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THE MECHANICS OF THE ATOM

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THE MECHANICS OF
THE ATOM

MAX BORN

PROFESSOR IN THE UNIVERSITY OF GOTTINGEN
TRANSLATED BY

J. W. FISHER, B.Sc., PH.D.

AND REVISED BY

D. R. HARTREE, PH.D.

LONDON

G. BELL AND SONS, LTD.
1927

Printed in Great Britain by
NKILL & Co., LTD., EDINBURGH.

DEDICATED IN ALL GRATEFULNESS

TO
MR HENRY GOLDMAN

OF NEW YORK

A FRIEND OF GERMAN LEARNING
WHO STANDS ALWAYS READY TO AID

PREFACE TO THE GERMAN EDITION

THE title " Atomic Mechanics," l given to these lectures which I
delivered in Gottingen during the session 1923-24, was chosen to
correspond to the designation " Celestial Mechanics." As the latter
term covers that branch of theoretical astronomy which deals with
the calculation of the orbits of celestial bodies according to mechanical
laws, so the phrase " Atomic Mechanics " is chosen to signify that the
facts of atomic physics are to be treated here with special reference
to the underlying mechanical principles ; an attempt is made, in
other words, at a deductive treatment of atomic theory. It may
be argued that the theory is not yet sufficiently developed to justify
such a procedure ; to this I reply that the work is deliberately con-
ceived as an attempt, an experiment, the object of which is to ascer-
tain the limits within which the present principles of atomic and
quantum theory are valid and, at the same time, to explore the ways
by which we may hope to proceed beyond these boundaries. In
order to make this programme clear in the title, I have called the
present book " Vol. I " ; the second volume is to contain a closer
approximation to the " final " mechanics of the atom. I know that
the promise of such a second volume is bold, for at present we have
only a few hazy indications as to the departures which must be made
from the classical mechanics to explain atomic phenomena. Chief
among these indications I include Heisenberg's conception of the
laws of multiplets and the anomalous Zeeman effect ; some features
of the new radiation theory of Bohr, Kramers, and Slater, such as
the notion of " virtual oscillators " ; the subsequent advances of
Kramers towards a quantum theory of dispersion phenomena ; as
well as some general considerations, which I have recently published,
relating to the application of the theory of perturbations to the
quantum theory. This mass of material, however, in spite of its

1 The German title " Atommechanik " corresponds to the title "Himinels-
mechanik" (celestial mechanics); the title "Mechanics of the Atom" appeared,
however, preferable for this book, although, in the text, the clumsier expression
atomic mechanics has often been employed.

Vlll THE MECHANICS OF THE ATOM

extensive range, is not nearly enough for the foundation of a
deductive theory. The second volume may, in Consequence, remain
for many years unwritten. In the meantime let its virtual ex-
istence serve to make clear the aim and spirit of this work.

This book is not intended for those who are taking up atomic prob-
lems for the first time, or who desire merely to obtain a survey of the
theoretical problems which it involves. The short introduction, in
which the most important physical foundations of the new mechanics
are given, will be of little service to those who have not previously
studied these questions ; the object of this summary is not an intro-
duction to this field of knowledge, but a statement of the empirical
results which are to serve as a logical foundation for our deductive
theory. Those who wish to obtain a knowledge of atomic physics,
without laborious consultation of original literature, should read
Sommerfeld's Atombau und Spektrallinien. 1 When they have mas-
tered this work they will meet with no difficulties in the present
volume, indeed a great deal of it will be already familiar. The fact
that many portions of this book overlap in subject-matter with
sections of Sommerfeld's is of course unavoidable, but, even in these
portions, a certain difference will be discernible. In our treatment
prominence is always given to the mechanical point of view ; details
of empirical facts are given only where they are essential for the
elucidation, confirmation, or refutation of theoretical deductions.
Again, with regard to the foundations of the quantum theory, there
is a difference in the relative emphasis laid on certain points ; this,
however, I leave for the reader to discover by direct comparison. My
views are essentially the same as those of Bohr and his school ; in
particular I share the opinion of the Copenhagen investigators, that
we are still a long way from a " final " quantum theory.

For the fact that it has been possible to publish these lectureq in
book form I am indebted in the first place to the co-operation of my
assistant, Dr. Friedrich Hund. Considerable portions of the text have
been prepared by him and only slightly revised by me. Many points,
which I have only briefly touched on in the lectures, have been
worked out in detail by him and expounded in the text. In this con-
nection I must mention, in the first place, the principle of the unique-
ness of the action variables which, in my opinion, constitutes the basis
of the present-day quantum theory ; the proof worked out by Hund
plays an important part in the second chapter ( 15). Further, the
account of Bohr's theory of the periodic system, given in the third

1 English translation of third edition, 1923, by H. L. Brose, Methuen & Co., Ltd.,
London.

PREFACE TO THE GERMAN EDITION IX

chapter, has, for the most part, been put together by Hund. 1 also
wish to thank oth$r collaborators and helpers. Dr. W. Heisenberg
has constantly helped us with his advice and has himself contributed
certain sections (as, for example, the last on the helium atom) ; Dr.
L. Nordheim has assisted in the presentation of the theory of per-
turbations, and Dr. H. Kornfeld has verified numerous calculations.

MAX BORN.

G6TTINGEN, November 1924.

AUTHOR'S PREFACE TO THE
ENGLISH EDITION

SINCE the original appearance of this book in German, the mechanics
of the atom has developed with a vehemence that could scarcely
be foreseen. The new type of theory which I was looking for as
the subject-matter of the projected second volume has already
appeared in the new quantum mechanics, which has been developed
from two quite different points of view. I refer on the one hand
to the quantum mechanics which was initiated by Heisenberg, and
developed by him in collaboration with Jordan and myself in Ger-
many, and by Dirac in England, and on the other hand to the wave
mechanics suggested by de Broglie, and brilliantly worked out by
Schrodinger. There are not two different theories, but simply two
different modes of exposition. Many of the theoretical difficulties
discussed in this book are solved by the new theory. Some may
be found to ask if, in these circumstances, the appearance of an
English translation is justified. I believe that it is, for it seems to
me that the time is not yet arrived when the new mechanics can
be built up on its own foundations, without any connection with
classical theory. It would be giving a wrong view of the historical
development, and doing injustice to the genius of Niels Bohr, to
represent matters as if the latest ideas were inherent in the nature
of the problem, and to ignore the struggle for clear conceptions
which has been going on for twenty-five years. Further, I can
state with a certain satisfaction that there is practically nothing in
the book which I wish to withdraw. The difficulties are always
openly acknowledged, and the applications of the theory to empirical
details are so carefully formulated that no objections can l)e made
from the point of view of the newest theory. Lastly, I believe that
this book itself has contributed in some small measure to the
promotion of the new theories, particularly those parts which have
been worked out here in Gottingen. The discussions with my
collaborators Heisenberg, Jordan and Hund which attended the

Xii THE MECHANICS OF THE ATOM

writing of this book have prepared the way for the critical step
which we owe to Heisenberg. *

It is, therefore, with a clear conscience that I authorise the English
translation. It does not seem superfluous to remark that this book
is not elementary, but supposes the reader to have some knowledge
of the experimental facts and their explanation. There exist excel-
lent books from which such knowledge can easily be acquired. In
Germany Sommerfeld's Atombau und SpeJctrallinien is much used :
an English translation has appeared under the title Atomic Structure
and Spectral Lines. I should like also to direct attention to Andrade's
book, The Structure of the Atom, in which not only the theories but
also the experimental methods are explained.

I desire to offer my warmest thanks to Professor Andrade for
suggesting an English edition of my book. I also owe my thanks
to Mr. Fisher, who prepared the translation in the first place ;
Professor Andrade, Professor Appleton and Dr. Curtis, who read it
over ; and finally to Dr. Hartree, who revised the translation, read
the proof-sheets, and made many helpful suggestions for elucidating
certain points. I also offer my sincere thanks to the publishers for
the excellent manner in which they have produced the book.

MAX BORN.

GETTING EN, January 1927.

NOTE

THE chief departures from the German text which have been made
by Professor Born or with his approval are (1) some modifications in
1,2 concerning the mechanism of radiation, in view of the experi-
ments of Geiger and Bothe, and of Compton and Simon, (2) a modi-
fication of the derivation, on the lines suggested by Bohr, of the
Rydberg-Ritz series formula in 26, and (3) various alterations in
24 and 30-32, made in view of the development of ideas and the
additional experimental data acquired since the German edition was
written.

D. R. H.

CONTENTS

INTRODUCTION

PHYSICAL FOUNDATIONS

PAGE

& 1. DEVELOPMENT OF THE QUANTUM THEORY OF AN OSCILLATOR

FROM THE THEORY or RADIATION 1

2. GENERAL CONCEPTION OF THE QUANTUM THEORY ... 6

3. THE CONCEPTIONS OF ATOMIC AND MOLECULAR STRUCTURE . 12

FIRST CHAPTER

THE THEORY OF HAMILTON AND JACOBI

4. EQUATIONS OF MOTION AND HAMILTON'S PRINCIPLE. . . 17

5. THE CANONICAL EQUATIONS 20

6. CYCLIC VARIABLES 24

7. CANONICAL TRANSFORMATIONS 28

8. THE HAMILTON- JACOBI DIFFERENTIAL EQUATION . . . 36

SECOND CHAPTER
PERIODIC AND MULTIPLY PERIODIC MOTIONS

9. PERIODIC MOTIONS WITH ONE DEGREE OF FREEDOM . . 45
10. THE ADIABATIC INVARIANCE OF THE ACTION VARIABLES AND THE

QUANTUM CONDITIONS FOR ONE DEGREE OF FREEDOM . 52

llf THE CORRESPONDENCE PRINCIPLE FOR ONE DEGREE OF FREEDOM 60

12. APPLICATION TO ROTATOR AND NON-HARMONIC OSCILL YTOR . 63

13. MULTIPLY PERIODIC FUNCTIONS 71

14. SEPARABLE MULTIPLY PERIODIC SYSTEMS .... 76
15. GENERAL MULTIPLY PERIODIC SYSTEMS. UNIQUENESS OF THE

ACTION VARIABLES 86

16. THE ADIABATIC INVARIANCE OF THE ACTION VARIABLES AND THE

QUANTUM CONDITIONS FOR SEVERAL DEGREES OF FREEDOM . 95
17. THE CORRESPONDENCE PRINCIPLE FOR SEVERAL DEGREES -OF

FREEDOM 99

18. METHOD OF SECULAR PERTURBATIONS 107

19. QUANTUM THEORY OF THE TOP AND APPLICATION TO MOLECULAR

MODELS 110

20. COUPLING OF ROTATION AND OSCILLATION IN THE CASE OF

DIATOMIC MOLECULES 122

XVI THE MECHANICS OF THE ATOM

THIRD CHAPTER

r
SYSTEMS WITH ONE RADIATING ELECTRON

PA.GB

21. MOTIONS IN A CENTRAL FIELD OF FORCE . . . .130

22. THE KEPLER MOTION 139

23. SPECTRA OP THE HYDROGEN TYPE 147

24. THE SERIES ARRANGEMENT OP LINES IN SPECTRA NOT OF THE

HYDROGEN TYPE 161

25. ESTIMATES OF THE ENERGY VALUES OF THE OUTER ORBITS IN

SPECTRA NOT OF THE HYDROGEN TYPE .... 155

26. THE RYDBERG-RITZ FORMULA 161

27. THE RYDBERG CORRECTIONS FOR THE OUTER ORBITS AND THE

POLARISATION OF THE ATOMIC CORE 165

28. THE PENETRATING ORBITS 169

29. THE X-RAY SPECTRA 173

30. ATOMIC STRUCTURE AND CHEMICAL PROPERTIES . . .180

31. THE ACTUAL QUANTUM NUMBERS OF THE OPTICAL TERMS . 183

32. THE BUILDING UP OF THE PERIODIC SYSTEM OF THE ELEMENTS . 191

33. THE RELATIVISTIC KEPLER MOTION 201

34. THE ZEEMAN EFFECT 207

35. THE STARK EFFECT FOR THE HYDROGEN ATOM . . .212

36. THE INTENSITIES OF LINES IN THE STARK EFFECT OF HYDROGEN . 220
37. THE SECULAR MOTIONS OF THE HYDROGEN ATOM IN AN ELECTRIC

FIELD 229

38. THE MOTION OF THE HYDROGEN ATOM IN CROSSED ELECTRIC AND

MAGNETIC FIELDS 235

39. THE PROBLEM OF Two CENTRES 241

FOURTH CHAPTER
THEORY OF PERTURBATION

40. THE SIGNIFICANCE OF THE THEORY OF PERTURBATIONS FOR THE

MECHANICS OF THE ATOM 247

41. PERTURBATIONS OF A NON-DEGENERATE SYSTEM . . . 249

42. APPLICATION TO THE NON-HARMONIC OSCILLATOR . . . * 257

43. PERTURBATIONS OF AN INTRINSICALLY DEGENERATE SYSTEM . 261

44. AN EXAMPLE OF ACCIDENTAL DEGENERATION .... 265

45. PHASE RELATIONS IN THE CASE OF BOHR ATOMS AND MOLECULES 269

46. LIMITING DEGENERATION 275

47. PHASE RELATIONS TO ANY DEGREE OP APPROXIMATION . . 282

48. THE NORMAL STATE OF THE HELIUM ATOM .... 286

49. THE EXCITED HELIUM ATOM 292

APPENDIX

I. Two THEOREMS IN THE THEORY OP NUMBERS .... 300

II. ELEMENTARY AND COMPLEX INTEGRATION 303

INDEX 313

THE MECHANICS OF
THE ATOM

INTRODUCTION

PHYSICAL FOUNDATIONS

1. Development of the Quantum Theory of the Oscillator
from the Theory of Radiation

BEFORE dealing with the mathematical theory of atomic mechanics
we shall give a brief account of its physical foundations. There are
two sources to be considered : on the one hand the theory of thermal
radiation, which led to the discovery of the quantum laws ; on the
other, investigations of the structure of atoms and molecules.

Among all the characteristics of the atom which can be inferred
from the physical and chemical properties of bodies, the radiation
phenomena are distinguished by the fact that they provide us with
the most direct information regarding the laws and structure of the
ultimate constituents of matter. The most universal laws of matter
are those manifested in such phenomena as are independent of the
nature of the particular substance with which we are dealing. This
constitutes the importance of KirchhoiFs discovery that the thermal
radiation in an enclosure is independent of the nature of the material
forming the walls of the enclosure, or contained in its interior. In an
enclosure uniformly filled with radiation in equilibrium with the
surroundings, the energy density, for a range of frequency dv, is
equal to pjdv, where p v is a universal function of v and the tempera-
ture T. From the standpoint of the wave theory the macroscopic
homogeneous radiation is to be regarded as a mixture of waves of
every possible direction, intensity, frequency, and phase, which is
in statistical equilibrium with the particles existing in matter which
emit or absorb light.

For the theoretical treatment of the mutual interaction between
radiation and matter it is permissible, by Kirchhoff s principle, to re-
place the actual atoms of the substances by simple models, so long
as these do not contradict any of the known laws of nature. The

2 THE MECHANICS OF THE ATOM

harmonic oscillator has been used as the simplest model of an atom
emitting or absorbing light ; the moving partible is considered to be
an electron, which is bound by the action of quasi-elastic forces to a
position of equilibrium at which a positive charge of equal magnitude
is situated. We thus have a doublet, whose moment (charge X
displacement) varies with time. H. Hertz showed, when investigat-
ing the propagation of electric waves, how the radiation from such a
doublet may be calculated on the basis of Maxwell's equations. It
is an even simpler matter to calculate the excitation of such an
oscillator by an external electromagnetic wave, a process which is
utilised to explain refraction and absorption in the classical theory of
dispersion. On the basis of these two results the mutual interaction
between such resonators and a field of radiation may be determined.
M. Planck has carried out the statistical calculation of this inter-
action. He found that the mean energy W of a system of resonators
of frequency v is proportional to the mean density of radiation p v ,
the proportionality factor depending on v but not on the tem-
perature T :

The complete determination of p v (T) is thus reduced to the determina-
tion of the mean energy of the resonators, and this can be found from
the laws of the ordinary statistical mechanics.

Let q be the displacement of a linear oscillator and %q the restoring
force for this displacement ; then p=mij is the momentum, and the
energy is

The force-coefficient x is connected with the angular frequency u and
the true frequency v I by the relation

^ ==a) 2 = ( 2 7rv) 2 .
m

According to the rules of statistical mechanics, in order to calculate
the mean value of a quantity depending on p and q the quantity
must be multiplied by the weighting factor e-* w , where /J=1/T, and
then averaged over the whole of the " phase space " (p, q) corre-
sponding to possible motions. Thus the mean energy becomes

1 In the following co will always be used to denote the number of oscillations or
rotations of a system in 2n sees, (the angular frequency), v will be used to denote the
number in 1 sec. (the true frequency).

PHYSICAL FOUNDATIONS

This can clearly also be written

where

W^logZ,

is the so-called partition function (Zustandsintegral). The evalua-
tion of Z gives

f* - f* -*
- 6 s ' P cM 6 *'dgi

J tt> J -f>

and since

we get

Hence

(%} W -T

(2) W_ r M.

This leads to the following formula for the density of radiation :

/q\ _ T,rv\

(3) P t . jf"^ 1 '

the so-called Rayleigh-Jeans formula. It is at variance not only with
the simple empirical fact that the intensity does not increase con-
tinually with the frequency, but also leads to the impossible con-
sequence that the total density of radiation

is infinite.

