energy of the
electrons and with that of the vibrational energy, but not with the angular
momentum K of the molecule. Hence the matrix of n is diagonal with respect
to all the quantum numbers except K and M K (where by M K we denote the
magnitude of the projection of the angular momentum K on a #-axis fixed
in space). Omitting all suffixes but these two, we write the matrix elements
in the form (K' 'M' 'k\o\KMk)- Their dependence on Mr is given directly
by the general formulae (29.7), (29.9), in which L and M must be replaced
by K and Mr, and the suffix n may be omitted. We denote the coefficients
in these formulae by n^,, for the present case, so that, for example,
(KM K \n z \KM K ) = «f .
§87 Matrix elements for the diatomic molecule 313
To calculate the quantities n%, we start from the equations n . £ = A,
n 2 = 1, written in the form (see (29.11))
\n+R.++\n-£.-+n z & z = A,
n+n-+n z 2 = 1,
and the commutation relation
n g n+— n+rig = 0.
Taking the diagonal matrix elements of these equations (the matrix elements
of K being determined from the general formulae (27.12) with K, M K in
place of L, M), we obtain, after some calculations which we here- omit, the
following formulae for the quantities required (H. Honl and F. London
1925):
H * = A Jw^) i W * _1 = (W * _l) * = iV[(K2 - A2)IK ^' (87 ' 1}
For A = these formulae give
n K = 0, n K ~i = WK;
K K v »
these correspond, as we should expect, to the matrix elements of a unit vector
for motion in a centrally symmetric field (see §29, Problem 1).
Next, let A be some vector physical quantity characterising the state of
the molecule when the nuclei are fixed.f Let us first consider this quantity
in the system of co-ordinates |, r\, £, which rotates with the molecule (the
£-axis coinciding with the axis of the molecule). The results of §29 cannot
here be applied in their entirety, since the angular momentum of the mole-
cule with respect to the system of co-ordinates £ rj, £ (i.e. the electron angular
momentum L) is not conserved ; only its ^-component A is conserved. The
results concerning the selection rule for the quantum number A (M in §29)
evidently remain fully valid, however. Thus the matrix elements of the vector
A that are not zero are
we denote by n the assembly of quantum numbers for the electron term, with
the exception of A.
If both the terms are S terms, we must also bear in mind the selection rule
arising from the symmetry with respect to reflection in a plane passing through
the axis of the molecule (the £-axis). Under such a reflection, the £-com-
ponent of an ordinary (polar) vector is unchanged, while that of an axial vector
changes sign. Hence we conclude that, for a polar vector, A c has non-zero
matrix elements only for the transitions S+ -» S+ and S~ -> S~, while for
t For example, the dipole moment or magnetic moment of the molecule.
314 The Diatomic Molecule §87
an axial vector the elements are non-zero for the transitions 2+ -> S~. We
need not discuss the components A ^, A v , since for these no transitions with-
out change of A are possible.
If the molecule consists of similar atoms, there is also a selection rule re-
garding parity. The components of a (polar) vector change sign under inver-
sion. Hence their matrix elements are non-zero only for transitions between
states of different parity (the reverse is true for an axial vector). In particular,
all the diagonal matrix elements of the components of a polar vector vanish
identically.
The question arises how the matrix elements (87.2) are related to those of
the same vector A in a fixed system of co-ordinates. In this system we can
again use the general formulae (29.7), (29.9), which give the dependence of
the matrix elements (nAKMK\A\n f A' K' M' k) on the quantum number Mr-
The coefficients in these formulae are naturally denoted by A^j^, ; we have
to relate these to the quantities (87.2).
It is seen from (87.2) that there are matrix elements diagonal with respect
to A only for the component along the axis of the molecule. Hence we can
write the equation
(nAKM K \A\nAK'M' K ) = {nAKM K \Ap\ri AK' M' K ).
In particular,
(nAKM K \A z \nAK'M K ) = (A^ A {KM K \n z \K'M K ).
Separating out the dependence on M K , we hence have
A nAK =n Kt A \*\ (87.3)
where the n%, are determined by formulae (87.1). Thus we have found some
of the relations which we desire.
To find the remaining relations (for the components non-diagonal with
respect to A), we notice that, since the quantity A refers to the molecule with
fixed nuclei, the operators Ag, A v , A^ evidently commute with the vector n.
The components of the vector A in the system x, y, z are linear combinations
of the components in the system |, r], £, the coefficients in these combinations
being functions of n x , n y , n z . Hence A x , A y , A z also commute with the vector
n. In particular
A z n z —n z 4 z = 0.
Taking from this equation the matrix elements for the transitions A,
K -> A— 1, K' with K' = K, K±\, we obtain three equations, from which
the dependence on K of the required quantities A^^E lK , can be found.
The coefficients in the resulting formulae can be related to the quantities
(87.2) by comparing the matrix elements of the scalar A 2 , these being calcu-
latedf in the two systems of co-ordinates x,y, z and £, rj, £. As a result,
t The calculation is conveniently performed directly from the formula (29.12), where we must
put A = B, replace L by K, and take n as w, A.
§87 Matrix elements for the diatomic molecule 315
we obtain the following final formulae :
A T.a-i.k ^m+^v^Vl^K+lXK+AXK-A+iyKiK+l)], n
A 1**-i.k-x = ~*<^+^CSui V[(K+A)(K+A-1)IK], [(87.4)
A 2tf-i!K = K^+iA^yiiK-AXK-A+iyK].
The components with A— 1 -* A are the complex conjugates of those given.
Finally, we must find how the formulae we have obtained should be modi-
fied for transitions between states with non-zero spin. Here it is important
to know whether the states belong to case a or to case b.
First, let the two states belong to case a. The unit vector n commutes
with the spin vector S, and hence its matrix is diagonal with respect to the
quantum numbers S and 2 (or, what is the same thing, S and Q., since
£1 = A + 2, and this matrix is diagonal with respect to A also). The quantum
numbers K and M K do not exist, and instead we have the total angular
momentum / and its projection M on the #-axis. Instead of the relation
n . K = A, which we used to derive (87.1), we now have n . J = Q.. Accord-
ingly, we again obtain the same formulae (87.1), except that K and A must
now be replaced by / and Q, respectively (we omit the diagonal suffix S).
The same is true for any orbital vector A (i.e. one which does not depend
on the spin). Such a vector commutes with S also, and hence its matrix is
diagonal with respect to S and 2 ; if we use the quantum number Q. in place
of 2, it changes together with A in the non-zero matrix elements (i.e. if
A' = A± 1, then Q' = Cl± 1). The formulae (87.3) and (87.4) are unchanged
except that we must add the suffixes Q. and Q', and everywhere (except the
suffixes) replace K and A by / and CI.
If the vector A depends on the spin, however, the selection rules are different.
The vector S commutes with the orbital angular momentum, and also with
the Hamiltonian, and hence its matrix is diagonal with respect to n and A;
we omit these suffixes. It is, however, not diagonal with respect to 2 (or CI).
The matrix elements of the components of S in the system $, rj, £ are deter-
mined by formulae (27.13), with S and 2 in place of L and M, and then the
transition to the system x, y, z is effected by the formulae (87.3), (87.4),
where we must everywhere (including the suffixes) replace K and A by / and
Q.
Now let both states belong to case b. The calculation of the matrix ele-
ments is here performed in two stages. First we consider the rotating mole-
cule without taking into account the addition of the spin to the angular
momentum K; the matrix elements are then determined by the same formulae
(87.1)-(87.4). The vector A is supposed orbital, so that, like n, it commutes
with S, and so the matrices are diagonal with respect to the quantum number
S, which we omit from the suffixes. The angular momentum K is then
316 The Diatomic Molecule §88
added to S to form the total angular momentum J, and the transition to the new
matrix elements is effected by the general formulae (109.3). The part of j x
in these formulae is here taken by S, that of j 2 by K, and we write n, A in
place of n±.
If one of the states belongs to case a and the other to case b, the calculation
of the matrix elements for transitions between the states is more involved ;
we shall not here pause to consider this problem, f
PROBLEMS
Problem 1. Determine the Stark splitting of the terms for a diatomic molecule having
a constant dipole moment, in the case where the term belongs to case a.
Solution. The energy of a dipole d in an electric field <f is — d . <f . From considera-
tions of symmetry, it is evident that the dipole moment of a diatomic molecule is directed
along its axis ; d = dn, where d is a constant. Taking the direction of the field as the sr-axis,
we obtain the perturbation operator in the form —dn z $.
Determining the diagonal matrix elements of n z in accordance with the formulae derived
above, we find that in case a the splitting of the levels is given by the formulaj
AE Mj = -£dMjQIJ(J+l).
Problem 2. The same as Problem 1, but for the case where the term belongs to case b
(and A # 0).
Solution. By the same method we have
AE M = —SdMjK .
2K(K+1)J(J+1)
Problem 3. The same as Problem 2, but for a *2 term.
Solution. For A = the linear effect is absent, and we must go to the second approxi-
mation of perturbation theory. In the summation in the general formula (38.9), it is sufficient
to retain only those terms which correspond to transitions between rotational components of
the electron term concerned; for other terms the energy differences in the denominators are
large. Thus we find
„*M KM k\< K - X > M K )\* UKMKlnzlK+^MK)]*
AiiAf =d z z { 1
K \ Ek—Ek-i Ek—Ek+i
where Ek = BK(K+1). A simple calculation gives
« 2 K(K+1)-3M K *
&E M =
* B 2K(K+l)(2K-l)(2K+3)
§88. A-doubling
The double degeneracy of the terms with A ^ (§78) is in fact only
approximate. It occurs only so long as we neglect the effect of the rotation
t See E. Hill and J. van Vleck, Physical Review 32, 250, 1928.
t It may seem that there is here a contradiction of the general assertion that there is no linear
Stark effect (§76). In fact, of course, there is no contradiction, since the presence of a linear Stark
effect is here due to the double degeneracy of the levels with CI ^ 0; the formula obtained is therefore
applicable provided that the energy of the Stark splitting is large compared with that of what is called
the A-doubling (§88).
§88 A-doubling 317
of the molecule on the electron state (and also the higher approximations with
respect to the spin-orbit interaction), as we have done throughout the above
theory. When the interaction between the electron state and the rotation is
taken into account, a term with A # is split into two levels close together.
This phenomenon is called A-doubling (E. Hill and J. H. van Vleck, and
R. de L. Kronig, 1928).
To consider this effect quantitatively, we again begin with the singlet
terms (S = 0). We have calculated (in §82) the energy of the rotational
levels in the first approximation of perturbation theory, determining the
diagonal matrix elements (i.e. the mean value) of the operator
B(r)(R-L)«.
To calculate the subsequent approximations, we must consider the elements
of this operator that are not diagonal with respect to A. The operators & 2
and L 2 are diagonal with respect to A, so that we need consider only the oper-
ator -2B&.L.
The calculation of the matrix elements of R . L is conveniently effected
by means of the general formula (29.12), in which we must put A = K,
B = L ; the parts of L and M are taken by K and M K , while in place of n we
must put n, A, where n denotes the assembly of quantum numbers (other
than A) which determine the electron term. Since the matrix of the vector
K, which is conserved, is diagonal with respect to «, A, while that of the
vector L contains non-diagonal elements only for transitions in which A
changes by unity (cf. what was said in §87 concerning an arbitrary vector A),
we find, using formulae (87.4),
(nAKMKlK.Lln^A-UKM^^^+iL^^VKK^AXK+l-A)^ (88.1)
There are no non-zero matrix elements corresponding to any greater change
in A.
The perturbing effect of the matrix elements with A -> A— 1 can cause
the appearance of an energy difference between states with ±A only in the
2Ath approximation of perturbation theory. Accordingly, the effect is pro-
portional to B 2A , i.e. to (mjM) 2A , where M is the mass of the nuclei and m
that of the electron. For A > 1, this quantity is so small that it is of no
interest. Thus the A-doubling effect is of importance only for II terms
(A = 1), which are considered below.
For A = 1 we must go to the second approximation. The corrections to
the eigenvalues of the energy can be determined from the general formula
(38.9). In the denominators of the terms in the sum occurring in this equa-
tion we have energy differences, of the form E nAK —E n , A _ 1K . In these
differences, the terms containing K cancel, since, for a given distance r be-
tween the nuclei, the rotational energy is the same quantity, B(r)K(K+l),
for all the terms. Hence the dependence on K of the required splitting Ai?
is entirely determined by the squared matrix elements in the numerators.
318 The Diatomic Molecule §88
Among these are the squared elements for transitions in which A changes
from 1 to and from to — 1 ; these both give, by (88.1), the same depen-
dence on K, and we find that the splitting of the 1 II term is of the form
AE = constant x K(K+ 1), (88.2)
where the constant is of the order of magnitude of B 2 ] e, e being the order of
magnitude of the differences between neighbouring electron terms.
Let us pass now to terms with non-zero spin ( 2 II and 3 II terms; higher
values of S are not found in practice). If the term belongs to case b, the
multiplet splitting has no effect on the A-doubling of the rotational levels,
which is determined as before by formula (88.2).
In case <z, however, the effect of the spin is important. Here each electron
term is characterised by the number Q. as Well as A. If we simply replace
A by —A, then Q = A+ 2 is changed, so that we obtain an entirely different
term. The levels with A, Q. and —A, — Q are mutually degenerate. This
degeneracy can here be removed not only by the effect, considered above, of
the interaction between the orbital angular momentum and the rotation of
the molecule, but also by the effect of the spin-orbit interaction. The con-
servation of the projection Q. of the total angular momentum on the axis of
the molecule is (if the nuclei are fixed) an exact conservation law, and so
cannot be destroyed by the spin-orbit interaction; the latter can, however,
change A and 2 (i.e. there are matrix elements for the corresponding transi-
tions) in such a way that Q. remains unchanged. This effect, alone or in
combination with the orbit-rotation interaction (which alters A but not 2),
may cause A-doubling.
Let us first consider the 2 II terms. For the 2 n i/2 term (A= 1, 2 =— \,
Q, = |), the splitting is obtained on taking into account simultaneously the
spin-orbit and orbit-rotation interactions, each in the first approximation.
For the former gives the transition A = 1, 2= — ^ -> A = 0, 2= J, and
then the latter converts the state A = 0, 2 = | into A = —1, 2 == \, which
differs from the initial state by the signs of A and Q. being reversed. The mat-
rix elements of the spin-orbit interaction are independent of the rotational
quantum number /, while the dependence of those for the orbit-rotation
interaction is determined by formula (88.1), in which (under the radical)
we must replace K and A by / and Q. Thus we have for the A-doubling
of a 2 II 1/2 term the expression
AE 1/2 = constant x {J+Q, (88.3)
where the constant is of the order of AB\ e. For a 2 n 3/2 term, on the other
hand, the splitting can be found only in higher approximations, so that in
practice &E 3/2 , = 0.
Finally, let us consider 3 II terms. For a 3 II term (A = 1, 2 = —1), the
splitting is obtained on taking into account the spin-orbit interaction in the
second approximation (because of the transitions A = 1, 2 = —1 ->- A = 0,
2 = -> A = — 1, 2=1). Accordingly, the A-doubling in this case is
§89 The interaction of atoms at large distances 319
entirely independent of /:
&E = constant ~ A 2 je. (88.4)
For a 3 I1 1 term, 2 = 0, and so the spin has no effect on the splitting; hence
we again have a formula like (88.2), but with K replaced by / :
AE X = constant x/(/+l). (88.5)
For a 3 II 2 term, higher approximations are needed, so that we can suppose
A£ 2 = 0.
One of the levels of the doublet resulting from A-doubling is always posi-
tive, and the other negative; we have already discussed this in §86. An
investigation of the wave functions of the molecule enables us to establish
the regularities of the alternation of positive and negative levels. Here we
shall give only the results of the investigation.f It is found that if, for some
value of J, the positive level is below the negative one, then in the doublet
for /+ 1 the order is opposite, the positive level being above the negative
one, and so on; the order varies alternately as the total angular momentum
takes successive values. We are speaking here of case a terms; for case b, the
same holds for successive values of the angular momentum K.
PROBLEM
Determine the A-splitting for a *A term.
Solution. Here the effect appears in the fourth approximation of perturbation theory.
Its dependence on K is determined by the products of the four matrix elements (88.1) for
transitions with change of A : 2 -> 1, 1 -»- 0, -> —1, —1 -> —2. This gives
AE = constant x(K-l)K(K+l)(K+ 2),
where the constant is of order of -B 4 /* 3 -
§89. The interaction of atoms at large distances
Let us consider two atoms in S states which are at a great distance from
each other (relative to their size), and determine the energy of their interaction.
In other words, we shall discuss the determination of the form of the electron
terms U n (r) when the distance between the nuclei is large.
To solve this problem we apply perturbation theory, regarding the two
isolated atoms as the unperturbed system, and the potential energy of their
electrical interaction as the perturbation operator. As we know from electro-
statics, the electrical interaction of two systems of charges at a large distance r
apart can be expanded in powers of 1/r, and successive terms of this expansion
correspond to the interaction of the total charges, dipole moments, quad-
rupole moments, etc., of the two systems. For neutral atoms, the total charges
are zero. The expansion here begins with the dipole-dipole interaction
(~ 1/r 3 ); then follow the dipole-quadrupole terms (~ 1/r 4 ), the quadrupole-
quadrupole (and dipole-octupole) terms (^ 1/r 5 ), and so on.
f This may be found in E. Wigner and E. WiTMER, Zeitschrift fur Physik 51, 859, 1928.
320 The Diatomic Molecule §89
In the first approximation of perturbation theory, the required energy of
the interaction of the atoms is determined as the diagonal matrix element of
the perturbation operator, calculated with respect to the unperturbed wave
functions of the system (expressed in terms of products of the unperturbed
functions for the atoms).f In S states, however, the diagonal matrix elements,
i.e. the mean values of the dipole, quadrupole, etc." moments, are zero ; this
follows at once from considerations of symmetry, since the distribution of
charges in an atom in the S state is spherically symmetrical on the average.
Hence each of the terms of the expansion of the perturbation operator in
powers of \\r gives zero in the first approximation of perturbation theory. J
In the second approximation it is sufficient to restrict ourselves to the dipole
interaction in the perturbation operator, since this decreases least rapidly
as r increases, i.e. to the term
V = [-d x . d 2 +3(d x . n)(d 2 . n)]/r\ (89.1)
where n is a unit vector in the direction joining the two atoms. Since the
non- diagonal matrix elements of the dipole moment are in general different
from zero, we obtain in the second approximation of perturbation theory a
non- vanishing result which, being quadratic in V, is proportional to 1/r 6 .
The correction in the second approximation to the lowest eigenvalue is, as we
know, always negative (§38). Hence we obtain for the interaction energy of
atoms in their normal states an expression of the form ||
U(r) = -constant/r 6 , (89.2)
where the constant is positive (F. London 1928).
Thus two atoms in normal S states, at a great distance apart, attract each
other with a force (— dU/dr) which is inversely proportional to the seventh
power of the distance. The attractive forces between atoms at large distances
are usually called van der Waals forces. These forces cause the appearance
of minima on the potential energy curves of the electron terms even for
atoms which do not form a stable molecule. These depressions, however,
are very shallow (being only tenths or even hundredths of an electron-volt
in depth) and lie at distances several times greater than the distances between
atoms in stable molecules.
If only one of the atoms is in the S state, the same result (89.2) is obtained
for the interaction energy, since, for the first approximation to vanish, it is
■f Here we neglect the exchange effects, which decrease exponentially with distance (see §62,
Problems).
% This, of course, does not imply that the mean value of the interaction energy of the atoms is pre-
cisely zero. It diminishes exponentially with distance, i.e. more rapidly than every finite power of 1/r,
and hence each term of the expansion vanishes. This occurs because the expansion of the interaction
operator in terms of the multipole moments involves the assumption that the charges of the two atoms
are at a large distance r apart, whereas in quantum mechanics the electron density distribution has
finite (though exponentially small) values even at large distances.
|| For brevity, we here and later omit the unimportant constant term in U(r), i.e. the value of U(co),
which is the sum of the energies of the two isolated atoms.
§89 The interaction of atoms at large distances 321
sufficient for the dipole (etc.) moment of only one atom to be zero. The
constant in the numerator of (89.2) here depends, not only on the states of
the two atoms, but also on their mutual orientation, i.e. on the value Q. of the
projection of the angular momentum on the axis joining the atoms.
If both atoms have non-zero orbital and total angular momenta, however,
the situation is changed. The mean value of the dipole moment is zero in
every state of the atom (§75). The mean values of the quadrupole moment in
states with L # 0, / # or \ are not zero, however. Hence the quadrupole-
quadrupole term in the perturbation operator gives a non-zero result even
in the first approximation, and we find that the interaction energy of the
atoms diminishes as the fifth, not the sixth, power of the distance :
U(r) = constant/r 5 . (89.3)
Here the constant may be either positive or negative, i.e. we may have either
attraction or repulsion. As in the previous case, this constant depends not
only on the states of the atoms, but also on the state of the system formed
by the two atoms.
A special case is the interaction of two similar atoms in different states.
The unperturbed system (the two isolated atoms) has here an additional de-
generacy due to the possibility of interchanging the states of the atoms.
Accordingly, the correction in the first approximation will be given by the
secular equation, in which the non-diagonal matrix elements of the perturba-
tion appear as well as the diagonal ones. If the states of the two atoms have
different parities, and angular momenta L differing by ± 1 or but not both
zero (the same restriction being placed on J), then the non-diagonal matrix
elements of the dipole moment for transitions between these states are in
general not zero. Hence an effect in the first approximation is obtained from
the dipole term in the perturbation operator. Thus the interaction energy
of the atoms is here proportional to 1/r 3 :
U(r) = constant/r 3 , (89.4)
where the constant may have either sign.
Usually, however, what is of interest is the interaction of the atoms aver-
aged over all possible states of the system which they form (for given states
of the atoms), including all possible orientations of the angular momenta
of the atoms.-|-
As a result of this averaging, all effects linear in the dipole or quadrupole
moment of each atom (i.e. all effects in the first approximation of perturba-
tion theory) vanish. The averaged interaction forces between atoms at large
distances therefore always follow the law (89.2)J.
t Such averaging is necessary, for example, in the problem of determining the interaction of atoms
in a gas.
t This law, derived on the basis of the non-relativistic theory, is Valid only so long as the retardation
of electromagnetic interactions is unimportant. For this to be so, the distance r between the atoms
must be small compared with c/cuon, where won are the frequencies of transitions between the ground
state and the excited states of the atom.
322 The Diatomic Molecule §90
PROBLEM
Derive a formula giving the van der Waals forces in terms of the matrix elements of dipole
moments for two like atoms in S states.
Solution. The answer is obtained by applying the general formula (38.9) of perturbation
theory to the operator (89.1). On account of the isotropy of the atoms in the S state it is
evident a priori that, on summation over all intermediate states, the squared matrix elements
of the three components of each of the vectors di and d2 give equal contributions, while the
terms which contain products of different components give zero. The result is
T7/ , 6 ^-a (d z )on 2 (d z )on' 2
U(r) = > ,
r«^2E Q -E n -E n >
n,n'
where Eo and E n are the unperturbed values of the energies of the ground state and excited
states of the atom. Since by hypothesis L = in the ground state, the matrix elements
(rfz)on are non-zero only for transitions to P states (L = l).t Using formulae (29.7), we bring
U(r) to the final form
U(r)= > ^ relM n V ,
3r6 ^ 2E -E nl -En'i
n,n'
where in the matrix suffixes nL the second suffix gives the value of L and the first represents
the assembly of the remaining quantum numbers which determine the energy level.J
§90. Pre-dissociation
A basic premise of the theory of diatomic molecules as given in this chapter
is the assumption that the wave function of the molecule falls into the product
of an electron wave function (depending on the distance between the nuclei
as a parameter) and a wave function for the motion of the nuclei. This sup-
position amounts to neglecting, in the exact Hamiltonian of the molecule,
certain small terms corresponding to the interaction of the nuclear and
electron motions.
When these terms are taken into account and perturbation theory is applied,
transitions between different electron states appear. || Physically, the transi-
tions between states of which at least one belongs to the continuous spectrum
are of particular importance.
Fig. 30 shows curves for the potential energy of two electron terms.ff The
energy E' (the lower dashed line in Fig. 30) is the energy of some vibrational
level of a stable molecule in the electron state 2. In state 1, this energy lies
in the range of the continuous spectrum. In other words, in passing from
state 2 to state 1 the molecule automatically disintegrates ; this phenomenon
is called pre-dissociation.%% As a result of pre-dissociation, the state of the dis-
crete spectrum corresponding to curve 2 has in reality a finite lifetime. This
t The spin of the atoms is not involved here.
% Calculation gives for the coefficient in U = — Ar~ 6 (with all quantities in atomic units) the values
A = 6 - 5 for two hydrogen atoms and A = 1*6 for helium atoms.
|( As well as the splitting of the levels by A-doubling (§88).
tf Strictly speaking, these curves must represent the effective potential energy Uj in some given
rotational states of the molecule.
{J Curve 1 may have no minimum at all if it corresponds to purely repulsive forces between the
atoms.
§90
Pre-dissociation
323
means that the discrete energy level is broadened, i.e. acquires a certain width
(see the end of §44).
Fig
If, on the other hand, the total energy E lies above the dissociation limit
in both states (the upper dashed line in Fig. 30), the transition from one
state to the other corresponds to what is called a collision of the second kind.
Thus the transition 1 -> 2 signifies the collision of two atoms, as a result of
which the atoms are left in excited states, and separate with diminished
kinetic energy (for r -> oo, curve 1 passes below curve 2; the difference
C/ 2 (oo) — U^oo) is the excitation energy of the atoms).
Because of the large masses of the nuclei their motion is quasi-classical.
The problem of determining the probability of the transitions under considera-
tion is therefore of the kind discussed in §52. From the general considera-
tions given there we can say that the transition probability will be mainly
determined by the point at which the transition could occur classically. f
Since the total energy of the system of two atoms (the molecule) is conserved
in the transition, the condition for it to be "classically possible" is that the
effective potential energies should be equal: Uji(r) = C/j 2 (r). On account
of the conservation of the total angular momentum of the molecule also, the
centrifugal energies are the same in the two states, and so this condition
means that the potential energies are equal:
Vl {r) = U 2 (r), (90.1)
the angular momentum not being involved at all.
If equation (90.1) has no real roots in the classically accessible region
(where E > Uji, Uj2), the transition probability according to §52 is expo-
nentially small.J Transitions occur with an appreciable probability only if
t Or else by the point r — at which the potential energy becomes infinite.
t A peculiar situation must occur in the case of a transition involving a molecular term which can
arise from two different pairs of atomic states (see the end of §85), i.e. when the potential energy curve
is, as it were, split into two branches with increasing distance. In this case the transition probability
should be considerably greater, but the problem has not yet been discussed in the literature.
324 The Diatomic Molecule §90
the potential energy curves intersect in the classically accessible region
(as shown in Fig. 30). Then the exponent in formula (52.1) is zero (and this
formula is therefore, of course, invalid) ; accordingly, the transition probability
is determined by a non- exponential expression which will be derived below.
The condition (90.1) can then be interpreted as follows. If the potential (and
total) energies are the same, so are the linear momenta. Hence the condition
(90.1) may also be written in the form
r i = r 2 , Pi = p*> (90.2)
where p is the momentum of the relative radial motion of the nuclei, and the
suffixes 1 and 2 refer to the two electron states. Thus we can say that the
distance between the nuclei and their relative momentum remain unchanged
at the instant when the transition occurs (this is called Franck and Condon's
principle). Physically, this is due to the fact that the electron velocities are
large compared with those of the nuclei, and "during an electron transition"
the nuclei cannot noticeably change their position or velocity.
It is not difficult to establish the selection rules for the transitions in ques-
tion. First of all, there are two obvious exact rules. The total angular
momentum / and the sign of the term (positive or negative; see §86) cannot
change in a transition. This follows at once from the fact that the conserva-
tion of the total angular momentum and of the behaviour of the wave function
under inversion of the co-ordinate system are exact laws for any (closed)
system of particles.
Next, the rule which forbids (for molecules composed of similar atoms)
transitions between states of unlike parity is very nearly accurate. For the
parity of the state is uniquely determined by the nuclear spin and the sign of
the term. The conservation of the sign of the term is an exact law, however,
while the nuclear spin is very nearly conserved by virtue of the weakness of
its interaction with the electrons.
The requirement that there should be a point of intersection of the potential
energy curves means that the terms must be of different symmetry (see §79).
Let us consider transitions occurring in the first approximation of perturba-
tion theory; the probability of transitions which occur only in higher approxi-
mations is relatively small. First of all, we notice that the terms in the
Hamiltonian which lead to the transitions in question are just those which
cause the A-doubling of the levels. Among these terms are, firstly, terms
representing the spin-orbit interaction. They are the product of two axial
vectors, of which one is of spin character (i.e. is composed of the operators
of the electron spins), and the other is of co-ordinate character ; we emphasise,
however, that these vectors are not simply the vectors S and L. Hence they
have non-zero matrix elements for transitions in which S and A change by
0, ±1. The case where AS and AA are both zero (and A # 0) must be
omitted, since the symmetry of the term would then be unchanged in the
transition. The transition between two 2 terms is possible if one of them is
§90 Pre-dissociation 325
a 2+ term and the other a S~ term; an axial vector has non-zero matrix
elements only for transitions between 2+ and 2~ (see §87).
The term in the Hamiltonian which corresponds to the interaction between
the rotation of the molecule and its orbital angular momentum is proportional
to j . L. Its matrix elements are non-zero for transitions with A A = ±1
without change of spin (only the ^-component of the vector, i.e. L v has ele-
ments with AA = 0, but L c is diagonal with respect to the electron states).
As well as the terms we have considered, there is also a perturbation due
to the fact that the operator of the kinetic energy of the nuclei (i.e. the operator
of differentiation with respect to the co-ordinates of the nuclei) acts, not only
on the wave function of the nuclei, but also on the electron function, which
depends on r as a parameter. The corresponding terms in the Hamiltonian
are of the same symmetry as the unperturbed Hamiltonian. Hence they can
lead only to transitions between electron terms of like symmetry, the prob-
ability of which is negligible in view of the non-intersection of these terms.