The formula (3) is valid only in the limiting case of small v (long
waves). W. Wien put forward a formula which represents correctly
the observed decrease in intensity for high frequencies. A forriiula
which includes both of these others as limiting cases was found by
Planck, first by an ingenious interpolation, and shortly afterwards
derived theoretically. It is

87T1/ 2 llV

where A is a new constant, the so-called Planck's Constant. Since

4 THE MECHANICS OF THE ATOM

it is the fundamental constant of the whole quantum theory, its
numerical value will be given without delay, viz. :

h =6-54. 10-27 erg sec.

Comparison of (4) with (1) shows that this radiation formula corre-
sponds to the following expression for the energy of the resonators :

(5) W=^-.

To derive this formula theoretically, a complete departure from
the principles of classical mechanics is necessary. Planck discovered
that the following assumption led to the required result : the energy
of an oscillator can take not all values, but only those which are multiples
of a unit of energy W .

According to this hypothesis of Planck, the integral formula for Z
is to be replaced by the sum

oo __f/W

(6) Z=2* .

n=0

The summation of this geometric series gives

1 e
From this it follows that

thus

(7) W=-^L.

6*T 1

This agrees with Planck's formula (5) if we put Vf =hv. This last
relation can be established with the help of Wien's displacement law,
which can be deduced from thermo-dynamical considerations com-
bined with the Doppler principle. Wien's law states that the density
of radiation must depend on the temperature and frequency in the
following way :

the energy of the resonator has therefore the form

W-rffg).

Comparison with (7) shows that W must be proportional to v.

PHYSICAL FOUNDATIONS 5

Einstein showed that the behaviour of the specific heat of solid
bodies furnished valuable support for Planck's bold hypothesis of
energy quanta. The crudest model of a solid consisting of N atoms
is a system of 3N linear oscillators, each of which more or less repre-
sents the vibration of an atom in one of the three directions of space.
If the energy content of such a system be calculated on the assump-
tion of a continuous energy distribution, we get from (2)

If we consider one gram molecule, then NA;=K, the absolute gas
constant, and we have the law of Dulong and Petit in the form

c v --=3R=5-9 calories per degree C.
rfj_

Experiment shows, however, that this is the case at high tempera-
tures only, while, for low temperatures, c v tends to zero. Einstein
took Planck's value (5) for the mean energy instead of the classical
one and obtained for one gram molecule :

JfeT
E=31lT- r .

ekr 1

This represents, with fair accuracy, the decrease in c v at low tem-
peratures for monatomic substances (e.g. diamond). The further
development of the theory, taking into account the coupling of the
atoms with one another, has confirmed Einstein's fundamental
hypothesis.

Whereas Planck's assumption of energy quanta for resonators is
well substantiated by this result, a serious objection may be brought
against his deduction of his radiation formula, namely, that the re-
lation (1) between the density of radiation p v and the mean energy W
of the resonators is derived from classical mechanics and electro-
dynamics, whereas the statistical calculation of W is based on the
quantum principle, which cannot be reconciled with classical con-
siderations. Planck has endeavoured to remove this contradiction
by the introduction of modified quantum restrictions ; but further
developments have shown that the classical theory is inadequate to
explain numerous phenomena, and plays rather the role of a limiting
case (see below), whereas the real laws of the atomic world are pure
quantum laws.

s/Let us recapitulate clearly the points in which the quantum
principles are absolutely irreconcilable with the classical theory.

6 THE MECHANICS OF THE ATOM

According to the classical theory, when a resonator oscillates, it
emits an electromagnetic wave, which carries a Way energy ; in conse-
quence the energy of the oscillation steadily decreases. But according
to the quantum theory, the energy of the resonator remains constant
during the oscillation and equal to n . Uv ; a change in the energy of the
resonator can occur only as the result of a process in which n changes
by a whole number, a " quantum jump."

A radically new connection between radiation and the oscillation
of the resonator must therefore be devised. This may be accom-
plished in two ways. We may either assume that the resonator does
not radiate at all during the oscillation, and that it gives out radia-
tion of frequency v only when a quantum jump takes place, there
being some yet unexplained process by which energy lost or gained by
the resonator is given to or taken away from the ether. The energy
principle is then satisfied in each elementary process. Or we may
assume that the resonator radiates during the oscillation, but retains
its energy in spite of this. The energy principle is then no longer
obeyed by the individual processes ; it can only be maintained on an
average provided that a suitable relation exists between the radia-
tion and the probabilities of transitions between the states of constant
energy.

The first conception was long the prevailing one ; the second
hypothesis was put forward by Bohr, Kramers, and Slater, 1 but new
experiments by Bothe and Geiger, 2 and by Compton and Simon, 3
have provided strong evidence against it. The investigations of this
book will, in general, be independent of a decision in favour of either
of these two assumptions. The existence of states of motion with
constant energy (Bohr's " stationary states ") is the root of the

problems with which we are concerned in the following pages.

2. General Conception ot the Quantum Theory

By consideration of Planck's formula W Q =hv, Einstein was led to
interpret phenomena of another type in terms of the quantum theory,
thus giving rise to a new conception of this equation which has proved
very fruitful. The phenomenon in question is the photoelectric
effect. If light of frequency v falls on a metallic surface, 4 electrons
are set free and it is found that the intensity of the light influences

1 Zeitschr. f. Physik, vol. xxiv, p. 69, 1924; Phil. Mag., vol. xlvii, p. 785, 1924.
* W. Bothe and H. Geiger, Zeitschr. f. Physik, vol. xxxii, p. 639, 1925.

3 A. H. Compton and W. Simon, Phys. Rev., vol. xxv, p. 306, 1925.

4 When the symbols v and v are employed concurrently, v always refers to the
frequency of the radiation, the symbol r to a frequency within the atom. (Trans-
lator's note.)

PHYSICAL FOUNDATIONS 7

the number of electrons emitted but not their velocity. The latter
depends entirely or- the frequency of the incident light. Einstein
suggested that the velocity v of the emitted electrons should be given
by the formula

which has been verified for high frequencies (X-rays), while for low
frequencies the work done in escaping from the surface must be taken
into consideration.

We have then an electron, loosely bound in the metal, ejected by
the incident light of frequency v and receiving the kinetic energy hv ;
the atomic process is thus entirely different from that in the case of
the resonator, and does not contain a frequency at all. The essential
point appears to be, that the alteration in the energy of an atomic
system is connected with the frequency of a light-wave by the
equation
(1) *P=W 1 -W fc

no matter whether the atomic system possesses the same frequency
v or some other frequency, or indeed has any frequency at all.
Planck's equation

W=n.W ; W =Ai/

gives a relation between the frequency of oscillation v of a resonator
and its energy in the stationary states, the Einstein equation (1)
gives a relation between the change in the energy of an atomic
system for a transition from one state to another and the frequency
v of the monochromatic light with the emission or absorption of
which the transition is connected.

Whereas Einstein applied this relation solely to the case of the
liberation of electrons by incident light and to the converse process,
viz. the production of light (or rather X-rays) by electronic bom-
bardment, Bohr recognised the general significance of this quantum
principle for all processes in which systems with stationary states
interact with radiation. In fact the meaning of the equation is in-
dependent of any special assumptions regarding the atomic system.
Since Bohr demonstrated its great fertility in connection with the
hydrogen atom, equation (1) has been called Bohr's Frequency
condition.

Taking into account the new experiments by Bothe and Geiger,
and by Compton and Simon, which have been mentioned above,
we have to assume that the frequency v is radiated during the
transition and the waves carry with them precisely the energy hv

8 THE MECHANICS OF THE ATOM

(light quantum) ; there is at present no theoretical indication of
the detailed nature of the transition process. *

If Bohr's frequency relation (1) be applied to the resonator we are
faced by alternatives which will now be considered. The change of
energy which takes place when the resonator passes from the state
with the energy njiv to that with the energy n 2 kv, viz. :

is, in general, a multiple of the energy quantum, hv, of the resonator.
According now to Bohr and Einstein, this change in energy must be
connected with the frequency ? of the emitted monochromatic radia-
tion by the equation

hv=(n 1 n 2 )hv.

This admits of two possibilities only : either we may require that, as
in the classical theory, the radiated frequency shall correspond with
that of the radiator, in which case only transitions between neigh-
bouring states, for which

i w a =l

are possible, or we may assume that the frequency of the radiation
differs from that of the resonator, being a multiple of it. In the latter
case the emitted light will not be monochromatic, on account of the
possibility of different transitions. The decision between these two
possibilities has been attained in the course of the further develop-
ment of Bohr's atomic theory, the conclusion being that the emitted
radiation is strictly monochromatic, with the frequency given by the
condition (1), but that the agreement between the frequency of the
radiation and the frequency of oscillation of the resonator (i.e.
n l n 2 =l) is brought about by an additional principle, which pro-
vides a criterion for the occurrence of transitions between the
different states, and is called the Correspondence Principle.

A fundamental difference between the quantum theory and the
classical theory is that, in the present stage of our knowledge of the
elementary processes, we cannot assign a " cause " for the individual
quantum jumps. In the classical theory, the transition from one
state to another occurs causally, in accordance with the differential
equations of mechanics or electrodynamics. The only connection in
which probability considerations find a place on the older theory is in
the determination of the probable properties of systems of many
degrees of freedom (e.g. distribution laws in the kinetic theory of
gases). In the quantum theory, the differential equations for the
transitions between stationary states are given up, so that in this
case special rules must be sought. These transitions are analogous

PHYSICAL FOUNDATIONS 9

to the processes of radioactive disintegration. All experiments go
to show that the radioactive transformation processes occur spon-
taneously and are not capable of being influenced in any way.
They appear to obey only statistical laws. It is not possible to
say when a given radioactive atom will disintegrate, but it is pos-
sible to say what percentage of a given number will disintegrate in
a given period ; or, what comes to the same thing, a probability
can be assigned for each radioactive transformation (which is
called a priori since we are not at present in a position to express
it in terms of anything more fundamental). We transfer this
conception to the states of an atomic system. We ascribe to each
transition between two stationary states an a priori probability.

The theoretical determination of this a priori probability is one of
the most fundamental problems of the quantum theory. The only
method of attack so far available is to consider processes in which the
energy transformed in the course of a single transition is small in com-
parison to the total energy, in which case the results of the quantum
theory must tend to agree with those of the classical theory. One
theorem based on this idea is the Correspondence Principle of Bohr
mentioned above ; here the transitions between states with large
quantum numbers (e.g. for large n in the case of a resonator) are
compared with the corresponding classical processes. The rigorous
formulation of this principle will be given later.

Another application of this idea occurs in a new derivation of
Planck's radiation formula ; this is due to Einstein, and has given
effective support to the ideas of the quantum theory and in particular
to Bohr's frequency condition.

In this case no assumptions are made regarding the radiating sys-
tem except that it possesses different stationary states of constant
energy. From these we select two with the energies W x and W 2
(W 1 >W 2 ), and suppose that, when statistical equilibrium exists,
atoms in these states are present in the numbers N x and N 2
respectively. Then, by Boltzmann's Theorem

N 2 <r*r w t -w.

- -. *T .

1 e kT

According to the classical theory, the mutual interaction between an
atomic system and radiation consists of three kinds of processes :

1 . If the system exists in a state of higher energy, it radiates energy
spontaneously.

2. The field of radiation gives up energy to or takes away energy

10 THE MECHANICS OF THE ATOM

from the system according to the phases and amplitudes of the waves
of which it is composed. We call these processes

(a) positive in-radiation, 1 if the system absorbs energy ;

(6) negative in-radiation (out-radiation), if it gives up energy.

In the cases 2 (a) and 2 (6) the contributions of the processes to the
alteration of the energy are proportional to the energy density of the
radiation.

In an analogous manner we assume for the quantum interaction
between radiation and atomic systems the three corresponding pro-
cesses. Between the two energy levels W x and W 2 there are then the
following transitions :

1. Spontaneous decrease in energy by transition from W l to W 2 .
The frequency with which this process occurs is proportional to the
number N x of the systems at the higher level W 1? but will also depend
on the lower energy state W 2 . We write for this frequency of
occurrence

2a. Increase in energy on account of the field of radiation (i.e.
transition from W 2 to Wj). We write in a corresponding way for its
frequency of occurrence

26. Decrease in energy on account of the radiation field (transition
from Wj to W 2 ) with the frequency of occurrence

We leave open the question whether the energy gained or lost by
the atomic system is subtracted from or given up to the radiation
during each individual process, or whether the energy principle is
satisfied statistically only.

Now the statistical equilibrium of the states Nj_ and N 2 requires
that

This gives

_____ _

P V jj W t -W 2

B 21 ^? B 12 B 21 e * T B 12
It is necessary now to make use of the frequency condition

a Emstrahlung is hero translated in-radjation, as there seems to be no exact
English equivalent. E. A. Milne (Phil. Mag., xlvii, 209, 1924) has already used in
English the terms " in-radiation " and " out-radiation " in this connection.

PHYSICAL FOUNDATIONS 11

in order that the formula for p v should be consistent with Wien's
displacement law. Then

__
ft-

At this stage Einstein makes use of the general consideration men-
tioned above, that the quantum laws must reduce to the classical
ones as a limiting case. Clearly in the present problem the limiting
case is that of high temperatures, where liv is small compared to AT.
In this case our formula (2) must become the Rayleigh-Jeans for-
mula (3) of 1, required by the classical theory (and verified by
experiment for high temperatures), namely,

c 3

Since, for large values of T, our p v becomes

A 12

"v L~ >

tt T> , T> A " ,

i >21~ i5 12 + i >21Tm-T . .

the agreement is possible only if

B 12 =B 21
and

Bia c* V
We arrive in fact at Planck's radiation formula

(3) P J JL.

Collecting our results together, we see that Planck's original formu-
lation of the quantum principles for the resonator embodies two
essentially different postulates :

1. The determination of the stationary states (constant energy) :
this is done in the case of the resonator by the equation

We shall generalise this equation later for any periodic system.
2. The Bohr Frequency condition

Av=W 1 -W 1 ,

which determines the frequency of the light emitted or absorbed in
the transition between two stationary states. The frequency v so
defined is positive for emission and negative for absorption.

12 THE MECHANICS OF THE ATOM

In addition to this there are certain statistical principles bearing
on the frequency of occurrence of the stationary states and of the
transitions between them, chief among which is the Correspondence
Principle already referred to.

3. -The Conceptions of Atomic and Molecular Structure

Having now considered the development of the special principles
underlying the quantum mechanics of the atom, we shall indicate
briefly the development of our knowledge regarding the material
substratum to which they apply.

The phenomena of electrolysis first led to the hypothesis of the
atomic structure of electricity. Subsequently the carriers of negative
electricity were detected in the free state as the cathode rays and
the j8-rays of radioactive substances. From the deviation of these
rays in electromagnetic fields, the ratio e/m, of charge to mass of
the particles, could be determined. It was found that

e/ m =5-31 . 10 17 E.S.U. per gram.

On the assumption that the same elementary quantum of elec-
tricity is concerned both here and in electrolysis (which can be
verified approximately by experiment), we are led to the conclusion
that the mass of these negative particles of electricity is about an
1830th part of that of a hydrogen atom. These carriers of negative
electricity are called electrons and it can be shown, by optical and
electrical experiments, that they exist as structural units in all
matter. By making use of the fact that it is possible to produce
on very small (ultra-microscopic) metal particles, and oil drops, a
charge of only a few electrons, and to measure it, very accurate
values have been found for the charge carried by an electron.
Millikan found

e=4-77. 10- 10 E.S.U.

Positive electricity has only been found associated with masses
of atomic magnitude. Positive rays have been produced and studied :
it will suffice to mention a-rays of radioactive substances, anode
rays and canal rays. The determination of e/m from deviation ex-
periments gave the mass of the a-particles to be that of the
helium atom ; for the particles of the anode rays the mass is that
of the atom of the anode material, while for the particles of the
canal rays the mass is that of an atom of the gas in the tube. We
must therefore assume that each atom consists of a positive particle,
at which is concentrated most of its mass, and of a number of
electrons. In the neutral atom the number of the elementary

PHYSICAL FOUNDATIONS 13

charges of the positive particle is equal to the number of electrons ;
positive ions result from loss of electrons, negative ions from capture
of extra electrons.

As regards the size of the electrons we can do nothing but make
doubtful theoretical deductions, which lead to an order of magnitude
of 10~ 18 cm. Lenard was the first to obtain definite conclusions
regarding the size of the positive particles, which he called dynamids
(Dynamiden). From experiments on the penetration of matter by
cathode rays, he found that only a vanishingly small fraction of the
space occupied by matter is impenetrable to fast cathode rays.
Subsequently Eutherford arrived at analogous conclusions, as the
result of experiments on the penetration of matter by a-rays. From
a study of the range and scattering of these rays he was able to
establish the fact that the linear dimensions of the positive particles,
which he named nuclei, are at least 10,000 times smaller than those
of an atom ; up to this limit the observed deviations can be ascribed
to Coulomb forces between the charged particles. The measure-
ments also provided information regarding the charge of the positive
particles, and gave for the number of the elementary charges about
half the value of the atomic weight ; the number of electrons in the
neutral atom must be the same. This result was supported by
investigations of the scattering of X-rays ; the amount of the scat-
tering depends principally, at any rate in the case of loosely bound
electrons, on their number only.

When we come to regard all the possible types of atoms we must
turn for guidance to the periodic system, which has been set up as
the result of chemical experience. By this, the elements are
arranged in an absolutely definite order ; the magnitudes of the
atomic weights give in the main the same order, but there are
certain discrepancies (e.g. argon and potassium). The result ob-
tained above, that the nuclear charge is approximately equal to
half the atomic weight, led van den Broek to the hypothesis that the
number of elementary nuclear charges is the same as the ordinal
number of the atom in the periodic system (atomic number or
order number). When, following Laue's discovery, X-ray spectro-
scopy had been begun by Bragg, van den Broek's hypothesis
was confirmed by Moseley's investigations on the characteristic
X-ray spectra of the elements. Moseley found that all elements
possess essentially the same type of X-ray spectrum, but that with
increasing atomic number the lines are displaced in the direction of
higher frequencies and, moreover, that the square root of the fre-
quency always increases by nearly the same amount from one

14 THE MECHANICS OF THE ATOM

element to the next. This established the fundamental character
of the atomic number, as contrasted with the atomic weight.
Further, the similarities of the X-ray spectra suggest the similarity
of certain features of atomic structure. If now we assume that
the structure of the atom, i.e. number and arrangement of its
electrons, is determined essentially by the nuclear charge, we are
led to the conclusion that there must be a close relationship be-
tween the nuclear charge and the atomic number ; in fact, with
the assumption that the two quantities are equal, the more precise
theory of the X-ray spectra, which we shall give later, leads to
Moseley's law.

Collecting together the results bearing on atomic structure, we have
then the following picture of an atom with the order number Z ; it
consists of a nucleus with the charge Z, 1 with which is associated
practically the whole mass of the atom, and (in the neutral state)
Z electrons. In the model atom imagined by Rutherford it is
supposed that these circulate round the nucleus in much the same
way as the planets round the sun, and that the forces holding them
together are essentially the electrostatic attractions and repulsions
of the charged particles.

But if, on the basis of these conceptions and the classical prin-
ciples, we now attempt to develop a mechanical theory of the atom,
we encounter the following fundamental difficulty : a system of
moving electric charges, such as is pictured in this model, would
continually lose energy owing to electromagnetic radiation and must
therefore gradually collapse. Further, all efforts to deduce the char-
acteristic structure of the series spectra on the basis of the classical
laws have proved fruitless.

Bohr has succeeded in overcoming these difficulties by rejecting the
classical principles in favour of the quantum principles discussed in
1 and 2. He postulates the existence of discrete stationary states,
fixed by quantum conditions, the exchange of energy between these
states and the radiation field being governed by his frequency con-
dition ( 1 ) , 2 . The existence of a stationary state of minimum energy,
which the atom cannot spontaneously abandon, provides for the
absolute stability of atoms which is required by experience. Further,

1 Later researches, chiefly by J. J. Thomson, Rutherford, Aston, and Dempster,
have shown that the nucleus itself is further built up of electrons and hydrogen
nuclei, called protons. As a consequence of these investigations the old hypothesis
of Prout regains significance, in a somewhat different form. The deviations of
the atomic weights from whole-number values, which previously ruled out this
hypothesis, can be accounted for by the conception of isotopes and energy-mass
variations. Not much is definitely known on the subject of nuclear mechanics,
and it will not be discussed in this book.

PHYSICAL FOUNDATIONS 15

in the case of the hydrogen atom, he succeeded in calculating the
energy levels by a rational generalisation of Planck's hypothesis,
in such a way that the frequency condition leads at once to the ob-
served spectrum (Balmer's formula), He has also given the prin-
ciples whereby the quantum conditions may be formulated in
more complicated cases ; this will be dealt with in the following
pages.

Bohr's fundamental concepts of discrete stationary states and a
quantum frequency condition receive their most direct confirmation
from a class of investigations initiated by Franck and Hertz, and sub-
sequently extended and refined by these and other investigators.
The fundamental idea of these experiments is that definite amounts
of energy can be communicated to atoms by bombarding them with
electrons of known velocity. As the velocity of the bombarding
electrons is increased, the abrupt occurrence of the stationary states
is indicated on the one hand by the sudden appearance of electrons
which have lost some of their incident energy, and on the other by
the sudden production of radiation of those frequencies which are
associated with transitions from the stationary state in question to
other stationary states of lower energy.

Analogous phenomena are observed in the domain of the X-rays,
where the occurrence of emission lines and absorption edges is bound
up with the attainment of definite energy levels, consequent on
electronic bombardment. In both the optical and the X-ray region
values for the constant h can be determined by means of the frequency
condition by measuring the energy supplied and the frequency of the
consequent radiation. These values are independent of the atom and
the particular quantum transition used to derive them, and are found
to be in good agreement with the values obtained from heat radiation
measurements.

Not only the structure of atoms but also their combination to form
molecules and extended bodies, and the laws of motion of the latter,
are governed by the same quantum laws. We may mention, for
example, the more precise development of the theory of specific heats
of solid bodies already referred to, and further the theory of the band
spectra of molecules, which we shall deal with in detail in this
book.

We give in conclusion a brief formulation of the ideas which have
led to Bohr's atomic theory. There are two observations which are
fundamental : firstly the stability of atoms, secondly the validity of
the classical mechanics and electrodynamics for macroscopic pro-
cesses. The application of the classical theory to atomic processes

16 THE MECHANICS OF THE ATOM

leads, however, to difficulties in connection with the stability. The
problem arises, therefore, of developing a mechanics of the atom free
from these contradictions. This new mechanics is characterised by
the fact that the classical continuous manifold of states is replaced
by a discrete manifold, defined by quantum numbers.

FIRST CHAPTER

THE THEORY OF HAMILTON AND JACOBI

4. Equations of Motion and Hamilton's Principle

NEWTON'S equations of motion for a system of free particles form the
starting-point for all the following considerations : l

d __

where m k denotes the mass of the Ath particle, v fc its velocity, and K k
the force acting on it. The product m fc v fr is called the impulse or
momentum.

In this form the equations (1) still hold if the mass is dependent
on the magnitude of the velocity, as is required by Einstein's rela-
tivity theory. *

In many cases the system of equations (1) is equivalent to a varia-
tion principle, known as Hamilton's Principle, viz. :

(2) I Ldt =stationary value.

Here L ig a certain function of the co-ordinates and velocities of all
the particles, and, in certain circumstances, also an explicit function
of the time, and equation (2) as an expression of Hamilton's Principle
is to be interpreted as follows : the configuration (co-ordinates) of the
system of particles is given at the times t t and t 2 and the motion is
sought (i.e. the co-ordinates as function of the time) which will take
the system from the first configuration to the second in such a way
that the integral will have a stationary value. 2 The chief advantage
of such a variation principle is its independence of the system of co-
ordinates.

1 Heavy type is used to indicate vector quantities. Vector products are indicated
by square brackets, and scalar products by round brackets or by absence of brackets.

9 It does not matter whether it is a maximum or a minimum or a saddle-point
value.

17 2

18 THE MECHANICS OF THE ATOM

Lagrange's equations 1

d SL_aL

(3) dtdF 1 T'fa l r

can be derived directly from the variation principle (2). We have
to determine L so that these equations agree with the Newtonian
equations (1).
If the forces K* have a potential U, i.e. if

we determine a function T* of the velocity components so that

0T*

The equations (1) can then be written in the form

d 0T*_8(-U)_

dt dJr k dx k
or

rfd(T*-U)_d(T*-U)_

rf/T d7^ dx~ k '

We put therefore in our variation principle (2)
(4) L=T*-U.

If, taking no account of the theory of relativity, we regard m k as
constant, T* is equal to the kinetic energy T. If we write, in accord-
ance with the special theory of relativity,

c/J '

where m is the " rest mass " and c is the velocity of light, we have
(for one particle)

(5) T*=m - eri-(l-(-) 1 ! 1 ],

L \ \ c / ) J

which, for the limiting case c oo , reduces to the expression w v 2 .

1 In the following we shall usually write down only the first of the three equations
corresponding to the co-ordinates x, y t z.

THE THEORY OF HAMILTON AND JACOBI 19

This function differs from the kinetic energy

(6) T=m c({l-(*) 2 j~ l -l],

which, of course, also reduces to ^m v 2 in the limiting case where
c=oo .

Besides a component K, which can be derived from a potential
U, the forces often contain a component K* depending on the
velocities (as in the case of magnetic forces acting on electric charges).
A function M is then determined so that

and the expression

(8) L=T*-U-M

is substituted in the variation principle (2).
The Lagrangian equations (3) then become

d #T* dU d 8M. 0M_

and our variation principle is, in fact, equivalent to the Newtonian
equations of motion

Hamilton's Principle is also valid when the particles are con-
strained in a manner defined by equations

/*(*!> 2/i> *i *> * - 0=0

between the co-ordinates. 1 In accordance with the rules of the
calculus of variations, additional forces of the form

must be added to the original forces, where the A^'s, which are
functions of the co-ordinates, are the " undetermined multipliers "
of Lagrange. These multipliers, together with the co-ordinates, are
to be regarded as unknowns ; the number of determining equations,
i.e. differential equations and equations of constraint, is then equal
to the number of unknowns.

As already mentioned, the chief advantage of Hamilton's Principle
is that it represents the laws of motion in a manner independent of
any special choice of co-ordinates. If a number of equations of con-

1 Such conditions, which do not involve the velocity components, are called
holonomous.

20 THE MECHANICS OF THE ATOM

straint be given, an equal number of co-ordinates can be eliminated
with their help. There remains then a certain number of indepen-
dent co-ordinates q^ 2 . . . q f .

The number /is known as the number of degrees of freedom. The
Lagrangian function will then be a function of the j's and of their
time derivatives ; the time may also appear explicitly :

k=L(?i, </!, y 2 , ^ 2 ... q r q f , t) ;
the variation principle (2) then leads to the Lagrangian equations

!!-- *-"/

These are also valid if the co-ordinates q k refer to arbitrarily moving,
or even deformed, systems of reference.

5. The Canonical Equations

Each of Lagrange's equations is a differential equation of the
second order. In many cases, particularly for work of a general
character, it is desirable to replace them by a system of twice as
many differential equations of the first order. The simplest way
of accomplishing this is to put q k =s k > and then to take these addi-
tional equations into account, treating the s k s, as well as the q k 's,
as unknown quantities. A much more symmetrical form is obtained
as follows :

In place of the q k 's the new variables

(1) ""

known as momenta, are introduced ; the Lagrangian equations (9)
of 4 now become

(2) p^ 9

fyk

where L is still to be regarded as a function of the q k s and q k 's.
Equations (] ) can now be expressed in a similar form by introducing
in place of the function L(7 1 r/ 1 . . . t) a new function H(^ 1 ;) 1 . . . t),
by means of a Legendre Transformation l

(3) H=

1 A Legendre Transformation transforms, in general, a function f(x, y) into a

function g(x, 2), where z ~* in such a way that the derivative of g with respect to
cy

the new variable z is equal to the old variable y Such transformations play a
considerable part in all branches of physics ; in thermodynamics, for example, the
energy and the free energy are related in the same way as two functions connected
by a Legendre Transformation.

THE THEORY OF HAMILTON AND JACOBI 21

If now we form the total differential

r\T QT *}T

k k k * k k *

the terms in dq k cancel out on account of (1). For the partial deriva-
tives of H(<npi t) with respect to p k and q k we have therefore

where the indices outside the brackets denote which variable is inde-
pendent. Now with the help of the new variables we can write (2)
and (4) (which is an expression of (1)) as follows :

This is the so-called canonical form of the equations of motion.
H(<7i, Pi, q z > p* t) is called the flamiltonian function. The
variables q k and p k are said -o be canonically conjugated.

The same equations are obtained if the momenta are defined by
(1) in the same way, and the function L in the variation principle (2),
i, is expressed in terms of H by means of equation (3). We have

(6)

J^?iPkQk'R(<IiPi $) cfe=stationary value,
^L k -1

for the same possible variations as before, i.e. variations for which
the configurations at fixed times t L and t 2 are themselves fixed ;
here the q k and p k are to be regarded as the functions required. It
is easily seen that the Lagrangian equations are equivalent to (5),
and it should be noted that the derivatives of the p k & do not occur
explicitly in the integrand ; for this reason only the values of the
q k s at the times ^ and t 2 can be prescribed as limiting conditions,
not those of the p k &.

All of these considerations remain valid if the function L, and
with it the function H, depends explicitly on the time t. The latter
case will occur, for example, either if external influences depending
on the time are present (U depending on t), or if, in the case of a
self-contained system, a system of reference is employed which itself

22 THE MECHANICS OF THE ATOM

performs a prescribed non-uniform motion. If, however, H does not
involve the time explicitly, we have

Substituting for q k and p k from the equations of motion (5), it follows
that

dR n

-*-*

so that

(7) &(PI<!I .) ==cons ^ an ^

is a first integral of the equations of motion (5).

We inquire now as to the mechanical significance of the quantity
H and consider first the case of the classical (non-relativistic)
mechanics. With any co-ordinates, in a stationary system of refer-
ence, the kinetic energy is a homogeneous quadratic function T 2 of
the velocities q k ; in moving co-ordinate systems additional linear
terms, and terms not involving q k> will occur, so that we can write :

Here T n denotes a homogeneous function of the nth degree of the
r/fc's, which may, moreover, depend on the q k 's. By Euler's Theorem
we have

thus
(8)

If we suppose that an ordinary potential energy U exists, in which
case

L=T-U,
we have

H=T 1 +2T 1 -(T +T 1 +T 1 )+U
= -T +T 2 +U.

In the case of a co-ordinate system at rest (T=T 2 )
(9) H-T+U

is the total energy. If the time does not occur explicitly in H, this
gives, in conjunction with equation (7), the law of the conservation
of energy.

THE THEORY OF HAMILTON AND JACOBI 23

In the case of moving co-ordinate systems, where T and T x are not
0, it may happen that H is independent of the time, thus

H=const.
is an integral, but not the energy integral.

Example. We consider a system of co-ordinates (|, rj) rotating with the
angular velocity (o round the axis of z. We transform to this from the sta-
tionary system (x, y) by means of the formula

x= f cos cut ry sin cut
\j % sin a>t-\- 1] cos cut

z=.

The kinetic energy then becomes

The momenta corresponding to the co-ordinates f and /? are then

so that we can also write

For H we obtain

If U is symmetrical around the axis of z, H does not contain the time explicitly
and is therefore constant. The integral

H= const.
is called the Jacobian Integral. It is, however, different from the energy

E=T+U= (

which is likewise constant.
From both integrals it follows that

E-H= const.
This gives the law of conservation of angular momentum. We have, in fact,

If we transform back to x, y, we have

E H= o>Sm(^ yx).

Z* THifi MJfiUJtiAJNlUS UJT THifi ATOM

We consider now the case of relativistic mechanics. By (4) and (5),
4, we have for a particle

L=T* U=
thus
(10) p

and

(11) =T+U,

so that in this case also H is the total energy. This result is indepen

dent of the co-ordinate system so long as this is at rest.

6. Cyclic Variables

Before dealing with the general theory of the integration of the
canonical equations, we will, first of all, consider some simple cases
If the Hainiltonian function H does not contain one co-ordinate, e.g
q l9 i.e. if

H^fptfapa . . . 0,
it follows from the canonical equations that

?>i=0

Thus we arrive immediately at one integral of these equations. The
co-ordinate q l is called, after Helmholtz, a cyclic co-ordinate (since il
often corresponds to a rotation about an axis).

Clearly this case always arises if the mechanical system is no1
affected by an alteration of the co-ordinate q l (e.g. by a single trans-
lation or rotation).

If, for example, a system of massive particles (iir 2 r n ) moves
under the action of mutual forces only, the potential energy will de-
pend solely on the differences

s 2 =r 2 i!, S 8 =r 8 r 1 , . . . s n r n r x .

We introduce as co-ordinates the components x^y^ of r x , and the
components 1^^ of these differences s fc . Since U is independent
of #i2/i2i, it follows that Px l p v j?z l are constant. Now the kinetic
energy is

2/* 2 +^ 2 ) (t=l, 2 . . . n),

THE THEORY OF HAMILTON AND JACOBI 25

Since

(&=2, 3 . . . n)

it follows that

(*=1, 2 . . . n).

The three integrals give therefore the principle of the conservation of
momentum.

Another important case is that in which the potential energy re-
mains unaltered by a rotation of the whole system about an axis fixed
in space. If (f> l9 <f> 2 . . . are the azimuths of the particles of the system
about this axis, we introduce as co-ordinates the magnitudes

and certain others, depending only on the relative positions of the
particles with respect to one another and to the axis (for example,
cylindrical co-ordinates r k , z k or polar co-ordinates r k> k ). Since the
Hamiltonian function does not depend on Oj, O a is a cyclic variable
(in this case in the true sense of the word) and the momentum p+ con-
jugate to it, is constant. Since

(*=2, 3 . . . n)
and

where r k is the distance from the axis, p^ has the value

3T 5T ,

ancf is therefore the angular momentum about the axis of sym-
metry.

If the massive particles move under the action of mutual forces
only, our considerations are valid for every fixed direction in space.
Since the quantity p^ is the component of the total angular momentum

in an arbitrary direction, and is always constant, the constancy of the
angular momentum follows.

It may happen that H depends only on the p k s, i.e.

. . .)