Let us go on to the actual calculation of the transition probability. For
definiteness, we shall consider a collision of the second kind. According to
the general formula (43.1), the required probability is given by the expression
2
w
h
= -T /xnuc,2*^)Xnuc,l dy
(90.3)
where x = f'/'nuc ('/w Denl g the wave function of the radial motion of the
nuclei) and V(r) is the perturbing energy; we have taken, as the quantity v in
(43.1), the energy E and integrated with respect to it. The final wave func-
tion Xnuc,2 must be normalised by the delta function of energy. The quasi-
classical function (47.4a), thus normalised, is
*"WiHi/* d '-4 (904)
a,
The normalising factor is determined by the rule given at the end of §21. The
wave function of the initial state can be written in the form
^-^""fi/**-*-}- (90 - 5)
It is normalised so that the current density is unity in each of the two travelling
waves into which the stationary wave (90.5) can be resolved; v x and v 2 are
the velocities of the relative radial motion of the nuclei. On substituting
these functions in (90.3), we obtain the dimensionless transition probability
w. It can be regarded as the transition probability for the nuclei to pass
twice the point r = r (the point of intersection of the levels). It must be
borne in mind that the wave function (90.5) corresponds, in a certain sense,
326 The Diatomic Molecule §90
to a double passage through this point, since it contains both the incident and
the reflected travelling waves.
The matrix element of V(r), calculated with respect to the functions
(90.4), (90.5), contains in the integrand a product of cosines, which can be
written in terms of the cosines of the sum and difference of the arguments.
On integrating near the point r = r where the terms intersect, only the
second cosine is important, so that
V(r) dr
4 1 f T 1 T , 1 C lV(r)dr
= id cos t Px dr p 2 dr —
The integral rapidly converges as we move away from the point of intersec-
tion. Hence we can expand the argument of the cosine in powers of £ = r — r
and integrate over £ from — oo to + oo (replacing the slowly varying coeffi-
cient of the cosine by its value at r = r ). Bearing in mind that, at the point
of intersection, p x = p 2 , we find
r r
where S is the value of the difference of the integrals at the point r = r .
The derivative of the momentum can be expressed in terms of the force
P= -dU/dr: differentiating the equation p^jlfi + U x = p^jljn + U 2 (where
H is the reduced mass of the nuclei), we have v x dpjdr—v^ dpjdr = ^—F^.
Thus
r r
Jftdr-J
Jf -pi
p % dr~S 9 +-±- -e,
2v
where v is the common value of v x and v 2 at the point of intersection. The
integration is effected by means of the well-known formula
oo
f cos(a+j8F) d£ = /^cos( a +^r),
and as a result we have
8nV 2 /S,
w = cos
hv\F 2 -F x \
<r*>
(90.6)
The quantity S Q jh is large and varies rapidly with the energy E. Hence,
on averaging over even a small interval of energy, the squared cosine can be
replaced by its mean value. As a result we obtain the formula
to = 4t7F 2 /^|F 2 -F 1 | (90.7)
§90
Pre-dissociation
327
(L. Landau 1932). All the quantities on the right-hand side of the equation
are taken at the point of intersection of the potential-energy curves.
In the application to pre-dissociation, we are interested in the probability
of the disintegration of the molecule in unit time. In this time, the nuclei
in their vibrations pass 2(o>/2tt) times through the point r = r (where co is
the angular frequency of the vibrations). Hence the required pre-dissociation
probability is obtained by multiplying w (the probability for a double passage)
by o>/2tt, i.e. it is
IV^lhv^-F^. (90.8)
The following remark must be made concerning these calculations. In
speaking of the intersection of terms, we have had in mind the eigenvalues
of the "unperturbed" Hamiltonian i? of the electron motion in the molecule ;
in this, the terms V which lead to the transitions concerned are not taken into
account. If we include these terms in the Hamiltonian, the intersection of
the terms becomes impossible, and the curves move apart slightly, as shown
in Fig. 3 1 . This follows from the results of §79 when regarded from a slightly
different point of view.
Let U JX (r) and U J2 (r) be two eigenvalues of the operator 3 (in which r
is regarded as a parameter). In the region near the point r where the curves
U JX {r) and Uj z (r) intersect, to determine the eigenvalues U(r) of the perturbed
operator i? +^ we must use the metn °d given in §79, as a result of which
we obtain the formula
u(r) = K^x+^+^n+^i V[K^i-^2+^ii-^2 2 ) 2 +l^i 2 | 2 ];
(90.9)
the matrix elements V Uf F" 22 , F 12 , like Uj-^ and U J2 , are functions of r. The
interval between the two levels is now
At/ = V[(^/l-^/2+^ll-^22) 2 + 4ir i2 |2].
(90.10)
328 The Diatomic Molecule §90
Hence it is clear that, if there are transitions between the two states (i.e. the
matrix element V X2 is not zero), the intersection of the levels disappears. The
least distance between the curves is now
A=2|F 12 |.
The formulae obtained above for the transition probability are applicable
only so long as the "divergence" of the curves is fairly small. If the latter
becomes considerable, the transition probability cannot be calculated by
ordinary perturbation theory.
To examine this problem, we use the following method (C. Zener 1932).
Let i/»i, «/r 2 be the wave functions of the electron states corresponding to the
"unperturbed" terms Ujx and U J2 , i.e. these functions are solutions of the
equations
#o^i = Ujifa, fi ip 2 = Uj 2 xft 2 .
Let us seek the solution of the perturbed wave equation
iH&P/dt =(# +P)Y
in the form
T = b x {t)^ x +b 2 {t)^ 2 . (90.11)
Substituting this expression in the wave equation, multiplying the latter
firstly by ip x and secondly by «/r 2 , and integrating, we obtain two equations
for the functions b x {t) and b 2 (t):
ihdbjdt = U Jx b x +Vb 2 , ifrdbjdt = U j2 b 2 +Vb x ; (90.12)
we here include V xx , V 22 in U JX , U J2 , and denote V X2 by V(r) simply, in agree-
ment with the notation in the preceding formulae. We consider the motion
of the nuclei quasi-classically. Accordingly, the variables r and * are related
by drjdt = v, where v is the classical velocity of the nuclei.
Near the point r where the curves of Uj^r) and U J2 (r) intersect (shown by
the dashed line in Fig. 31), we can expand Uj x and U J2 as series of powers of
£ — r—r , writing
V Jx = Uj-F Jx £, U J2 = Uj-Fj^ (90.13)
where Uj is the common value of U JX and U J2 at the point r = r , and we
have introduced the notation Fj = -{dUjldr) u . Introducing also new un-
knowns a x , a % in place of b x , b 2 by means of
b x = ajerVmvj, b 2 = a 2 e-QW u J f , (90.14)
and replacing the differentiation with respect to t by one with respect to $
(d/dt = t;d/d£), we obtain from equations (90.12)
ihadajdg = -F Jx £ ai +Va 2 , ihv da 2 /d$ = - Fj 2 fr 2 + Va lt (90.15)
§90
Pre-dissociation 329
where v and V may be taken with sufficient accuracy to have their values at
the point of intersection.
If we solve equations (90.15) with the boundary condition a x = 1, a % —
as £j -> +oo, then |0i(-oo)| 2 determines the probability that, as the nuclei
pass through the point £ = 0, the molecule remains in the electron state lf
indicating a transition from the curve 12' to the curve 21' (see Fig. 31).
Similarly, \a^ - oo) | 2 = 1 - \a x { - oo) | 2 is the probability of a transition to the
electron state «^ 2 , i.e. the probability that the molecule remains on the curve
12'. The transition from curve 1 to curve 2 (as £ -> + oo) in a double passage
through the point of intersection can be effected in two ways: either by
1 -+ V -> 2 (as the nuclei approach, the transition from curve 12' to curve
21' occurs, and as they recede the molecule remains on the curve 21'), or by
1 _> 2' -> 2. Hence the required probability for such a transition is
W =2K(-oo)|*{lH*i(-°°)l 2 }-
We shall not pause here to explain the manner of solution of equations (90.15)
(they reduce to a single equation of the second order, which can be solved by
Laplace's method), but give only the final result :f
(the difference Fj^—F^ is replaced by the equal difference F 2 —F x ). Thus
w = 2e- 2vV ^ F t- F ^{l—e- 2nV ^ nvlF *- F ^}. (90.16)
We see that the probability of the transition in question is small in two
limiting cases, when V is fairly small and fairly large. For V 2 <^ hv\F 2 —F x \,
formula (90.16) becomes (90.7).
Finally, let us consider the phenomenon, akin to pre-dissociation, of what
are called perturbations in the spectra of diatomic molecules. If two discrete
molecular levels E x and E 2 corresponding to two intersecting electron terms
are close together, the possibility of a transition between the two electron
states results in a displacement of the levels. According to the general
formula (79.4) of perturbation theory, we have for the displaced levels the
expression
K£i+£ 2 )± V[m-E z y+\ v 12tRUC \*], (90.17)
where Fi 2fnuc is the matrix element of the perturbation for the transi-
tion between the molecular states 1 and 2; the matrix elements V llnuc and
Voo must, of course, be included in & and E 2 . From this formula we see
that the two levels are moved apart, being displaced in opposite directions
(the higher level is raised and the other lowered). The amount of the dis-
placement is the greater, the smaller the difference \E X —E 2 \.
The matrix element r i2nuc is calculated in exactly the same way as for
determining the probability of a collision of the second kind. The only
t See C. Zener {Proceedings of the Royal Society A 137, 696, 1932).
330 The Diatomic Molecule §90
difference is that the wave functions Xnuc§1 and Xnuc2 belong to the discrete
spectrum, and hence must be normalised to unity. ' According to (48.3) we
have for these functions
w ~y;? "fiiJ* dr -H
A comparison with formulae (90.3) to (90.5) shows that the matrix element
Pi2,nuc ner e considered is related to the transition probability w for a double
passage through the point of intersection by
I ^i2,nud 2 = w(^<o 1 /27r)(^co 2 /27r). (90.18)
PROBLEMS
Problem 1. Determine the total effective cross-section for collisions of the second kind,
as a function of the kinetic energy E of the colliding atoms, for transitions pertaining to the
spin-orbit interaction.
Solution. On account of the quasi-classical motion of the nuclei, we can introduce the
concept of the impact parameter p (the distance at which the nuclei would pass if there were
no interaction between them) and define the effective cross-section da as the product of the
"target area" 2np dp and the transition probability w(p) per collision, f The total effective
cross-section a is obtained by integrating with respect to p.
For spin-orbit interaction, the matrix element V(r) is independent of the angular momen-
tum M of the colliding atoms. We write the velocity v at the point r = r , where the curves
intersect, in the form
v = ViWME-U-MiptirS)-] = VK2/ix)(E-U-fElr *)].
Here U is the common value of U x and U 2 at the point of intersection, p. is the reduced mass
of the atoms, and the angular momentum M = ppvao, where v«> is the relative velocity of the
atoms at infinity. The zero of energy is chosen so that the interaction energy of the atoms in
the initial state is zero at infinity; then E = ip,v m *. Substituting this expression in (90.7),
we find
8tt 2 F 2 pdp
da = 2vp dp . w =
*l*r-*il V[2(E-U-p*E/r *)lp.]
ust be taken from zero uj
ave
The integration with respect to p must be taken from zero up to the value for which the
velocity v vanishes. As a result we have
HlFt-FJl E
Problem 2. The same as Problem 1, but for transitions pertaining to the interaction be-
tween the rotation of the molecule and its orbital angular momentum.
t Cf. Mechanics, §18.
§90 Pre-dissociation 331
Solution. The matrix element V is of the form V(r) = MD\[ir % , where D(r) is the matrix
element of the electron orbital angular momentum. By the same method as in Problem 1 we
obtain
16V27r 2 Z> 2 (E-U)*'*
3hVn\F*-Fi\ E '
Problem 3. Determine the transition probability for energies E close to the value Uj
of the potential energy at the point of intersection.
Solution. For small values of E-Uj, formula (90.7) is inapplicable, since the velocity
v of the nuclei cannot be regarded as constant near the point of intersection, and hence it
cannot be taken outside the integral as it was in deriving (90.7).
Near the point of intersection we replace the curves of Uj u Uj t by the straight lines (90.13).
The wave functions Xnuc,i and Xn™,2 in this region are wave functions of one-dimensional
motion in a homogeneous field (§24). The calculations are conveniently effected by means
of wave functions in the momentum representation. The wave function normalised by the
delta function of energy is of the form (see §24, Problem)
a 9 =
V{2*h\F Jz \)
exp \-^ri( E - u j)P~P z l 6 ri } •
while the wave function normalised to unit current density in the incident and reflected waves
is obtained by multiplying by \/(2irH) :
1 ( i
■* --^n ex ^\jw- [{E - Uj)p - pzl6tl]
V\ F Ji\ WJi
On integrating, the perturbing energy (matrix element) V may again be taken outside the
integral, replacing it by its value at the point of intersection;
°°
2tt\ r 2
to = — \V c^a^dp
h • J
— 00
As a result we obtain
47rF 2 (2/Li) 2 / 3
^3 r /2/A 1/3 / 1 1 \ 2/3 "l
where $(£) is the Airy function (see §b of the Mathematical Appendices). For large E—Uj,
this formula reduces to (90.7).
CHAPTER XII
THE THEORY OF SYMMETRY
§91. Symmetry transformations
The classification of terms in the polyatomic molecule is fundamentally
related to its symmetry, as in the diatomic molecule. Hence we shall begin
by examining the types of symmetry which a molecule can have.
The symmetry of a body is determined by the assembly of all those re-
arrangements after which the body is unaltered; these rearrangements are
called symmetry transformations. Any possible symmetry transformation can
be represented as a combination of one or more of the three fundamental
types of transformation. These three essentially different types are: the
rotation of the body through a definite angle about some axis, the reflection
of it in some plane, and the parallel displacement of the body over some
distance. Of these, the last evidently is applicable only to an infinite medium
(a crystal lattice). A body of finite dimensions (in particular, a molecule)
can be symmetrical only with respect to rotations and reflections.
If the body is unaltered on rotation through an angle lirjn about some
axis, then that axis is said to be an axis of symmetry of the nth order. The
number n can take any integral value: n = 2, 3, ... . The value n = 1
corresponds to a rotation through an angle of 2n or, what is the same thing,
of 0, i.e. it corresponds to an identical transformation. We shall symbolically
denote by C n the operation of rotation through an angle 2rt\n about a given
axis. Repeating this operation two, three, ... times, we obtain rotations
through angles 2(2?r/«), 3(27r/«), ... , which also leave the body unaltered;
these rotations may be denoted by C n 2 , C n 3 , .... It is obvious that, if p
divides n,
C n p = C n/P . (91.1)
In particular, performing the rotation n times, we return to the initial position,
i.e. we effect an identical transformation. The latter is customarily denoted
by E, so that we can write
C n n = E. (91.2)
If the body is left unaltered by a reflection in some plane, this plane is said
to be a plane of symmetry. We shall denote by the symbol a the operation
of reflection in a plane. It is evident that a double reflection in the same
plane is the identical transformation :
<r 2 = E. (91.3)
332
§91 Symmetry transformations 333
A simultaneous application of the two transformations (rotation and
reflection) gives what are called the rotary-reflection axes. A body has a
rotary-reflection axis of the nth order if it is left unaltered by a rotation
through an angle lirjn about this axis, followed by a reflection in a plane
perpendicular to the axis (Fig. 32). It is easy to see that this is a new form
1
oP
Kl
Fig. 32
of symmetry only when n is even. For, if n is odd, an w-fold repetition of the
rotary-reflection transformation would be equivalent to a simple reflection
in a plane perpendicular to the axis (since the angle of rotation is 2tt, while
an odd number of reflections in the same plane amounts to a simple reflection).
Repeating this transformation a further n times, we have as a result that the
rotary-reflection axis reduces to the simultaneous presence of an axis of
symmetry of the nth order and an independent plane of symmetry perpen-
dicular to this axis. If, however, n is even, an w-fold repetition of the rotary-
reflection transformation returns the body to its initial position.
We denote the rotary-reflection transformation by the symbol S n .
Denoting by a h a reflection in a plane perpendicular to a given axis, we can
put, by definition,
S n = C n c h = a h C n \ (91.4)
the order in which the operations C n and a h are performed clearly does not
affect the result.
An important particular case is a rotary-reflection axis of the second
order. It is easy to see that a rotation through an angle tt, followed by a
reflection in a plane perpendicular to the axis of rotation, is the inversion
transformation, whereby a point P of the body is carried into another point
P', lying on the continuation of the line which joins P to the intersection O
of the axis and the plane, and such that the distances OP and OP' are the same.
A body symmetrical with respect to this transformation is said to have a
centre of symmetry. We shall denote the operation of inversion by /, so that
we have
I=S 2 = C 2 c h . (91.5)
It is also evident that Ia h = C 2 , 7C 2 = a h \ in other words, an axis of the
334 The Theory of Symmetry §91
second order, a plane of symmetry perpendicular to it and a centre of sym-
metry at their point of intersection are mutually dependent: if any two of
these elements are present, the third is automatically present also.
We shall now give various purely geometrical properties of rotations and
reflections which it is useful to bear in mind in studying the symmetry of
bodies.
A product of two rotations about axes intersecting at some point is a
rotation about some third axis also passing through that point. A product
of two reflections in intersecting planes is equivalent to a rotation ; the axis
of this rotation is evidently the line of intersection of the planes, while
the angle of rotation is easily seen, by a simple geometrical construction, to
be twice the angle between the two planes. If we denote a rotation through
an angle about an axis by C(<f), and reflections in two planes passing through
that axis by the symbols! a v and a' v , the above statement can be written as
a v a' v = C(2cf>), (91.6)
where <j> is the angle between the two planes. It must be noted that the order
in which the two reflections are performed is not immaterial. The trans-
formation a v o' v gives a rotation in the direction from the plane of a' v to
that of a v ; on interchanging the factors we have a rotation in the oppo-
site direction. Multiplying equation (91.6) on the left by a v , we obtain
a' v = G v C(2<f>); (91.7)
in other words, the operation of rotation, followed by reflection in a plane
passing through the axis, is equivalent to a reflection in another plane
intersecting the first at half the angle of rotation. In particular, it follows
from this that an axis of symmetry of the second order and two mutually
perpendicular planes of symmetry passing through it are mutually dependent ;
if two of them are present, so is the third.
We shall show that the product of rotations through an angle tt about two
axes intersecting at an angle <f> (Oa and Ob in Fig. 33) is a rotation through
p'
Fig. 33
"f The suffix v customarily denotes a reflection in a plane passing through a given axis (a "vertical'
plane), and the suffix h a reflection in a plane perpendicular to the axis (a "horizontal" plane).
§92 Transformation groups 335
an angle 2^ about an axis perpendicular to the first two (PP' in Fig. 33).
For it is obvious that the resulting transformation is also a rotation; after
the first rotation (about Oa) the point P is carried into P', and after the second
(about Ob) it returns to its original position. This means that the line PP r
remains fixed, and is therefore an axis of rotation. To determine the angle
of rotation, it is sufficient to note that, in the first rotation, the axis Oa
remains fixed, while after the second it takes the position Oa\ which makes
an angle 24 with Oa. In the same way we can see that, when the order of the
two transformations is reversed, we obtain a rotation in the opposite direction.
Although the result of two successive transformations in general depends
on the order in which they are performed, in some cases the order of opera-
tions is immaterial : the transformations commute. This is so for the following
transformations :
(1) Two rotations about the same axis.
(2) Two reflections in mutually perpendicular planes (equivalent to a
rotation through it about their line of intersection).
(3) Two rotations through rr about mutually perpendicular axes (equivalent
to a rotation through -it about the third perpendicular axis).
(4) A rotation and a reflection in a plane perpendicular to the axis of
rotation.
(5) Any rotation or reflection and an inversion with respect to a point
lying on the axis of rotation or in the plane of reflection; this follows
from (1) and (4).
§92. Transformation groups
The set of all the symmetry transformations for a given body is called its
symmetry transformation group (or simply its symmetry group). Hitherto we
have spoken of these transformations as geometrical rearrangements of the
body. However, in quantum-mechanical applications it is more convenient
to regard symmetry transformations as transformations of the co-ordinates
which leave the Hamiltonian of the system in question invariant. It is obvious
that, if the system is left unaltered by some rotation or reflection, the cor-
responding transformation does not change its Schrodinger's equation.
Thus we shall speak of a transformation group with respect to which a given
Schrodinger's equation is invariant.!
f This point of view enables us to include in our considerations not only the rotation and reflection
groups discussed here, but also other types of transformation which leave Schrodinger's equation
unaltered. These include the interchange of the co-ordinates of identical particles forming part of the
system considered (a molecule or atom). The set of all possible permutations of identical particles
in a given system is called its permutation group (we have already met these permutations in §63).
The general properties of groups given below apply to permutation groups also; we shall not pause
to study this type of group in more detail here.
The following remark should be made concerning the notation which we use in this chapter. Sym-
metry transformations are essentially operators just like those which we consider all through the book
(in particular, we have already considered the inversion operator in §30). They ought, therefore, to
be denoted by letters with circumflexes. We do not do this, in view of the generally accepted notation,
and because this omission cannot lead to misunderstandings in the present chapter. For the same
reason we denote the identical transformation by the customary symbol E, and not by 1, which would
correspond to the notation in the other chapters.
336 The Theory of Symmetry §92
Symmetry groups are conveniently studied with the help of the general
mathematical techniques of what is called group theory, the fundamentals of
which we shall explain below. At first we shall consider groups, each of which
contains a finite number of transformations (known as finite groups). Each of
the transformations forming a group is said to be an element of the group.
Symmetry groups have the following important properties. Each group
contains the identical transformation E (called the unit element of the group).
The elements of a group can be "multiplied" by one another; by the product
of two (or more) transformations we mean the result of applying them in
succession. It is obvious that the product of any two elements of a group is
also an element of that group. For the multiplication of elements we have
the associative law (AB)C — A(BC), where A, B, C are elements of a group.
There is evidently no general commutative law; in general, AB ^ BA.
For each element A of a group there is in the same group an inverse element
A -1 (the inverse transformation), such that AA~ X = E. In some cases an
element may be its own inverse ; in particular, E _1 = E. It is evident that
mutually inverse elements A and A -1 commute.
The element inverse to the product AB of two elements is
(AB)- 1 = Br*-A~\
and similarly for the product of a greater number of elements ; this is easily
seen by effecting the multiplication and using the associative law.
If all the elements of a group commute, the group is said to be Abelian.
A particular case of Abelian groups is formed by what are called cyclic groups.
By a cyclic group we mean a group, all of whose elements can be obtained by
raising one of them to successive powers, i.e. a group consisting of the
elements
A, A\ A\ ... , A n = E,
where n is some integer.
Let G be some group.f If we can separate from it some set of elements
H such that the latter is itself a group, then the group H is called a sub-group
of the group G. A given element of a group may appear in several of its
sub-groups.
By taking any element A of a group and raising it to successive powers,
we finally obtain the unit element (since the total number of elements in the
group is finite). If n is the smallest number for which A n = E, then n is
called the order of the element A, and the set of elements A, A 2 , ... , A n = E
is called the period of A. The period is denoted by {A} ; it is itself a group, i.e.
it is a sub-group of the original group, and is cyclic.
In order to find whether a given set of elements of a group is a sub-group
of it, it is sufficient to find whether, on multiplying any two of its elements,
we obtain another element of the set. For in that case we have, together with
t We shall denote groups by bold italic letters.
§92 Transformation groups 337
each element A, all its powers, including A n ~ x (where n is the order of A),
which is the inverse of A (since A n ~ x A = A n = E); and there will obviously
be a unit element.
The total number of elements in a group is called its order. It is easy
to see that the order of a sub-group is a factor of the order of the whole group.
To show this, let us consider a sub-group H of a group G, and let G x be
some element of G which does not belong to H. Multiplying all the elements
of H (on the right, say) by G ly we obtain a set (or complex, as it is called)
of elements, denoted by HG X . All the elements of this complex clearly belong
to the group G. However, none of them belongs to H; for, if for any two
elements H a , H b belonging to H we had H a G x = H b , it would follow that
G x = H a ~ x H b , i.e. G x would also belong to the sub-group H, which is
contrary to hypothesis. Similarly we can show that, if G 2 is an element of G
not belonging to H or to HG X , none of the elements of the complex HG 2
will belong to H or to HG V Continuing this process, we finally exhaust
all the elements contained in the finite group G. Thus all the elements are
divided among the complexes
H, HG lt HG 2 , ... , HG m
(where m is some integer), each of which contains h elements, h being the
order of the sub-group H. Hence it follows that the order g of the group G
is g = km, and this proves the theorem.
If the order of a group is a prime number, it follows at once from the
above that the group has no sub-groups (except itself and E). The converse
theorem is also valid: a group having no sub-groups is of prime order and in
addition must be cyclic (since otherwise it would contain elements whose
period would form a sub-group).
We shall now introduce the important concept of conjugate elements. Two
elements A and B are said to be conjugate if
A = CBC- 1 ,
where C is also an element of the group ; multiplying this equation on the
right by C and on the left by C _1 , we have the converse equation B = C~ X AC.
An important property of conjugate elements is that, if A is conjugate to B,
and B to C, then A is conjugate to C; for, if B = P- X AP, C = Q~ X BQ
(P and Q being elements of the group), it follows that C = (P0~M(P0.
For this reason we can speak of sets of conjugate elements of a group.
Such sets are called classes of the group. Each class is completely determined
by any one element A of it ; for, given A, we obtain the whole class by forming
the products GAG~ X , where G is successively every element of the group
(of course, this may give each element of the class several times). Thus we
can divide the whole group into classes ; each element of the group can clearly
appear in only one class. The unit element of the group is a class by itself,
since for every element of the group GEG~ X = E. If the group is Abelian,
each of its elements is a class by itself; since all the elements, by definition,
338 The Theory of Symmetry §93
commute, each element is conjugate only to itself. We emphasise that a
class of a group (not being E) is not a sub-group of it; this is evident from the
fact that it does not contain a unit element.
All the elements of a given class are of the same order. For, if n is the
order of the element A (so that A n = E), then for a conjugate element
B = CMO 1 we have {CAC- X ) n = CA»C~ X =* E.
Let H be a sub-group of G, and G x an element of G not belonging to H.
It is easy to see that the set of elements GiHG-f 1 has all the properties of a
group, i.e. it also is a sub-group of the group G. The sub-groups H and
G x HG-f x are said to be conjugate; each element of one is conjugate to one
element of the other. By giving G t various values, we obtain a series of
conjugate sub-groups, which may partly coincide. It may happen that all
the sub-groups conjugate to H are H itself. In this case H is called a normal
divisor of the group G. Thus, for example, every sub-group of an Abelian
group is clearly a normal divisor of it.
Let us consider a group A with n elements A, A', A", ... , and a group
B with m elements B, B', B", ... , and suppose that all the elements of A
(apart from the unit E) are different from those of B but commute with
them. If we multiply every element of group A by every element of group B,
we obtain a set of nm elements, which also form a group. For, for any two
elements of this set we have AB . A'B' = AA' . BB' = A"B", i.e. another
element of the set. The group of order nm thus obtained is denoted by
A xB, and is called the direct product of the groups A and B.
Finally, we shall introduce the concept of the isomorphism of groups.
Two groups A and B of the same order are said to be isomorphous if we can
establish a one-to-one correspondence between their elements, such that, if
the element B corresponds to the element A, and B' to A', then B" = BB'
corresponds to A" = AA' . Two such groups, considered in the abstract,
clearly have identical properties, though the actual meaning of their elements
may be different.
§93. Point groups
Transformations which appear in the symmetry group of a body of finite
dimensions (in particular, a molecule) must be such that at least one point of
the body remains fixed when any of these transformations is applied. In
other words, all axes and planes of symmetry of a molecule must have at least
one common point of intersection. For a successive rotation of the body
about two non-intersecting axes or a reflection in two non-intersecting planes
results in a translation of the body, which obviously cannot then be left
unaltered. Symmetry groups having the above property are called point groups.
Before going on to construct the possible types of point group, we shall
explain a simple geometrical procedure whereby the elements of a group may
be easily divided into classes. Let Oa be some axis, and let the element A
of the group be a rotation through a definite angle about this axis. Next, let G
§93 Point groups 339
be a transformation (rotation or reflection) in the same group, which on being
applied to the same axis Oa carries it to the position Ob. We shall show that
the element B = GAG' 1 then corresponds to a rotation about the axis Ob
through the same angle as that of the rotation about Oa to which the element
A corresponds. For, let us consider the effect of the transformation GAG' 1
on the axis Ob itself. The transformation G _1 inverse to G carries the axis
Ob to the position Oa, so that the subsequent rotation A leaves it in this
position; finally, G carries it back to its initial position. Thus the axis Ob
remains fixed, so that B is a rotation about this axis. Since A and B belong
to the same class, their orders are the same; this means that they effect
rotations through the same angle.
Thus we reach the result that two rotations through the same angle belong
to the same class if there is, among the elements of the group, a transformation
whereby one axis of rotation can be carried into the other. In exactly the same
way, we can show that two reflections in different planes belong to the same
class if some transformation in the group carries one plane into the other.
The axes or planes of symmetry whose directions can be carried into each
other are said to be equivalent.
Some additional comments are necessary in the case where both rotations
are about the same axis. The element inverse to the rotation C n k (k =
1, 2, ... , n — 1) about an axis of symmetry of the «th order is the element
C n ~ k = C n n ~ k , i.e. a rotation through an angle (n—k)2irjn in the same
direction or, what is the same thing, a rotation through an angle 2hn\n in
the opposite direction. If, among the transformations in the group, there is
a rotation through an angle it about a perpendicular axis (this rotation reverses
the direction of the axis under consideration), then, by the general rule
proved above, the rotations C n k and C n ~ k belong to the same class. A
reflection a n in a plane perpendicular to the axis also reverses its direction ;
however, it must be borne in mind that the reflection also changes the direction
of rotation. Hence the existence of a h does not render C n k and C n ~ k conju-
gate. A reflection <j v in a plane passing through the axis, on the other hand,
does not change the direction of the axis, but changes the direction of rota-
tion, and therefore C n ~ k = o v C n k cr v , so that C n k and C n ~ k belong to the same
class if such a plane of symmetry exists. If rotations about an axis through the
same angle in opposite directions are conjugate, we shall call it bilateral.
The determination of the classes of a point group is often facilitated by the
following rule. Let G be some group not containing the inversion /, and C i
a group consisting of the two elements / and E. Then the direct product
G x C i is a group containing twice as many elements as G ; half of them are the
same as the elements of the group G, while the remainder are obtained by
multiplying the latter by /. Since / commutes with any other transformation
of a point group, it is clear that the group GxC t contains twice as many
classes as G ; to each class A of the group G there correspond the two classes
A and AI in the group G x C t . In particular, the inversion I always forms
a class by itself.
340 The Theory of Symmetry §93
Let us now go on to enumerate all possible point groups. We shall con-
struct these by starting from the simplest ones and adding new elements of
symmetry. We shall denote point groups by bold italic Latin letters with
appropriate suffixes.