26 THE MECHANICS OF THE ATOM

In this case the canonical equations admit of immediate integration.
We have

Here the co fc 's are constants characteristic of the system and a fc , fi k are
constants of integration. It will be seen from this that a mechanical
problem is solved as soon as we have found co-ordinates for which the
Hamiltonian function depends only on the canonically conjugated
momenta. The methods treated in this book will usually follow
this course. In general, such variables cannot be found by a simple
point transformation of the q k 's into new co-ordinates, but rather the
totality (qk,pk) of the co-ordinates and momenta must be transformed
to new conjugated variables.
. We shall, however, first consider some more examples.

1. The Rotator. By this we understand a rigid body which can rotate about
an axis fixed in space. If ^ denotes the angle of rotation and A the moment
of inertia about the axis, then

and the momentum corresponding to < is

p=^A<f>.
For motion under no forces (U~0).

(1) H-T-1*

<f) is therefore cyclic, and consequently

p= constant,
and

<=co=-, <j>wt\-p.
A.

The motion under no forces is therefore a uniform rotation about the axis.

2. The Symmetrical Top. If A^ denotes the moment of inertia about an axis
perpendicular to the axis of symmetry (z), A z the moment of inertia about the
axis of symmetry, and d^, d v , d z the components of the angular velocity in the
system of reference (x, y, z) rigidly fixed in the body, then

We introduce as co-ordinates the Eulerian angles 0, <f>, defined as follows :
Rectangular axes x, y, z are taken fixed in space ; is the angle between the
axis of symmetry (z) and the z-axis, < is the angle between the #-axis and
the nodal line (line of intersection between the (x, y) plane and the plane
(x, y)) 9 and is the angle between the a:-axis and nodal line. The components
of the angular velocity will then be

THE THEORY OF HAMILTON AND JACOBI 27

da,= cos <-h^ sin 6 sin #,
(2) dy=0 sin </>$ sin 6 cos ^

and the kinetic energy

T+KAa.(0 2 +0 2 sin 2 6)+A n (j+t cos 0) 2 ].
The momenta corresponding to 0, <f>, are
0T .

p ==( A x sin 2 | A, cos 2

cty

In order to make the physical significance of these momenta clear, we use (2)
to replace the 0, </>, $ by the components of d : then

sin <

2y=A3(da sin <f> d z cos <^) sin 0+A z d 2 cos 0,

in which (d^ cos ^+ d^ sin ^) clearly denotes the angular velocity about the
nodal line and (d^sin^ dycos^) the angular velocity about a perpendicular
direction in the (#, y) plane. We see then from the equations, that

p is the angular momentum about the nodal line,

p , is the angular momentum about the axis of symmetry,

p is the angular momentum about the direction z fixed in space.

For motion under no forces (U=0), a simple calculation gives

J V"*v*' efl V1i V

sine yJ + 2A;

In this expression ^ and /> do not appear; they are therefore cyclic, and
consequently

p = constant, p. = constant.

Since we have in addition the principle of the conservation of the total angular
momentum at our disposal the integration can be completely carried out. We
can take the hitherto arbitrary axis of z in the direction of the resultant angular
momentum. Since the nodal line is perpendicular to this, the angular
momentum about the nodal line will be

The canonical equations give firstly

0= constant,
and then

28 THE MECHANICS OF THE ATOM

which leads to

(P+-P+ cos 0)(JV~*V cos 0H -
Since p. is essentially greater than or equal to p , , it follows that

jy-fy cos 0=0,

as can also be seen immediately. The Hamil toman function now takes the
simple form

i/t and (f> therefore execute uniform rotations with the angular velocities
. 8H 1 1

The motion under no forces of the symmetrical top therefore consists of a
uniform rotation about the axis of symmetry, together with a uniform pre-
cession of this axis about the direction of the resultant angular momentum.

7. Canonical Transformations

As already mentioned, the integration of the equations of motion
can be effected by introducing new co-ordinates having a cyclic char-
acter if such can be found. We shall therefore quite generally seek
a transformation

Pk=Pk(qi<l2

such that the new variables again satisfy the canonical equations of
motion. For this to be the case it is necessary and sufficient that the
variation principle (6) of 5

fa**- ^~ x^'i p=stationary value

k J

shall transform into

(\ 2iP*#*~"~-^(?iPi |d=stationary value.
This will be the case if, and only if, the difference of the integrands is

the complete derivative of a function of 2f of the old and new

at .

variables and of the time ; for, if V be regarded as a function of the q k
an.d q k , the values of V at the limits of integration will be fixed.
According now as we take V to be a function of q k , q k , t, or of q k , p k9 1,
or of q k , p k> t, or finally of p k , p k , t, we obtain four principal forms for
canonical transformations.

THE THEORY OF HAMILTON AND JACOBI 29

We choose therefore an arbitrary function V(y 1 , q^ . . . t). The
condition

is fulfilled, if the coefficients of q k and q k , and of terms independent of
these quantities, are the same on both sides, that is if

(1) P* =

H=H~
Since in general the y fc 's can be calculated from the equations of the

second line, and then the p k s can be calculated from the equations of
the first line as functions of the q k and p k> the system (1) replaces the
equations of transformation.

Again, in order to obtain a canonical transformation by means of
an arbitrary function V(g r 1 , p . . . t), we write our condition in the
form :

or, what comes to the same thing :

2P*7*- H (?i> Pi = ~2^*"H(y 1 , P! . . .

A; *

+|v( ?1 , p, .

Comparison of the coefficients of q k and p k gives

These equations can also be regarded as equations of transformation.

30 THE MECHANICS OF THE ATOM

The third form we obtain by simply interchanging the old and new
variables, and replacing V by V, in order to obtain the simplest
possible correspondence between the four forms. We obtain :

o
?*=---

(3) J=-

Finally, in order to arrive at the fourth form, we write the condition
in the form :

* -Hfo, ^ . . . t)

or :

and obtain :

(4) fc^vto, ft . . . 0,

We can express all four forms at once in the following manner : In
the arbitrary function V(a? 1? x l9 x 2 , x 2 . . . t) let x k be one of the
variables q k and p k , x k one of the variables q k> p k ; then the equations

, av

yk=t ,
fa*

(5) y fc =T s-i

ox k

give a canonical transformation. Here j/ fc is conjugated to x k and y k
to fc ; the upper sign applies to the case where the differentiation is
taken with respect to a co-ordinate and the lower one to the case of

THE THEORY OF HAMILTON AND JACOBI 31

differentiation with respect to a momentum. The function V we
shall call the generating function, or shortly, the generator, of the
canonical transformation.

Further, it must be emphasised that the canonical property of a
transformation depends in no way on the special mechanical pro-
blem ; if a transformation is canonical, it remains so for every form
of the function H. We now give some transformations which we
shall need later :

The function

leads to the identical transformation

The function

gives, after solving (2) for p k and q k9 the transformation

/ 6) 21=71 Pl=PlP2

q2=<J2 : fqi P2=P2

and the function

V =qii>i
leads to

A transformation for three pairs of variables is provided by

V=9&i+ViP*+9iP*+9&*+9&*+W*
namely

?1='/1 Jl=Pl+P2+P3

In all of these examples the co-ordinates and the momenta are kept
separate in the transformation. The general necessary and sufficient
condition for this is clearly, that V shall be linear in the q's and p's :

i,k

This function gives
(9)

32 THE MECHANICS OF THE ATOM

If the constants j8 rf and y t are zero we have a transformation which
transforms the q k 's and p k & linearly and homogeneously into the
q k and p k , viz. :

k M I

The necessary and sufficient condition that the q k s shall transform
among themselves is the linearity of V in the p^'s. In fact

provides the transformation :

_
do) p *~

?*=/*(?! ?2 ' )

Linearity of V in the # fc 's gives, on the other hand, a transformation
of the momenta between themselves ;

leads to

J>*=/*(pi, Pa )
(11) -

Tt appears if the variables of the one kind transform among them-
selves, the new variables of the second kind will be linear functions
of the old variables of the second kind, the coefficients of which will
be determined functions of the variables of the first kind, and the
free terms arbitrary functions of the variables of the first kind.

Transformations of the co-ordinates among themselves which are
frequently employed are those which transform rectangular co-
ordinates into cylindrical or polar co-ordinates, and also those which
correspond to rotations of the co-ordinate system.

The function

V^^r cos </>+p y r sin <+p/

transforms rectangular co-ordinates into cylindrical co-ordinates.
It gives

x r cos <f> p r p x cos <j> +p v sin <f>

(12) y =r sin <f> p^ = p x r sin <f> +p v r cos <{>

Z= Z P;=P Z .

The expression

p* 2 +Pt, 2
then becomes

THE THEORY OF HAMILTON AND JACOBI 33

In transforming to spatial polar co-ordinates we take
V=p x r cos <f> sin 0+p y r sin <f> sin Q+pf cos 6.
This function leads to the transformation
x=rcoa <sin 8
y=rsin ^sin

(13)

Prp x cos ^ sin 0+y,, sin ^ sin 0+2^ cos 6
p^=^px r sin < sin 6 +jv cos < sin
p e =p x r cos ^ cos 0+p v r sin ^ sin 0p t r sin

and transforms the expression

into

2 I M 2 I

A rotation of the rectangular co-ordinate system (x, y, z) involves
a linear transformation of the co-ordinates with constant coefficients.
The momenta transform then contravariantly. In this case, where
the coefficients a ik defining the rotation fulfil the conditions

<<=*)
(+*)

the contra variant transformation is equivalent to the original one.
The momenta transform like the co-ordinates ; we have

(14) y^==a

9 z
and

We give two further transformations, for which V depends on q k and
q k . The function

V=M*

k
gives, by (1),

?*=-?*
ft=ft-

It therefore interchanges co-ordinates and momenta.
A transformation frequently employed is given by

34 THE MECHANICS OF THE ATOM

it leads to

(15)

y=V2p cos q

and transforms the expression q*+p 2 into 2p. The somewhat more
general function

(16) V = ~o>j a cotg

gives

(16') *

p=V'2nia)p cos q
and transforms

mco 2 2

2
into co/J.

We shall illustrate now, by means of an example, how the canonical
substitutions can be used to integrate the equations of motion.

Linear Harmonic Oscdlalor. In this case

where q denotes the displacement, m the mass, and # the elastic constant.
Introducing tho momentum

p=mq
and putting

m
we get

-S^T^

The transformation last mentioned (16) applies then to this case. We call the
new variables </> and a and write :

(18) </

j> - V 2meoa cos ^.

The Hamiltonian function then becomes

H=wa;
and the equations of motion give

THE THEORY OF HAMILTON AND JACOBI 35

a= constant

The displacement q will therefore be given by

sin (cot + ft),
nuo

The canonical transformations are characterised by the fact that
they leave invariant the form of the equations of motion, or the sta-
tionary character of the integral [(6) of 5] expressing Hamilton's
principle. This raises the question whether there are still other
invariants in the case of canonical transformations. This is in fact
the case, and we shall give here a series of integral invariants intro-
duced by PoincarS. 1

We can show that the integral

(19)

taken over an arbitrary two-dimensional manifold of the 2/-dimen-
sional (p, q) space, is such an invariant. If we represent the two-
dimensional manifold by taking p k and q k as functions of two para-
meters u and v, then

~du ~du 7 7
dudv.

dv dv
We prove the invariant character of J by showing that

du du

~dv !)v

provided that p k and q k are derived from q k) p k by a canonical trans-
formation. We write the transformation in the form (2)

2*=

du
3p*

du
~dv

-2

av( gl ,

1 H. Poincar6, Mtthode* nouvelles de la mtcanique clleste, vol. iii, ch. xxii-xxiv
(Paris, 1899) ; proof of the invariance by E. Brody, Zeitechr. f. Physik, vol. vi,
p. 224, 1921.

36 THE MECHANICS OF THE ATOM

and replace q k , p k by q ky p k with the help of the first equation ; then

du du

~dv ~dv

-2

G v wpt vq k
~dq k dp t ' Hu Hu

dv dv

du du
dp t dq k
~dv ~dv
Interchanging the indices k, i, this becomes

dp k dq t

~du du

&Pk &h

dv dv

and we now transform q k , p k into q k , p k by means of the second equa-
tion of transformation ; the integrand becomes equal to

T - - 3q,

<Zjfi<n,fin r ' du

&Pk ^ v-v <W
dv ~dp k dqi' dv

proving the invariance of the integral (19).
The invariance of the integral

du du

dv dv

in which every combination of two indices occurs in the integrand,
may be proved in a precisely similar way. The same holds true for

and so on. The last integral of the series is

J/H" - Jtfp< dpfei . . . dq f .

The volume in the phase space is consequently invariant with respect
to a canonical transformation.

8. The Hamilton-Jacob! Differential Equation

The idea underlying the method of integration which is so par-
ticularly suited to the problems of atomic mechanics (just as it is to
those of celestial mechanics) will be clear from the example of the

THE THEORY OF HAMILTON AND JACOBI 37

oscillator given in 7. Although it appears very awkward in this
case, yet, on the other hand, it is powerful enough to lead to the
required end even for some quite complicated (particularly periodic)
motions. We shall now give a general formulation for the case in
which the Hamiltonian function does not contain the time explicitly :
We endeavour to transform the variables q kt p k , by means of a
canonical transformation, into new variables </> k , a k in such a way
that the Hamiltonian function depends only on the quantities a fc ,
which correspond to the momenta. For this purpose the most suit-
able form of the canonical transformations is (2), 7. We seek there-
fore to determine a function

such that, by means of the transformation

jP*=p-S(gift . . . at, a, . . .)
(1) **

<*= S(gri>?2 . . . C4, a a . . .)
ca k

H is transformed into a function

depending only on the a k s. The < fc 's are then cyclic variables and
the equations of motion lead at once to the solution

a^constant

(2) , aw

The determination of the function S can be made to depend on the
solution of a partial differential equation of the first order. A par-
ticular case of some importance is given by taking W equal to 04 ;

let each p k in H be replaced by the corresponding , then S has to

a?*
satisfy the condition

/ox

(3)

This equation is known as the Hamilton- Jacobi differential equation.
The problem is now to find a complete solution, i.e. a solution which
involves a x and /I other constants of integration a 2 , a 3 . . . a/,
apart from the purely additive constant in S. This function S
provides a transformation (1) of the kind desired ; at the same time

38 THE MECHANICS OF THE ATOM

the following special relations hold,
dW

>! = =1 J ^2=^3= =Wf=().

C/djL

The solution of the equations of motion will then be given by the
solution of (1) in terms of q k and p k , if the substitutions

(4)

are made.

The problem of solving the system of 2/ ordinary differential equa-
tions of the first order, i.e. the canonical equations, is therefore equi-
valent to that of finding a complete solution of the partial differential
equation (3) (/being greater than 1). This is a special case of general
theorems on the relation between ordinary and partial differential
equations.

For many purposes it is more advantageous not to single out one of
the a's, as has just been done. A canonical transformation may be
carried out, in which the a fc 's transform into a like number of new
variables, which we shall also call 04 . . . a/, in such a way that the
</) k s do not enter into the relations between the old and new a fc 's.
a x is transformed into

W(a 1} a 2 . . . a,).

According to a theorem proved in 7 (equation (11)), new variables
(f> k can be introduced, which are conjugated with the a k s and are
linear functions of the old (f> k s with coefficients depending only on the
constants a k . Thus the new </) k s are likewise linear functions of the
time and the equations of motion hold in the form (2).

The function S may be regarded as a solution of the differential
equation

depending on/ constants 04 . . . a/, between which and W a relation

W=WK . . . a f )

exists. By (5) the transformation (1) transforms the function H
into the function W(a x . . . a^), and we have here also

THE THEORY OF HAMILTON AND JACOBI 39

An important property of the function S can be derived from (1),
namely, that for a path defined by fixed values of the a fc 's

i* ^ aS
(io= > - e

k 9k

S is therefore the line integral
(6) S

taken along the path, where Q denotes a fixed and Q a moving
point of the path.

In the case of classical mechanics, and for a system of co-ordinates
at rest, this integral has a simple significance. For in this case we
have (see (8), 5)

and thus
(7)

In the case of the theory of relativity, if we take a single particle,
2T must be replaced by

It will be seen that in both cases S is a function continuously in-
creasing with the time, it is called the Principal Function of the
system.

We will now consider the simplest case, namely, that of one degree
of freedom. Then the differential equation (5) becomes an ordinary
one

Solving for p as a function of q and W, and integrating with respect
to q, since

y= %-

we find

This can also be regarded as a special case of the general formula (6).
The function S determined in this way, which contains no constants

40 THE MECHANICS OF THE ATOM

apart from W, provides the general solution of the equations of
motion ; we have

which, on solution, gives 9 as a function of the time with the con-
stants of integration W and t .
For co-ordinate systems at rest T has the form

where //, denotes mass, moment of inertia, or some such quantity.
We have then

so the solution for p in terms of q and W is

(8) P

and

-*.- /~tf-f=.
V 2J fc V/W-U(fl

(9) I

Example 1 . Particle falling freely or projected vertically. Here g denotes the
height of the moving body and /LI the mass. The potential energy is

where g is the constant of gravitational acceleration. Then we have

W
where g is taken equal to ; q obviously denotes the maximum height

attained, and t the instant at which it is attained. On solving for q we obtain
the well-known formula

Example 2. The Pendulum. Here q denotes the angular displacement and
/*= A the moment of inertia of the pendulum. The potential energy, reckoned
from g=7r/2 as zero, is

U= Dcosg.

THE THEORY OF HAMILTON AND JACOBI 41

We find

(10) p= V2A VW-hD cos q,

-j D cos q 2 \/W |D- 20 sin 2
and if we put

then

, o _U A r <*?