!• c n groups
The simplest type of symmetry has a single axis of symmetry of the wth
order. The group C n is the group of rotations about an axis of the nth order.
This group is evidently cyclic. Each of its n elements forms a class by itself.
The group C x contains only the identical transformation E, and corresponds
to the absence of any symmetry.
U- s zn groups
The group S 2n is the group of rotary-reflections about a rotary-reflection
axis of even order 2n. It contains 2n elements and is evidently cyclic. In
particular, the group S 2 contains only two elements, E and /; it is also denoted
by C^ We may note also that, if the order of a group is a number of the form
2n = 4^+2, inversion is among its elements; it is clear that (S 4p+2 ) 2p+1
= C 2 a h = I. Such a group can be written as a direct product S^ p+2 — C 2p+1
xC t ; it is also denoted by C 2p+1 4 .
In » c nh groups
These groups are obtained by adding to an axis of symmetry of the «th
order a plane of symmetry perpendicular to it. The group C nh contains
2m elements: n rotations of the group C n and n rotary-reflection trans-
formations C n k a h , k = 1, 2, ... , n (including the reflection C n n a h = a h ).
All the elements of the group commute, i.e. it is Abelian; the number of
classes is the same as the number of elements. If n is even (n = 2p), the group
contains a centre of symmetry (since C 2p p a h = C 2 a h = /). The simplest
group, C lh , contains only two elements, E and a h \ it is also denoted by C s .
IV - c nv groups
If we add to an axis of symmetry of the «th order a plane of symmetry
passing through it, this automatically gives another n — 1 planes intersecting
along the axis at angles of 7r/«, as follows at once from the geometrical
theorem f (91.7) stated in §91. The group C nv thus obtained therefore con-
tains 2« elements : n rotations about the axis of the «th order, and n reflections
a v in vertical planes. Fig. 34 shows, as an example, the systems of axes and
planes of symmetry for the groups C 3V and C iv .
To determine the classes, we notice that, because of the presence of planes
of symmetry passing through the axis, the latter is bilateral. The actual
distribution of the elements among the classes depends on whether n is
even or odd.
f It is easy to see that, in a finite group, there cannot be two planes of symmetry intersecting at an
angle which is not a rational fraction of 2w. If there were two such planes, it would follow that there
were an infinite number of other planes of symmetry, intersecting along the same line and obtained by
reflecting one plane in the other ad infinitum. In other words, if there are two such planes, there must
be complete axial symmetry.
§93
Point groups
341
If n is odd (n = 2/> + l), successive rotations C 2p+1 carry each of the
planes successively into each of the other 2p planes, so that all the planes of
symmetry are equivalent, and the reflections in them belong to a single class.
Among rotations about the axis there are 2p operations apart from the identity,
and these are conjugate in pairs, forming p classes each of two elements
(C 2p +i fc and C 2p +r k , k = 1, 2, ... ,p); moreover, E forms an extra class.
Thus there arep+2 classes altogether.
X
l/Sl
^
C 4¥
Fig. 34
If, on the other hand, n is even (n = 2p), only every alternate plane can
be interchanged by successive rotations C 2p ; two adjacent planes cannot be
carried into each other. Thus there are two sets of p equivalent planes, and
accordingly two classes of p elements (reflections) each. Of the rotations
about the axis, C 2p 2p = E and C 2p v = C 2 each form a class by themselves,
while the remaining 2p— 2 rotations are conjugate in pairs and give another
p— 1 classes, each of two elements. The group C 2p „ thus has />+3 classes
altogether.
V. D n groups
If we add to an axis of symmetry of the nth order an axis of the second
order perpendicular to it, this involves the appearance of a further n — 1
such axes, so that there are altogether n horizontal axes of the second order,
intersecting at angles irjn. The resulting group D n contains 2n elements:
n rotations about an axis of the nth order, and n rotations through an angle
77 about horizontal axes (we shall denote the latter by U 2 > reserving the notation
C 2 for a rotation through an angle it about a vertical axis). Fig. 34 shows, as
an example, the systems of axes for the groups D 3 and D 4 .
In an exactly similar manner to case IV, we may verify that the axis of the
nth order is bilateral, while the horizontal axes of the second order are all
equivalent if n is odd, or form two non-equivalent sets if n is even. Con-
sequently, the group D 2p has the following/) +3 classes: E, 2 classes each of
p rotations U 2 , the rotation C 2 , and p — 1 classes each of two rotations about
the vertical axis. The group D 2p+1 , on the other hand, has p+2 classes:
E, 2p + 1 rotations C/ 2 , and p classes each of two rotations about the vertical
axis.
342
The Theory of Symmetry
§93
An important particular case is the group D 2 . Its system of axes is
composed of three mutually perpendicular axes of the second order. This
group is also denoted by V.
VI - D nh groups
If we add to the system of axes of a group D n a horizontal plane of sym-
metry passing through the n axes of the second order, n vertical planes
automatically appear, each of which passes through the vertical axis and one
of the horizontal axes. The group D nh thus obtained contains 4« elements;
besides the 2n elements of the group D n , it contains also n reflections a v
and n rotary-reflection transformations C n k a h . Fig. 35 shows the system
of axes and planes for the group D 3h .
^
h ^V
1/\J
u 3h
The reflection a h commutes with all the other elements of the group;
hence we can write D nh as the direct product D nh = D n xC s , where C s is
the group consisting of the two elements E and a h . For even n the inversion
operation is among the elements of the group, and we can also write
D 2j >,fc = Z> 23 ,xCV
Hence it follows that the number of classes in the group D nh is twice the
number in the group D n . Half of them are the same as those of the group
D n (rotations about axes), while the remainder are obtained by multiplying
these by a h . The reflections a v in vertical planes all belong to a single class
(if n is odd) or form two classes (if n is even). The rotary-reflection trans-
formations a h C n k and a h C n ~ k are conjugate in pairs.
VI1 - D nd groups
There is another way of adding planes of symmetry to the system of axes
of the group D n . This is to draw vertical planes through the axis of the nth
order, midway between each adjacent pair of horizontal axes of the second
order. The adding of one such plane again involves the appearance of another
(»— 1) planes. The system of axes and planes of symmetry thus obtained
determines the group D nd . Fig. 35 shows the axes and planes for the groups
D 2d andD 3d .
§93 Point groups 343
The group D nd contains An elements. To the 2« elements of the group
D n are added n reflections in the vertical planes (denoted by a d — the "diago-
nal" planes) and n transformations of the form G = U 2 a d . In order to
ascertain the nature of these latter, we notice that the rotation U 2 can, by
(91.6), be written in the form U 2 = a h a v , where a v is a reflection in the verti-
cal plane passing through the corresponding axis of the second order. Then
G = a n a v a d (the transformations a v , a h alone are not, of course, among
the elements of the group). Since the planes of the reflections a v and a d
intersect along an axis of the wth order, forming an angle (2&+l)7r/2w,
where &=l,...,(n— 1) (since here the angle between adjacent planes is
tt/2w), it follows that, by (91.6), we have a v a d = C 2n 2k+1 . Thus we find that
G = a h C 2n 2k+1 = S 2n 2k+1 , i.e. these elements are rotary-reflection trans-
formations about the vertical axis, which is consequently not a simple axis
of symmetry of the nth. order, but a rotary-reflection axis of the 2»th order.
The diagonal planes reflect two adjacent horizontal axes of the second
order into each other ; hence, in the groups under consideration, all axes of
the second order are equivalent (for both even and odd n). Similarly, all
diagonal planes are equivalent. The rotary-reflection transformations
S2n 2k+1 and S2n~ 2k ~ 1 are conjugate in pairs.f
Applying these considerations to the group D 2p d , we find that it contains
the following 2p+3 classes: E, the rotation C 2 about the axis of the nth
order, (p— 1) classes each of two conjugate rotations about the same axis,
one class of the 2p rotations U 2 , one class of 2p reflections a d , and^ classes
each of two rotary-reflection transformations.
For odd n (= 2p+\), inversion is among the elements of the group; this
is seen from the fact that, in this case, one of the horizontal axes is perpen-
dicular to a vertical plane. Hence we can write D 2p+ld = D 2p+1 xC it
so that the group D 2p+1 d contains 2p+4 classes, which are obtained at
once from the p+2 classes of the group D 2p+1 .
VIII. The group T (the tetrahedron group)
The system of axes of this group is the system of axes of symmetry of a
tetrahedron. It can be obtained by adding to the system of axes of the group
V four oblique axes of the third order, rotations about which carry the three
axes of the second order into one another. This system of axes is conveniently
represented by showing the three axes of the second order as passing through
the centres of opposite faces of a cube, and those of the third order as the
spatial diagonals of the cube. Fig. 36 shows the position of these axes in a
cube and in a tetrahedron (one axis of each type is shown).
The three axes of the second order are mutually equivalent. The axes of
the third order are also equivalent, since they are carried into one another by
f For we have
"A* 1 ** 1 ** = WV** 1 ** = WV*«<r, = a/An" 2 *- 1 = s 2n - 2 *-i
344
The Theory of Symmetry
§93
Fig. 36
the rotations C 2 , but they are not bilateral axes. Hence it follows that the
twelve elements in the group T are divided into four classes : E, the three
rotations C 2 , the four rotations C 3 and the four rotations C 3 2 .
IX. The group T d
This group contains all the symmetry transformations of the tetrahedron.
Its system of axes can be obtained by adding to the axes of the group T planes
of symmetry, each of which passes through one axis of the second order and
two of the third order. The axes of the second order thereby become rotary-
reflection axes of the fourth order (as in the case of the group D 2d ). This
system is conveniently represented by showing the three rotary-reflection
axes as passing through the centres of opposite faces of a cube, the four
axes of the third order as its spatial diagonals, and the six planes of symmetry
§93
Point groups
345
as passing through each pair of opposite edges (Fig. 37 shows one of each
kind of axis and one plane).
Since the planes of symmetry are vertical with respect to the axes of the
third order, the latter are bilateral axes. All the axes and planes of a given
kind are equivalent. Hence the 24 elements of this group are divided into
the following five classes: E, eight rotations C 3 and C 3 2 , six reflections in
planes, six rotary-reflection transformations 5 4 and 5 4 3 , and three rotations
C — S 2
X. The group T h
This group is obtained from T by adding a centre of symmetry: T h
= TxCp As a result, three mutually perpendicular planes of symmetry
appear, passing through each pair of axes of the second order, and the axes of
the third order become rotary-reflection axes of the sixth order (Fig. 38
shows one of each kind of axis and one plane).
/ ' A
'A
(~\-
K
n__
7
7
4-^
V-j,'
Fig. 38
The group contains 24 elements divided among eight classes, which are
obtained at once from those of the group T.
XI. The group O (the octahedron group)
The system of axes of this group is the system of axes of symmetry of a
cube: three axes of the fourth order pass through the centres of opposite
346
The Theory of Symmetry
§93
faces, four axes of the third order through opposite corners, and six axes of the
second order through the midpoints of opposite edges (Fig. 39).
It is easy to see that all the axes of a given order are equivalent, and each
of them is bilateral. Hence the 24 elements are divided among the following
five classes: E, eight rotations C 3 and C 3 2 , six rotations C 4 and C 4 3 , three
rotations C 4 2 and six rotations C 2 .
XII. The group O h
This is the group of all symmetry transformations of the cube.f It is
obtained by adding to the group O a centre of symmetry: O h = OxC^
The axes of the third order in the group O are thereby converted into rotary-
reflection axes of the sixth order (the spatial diagonals of the cube); in
addition, another six planes of symmetry appear, passing through each pair
of opposite edges, and three planes parallel to the faces of the cube (Fig.
40). The group contains 48 elements divided among ten classes, which
Fig. 40
can be at once obtained from those of the group O ; five classes are the same
as those of the group O, while the remainder are : /, eight rotary-reflection
transformations S 6 and 5 6 5 , six rotary-reflection transformations C^a^
C 4 3 a h about axes of the fourth order, three reflections a h in planes horizontal
with respect to the axes of the fourth order, and six reflections a v in planes
vertical with respect to these axes.
XIII, XIV. The groups Y, Y h {the icosahedron groups)
These groups are of no physical interest, since they do not occur in
Nature as symmetry groups of molecules. Hence we shall here only mention
that Y is a group of 60 rotations about the axes of symmetry of the icosa-
hedron (a regular solid with twenty triangular faces) or of the pentagonal
dodecahedron (a regular solid with twelve pentagonal faces); there are six
axes of the fifth order, ten of the third and fifteen of the second. The group
Y h is obtained by adding a centre of symmetry: Y h = YxC it and is the
f The groups T, T d , T h , O, 0% are often called cubic.
§94 Representations of groups 347
complete group of symmetry transformations of the above-mentioned poly-
hedra.
This exhausts all possible types of point group containing a finite number
of elements. In addition, we must consider what are called continuous point
groups, which contain an infinite number of elements. This we shall do in
§98.
§94. Representations of groups
Let us consider any symmetry group, and let fa be some one-valued func-
tion of the co-ordinates."}* Under the transformation of the co-ordinate
system which corresponds to an element G of the group, this function is
changed into some other function. On performing in turn all the g transfor-
mations in the group {g being the order of the group), we in general obtain
g different functions from fa. For certain fa, however, some of these functions
may be linearly dependent. As a result we obtain some number f(<g) of
linearly independent functions fa, fa, ... , ifj f , which are transformed into
linear combinations of one another under the transformations belonging to
the group in question. In other words, as a result of the transformation G,
each of the functions fa (i = 1, 2, 3, ... ,/) is changed into a linear combina-
tion of the form
where the G ik are constants depending on the transformation G. The array
of these constants is called the matrix of the transformation. J
In this connection it is convenient to regard the elements G of the group
as operators acting on the functions tft^ so that we can write
* = S«; (94.1)
the functions fa can always be chosen so as to be orthonormal. Then the
concept of the matrix of the transformation is the same as that of the matrix
of the operator, in the form defined in §11:
G ik =jfa*Gfadq. (94.2)
To the product of two elements G and H of the group there corresponds
the matrix obtained from the matrices of G and H by the ordinary rule
of matrix multiplication (11.12):
{GH) ik = ZG a H lk . (94.3)
The set of matrices of all the elements in a group is called a representation
t In the configuration space of the physical system concerned.
t Since the functions i[n are assumed one-valued, a definite matrix corresponds to each element of
the group.
348 The Theory of Symmetry §94
of the group. The functions tf/ 1} ... , tf/ f with respect to which these matrices
are defined are called the basis of the representation. The number/ of these
functions gives what is called the dimension of the representation.
Let us consider the integral j \ip\ 2 dq, where \\s is some function of the
co-ordinates. Since the integral is taken over all space, it is evident that
its value is unchanged by any rotation or reflection of the co-ordinate system.
Hence, for any symmetry transformation G, we can write
f (0*<P*)(OiJj) dq = f 0*0 dq.
Introducing the transposed operator 0, we have
f (0*ift*)(Otf>) dq = ! 0(5(5*0* dq = ! 0*0 dq,
whence, from the fact that is arbitrary, it follows that 00* = 1, or
0* = 0~\
i.e., the operators are unitary.
Thus a representation of a symmetry group in terms of orthonormal
base functions is unitary, i.e. the group is represented by unitary matrices.
Suppose that we perform on the system of functions 0i, . . . , 0/ the linear
unitary transformation
0' f =£0*. (94.4)
This gives a new system of functions 0'i, . . . , 0'/, which are also orthonormal
(see §12).| If we now take, as the basis of the representation, the functions
ifj'i, we obtain a new representation of the same dimension. Such representa-
tions, obtained from one another by a linear transformation of their base
functions, are said to be equivalent ; it is evident that they are not essentially
different.
The matrices of equivalent representations can be simply expressed in terms
of one another. According to (12.7), the matrix of the operator in the new
representation is the matrix of the operator
0' =S-iO§ (94.5)
in the old representation.
The sum of the diagonal elements (i.e. the trace) of the matrix representing
an element G of a group is called its character; we shall denote it by x(G).
It is a very important result that the characters of the matrices of equivalent
f It may also be noted that the unitary transformation leaves invariant the sum of the squared
moduli of the functions: using (12.6), we have
E|0' i\ 2 = J ZSk t faS li fp
= S^i0i«8jM = S|$t| 2 .
§94 Representations of groups 349
representations are the same (see the end of §12). This circumstance gives
particular importance to the description of group representations by stating
their characters: it enables us to distinguish at once the fundamentally
different representations from those which are equivalent. Henceforward
we shall regard as different representations only those which are not
equivalent.
If we take 5 in (94.5) to be that element of the group which relates the
conjugate elements G and G', we have the result that, in any given represen-
tation of a group, the characters of the matrices representing elements of the
same class are the same.
The identical transformation corresponds to the unit element E of the
group. Hence the matrix representing the latter is diagonal in every represen-
tation, and the diagonal elements are unity. The character x( E ) is con-
sequently just the dimension of the representation :
X(E) =/• (94.6)
Let us consider some representation of dimension/. It may happen that,
as a result of a suitable linear transformation (94.4), the base functions
divide into sets of f v f 2 , ... functions (/i4-/ a +- =/)> in such a way that,
when any element of the group acts on them, the functions in each set are
transformed only into combinations of themselves, and do not involve
functions from other sets. In such a case the representation in question is
said to be reducible.
If, on the other hand, the number of base functions that are transformed
only into combinations of themselves cannot be reduced by any linear trans-
formation of them, the representation which they give is said to be irreducible.
Any reducible representation can, as we say, be decomposed into irreducible
ones. This means that, by the appropriate linear transformation, the base
functions divide into several sets, of which each is transformed by some
irreducible representation when the elements of the group act on it. Here
it may be found that several different sets transform by the same irreducible
representation; in such a case this irreducible representation is said to be
contained so many times in the reducible one.
Irreducible representations are an important characteristic of a group, and
play a fundamental part in all quantum-mechanical applications of group
theory. We shall give the chief properties of irreducible representations.f
It may be shown that the number of different irreducible representations
of a group is equal to the number r of classes in the group. We shall distin-
guish the characters of the various irreducible representations by indices ;
the characters of the matrices of the element G in the representations are
x d)(G), X < 2 >(G), ... , X <->(G).
t The proof of these properties may be found in books on group theory, for example E. Wigner,
Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press,
New York 1959 (Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren,
Vieweg, Brunswick 1931); H. Weyl, The Theory of Groups and Quantum Mechanics, Methuen,
London 1931.
350 The Theory of Symmetry §94
The matrix elements of irreducible representations satisfy a number of ortho-
gonality relations. First of all, for two different irreducible representations
the relations
SGW tt GW> Im *=0 (94.7)
hold, where a and /J (a # j8) refer to the two irreducible representations, and the
summation is taken over all the elements of the group. For any irreducible
representation the relations
|GW=^A,» (94.8)
Ja.
hold, i.e. only the sums of the squared moduli of the matrix elements are not
zero:
§|G<«y 2 =£// a .
The relations (94.7), (94.8) can be combined in the form
S &«\ k GV» lm * = fs^Am- (94.9)
JOL
In particular, we can obtain from this an important orthogonality relation
for the characters of the representations. Summing both sides of equation
(94.9) over equal values of the suffixes i, k and /, m, we have
S x («)(G) x ^(G)*=^S a/? . (94.10)
For a = j8 we have
glx (a, (G)l 2 =£,
i.e. the sum of the squared moduli of the characters of an irreducible represen-
tation is equal to the order of the group. We may notice that this relation
can be used as a criterion of the irreducibility of a representation; for a
reducible representation, this sum is always greater than g (it is ng, where n
is the number of irreducible representations contained in the reducible
one).
It also follows from (94.10) that the equality of the characters of two
irreducible representations is not only a necessary but also a sufficient con-
dition for them to be equivalent.
Since the characters of elements of the same class are equal, the sum
(94.10) actually contains only r independent terms, and can be written in
the form
^gcx^{C) X ^{C)*=go^ (94.11)
§94 Representations of groups 351
where the summation is over the r classes of the group (arbitrarily denoted
by C) and^c is the number of elements in class C.f
The relation (94.10) enables any reducible representation to be very easily
decomposed into irreducible ones if the characters of both are known. Let
x(G) be the characters of some reducible representation of dimension /,
and let the numbers a (1) , a {2) , ... , a {r) indicate how many times the corres-
ponding irreducible representations are contained in it, so that
£ d% =/, (94.12)
p-i
where fp are the dimensions of the irreducible representations. Then the
characters x(G) can be written
X (G) = 2 a^x^(G). (94.13)
p -l
Multiplying this equation by x (a) (^)* an & summing over all G, we have by
(94.10) i
a*">=-Sx(G) x («>(G)*. (94.14)
§ °
Let us consider a representation of dimension f — g, given by the g
functions 0$, tft being some general function of the co-ordinates (so that
all the g functions Oi/j obtained from it are linearly independent); such a
representation is said to be regular. It is clear that none of the matrices of
this representation will contain any diagonal elements, with the exception of
the matrix corresponding to the unit element ; hence x(G) = for G =£ E,
while x{E) = S- Decomposing this representation into irreducible ones, we
have for the numbers a^\ by (94.14), the values a™ = (llg)gf (0L) =/ (a) , i.e.
each irreducible representation is contained in the reducible one under
consideration as many times as its dimension. Substituting this in (94.12),
we find the relation
/i 2 +/ 2 2 +...+/ r 2 =iT, (94-15)
the sum of the squared dimensions of the irreducible representations of a
group is equal to its order .J Hence it follows, in particular, that for Abelian
groups (where r = g) all the irreducible representations are of dimension one
(/i=/ a = ..•= /,= !)•
t Since the number of irreducible representations is equal to the number of classes, the quantities
f ac = ■\/( < gclg)x^XC) form a square matrix of r 2 quantities.
The orthogonality relations for the first suffix (2/ac//3c* = Sa/3) then automatically give those for
the second suffix: S/ac/ac'* = hcc'. Hence, besides (94.11), we have
Sx (a) (C)x (a) (C")* = — Sec-. (94.11a)
a gc
% It may be mentioned that, for point groups, equation (94.15) for given r and g can in practice
be satisfied in only one way by a set of integers / 1( ... ,/ r .
352 The Theory of Symmetry §94
We may also remark, without proof, that the dimensions of the irreducible
representations of a group divide its order.
Among the irreducible representations of any group there is always a
trivial one, given by a single base function invariant under all the trans-
formations in the group. This one-dimensional representation is called the
unit representation ; in it, all the characters are unity.
Let us consider two different systems of functions $t (a) , ... , if) f (a) and
ti^K ... , "A/g^, which form two irreducible representations of a group.
By forming the products «A < (a) «/'jfc (/?) we obtain a system off a fp new functions,
which can serve as the basis for a new representation of dimension f u fp.
This representation is called the direct product of the other two; it is irreduc-
ible only if f a orfp is unity. It is easy to see that the characters of the direct
product are equal to the products of the characters of the two component
representations. For, if
I m
then
hence we have for the characters, which we denote by (x^Xx^X^D*
(x (a) Xx^)(<?) = xgjcog^ = 2 G w s G tt (0,
i.e.
( x «*) Xx (/?))(G) = x m (G)x^(G). (94.16)
The two irreducible representations so multiplied may, in particular, be
the same; in this case we have two different sets of functions if/ ly ... , iftf
and <f) x , ... , <f) f giving the same representation, while the direct product of
the representation with itself is given by the f 2 functions ifti<f> k , and has the
characters
(xXxXG) = b((G)] 2 -
This reducible representation can be at once decomposed into two represen-
tations of smaller dimension (although these are, in general, themselves
reducible). One of them is given by the i/(/+l) functions ^&+^a;<^> the
other by the \f{f— 1) functions 0^&— «Afc<^, i ¥=■ k; it is evident that the func-
tions in each of these sets are transformed only into combinations of them-
selves. The former is called the symmetric product of the representation with
itself, and its characters are denoted by the symbol [x 2 J(G);the latter is
called the antisymmetric product, and its characters are denoted by {x 2 } (G).
To determine the characters of the symmetric product, we write
G(rpi<f> k +*p k $i) =£ G M G r ?nfc (j/f,^ m +^ tn ^ J )
= I ^(GuGrnk+GmiGt^ifiicfin+tfimfa).
§94 Representations of groups 353
Hence we have for the character
[X 2 ](G) = \ §(<?«<?»+<?*<?«).
But S(? w =x(G), and £ G ik G ki = x(G 2 ) ; thus we finally obtain the
formula
\X*](G) = ttbc(G)?+x(G*)}> (94.17)
which enables us to determine the characters of the symmetric product of a
representation with itself from the characters of the representation. In an
exactly similar manner, we find for the characters of the antisymmetric
product the formula
(X 2 }(G) = m(G)]*-x(G*)}. (94.18)
If the functions tft i and <^ are the same, we can evidently construct from
them only the symmetric product, formed by the squares ^ 2 and the pro-
ducts iff^k, i ¥" k. In applications, symmetric products of higher orders are
also encountered; their characters may be obtained in a similar manner.
An important property of direct products is the following. The resolution
of the direct product of two different irreducible representations into irre-
ducible parts never involves the unit representation, but the direct product
of an irreducible representation with itself (that is, of course, its symmetric
part) always contains (and only once) the unit representation. In order to
know whether the unit representation is present in the representation
(94.16), we simply sum its characters with respect to G and divide the result
by the order g of the group, in accordance with (94.14). From (94.10) it is
seen that this gives zero if a ^ j3 and unity if a = j8.
For applications it is useful to know a formula which enables us to represent
an arbitrary function iff as a sum of functions transformed by the irreducible
representations of the group, i.e. in the form
= SE&< a >, (94.19)
where the functions «/f i (a) (i = 1, 2, ... ,/ a ) are transformed by the ath
irreducible representation. The problem consists in determining the func-
tions < < a > from the function tft, and is solved by the formula
rfjfoo = {5L 2 Gi^Gj*. (94.20)
8 G
To prove this, it suffices to show that the expression on the right-hand
side of the equation reduces to ^r 4 (a) identically if we put ijj = ^ (a) , and
to zero if we put tft = ip k W with k # i or j8 # a; both these results follow
at once from the orthogonality relations (94.7), (94.8), putting G$ k W
= S G lk ^^\
354 The Theory of Symmetry §95
If we substitute (94.20) in (94.19) and effect the summation over t, we
obtain a simpler expansion of the arbitrary function xjj into functions ^t (a)
belonging to the various irreducible representations, but not to any definite
rows of these representations :
= 2 «A (a) , </-«*> = — S x < a >(G)*&A. (94.21)
Finally, we shall make a few remarks regarding the irreducible represen-
tations of a group which is the direct product of two other groups (not
to be confused with the direct product of two representations of the same
group). If the functions «/r f (a) give an irreducible representation of the group
A, and the functions <f> k W give one of the group B, the products ^ A (/?) ^ f (ot)
are the basis of an / a /^-dimensional representation of the group A xB, and
this representation is irreducible. The characters of this representation are
obtained by multiplying the corresponding characters of the original represen-
tations (cf. the derivation of formula (94.16)); to an element C= AB of
the group A xB there corresponds the character
X(Q= X M(A) X W(B). (94.22)
Multiplying together in this way all the irreducible representations of the
groups A and B, we obtain all the irreducible representations of the group
AxB.
§95. Irreducible representations of point groups
Let us pass now to the actual determination of the irreducible represen-
tations of those point groups which are of physical interest. The great
majority of molecules have axes of symmetry only of the second, third,
fourth or sixth order. Hence it is unnecessary to consider the icosahedron
groups Y, Y h ; we shall examine the groups C n , C nhi C nv , D n , D nh only for
the values n = 1, 2, 3, 4, 6, and the groups S 2ny D nd only for n = 1, 2, 3.
The characters of the representations of these groups are shown in Table 7.
Isomorphous groups have the same representations and are given together.
The numbers in front of the symbols for the elements of a group in the upper
rows show the numbers of elements in the corresponding classes (see §93).
The left-hand columns show the conventional names usually given to the
representations. The one- dimensional representations are denoted by the
letters A, B, the two-dimensional ones by E, and the three-dimensional
ones by F; the notation E for a two-dimensional irreducible representation
should not be confused with the unit element of a group.f The base functions
of A representations are symmetric, and those of B representations antisym-
metric, with respect to rotations about a principal axis of the nth. order.
t The reason why two complex conjugate one-dimensional representations are shown as one
two-dimensional one is explained in §96.
§95
Irreducible representations of point groups
355
The functions of different symmetry with respect to a reflection an are
distinguished by the number of primes (one or two), while the suffixes
g and u show the symmetry with respect to inversion. Beside the symbols
for the representations are placed the letters x, y, z; these show the repre-
sentations by which the co-ordinates themselves are transformed (with a view
to later quantum-mechanical applications). The z-axis is always taken along
Table 7
Characters of irreducible representations of point groups
c x
E
c t
c 2
C s
E
E
E
c 2
a
c 3
E
C 3 C 8
A
1
A\z
1
1 1
E;x±iy j
1
1
e e 2
A g
A;z A';x,y
1
1
e 2 e
Au\
x,y,z
B;x,y A"\z
1
-1
(-■2h
E
c 2
Oh
/
c 2e
E
c 2
a v
O'v
V^D 2
E
c 2 *
c 2 y
c 2 *
Ag
A l ;z
A
1
1
l
1
Bg
B 2 ;y
B 3 ;x
1
-1
-l
1
Au\
z
A 2
B t ;z
1
1
-l
-1
Bu\
X,
y
B x ;x
B 2 ;y
1
-1
l
-1
^3«
E
2C S
3<7„
D z
E
2C S
3C/ 2
A t ;z
A 1
1
1
1
A 2
A 2 \ z
1
1
-1
E;x,
y
E;x,y
2
-1
c.
E d C 2 C 4 S
s t
E S 4 C 2 S 4 *
A\z A
1111
B B;z
1-11 -1
E;x±iy E;x±iy \
1 i -1 -i
1 -i -1 i
C t
E C 6
c 3
c 2
C 3 2
cv
A;z
1
1
1
1
1
B
-1
1
-1
1
-1
f
O) 2
— m
1
CO 2
— CO
E 1
I
— a>
u,*
1
— CO
CO 2
CO
CO 2
-1
— CO
-co 2
E 2 ; x
±iy
1
-to 2
— CO
-1
to*
CO
356
The Theory of Symmetry
Table 7 — continued
§95
c 4c
E
c 2
2C 4
2<7„
2a' \
D t
E
c %
2C 4
2U 2
2U\
D 2d
E
c 2
2S t
2U 2
2a a
A x \z
A t
A 1
1
1
1
1
1
A 2
A 2 \ z
A 2
1
1
1
-1
-1
B 1
B 1
B 1
1
1
-1
1
-1
B 2
B 2
B 2 ;z
1
1
-1
-1
1
E;x,y
E;x,y
E;x,y
2
-2
D*
E
c 2
2C 8
2C e
3U 2
3U' 2
^60
E
c 2
2C 8
2C 6
3ff„
3ct'„
D ah
E
Oh
2C 8
2S 3
3U 2
3cr'„
A x
Aaz
AS
1
1
1
1
1
1
A 2 \z
A %
A,'
1
1
1
1
-1
-1
B 1
B 2
A,"
1
-1
1
-1
1
-1
B 2
B,
A 2 ;
z
1
-1
1
-1
-1
1
E 2
E 2
E';x
y
2
2
-1
-1
E x ;x,y
Ei;x,y
E"
2
-2
-1
1
o
E
8C 3
3C 2 6C 2 6Ct
T
E
3C a
4C 8
4C 3 2
T d
E
8C 8
3C 2 6a,i 6S^
A
1
1
1
1
A,
A 1
1
1
1 1 1
* {
1
1
e
e*
A 2
A 2
1
1
1 -1 -1
I
1
1
e*
e
E
E
2
-1
2
F; x, y, z
3
-1
F 2
Ftix
F 2 ;x,
y,z Fj.
y,z
3
3
-1 1 -1
-1 -1 1
§95 Irreducible representations of point groups 357
the principal axis of symmetry. The letters e and a> denote
e = «**/», to = e 2ffi/6 = -co 4 .
The simplest problem is to determine the irreducible representations for
the cyclic groups (C n , S n ). A cyclic group, like any Abelian group, has only
one-dimensional representations. Let G be a generating element of the group
(i.e. one which, on being raised to successive powers, gives all the elements
of the group). Since G e = E (where g is the order of the group), it is clear
that, when the operator acts on a base function ip, the latter can be multi-
plied only by l 1 /*, i.e.f
Cty^ftvaf (A = 1,2, ...,£).
The group C 2 n (and the isomorphous groups C 2v and D 2 ) is Abelian, so
that all its irreducible representations are one- dimensional, and the characters
can only be ± 1 (since the square of every element is E).
Next we consider the group C 3V . As compared with the group C 3 , the
reflections a v in vertical planes (all belonging to one class) are here added.
A function invariant with respect to rotation about the axis (a base function
of the representation A of the group C 3 ) may be either symmetric or anti-
symmetric with respect to the reflections a v . Functions multiplied by €
and e 2 under the rotation C 3 , on the other hand (base functions of the com-
plex conjugate representations £), change into each other on reflection. J
It follows from these considerations that the group C sv (and D 3 , which is
isomorphous with it) has two one-dimensional irreducible representations
and one two-dimensional, with the characters shown in the table. The fact
that we have indeed found all the irreducible representations may be seen
from the result l 2 + l 2 +2 2 = 6, which is the order of the group.
Similar considerations give the characters of the representations of other
groups of the same type (C 4 „, C 6V ).
The group T is obtained from the group V by adding rotations about
four oblique axes of the third order. A function invariant with respect to
transformations of the group V (a basis of the representation A) can be
multiplied, under the rotation C 3 , by 1, e or e 2 . The base functions of the
three one- dimensional representations B ly B 2 , Bz of the group V change into
one another under rotations about the axis of the third order (this is seen,
for example, if we take as these functions the co-ordinates x, y, z themselves).
Thus we obtain three one-dimensional irreducible representations and one
three-dimensional (l 2 + l 2 + l 2 + 3 2 = 12).
Finally, let us consider the isomorphous groups O and T d . The group
T d is obtained from the group T by adding reflections a d in planes each of
Which passes through two axes of the third order. A base function of the
f For the point group C n we can, for example, take as the functions ift the functions e tk *, k = 1,
2, ... ,n, where <f> is the angle of rotation about the axis, measured from some fixed direction.
' % These functions may, for example, be taken as fa == c'*, fa = e~i<l>. On reflection in a vertical
plane, <f> changes sign.
358
The Theory of Symmetry
§96
unit representation A of the group T may be symmetric or antisymmetric
with respect to these reflections (which all belong to one class), and this
gives two one- dimensional representations of the group T d . Functions
multiplied by e or e 2 under a rotation about an axis of the third order (the
basis of the complex conjugate representations E of the group T) change into
each other on reflection in a plane passing through this axis, so that one
two-dimensional representation is obtained. Finally, of three base functions
of the representation F of the group T, one is transformed into itself on
reflection (and can either remain unaltered or change sign), while the other
two change into each other. Thus we have altogether two one-dimensional
representations, one two-dimensional and two three-dimensional.
The representations of the remaining point groups in which we are inter-
ested can be obtained immediately from those already given, if we notice
that the remaining groups are direct products of those already considered
with the group C i (or C s ) :
^3h = C'sXCg
Ceh = CeXC,-
D 2h = D 2 xC €
D, h = D 4 xQ
"6 = ^3XC t -
D 3d = D^xCt
D 6h = D 6 xC f
T h = TxC t
o h =OxCi
Each of these direct products has twice as many irreducible representations
as the original group, half of them being symmetric (denoted by the suffix g)
and the other half antisymmetric (suffix u) with respect to inversion. The
characters of these representations are obtained from those of the representa-
tions of the original group by multiplying by + 1 (in accordance with the
rule (94.22)). Thus, for instance, we have for the group Dza the repre-
sentations :
D 3d
E
2C 3
3*7 2
I
25 6
3(7,,
A Xo
1
1
1
1
A 2 g
1
-1
1
-1
K
2
— 1
2
— 1
■A-iu
1
1
-1
— 1
-1
A 2u
1
-1
-1
— 1
1
E u
2
— 1
-2
§96. Irreducible representations and the classification of terms
The quantum-mechanical applications of group theory are based on the
fact that the Schrodinger's equation for a physical system (an atom or
§96 Irreducible representations and the classification of terms 359
molecule) is invariant with respect to symmetry transformations of the system.
It follows at once from this that, on applying the elements of a group to a
function satisfying Schrodinger's equation for some value of the energy (an
eigenvalue), we must again obtain solutions of the same equation for the same
value of the energy. In other words, under a symmetry transformation the
wave functions of the stationary states of the system belonging to a given
energy level transform into linear combinations of one another, i.e. they give
some representation of the group. An important fact is that this representa-
tion is irreducible. For functions which are invariably transformed into linear
combinations of themselves under symmetry transformations must belong
to the same energy level ; the equality of the eigenvalues of the energy cor-
responding to several groups of functions (into which the basis of a reducible
representation can be divided), which are not transformed into combinations
of one another, would be an utterly improbable coincidence (provided that
there is no special reason for such equality ; see below).
Thus, to each energy level of the system, there corresponds some irreduc-
ible representation of its symmetry group. The dimension of this represen-
tation determines the degree of degeneracy of the level concerned, i.e. the
number of different states with the energy in question. The fixing of the
irreducible representation determines all the symmetry properties of the
given state, i.e. its behaviour with respect to the various symmetry trans-
formations.
Irreducible representations of dimension greater than one are found only
in groups containing non-commuting elements; Abelian groups have only
one- dimensional irreducible representations. It is apposite to recall here that
the relation between degeneracy and the presence of operators which do not
commute with one another (but do commute with the Hamiltonian) has
already been found above from considerations unrelated to group theory
(§10).
The following important reservation should be made regarding all these
statements. As has already been pointed out (§18), the symmetry (valid in
the absence of a magnetic field) with respect to a change in the sign of the
time has, in quantum mechanics, the result that complex conjugate wave
functions must belong to the same eigenvalue of the energy. Hence it follows
that, if some set of functions and the set of complex conjugate functions give
different irreducible representations of a group, these two complex conjugate
representations must be regarded, from the physical point of view, as forming
together a single representation of twice the dimension. In the preceding
section we had examples of such representations. Thus the group C 3 has
only one-dimensional representations; however, two of these are complex
conjugates, and correspond physically to doubly degenerate energy levels.
(In the presence of a magnetic field there is no symmetry with respect to a
change in the sign of the time, and hence complex conjugate representations
correspond to different energy levels.)
Let us suppose that a physical system is subjected to the action of some
360 The Theory of Symmetry §96
perturbation (i.e. the system is placed in an external field). The question
arises to what extent the perturbation can result in a splitting of the degener-
ate levels. The external field has itself a certain symmetry.-}- If this symmetry
is the same as or higher J than that of the unperturbed system, the symmetry
of the perturbed Hamiltonian & = fi + f is the same as the symmetry
of the unperturbed operator fi . It is clear that, in this case, no splitting
of the degenerate levels occurs. If, however, the symmetry of the pertur-
bation is lower than that of the unperturbed system, then the symmetry of
the Hamiltonian i? is the same as that of the perturbation V. The wave
functions which gave an irreducible representation of the symmetry group
of the operator i? will also give a representation of the symmetry group
of the perturbed operator #, but this representation may be reducible, and
this means that the degenerate level is split.
We shall show by means of an example how the mathematical techniques
of group theory enable us to solve the problem of the splitting of any given
level.
Let the unperturbed system have symmetry Ta, and let us consider a
triply degenerate level corresponding to the irreducible representation F 2
of this group. The characters of this representation are
E 8C 3 3C 2 6a d 6S4
3 0-11-1
Let us assume that the system is subjected to the action of a perturbation
with symmetry Cs v (with the third-order axis coinciding with one of those of
the group Ta). The three wave functions of the degenerate level give a
representation of the group Cz v (which is a sub-group of the group T a ), and
the characters of this representation are equal to those of the same elements
in the original representation of the group Ta, i.e.
E 2C3 3g v
This representation, however, is reducible. Knowing the characters of the
irreducible representations of the group Czv, it is easy to decompose it into
irreducible parts, using the general rule (94.14). Thus we find that it consists
of the representations A\ and E of the group C% v . The triply degenerate
level F2 is therefore split into one non- degenerate level A\ and one doubly
degenerate level E. If the same system is subjected to the action of a per-
turbation of symmetry Czv, which is also a sub-group of the group Ta, then
t For example, in the case of the energy levels of the d and/ shells of ions in a crystal lattice which
interact slightly with the surrounding atoms, the perturbation (the external field) is the field acting
on an ion due to the other atoms.
% If a symmetry group H is a sub-group of the group G, we say that H corresponds to a lower
symmetry and Gtoa higher symmetry. It is evident that the symmetry of the sum of two expressions,
one of which has the symmetry of G and the other that of H, is the lower symmetry, that of H.
§97 Selection rules for matrix elements 361
the wave functions of the same level F 2 give a representation with characters
3 ~i i r
Decomposing this into irreducible parts, we find that it contains the repre-
sentations Ai, Bi, B 2 . Thus in this case the level is completely split into three
non- degenerate levels.
§97. Selection rules for matrix elements
Group theory not only enables us to carry out a classification of the terms
of any symmetrical physical system, but also gives us a simple method of
finding the selection rules for the matrix elements of the various quantities
which characterise the system.
This method is based on the following general theorem. Let «/»/«> be one
of the base functions of an irreducible (non-unit) representation of a symmetry
group. Then the integral of this function over all spacef vanishes identically :
f ,/,.(«) d? =0. (97.1)
The proof is based on the evident fact that the integral over all space is
invariant with respect to any transformation of the co-ordinate system,
including any symmetry transformation. Hence
J #* dq = J &A/ a > dq = J* S Gtpty,?* dq.
We sum this equation over all the elements of the group. The integral on
the left is simply multiplied by g, the order of the group, and we have
g^Wdq-ZJ^ZG^dq.
However, for any non-unit irreducible representation we have identically
S G k M = 0;
G k%
this is a particular case of the orthogonality relations (94.7), when one of
the irreducible representations is the unit representation. This proves the
theorem.
If j/r is a function belonging to some reducible representation of a group,
the integral J «/» dq will be zero except when this representation contains the
unit representation. This theorem is a direct consequence of the previous
one.
t That is, the configuration space of the physical system concerned.
362 The Theory of Symmetry §97
Let / be the operator of some scalar physical quantity. By definition,
it is invariant with respect to all symmetry transformations. Its matrix
elements are the integrals
j *f,M*fy k W dq, (97.2)
where the indices a, j8 distinguish different terms of the system, and the
suffixes j, k denumerate the wave functions of states belonging to the same
degenerate term. We denote the irreducible representations of the symmetry
group of the system concerned that are given by the functions >> and if/ k W
by the symbols D™ and W>. The products faM fc W give the representa-
tion D<«>xZ)W; since the operator /itself is invariant with respect to all
transformations, the whole expression in the integrand belongs to this
representation. The direct product of two different irreducible represen-
tations, however, does not contain the unit representation (see §94), whilst
the direct product of an irreducible representation with itself always contains
the unit representation, and only once; the integrals (97.2) are a constant
(independent of i and k) times 8 ik . Thus we reach the conclusion that, for a
scalar quantity, the matrix elements are non-zero only for transitions between
states of the same type (i.e. belonging to the same irreducible representation).
This is the most general form of a theorem of which we have already met
several particular cases.
Let us next consider some vector physical quantity A. The three compo-
nents A x , A y , A z transform into linear combinations of themselves under
symmetry transformations, and therefore give some representation of the
symmetry group, which we shall denote by| D A . The products «A i (a) A^ & ( ^ )
give the representation D < - a) xD A xD^ ) ; the matrix elements are non-zero
if this representation contains the unit representation. In practice, it is more
convenient to decompose into irreducible parts the direct product Z) (os) x D A ;
this gives us immediately all the types Z) W of states for transitions into which
(from a state of type D (a) ) the matrix elements are not zero.
The diagonal matrix elements (unlike those for transitions between diff-
erent states of the same type) require special consideration. In this case we
have only one system of functions ^//>>, not two different ones, and their
products in pairs give the symmetric product [Z) (a > 2 ] of the representation
Z> (a) with itself, not the direct product Z>( a >xD< a >. Hence the presence of
diagonal matrix elements of a vector quantity means that the unit represen-
tation is present in the decomposition of the product [Z) (a)2 ] XD A , or, what
is the same thing, that D A is present in [Z) (a)2 ]|.
Similarly we can find the selection rules for the matrix elements of a tensor.
Examples of the application of these rules are given in the following Problems.
f In general Da is different for polar and axial vectors.
J We did not make this remark in considering the scalar quantity/, seeing that the symmetric pro-
duct [D*") 2 ], like the direct product Z)( a >xZ)< a ), always contains the unit representation. Hence the
diagonal matrix elements of a scalar quantity are, in general, different from zero.
§97 Selection rules for matrix elements 363
PROBLEMS
Problem 1. Find the selection rules for the matrix elements of a polar vector when
symmetry O is present.
Solution. The components of a vector are transformed by the irreducible representation
F x . The decompositions of the direct products of F x with the other representations of the
group O are
F x xA x = F ly F ± xA 2 = F 2 , F ± xE = F x +F 2 ,
F x xF x = A x +E+F x +F ty F x xF 2 = A^E+F^F^
(1)
Hence the non-zero non-diagonal matrix elements are those for the transitions
F 1 <^. A lf E, F ly F 2 , F 2 4-> A 2 , E, F 2 .
The symmetric products of the irreducible representations of the group O are
W] = W] = A, [E 2 ] = A x +E, [*?] = [Ff] = A x +E+F v (2)
F x is contained in none of these; hence the diagonal matrix elements vanish.
Problem 2. The same as Problem 1, but for symmetry Z> 3 <f.
Solution. The ^-component of the vector is transformed by the representation Am, the
* and y components by E u - We have
E u xA lg = E u xA 2g = E u , E U XA 1U = E u xA 2u = E g ,
E u xE u = A lg +A 2g +E g , E u xE g = A lu +A 2u +E u .
Hence the non-diagonal matrix elements oiA x ,A v are non-zero for the transitions E u <— > A ig ,
A 29 , Eg',E g <-+ A 1U , A 2U . In the same way we find the selection rules for the matrix elements
of At : A ig <-> A w , A 2B <-» A 1U , Eg <-> E u .
The symmetric products of the irreducible representations are
[A 1B 2 ] = [Au 2 ] = W] = [AJ] = A u ,
[E g 2 ] = [E u *] = E g +A lg .
These do not contain either A 2U or E u \ hence the diagonal matrix elements vanish for both
At and A x , A y .
Problem 3. Find the selection rules for the matrix elements of a symmetrical tensor A ik
of rank two (with Axx+A yy +Azz = 0) when symmetry O is present.
Solution. The components A xy , A X z, A yz are transformed by F % . Decomposing the
direct products of F 2 with all the representations of the group O, we find the selection rules
F l < r ^A i , E, F lt F 2 ; F 2 <->A U E, F lt F 2 . The diagonal matrix elements exist (as we see from
(2), Problem 1) for the states F x and F 2 .
The sums A xx + sA yy + s*Azz, A xx + e?A yy + sA zz (e = e 2 "*/ 3 ) are transformed by the
representation E. The selection rules for the non-diagonal elements are E<- >A lt A it E;
Fi^-^Fi, F 2 ; F 2 <r^F 2 . The diagonal elements are non-zero for the states E, F lt F 2 .
Problem 4. The same as Problem 3, but for symmetry D 3d .
Solution. Azz is transformed by A ig , i.e. A Z z behaves as a scalar. The components
A xx — A yy and A xy are transformed by E g ; the same is true of the components A X z, A yz .
Decomposing the direct products of Eg with all the representations of the group Dad, we
find the selection rules for the non-diagonal matrix elements ; E g <— > Aig, A 2g , E g ; E u <— > A 1U ,
A 2U , E u . The diagonal elements are not zero (as we see from (2), Problem 2) only for the
states Eg and E u .
364 The Theory of Symmetry §98
§98. Continuous groups
As well as the point groups enumerated in §93, there exist also what are
called continuous point groups, having an infinite number of elements. These
are the groups of axial and spherical symmetry.
The simplest axial symmetry group is the group C m , which contains rota-
tions C(<f>) through any angle <f> about the axis of symmetry ; this is called the
two-dimensional rotation group. It may be regarded as the limiting case of
the groups C n as n -> oo. Similarly, as limiting cases of the groups C nh ,
c nv> D n, D nh we obtain the continuous groups C mht C,,, D MI D xh .
A molecule has axial symmetry only if it consists of atoms lying in a straight
line. If it meets this condition, but is asymmetric about its midpoint, its
point group will be the group C^, which, besides rotations about the axis,
contains also reflections a v in any plane passing through the axis. If, on
the other hand, the molecule is symmetrical about its midpoint, its point
group will be D ooh = C mv xC t . The groups C ro , C wft , D^ cannot appear as
the symmetry groups of a molecule.
The group of complete spherical symmetry contains rotations through any
angle about any axis passing through the centre, and reflections in any plane
passing through the centre; this group, which we shall denote by K h , is
the symmetry group of a single atom. It contains as a sub-group the group
K of all spatial rotations (called the three-dimensional rotation group, or simply
the rotation group). The group K h can be obtained from the group K by
adding a centre of symmetry (K h — K x C f ).
The elements of a continuous group may be distinguished by one or more
parameters which take a continuous range of values. Thus, in the rotation
group, the parameters might be the two angles determining the direction of
the axis, and the angle of rotation about this axis.
The general properties of finite groups described in §92, and the concepts
appertaining to them (sub-groups, conjugate elements, classes, etc.), can be
at once generalised to continuous groups. Of course, the statements which
directly concern the order of the group (for instance, that the order of a sub-
group divides the order of the group) are no longer meaningful.
In the group C oov all planes of symmetry are equivalent, so that all reflec-
tions a v form a single class with a continuous series of elements ; the axis of
symmetry is bilateral, so that there is a continuous series of classes, each
containing two elements C(±<£). The classes of the group D ooh are obtained
at once from those of the group C mVi since D ooh = C aav xC i .
In the rotation group K, all axes are equivalent and bilateral; hence the
classes of this group are rotations through an angle of fixed absolute magnitude
\<f>\ about any axis. The classes of the group K h are obtained at once from
those of the group K.
We have already found, in essence, the irreducible representations of the
three-dimensional rotation group (without using the terminology of group
theory), when determining the eigenvalues and eigenfunctions of the total
§98 Continuous groups 365
angular momentum. For the angular momentum operator is (apart from a
constant factor) the operator of an infinitely small rotation, and its eigenvalues
characterise the behaviour of the wave functions with respect to spatial
rotations. To a valued of the angular momentum there correspond 2/ + 1
different eigenfunctions ip] m > differing in the values of the component m of
the angular momentum and all belonging to one (2/ + l)-fold degenerate
energy level. Under rotations of the co-ordinate system, these functions are
transformed into linear combinations of themselves, and thus give irreducible
representations of the rotation group. Thus, from the group-theory point of
view, the numbers j number the irreducible representations of the rotation
group, and one (2/+ 1)- dimensional representation corresponds to each j.
The number j takes integral and half- integral values, so that the dimension
2/+ 1 of the representation takes all the integral values 1, 2, 3, . . . .
The base functions of these representations have been, in essence, investi-
gated in Chapter VIII. The basis of a representation of given/ is formed by
the 2j+ 1 independent components of a symmetrical spinor of rank 2/ (which
are equivalent to the set of 2j+ 1 functions fym).
The irreducible representations of the rotation group which correspond
to half-integral values of j are distinguished by an important property.
Under a rotation through 2tt, the base functions of the representations
change sign (being components of a spinor of odd rank). Since, however,
a rotation through 2n is the same as the unit element of the group, we
reach the result that representations with half-integral j are, as we say,
two-valued; to each element of the group (a rotation through an angle
cf>, ^ ^ 2tt, about some axis) there correspond in such a representation
not one but two matrices, with characters differing in sign.f
An isolated atom has, as we have already remarked, the symmetry K h
= KxC t . Hence, from the group-theory point of view, there corresponds
to each term of the atom some irreducible representation of the rotation
group K (determining the value of the total angular momentum J of the atom)
and an irreducible representation of the group C^ (determining the parity
of the state). J
When the atom is placed in an external electric field, its energy levels are
split. The number of different levels resulting and the symmetry of the
corresponding states can be determined by the method described in §96.
t It must be mentioned that "two-valued representations" of a group are not representations in
the true sense of the word, since they are not given by one-valued base functions; see also §99.
J Moreover, the Hamiltonian of the atom is invariant with respect to interchanges of the electrons.
In the non-relativistic approximation, the co-ordinate and spin wave functions are separable, and we
can speak of representations of the permutation group that are given by the co-ordinate functions. If
the irreducible representation of the permutation group is given, the total spin S of the atom is deter-
mined (§63). When the relativistic interactions are taken into account, however, the separation of the
wave functions into co-ordinate and spin parts is not possible. The symmetry with respect to simul-
taneous interchange of the co-ordinates and spins of the particles does not characterise the term,
since Pauli's principle admits only those total wave functions which are antisymmetric with respect
to all the electrons. This is in accordance with the fact that, when the relativistic interactions are
taken into account, the spin is not, strictly speaking, conserved; only the total angular momentum /
is conserved.
366 The Theory of Symmetry §98
It is necessary to decompose the (27+l)-dimensional representation of the
symmetry group of the external field (given by the functions ifj JM ) into
irreducible representations of this group. This requires a knowledge of the
characters of the representation given by the functions iJjjm-
By a rotation through an angle <£ about an axis the wave functions iftj M
are, as we know, multiplied by e iM $, where M is the component of the angular
momentum along that axis. The transformation matrix for the functions
i/jjm will therefore be diagonal, with character
^4 e U J+l)$ _ e -u<t>
> lJ) (<f>) = y e iM <t> =
m=-J
e W—\
or
sin(7+A)(£
X {J) (<1>) = . J . (98.1)
sinf<p
With respect to inversion I, all the functions iff JM with different M behave
in the same way, being multiplied by + 1 or - 1 according as the state of the
atom is even or odd. Hence the character
X {J) (I)= ±(27+1). (98.2)
Finally, the characters corresponding to reflection in a plane a and rotary
reflection through an angle <f> are found by writing these symmetry trans-
formations as
CT = /C 2 , S((f>)= IC(7T + <f>).
Let us pause to consider also the irreducible representations of the
axial symmetry group C mv . This problem has, in essence, been solved when
we ascertained the classification of the electron terms of a diatomic molecule
having this symmetry C aav (i.e. when the two atoms are different). To the
terms 0+ and 0~ (with Q. = 0) there correspond two one-dimensional
irreducible representations : the unit representation A\ and the representation
A2, in which the base function is invariant under all rotations and changes
sign under reflections in planes a Vy while to the doubly degenerate terms with
Q, = 1, 2, ... there correspond two-dimensional representations denoted by
Ei, E2, .... Under a rotation through an angle <f> about the axis, the base
functions are multiplied by e ±iQ ^, while on reflection in planes a v they change
into each other.
The irreducible representations of the group D^ h = C oav xC i are ob-
tained at once from those of the group C oav (and correspond to the classifi-
cation of the terms of a diatomic molecule composed of like nuclei).
If we take half-integral values for Q., the functions e ±in ^ give two-valued
§99 Two-valued representations of finite point groups 367
irreducible representations of the group C^ v , corresponding to the terms of
the molecule having half-integral spin.f
§99. Two-valued representations of finite point groups
To the states of a system with half-integral spin (and therefore half-
integral total angular momentum) there correspond two-valued represen-
tations of the point symmetry group of the system. This is a general property
of spinors, and therefore holds for both continuous and finite point groups.
The necessity thus arises of finding the two-valued irreducible representa-
tions of finite point groups.
As we have already remarked, the two-valued representations are not
really true representations of a group. In particular, the relations discussed
in §94 do not apply to them, and where all irreducible representations were
considered in these relations (for example, in the relation (94.15) for the sum
of the squared dimensions of the irreducible representations), only the true
one-valued representations were meant.
To find the two- valued representations, it is convenient to employ the
following artifice (H. Bethe 1929). We introduce, in a purely formal
manner, the concept of a new element of the group (denoted by Q) ; this is a
rotation through an angle of 2tt about an arbitrary axis, and is not the unit
element, but gives the latter when applied twice : Q 2 = E. Accordingly,
rotations C n about the axes of symmetry of the «th order will give identical
transformations only after being applied 2w times (and not n times) :
C n n = Q, C n *" = E. (99.1)
The inversion /, being an element which commutes with all rotations,
must give E as before on being applied twice. A twofold reflection in a plane,
however, gives Q, not E:
g 2 = Q, c* = E; (99.2)
this follows, since the reflection can be written in the form a h = IC % . As a
result we obtain a set of elements forming some fictitious point symmetry
group, whose order is twice that of the original group ; such groups we shall
call double point groups. The two-valued representations of the actual point
group will clearly be one-valued (i.e. true) representations of the correspond-
ing double group, so that they can be found by the usual methods.
The number of classes in the double group is greater than in the original
group (but not, in general, twice as great). The element Q commutes with all
f Contrary to the result for the three-dimensional rotation group, it would here be possible, by a
suitable choice of fractional values of CI, to obtain not only one- valued and two-valued representations,
but also those of three or more values. However, the physically possible eigenvalues of the angular
momentum, which is the operator of an infinitely small rotation, are determined by the representations
of the aforementioned three-dimensional rotational group. Hence the three (or more)-valued repre-
sentations of the two-dimensional rotation group (and of any finite symmetry group), though mathe-
matically determinate, are without physical significance.
368 The Theory of Symmetry §99
the other elements of the group,f and hence always forms a class by itself. If
the axis of symmetry is bilateral, the elements C n k and C n 2n ~ k = QC n n ~ k are
conjugate in the double group. Hence, when axes of the second order are
present, the distribution of the elements among classes depends also on
whether these axes are bilateral (in ordinary point groups this is unimportant,
since C 2 is the same as the opposite rotation C 2 -1 ).
Thus, for instance, in the group T the axes of the second order are equiva-
lent, and each of them is bilateral, while the axes of the third order are equiva-
lent but not bilateral. Hence the 24 elements of the double group J T' are
distributed in seven classes : E, Q, the class of three rotations C 2 and three
C 2 Q, and the classes 4C 3 , 4C 3 2 , 4C 3 £, 4C 3 2 £.
The irreducible representations of a double point group include, firstly,
representations which are the same as the one- valued representations of the
simple group (a unit matrix corresponding to both Q and E)\ secondly,
the two-valued representations of the simple group, a negative unit matrix
corresponding to Q. It is these latter representations in which we are now
interested.
The double groups C n ' (n = 1, 2, 3, 4, 6) and 5 4 ', like the corresponding
simple groups, are cyclic. || All their irreducible representations are one-
dimensional, and can be found without difficulty as shown in §95.
The irreducible representations of the groups D n ' (or C nv ', which are
isomorphous with them) can be found by the same method as for the cor-
responding simple groups. These representations are given by functions
of the form e ±ik ^ y where <j> is the angle of rotation about an axis of the nth
order, and k is given half-integral values (the integral values correspond to
the ordinary one-valued representations). Rotations about horizontal axes
of the second order change these functions into one another, while the rota-
tion C n multiplies them by e ±2 " ik / n .
It is a little less easy to find the representations of the double cubic groups.
The 24 elements of the group T" are divided among seven classes. Hence
there are altogether seven irreducible representations, of which four are
the same as those of the simple group T. The sum of the squared dimensions
of the remaining three representations must be 12, and hence we find that
they are all two-dimensional. Since the elements C 2 and C 2 Q belong to the
same class, x(C 2 ) = xiQQ) = ~ x(Q)> whence we conclude that x(Q) =
in all three representations. Next, at least one of the three representations
must be real, since complex representations can occur only in conjugate
pairs. Let us consider this representation, and suppose that the matrix of
the element C 3 is brought to diagonal form, with diagonal elements a 1} a^.
Since C 3 3 = Q, a x 3 = a 2 3 = — 1. In order that x(Cs) — a i+ a z ma y De rea l>
we must take a x = e ni / 3 , a 2 = r^/ 3 . Hence we find that x(C 3 ) = 1, x(Q a )
"f This is obvious for rotations and inversion; for a reflection in a plane, it follows since the reflec-
tion can be represented as the product of an inversion and a rotation.
J We distinguish the double groups by primes to the symbols for the ordinary groups.
|| The groups S 2 ' = C/, S 6 ' = C 8 ,', however, which contain the inversion J, are Abelian but not
cyclic.
§99 Two-valued representations of finite point groups 369
= a ^-\-a£ = — 1. Thus one of the required representations is obtained. By
comparing its direct products with the two complex conjugate one-dimensional
representations of the group T, we find the other two representations.
By means of similar arguments, which we shall not pause to give here, we
may find the representations of the group O'. Table 8 gives the characters
of the representations of the double groups mentioned above. Only those
representations are shown which correspond to two-valued representations
of the ordinary groups. The isomorphous double groups have the same
representations.
The remaining point groups are isomorphous with those we have con-
sidered, or else are obtained by direct multiplication of the latter by the group
C t , so that their representations do not need to be specially calculated.