2 -DJo / . a <y

A/ sin 2 sin 2 -

Sm 2 8m 2

The solution of this equation, which involves an elliptic integral, gives q as
a function periodic in time, and oscillatiag between -\-a and a. For suffi-
ciently small values of a we can write

and obtain the solution in a simple form. We have

Clearly all problems for which every co-ordinate, with the excep-
tion of one, is cyclic, reduce to the case of one degree of freedom. Let

H=H(y 1 ,y 1 ,y a . . .p f ),
the solution will then be represented by

p 2 =a 2 . . . p f =a f
and

S=J>i (?i, W, a 2 . . . a f )dq l9
where p t is found by solving
(11)
Therefore

, o, . . .

W, o, . . . a / )d ?1 (t=2, 3 . . ./).

Example 3. Projectile Motion. Let g x =2 be the vertical co-ordinate,
reckoned positively upwards, and q*=x, q z y the horizontal co-ordinates, then

42 THE MECHANICS OF THE ATOM

and

W=K
Since x and y are cyclic variables, we put

2>a5=2

and obtain

p.= [2m( W- m</z) - a 2 2 -

f* ""k _ / 2 /

/ ~^ == J, [2m(W-^)-a 2 2 -a 3 ^ = ""^
being given by

2wW- a 2 2 a 3 2 =
It follows from this that

The two other equations of motion follow most simply from

mx=p x = a a , my=p y = a s .
We find

Elimination of t from the three equations of motion gives the equations of
the path, which is, of course, a parabola :

These results could also have been found from the second of equations (11'),
without making use of relations involving the time.

Example 4. Heavy Symmetrical Top. In 6 we found for the kinetic energy

and now, in addition to this, let there be the potential energy

U=Dcos0,
so that

Since <f> and ^ are oych'c variables, we have

THE THEORY OF HAMILTON AND JACOBI

(12)
and

In the equation for t we put cos 0= M, and obtain

(13, ' frf<t

where

this is a cubic in u, so the solution of (13) involves elliptic integrals.

The Eulerian angles <f> and may be expressed by similar elliptic integrals. If,
for example, we solve the equations (3), 6, for $ and ^, we obtain, taking (12)
into account,

/I 1 \ <V ( h c>os

^A z A X J i

- (Z 3 (JL 2 COS

A x sin 2

and

(14)

The evaluation the integral type (13) gives t/=i i os as a periodic function
of the time. It oscillates backwards and forwards between two zero points
of F, which enclose an interval in which F is positive. If a 2 is not precisely
equal to a 3 , we have

and

both negative. If a motion is to be possible at all it follows that, somewhere in

the interval (1, +1), F must not be

negative ; it has then two zero points u

and w a which may coincide. If the zero

points are different, it means that the

point of intersection of the axis of the top

with a sphere, described about the centre

of the top, oscillates backwards and for-

wards between two parallel circles. It

describes a curve shown in fig. 1. In the

case of the double root our equations (13)

and (14) fail, but the motion can be easily

calculated in an elementary manner : is then a constant, and we have

the case of a regular precession.

FIG. 1.

44 THE MECHANICS OF THE ATOM

A general rule for the rigorous solution of the Hamilton-Jacobi
differential equation (5) cannot be given. In many cases a solution
is obtained on the supposition that S can be represented as the sum
of /functions, each of which depends on only one of the co-ordinates
q (and, of course, on the integration constants a 1 . . . a/) :

(15) 8=8^0 +...+SX?,).

The partial differential equation (5) then resolves into / ordinary

differential equations of the form

or, if we solve for -,
dq k

The differential equation (5) is said in this case to be soluble by
separation of the variables, or, for short, to be separable.

The case dealt with above, where all co-ordinates with the excep-
tion of one (y t ) are cyclic, can be regarded as a special case of this.
We make the hypothesis

and the differential equation becomes

/ as ss\

"("*;*)-

which agrees exactly with (11).

SECOND CHAPTER

PERIODIC AND MULTIPLY PERIODIC MOTIONS

9. Periodic Motions with One Degree of Freedom

WE have seen that, in the case of systems of one degree of freedom,
new variables <, a can be introduced in place of the variables q, p,
such that a is constant and <f> is a linear function of the time. The
variables </> and a are not, however, determined uniquely in this
way ; we can in fact replace a by an arbitrary function of a, whilst
</) is multiplied by a factor dependent on a.

For periodic motions it is an advantage to make a perfectly definite
choice of <f> and a. Now there are two kinds of periodicity. Either
different values of q correspond to different positions of the system
and q and p are periodic functions of the time, and also of the
variable ^ which is linearly connected with the time, in which case
there is a quantity o> such that

for all values of q ; or else the configuration of the system is the
same for any two values of q differing by a constant quantity,
which we shall take to be 2?r. This increase in q of amount 2?r
always takes place during the same time, and then

) =q((/>) +27T.

In the first case we speak of libration, in the second of rotation.
Examples of these are the oscillating pendulum and the rotating
pendulum respectively (see below).

In both cases we shall choose ^ in a particular way, namely, in such
a way that it increases by 1 during one period of the motion, in which
case we shall denote it by w. Let the corresponding conjugated
variable be J. We call w an angle variable and J an action variable.

If we consider S to be a function of q and J, then

46 THE MECHANICS OF THE ATOM

(cf. (1), 8, remembering that w and J are particular examples of
the quantities there written < and a), and the differential quotient
of w along the path is

That the period of w shall be 1 therefore implies that

a fas

where the symbol $ denotes that the integration is to be extended
over one period, i.e. in the case of libration, over one back and forward
motion of q, and in the case of a rotation, over a path of length 27r.
We can clearly satisfy this requirement by putting

(i)

or, in other words, by making J equal to the increase of S during one
period. 1

The variables w, J may therefore be introduced in the following
way. If H is given as a function of some canonical variables q, p, the
action function

S=S(y, a)

is determined by integration of the Hamilton-Jacobi equation, and
the integral

J=(T) dq

J fy

is calculated as a function of a or W. J is then introduced into S in
place of a or W.

By means of the transformation

=s J + constant would also satisfy the condition. The general transformation
(</>, a)->(w, J)

which satisfies the periodicity conditions postulated contains in fact another arbi-
trary constant in addition to the phases constant for w. Its generator is

The method for determining J given above is equivalent to putting q =0 ; it is
particularly usefu] in the quantum theory.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 47

p and q will become periodic functions of w with the period 1, and
H will be a function W of J alone. From the canonical equations it
follows that for any one possible motion of the system

J=constant
and

(3) W

Since we have chosen w so that it increases by 1 during each period
of the motion, it follows that W is a function which increases con-
tinuously with J ; v must be a positive number, it is equal to the
number of periods in unit time, or the frequency of the motion.

If the variable <f> conjugate to a is already known, J can be found
from the equation

The equations of transformation are then

A consequence of the above determination of J as the increase in S
during one period, is that the function

(4) S*=S-wJ

is a periodic function of w with the period 1. Conversely this require-
ment may also be used for the unique determination of the magnitude
J, which is fixed except for an additive constant by $dw=l, in which
case equation (1) is obtained. In place of S the function S* can be re-
garded as the generator of the canonical transformation which trans-
forms q and p into w and J . Comparing the transformation equations
(2) with equations (2 )of 7, it will be seen that S satisfies the equation

9
whence

and this implies that S* is the generator of the transformation

THE MECHANICS OF THE ATOM

The calculation of the integral J necessitates study of the connec-
tion between q and p as given by the equation
(6) HfopHW.

Let this relation be represented by a family of curves with the para-
meter W in the (p, q) plane. The cases of libration and rotation are
then represented by two typical figures (figs. 2 and 3).

FIG. 2.

In the case of libration a closed branch of the curve (6) must exist,
and J denotes the enclosed area, which, by (19), 7, is a canonical
invariant.

For rotation, p must be a periodic function of q with period 2?r, and
J denotes the area between the curve, the #-axis, and two ordinates
at a distance 2?r apart.

For the purpose of illustration we shall deal with the case of
classical mechanics on the assumption of a co-ordinate system at rest.
By (8), 8,

In order that libration may occur, the expression under the square
root must have two zeros, q' and q" between which it is positive ; then
p vanishes only at the limits of the interval (q' 9 q"). In order that a
closed loop may be formed from the two branches of the curv$ (6), it

dp
is further necessary that -=- shall be infinite at q' and y". Now

the condition is therefore fulfilled, provided is not at the same

time zero, i.e. provided g', q" are simple roots of the expression under
the root sign. In this case the resulting curve, which is symmetrical
about the q axis, will be traversed completely, and always in the same
sense. Then, by (8), 5,

PERIODIC AND MULTIPLY PERIODIC MOTIONS 49

and thus pdq is always positive ; therefore on the outward journey

(dq>0) the upper branch (p>0) will be traversed, and in the return

journey (dq<Q) the lower branch (p<Q)

will be followed. The co-ordinate q tra-

verses the whole region between the zero

points q' and q" ; these zero points form

the limits of libration.

If W be varied, the corresponding
curves lie within one another, without
intersecting. If W be decreased, the
zero points move towards one another Fia * 4>

and converge to a point, provided no new zero points occur be-
tween them. This point we call the libration centre ; at it

It corresponds to a state of stable equilibrium of the system, since the
movement resulting from slight alteration of the initial conditions re-
mains in its vicinity. If new zero points occur between q' and q", they
coincide at their first appearance as W is decreased, and here too

"-0.
dq

In this case the state of equilibrium is unstable, since, for a small
variation of W, the motion does not remain in the immediate vicinity
of the equilibrium position.

If W be increased it may happen that at q' or q" the derivative

d\J

vanishes, in which case we again have a condition of unstable

equilibrium. For such values of W it may also happen that the
motioi? approaches the state of unstable equilibrium asymptotically
with the time. The motion is then said to be one of limitation.

In order that rotational motion may occur, U must first of all be
periodic in q, and we assume as period 2?r ; further, the quantity
under the root sign must always be positive.

In order to illustrate these ideas we consider the pendulum, for
which all three possibilities rotation, libration, and limitation
occur. We have (see (10), 8)

cos q D>0 ;

the curves (6) have therefore the form shown in fig. 5.
For

50 THE MECHANICS OF THE ATOM

the curves contract on the libration centre =0. For

-D<W<D
we have libration between the limits

For

j*=-coB- 1 (--D/W).
W>D,

on the other hand, we have rotation, the pendulum rotating always
in the same direction. In the limiting case

it approaches asymptotically the position 377.

7 1

FIG. 5.
In this case the integral

(7) J=(()V2A\/W+D cos qdq

for the libration motion is an elliptic integral. Only in the case when
the libration limits lie close together (on the two sides of the libration
centre) can it be approximated to by a simple integral. The calcula-
tion then corresponds to that for the linear harmonic oscillator, to
which we now turn.

Example. Linear Harmonic Oscillator. In 7 we have already found the
variables </> and a, and according to (18), 7, q has the period c5=27i in </>. The
variables w and J are introduced in accordance with the formulae.

and

ad</)

PERIODIC AND MULTIPLY PERIODIC MOTIONS 51

where

The motion will now be represented by

q= - ( ) sin 2nw

2n \mv/

p (2m vJ)* cos 2nw.
The energy becomes

from which the relation

is at once evident.

In order to show how the change to angle and action variables can be made
without a knowledge of < and a, we shall once again carry out the calculations
for the oscillator, starting out from

H =.+?y.

2m 2
If we put this expression equal to W, then

where, for shortness, we write

_ __ . fi
MO) 2

From this it will be seen that the libration limits are situated at q = + and
q a . We calculate the integral

by introducing the auxiliary variable 0, by means of the equation

q= a sin <
</> goes frern to 2n during one period of the motion. We obtain

J=^f%os W = 2 -?W
(*> Jo o>

and, consequently, the energy or the Hamiltonian function is given by

(8) W=H=vJ,

where we have put

a)2nv.

To express the co-ordinate q in terms of the new variables w, J, and hence in
terms of the time, we do not need to calculate S itself. We have

in which p is to be considered as a function of q and J :
p= (2mvJ

52 THE MECHANICS OF THE ATOM

We obtain

mvdq I . _ 1 /2jr 2 vm\*

where

We then have for ^
( 10) #

For the pendulum with small amplitude the corresponding formulae are :

10. The Adiabatic Invariance of the Action Variables and
the Quantum Conditions for One Degree of Freedom

Now that we have considered in detail the mechanics of periodic
systems with one degree of freedom we can pass on to the question
how, and how far, the mechanical principles may be applied to the
mechanics of the atom, the chief characteristic of which is the
existence of discrete stationary states.

We have a typical example of this application in Planck's treat-
ment of the simple linear oscillator (see 1). The stationary states
were defined there by the condition that the energy should have only
the discrete values

(1) W=n.hv (n=fi y 1, 2 . . .)

The question now occurs whether it is possible to deal in a similar
way with the general case of periodic systems of one degree of
freedom.

In the development of the mechanics of the atom the method of
discovery has been to retain the classical mechanics as far as possible.
Planck's theory of the oscillator, for example, is based on the view
that the motion of the vibrating particle takes place entirely in
accordance with the classical principles. Not all motions, however,
with arbitrary initial conditions, i.e. values of the energy, are equally
permissible ; certain motions, characterised by the energy values (1),
occupy a preferential position in the interaction with radiation, on
account of a certain inherent " stability " ; these motions con-
stitute the " stationary states."

The endeavour to retain the classical mechanics as far as possible

PERIODIC AND MULTIPLY PERIODIC MOTIONS 53

having proved to be a fertile method, we take as our first require-
ment that the stationary states of an atomic system shall be cal-
culated, as far as possible, in accordance with the laws of classical
mechanics, but the classical theory of radiation is disregarded. For
this requirement to be fulfilled it is essential that the motion shall
be of such a nature that the term " state " is applicable to it. This
would not be the case, for example, if the path went off to infinity or
if it approached a limiting curve asymptotically. In the case of
periodic motions, however, the system may well be said to be in a
definite state. We shall see later that there is still a further class
of motions, the so-called multiply periodic systems, to which the
same applies. On the other hand, the development of the quantum
theory has shown that these probably exhaust the types of motion
for which classical mechanics gives a valid description of the station-
ary states ; we shall restrict ourselves in this book essentially to this
domain.

The next question concerns the manner in which the stationary
motions are to be selected from the continuous manifold of those
mechanically possible motions. We shall first try to give an answer
to this in the case of periodic systems with one degree of freedom.

At first sight we might be inclined to apply to the general case the
formula (1) established for the oscillator. Since, in general, v is a
function of W, a transcendental equation would have to be solved to
determine W. This method of procedure must, however, be rejected ;
it leads in certain instances to results which are in contradiction
with observation (e.g. in the case of diatomic molecules, the atoms
of which are coupled non-harmonically with one another) and, further,
it cannot be sustained theoretically.

The quantum conditions by means of which the stationary orbits
are selected can be expressed in the form that a certain mechanically
defined magnitude is an integral multiple of Planck's constant h. In
the case of the oscillator this magnitude is W/i/ ; the question is, what
is to take the place of this quantity in the case of other systems ?

We now examine the conditions to be satisfied by a magnitude in
order that it may be " quantised ' ' in this manner. In the first place
it must be uniquely determined and independent of the co-ordinate
system. This, however, would do but little to narrow the choice, and,
if nothing more was known, the results of a comparison of the theory
with observation would be our sole guide. In this connection Ehren-
fest has, however, done much for the development of the quantum
theory by advancing a postulate which makes possible a purely
theoretical determination of the quantum magnitudes.

54 THE MECHANICS OF THE ATOM

The novelty of Ehrenfest's idea lies in regarding the atoms not as
isolated systems, but as subject to external influences. We have pos-
tulated above that classical mechanics shall be valid for isolated sys-
tems in the stationary states ; following Ehrenfest, we now require
that classical mechanics shall also be retained as far as possible in the
presence of external influences.

We must now investigate to what extent this is possible without
coming into conflict with the principles of the quantum theory.
According to these the magnitude to be quantised can change only
by integral multiples of h. If, therefore, an external influence is not
sufficient to cause an alteration of magnitude h, the quantum magni-
tude must remain absolutely unaltered.

It is first necessary to find the conditions which determine whether
the external influence is capable of causing such an alteration (known
as a quantum transition or jump) or not. It is known from experi-
ence that quantum transitions can be caused by light and by mole-
cular impacts. In these cases we have to deal with influences which
vary very rapidly. If we consider, on the other hand, actions which
change very slowly slowly, that is to say, in comparison with the
processes occurring within atomic systems- e.g. the switching on of
electric or magnetic fields, experience teaches us that in this case no
quantum transitions are excited ; neither emission of light nor other
processes associated with quantum transitions are observed in such
cases.

The quantum transitions certainly take place in a non-mechanical
manner. The maintenance of the classical mechanics, required by
Ehrenfest in the case of external influences, is then possible only if no
quantum transitions are excited by these influences, i.e. only in the
case of processes which vary very slowly.

Ehrenfest calls this postulate, that, in the limiting case of infinitely
slow changes, the principles of classical mechanics remain valid, the
adiabatic hypothesis, by analogy with the terminology of thermo-
dynamics * ; Bohr speaks of the principle of mechanical transform-
ability.

This postulate severely restricts the arbitrariness in the choice of
the magnitude to be quantised. For now only those quantities are
to be taken into account which, according to the laws of classical
mechanics, remain invariant for slow variations of external influ-
ences ; following Ehrenfest, we name them " adiabatic invariants."