For the same reasons as for ordinary representations, two complex con-
jugate two-valued representations must be regarded, from the physical
point of view, as one representation of twice the dimension. It is necessary
to pair one-dimensional two-valued representations even when they have
real characters. For (see §60) in systems with half-integral spin, complex
conjugate wave functions are linearly independent. Hence, if we have a
two- valued one- dimensional representationf with real characters (given
by some function if/), then, although the complex conjugate function i/j* is
transformed by the same representation, we can nevertheless see that ifj and
^r* are linearly independent. Since, on the other hand, the complex conjugate
wave functions must belong to the same energy level, we see that in physical
applications this representation must be doubled.
PROBLEM
Determine how the levels of an atom (with given values of the total angular momentum J)
are split when it is placed in a field having the cubic symmetry J O.
Solution. The wave functions of the states of an atom with angular momentum / and
various values Mj give a (2/+l)-dimensional reducible representation of the group O, with
characters determined by the formula (98.1). Decomposing this representation into irreduc-
ible parts (one-valued for integral / and two-valued for half-integral /), we at once find the
required splitting (cf. §96). We shall list the irreducible parts of the representations corres-
ponding to the first few values of J:
7=0 A 1
1/2 E{
1 *"i
3/2 G'
2 E+F 2
5/2 E 2 '+G'
3 A 2 +F 1 +F 2
t Such representations are found in the group C n ' for odd n; the characters are x(C„*) = (—!)*•
% For example, an atom in a crystal lattice. The presence or absence of a centre of symmetry in
the symmetry group of the external field is immaterial to this problem, since the behaviour of the
wave function on inversion (the parity of the level) is unrelated to the angular momentum J.
370
The Theory of Symmetry
§99
Table 8
Two-valued representations of point groups
A'
E
Q
C 2 <*>
C 2 <*>0
C 2 <*>
C 2 <">£>
c 2 <*>
C 2 < 2 >£>
E'
2
-2
*>»'
E
Q
c 3
c 3 2
c 3 p
3C/ 2
3C/ 2
.,|
1
1
-l
-l
-1
-1
1
1
i
— i
—i
i
E % '
2
-2
1
-1
D*
E
c 2
C 2 Q
c 3
CJQ
c 3 2
C 3 Q
c 6
C 6 6
c 6 5
C 6 £
3C/ 2
3C7 2
3C7' 2
3C7' 2
Ex'
2
-2
1
-1
V3
-V3
E % '
2
-2
1
-1
-V3
V3
E s '
2
-2
-2
2
d;
E
c 2
C 2
c 4
C7Q
C 4 3
C 4 Q
2£/ 2
2TJ\
2U\Q
E x '
2
-2
V2
-V2
E 2 '
2
-2
-V2
V2
T
E
4C,
4C 3 2
4C 3 Q
4C 3 2
3C 2
3C 2 Q
E'
2
-2
1
-1
-1
1
°'l
2
-2
e
-e 2
— s
e 2
{
2
-2
e 2
— s
-e 2
s
O'
E
Q
4C 3
4C 3 *£>
4C 3 2
4C 3 £>
3C 4 2
3C 4 2
3C 4
3C 4 3
3C 4 3
3C 4
6C 2
ec 2 Q
E^
2
-2
1
-1
V2
-V2
E t '
2
-2
1
-1
-V2
V2
G'
4
-4
-1
1
CHAPTER XIII
POLYATOMIC MOLECULES
§100. The classification of molecular vibrations
In its applications to polyatomic molecules, group theory first of all resolves
at once the problem of the classification of their electron terms, i.e. of the
energy levels for a given situation of the nuclei. They are classified according
to the irreducible representations of the point symmetry group appropriate
to the configuration of the nuclei. Here, however, we must emphasise what
is really obvious, that the classification thus obtained belongs to the definite
nuclear configuration considered, since the symmetry is in general destroyed
when the nuclei are displaced. We usually discuss the configuration cor-
responding to the equilibrium position of the nuclei. In this case the classi-
fication continues to possess a certain amount of meaning even when the
nuclei execute small vibrations, but of course becomes meaningless when
the vibrations can no longer be regarded as small.
In the diatomic molecule this question did not arise, since its axial sym-
metry is of course preserved under any displacement of the nuclei. A
similar situation occurs for triatomic molecules also. The three nuclei
always lie in a plane, which is a plane of symmetry of the molecule. Hence
the classification of the electron terms of the triatomic molecule with respect
to this plane (wave functions symmetric or antisymmetric with respect to
reflection in the plane) is always possible.
For the normal electron terms of polyatomic molecules there is an empirical
rule according to which, in the great majority of molecules, the wave function
of the normal electron state is completely symmetrical (this rule, for diatomic
molecules, has already been mentioned in §78). In other words, the wave
function is invariant with respect to all the elements of the symmetry group
of the molecule, i.e. it belongs to the unit irreducible representation of the
group.
The application of the methods of group theory is particularly significant
in the investigation of molecular vibrations (E. Wigner 1930). Before
beginning a quantum-mechanical investigation of this problem, a purely
classical discussion of the vibrations of the molecule is necessary, in which
it is regarded as a system of several interacting particles (the nuclei).
A system of N particles (not lying in a straight line) has 3N— 6 vibrational
degrees of freedom; of the total number of degrees of freedom 3N, three
correspond to translational and three to rotational motion of the system as
a wholef. The energy of a system of particles executing small vibrations
t See Mechanics, §§23, 24. If all the particles lie in a straight line, the number of vibrational degrees
of freedom is 3N— 5 ; in this case, only two co-ordinates correspond to rotation, since it is meaningless
to speak of the rotation of a linear molecule about its axis.
13 371
372 Polyatomic Molecules §100
can be written
E = * Be "^"fc+i 5 kikUiUk > (100. 1)
where m ik , k ik are constant coefficients, and the u t are the components of
the vector displacements of the particles from their equilibrium positions
(the suffixes *, k denumerate both the components of the vector and the
particles). By a suitable linear transformation of the quantities u t , we can
eliminate from (100.1) the co-ordinates corresponding to translational motion
and rotation of the system, and take the vibrational co-ordinates in such a
way that both the quadratic forms in (100.1) are transformed into sums of
squares. Normalising these co-ordinates so as to make all the coefficients
in the expression for the kinetic energy unity, we obtain the vibrational energy
in the form
E = i s QJ+i S <o a 2 f QJ. (100.2)
The vibrational co-ordinates Q ai are said to be normal; the co a are the fre-
quencies of the corresponding independent vibrations. It may happen that
the same frequency (which is then said to be multiple) corresponds to several
normal co-ordinates ; the suffix a to the normal co-ordinate gives the number
of the frequency, and the suffix i = 1, 2, ... ,/ a numbers the co-ordinates
belonging to a given frequency (J^ being the multiplicity of the frequency).
The expression (100.2) for the energy of the molecule must be invariant
with respect to symmetry transformations. This means that, under any
transformation belonging to the point symmetry group of the molecule, the
normal co-ordinates Q ai , i = 1, 2, ... ,f a (for any given a) are transformed
into linear combinations of themselves, in such a way that the sum of the
squares £ Q ai 2 remains unchanged. In other words, the normal co-ordinates
i
belonging to any particular eigenfrequency of the vibrations of the molecule
give some irreducible representation of its symmetry group ; the multiplicity
of the frequency determines the dimension of the representation. The
irreducibility follows from the same considerations as were given in §96 for
the solutions of Schrodinger's equation. The equality of the frequencies
corresponding to two different irreducible representations would be an
improbable coincidence. An exception is again formed by the irreducible
representations with complex conjugate systems of characters. Since the
normal co-ordinates are by their physical nature real quantities, two complex
conjugate representations correspond physically to one eigenfrequency of
twice the multiplicity.
These considerations enable us to carry out a classification of the eigen-
vibrations of a molecule without solving the complex problem of actually
determining its normal co-ordinates. To do so, we must first find (by the
method described below) the representation given by all the vibrational
co-ordinates together, which we shall call the total representation; this
§100 The classification of molecular vibrations 373
representation is reducible, and on decomposing it into irreducible parts we
determine the multiplicities of the eigenfrequencies and the symmetry
properties of the corresponding vibrations. Here it may happen that the
same irreducible representation appears several times in the total represen-
tation; this means that there are several different frequencies of the same
multiplicity and with vibrations of the same symmetry.
To find the total representation, we start from the fact that the characters
of a representation are invariant with respect to a linear transformation of the
base functions. Hence they can be calculated by using as base functions not
the normal co-ordinates, but simply the components u t of the vectors of the
displacements of the nuclei from their equilibrium positions.
First of all, it is evident that, to calculate the character of some element
G of a point group, we need consider only those nuclei which (or, more exactly,
whose equilibrium positions) remain fixed under the given symmetry trans-
formation. For if, under the rotation or reflection G in question, nucleus 1
is moved to a new position, previously occupied by a similar nucleus 2,
this means that under the operation G a displacement of nucleus 1 is trans-
formed into a displacement of nucleus 2. In other words, there will be no
diagonal elements in the rows of the matrix G ik which correspond to this
nucleus (i.e. to its displacement u { ). The components of the displacement
vector of a nucleus whose equilibrium position is not affected by the operation
G, on the other hand, are evidently transformed into combinations of them-
selves, so that they may be considered independently of the displacement
vectors of the remaining nuclei.
Let us first consider a rotation C{<j>) through an angle <£ about some sym-
metry axis. Let u x , u y , u z be the components of the displacement vector
of some nucleus, whose equilibrium position is on the axis, and hence is
unaffected by the rotation. Under the rotation these components are
transformed, like those of any ordinary (polar) vector, according to the for-
mulae (the #-axis being the axis of symmetry)
u 'x = u x cos<£+#„ sin<£,
u' y = —u x sm<f>-\-u v cos(f> t
u\ =11,.
The character, i.e. the sum of the diagonal terms of the transformation matrix,
is 1+2 cos<f>. If altogether N c nuclei lie on the axis in question, the total
character is
N c (l+2cos<f>). (100.3)
However, this character corresponds to the transformation of all the 3iV
displacements u t ; hence it is necessary to separate the part corresponding to
the transformations of translation and (small) rotation of the molecule as a
whole. The translation is determined by the displacement vector U of the
centre of mass of the molecule ; the corresponding part of the character is
374 Polyatomic Molecules §100
therefore 1+2 cos^. The rotation of the molecule as a whole is determined
by the vector 8SI of the angle of rotation.f The vector 8£l is axial, but with
respect to rotations of the co-ordinate system an axial vector behaves like
a polar vector. Hence a character of 1 +2 cos <j> also corresponds to the vector
8€l. Altogether, therefore, we must subtract from (100.3) a quantity
2(1 +2 cos<£). Thus we finally have the character x(C) of the rotation C(<f>)
in the total vibrational representation :
X (C) = (7Vc-2)(l+2 cos<£). (100.4)
The character of the unit element is evidently just the total number of
vibrational degrees of freedom: x( E ) = 3JV-6 (as is obtained from (100.4)
when N c = N, <f> = 0).
In an exactly similar manner, we calculate the character of the rotary-
reflection transformation S(<f>) (a rotation through an angle <f> about the
^-axis and a reflection in the xy-plane). Here a vector is transformed accord-
ing to the formulae
u' x = u x cos0+Mj, sin<£,
u' y = — u x sm(f>+Uy COS(f>,
u'z = — u z,
to which there corresponds a character — 1+2 cos <f>. Hence the character
of the representation given by all the 3AT displacements u i is
iV,s(-l+2cos<£), (100.5)
where N s is the number of nuclei left unmoved by the operation S(<j>) ; this
number is evidently either none or one. To the vector U of the displacement
of the centre of mass there corresponds a character —1+2 cos<f>. The vector
8SI, being an axial vector, is unchanged by an inversion of the co-ordinate
system ; on the other hand, the rotary-reflection transformation S(<f>) can be
represented in the form
S(cf>) = C(<f>)a h = C{4>)C 2 I = C(tt+0)/,
i.e. as a rotation through an angle 7T+0, followed by an inversion. Hence
the character of the transformation S(<f>) applied to the vector 8SI is equal
to the character of the transformation C{tt-\-(J>) applied to an ordinary vector,
i.e. it is 1 +2 cos (it +<£) = 1 — 2 cos <f>. The sum ( — 1 +2 cos <f>) +(1 —2 cos cf>)
= 0, so that we reach the conclusion that the expression (100.5) is equal to
the required character x(S) of the rotary-reflection transformation S((f>) in
the total representation :
x (5) = Ns(-l+2 cos<£). (100.6)
In particular, the character of reflection in a plane (<£ = 0) is x( CT ) = N ai
while that of an inversion (<f> = rr) is xCO = — 3AT 7 .
f As is well known, the angle of a small rotation can be regarded as a vector 8SI, whose modulus is
equal to the angle of rotation and which is directed along the axis of rotation in the direction deter-
mined by the corkscrew rule. The vector 8£l so denned is clearly axial.
§100 The classification of molecular vibrations 375
Having thus determined the characters x °f tne total representation, we
have only to decompose it into irreducible representations, which is done by
formula (94.14) and the character tables given in §95 (see the Problems at the
end of the present section).
To classify the vibrations of a linear molecule there is no need to have
recourse to group theory. The total number of vibrational degrees of freedom
is 3iV— 5. Among the vibrations, we must distinguish those in which the
atoms remain in a straight line, and those where this does not happen, f The
number of degrees of freedom in the motion of N particles in a straight line
is N; of these, one corresponds to the translational motion of the molecule
as a whole. Hence the number of normal co-ordinates of the vibrations
which leave the atoms in a straight line is N— 1; in general, N— 1 different
eigenfrequencies correspond to them. The remaining (3iV — 5)— (N— 1)
= 22V — 4 normal co-ordinates relate to vibrations which destroy the col-
linearity of the molecule; to these, there correspond N—2 different double
frequencies (two normal co-ordinates, corresponding to the same vibrations
in two mutually perpendicular planes, belong to each frequency). J
PROBLEMS
Problem 1. Classify the normal vibrations of the molecule NH 3 (an equilateral triangular
pyramid, with the N atom at the vertex and the H atoms at the corners of the base; Fig. 41).
Solution. The point symmetry group of the molecule is C Sv . Rotations about an axis
of the third order leave only one atom (N) fixed, while reflections in planes each leave two
atoms fixed (N and one H). From formulae (100.4), (100.6) we find the characters of the
total representation :
E lCo 3 g„
6 2
Decomposing this representation into irreducible parts, we find that it contains the repre-
sentations A x and E twice each. Thus there are two simple frequencies corresponding to
vibrations of the type A u which conserve the complete symmetry of the molecule (what are
called totally symmetric vibrations), and two double frequencies with corresponding normal
co-ordinates which are transformed into combinations of each other by the representation E.
Problem 2. The same as Problem 1, but for the molecule H 2 (Fig. 42).
Solution. The symmetry group is C 2v . The transformation C 2 leaves the O atom fixed ;
the transformation o„ (a reflection in the plane of the molecule) leaves all three atoms fixed ;
f If the molecule is symmetrical about its centre, a further characteristic of the vibrations appears ;
see Problem 10 at the end of this section.
J Using the notation for the irreducible representations of the group C KV (see §98), we can say
that there are N— 1 vibrations of the type A lt and N — 2 of the type £^.
376
Polyatomic Molecules
§100
Fig. 42
the reflection a' v leaves only the O atom fixed. The characters of the total representation are
E Co a„ a'.,
3 13 1
This representation divides into the irreducible representations 2A X , \B lt i.e. there are two
totally symmetric vibrations and one with the symmetry given by the representation B 1 ; all
the frequencies are simple. Fig. 42 shows the corresponding normal vibrations.
Problem 3. The same as Problem 1, but for the molecule CHC1 3 (Fig. 43a).
Solution. The symmetry group of the molecule is C 3v . By the same method we find
that there are three totally symmetric vibrations A x and three double vibrations of the type E.
Problem 4. The same as Problem 1 , but for the molecule CH 4 (the C atom is at the centre
of a tetrahedron with the H atoms at the vertices; Fig. 43b).
Solution. The symmetry of the molecule is T d . The vibrations are 1A U IE, 2F 2 .
Problem 5. The same as Problem 1, but for the molecule C 6 H 6 (Fig. 43c).
Solution. The symmetry of the molecule is D th . The vibrations are 2A ig , lA zg , lA iU ,
lB ig , lBiu, \B %g , 3B2U, lExg, oEiu, ^E^g, 2E 2 U-
Problem 6. The same as Problem 1, but for the molecule OsF 8 (the Os atom is at the
centre of a cube with the F atoms at the vertices; Fig. 43d).
Solution. The symmetry of the molecule is O^. The vibrations are lA ig , XA^u, \Eg,
1E U , 2i ? 1 u> 2F 2 g, 1jF2U.
Problem 7. The same as Problem 1 , but for the molecule UF « (the U atom is at the centre
of an octahedron with the F atoms at the vertices; Fig. 43e).
Solution. The symmetry of the molecule is Of,. The vibrations are lA ig , \E g , 2F 1U ,
§100
The classification of molecular vibrations
» H H
377
(a)
(d)
y
y
F.
Os
o
V
y
A
(g)
Fig. 43
Problem 8. The same as Problem 1, but for the molecule QHg (Fig. 43 f).
Solution. The symmetry of the molecule is D M . The vibrations are ZA xg , 1A 1U , 2A iU ,
"SEg, 2Eu-
Problem 9. The same as Problem 1, but for the molecule C 2 H 4 (Fig. 43g; all the atoms
are coplanar).
Solution. The symmetry of the molecule is D 2h . The vibrations are 3A ig , IA 1U , 2B ig ,
\B 1U , 2B au , 1B 2 g, 2B 2U ', the axes of co-ordinates are taken as shown in the figure.
Problem 10. The same as Problem 1, but for a linear molecule of N atoms symmetrical
about its centre.
Solution. To the classification of the vibrations of a linear molecule considered in the
text, we must add the classification from the behaviour with respect to inversion in the centre.
There are two distinct cases, according as JV is even or odd.
If N is even (JV = 2p), there is no atom at the centre of the molecule. On giving to the
p atoms in one half of the molecule independent displacements along the line, and to the re-
maining p atoms equal and opposite displacements, we find that p of the vibrations leaving
the atoms in line are symmetrical with respect to the centre, while the remaining (2p — 1) — p =
p—\ vibrations of this type are antisymmetrical. Next, p atoms have 2p degrees of freedom
for motions in which the atoms do not remain in line. On giving equal and opposite displace-
ments to symmetrically placed atoms, we should obtain 2p symmetrical vibrations; of these,
however, the two corresponding to a rotation of the molecule must be removed. Thus there
are p — 1 double frequencies of vibrations which bring the atoms out of line and are symmetri-
cal about the centre, and the same number [(2p— 2)— (p— 1) = p— 1] which are antisym-
metrical. Using the notation for the irreducible representations of the group D<x> h (see
378 Polyatomic Molecules §101
the end of §98), we can say that there are p vibrations of the type A ig and p—\ of the types
■A-xu, E\g, Em-
it N is odd (N = 2p + l), similar arguments show that there are p vibrations of each of the
types A ig , A 1U , E 1U and p — 1 of the type E ig .
§101. Vibrational energy levels
From the viewpoint of quantum mechanics, the vibrational energy of a
molecule is determined by the eigenvalues of the Hamiltonian
f f
&* = | S .2 4< 2 +i J aJ&QJ, (101.1)
where P^i = —^I^Qoci are tne momentum operators corresponding to the
normal co-ordinates Q ai . Since this Hamiltonian falls into the sum of in-
dependent terms K^t^+^Ga* 2 )* tne ener gy levels are given by the sums
ES* = h S co a S K,+i) = S /KK+iU (101.2)
where ^ a = S ^ ai , and / a is the multiplicity of the frequency a> a . The wave
functions are given by the products of the corresponding wave functions for
linear harmonic oscillators :
0=n0„ (101.3)
a
where «/r a = constant xexp{-|c a 2 S ^ ai 2 }n H v Jc„<Qaa)* (101.4)
where i/„ denotes the Hermite polynomial of order v, and c a = ^(wjh).
If there are multiple frequencies among the a> a , the vibrational energy levels
are in general degenerate. The energy (101.2) depends only on the sums
v a = S v ai . Hence the degree of degeneracy of the level is equal to the
number of ways of forming the given set of numbers v a from the v a{ . For a
single number v a it isj-
(f«+/«-i)W(/«-i)i.
Hence the total degree of degeneracy is
n —
(101.5)
>.!(/«-l)l
For double frequencies, the factors in this product are v x + l, while for triple
frequencies they are i(^ a +l)(^ a +2).
It must be borne in mind that this degeneracy occurs only so long as we
consider purely harmonic vibrations. When terms of higher order in the
normal co-ordinates are taken into account in the Hamiltonian {anharmonic
vibrations), the degeneracy is in general removed, though not completely
(see §104 for a further discussion of this point).
t This is the number of ways in which » a balls can be distributed among / a urns.
§101 Vibrational energy levels 379
The wave functions (101.3) belonging to the same degenerate vibrational
term give some representation (in general reducible) of the symmetry group
of the molecule. The functions belonging to different frequencies are
transformed independently of one another. Hence the representation given
by all the functions (101.3) is the product of the representations given by the
functions (101.4), so that we need consider only the latter.
The exponential factor in (101.4) is invariant with respect to all the
symmetry transformations. In the Hermite polynomials, the terms of any
given degree are transformed only into similar terms; a symmetry trans-
formation evidently does not change the degree of any term. Since, on the
other hand, each Hermite polynomial is completely determined by its highest
term, it follows that it is sufficient to consider only the highest term, writing
/«
T^HiJc&rt) = constant x& a M>e* r «' - Qocf /«'«+
+ terms of lower degree.
The functions for which the sum v a = 2 v ai has the same value belong
to the same term. Thus we have a representation given by the products of
v a quantities Q ai ; this is just the symmetric product (see §94) of the irreducible
representation given by the Q ai with itself v a times.
For one-dimensional representations, the finding of the characters of their
symmetric products with themselves v times is trivial :f
Xv (G) = [x(G)]».
For two- and three-dimensional representations it is convenient to use the
following mathematical device. J The sum of the squared base functions of
an irreducible representation is invariant with respect to all symmetry
transformations. Hence we can formally regard these functions as the com-
ponents of a vector in two or three dimensions, and the symmetry transfor-
mations as some rotations (or reflections) applied to these vectors. We
emphasise that there is in general no relation between these rotations and
reflections and the actual symmetry transformations, the former depending
(for any given element G of the group) also on the particular representation
considered.
Let us consider two-dimensional representations more closely. Let x(^)
be the character of some element of the group in the two-dimensional repre-
sentation concerned, with x(G) # 0. The sum of the diagonal elements of
the transformation matrix for the components x, y of a two-dimensional
vector on rotation through an angle <j> in a plane is 2 cos <f>. Putting
2cos<f> = X (G), (101.6)
we find the angle of the rotation which formally corresponds to the element
f We use the notation X v (&) * n place of the cumbersome [x c ] (G).
X It was applied to this problem by A. S. Kompaneets (1940).
380 Polyatomic Molecules §102
G in the irreducible representation considered. The symmetric product of
the representation with itself v times is the representation whose basis is
formed by the v + 1 quantities *", x^^y, ... , y v . The characters of this
representation aref
Xv (G) =sin(*>+l)#5in0. (101.7)
The case where x(G) = requires special consideration, since a zero charac-
ter corresponds both to a rotation through \n and to a reflection. If
x(G 2 ) = —2, we have a rotation through \tt, and for Xv(G) we obtain
X*(G) = — *[l + (— 1)']. (101.8)
If x(G 2 ) — 2, on the other hand, x(G) must be regarded as the character
of a reflection (i.e. a transformation x -> x, y -> — y) ; then
xv(G)=m+(-m- (ioi.9)
We can similarly obtain the formulae for the symmetric products of three-
dimensional representations. The finding of the rotation or reflection which
formally corresponds to an element of the group in a given representation is
easily accomplished with the aid of Table 7. This is the transformation which
corresponds to the given x(G) in that isomorphous group in which the
co-ordinates are transformed by the representation in question. Thus, for
the representation F t of the groups O and T d we must take a transformation
from the group O, but for the representation F 2 we must take one from the
group T d . We shall not pause here to derive the corresponding formulae for
the characters Xv(G)-
§102. Stability of symmetrical configurations of the molecule
For a symmetrical position of the nuclei, an electron term of the molecule
may be degenerate, if there are among the irreducible representations of the
symmetry group one or more whose dimensions exceed unity. We may ask
whether such a symmetrical configuration is a stable equilibrium configura-
tion of the molecule. Here we shall entirely neglect the effect of spin (if any),
since this effect is usually insignificant in polyatomic molecules. The degene-
racy of the electron terms of which we shall speak is therefore only the "orbi-
tal" degeneracy, and is unrelated to the spin.
If the configuration in question is stable, the energy of the molecule as a
function of the distances between the nuclei must be a minimum for the given
position of the nuclei. This means that the change in the energy due to a small
displacement of the nuclei must contain no terms linear in the displacements.
*|* For purposes of calculation it is convenient to take the base functions in the form
(x+iy) v , {x+iyY'^x—iy), ... , (x— iy) v ;
the matrix of the rotation is then diagonal, and the sum of the diagonal elements takes the form
§102 Stability of symmetrical configurations of the molecule 381
Let fi be the Hamiltonian of the electron state of the molecule, the
distances between the nuclei being regarded as parameters. We denote by i?
the value of this Hamiltonian for the symmetrical configuration considered.
The quantities defining the small displacements of the nuclei can be taken
as the normal vibrational co-ordinates Q^. The expansion of A in powers
of the Q ai is of the form
tf = # + S V a & ai +J i W 0liM Q 0li Q fik + ... . (102.1)
The expansion coefficients V, W, ... are functions only of the co-ordinates of
the electrons. Under a symmetry transformation, the quantities Q ai are
transformed into combinations of one another, and the sums in (102.1) are
changed into other sums of the same form. Hence we can formally regard
the symmetry transformation as a transformation of the coefficients in these
sums, the Q ai remaining unchanged. Here, in particular, the coefficients
V ai (for any given a) will be transformed by the same representation of the
symmetry group as the corresponding co-ordinates Q ai . This follows at
once from the fact that, by virtue of the invariance of the Hamiltonian under
all symmetry transformations, the group of terms of any given order in its
expansion must be invariant also, and in particular the linear terms must be
invariant.!
Let us consider some electron term E Q which is degenerate in the sym-
metrical configuration. A displacement of the nuclei which destroys the
symmetry of the molecule generally results in a splitting of the term. The
amount of the splitting is determined, as far as terms of the first order in
the displacements of the nuclei, by the secular equation formed from the
matrix elements of the linear term in the expansion (102.1),
V,
-g&iJlM^d* (102.2)
where tp p , tp a are the wave functions of electron states belonging to the
degenerate term in question (and are chosen to be real). The stability of the
symmetrical configuration requires that the splitting linear in Q should be
zero, i.e. all the roots of the secular equation must vanish identically. This
means that the matrix V pa must itself be zero. Here, of course, we must
consider only those normal vibrations which destroy the symmetry of the
molecule, i.e. we must omit the totally symmetric vibrations (which corres-
pond to the unit representation of the group).
Since the Q ai are arbitrary, the matrix elements (102.2) vanish only if
all the integrals
t Strictly speaking, the quantities F" a ,- must be transformed by the representation which is the com-
plex conjugate of that by which the Q a i are transformed. However, as we have already pointed out,
if two complex conjugate representations are not the same, they must physically be considered to-
gether as one representation of twice the dimension. The above remark is therefore unimportant.
382 Polyatomic Molecules §102
vanish. Let Z) el be the irreducible representation by which the electron
wave functions ip p are transformed, and D a the same for the quantities V ai ;
as we have already remarked, the representations D a are those by which the
corresponding normal co-ordinates Q ai are transformed. According to the
results of §97, the integrals (102.3) will be non-zero if the product
[£)(ei)2j xD a contains the unit representation or, what is the same thing, if
|7)(ei)2j contains D a . Otherwise all the integrals vanish.
Thus a symmetrical configuration is stable if the representation [Z> (el > 2 ]
does not contain any (except the unit representation) of the irreducible
representations D a which characterise the vibrations of the molecule. This
condition is always satisfied for non-degenerate electron states, since the sym-
metric product of a one- dimensional representation with itself is the unit
representation.
Let us consider, for instance, a molecule of the type CH 4 , in which one
atom (C) is at the centre of a tetrahedron, with four atoms (H) at the vertices.
This configuration has the symmetry T d . The degenerate electron terms
correspond to the representations E, F lt F 2 of this group. The molecule has
one normal vibration A x (a totally symmetric vibration), one double vibration
E, and two triple vibrations F 2 (see §100, Problem 4). The symmetric pro-
ducts of the representations E, F^ F 2 with themselves are
[£"] = A ± +E, [Ffl = [FJ] = A,+E+F 2 .
We see that each of these contains at least one of the representations E, F 2 ,
and hence the tetrahedral configuration considered is unstable when there
are degenerate electron states.
This result constitutes a general rule (H. A. Jahn and E. Teller 1937). A
more thorough investigation! than that given above, considering all possible
types of symmetrical configuration of the nuclei, shows that, when there is a
degenerate electron state, any symmetrical position of the nuclei (except when
they are collinear) is unstable. As a result of this instability, the nuclei
move in such a way that the symmetry of their configuration is destroyed, the
degeneracy of the term being completely removed. In particular, we can
say that the normal electron term of a symmetrical (non-linear) molecule
can only be non-degenerate.
As we have just mentioned, the linear molecules alone form an exception.
This is easily seen, without using group theory. A displacement of a nucleus
whereby it moves off the axis of the molecule is an ordinary vector with £
and r] components (the £-axis being along the axis of the molecule). We
have seen in §87 that such vectors have matrix elements only for transitions
in which the angular momentum A about the axis changes by unity. On the
other hand, to a degenerate term of a linear molecule there correspond states
with angular momenta A and —A about the axis (A ^ 1). A transition be-
tween them changes the angular momentum by at least 2, and therefore the
t See H. A. Jahn and E. Teller, Proceedings of the Royal Society A 161, 220, 1937.
§103 Quantisation of the rotation of a rigid body 383
matrix elements always vanish. Thus the linear position of the nuclei in the
molecule may be stable, even if the electron state is degenerate.
§103. Quantisation of the rotation of a rigid body
The investigation of the rotational levels of a polyatomic molecule is often
hampered by the necessity of considering the rotation simultaneously with
the vibrations. As a preliminary example, let us consider the rotation of a
molecule as a solid body, i.e. with the atoms "rigidly fixed".
Let £, r], I be a system of co-ordinates with axes along the three principal
axes of 'inertia of a rigid body, and rotating with it. The corresponding
Hamiltonian is obtained by replacing the components J ^ J v J^ of the angular
momentum of the rotation, in the classical expression for the energy, by the
corresponding operators :
g =W (ll+ll + Jl\ (103.1)
\ I A Ib 1C /
where I A , I B , I c are the principal moments of inertia of the body.
The commutation rules for the operators J^J V ,J C of the angular momen-
tum components in a rotating system of co-ordinates are not obvious, since
the usual derivation of the commutation rules relates to the components
Jx> Jv> Jz in a fixed system of co-ordinates. They are, however, easily
obtained by using the formula
(J.a)(J.b)-(J.b)(J.a) = -/J.axb, (103.2)
where a, b are any two commuting vectors which characterise the body in
question. This formula is easily verified by calculating the left-hand side
of the equation in the fixed system of co-ordinates *, y, z, using the general
rules for the commutation of angular momentum components with one
another and with the components of an arbitrary vector.
Let a and b be unit vectors along the £ and y axes. Then axb is a umt
vector along the £-axis, and (103.2) gives
UJ,-UJi = -ft- < 103 - 3 >
Two other relations are obtained similarly. Thus we reach the result that
the commutation rules for the operators of the angular momentum com-
ponents in the rotating system of co-ordinates differ from those in a fixed
system only in the sign on the right-hand side of the equation. Hence it
follows that all the results which we have previously obtained from the com-
mutation rules, relating to the eigenvalues and matrix elements, hold for
j.j y j also, with the difference that all expressions must be replaced by
their complex conjugates. In particular, the eigenvalues of / ? (as of J z )
are the integers k = —J, ...,/•
The finding of the eigenvalues of the energy of a rotating body (a top,
384 Polyatomic Molecules §103
as it is called) is simplest for the case where all three principal moments of
inertia of the body are equal: I A = I B = I C = I ( a spherical top). This
holds for a molecule in cases where it has the symmetry of one of the cubic
point groups. The Hamiltonian (103.1) takes the form
fi = frp/21,
and its eigenvalues are
E = &J(J+1)I2I. (103.4)
Each of these energy levels is degenerate with respect to the 2J+ 1 directions
of the angular momentum relative to the body itself (i.e. with respect to the
values of J%= A).|
There is also no difficulty in calculating the energy levels in the case where
only two of the moments of inertia of the body are the same: I A — I B ^ I
(a symmetrical top). This holds for molecules having one axis of symmetry of
order above the second. The Hamiltonian (103.1) takes the form
# = k 2 (j£ 2 +J v 2 )I2I A +H*Jfl2I c
= Wl2I A+ &(L-l>)]t. (103 . 5)
Hence we see that, in a state with given values of/ and k, the energy is
E .^-JU+Vi+wQ-Ly, (103.6)
which determines the energy levels of a symmetrical top.
The degeneracy with respect to values of k which occurred for a spherical
top is here partly removed. The values of the energy are the same only for
values of k differing in sign alone, corresponding to opposite directions of
the angular momentum relative to the axis of the top. Thus the energy
levels of a symmetrical top are (for k # 0) doubly degenerate.
For I A # I B # I c (an asymmetrical top), the calculation of the energy
levels in a general form is impossible. The degeneracy with respect to the
directions of the angular momentum relative to the body is here removed
completely, so that 2J+ 1 different non- degenerate levels correspond to any
given /. The calculation of these levels involves the solution of Schrod-
inger's equation in matrix form; this amounts to solving a secular equation
of degree 2J+ 1, formed from the matrix elements Hj\. with the given value
of/ (O. Klein 1929). The matrix elements Hj% are defined with respect to
the wave functions ip Jk of states in which the absolute value and the
^-component of the angular momentum have definite values (but the energy
has no definite value). On the other hand, in the stationary states of an
t Here and subsequently we ignore the (2/+l)-fold degeneracy with respect to the directions of
the angular momentum relative to a fixed co-ordinate system. This degeneracy always occurs and is
not physically important. If it is included, the total degree of degeneracy of the enenrv levels of a
spherical top is (27+ 1) 2 .
§103 Quantisation of the rotation of a rigid body 385
asymmetrical top the projection J^ of the angular momentum has, of course,
no definite values, i.e. no definite values of k can be assigned to the energy
levels.
The operators ] ^ J v have matrix elements only for transitions in which
k changes by unity, while J^ has only diagonal elements (see formulae (27.13),
in which we must write /, k instead of L, M). Hence the operators J^ 2 ,
J 2 , /j 2 , and therefore fi, have matrix elements only for transitions with
k -> k or &±2. The absence of matrix elements for transitions between
states with even and odd k has the result that the secular equation of degree
2y + l immediately falls into two independent equations of degrees J and
/ + 1. One of these contains matrix elements for transitions between states
with even k, and the other contains those for transitions between states
with odd k. Each of these equations, in turn, can be reduced to two equations
of lower degree. To do this, we must use the matrix elements defined, not
With respect to the functions iff J]e , but with respect to the functions
«A/fc + = (fo*+fo-*)/V 2 . / 103 7 n
for = (fo*-fo-*)/\/2 (**0);
the function ifi J0 is, of course, the same as ifi J0 +. Functions differing in the
index + and — are of different symmetry (with respect to a reflection in a
plane passing through the £-axis, which changes the sign of k), and hence
the matrix elements for transitions between them vanish. Consequently we
can form the secular equations separately for the + and — states.
The Hamiltonian (103.1), and the commutation rules (103.3), have an
unusual symmetry: they are invariant with respect to a simultaneous change in
sign of any two of the operators J^J V , J v This symmetry formally corres-
ponds to the group D 2 . Hence the levels of an asymmetrical top can be
classified in accordance with the irreducible representations of this group.
Thus there are four types of non-degenerate level, corresponding to the
representations A, B ly B 2 , B z (see Table 7).
It is easy to establish which states of the asymmetrical top belong to each
of these types. To do so, we must find the symmetry properties of the func-
tions (103.7), where the functions fo fc may be taken as simply the complex
conjugate functions to the eigenfunctions of the angular momentum (which
have the same symmetry; the complex conjugates are taken because of the
change in sign on the right-hand sides of the commutation relations (103.3)) :
where and <f> are spherical angles in £rj£-space. A rotation through an
angle it about the £-axis (i.e. the symmetry operation C 2 (0 ) multiplies this
function by (—1)*:
C 2 <0: <A/*^(-l)*fo*.
The operation C 2 ^> may be regarded as the result of successively per-
forming an inversion and a reflection in the |£-plane; the first operation
386
Polyatomic Molecules
§103
multiplies fok by (~l) J , and the second (a change in sign of <f>) is equivalent
to changing the sign of k. Using the definition (28.6) of the functions
©j,-fc> we therefore have
C 2 ( *>: <l>Jk->{-\) J+ Hj,-ic.
Finally, the operation C 2 ® = C2<'>C 2 ^> gives
Using these transformation rules, we find that the states corresponding
to the functions (103.7) belong to the following types of symmetry:
4>Jk
.+1
A
B!
B 2
\
$Jk <
(103.8)
' J even, k even
J even, k odd
J odd, k even
K J odd, k odd
f J even, k even i?i
/ even, k odd 2?2
y odd, k even ^4
/ odd, k odd U3
By simple counting it is easy to find the number of states of each type for
a given value of/. The following numbers of states correspond to the types
A and each of B h jB 2 , B3:
A Bi, B2, B$
*7+i
J even
u
(103.9)
J odd ij-i |7+|
Finally, we may add some remarks concerning the calculation of the matrix
elements of various physical quantities characterising the top (or molecule).
These are, of course, matrix elements with respect to the true rotational wave
functions of the stationary states of the molecule (its electron state and
vibrational state remaining unchanged).
The wave functions of the symmetrical top are essentially the same as the
angular part of the wave function of a diatomic molecule (with appropriate
changes in the notation for the quantum numbers) : if the rotation is de-
scribed by means of the Eulerian angles d, <f>, $ (Fig. 20, §58), the wave
function of the state with quantum numbers k, J, Mj is
fojMj = —e iM ->*eiW® lcJM (e),
Ltt
(103.10)
where @ are the functions calculated in §82, Problem.
The dependence of the matrix elements for a symmetrical top on the
quantum numbers / and Mj is given by the formulae derived in §87 for a
§103 Quantisation of the rotation of a rigid body 387
diatomic molecule (of zero spin). In formulae (87.1) — (87.4) K, Mr and A
must be replaced respectively by the angular momentum / of the top, its
projection Mj on the fixed .s-axis, and the projection k on the moving
£-axis.
The calculation of the matrix elements for the asymmetrical top is more
complex. The solution of the secular equation derived from the matrix
H J j\ gives linear combinations of the functions i/jjjc defined above which
diagonalise the Hamiltonian. Replacing ifij^ in these combinations by the true
wave functions (103.10) of the symmetrical top (with the same values of Mj
in all these functions), we obtain the true wave functions of the stationary
states of the asymmetrical top, described by values of the pair of numbers
J, Mj. As a result, the calculation of the matrix elements of any quantity for
the asymmetrical top reduces to that of the matrix elements for the sym-
metrical top, which are already known.
For the asymmetrical top there are selection rules for the matrix elements
of transitions between states of the types A, B lt B 2 , B z \ these rules are easily
obtained from symmetry considerations in the usual way. Thus, for the
components of a vector physical quantity A we have the selection rules
for ,4 g : A<->B 3 ®, B^^B^\
forA v : A++Bf», B^+^B^,
for At : A<-> B^>, Bf*<-> B^.
For clarity we show, as an index to the symbol for the representation, the
axis about which a rotation has the character +1 in the representation
concerned.
PROBLEMS
Problem 1. Calculate the matrix elements H J j\, for an asymmetrical top.
Solution. From formulae (27.13) we find
Ui%=Ur,% =*[/(/+ 1)"* 2 ]*
: ( / 2)*+2 =
'ft+2 we 'Jc w ' 'Jfe+2
w)L - wr = -w>l = -ujr*
for brevity, we everywhere omit the diagonal suffixes /, / of the matrix elements. Hence
we have for the required matrix elements of H:
1 1
h*= ^(j+j^) U(J+i)-k*]+hWi2i c ,
H l + z = H T =i h2 (^-J^Vi(J-k)(J-k-l)(J+k+l)(J+k+2)].
388
Polyatomic Molecules
§103
The matrix elements with respect to the functions (103.7) are expressed in terms of the
elements in formulae (1) by
#£ = #1- =#*(*#!). &± = &+&,, &-=&-&,
*+
1+
(fc+2)+ (fc+2)- fc+2 V r h 2+ V 2
(2)
PROBLEM 2. Determine the energy levels for an asymmetrical top with / = 1.
Solution. The secular equation, of the third degree, falls into three linear equations.
One of these is E x = H°+ whence
E ± = W
(i + r}
(3)
From this we can at once write down the other two energy levels, since it is obvious that the
three moments of inertia I a, Ib, Ic enter the problem in a symmetrical manner. Hence it is
sufficient simply to replace the moments of inertia I a, Ib once by 1 A , Ic and once by Ib, Ic-
Thus
E *= ih ih + r} E *= ih ir B + T}
W
The levels E u E 2 , E 3 belongf to the types B u B 2 , B 3 respectively, if I A , I B , I c are the moments
of inertia about the $, 17, £ axes.
Problem 3. The same as Problem 2, but for / = 2.
Solution. The secular equation, of the fifth degree, falls into three linear equations and
one quadratic. One of the linear equations is of the form E t = H2~, whence
2k 2 / 1 1 \
(5)
a level of the type B x . Hence we at once conclude that there must be two other levels, of the
types B 2 and B a :
2h 2 / 1 1 \
The equation of the second degree is
2h 2 / 1 1 \
E °=lu +ih iT B + T}
H°+-E FP+
0+ 0+
#2+
0+
H*+-E
2+
= 0.
(6)
Solving this, we obtain
** - *(H + ^) ± ^[(^rt) , - 3 Gs; H 7i + i)]- (7)
These levels belong to the type A.
Problem 4. The same as Problem 2, but for / = 3.
f This follows at once from considerations of symmetry. The energy E lt for instance, is symmetri-
cal with respect to the moments of inertia I a and Ib, and this property belongs to the energy of a state
whose symmetry about the £ and 17 axes is the same, i.e. a state of the type B\.
§104 Interaction between the vibrations and the rotation of the molecule 389
Solution. The secular equation, of the seventh degree, falls into one linear equation and
three quadratic. The linear equation is of the form E t = H%2, whence
/J_ 1 1\
\T A + T B + TJ'
E ^ 2h \T A + T B + T c y < 8)
a level of the type A. One of the quadratic equations is equation (6) of Problem 3, with a
different value of J. Its roots are
5h 2 /l 1\ h 2 ,r / 1 1\ 2 1 1 1 In
E ™ ^ T\Ta + T b ) + T c ± h *J L\T A ~T B ) + ^IaTb'JaTc'JbTcS^
a level of the type B^. The remaining levels are obtained from these by permuting I a, Ib
and Ic.
Problem 5. Determine the splitting of the levels of a system having a quadrupole
moment, in an arbitrary external electric field.
Solution. Taking as co-ordinate axes the principal axes of the tensor d 2 <f>/8xi8xjc (see
§75, Problem 4), we bring the quadrupole part of the Hamiltonian of the system to the form
8 = Ajj + Bjf + CjA A + B+C = 0.
Owing to the complete formal analogy between this expression and the Hamiltonian (103.1),
the problem under consideration is equivalent to that of finding the energy levels of an
asymmetrical top, the only difference being that here the sum of the coefficients A + B + C = 0,
and the angular momentum can have half-integral values also. For these the calculations must
be done afresh by the same method, but for integral J we can use the results of Problems 2
to 4, obtaining the following values for the energy displacement AE for the first few values
of J:
7=1: AE= -A,-B,-C;
7=3/2: AE = ± VP(^ 2 +5 2 +C 2 )/2];
7=2: AE = 3A,3B,3C,±^/[6(A2 + B2 + C2)].
For J = 3/2 the energy levels remain doubly degenerate, in accordance with Kramers'
theorem (§60).
§104. The interaction between the vibrations and the rotation of the
molecule
Hitherto we have regarded the rotation and the vibrations as independent
motions of the molecule. In reality, however, the simultaneous presence of
both motions results in a peculiar interaction between them (E. Teller,
L. Tisza and G. Placzek 1932-33).
Let us start by considering linear polyatomic molecules. A linear molecule
can execute vibrations of two types (see the end of §100): longitudinal vibra-
tions with simple frequencies and transverse ones with double frequencies.
We shall here be interested in the latter. A molecule executing transverse
vibrations has in general some angular momentum. This is evident from
simple mechanical considerations,-}- but it can also be shown by a quantum-
mechanical discussion. The latter also enables us to determine the possible
values of this angular momentum in a given vibrational state.
f For example, two mutually perpendicular transverse vibrations with a phase difference of in
can be regarded as a pure rotation of a bent molecule about a longitudinal axis.
390 Polyatomic Molecules §104
Let us suppose that some one double frequency a> a has been excited in
the molecule. The energy level with the vibrational quantum number v a
is (z; a +l)-fold degenerate. To this level there correspond the v a +l wave
functions
Kdfa, = constant x e-c^Q^+Q.^H.J^Q^H^^Q^),
where ^ a i+^ a2 = ^ a > or any independent linear combinations of them.
The total degree (in Q al and Q a2 together) of the polynomial by which the
exponential factor is multiplied is the same in all these functions, and is equal
to v a . It is evident that we can always take, as the fundamental functions,
linear combinations of the functions ift v v of the form
^ai^ai
*,*. = constant x e -ca'(Q ai J +Q a , 8 )/2 [{Q al +iQ^r^)l^Q^-iQ^^)l2 + ... j.
(104.1)
The square brackets contain a determinate polynomial, of which we have
written out only the highest term. / a is an integer, which can take the v a -\-l
different values v a , v a —2,v (X —4;..., —v^.
The normal co-ordinates Q al , Q a2 of the transverse vibration are two
mutually perpendicular displacements off the axis of the molecule. Under a
rotation through an angle cf> about this axis, the highest term of the poly-
nomial (and therefore the whole function ip v z ) is multiplied by
gi0(i> a +7 a )/2£-i<Mv a -Z a )/2 _ gi* a #.
Hence we see that the function (104.1) corresponds to a state with angular
momentum / a about the axis.
Thus we reach the result that, in a state where the double frequency a> a
is excited (with quantum number v a ), the molecule has an angular momentum
(about its axis) which takes the values
l« = »«, ^~ 2 > ®«-4» - . -»«• (104.2)
This is called the vibrational angular momentum of the molecule. If several
transverse vibrations are excited simultaneously, the total vibrational angular
momentum is equal to the sum S/ a . On being added to the electron orbital
angular momentum, it gives the total angular momentum / of the molecule
about its axis.
The total angular momentum / of the molecule cannot be less than the
angular momentum about the axis (just as in a diatomic molecule), i.e. J
takes the values
In other words, there are no states with J = 0, 1, ... , |/| — 1.
For harmonic vibrations, the energy depends only on the numbers v^ and
not on l a . The degeneracy of the vibrational levels (with respect to the
§104 Interaction between the vibrations and the rotation of the molecule 391
values of / a ) is removed by the presence of anharmonic vibrations. The
removal is not complete, however: the levels remain doubly degenerate, the
same energy belonging to states differing by a simultaneous change of sign of
all the / a and of /. In the next approximation (after that of harmonic motion),
a term quadratic in the angular momenta l a , of the form S g a /J fi (the g ufi
being constants), appears in the energy. This remaining double degeneracy
is removed by an effect similar to the A-doubling in diatomic molecules.
When we turn to non-linear molecules, we must first of all make the
following remark, which has a purely mechanical significance. For an arbi-
trary (non-linear) system of particles, the question arises how we can at all
separate the vibrational motion from the rotation; in other words, what we are
to understand by a "non-rotating system". At first sight it might be thought
that the vanishing of the angular momentum,
S mrxv = (104.3)
(the summation being over the particles in the system), could serve as a
criterion of the absence of rotation. However, the expression on the left-
hand side is not the complete derivative, with respect to time, of any function
of the co-ordinates. Hence the above equation cannot be integrated with
respect to time in such a way as to be formulated as the vanishing of some
function of the co-ordinates. This, however, is necessary if a reasonable
definition of the concepts of "pure vibrations" and "pure rotation" is to be
possible.
As a definition of the absence of rotation, we must therefore use the
condition
S mr xv = 0, (104.4)
where r are the radius vectors of the equilibrium positions of the particles.
Putting r = r +u, where u are the displacements in small vibrations, we
have v = f = u. The equation (104.4) can be integrated with respect to
time, giving
2wr xu=0. (104.5)
The motion of the molecule will be regarded as a combination of the purely
vibrational motion, in which the condition (104.5) is satisfied, and the
rotation of the molecule as a whole.f
Writing the angular momentum in the form
2 mrxv = S mr Xv+ 2 muxv,
we see that, in accordance with the definition (104.4) of the absence of
rotation, the vibrational angular momentum must be understood as the sum
Smuxv. However, it must be borne in mind that this angular momentum,
f The translational motion is supposed removed from the start, by choosing a system of co-ordinates
in which the centre of mass of the molecule is at rest.
392 Polyatomic Molecules §104
being only a part of the total angular momentum of the system, is not con-
served. Hence only a mean value of the vibrational angular momentum can
be ascribed to each vibrational state.
Molecules having no axis of symmetry of order above the second belong to
the asymmetrical-top type. In a molecule of this type, all the frequencies
are simple (their symmetry groups have only one-dimensional irreducible
representations). Hence none of the vibrational levels is degenerate. In any
non-degenerate state, however, the mean angular momentum vanishes (see
§26). Thus, in a molecule of the asymmetrical-top type, the mean vibra-
tional angular momentum vanishes in every state.
If, among the symmetry elements of the molecule, there is one axis of order
higher than the second, the molecule is of the symmetrical-top type. Such a
molecule has vibrations with both simple and double frequencies. The mean
vibrational angular momentum of the former again vanishes. To the double
frequencies, however, there corresponds a non-zero mean angular momentum
component along the axis of the molecule.
It is easy to find an expression for the energy of the rotational motion of the
molecule (of the symmetrical-top type), taking into account the rotational
angular momentum. The operator of this energy differs from (103.5) in that
the rotational angular momentum of the top is replaced by the difference
between the total (conserved) angular momentum J of the molecule and its
vibrational angular momentum J(») :
tfrot = — — (J- K> 2 +P 2 (^ i-XJs- Ja 2 - (104.6)
The required energy is the mean value H TO t. The terms in (104.6) which
contain the squared components of J give a purely rotational energy which is
the same as (103.6). The terms which contain the squared components of
J(f) give constants independent of the rotational quantum numbers and may
be omitted ; the terms which contain products of components of J and JW
constitute the interaction here considered between the vibrations of the
molecule and its rotation. This is called the Coriolis interaction (since it
corresponds to the Coriolis forces in classical mechanics). In averaging these
terms it must be borne in mind that the mean values of the transverse (£,17)
components of the vibrational angular momentum are zero. The energy of
the Coriolis interaction is therefore
Ecox=-h*kk v IIc, (104.7)
where the integer k is, as in §103, the component of the total angular momen-
tum along the axis of the molecule, and k v = 1^ is the mean value of
the component of the vibrational angular momentum for the vibrational
state concerned; k v , unlike k, is not an integer.
Finally, let us consider molecules of the spherical-top type. These include
molecules whose symmetry is that of any of the cubic groups. Such molecules
§104 Interaction between the vibrations and the rotation of the molecule 393
have simple, double and triple frequencies (there being one-, two- and three-
dimensional irreducible representations of the cubic groups). The degeneracy
of the vibrational levels is, as usual, partly removed by the presence of
anharmonic motion; when these effects have been taken into account there
remain, apart from the non-degenerate levels, only doubly and triply degene-
rate levels. Here we shall discuss these levels that are split by the presence of
anharmonic motion.
It is easy to see that, for molecules of the spherical-top type, the mean
vibrational angular momentum is zero not only in the non- degenerate
vibrational states but also in the doubly degenerate ones. This follows from
simple considerations based on symmetry properties. The mean angular
momentum vectors in two states belonging to the same degenerate energy
level must be transformed into each other in all possible symmetry trans-
formations of the molecule. None of the cubic symmetry groups, however,
allows the existence of two directions transformed only into each other;
only sets of three or more directions are so transformed.
From these arguments it follows that, in states corresponding to triply
degenerate vibrational levels, the mean vibrational angular momentum is
non-zero. After averaging over the vibrational state, this angular momentum
is represented by an operator whose matrix elements correspond to transitions
between three mutually degenerate states. In accordance with the number of
these states, this operator must have the form £l, where 1 is the operator of
an angular momentum of unity (for which 2/+ 1 = 3) and £ is a constant
characterising the vibrational level in question. The Hamiltonian of the
rotational motion of the molecule is
tfrot^WXJ-H
and, after averaging, becomes the operator
fc2 fe2-^ — ffi
# rot = -J2+ J<*>2_ £2j.i. (104.8)
21 21 I
The eigenvalues of the first term give the ordinary rotational energy (103.4);
the second term gives an unimportant constant, which does not depend on the
rotational quantum number. The last term in (104.8) gives the desired energy
of the Coriolis splitting of the vibrational level. The eigenvalues of the
quantity J . 1 are calculated in the usual way ; it has (for a given J) three
different values corresponding to the values /+1, /— 1, / of the vector
J + l. The result is
EcoT iJ+1) =-hHJII, ^
j? Cor w-i) = hH(J+ 1)//, (104.9)
Ecot (J) =h%II. J
394 Polyatomic Molecules §105
§105. The classification of molecular terms
The wave function of a molecule is the product of the electron wave func-
tion, the wave function of the vibrational motion of the nuclei, and the rota-
tional wave function. We have already discussed the classification and types
of symmetry of these functions separately. It now remains for us to examine
the question of the classification of molecular terms as a whole, i.e. of the
possible symmetry of the total wave function.
It is clear that, if the symmetry of all three factors with respect to some
transformation is given, the symmetry of the product with respect to that
transformation is determined. For a complete description of the symmetry of
the state, we must also specify the behaviour of the total wave function when
the co-ordinates of all the particles in the molecule (electrons and nuclei)
are inverted simultaneously. The state is said to be negative or positive,
according as the wave function does or does not change sign under this
transformation.-}-
It must be remembered, however, that the characterisation of the state
with respect to inversion is significant only for molecules which do not
possess stereoisomers. If stereoisomerism is present, the molecule assumes
on inversion a configuration which can by no rotation in space be made to
coincide with the original configuration; these are the "right-hand" and
"left-hand" modifications of the substance.J Hence, when stereoisomerism
is present, the wave functions obtained from each other on inversion belong
essentially to different molecules, and it is meaningless to compare them. ||
We have seen in §86 that, for diatomic molecules, the spin of the nuclei
exerts an important indirect effect on the arrangement of the molecular
terms by determining their degree of degeneracy, and in some cases entirely
forbidding levels of a certain symmetry. The same is true for polyatomic
molecules. Here, however, the investigation of the problem is considerably
more complex, and requires the application of the methods of group theory
to each particular case.
The idea of the method is as follows. The total wave function must con-
tain, besides the co-ordinate part (the only part we have considered so far), a
spin factor, which is a function of the projections of the spins of all the nuclei
on some chosen direction in space. The projection a of the spin of a nucleus
takes 2/+1 values (where i is the spin of the nucleus); by giving to all the
oj, a 2 , ... , a N (where N is the number of atoms in the molecule) all possible
values, we obtain altogether (2^ + 1) (2/ 2 +l) ... (2^ + 1) different values of
the spin factor. In each symmetry transformation, certain nuclei (of the same
kind) change places, and if we imagine the spin values to "remain fixed",
the transformation is equivalent to an interchange of spin values among
t We use the same customary, though unfortunate, terminology as for diatomic molecules (§86).
% For stereoisomerism to be possible, the molecule must have no symmetry element pertaining to
reflection (i.e. no centre of symmetry, plane of symmetry, or rotary-reflection axis).
|| Strictly speaking, quantum mechanics always gives a non-zero probability for the transition from
one modification to the other. This probability, however, which relates to the passage of nuclei
through a barrier, is so small that the phenomenon can always be neglected.
§105 The classification of molecular terms 395
the nuclei. Accordingly, the various spin factors will be transformed into
linear combinations of one another, thus giving some representation (in
general reducible) of the symmetry group of the molecule. Decomposing
this into irreducible parts, we find the possible types of symmetry for the
spin wave function.
A general formula can easily be written down for the characters x sp (^)
of the representation given by the spin factors. To do this, it is sufficient
to notice that, in a transformation, only those spin factors are unchanged in
which the nuclei changing places have the same a a \ otherwise, one spin
factor changes into another and contributes nothing to the character. Bearing
in mind that a a takes 24+1 values, we find that
Xs P (G0=II(2* a +l), (105.1)
where the product is taken over the groups of atoms which change places
under the transformation G considered (there being one factor in the product
from each group).
We are, however, interested not so much in the symmetry of the spin
function as in that of the co-ordinate wave function (by which we mean its
symmetry with respect to interchanges of the co-ordinates of the nuclei, the
co-ordinates of the electrons remaining unchanged). These two symmetries
are directly related, however, since the total wave function must remain
unchanged or change sign when any pair of nuclei are interchanged, accord-
ing as they obey Bose statistics or Fermi statistics (in other words, it must be
multiplied by ( — 1) 2 \ where i is the spin of the nuclei that are interchanged).
Introducing the appropriate factor in the characters (105.1), we obtain the
system of characters x(G) for the representation containing all the irreducible
representations by which the co-ordinate wave functions are transformed:
X (G) = n(2f.+ l)(-l)^».-« (105.2)
where n a is the number of nuclei in each group which change places under the
transformation in question. Decomposing this representation into irreducible
parts, We obtain the possible types of symmetry of the co-ordinate
wave functions of the molecule, together with the degrees of degeneracy of
the corresponding energy levels (here and later we mean the degeneracy with
respect to the different spin states of the system of nuclei).f
Each type of symmetry of the states is related to definite values of the
total spins of the groups of equivalent nuclei in the molecule (i.e. groups of
nuclei which change places under the various symmetry transformations of
the molecule). This relation is not one-to-one; each type of symmetry of
states can, in general, be brought about with various values of the spins of
equivalent groups. The relation can also be established, in any particular
case, by means of group theory.
t The degree of degeneracy of the level in this respect is often called its nuclear statistical weight;
see the last footnote to §86.
396 Polyatomic Molecules §105
As an example, let us consider a molecule of the asymmetrical-top type, the
ethylene molecule C^H^ (Fig. 43g), with the symmetry group D 2h . The
index to the chemical symbol indicates the isotope to which the nucleus
belongs; this indication is necessary, since the nuclei of different isotopes
have different spins. In this case, the spin of the H 1 nucleus is \, while the
C 12 nucleus has no spin. Hence we need consider only the hydrogen atoms.
We take the system of co-ordinates shown in Fig. 43g; the z-zxis is per-
pendicular to the plane of the molecule, while the #-axis is along the axis
of the molecule. A reflection in the ry-plane leaves all the atoms fixed,
while other reflections and rotations interchange the hydrogen atoms in
pairs. From formula (105.2) we have the following characters of the represen-
tation :
E <j(xy) o(xz) a{yz) I C 2 (x) C 2 (y) C 2 (z)
16 16 4 4 4 4 4 4
Decomposing this representation into irreducible parts, we find that it con-
tains the following irreducible representations of the group D ih : 7A g , 3B lg ,
3B 2u , 3B 3u . The figures show the number of times each irreducible represen-
tation appears in the reducible one; these numbers are also the nuclear
statistical weights of the levels with the corresponding symmetry.f
The classification of the states of the ethylene molecule thus obtained
relates to the symmetry of the total (co-ordinate) wave function, including
the electron, vibrational and rotational parts. Usually, however, it is of
interest to arrive at these results from a different point of view. Knowing
the possible symmetries of the total wave function, we can find at once which
rotational levels are possible (and with what statistical weights) for any
prescribed electron and vibrational state.
Let us consider, for instance, the rotational structure of the lowest vibra-
tional level (that for which the vibrations are not excited at all) of the normal
electron term, assuming that the electron wave function of the normal
state is completely symmetrical (as is the case for practically all polyatomic
molecules). Then the symmetry of the total wave function with respect to
rotations about the axes of symmetry is the same as the symmetry of the
rotational wave function. Comparing this with the results obtained above, we
therefore conclude that in the ethylene molecule the rotational levels of the
types A and B ± (see §103) are positive with statistical weights 7 and 3,
while those of the types B 2 and B3 are negative with statistical weight 3.