1 Proc. Kon. Akad. Amsterdam, vol. xvi, p. 591, 1914, and Ann. d. i'hystk, vol. li,
p. 327, 1916. Ehrenfest found his " adiabatic hypothesis " in an altogether different
way, namely, by an examination of the statistical foundations of Planck's radiation
formula.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 55

In order to make clear the conception of adiabatic invariance, we
consider the example of a simple pendulum consisting of a bob of
mass m on a thread whose length I is slowly decreased by drawing
the thread up through the point of suspension. This shortening
causes an alteration of the energy W and the frequency v of the
pendulum ; we can show, however, that for small oscillations the
magnitude W/v remains invariant.

The force which keeps the thread of the pendulum taut consists of a
gravity component mg cos </>, and the centrifugal force ml</>*, the
angular displacement being <. The work done then during a shorten-
ing of the thread is

A = $mg cos <l>dl $ml(J> 2 dL

If this shortening occurs so that its progress in time has no relation to
the period of oscillation, and sufficiently slowly for us to be able to
ascribe an amplitude to each single period, we can write

dA = mg cos <fxll v

where the bar denotes an average taken over one period. For small
oscillations we can write

cos <=-

If this is substituted, rfA resolves into an expression mgdl, which
represents the work done in raising the position of equilibrium of
the bob of the pendulum, and a second expression

which denotes the energy communicated to the oscillation . The mean
values of the kinetic and potential energy of the pendulum are the
same, and are thus equal to half the total energy W :

W m T" m

T-2^-2^ 1 '
Substituting, we have

Since now the frequency v is proportional to , and therefore

dv dl

56 THE MECHANICS OF THE ATOM

it follows that

This differential equation expresses the way in which the energy of
oscillation is connected with the frequency for an adiabatic shorten-
ing, and it follows by integration that

=constant
v

as asserted.

A similar argument applies when v is slowly varied by some other
external influence. Since the harmonic oscillator is mathematically
equivalent to a pendulum with an infinitely small amplitude of oscilla-
tion, W/F is constant in that case also ; Planck's quantum condition

(1) is consequently in agreement with the adiabatic hypothesis. It
can be shown, on the other hand, that for other periodic systems of
one degree of freedom W/v is not an adiabatic invariant.

We remember that, according to (8), 9, in the case of the harmonic
oscillator the magnitude W/v is also the action variable J. This
suggests

(2) 3=nh

as a general quantum condition for systems of one degree of freedom.
The quantity J fulfils the requirement of uniqueness, since it is inde-
pendent of the co-ordinate system (on account of the invariance of
tfdpdq, cf . 7), and we shall show now that it is an adiabatic invariant.

The general proof of the theorem of the adiabatic invariance (or,
as Bohr calls it, of the mechanical transformability) of the action
variables was carried out by Burgers x and Krutkow, 2 who at the
same time treated the case of several degrees of freedom. 3

We think of a mechanical system of one degree of freedom subject
to an external influence. This can be expressed by introducing in
the equation of motion, in addition to the variables, a parameter
a(t) depending on the time. We consider now an adiabatic variation
of the system to be such that it has firstly no relation to the period
of the undisturbed system, and secondly, that it takes place suffi-
ciently slowly for a to be regarded as indefinitely small. We assume
further that, for a certain range of values of a, the motion for con-

1 J. M. Burgers, Ann. d. Physik, vol. lii, p. 195, 1917.

2 S. Krutkow, Proc. Kon. Akad. Amsterdam, vol. xxi, p. 1112, comm. 1919.

8 Other proofs on more general assumptions have been given by M. v. Laue, Ann.
d. Physik, vol. Ixxvi, p. 619, 1926, and P. A. M. Dirac, Proc. Roy. Soc., vol. cvii,
p. 725, 1925.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 57

stant a is periodic, and that we can introduce angle and action vari-
ables w and J. We then have the theorem :

The action variable J is adiabatically invariant, provided the
frequency does not vanish.

The Hamiltonian function

tt(p, q, a(t))

is dependent on the time ; the energy therefore is not constant,
but the canonical equations

.an .an

* = a/ P== 8q
are still valid.

We imagine now the canonical transformation carried out which
transforms, for constant a, the variables q, p into the angle and action
variables w, J. It is useful to write the transformation in the form
(c/. (1), 7, and (5), 9)

as*

_
~

The function S* depends on the parameter a in addition to the vari-
ables q and w ; S* is therefore dependent on the time and, by (1),
7, H becomes

The transformed canonical equations are therefore

an a/as*\

j = _l 1 -^).

Since H depends on the action variables only

___^/as*\ _^>_/as*\

in which the differentiation with respect to t and a is to be carried
out for fixed values of q and w, and the differentiation with respect
to w for fixed J and a. The change of J in a time interval (t l9 2 )
will be

58 THE MECHANICS OF THE ATOM

Since the variation of a is supposed slow and not connected with
the period of the system, a can be brought outside the integral sign.
We shall carry out the proof of the invariance of J by showing that

is of the order of magnitude d(t 2 t l ) ; for from this it follows that
in the limit of infinitely slow variation (d->0) and for finite d(t 2 t 1 ),
the variation of J becomes zero.
Since (by 9) S* is a periodic function of w, the same is true of

; this remains true if we introduce the variables w, J, a. The
da

integrand of (3) is therefore a Fourier series

without a constant term (this we signify by the dash on the summa-
tion sign). If we write w as a function of the time, the integral to be
estimated becomes :

The integrand is no longer exactly periodic in t as the A T 's, v, and S
depend on #, which varies with t ; however, in the neighbourhood
of a certain instant of time, which we can take as =0, the A T 's v 9
and 8 can be expanded in powers of the alteration in a from its value
at =0 ; this alteration is small, as the expansion is not going to be
used for values of t greater than the periodic time T, and a is to be
taken so small that the variation of a in a period of the undisturbed
motion is small. Indicating differentiation with respect to a by a
dash, and values of the A T 's, v, and S for the value of a at t==0 by
suffixes zero, the integrand then becomes

(4)

If we integrate this expression over a period of the first term, we
obtain expressions of the order of magnitude dT and dT 2 . We now
imagine the expansion (4) carried out at the beginning of the interval
(t l9 1 2 ) and the integral taken over one period of the first term. We
then imagine a new expansion (4) carried out at the beginning of the
remaining interval and the integral taken once more over one period
of the first term. We continue this process until the interval (t^ t 2 )

PERIODIC AND MULTIPLY PERIODIC MOTIONS 59

is all used up. The last integral will, in general, not be taken over
a full period ; it has a finite magnitude even when t^t l is inde-
finitely great. It is seen that if T remains finite over the whole range
of integration, i.e. if v does not vanish, the whole integral will be of
the order of magnitude d(t 2 tj).

We have proved by this the adiabatic invariance of J. On the
basis of this invariance, and the special result in the case of the
oscillator, we are led to the choice of J as the quantity to be quantised
in general. This assumption has been confirmed by the further
development of the quantum theory. We state it in the following
way :

Quantum Condition. In the stationary states of a periodic system
with one degree of freedom the action variable is an integral multiple
of h:

The energy steps, as functions of the quantum number n, are also
determined by this quantum condition. 1

The experimental method of electron impact, mentioned in the
introduction, enables the energy levels of the atomic systems to be
determined in a purely empirical manner. Comparison of these
determinations with the theoretical values provides a test of the
foundations of the quantum theory as far as they have hitherto been
developed.

As mentioned in the introduction, the interaction of the atomic
systems with the radiation is governed by a further independent
quantum principle, Bohr's frequency condition,

which determines the frequencies of the emitted and absorbed light.
W (1) and W (2) denote here the energies of two stationary states and v
the frequency of the light, the emission or absorption of which is
coupled with the transition of the system from the state 1 to the state
2. In the case of emission (W (1) > W (2) ) our formula gives a positive i>,
in the case of absorption (W (1) <W (2) ) a negative v.

A much more rigorous test of the quantum rules is made possible by
applying Bohr's frequency condition to the frequencies of spectral
lines.

1 This quantum condition was given first in geometrical form by M. Planck,
Vorlesungen iiber die Theorie der Wdrmestrahlung, first edition, 1906, 160. It is
to be found also in P. Debye, Vortrage uber kinetische Theorie der Materie und, der
(Wolfskehl Congress), p. 27, 1913.

60 THE MECHANICS OF THE ATOM

11. The Correspondence Principle for One Degree
of Freedom

The fundamental postulate of the stability of atoms referred to in
the introduction is satisfied by the two principles of atomic mechanics
given in 10. We now inquire to what extent they are in agreement
with the other fundamental postulate, that the classical theory shall
appear as a limiting case of the quantum theory.

Planck's constant h occurs as a characteristic magnitude in both
quantum principles, and is a measure of the separation of the quan-
tum states. Our requirement signifies that the quantum laws shall
tend into the classical ones as limits as A-^0 ; the discrete energy
steps then converge to the continuum of the classical theory. The
frequency condition requires special examination : we have to see
if the frequencies calculated by it agree in the limit with those to be
expected from the classical theory.

The radiation from a system of electrically charged particles with
charges e k at the points r fc is determined, according to the classical
theory, by the electric moment

If the energy radiated in the course of one period is small, the damp-
ing may for the present be neglected. For a system of one degree of
freedom, such as we consider here, the rectangular co-ordinates of the
charged particles will be periodic functions of

with the period 1. Since the same will hold for p, each component of
the electric moment may be developed in a Fourier series of the type

The C T 's are complex numbers ; since, however, the electric moment is
real, the C T 's and C_ T 's must be conjugate complex quantities.

On this basis the time variation of the electric moment can be con-
sidered as a superposition of harmonic oscillations with the frequency
TV ; the amplitudes of the corresponding partial oscillations of the
moment are given by the values of | C T | and their energies are pro-
portional to the values of | C T | 2 . According to the classical theory
the r-th oscillation component would give rise to a radiation of
frequency * 0W_dW

(1) * =TV - T -8J ~djTr _

1 Since T as well as i always occurs in the Fourier expansion, the sign of the
expression for the classical radiation frequency has no significance,

PERIODIC AND MULTIPLY PERIODIC MOTIONS 61

We compare with this the quantum frequency 1

If the quantum number n decreases by r in the quantum transition
under consideration, then

AJ=J 2 ~J 1 =(^ 2 n^h, rh,
so that we can write

~ AW

( ' "* W

If we proceed to the limit A->0, or AJ/r->0, then (2) and (1) become
identical.

For the case of a finite h we can state the relation between the two
frequencies (1) and (2) as follows :

The quantum theory replaces the classical differential coefficient by
a difference quotient. We do not proceed to the limit of infinitely
small variations of the independent variables, but stop at finite in-
tervals of magnitude h.

The transition between two neighbouring quantum states for which
r=l is associated with, or " corresponds " to, the classical funda-
mental vibration ; a transition in which n changes by r corresponds
to the classical rth overtone VTV.

This relation between classical and quantum frequencies forms the
substance of Bohr's correspondence principle.

According to this correspondence the quantum frequency v is, in
general, different from the classical frequency TV. If, instead of pro-
ceeding to the limit #->0, we go to the limiting case of large quantum
numbers n, and consider only such changes of n as are small com-
pared with the value of n itself, then, on account of the monotonic
character 2 ( 9) of the function W(J), the difference quotient will
very nearly coincide with the differential coefficient and we obtain
the approximately correct equation

v=^rv=(n l n 2 )v, (n l large, n 1 w 2 small compared to n t ).

If n 1 n 2 is no longer small in comparison witt n l9 then the agree-
ment between the classical and quantum frequencies will not be so
good. For a given n t the correspondence between the frequencies in
the case of emission (n^>n^ has a limit, inasmuch as r=n l w 2 can-
not be greater than n v

1 Positive v in the expression for the frequency given by the quantum theory
denotes emission, negative v absorption.

8 A function of one variable is said to be monotonic when its differential coeffi-
cient has the same sign for all relevant values of the independent variable.

62 THE MECHANICS OF THE ATOM

The two quantum principles hitherto given do not, however, pro-
vide a complete description of the radiation processes. A light wave
is characterised not only by a frequency, but also by intensity, phase,
and state of polarisation. The quantum theory is at present unable
to give exact information with regard to these features. Bohr has,
however, shown that it is possible, by extending the correspondence
principle from frequencies to amplitudes, to make at any rate ap-
proximate estimates regarding the intensity and polarisation.

In order that, in spite of the totally different mechanism of radia-
tion, quantum theory and classical theory may give, in the limit-
ing case of large quantum numbers (or in limit A->0), radiations
with the same distribution of intensity among the component
oscillations, it must be assumed that in this limiting case the Fourier
coefficients C T represent the amplitudes of the emission governed by
the quantum theory. Thus the values of C T must be related to the
probabilities of the transitions necessary in order that the energy
principle may remain valid. By considering the different components
of the electric moment p a determination of the polarisation proper-
ties can be made at the same time as that of the intensities.

The case C T =0 is of especial importance, for here in the classical
case there is no emission of the corresponding frequency, so the corre-
sponding quantum transition should not occur. Since, however, the
correspondence principle only gives a relation between radiation
phenomena on the classical and quantum theories the results de-
duced from it concerning the possibility of quantum transitions hold
only in those cases where the atomic system is interacting with radia-
tion. They need not hold for impacts between atomic systems.

On the basis of the correspondence principle we can deal effectively with the
difficulties which we have met with in the introduction ( 1, 2) in the case of the
resonator. The expression for the displacement q as a function of tho angle
variables is by (9), 9 :

J \*

sm xnw ;

=( 2 __

this is clearly a Fourier series with only the one term T= 1, according as we
take the positive or negative root. According to the correspondence principle,
therefore, the quantum number can, in the case of the resonator, change by 1
only, giving

v=v.

The correspondence principle leads then to the result that a resonator behaves
on the quantum theory, as far as the frequency of its radiation is concerned,
exactly as it would do according to the classical theory. In the case of other
atomic systems, however, we shall see that this is by no means the case.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 63

12. Application to the Rotator and to the Non-harmonic

Oscillator

]. THE ROTATOR. By (1), 6, the Hamiltonian function is

where p is the momentum conjugated to the angle of rotation 0, and
signifies angular momentum. In this case

J =^>pd(f>=

since the system assumes the same aspect each time < increases by
27T. The energy, as a function of the action variable, and then of
the quantum number m, becomes

(1) W=H

v '

and the angle variable is

where

This calculation can be applied to the motion of diatomic molecules
and concerns two classes of phenomena : the theory of the rotation
band spectra of polar molecules and the theory of the specific heats
of gases. The simplest model of a diatomic molecule is that known
as the dumb-bell model ; the two atoms are regarded as massive par-
ticles at a fixed distance I apart, and it is assumed that the structure
rotates^ with moment of inertia A, about an axis perpendicular to the
line joining the atoms. A rigorous foundation for these assumptions
(i.e. the neglect of the rotation about the axis joining the atoms and
the assumption of a rigid separation) and their replacement by more
general assumptions will be given later.

(a) THEORY OF ROTATION BAND SPECTRA. We assume that the
molecule has an electric moment (e.g. we regard HC1 as a combina-
tion of the H+ and Cl~ ions), in which case it would, according to the
classical theory, radiate light of frequency

0J 47T 2 A

Overtones do not occur. If the particles have the charges e and e,

64 THE MECHANICS OF THE ATOM

the components of the electric moment p in the plane of the rotation
are :

P*=(a?a x l )=el cos 2irw

P=2 /i = e l sin

in which the two signs correspond to the two possible directions of
rotation. The expressions for the components of p in terms of w con-
tain therefore only one Fourier term each, r=I or T= 1.

We should expect that such a molecule, possessing an electric mo-
ment, would radiate according to the quantum theory ; the quantum
frequencies will, however, differ from the classical ones. The energies
of the stationary states are given by ( 1 ) . Since only one Fourier term
occurs, in the motion the quantum number can change by +1 or 1
only, and the Bohr frequency condition gives therefore for the emis-
sion (m+l)->m :

If this formula be compared with that for the classical frequency

it will be seen from the relation

1
2m

that the relative difference between the two frequencies will be the
smaller the greater the value of m.

Except for a small additive constant difference in the frequencies,
the classical theory and the quantum theory both lead to essentially
the same results in this case ; each gives a system of equidistant lines
in the emission and absorption spectrum. This is the simplesfccase of
the empirical band formula first found by Deslandres. It is easy to
see that these lines are to be sought for in the infra-red. In the case of
HC1, for instance, the light H atom of mass 1*65 X lO^ 24 gm. essentially
rotates about the much heavier 01 atom at a distance of the order of
magnitude of all molecular separations, say a Angstrom units or
a . 10" 8 cm., a being of the order of 1. The moment of inertia will
then be

A=a 2 . 1-65 X 10- 40 gm. cm. 2 ,

the frequency of the first line

5X10 11

v sec

-i

PERIODIC AND MULTIPLY PERIODIC MOTIONS 65

and the wave-length

A=-=0-06a 2 cm.
v

Since a is of the order 1, we have to deal with lines on the farther side
of the optically attainable infra-red. These pure rotation bands have
been observed in the case of water vapour (for example). In the case
of a large number of gases, bands have been found which are due to
the combined action of the nuclear oscillations and rotation ; these
exhibit the same type of equidistant lines, but are situated in the
region of much shorter wave-lengths. We shall deal with the theory
of them further on ( 19, 20).