As with diatomic molecules (see the end of §86), owing to the extreme
weakness of the interaction between the nuclear spins and the electrons,
transitions between states of different nuclear symmetry in the ethylene
molecule do not occur in practice. Hence molecules in such states behave
like different modifications of the substance. Thus ethylene C 12 2 H 1 4 has
four modifications, with nuclear statistical weights 7, 3, 3, 3.
t The relation between the symmetry of states and the values of the total spin of the four hydrogen
nuclei in the ethylene molecule is derived in Problem 1.
§105 The classification of molecular terms 397
In reaching this conclusion it is important that states with different
symmetry belong to different energy levels (the intervals between which are
large compared with the interaction energy of nuclear spins). The conclusion
is therefore invalid for molecules in which there exist states of different
nuclear symmetry belonging to the same degenerate energy level.
Let us consider another example, the ammonia molecule N 14 H 1 3 , of the
symmetrical-top type (Fig. 41), whose symmetry group is C 3v . The spin of
the nucleus N 14 is 1, and that of H 1 is £. Using formula (105.2), we find the
characters of the representation of the group C 3v in which we are interested:
E 2Cq 3 <7„
24 6 -12
It contains the following irreducible representations of the group C 3v : 12A 2 ,
6E. Thus two types of level are possible; their nuclear statistical weights
aref 12 and 6.
The rotational levels of a symmetrical top are classified (for a given /)
according to the values of the quantum number k. Let us consider, as in the
previous example, the rotational structure of the normal electron and
vibrational state of the NH 3 molecule (i.e. we suppose the electron and vibra-
tional wave functions to be completely symmetrical). In determining the
symmetry of the rotational wave function, we must bear in mind that it is
meaningful to speak of its behaviour only with respect to rotations about axes.
Hence we replace the planes of symmetry by axes of symmetry of the second
order perpendicular to them, a reflection in a plane being equivalent to a
rotation about such an axis, followed by an inversion. In the present case,
therefore, instead of the group C 3v we have to consider the isomorphous point
group D 3 .
The rotational wave functions with k = ±\k\ are multiplied by e ±2lTiW / z
respectively under a rotation C 3 about a vertical axis of the third order,
while under a rotation U 2 about a horizontal axis of the second order they
change into each other, thus giving a two-dimensional representation of the
group D s . If \k\ is not a multiple of three, this representation is irreduc-
ible; it is E. The representation of the group C 3V corresponding to the total
wave function is obtained by multiplying the character x(U 2 ) by 1 or — 1,
according as the term is positive or negative. Since, however, in the repre-
sentation E we have x(^) = 0> we obtain the same representation E in
either case (but this time as a representation of the group C 3v , and not D 3 ).
Bearing in mind the results obtained above, we thus conclude that, when \k\
is not a multiple of three, both positive and negative levels are possible, with
nuclear statistical weights of 6 (the symmetry of the total co-ordinate wave
function being of the type E).
When \k\ is a multiple of three (but not zero), the rotational functions
t A total spin of the hydrogen nuclei of 3/2 corresponds to the terms of symmetry A 2 , and one of
1/2 to those of symmetry E.
398 Polyatomic Molecules §105
give a representation (of the group D 3 ) with characters
E 2C 3 3U 2
2 2
This representation is reducible, and divides into the representations A x , A%.
In order that the total wave function should belong to the representation A 2 of
the group C 3 „, the rotational level A x must be negative and A % positive.
Thus, when \k\ is a multiple of three and not zero, both positive and negative
levels are possible, with nuclear statistical weights of 12 (levels of the type
A 2 ).
Finally, only one rotational function corresponds to an angular momentum
component k = 0; it gives a representation with charactersf
E 2C, 3U 9
1 1 (~iy
If the total wave function has the symmetry A 2 , its behaviour with respect to
inversion must therefore be given by the factor ( — 1) J+1 . Thus, for k = 0,
levels with even and odd / can only be negative and positive respectively;
the statistical weight is 6 in either case (levels of the type A 2 ).
Summarising these results, we have the following table of possible states
for various values of the quantum number k for the normal electron and
vibrational term of the molecule N 14 H 1 3 (the symbols + and - denoting
positive and negative states).
+ -
| k | not a multiple of 3 6E 6E
| A | a multiple of 3 12^4 2 12^ 2
h _ n / J even _ 6A *
* ~ U \ J odd 6A 2
For given J and k, the energy levels of the NH3 molecule are in general
degenerate (see also the table for ND 3 in Problem 3). This degeneracy is
partly removed by a peculiar effect due to the flat shape of the ammonia
molecule and the small mass of the hydrogen atoms. By a fairly small vertical
displacement of the atoms in this molecule a transition can be brought about
between two configurations obtained from one another by a reflection in a
plane parallel to the base of the pyramid (Fig. 44). These transitions cause a
splitting of the levels, separating positive and negative levels (an effect
similar to the one- dimensional case considered in §50, Problem 3). The
magnitude of the splitting is proportional to the probability of passage of the
atoms through the "potential barrier" separating the two configurations of
the molecule. Although this probability is comparatively high in the ammonia
t On rotation through an angle w, the eigenfunction of the angular momentum with magnitude J
and component zero is multiplied by (— \) J .
§105 The classification of molecular terms 399
molecule, owing to the above-mentioned properties, the splitting is still
small (1 x 10- 4 eV).
An example of a molecule of the spherical-top type is discussed in
Problem 5.
C2(y)
C 2 (x)
1
1
2
2
PROBLEMS
Problem 1. Find the relation between the symmetry of the state of the C 12 2H 1 4 molecule
and the total spin of the hydrogen nuclei in the molecule.
Solution^ The total spin of the four H 1 nuclei can take the values / = 2, 1, 0, and its
component Mi takes values from 2 to —2. Let us consider the representations given by the
spin factors for each value of Mi, beginning with the largest.
The value Mi = 2 corresponds to only one spin factor, in which all the nuclei have a spin
component + i- The value Mi = 1 corresponds to four different spin factors differing as
regards the nucleus which has spin component -£. Finally, the value Mi = is given
by six spin factors, depending on the pair of nuclei which have spin components — \. The
characters of the three corresponding representations are as follows:
E a(xy) a(xz) o(yz) I Cz(z)
Mi = 2 1 1 1 1 1 1
Mi = \ 4 4
Mi = 6 6 2 2 2 2
The first of these representations is the unit representation A g ; since the value Mi = 2
can occur only for J = 2, we conclude that a state with symmetry A g corresponds to spin
1 = 2.
The value Mi = 1 can occur for both / = 1 and / = 2. Subtracting the first representa-
tion from the second and decomposing the result into irreducible parts, we find that states
Big, Bz u , Bzu correspond to spin 1 = 1.
Finally, the value Mi = can occur in all cases where Mi = 1 is possible, and also for
1 = 0. Subtracting the second representation from the third, we find two states Ag corres-
ponding to spin 1 = 0.
Problem 2. Determine the types of symmetry of the total (co-ordinate) wave functions,
and the statistical weights of the corresponding levels, for the molecules C 12 2 H%, C'^H 1 ^
N 14 2 14 4 (all these molecules are of the same form; the spins are i(H*) = 1, i(C 13 ) = £,
t(N 14 ) = 1).
Solution. By the method shown in the text for the molecule C^H^, we find the follow-
ing states (the axes of co-ordinates being taken the same as above) :
Molecule + —
C 12 2 H 2 4 27 A g , lSBig lSBzu, WBau
Ci3 2 H 1 4 16A g , \2Big \2Bzu, 2\Bzu
Ni4 2 oi 6 4 6A g 3i?3u
f A method of solving problems of this kind, based on the theory of permutation groups,
is given by E. G. Kaplan, Soviet Physics JETP 37(10), 747, 1960.
400 Polyatomic Molecules §105
Problem 3. The same as Problem 2, but for the molecule N 14 H a 3 .
Solution. In the way shown in the text for the molecule N 14 !! 1 ., we find the states 30.4,,
3A 2 , 2AE.
In the normal electron and vibrational state, the following terms are possible for various
values of the quantum number k :
\k\ not a multiple of 3
\k\ a multiple of 3
* = «{ JST
Problem 4. The same as Problem 2, but for the molecule C^H 1 , (see Fig. 43f; the
symmetry is D a d ).
Solution. The possible states are of the types 7A ig , 1A 1U , 3A ig , 13A iV , 9E g , 11E U .
In the normal electron and vibrational state, the following levels are obtained:
-r-
24£
2AE
30A U 3A 2
30A lt 3A %
30A t
3A 2
3A %
30A t
+
—
\k\ not a multiple of 3
9E g
UE U
\k\ a multiple of 3
* A\ g , 3A ig
1A 1U , 13A, 4
j J even
i /odd
7A ig
\A\u
3A 2g
13 A%u
k =
Problem 5. The same as Problem 2, but for the methane molecule C^Wi (the C atom
is at the centre of a tetrahedron with the H atoms at the vertices).
Solution. The molecule is of the spherical-top type, and has the symmetry T&- Follow-
ing the same method, we find that the possible states are of the types 5Az, IE, 3Fi (cor-
responding to a total spin of the molecule of 2, 0, 1 respectively).
The rotational states of a spherical top are classified according to the values of the total
angular momentum /. The 2/4-1 rotational functions belonging to a particular value of /
give a (2/ + l)-dimensional representation of the group O, which is isomorphous with the
group Tdl it is obtained from the latter by replacing all planes of symmetry by axes of the
second order perpendicular to them. The characters in this representation are given by
formula (98.1). Thus for example, for J = 3 we obtain a representation with characters
E 8C 3 6C 2 6C 4 3C 4 2
1 -1 _i _i
This contains the following irreducible representations of the group O: A if F u F 2 . Again
considering the rotational structure of the normal electron and vibrational term, we therefore
conclude that, for J = 3, the states with a symmetry Az of the total wave function can only
be positive, while those of type F x can be either positive or negative. For the first few values
of/ we thus obtain the following states (which we write together with their statistical weights) :
+ -
/ = — 5A 2
/=1
/ = 2
/ = 3
/ = 4
IE
IE, 3F X
5A 2 , 3F t
3F X
IE, 3F X
SA 2 , IE, 3F X
CHAPTER XIV
ADDITION OF ANGULAR MOMENTA
§106. 3/-symbols
The rule of addition of angular momenta deduced in §31 gives the possible
values of the total angular momenta of a system consisting of two particles
(or more complex components) with angular momenta j\ and j%.-\ This rule
is in fact closely related to the properties of wave functions with respect to
spatial rotations, and follows immediately from the properties of spinors
considered in Chapter VIII.
The wave functions of particles with angular momenta j± and j% are sym-
metrical spinors of ranks 2/i and 2/2, and the wave function of the system is
their product,
0(i) ht^mH^-. (106.1)
2/1 2/2
Symmetrising this product with respect to all the indices, we obtain a sym-
metrical spinor of rank 2(/i +72), corresponding to a state with total angular
momentum 71 +7*2. If we contract the product (106.1) with respect to one
pair of indices, of which one must belong to if/W and the other to tpW (since
otherwise the result is zero), the symmetry of each of the spinors j/t* 1 ) and
«/r( 2 > shows that it does not matter which indices are taken from A, /*, ... and
p, cr, ... . After symmetrisation we obtain a symmetrical spinor of rank
2(/i+/2—l), corresponding to a state with angular momentum j\ +J2— 1 •%
Continuing this process, we find, in agreement with the rule already known,
that/ takes values from j\ +j% to |/i— /2I, each occurring once.
For a complete solution of the problem of the addition of angular momenta,
we must also consider the problem of constructing the wave function of a
system with a given total angular momentum from those of its two component
particles.
t Strictly speaking, we shall always be considering (without explicitly mentioning the fact each
time) a system whose parts interact so weakly that their angular momenta may be regarded as con-
served in a first approximation.
All the results given below apply, of course, not only to the addition of the total angular momenta
of two particles (or systems) but also to the addition of the orbital angular momentum and spin of
the same system, assuming that the spin-orbit coupling is sufficiently weak.
% To avoid misunderstanding, the following comment is useful. The wave function of a system
of two particles is always a spinor of rank 2{ji-\-J2), and this is in general not equal to 2j, where / is
the total angular momentum of the system. Such a spinor may, however, be equivalent to a spinor of
lower rank. For example, the wave function of a system of two particles with angular momenta
/ 1 =ji = £ is a spinor of rank two ; but if the total angular momentum j = 0, this spinor is anti-
symmetrical, and therefore reduces to a scalar. In general, the total angular momentum / determines
the symmetry of the spinor wave function of the system : this is symmetrical with respect to 2j indices
and antisymmetrical with respect to the remainder.
401
402 Addition of Angular Momenta §106
Let us begin with the simple case of the addition of two angular momenta
to give a zero total angular momentum. Here we must evidently have
/i = j2 and angular momentum components m\ = — m^. Let tftj m be the
normalised wave functions of the states of one particle with angular momentum
j and component thereof m (in the non-spinor representation). The required
wave function To of the system is the sum of the products of the wave
functions of the two particles with opposite values of m :
T =
2 (-iy- m (1 W (2) y,-m, (106.2)
where/ is the common value of/i and 72- The factor preceding the sum is
due to the normalisation. The coefficients in the sum must all have the
same absolute value, since all values of the components m of the angular
momenta of the particles must be equally probable. The sequence of signs in
(106.2) is easily found by means of the spinor representation of the wave
functions. In spinor notation the sum in (106.2) is a scalar (the total angular
momentum of the system being zero)
^•"^V..* (106.3)
formed from two spinors of rank 2/. Using this, we find the signs in (106.2)
directly from (57.3).
It should be borne in mind, however, that in general only the relative
signs of the terms in the sum (106.2) are determinate, while the sign of the
whole sum may depend on the "order of addition" of the angular momenta.
For, if we lower all spinor indices (j+m ones and/— m twos) in j/^ 1 ) and
raise them in j/»( 2 >, the scalar (106.3) is multiplied by (— l) 2 ^, and therefore
changes sign when/ is half-integral.
Next we consider a system with zero total angular momentum consisting of
three particles with angular momenta /1, j'2, 7*3 and components thereof mi,
m2, m%. The condition for the total angular momentum to be zero is that
m>\ + W2 + m% = and j\, j'2, js have values such that each of them can be
obtained by vector addition of the other two, i.e. geometrically /1, j'2, jz must
be the sides of a closed triangle. In other words, each of them lies between
the difference and the sum of the other two :
|/i -/2I </3 <ji+J2, etc.
It is evident that the algebraic sum /1 4-/2 +/s is an integer.
The wave function of the system under consideration is the sum
Wo= 2 ( Jl J2 ^VW^m^W (106.4)
•*— ' \7Wi 7W2 mzt
WJl,Wl2,Wl3
taken over the values of each rm from — j\ toji. The coefficients in this formula
§106 3j-symbols 403
are termed Wigner 3j-symbols. By definition they are non-zero only if
W1 + W2 + W3 = 0.
When the suffixes 1, 2, 3 are permuted, the wave function (106.4) can
change only by an unimportant phase factor. The 3j-symbols can in fact be
defined as purely real quantities (see below), and then the indeterminacy of
To can consist only in its sign as a whole being indefinite (as is true of the
function (106.2) also). This means that interchanging the columns of a
3/-symbol can either leave it unchanged or change its sign.
The most symmetrical way of defining the coefficients in the sum (106.4),
which is the definition generally used for the 3/-symbols, is as follows. In
spinor notation, T is a scalar formed by contracting the product of the three
spinors j/fW^-, 0<2)*/«- f 0(3) V... w ith respect to all pairs of indices belonging
to two different spinors. In each pair belonging to particles 1 and 2 the spinor
index will be written superior with ifjM and inferior with 0< 2 ) ; in a pair belong-
ing to particles 2 and 3, superior with j/»< 2 > and inferior with </f< 3 > ; and in a pair
belonging to particles 3 and 1, superior with ifj® and inferior with 0W. It
is easily seen that the total number of pairs of each kind is/i +72 —7*3,72 +73 —ji,
71+73—72 respectively. This rule determines uniquely the sign of To.
It is evident that, with this definition, cyclic interchange of the indices
1, 2 and 3 leaves T unchanged. This means that the 37-symbol is unchanged
when its columns are cyclically permuted. Interchange of any two indices
is easily seen to require the raising of the lower indices and lowering of the
upper indices in all 71 +72 +73 pairs. This means that T is multiplied by
(_l)A+/2+i 3 ; in other words, the 37-symbols have the property
lh h h\ _ ( _ l)h+h+l ,(h h h\ etc _ (1065)
W2 mi mz/ wi mz m 3 )
i.e. they change sign when two columns are interchanged if7i+72+73 is odd.
Finally, we easily see that
/ h h h\ { _ 1)h+M /h h h\ {106 . 6)
\ — mi —mi —mz) Wi m% mz)
a change in the sign of the z- component of each angular momentum can
be regarded as the result of a rotation through an angle -n- about the jy-axis,
and this is equivalent to raising all the lower spinor indices and lowering all
the upper ones (see (58.11)).
From (106.4) we can derive an important formula which gives the wave
function Yj m of a system consisting of two particles and having given values
of 7 and m. To do so, we consider the particles 1 and 2 together as one system.
Since the angular momentum j of this system together with the angular
momentum J3 of particle 3 gives a total angular momentum of zero, we must
have/ = 73, m = —mz. According to (106.2) we can then write
To = J- ix 2 ( - 1 )'- w, *W (8) /. -m- (106.7)
14
404 Addition of Angular Momenta §106
This formula is to be compared with (106.4) (in which we replace jz, mz by
/, — m). Here, however, we must first take into account the fact that the
rule for constructing the sum in (106.7) according to (106.3) does not corres-
pond to the rule for constructing the sum (106.4): to bring (106.7) to the
form (106.4) we must, as is easily seen, interchange pairs of upper and lower
indices corresponding to particles 1 and 3. This leads to an additional factor
(-\)h-h+h. The result is
W jm = (-1)W.+«V(2/ + 1) y ( Jl 32 J VW"W (106.8)
*— ' wi t»2 — tn/
where the summation over m\ and m^ is subject to the condition m± + mi = m.
Formula (106.8) gives the required expression for obtaining the wave
function of a system from those of its two particles, which have definite
angular momenta j\ and }<i> It can be written in the form
Vjm= 2 0m tt) W (2) 'a»»2 (m 2 = m-m 1 ). (106.9)
mum* numa
The coefficients!
c L m w =(-iy^^v(2y+i)(- /1 n J ) (106.10)
form the matrix of the transformation from the complete orthonormal set of
(2/i+l)(2/2+l) wave functions of states with definite mi, mi to the similar
set with definite/, m (for given values oijijjz). As we know (see §12), such a
matrix is unitary. Hence we can immediately write down the inverse trans-
formation:
h+it
^W^y.-, = 2 cs 'T mzX¥ ^^^> ( 106 - n )
i=\h-h\ mimz
where we have also used the fact that the coefficients C are real. According
to the general rules of quantum mechanics, the squares of the coefficients in
the expansion (106.11) give the probability for the system to have any par-
ticular value of/ (for given /i, mi and 72, mi).
The unitarity of the transformation (106.9) means that its coefficients
satisfy certain orthogonality conditions. According to formulae (12.5) and
(12.6)
2 0m 0'm'
miwi2 minis
mi,t»2
t Called vector addition coefficients or Clebsch-Gordan coefficients. The notation Cj m,) 2 m 2 or
(jlJ2J T n\jljzmim2) is also used in the literature.
§106 3j-symbols 405
- (2;+i) 2 ( h h j )( hh f )
*—> \mi m 2 — ml \m\ m 2 —m I
2CJm Qim
miffls mi'rm
j,m
2mv( hh i )( h . h .
*—< \m± m 2 —ml \m± m% — ml
= 8 8 . (106.13)
JBlWll' 77127712'
The explicit general form of the 3j- symbols is quite lengthy. It can be
written asf
(h h h\ _ r O'i +h -h) l -(ji ~h +js)i( -ji +h +73)1 -1 1/2
Vmi m2 m3/ L ( J i -h/2 -+-J3 + 1 ) - J
x [(ji + wi)!(ji - mi)!(j 2 + m 2 )!(j2 - m 2 )\(js + m s)Kh ~ »*3)!] 1/2 x
( — \\z+h-h-m*
*!(ji +72 -;s - *)!(ji - w i - *)Ki2 + m 2 - ar)!(j3 -72 + «i + *)!(js -/1 - ^ 2 +*) !
(106.14)
The summation is over all integers z but, since the factorial of a negative
number is infinite, the sum contains only a finite number of terms. The
coefficient of the sum is obviously symmetrical in the suffixes 1, 2, 3; the
symmetry of the sum itself appears if the values of the summation variable
z are interchanged.
Besides the symmetry properties (106.5) and (106.6), which follow imme-
diately from the definition of the 3/-symbols, the latter also have other
symmetry properties, though the derivation of these is more complex and
will not be given here.J The properties in question can be conveniently
formulated in terms of a three-by-three array of numbers derived from the
parameters of the 3/-symbol as follows :
. . p/2 +73-.71 73+71-72 71 + 72-73-1
( ) = I h— m i j'2-m h—mz I; (106.15
\mi m 2 mz I I I
I— j'i 4- m-t Jo 4- mo 11 4- m* -■*
7l + »*l 72 + ^2 73 + W3
t The coefficients in (106.9) were first calculated by E. Wigner (1931). Their symmetry properties
and the symmetrical expression (106.14) were first derived by G. Racah (1942). The most direct
method of calculation is probably to go immediately from the spinor representation of *Fo (approp-
riately normalised) to the representation in the form of the sum (106.4) by means of the correspon-
dence formula (58.1); it may be noted that, since the coefficient in this formula is real, so also must be
the 3/-symbols. Another derivation is given by A. R. Edmonds, Angular Momentum in Quantum
Mechanics, Princeton 1957. The table of 3/-symbols given below is also taken from Edmonds' book.
{ See T. Regge, 72 nuovo cimento [10] 10, 544, 1958.
406 Addition of Angular Momenta §106
the sum of the numbers in each row and each column of this array is/i +72 +73«
Then (1) interchange of any two columns of the array multiplies the 3/-
symbol by (—\)h+U+h (the same property as that given by (106.5)); (2) the
same is true for interchange of any two rows (for the two lower rows, the same
property as that given by (106.6)); (3) the 3/-symbol is unchanged when the
rows and columns of the array are interchanged.
Some of the simpler formulae for particular cases will be given here. The
value
(3 3 °\ 1
r i = (-iy-» (106.16)
corresponds to formula (106.2). The formulae
fh h h+j
h/2 \
j = (-i)h-h
[(2ji)\(2J2)l(ii+J2 + mi + m 2 )Kh+3'2-mi-m 2 )\ ~| 1/2 ^ .„.
, (106.17)
(2/i + 2j 2 + l)!(ji + wn)!(ji - mi)!(; 2 + m 2 )!0' 2 - ^2)U
V/i — 71 — ms mzl
[(2/i) ! ( -h +32 +J3)Kh +h + m 3)Kh - *»s)! "] ]
( 71 +72 + 73 + 1)!( Jl -72 + /s)Kii + 72 -h)K ~/l +7*2 - "* 3 )!( j 3 + m 3 )! J
Vi -7i-
( 2 7i) ! ( -7*1 +7*2 +73)!(ii +72 + ^3)!(j3 - «s)! "1 1/2
. (jl +72 +73 + l)!(j'i -72 +73)Kii +7*2 -ja)K -jl +h ~ ™3)KJ3 + «b)L
are obtained directly from (106.14). The derivation of the formula
' (ji +72 -J3)K h-h+J3) K -j\ +72 +73)! ~| 1/2
(2?+ 1)1
/7i 72 73\ = r Qi+72-73)!0i-72+73)!(-7i+72+73)H ]
\0 0/ L (2/>+l)! J
p\
- , (106.18)
(p-h)KP-h)KP-33)\
where 2p = j\ +72 +73 is even, requires a number of additional calculations ;f
when 2p is odd, this 3/-symbol is zero owing to the symmetry property
(106.6).
Table 9 gives for reference the values of the 3/-symbols for 73 = ^, 1, § , 2.
For each js the minimum number of 3/-symbols are shown from which
the remainder may be obtained by means of the relations (106.5), (106.6).
Table 9
Formulae for 3j -symbols
(' +i '' *\ = (-nn-1/.r J '- m+i T
\ m -m-i i/ L(2/+1)(2;+2)J
(-D-f '' ')
\m —m—ms W3/
t See Edmonds' book already quoted.
§106 3j-symbols 407
\«8
n
V«3
y+i
^3
n
j+
j+4
2m
[2/(2/+l)(2/+2)]i/2
2(y+m+l)(y-m+l)-ji/ 2
r 2Q+m+ix;-m+in
;+ L(2/-+l)(2/-+2)(2y+3) J
F(/-m)(y+m+l)-| 1/2
2y(2y+i)(2y+2) J
_ r U-™)U- m + 1 ) 1 1/2
L(2y+i)(2y+2)(2y+3)J
(-ly-m+l^p 1 J ? \
\m —m—ms m%)
r /— m+h ~| 1/2
y+i -(y+3m+f)
3 2 U 2 l2/(2/+l(2y+2)(2y+3)J
2y(2y+i)(2y+2)(2y+3).
r 3(y-m+ixy-m+f)(y+m+§) -ji /2
' ' L (2y+i)(2y+2)(2y+3)(2y+4) J
ma I
_ r 3(y-m-|)(y-m+|)(y+m+|) -|i /2
L 2y(2y+i)(2y+2)(2y+3) J
. a r (y-m-i)0-m+|)(y-m+f) -ji/2
;+ " L (2y+i)(2y+2)(2y+3)(2y+4) J
(-D-f j 2 )
\W3
11 . 2[3m2-y(y+l)]
[(2y- 1)27(2; + l)(2y+2)(2y+ 3)F 2
;+ m L2y(2y+ l)(2/+2)(2/ + 3)(2/+4)J
r6(y+m+2)(y+w+l)(y-m+2)(y-w+l)-] 1 / 2
3+2 L (2y + 1)(2; + 2)(2j+ 3)(2y+ 4)(2/+ 5) J
408 Addition of Angular Momenta §107
^^3 1
1" 6(j+m+l)(j-m) -ji /2
m) l{2j- l)2j(2j+ l)(2/+2)(2/+3)J
n
-2 ( y + 2, + 2)r (i-^DO--^) _1 1/2
12/(2/+ l)(2/+2)(2/+3)(2/+4)J
3 L (2y+l)(2/+2)(2y+3)(2y+4)(2y+5) J
\w 3 2
;i
i+i
6(y-w-i)(y-w)(y+m+i)(y+m+2)-i 1 / 2
(2;-l)2/(2y+l)(2; + 2)(2; + 3) J
■(y-m-l)(y-m)(y-m+l)(y+m+2)"] 1 / 2
2y(2y+i)(2y+2)(2y+3)(2y+4)
[
L 2y(2y+i)(2y+2)(2y+3)(2y+4) J
• 2 r (y-m-l)(y-m)(y-m+l)(y-m+2) ni/2
;+ L (2/+l)(2; + 2)(2;+3)(2/+4)(2/+5) J
PROBLEM
Determine the angle dependence of the wave functions of a particle with spin J in states
with given values of the orbital angular momentum /, the total angular momentum j and
component thereof m.
Solution. The problem is solved by the general formula (106.8), in which ^'must
be taken as the eigenfunctions of the orbital angular momentum (i.e. the spherical harmonic
functions Yim t ), and t/r< 2 > as the spin wave function x( a ) (where a = ±£):
y, w = (-1)m^i/V(2;+1)T( l * 3 )Yi, m - (T x(<y).
*-~ ' \m—a a —ml
<r
Substituting the values of the 3/-symbols, we obtain
7j+m ij—tn
-TT-xCi) Yi, m -U2 + J —rx(-i) Yl,m+l/2,
lj—m+1 fj+m+1
*i- 1/a> „ = _^_— ^)y, >w _ 1/2+ ^__ x (- i) y ZiWl+1/2 .
§107. Matrix elements of tensors
In §29 formulae have been obtained which give the matrix elements of a
vector physical quantity in terms of the value of the angular momentum
component. These formulae are really a particular case of the corresponding
general formulae for an irreducible (see §58) tensor of any rank.f
t The analysis of the problems discussed in §§107-109, and most of the results given, are due to
G. Racah (1942-1943).
§107 Matrix elements of tensors 409
The set of 2k + 1 components of an irreducible tensor of rank k (an integer)
are equivalent, as regards their transformation properties, to a set of 2k +\
spherical harmonic functions Y* a , q = —k,...,k (see the last footnote to
§58). This means that, by means of appropriate linear combinations of the
components of the tensor, we can obtain a set of quantities which are trans-
formed under rotations as the functions Y k q. A set of such quantities, which
will be denoted here byfjcq, is called a spherical tensor of rank k.
For example, k = 1 for a vector, and the quantities f\ q are related to the
components of the vector by the formulae
_ i
/io = ia z , /i,±i = + — {a x ±ia y )\ (107.1)
V 2
cf. (58.2a). The corresponding formulae for a tensor of rank two are
/20 = — vINzz. h,±i= ±(a xz ±ia vz ),
H,±2 = — \{C-xx — ayy±2ia X y),
with a xx +a yy +a Z z — 0.
The construction of tensor products from two (or more) spherical tensors
fktfo fk t q t is effected in accordance with the rules for addition of angular
momenta, with k\, k% formally representing the "angular momenta" corres-
ponding to these tensors. Thus from two spherical tensors of ranks ki and
&2> one can form spherical tensors of ranks K = &i + &2, —, \ki — k%\ by means
of the formulae
(higk^KQ = £ C Qi9Jk 1 q 1 gk i
t t q 2
QuQi
1 2 J/^.*; (107.3)
*.*. qi q2 ~ Q}
cf. (106.9). The scalar product of two spherical tensors of the same rank k is,
however, usually denned as
(fkgk)00 = 2 (- \)^fkqgk,-q, (107.4)
which differs from the definition according to (107.3) with K = Q = by
the factor \/(2k+ 1); cf. (106.2).f This definition can also be written in the
form
(fkgk)00 = ^fkqgkq*
if we note that g kq * = ( - \) k ^g k -q (cf. (28.8))4
t If A and B are two vectors corresponding to the spherical tensors fi q and giq according to formulae
(107.1), then (/i#i)oo = A . B.
t It is assumed that the original irreducible tensors represent real physical quantities, so that only
the spherical harmonics in the fyq are complex.
410 Addition of Angular Momenta §107
The representation of physical quantities in the form of spherical tensors is
particularly convenient for the calculation of their matrix elements, since it
allows the direct application of the results of the theory of addition of angular
momenta.