(b) HEATS OF ROTATION OF DIATOMIC GASES. The dumb-bell
molecular model leads also, as is well known, to the correct result
in the theory of specific heats at high temperatures. Three trans-
lational and two rotational degrees of freedom are ascribed to such
a model ; the rotation about the line joining the atoms is not
counted. According to the theorem of equipartition of energy,
which is deduced by applying statistical mechanics to classical
systems, the mean energy |AT is associated with every degree
of freedom without potential energy, and consequently the total
energy -|T would be associated with the five degrees of freedom
mentioned ; the molecular heat is therefore !>R. Now Eucken l has
shown experimentally that the molecular heat of hydrogen decreases
with decreasing temperature ; for T=40 abs. it reaches the value
|R and subsequently remains constant. Hydrogen changes then,
in a sense, from a diatomic to a monatomic gas ; its rotational
energy disappears with decreasing temperature. Ehrenfest 2 has
given the elementary theory of this phenomenon. The mean energy
of a rotator, which can exist only in the quantum states (1), is

w,=^r
2

m=i

where

If the values (1) be substituted for W m we shall have

1 A Eucken, Sitzungsber. d. Prtusa. Akad. d. Ww., p. 141, 1912; see also K.
Scheel and W. Heuse, Ibid., p. 44, 1913 ; Ann. d. Phytiik, vol. xl, p. 473, 1913.

2 P. Ehrenfest, Verhandl d, Deutsch. Physical. Get., vol. xv, p. 451, 1913.

66 THE MECHANICS OF THE ATOM

m=0

where

Ehrenfest calculates the heats of rotation by assuming for the mean
energy of a molecule twice the mean energy of one of our rotators,
because the molecule has two perpendicular axes about which it can
rotate. The heat of rotation per gram molecule is then

*- f

We examine the behaviour of this expression for low and high tem-
peratures.

For small values of T we have large a ; thus er* is very small, and
the series for Z may therefore be broken off after the first two terms :

Z-l-Hr*
log 7=0-*,
consequently

c^RaV-*,

and this expression tends to zero with decreasing T (increasing cr).

For large values of T, o is small, and then the sum in the expression
for Z may be replaced by an integral

Z=J e-* m \lm=-lj-

log Z= % log or+coristant,

consequently t

c r =R.

The heat of rotation gives rise therefore, with increasing tempera-
ture, to an increase of the total molecular heat from 211 to R.

Ehrenfest's theory can, of course, give only a rough approximation
to the actual state of affairs, since the two rotational degrees of
freedom are not independent of one another. A more rigorous in-
vestigation must take account of the motion of the molecules in
space. 1

2. THE NON-HARMONIC OSCILLATOR. We shall consider the case
of a linear oscillator of slightly non-harmonic character, i.e. with a

1 See the detailed treatment by F. Eeiche, Ann. d. Physik, vol. Iviii, p. 657, 1919,
or see C. G. Darwin and R. H. FoVler, Phil. Mag., vol. xliv, p. 472, 1922.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 67

system of one degree of freedom for which the Hamiltonian function
is given by

(3) H = ri +- t o

where a is small.

Our first object will be to find the relation between the action vari-
able J and the energy \V in the form of an expansion in powers of a.

We have

where

We write this in the form

For small values of a, two roots, which we take as e l and e 2 , lie in the

I~2W

neighbourhood of ib . / - -, and the motion takes place between
/ V mo) 2

them ; the third root, e 3 , is large compared to e l? e 2 , and has the
opposite sign from a (af(q) must be positive for values of q lying
between e^ and e 2 ). We write therefore

(5)

and obtain the following expansion for J :

y=*i^J.~3 l ~J n +...};
where

We transform these integrals by means of the substitution (cf.
Appendix II)

(6)

If g oscillates from one the libration limits e to the other e 2 , and

77 . 77

back, increases from ~ to +27r. We then find :

68 THE MECHANICS OF THE ATOM

e 2 _ e 2p2ff / e _ e \ 2/27r "|

+ - - - sin ^ cos 2 ^ +( -^T - ) sin 2 iA cos 2 ^

2 J \ 2 1 J J

or, on inserting the values of the integrals,

e i) a -* c ia] ' J o-

To determine the roots e^ and e 2 we write q as a power series in a
and then find for what values of the coefficients the polynomial/ (q)
vanishes. We thus find

where

_5
2

To obtain the third root we find for what values of the coefficients in

the function /(g) vanishes. We thus find

If W . \

(8) e s =-a + -a 2 + . . . , a

. . . , .

If we now introduce these expressions into the equations for J ,
and J 2 , we obtain, after a somewhat lengthy calculation,

W/ 15 W

If we further substitute the first approximation

W W( >T T

W = -J=VoJ

for W within the brackets, we get finally

PERIODIC AND MULTIPLY PERIODIC MOTIONS 69

.
It will thus be seen that the frequency v= - is not V Q , but, to this

u J

degree of approximation,

In the case of radiation from an atomic system which may be repre-
sented approximately by a non-harmonic oscillator it becomes of
importance to determine which transitions between the energy steps
given by (9) are permissible according to the correspondence principle.
In order to find this, we calculate q as a function of the angle variable
w. The latter is given by

_aS rdp _ fm dWc dq

w ~w jan v 2^ rfjj

and thus, from the expansion (5), we have, to the order required,

/ m d\V r dq /

V *2ae 3 dJ J V(e 1 q)(q e 2 )\

__ /~~ m ~ (AV /K l " ^
V '2ae^ dJ\ "2c.

The integrals

.= f

t = __

l J V^-qKq-

may also be calculated by means of the substitution (6), their values
being

If now we substitute the values (7) and (8) found above for e l9 e 2 ,
we get

...

]

If we now neglect the terms in a 2 , we can put

and obtain

(10) w=J-L +a / 2l/ J cos

; 27r V 2w 6 m 3

70 THE MECHANICS OF THE ATOM

It follows from (6) and (7) that, neglecting terms in a 2 ,

q=aq 1 +q () smifj )
where sin iff may be calculated from (10). To the same order we get

?o 2

q=q Q sin %TTW a- - -(3 +
*

and finally
( 1 1 ) ( L =^

in 2nw <ir (3 +
2

The deviation of the co-ordinate q from its value in the case of the
harmonic oscillator (# 0) is of the order a, whereas the energy
difference is of the order a 2 . The mean value of the co-ordinate will
not be zero, but to our degree of approximation will be given by

Jn the case of the non-harmonic oscillator, therefore, the co-ordinate
oscillates about a mean position differing from the position of equi-
librium. The oscillation is not harmonic, for overtones occur, the
first of which has an amplitude of the order a.

On the basis of the correspondence principle, the appearance of
overtones in the motion of the system implies that quantum transi-
tions are possible for which the quantum number alters by more
than one unit. The probability of an alteration in the quantum
number of 2 is of the order a 2 (i.e. the square of the amplitude of the
corresponding oscillation).

The fact that the mean value of the displacement does not vanish,
but increases in proportion to the energy, has been used by Boguslaw-
ski 1 in explaining the phenomena of pyroelectricity. He imagines
the (charged) atoms of a polar crystal bound non-harmonically in
equilibrium positions, so that with increasing temperature (i.e.
energy) a mean electric moment will arise. In his first calculation
Boguslawski took for the mean energy the classical value JfeT but later
introduced the quantum theory by using for the mean energy Planck's
resonator formula ((5), 1).

1 S. Boguslawski, Physikal. Zeitschr., vol. xv, pp. 283, 569, 805, 1914. The problem
of the non-harmonic oscillator was first considered by Boguslawski, in an attempt to
explain pyroelectricity by means of the quantum theory. The phase integral is
actually a period of the elliptic function belonging to f(q) and may be represented
exactly by means of hypergeometric functions. In the physical application,
Boguslawski restricts himself to small values of a, and arrives at the same final
formula as that given in the text.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 71

The theory of the non-harmonic oscillator finds a further applica-
tion in the explanation of the increase in the specific heat of solid
bodies at high temperatures above Dulong and Petit's value, 1 and
also in the explanation of band spectra (see 20).

13. Multiply Periodic Functions

Before we can proceed to apply our results to systems of several
degrees of freedom we must introduce the conception of multiply
periodic functions, and examine some of their properties.

Definition 1.- A function F(x x . . . x f , y 1 . . .) is periodic in the
variables x . . . x f , with the period a> having the components

>!, 0> 2 . . . 0) f ,

(1) F^J+O)!, x 2 +oj 2 . . . x / +a / )=F(a 1> x 2 . . . x f )

identically in x t . . . x f .

If x l9 x 2 ... x f be considered as co-ordinates in /-dimensional space,
each period corresponds to a vector in this space.

If in (1) (x l9 x 2 . . . x f ) be replaced by (XiS> l9 x 2 & 2 . . . x f a) f ),
and this operation be repeated indefinitely, the truth of the following
theorem will be found to hold.

Theorem 1. A function which has the period o> has also the
period TO), i.e. the period with the components TO^, To> 2 . . . TO)/,
where r is an arbitrary integer (positive or negative).

If the function F has the period a/, in addition to o>, it will be seen,
by replacing (x l9 x 2 . . . x f ) in (1) by (Xi+tii, x 2 +& 2 . . . x f +a>/),
that the following also holds.

Theorem 2. The vectorial sum o>+o/ of two periods o> and a/, i.e.
the vector having the components

is likewise a period.

By combining the theorems 1 and 2 we have the general
Theorem 3. If a function has several periods

then every integral linear combination of these periods

1 M. Born and E. Brody, Zeitechr. /. Physik, vol. vi, p. 132, 1921 ; for detailed
list of literature, see M. Born, Atomtheorie des fcsten Znstandes, Leipzig, 1923,
p. 698.

72 THE MECHANICS OF THE ATOM

(2)

* k

is likewise a period.

Definition 2. Two points (Xj . . . #/) and (x,' . . . x/) are said to
be equivalent if the vector joining them is of the form Z T *^ W '

In order to eliminate trivial exceptional cases we add the con-
dition :

Condition. The function F shall possess no infinitely small
periods, i.e. none, for which the length of the representative vector is
smaller than any arbitrary number.

We shall consider now two periods o> and Ao>, represented by
parallel vectors, in which case A must be a rational number, other-
wise the period (r-fr'A) . & could, by a suitable choice of the in-
tegers r and r', be made arbitrarily small. 1

If now q is the smallest denominator by means of which A may be
expressed in the form p/q, p and q being integers, then &lq is likewise
a period, for by a theorem in the theory of numbers we can always
find two integers r and r', so that

and so

We see now that we can express each period whose vector has a
certain direction as an integral multiple of a certain minimum one.

From this theorem may be deduced a generalisation which is
valid for all periods of a function F. In order to derive it we shall
suppose all the periods arranged in order according to the magnitude
of their vectors :

(3) | o> | <; | a>' | <L | o>" <L .

We select the first period of this series together with the next one
having a vector in another direction. These two periods, which we
now call a)W and a>< 2 >, define a parallelogram mesh in the plane of
the corresponding vectors, with this property, that each vector
which joins two points of intersection of the net also represents a
period.

In this way we can account for all periods whose vectors lie in
this plane, for if there were a vector <, the end point of which did
not coincide with a mesh point (see fig. 6), then there would be a
mesh point at a distance less than | o>( 2 ) | from that end point. If w

1 See Appendix I.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 73

were a period, then the vector represented by this separation would
likewise be a period ; its magnitude would, however, be smaller than
| oV 2 ) |, which is contrary to supposition.

To a>W and o>< 2 > we now add the immediately succeeding period in
the series (3), whose vector does not lie in the plane defined by
and o>< 2 ), and call it dV 3 ). .

These three periods deter- ^/ ^/ /

mine in this way a three-
dimensional lattice, pos-
sessing the property that /?;
each vector joining two #/^
lattice points corresponds /

to a period. In this way / / / >

all periods are accounted / / ~T

for, whose vectors lie /

in the three-dimensional

space defined by ait 1 ), aV 2 ), aV 3 ). If we continue this procedure until
all the periods are exhausted, which must happen when the co-space
becomes /-dimensional, if not before, we shall have proved the
following theorem.

Theorem 4. For each periodic function V(x 1 .../, y l . . .) of
x 1 . . . x f there is a system of periods co^, oV 2 ) . . . oW with the
property that every period a> of the function F can be expressed in
the form

The highest possible value of g, the number of the periods, is equal
to the number of variables/.

Definition 3. A system of periods, possessing the property men-
tioned in theorem 4, is called a fundamental period system.

We have represented all periods of F by means of a ^-dimensional
lattice. For this, of course, only the lattice points are essential, and
not the vectors joining them. If the system wW, &W . . . a>W were
replaced by another system, with the same number (g) of periods

giving the same points of intersection, the new system of periods
oV 1 )', oV 2 )'. . . coW would be equally suitable for the representation

74 THE MECHANICS OF THE ATOM

of the periods of F. The coincidence of the lattice points in the two
systems is achieved if, and only if, the determinant of the r ik s has the
value 1. This determinant represents the ratio of the cell volumes
of the two lattices. Thus we have :

Theorem 5. All fundamental period systems of a function are
connected by integral linear transformations with determinants 1.

In the following we only consider functions for which the number
of periods in the fundamental system is equal to the number /of the
variables in which the periodicity holds. We consider therefore only
functions of periodicity/.

In place of the co-ordinate system x t . . . x f , we introduce in our
/-dimensional space a new co-ordinate system^ . . . w f , whose axes
are parallel to the vectors corresponding to a fundamental period
system for which these vectors form the units ; then the function F,
expressed as a function of the w's, has the fundamental period system

(1, 0, ... 0)

(0, 1,0... 0)

(5) (0, 0, 1 ... 0)

(0, 0, ... 1).

In this case, F is said to have the " fundamental period 1." This
leads us then to

Theorem 6.- By means of a linear transformation of the variables
in which a function is periodic, it may be made to have the funda-
mental period 1.

We shall now see to what extent this co-ordinate system w l9 w 2 . . .
is still arbitrary. First it is clear that by means of a transformation

WzWz+ifj^w^Wz . . . w f , y lt ?/ 2 . . .)

(6)

Wf^Wf+l/ljiWjWs . . . W f , UHJz . . .),

in which each is periodic in all the w k s with the period 1 in each,
the periodic properties of V(x l . . . x f) y t . . .) will not be altered. The
lattice points of the w-co-ordinate system pass to the lattice points
of the w-co-ordinate system by means of a simple displacement.
Further, it is evident that the given transformation is the only one
for which this transference from one set of intersections to the other
is the result of a simple displacement. On passing, for example, from
a point in the w-space to an equivalent one, each w k increases by a
whole number. The w k must increase by the same whole number
when we carry out the similar transition in the w fc -space. The differ-

FEB10DIC AttD MtJLtlPLY PERIODIC MOTIONS 75

ences w k w k must therefore have the same value for all equivalent
points, i.e. they are periodic in the w k and in the w k .

Now there are still other transformations for which the correlation
of the lattice points with values of the w k s will be varied, but for which
lattice point will still coincide with lattice point. To each of the fun-
damental period systems in the x k , referred to in theorem 5, there
corresponds, for example, such a transformation ; these are the integ-
ral homogeneous linear transformations with the determinant 1-

Let us suppose that the most general transformation, which trans-
forms the periodicity lattice into itself, be resolved into such a linear
one and also another transformation. This second transformation
must be of the form (6). The most general transformation is therefore

Theorem 7. All systems of variables, in which a function of
periodicity / has the fundamental period 1 , are connected by trans-
formations of the form (7), where the r ik are whole numbers, the
system of which has the determinant 1 and the t/j t are periodic in
the w k with the period I. 1

The function F may be written very simply with the help of the
variables w l . . . w f . It may be expressed as a Fourier series

(8) V(w l . . . w f )= 2 Cv. r e' 2n ' (TlWl f T w '' f ' T f w f\
'.-v *> TlT *'" f

1 This theorem may be proved analytically as follows : we seek a transformation

fc=4K^2 "jiVi ' ' ')
for which the periodicity of the function

Vfawi . . . w f , y l . . .) = FKw a . . . iv ff $! . . .)
is preserved in the first / variables. If we put

4K+1, w> 2 ">/ 2/i ) =w k ',
then

F(w 1 V 2 / . . . w f ' 9 yi . . .)=-#(#! + 1, w 2 . . . w r y v . . .)

= y(w lt w^. . ,w f1 y 1 . . .)-=F(?/' 1 w> 2 . . . w r y l . . .).
This means, however, that w k ' and w k differ by a whole number :

4(^i + 1 w t ...w f9 y l .. .)=f k (u>i, 3> 9
We likewise conclude that

4(^i i + l w /f 2/i . . .)=-/ fc (Wi,
This, however, is possible only if f k j s O f the form :

4(u>i . . . w f , fr . . .)=^r kl
where $ k is periodic in the w with the period 1.

76 THE MECHANICS OF THE ATOM

which, for conciseness, we write
(8') F(w)

If the function F be multiplied by ?-^W w ) and integrated over a
unit cube of the w-space, we get

The coefficients of the Fourier expansion may therefore be obtained

in the form

(9) (\^(w)e~ >M ^dw

from the function F.

If the function F(w) is real, C Tl . T/ and C- r , . . . - T/ are conjugate
complex quantities.

14. Separable Multiply Periodic Systems

Our next problem is to extend the results found for a system with
one degree of freedom to systems with several degrees of freedom.

In the case of absolutely general systems there is no object in
introducing angle and action variables, since these are associated
with the existence of periodic properties.