By the definition of the matrix elements we have
hdnm= I UtXjL m '**'i'm>, (107.5)
n'j'm'
where the ijs n j m are the wave functions of the stationary states of the system,
described by its angular momentum /, component thereof m and the set of
the remaining quantum numbers n. As regards transformation properties,
the functions on the right-hand and left-hand sides of equation (107.5)
correspond to the respective sides of equation (106.11). Hence we imme-
diately obtain the selection rules: for given/, m the matrix elements are zero
except for values/', rri which satisfy the "angular momentum addition rule"
j' = j + k, with m' = m + q.-f
Next, it follows from the same transformation correspondence that the
coefficients in the sum (107.5) must be proportional to the coefficients in
(106.11). Accordingly we write the matrix elements in the form
{ha)lt m ' = »•*(-!)'»«-»'( f k 3 \h) n nT> ( 107 - 6 )
J \ — m q ml
where / m ax is the greater of/ and/', and the (fk)*'f 'are quantities independent
of m, m' and q, called "reduced" matrix elements. % This formula gives the
solution to the problem considered. In particular, for k = 1 we obtain
formulae (29.7) and (29.9) for the matrix elements of a vector quantity.
The operators fkq are related by
/* ff = (-l)fl-tf*,- fl +. (107.7)
The equation
V*X% M ' = (" l)^(A.-(r)??«' (1° 7 - 8 )
therefore holds for their matrix elements. Substituting (107.6) and using the
properties (106.5) and (106.6) of the 3/-symbols, we obtain for the reduced
matrix elements the "Hermitian" relation
<Ml? = ifr)$- (i° 7 - 9 )
The matrix elements of the scalar (107.4) are diagonal in/ and m. According
to the rule of matrix multiplication,
q n"j"m"
t Hence, in particular, there follow at once the rules given in §29 for the matrix elements of a
vector.
t The notation for these differs from that for the original matrix elements in that the angular
momentum components do not appear in the indices.
§107 Matrix elements of tensors 41 1
Substituting here the expressions (107.6) and effecting the summation over q
and m" by means of the orthogonality relation (106.12) for the 3/-symbols, we
obtain
ka&jook;? = ^ 2 (A)W-(»C- ( 107 - 10 )
n j
Similarly, we easily obtain the following formulae for the sums of the
squared matrix elements:
2 ic/w;;j,' w 'i 2 = ^ric/tt:/'! 2 . (io7.ii)
m,m' Z«+i
In the first of these the summation is over q and m' for a given value of m,
and in the second it is over m and m' for a given value of q (in every case,
m' = m + q).
For reference purposes we may consider the case where the quantities feq
are the spherical harmonic functions Y^q themselves, and calculate their
matrix elements for transitions between the states of one particle with integral
orbital angular momenta l\ and h. In other words, we have to calculate
the integrals
(Yi m )U™i = [Y *Y 7 Y lm do. (107.13)
Besides the selection rules corresponding to the rule of addition of angular
momenta (1 + 12 = li), there is also a rule for these matrix elements whereby
the sum /+/1 + /2 must be even. This is due to the conservation of parity,
according to which the product ( — l)*i+*a of the parities of the two states must
be the same as the parity (— 1)* of the physical quantity considered (see §30).
From (107.6) we have
(Y lM )l£i = (-1Y-.-** ( h l /2 Wi){;. (107.14)
2 2 \ — mi m mil 2
To determine the quantities (Yi) 1 ^ it is sufficient to calculate the matrix
element for m± = m^ = m = 0. The integral (107.13) reduces in this case
to the integral of the product of three Legendre polynomials \\
}Pi^)Pi^)Pi^) df* = 2 ^ Q J . (107.15)
t See E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, University Press, Cambridge
1931, p. 87.
412 Addition of Angular Momenta §108
The result is
(^,- = (±.)" + "( 00 JL ^ J , (107.16)
the upper and lower signs of i referring to the cases h ^ h and l\ ^ h
respectively.
§108. 6/-symbols
In §106 we have defined 3/-symbols as the coefficients in the sum (106.4)
which represents the wave function of a system of three particles with zero
total angular momentum. As regards the transformation properties under
rotations, this sum is a scalar. Hence it follows that the set of 3/-symbols
with given values of j\, j'2, jz (and all possible mi, m%, mz) may be regarded as
a set of quantities which are transformed under rotations according to a law
contragredient to that for the products ip jjmfl* Umjl* y 3 m 3 > so that the sum
as a whole is invariant.
From this viewpoint we may put the problem of constructing a scalar
consisting of 3/-symbols only. This scalar must depend only on the numbers
y, and not on the numbers m, which are altered by rotations. In other words,
it must be expressible in terms of sums over all the numbers m. Each such
sum consists of a "contraction" of the product of two 3/-symbols according
to the formula
y { _ iy -J J "\f J • -V (mi)
^—' \m . ./ \ — m . ./
TO
cf. the method of constructing the scalar (106.2).
Since in each "contraction" a pair of numbers m is involved, we must
consider products of an even number of 3/'-symbols in constructing the
complete scalar. The contraction of the product of two 3y-symbols gives,
owing to their orthogonality, the trivial result
2 th h h\( h h h\, „,,,,,
\mi ni2 mz/K — mi —m% —mz)
. j ( Ji h *y_u
m\,mi,m%
here we have used the equation mi + mz + mz = and formulae (106.6) and
(106.12). The smallest number of factors needed to form a non-trivial scalar
is therefore four.
In each 3/-symbol the three numbers j form a closed triangle. Since
each numbery must appear in the "contraction" in two 3/-symbols, it is clear
§108 bj-symbols 413
Fig. 45
that in the construction, of a scalar from the products of four 3/-symbols
there are six numbers/ forming the edges of an irregular tetrahedron (Fig. 45),
one face of which corresponds to each 3/-symbol. In defining the required
scalar it is customary to use a certain condition as regards the contraction
process, given by the formula
[*"|.7 ( .i)«wf h h j3 ) x
\hJ5Je) r- 1 \— «i — iii2 — ma/
all ?re
X
Ik h h\(H h M( h h h\
Wi — W5 W6/ W4 mi — me/ \ — W4 m^ mz/
The summation here is over all possible values of all the numbers m ; however,
since the sum of the three m in every 3/-symbol must be zero, only two of the
six m are in fact independent. The quantities defined by the formula (108.2)
are called dj-synibols or Racah coefficients.
From the definition (108.2), using the symmetry properties of the 3/-
symbols, we easily see that a 6/-symbol is unchanged by any permutation of
its three columns, and in any pair of columns the two numbers can be
simultaneously interchanged. Owing to these symmetry properties, the
sequence of numbers /i, . . ,j$ in the 6/-symbol can be put in 24 equivalent
forms.f In addition, the 6/-symbols have another, less evident, symmetry
property which states an equality between symbols with different sets of
numbers/: J
\ji hk) I j
f/i hih +h +h -h) Mj* +h +h -js)\
1/4 h y«i U Kj'2 +h +h -ja) iO'3 +h +h -hV
We may mention a useful relation between the 67- and 3/-symbols which can
t If we regard the tetrahedron in Fig. 45 as being regular, the 24 equivalent permutations of the
numbers j can be obtained by means of the 24 symmetry transformations (rotations and reflections)
of the tetrahedron.
t See T. Regge, II nuovo cimento [10] 11, 116, 1959.
414 Addition of Angular Momenta §108
be derived from the definition (108.2):
2 1 h h h\
\mi -ms m§/
m4,ms,mt
X
th h k\i k h h\ _ ih h »\ihhh\ (1084)
W4 m2 —me/K — m^ m§ m^J \nti m^ ms/[JAJ5J6)
The expression which is summed on the left-hand side of the equation differs
from that in (108.2) by the absence of one 3/-symbol. We can therefore say
that the sum in (108.4) is represented by the tetrahedron (Fig. 45) without
one of its faces; this determines the difference of the sum from a scalar.
In other words, as regards transformation properties it corresponds to one
3/-symbol, the one on the right-hand side of equation (108.4), to which it
must be proportional. The proportionality coefficient (the 6/-symbol on the
right-hand side of the equation) is easily found by multiplying both sides by
fh h h\
\m1m2m3
J and summing over the remaining numbers tn\, m^ m%.
The 67-symbols arise naturally in connection with the following problem
concerning the addition of three angular momenta.
Let three angular momenta j\, 72, 73 be added to give a resultant angular
momentum J. If the value of J (and of its component M) is given, the state
of the system is not yet uniquely determined, but depends also on the manner
of addition of the angular momenta (or, as we say, on their coupling scheme).
For example, let us consider two such coupling schemes: (1) first the
angular momenta j\ and 72 are added to give a total angular momentum j'12,
and then/12 and 73 are added to give the final angular momentum J; (2) the
angular momenta 72 and jz are added to give 723, and then 723 and 71 to give J.
The former scheme corresponds to states in which the quantity 712 (as well
asj\,J2,J3, J, M) has a definite value; their wave functions will be denoted
by T; 12 jm (omitting for brevity the repeated suffixes j\, j'2, 73)- Similarly, the
wave functions of the second coupling scheme are denoted by T/ 28 jm-
In both cases the values of the "intermediate" angular momentum (/12 or
723) are in general not unique, so that (for given J and M) we have two different
sets of wave functions differing in the values of 7*12 or 723- According to
the general rules, functions of these two sets are related by a certain unitary
transformation:
W h3 j M = S S jl2h y Jlz j M . (108.5)
It is evident from physical considerations that the coefficients in this
transformation are independent of the number M : they must be independent
of the orientation of the whole system in space. Thus they depend only on
§108 6j-symbols 415
the values of the six angular momenta /i, 72, J3,ji2,j23, J, not on their com-
ponents, i.e. are scalar quantities (in the sense defined above). The actual
calculation of these coefficients is easily effected as follows.
By a repeated application of formula (106.9) we find
mum*
mi,m,2,m3
mi,m2,m3
mi*
According to the general rules, the coefficients in the transformation (108.5)
are given by the integrals
s h.U* = J ^hzJM^hJM dq
= 7 ci M m ci M m c^z i2 C h3m * 3 >
mxA.tn, m * m * "^ M2 ™*™>
mi2,m23
here we have used the fact that the functions ifijm are orthonormal. The sum
on the right-hand side is taken for a fixed M, but the result is actually
independent of M (for the reason already mentioned). The summation can
therefore be extended over the values of M if the sum is multiplied by a factor
1/(27+ 1). Expressing the coefficients C in terms of 3/-symbols by (106.10),
we obtain the following expression :
S JlMt „ = (-l)M> + h + J V[{ 2j 12 + 1)(2; 23 + 1)]P 3 ) ;i2 }. (108.6)
\)Z J J23)
The relation between the 67-symbols and the transformation coefficients in
(108.5) makes it easy to derive some useful formulae for the sums of products
of 6/- symbols.
First of all, since the transformation (108.5) is unitary and its coefficients
are real, the relation
2»-+W+l)M{**'4-W (108.7)
i
holds.
Next, let us consider the three coupling schemes of three angular momenta,
with intermediate sums 712, 723 and 731 respectively. The coefficients (108.6)
of the corresponding transformations are related, according to the matrix
multiplication rule, by
416 Addition of Angular Momenta §108
Substituting here (108.6) and renumbering the suffixes we have
y (_ 1)W>+ , (2 / + 1)P * % k h {\~ l h h j % (108.8)
V VI 75 J I V 2 75 W U4 75 JQ)
3
Finally, by considering the various coupling schemes of four angular
momenta, we can derivef the following addition formula for the products of
three 6/-symbols:
2
(74 72 M(J2 7i 7sH74 73 75
(j© 78 7 A77 7 79^77 7*8 7
(-l)^(2i + l)( J4; . 27 f
1/9 78 7 J Ij . . ...
( U 19. H\ I 1R u iz\
(108.9)
i
_ (7*i 72 h\[k 7i 75
I74 75 h)\jl 7*8 79
(L. C. Biedenharn, and J. P. Elliott, 1953).
For reference we shall give some explicit formulae for the 6/-symbols.
In the general case, a 6/-symbol can be written as the following sum :
71 72 73l
\JA 75 JV
(-1)^+1)!
x N X
^ (* -7*1 -72 -7*3)K^ -7*1 -75 — 7*6)!(^ -7*4 -72 ->6)!(* -74 -75 -73)!
z
x _ , (108.10)
(ji +72 +74 +75 - z)Kh +73 +75 +76 - *)Kh +71 +76 +74 - *)*
where
r(a + b-c)\(a-b + c)\(-a + b+c)\ "I 1 / 2
A«=L («+»+«+!). J '
and the sum is taken overall positive integers z for which none of the factorials
in the denominator has a negative argument.
Table 10 gives the values of the 67- symbols for cases where one of the
parameters is 0, £ or 1.
Finally, we shall make a few remarks concerning the higher-order scalars
constructed from the 3y-symbols.
The next in complexity after the 67-symbols is a scalar formed by contract-
ing products of six 3/-symbols. These 3/-symbols contain 18 numbers j
equal in pairs, and so the resulting scalar depends on 9 parameters j. It
is called a 9j-symbol and is defined as follows J (E. Wigner 1951):
t See Edmonds' book quoted in §106.
J According to the general rule of contraction (108.1) it would be necessary to write the arguments
m in the last three 3./"-symbols with the minus sign and to include in the summand a factor (— l)S^ _m) .
However, by using the property (106.6) of the 3/-symbols and the fact that in this case, as is easily seen,
the sum 2w of all the nine numbers m is zero, we have the definition (108.11).
§108 (yj-symbols 417
Table 10
Formulae for 6j-symbols
(a b c\ (-l) s
(a b c\
(o c b)~
V[(26+l)(2c+l)]
s = a + b+c
a b c \ r (s-2b)(s-2c+l) -ji /2
i c _i h +%) L(2b+l)(2b+2)2c(2c+l)J
\i c _i j_|j v |_26(26+l)2c(2c+l)J
a 6 c \ J s(*+l)(*-2fl-l)(s-2a) -[i /2
1 c _i b-V ( ~ ^L (26- 1)26(26+ l)(2c-l)2<;(2c+l) J
a 6 c \ r 2{s+l)(s-2a)(s-2b)(s-2c+l) "l 1 / 2
1 c-1 b ] ~ ^ L26(26+l)(26 + 2)(2c-l)2c(2c+l)J
a 6 c | r(s-26-l)(s-26)(s-2<:+l)(s-2c+2)-|i /2
1 c-1 b+l) = ( ~ 1)S L(26 + 1)(26 + 2)(26 + 3)(2c-l)2c(2c+ 1) J
a b c \ 2[b(b+l) + c(c+l)-a(a+l)]
1 = (-l) s+1 —
[1 c b ) K ' [26(26+ l)(26 + 2)2c(2c+l)(2c+2)]i/2
fjll jl2 jl3
. . . I _ ^T (hi /l2 ;'l3 \ /m 722 .723 \ /J31 732 ]33 \
I Z—i Vmii mi2 m\z) \m%\ ni22 m^l W31 mz2 77*33/
K J31J32J3V allOT
/Jll 721 M\/jl2 ;*22 J32\/jl3 723 7*33 \
Vmii W21 W31/ \mi2 W22 ^32/ W13 ^23 W33/
This quantity can also be written as the sum of products of three 6/-symbols:
!/ll 7l2 713 \
• • • V, n , v1 . , x (ill J2l J3l\ (jl2 J22 J32\ (jl3 723 733^
J21J22J23 = )(-TO + l). . . . ..... .(108.12)
. . . Z-' U32 733 j ) I.721 ; ;23/ \j ju J12)
J3I J32 J33 J }
The equivalence of (I08.ll) and (108.12) can be seen by substituting in
(108.12) and the definition (108.2) and using the orthogonality properties of
the 3/-symbols.
The 9/-symbol has a high degree of symmetry, which follows directly from
the definition (I08.ll) and the symmetry properties of the 3/-symbols. It is
418 Addition of Angular Momenta §109
easily seen that when any two rows or columns are interchanged the 9/-symbol
is multiplied by ( — l) 2 ^'. Moreover, the 9/-symbol is unaltered by transposi-
tion, i.e. interchange of rows and columns.
Scalars of still higher orders depend on a still larger number of parameters
j. It is evident that this number will always be a multiple of three (3nj-
symbols). We shall not pause to discuss their properties, but merely mention
that for every n > 3 there is more than one type of 3nj- symbol, and these
do not reduce to one another. For example, there are two different types of
12/-symbol.t
§109. Matrix elements for addition of angular momenta
Let us again consider a system consisting of two parts (referred to as sub-
systems 1 and 2), and let/^ 1 * be a spherical tensor pertaining to sub-system
1. Its matrix elements with respect to the wave functions of this sub-system
are given, according to (107.6), by the formula
(ni'ji'nn'lfk^lmjmn) = i*(-l)ii.-~-»iY n Jl )(n 1 'j 1 '\f k ^\n 1 j 1 ). (109.1)
\ — 7»i q mj
The question arises of calculating the matrix elements of these quantities
with respect to the wave functions of the system as a whole. We shall show
how they may be expressed in terms of the same reduced matrix elements
as appear in the expression (109.1).
The states of the system as a whole are denned by the quantum numbers
hi J2i Ji M, «i, «2 (where J and M are the angular momentum and its com-
ponent for the whole system). Since fkq^ refers to sub-system 1, its operator
commutes with the angular momentum operator of sub-system 2. Its matrix
is therefore diagonal with respect to/2; it is also diagonal with respect to the
remaining quantum numbers n% of this sub-system. These indices ht n % WU< 1
be omitted, for brevity, and the required matrix elements will be written as
(tii'n'J'M'lh^ln^JM).
According to (107.6), their dependence on the number M is given by
{m'h'J'M'lftMn&JM) = i*(-l)'— -"' (_ M , ^jfa'h'J'lfiPln&J).
q (109.2)
To establish the relation between the reduced matrix elements on the right-
hand sides of (109.1) and (109.2) we write, using the definition of the matrix
t A more detailed account of the theory of 9/-symbols and of the properties of 3w/'-symbols is given
in Edmonds' book quoted in §106 and by A. P. Yutsis, I. B. Levinson and V. V. Vanagas, Mathe-
matical Apparatus of the Theory of Angular Momentum, Oldbourne Press, London 1963 (Mate-
maticheshii apparat teorii momenta kolichestva dvizheniya, Vilnius 1960).
§109 Matrix elements for addition of angular momenta 419
elements,
= 2 (-l).W 1+ M'-M V[(2 ^ +1)(2 y +1)] (^ * _g x
M!l,ffll'
(/l 7*2 J \
J («i';i'«i'|/*a (1) l n ih m i)-
m\ m2 —Ml
Substituting here (109.1) and (109.2) and comparing the resulting relation
with formula (108.4), we see that the ratio of the reduced matrix elements in
(109.1) and (109.2) must be proportional to a certain 6/-symbol. A careful
comparison of the two relations mentioned leads to the final formula
(ni'ji'J'\f k a) \nijiJ) = (-iy 1 . m «+^+^-+V[(2-/+l)(2i'+l)]x
xP' 7 ; 1("i7i1/fc (1) l"iji)> (109.3)
\J Ji k)
where 7'max is the greater of j\, ji, and / mln the smaller of J, J'. A similar
formula for the reduced matrix elements of the spherical tensor pertaining
to the second sub-system is
(n 2 'h'J'\fic {2) \n2i2J) = (-l)^-.-^— +»V[(2^+lX2^'4-l)]x
xP J Jl )(n2ti\f k ™\n2J2). (109.4)
\J J2 k)
The lack of complete symmetry between the expressions (109.3) and (109.4)
(in the exponent of - 1) is due to the dependence of the phase of the wave
functions on the order of addition of the angular momenta. The difference
must be borne in mind when calculating matrix elements for both sub-systems
simultaneously.
We shall also derive a useful formula for the matrix elements, with respect
to the wave functions of the whole system, of a scalar product (see the defini-
tion (107.4)) of two spherical tensors of the same rank k pertaining to different
sub-systems (and therefore commuting). According to (107.10), these matrix
elements are given in terms of the reduced matrix elements of each tensor
(with respect to the wave functions of the whole system) by
(nMjiWJMlWUfrtofalnwijzJM)
= — ^ (»iyi^lA m l»iJi^ WB^I/^lnj^T),
<J
where we have used the fact that the matrix of a quantity pertaining to one
420 Addition of Angular Momenta §109
sub-system is diagonal with respect to the quantum numbers of the other
sub-system. Substituting (109.3) and (109.4) and using the summation
formula (108.8), we obtain the desired formula expressing the matrix elements
of the scalar product in terms of the reduced matrix elements of each tensor
with respect to the wave functions of the corresponding sub-systems:
(niWjitiJM\(f k ay k M) 0Q \ ni n2JiJ2JM)
= (-l)h, ml » + j 2 , m „ + jf J2 3 ] y^'hy^ln^Xn^y^ln^). (109.5)
l* Jl J2 i
CHAPTER XV
MOTION IN A MAGNETIC FIELD
§110. Schrbdinger's equation in a magnetic field
In non-relativistic theory, a magnetic field can be regarded only as an
external field. The magnetic interactions between particles are a relativistic
effect, and a consistently relativistic theory is needed if they are to be taken
into account.
The Hamiltonian of a system of particles in a magnetic field can be obtained
from the classical expression for the corresponding Hamilton's function.
As is known from electrodynamics, this function is of the form
2 (pa— *aA a /c) 2
2m a
g-y "". " +Pfoy,»),
where p a is the generalised momentum of the ath particle, A the vector
potential of the magnetic field at the point occupied by this particle, and
U(x, y, z) the potential energy of the interaction of the particles (or their
energy in the external electric field).f For brevity, we shall write the formulae
below for a single particle.
If the particle has no spin, the transition to quantum mechanics can be
made in the usual manner ; the momentum p must be replaced by the operator
p = —ih\J, and we obtain the Hamiltonian
fi = ™ '-L+Ufay.z). (110.1)
2m
If, on the other hand, the particle has a spin, this procedure does not
suffice. This is because the spin interacts directly with the magnetic field.
In the classical Hamilton's function this interaction does not appear, since
the spin, which is a purely quantum effect, vanishes when we pass to the
limit of classical mechanics. To a particle with a spin we must also ascribe an
intrinsic magnetic moment ; from considerations of symmetry it is evident that
this moment is parallel or antiparallel to the spin. We write it in the form
pi = /*§/*, (110.2)
where I is the spin operator and p is a constant characterising the particle.
This form corresponds to the usual terminology, according to which the
t See, for instance, The Classical Theory of Fields, §16. We here denote the generalised momentum
by p and not by P.
421
422 Motion in a Magnetic Field §110
value fi, of the magnetic moment of a particle is taken to be the maximum value
of its projection, i.e. the value of fx z = po/s when a = s. The correct
expression for the Hamiltonian of the particle is obtained by adding to (1 10.1)
an additional term - p. . 34?, which corresponds to the energy of the magnetic
moment ji in the field 2tf. Thus the Hamiltonian of a particle having a spin
and in a magnetic field is
A = ®-eAlcy*l2m-(tils)i.jP+U(x,y,z). (110.3)
In expanding the square (p-eA/c)\ we must bear in mind that p does not
in general commute with the vector A, which is a function of the
co-ordinates. Hence we must write
tt = p 2 /2m-(e/2mc)(A.^+p.A)+ e 2A2/2mc2-( J[i /^§.^+C/. (110.4)
According to the rule (16.4) for the commutation of the momentum operator
with any function of the co-ordinates, we have
p.A-A.p = —ih divA. (110.5)
Thus p and A commute if div A = 0. This holds, in particular, for a homo-
geneous field, if its vector potential is expressed in the form
A=£jfxr. (110.6)
The equation fiifj = Eip for the eigenvalues of the operator (110.3) is
a generalisation of Schrodinger's equation to the case where a magnetic
field is present. The properties of the operator § have been considered in
detail in Chapter VIII. The wave functions on which the operator (110.3)
acts are symmetrical spinors of rank 2s.
The wave functions of the stationary states of a particle in a magnetic
field are not uniquely defined, because the choice of the vector potential is
not unique; the latter is defined only to within the gradient of an arbitrary
function/. If we make the substitution
A-*A+V/(*o'.*), (110.7)
the magnetic field is unaffected. It is therefore clear that the transformation
(110.7) cannot essentially change the eigenfunctions of the operator (110.3);
in particular, the squared modulus |^| 2 must remain unchanged. In fact,
it is easy to see that, on making together with (110.7) the substitution
ifj -> ift exp[i(elhc)f(x t y, *)], (1 10.8)
we return to the original equation.f Thus the phases of the wave functions
t If we are concerned with a system of several particles, the transformation of the wave function is
ifj -> i/j expi(t/hc) S e a f(x a ,y a , z a )},
the summation being over the particles forming the system.
§110 Schrodinger's equation in a magnetic field 423
of the stationary states are determined only to within an arbitrary function
of the co-ordinates. This non-uniqueness, however, does not affect any
quantity having a physical significance; the vector potential does not appear
explicitly in the definition of such a quantity.
In classical mechanics, the generalised momentum of a particle is related to
its velocity by
m\ = p— eA/c.
In order to find the operator v in quantum mechanics, we have to commute
the vector r with the Hamiltonian. A simple calculation gives the result
wv = p-eAjc, (110.9)
which is exactly analogous to the classical expression. For the operators of
the velocity components we have the commutation rules
{v x , %} = i(ehlm 2 c)J^ z , \
{%^}=#/« 2 ^, J (no.io)
{»«,»*} = i(ehlm 2 c)Jify, J
which are easily verified directly. We see that, in a magnetic field, the
operators of the three velocity components of a (charged) particle do not
commute. This means that the particle cannot simultaneously have definite
values of the velocity components in all three directions.
The constant fijsh gives the ratio of the intrinsic magnetic moment of the
particle to its spin angular momentum hs. As is well known, f for the ordinary
(orbital) angular momentum this ratio is ejlrnc. The coefficient of propor-
tionality between the intrinsic magnetic moment and the spin of the particle
is not the same. For an electron it is -\e\jmc, i.e. twice the usual value.! The
intrinsic magnetic moment of the electron (spin |) is consequently
Ho = \e\fylmc = 0-927X 10" 20 erg/gauss. (110.11)
This quantity is called the Bohr magneton.
The magnetic moment of heavy particles is customarily measured in
nuclear magnetons, defined as eh/lmc with m the mass of the proton. The
intrinsic magnetic moment of the proton is found by experiment to be
2-793 nuclear magnetons, the moment being parallel to the spin. The magnetic
moment of the neutron is opposite to the spin, and is 1 -913 nuclear magnetons.
In motion in a magnetic field, the symmetry with respect to time reversal
occurs only if the sign of the field 2tf (and of the vector potential A) is changed.
This means (see §§18 and 60) that Schrodinger's equation ify = ^ must
keep the same form when we take complex conjugates and change the sign of
t See The Classical Theory of Fields, §44.
J This value of jko can be obtained theoretically from DlRAC's relatmstic wave equation.
424 Motion in a Magnetic Field §111
3tf. This is immediately evident for all terms in the Hamiltonian (110.4)
except - s .Jtif. The term - s .2? i/j in Schrodinger's equation becomes
s* . J0?0* under the transformation in question, and at first sight this destroys
the required invariance, since the operator s* is not the same as -§. It must
be remembered, however, that the wave function is in reality a spinor j^-,
and on time reversal a contravariant spinor must be replaced by a covariant
one (see §60), so that in Schrodinger's equation the term - § . 3^^- is
replaced by §* . «3^ A/t .... It is easily seen by means of the definitions (57.4),
(57.5) that the result of the action of the operator §* on the components of
the covariant spinor has the opposite sign to that of the operator s on the
components of the contravariant spinor. The operation of time reversal
therefore leads to a Schrodinger's equation for the components A which
is of the same form as the original equation for the components j/t a <"-.
§111. Motion in a uniform magnetic field
Let us determine the energy levels of a particle in a constant uniform
magnetic field (L. D. Landau 1930). The vector potential of the uniform
field is conveniently taken here not in the form (110.6), but as
A x = -Jt?y, Ay =A z =0 (lll.l)
(the sr-axis being taken in the direction of the field).
The Hamiltonian (110.3) becomes
1 .ll + M
2m KJr * ' ' 2m 2m
H=~ (p x +e*yW+Z!-+£. G*/,)W. (111.2)
First of all, we notice that the operator s e commutes with 8, and the
coefficient of s z in (111.2) is a constant, independent of the co-ordinates.
From the first of these facts it follows that s z is conserved; from the second,
that the spin and co-ordinate variables in Schrodinger's equation are
separable. In other words, the eigenfunctions can be represented as products
of co-ordinate functions with spin functions which correspond to definite
values a of the component of the spin. For the co-ordinate function ifj we
have the equation
The operator (111.2) does not contain the co-ordinates x and z explicitly.
The operators p x and f> z therefore commute with i?, i.e. the x and z com-
ponents of the generalised momentum are conserved. We accordingly seek
if) in the form
tf, — fti imp x x+p z z) x (yj # (111.4)
The eigenvalues p x and/^ take all values from —00 to + 00. The momentum
§111 Motion in a uniform magnetic field 425
p z is related to the velocity by p z = mv z (see (110.9)). Thus the velocity of
the particle in the direction of the field can take any value; we can say that
the motion along the #-axis is "not quantised". The physical significance
of p x will be explained below.
Substituting (111.4) in (111.3), we obtain the following equation for the
function x(j) :
2ml Pz 2 SeJF\ 2 )
where we have introduced the notation
y — —cpxletf'.
This equation is formally identical with SchrSdinger's equation (23.6) for a
linear oscillator, oscillating with frequency co = \e\3^jmc about the point
y = y . Hence we conclude immediately that the constant E+(fijs) otf -
-\pf\m, which takes the part of the oscillator energy, can have the values
(n + %)hu), where n is any integer.
Thus we obtain the following expression for the energy levels of a particle
in a uniform magnetic field:
E = {n+^)\e\HJiflmc-\-p z 2 l2m-( H .js)a^'. (111.5)
The corresponding wave functions are (apart from a normalising constant)
$ = #/w^#e-^^S'™H»W(W*lch)(y-yo)], ( m - 6 )
where the H n are Hermite polynomials.
The energy given by the first term in (111.5) corresponds to the motion in
the ay-plane. In classical mechanics this is a motion in a circle about a fixed
centre. The quantity y , which is conserved, corresponds to the classical y
co-ordinate of the centre of the circle. The quantity x = cpJeJ^+x is
also conserved; it is easy to see that its operator commutes with the
Hamiltonian (111.2). This quantity x corresponds to the classical x co-
ordinate of the centre of the circle.f The operators £ and y , however, do
not commute. In other words, the co-ordinates x and jo cannot take definite
values simultaneously.
Since (111.5) does not contain the quantity p x , which assumes a continuous
sequence of values, the energy levels are continuously degenerate. Howeve