We consider first the simple case in which the Hamiltonian func-
tion of the system resolves into a sum of terms, each of which con-
tains only one pair of variables q k , p k :

(1) R=R l (q 1 ,p 1 )+...+R f (q f , Pf ).

The Hamilton- Jacob! equation is solved by separation of the vari-
ables on putting

where the relation

Wl+ . .

holds between the W fc . It is seen that the motion corresponds com-
pletely with that of / independent systems, each of which has one
degree of freedom. We consider now the case where the variation
of each of the co-ordinates q k is periodic in time. The correct
generalisation of the earlier considerations is to define the action
variables by

to express the function S*. in terms of q k and J^, and to put
(2) .-S.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 77

Example : Spatial Oscittator. A massive particle is restrained by any set of
forces in a position of stable equilibrium (e.g. a light atom in a molecule other-
wise consisting of heavy, and therefore relatively immovable atoms). The
potential energy is then, for small displacement, a positive definite quadratic
function of the displacement components. The axes of the co-ordinate system
(x 9 y, z) can always be chosen to lie along the principal axes of the ellipsoid
corresponding to this quadratic form. The Hamiltonian function is then

(3) H= (p x 4V+P* 2 )+J^ 2 +~vy+>, 2 z i ).

It has therefore the form of (1) above, so the motion may be considered as the
resultant of the vibrations of three linear oscillators along the co-ordinate axes.
We have then, by (9) and (JO), ( .) :

(4)

sin 2nw z P z =

where

The energy has the value

( 5 ) W^JoH-VVl v z .] z .

The motion is of an altogether different type according to whether integral
linear relations

T x v x I Vv 1-V=

exist between the v or not. We assume first that such relations do not exist.
We can prove quite generally (see Appendix 1) that in such cases the path
traverses a region of as many dimensions as there are degrees of freedom ; it
approaches indefinitely close to every point in this region. In the case of the
spatial oscillator this region is a rectangular prism parallel to the axes having
sides of lengths

- Vj *> A-- . vj;, -- . vr z

x V * v * z

(spatial Lissajous-figure).

In order to see what special cases may arise when the v's are commensurable
with one another, we consider the simple case where v x v y . This occurs when
the ellipsoid corresponding to the potential energy possesses rotational sym-
metry about the z-axis. The curve representing the path is situated then on
an elliptic cylinder enclosing the z-axis. Corresponding to a given motion we
no longer have uniquely determined values of J^ and J v . for we can rotate
the co-ordinate system arbitrarily about the z-axis, whereby the sides perpen-
dicular to the z-axis of the rectangular prism touching the path will be varied.
J z , on the other hand, remains uniquely determined as the height of the
elliptic cylinder on which the path is situated (if no other fresh commensura-
bility exists). Since the energy is

78 THE MECHANICS OF THE ATOM

(6) W=(J.+

only the sum J x -}- J y is determined by the motion.

If, finally, all three frequencies are equal, the motion is confined to an ellipse
and none of the three J's are now uniquely determined, since the co-ordinate
system may still be arbitrarily rotated. The energy is

(7) W=v(J x +J v !-.!,),

the sum of the .J's will therefore remain unaltered by such a rotation.

If now we ask what are the quantum conditions for such a system of several
degrees of freedom, the obvious suggestion is to put

(8) J*=**A.

In the ease of the oscillator with two equal frequencies v x ~v y the conditions
J x n x hj J v = n y h

are clearly meaningless. Lf, for instance, we have a motion, for which J x and
J y are integral multiples of h for any position of the x- and //-axes, we can
always rotate the co-ordinate system so that this property is destroyed. The
sum ,] x -fF y , on the other hand, remains integral, so that the condition

(i>) .] x +J v ^nh

would still be significant. Since in the expression for the energy J a and J v occur
in this combination only, this quantum condition would not define the path
uniquely, but would fix the energy. The condition

(10) J Z =,A

retains its significance. The example shows, therefore, that only so many
quantum conditions may be prescribed as there are different periods.
If all three frequencies coincide there remains only the one condition

(11) J.+J^+J^nA
left. This again fixes the energy uniquely.

We shall now examine more closely the manner in which the action variables
alter when, in the case of v x v y , the co-ordinate system is rotated. Let the
action variables J xt J y correspond to the rectangular co-ordinates x, y, and the
action variables J 7 , J^ to the co-ordinates

x~x cos a y sin a
yxsin a -\-y cos a.

If we express the barred co-ordinates and momenta occurring in

m _

in terms of those not barred (the momenta transform just like the co-
ordinates) we get

+ cos2 a+ 2+v sin8 a

( p x p y -\-mca 2 xy J sin a cos a,

PERIODIC AND MULTIPLY PERIODIC MOTIONS 79

-|- ( p x p y - 1 maAcy j sin a cos a.
\w /

The coefficients of cos 2 a and sin 2 a are clearly the magnitudes v,l x and vJ tf .
The coefficients of sin a cos a we determine from the transformation equations
(4) and obtain

Jj= Jjc cos2 Ct 1~ Jy sm2 a ~~ ^ ^3 x ')y cos ( W V~~ w y) s * n a eos a >
J-= J^ sin 2 a-f J tf cos 2 a j - 2 V/J^J^ cos ( M^ t0 y ) sin a cos a ;

where, in our case w x ~w y is a constant since ?' a . = J' v . The constants J^, J y are
thus transformed into Jj, Jy, which are also constants.

The transformation which transforms the angle and action variables, corre-
sponding to a rectangular co-ordinate system, into those associated with another
rectangular co-ordinate system, is not one which transforms the angle and
action variables among themselves. In fact, the constant difference of the
angle variables appears in the transformation eq nations for the J. We shall
meet with a similar state of affairs in the case of a second example, and later
quite generally in the case of degeneration.

It may happen that the Hamiltonian function does not consist of
a sum of terms depending on only one pair of variables q k p k > but
that the Hamilton-Jacob! equation may be solved by separation of
the variables, i.e. on the assumption that

(11') S=S 1 (

Then

is a function of q k alone. We now suppose that each of the co-
ordinates q k behaves in the same way as we assumed above ( 9) in
the case of systems of one degree of freedom, i.e. either that q k
oscillates to and fro periodically in time, between two fixed limits
(case of libration), or that the corresponding p k is a periodic function
of q k (case of rotation). Since each integral

(12) J*=#>*#fc

taken over a period q k is constant, we can introduce the J^ here as
constant momenta in place of a t a 2 .... The function H depends
then only on the J fc 's ; S may be expressed as a function of the y fc 's
and of the J fc 's. Instead of the q k s, the quantities w k , conjugate to
the J fc 's, will now be introduced ; they are related to the q k s by
means of the equations

as ^ as,

(13) '

80 THE MECHANICS OF THE ATOM

We will now prove that the variables w k> J k , introduced in this way,
have similar properties to w and J for one degree of freedom, namely,
that the q k n are multiply periodic functions of the w k s with the
fundamental period system

(1, 0, ... 0)

(0, 1, ... 0)

(0, 0, 1 ... 0)

(0, 0, ... 1).

We wish to find the change in w k during a to-and-fro motion, or
in the course of one revolution, of the co-ordinate q h , when the other
co-ordinates remain unaltered. This change will be :

Now, by partial differentiation of equation (13)
so by integration

If we fix our attention on the functions qi(w . . . w f ), and increase
w k by 1, while the other w's remain unaltered, y fc goes through one
period ; the remaining y's may also depend on w k , but they return
to the initial values without going through a complete period (if, for
example, qi went through a complete period Wi would increase by 1).
This proves the theorem stated above, concerning the periodic pro-
perties of the q k $ in the w k s.

It may happen that a particular q does not depend on all the w k s,
that is, it may not have the full periodicity/, but the system of all
the j's taken together depends of course on all the W A 'S.

In our treatment of the spatial oscillator, for example, each co-ordinate
depended on one w only.

In every case q k may be expressed as a Fourier series in the form
(U) ?Jfc =2 C r W ^ 2jrl(TW)

(see (8) and (8'), 13, for the abbreviated notation adopted). We
obtain the w's as functions of the time from the canonical equations

OTT

(16) tf*

PERIODIC AND MULTIPLY PERIODIC MOTIONS 81

Written as a function of t :

q k =y\CW . e 2 T'Ky i (rf)^
where

0"')=T 1 V 1 +T 2 l> 2 + . . . +T f Vf

( T 8)=T 1 8 1 +T 2 8 2 +

so that q k is not in general periodic ; it Ls only periodic when, and
only when, (/I) rational relations exist between the v's (for example
when all the i/'s are equal). Periodicity of the motion signifies, there-
fore, that all individual periods (l/v k ) have a common multiple (l/i>,
say), i.e. that a relation

with integral T>"S, exists. This is equivalent, however, to (/I)
rational relations between the v k &. Conversely (/I) independent
linear homogeneous equations with integral coefficients

TIZV* I - - -

determine the ratios of the i/ fc 's ; these ratios are rational, it is there-
fore possible to choose v so that

Vk =f r k 'v,

the TJ/'S being integers. The Fourier representation of the co-ordin-
ates q k assumes in this case the form

q h = y\C w e 2lTl[(T > T *' hT ' Ti/ h ' T / T / )lV ' (T5)1 .

Here again the periodicity will at once be recognised.

In the non-periodic case the motion is analogous to that which in
two dimensions is called a Lissajous-motion, the path being closed
only in the event of a rational relation between the v k a. We consider
the path in the w-space, confined to a standard unit cell of the period
lattice (see 13) by replacing every point on the actual path by the
equivalent point in the standard cell. If there are no linear integral
relations between the v k n 9 this path in the w-space approaches in-
definitely near to each point in the standard cell (as proved in
Appendix I). The representation of the g-space in the w-space is
continuous ; so in this case the path in the ^-space approaches in-
definitely close to every point of an /-dimensional region.

Astronomers call such motions conditionally periodic.

82 THE MECHANICS OF THE ATOM

Since the function S increases by J fc each time the co-ordinate q k
traverses a period while the other gr's remain unaltered, it follows that
the function

(16) S*=S-2>*J*

is a multiply periodic function of the w's with the fundamental period
1. For, if w k alters by 1, and the other w's remain the same, q k tra-
verses one period and the remaining </'s return to their original values,
without having completed a period, i.e. S increases by J ft and S*
remains unaltered.

S* may be regarded as the generator of a canonical transforma-
tion instead of S. The equation

is in fact equivalent to

-" *" (H

and this gives the transformation

(17) * ?,

Pk ~ fyk

From this wo can deduce a simple expression for the mean kinetic
energy in the case of iion-rektivistic mechanics. We have (cf. (8), 5)

If we choose the time interval (t l9 t 2 ) suSiciently long, it follows
that

- 1
2T=
(18) tz-

The integrals J k (12) introduced here appear to be suitable for the
formulation of quantum conditions in the form J k =n k h. By defini-
tion, however, they are associated with a co-ordinate system (g, p)
in which the Hamilton-Jacobi equation is separable ; it is therefore
essential that we should next examine the conditions under which
this co-ordinate system is uniquely determined by the condition of

PERIODIC AND MULTIPLY PERIODIC MOTIONS

separability. We shall therefore see if there are point transforma-
tions (i.e. transformations of the co-ordinates among themselves)
which transform the set of variables in which the Hamilton- Jacobi
equation is separable into another such set.

Let us suppose that there is a co-ordinate system, in which the
Hamilton-Jacobi equation of the motion under consideration is
separable. We suppose further, that no commensurabilities inde-
pendent of the initial conditions, or, as we say, " identical " com-
mensurabilities, exist between the periods. We can then choose the
initial conditions so that the path does not close. If one variable
q k performs a libration, the motion is confined between two (/!)-
dimensional planes g^const., which are touched successively. If,
however, q k performs a rotation, it may be confined to the region
to a> k , where w k is the corrc-
spending period, by displacing
the parts of the path in the
intervals

back to the interval (0, a> k ).
The whole path is confined
then to the interior of

\ I

Fio. 7.

/-dimensional " parallelepiped " orientated in the direction of the
co-ordinate axes. The (/ l)-dimensional planes bounding the
parallelepiped have a significance independent of the co-ordinate
system.

By varying the initial conditions we can alter the dimensions of
the parallelepiped and so displace the invariable planes. It follows
that in this case (i.e. no identical commensurabilities) the directions
in the/-dimensional y-spacc which arc the axes of the co-ordinates in
which the Hamilton-Jacobi equation is separable have an absolute
significance, and that only the scale of each individual variable can
be altered.

Hence in the absence of identical commensurabilities all systems
of co-ordinates, in which separation of the variables is possible, are
connected by a transformation of the form

?*=/*(?*)

The associated momenta transform, by (10), 7, according to the
equation

df k

We thus have

84 THE MECHANICS OF THE ATOM

=fop k ~dq k +ft)g k dq k .
J dq k J'

The second integral on the right-hand side vanishes (on account of
the closed path of integration), and the first integral becomes

$p k dq k .
Thus the integrals J fc are really uniquely determined.

In the case of the spatial oscillator the path fills, in the general case, a
parallelepiped. In the absence, then, of identical commensurabilities, the rect-
angular co-ordinates, or functions of them, are the only separation variables,
and the integrals J x , J y9 and .7 z have an absolute significance.

If identical commensurabilities exist, the path does not occupy all
the space of the parallelepiped and the co-ordinate directions need
no longer possess an absolute significance. The J fc also need not be
uniquely determined.

In the ease of the spatial oscillator with v x =v y9 we could rotate the co-
ordinate system arbitrarily about the z-axis without destroying the property
of separation in x, y, z co-ordinates. We obtained, in the various co-ordinate
systems, different J^'s and J v 's. Further, rectangular co-ordinates are not the
only ones for which the oscillator may be treated by the separation method.

In order to show this and at the same time to give an example of the solution
of the Hamilton-Jacobi equation by separation, in a case where it does not
resolve additively (i.e. is not of the form (1)), we shall use cylindrical co-
ordinates in treating the spatial oscillator for which v x v y v. The canonical
transformation (12), 7 :

x r cos <j> p r = p x cos (/> | p y sin <f>
y= r sin <f> p^= - p x r sin <f> f p y r cos <f>

Z=Z Pz=Pz

transforms the Hamiltonian function into

Wo try to solve the Hamilton-Jacobi equation

on the assumption that

S=S
Since <f> is a cyclic co-ordinate,

s,=V-

If now we collect together the terms dependent on z and put them equal to a
constant, which we denote by m 2 a> z 2 a z z , we get :

PERIODIC AND MULTIPLY PERIODIC MOTIONS 85

and, for the terms depending on r there remains

.drl ' r
Two of the three action integrals may be evaluated at once ( J z by introducing

the auxiliary variable y^sin" 1 as in 9) ; we lind :

/ln . r jl mco 2 m 2 (t) 2 J r

( *")

J z = mu) z <j>(d z 2 z 2 }ldz nmo) z a z 2 .
On substituting r 2 jc, the first integral takes the form

J r = (h[ \-2bx
where

m 2 co 2 ma> z

This integral may be evaluated by the method given in the Appendix. We get
(cf. (5) in Appendix II)

J "' ,. , , . / . / * "
r-

By expressing a and a z here in terms of J^ and J z , we get for the energy

(O (i)~

(20) W=v(2J r l-J^) ^v t 3 t , ==-, v t ^.

It will bo seen from the equations (J9) that J r and J, have a completely
different meaning from the quantities J x and J v , derived by separation in
rectangular co-ordinates ; J,, for example, is now 2jE times the angular
momentum about the z-axis. J z , however, has the same meaning as before ;
also, the factor of v, namely 2.J r -f-'J,, has the same significance as the
former .1^+ J y (it is l/v times the energy of an oscillator for which J z is
zero). In this case, therefore, a meaning could be attached to the quantum
conditions

2J r -fJ =wA

7 4>

,] z n z h.

The restriction of J r and J , individually by such conditions would, on the
other hand, lead to quantum motions altogether different from those arising
from the corresponding restriction of 3 X and J y in the case of a certain rect-
angular co-ordinate system.

We now consider more closely the connection between the w x , w y , J x , J v
and the w r , w^ J r , J ..

We have

where

86 THE MECHANICS OF THE ATOM

is the component of angular momentum about the z- axis. If x and y are ex-
pressed here in terms of the angle and action variables by (!)), 9, we find

(21 ) J == - ^J x J y sin 2ji(iv x -w y ).

Here w x tc y () x b y is a constant. On the other hand

'>* i <\

is equal to the variable M;.--~, conjugated with J . '1 he value of J r is found

<p 277 v

from the equation

2.) r +J^J*-hV
and is given by

Finally, the equation for w r may be obtained by calculating w r from J r and
J ^ with the hel]> of the equations of motion and substituting for these quantities
the values found above.

The transformation which connects the system ot variables w r ' /; </> 'r f ^ with
the system w x w y J^Jy i >s no * ono which transforms the w's and the J's among
themselves. In fact, the constant difference w x w y enters into tin* relations
between il^J r and Jg'I y . Wo shall see that this is a characteristic of every
degenerate system (see 15 for definition of degenerate system).

15. General Multiply Periodic Systems. Uniqueness of
the Action Variables

Hitherto we have applied the quantum theory only to mechanical
systems whoso motion may be calculated by separation of the
variables. We proceed now to deal in a general manner with the
question of when it is possible to introduce the angle and action
variables w k and J 7c so admirably suited to the application of the
quantum theory. For this purpose it is necessary, in the first place,
to fix the J's by suitable postulates so that only integral linear trans-
formations with the determinant 1 are possible ; for it is only
in such cases that the quantum conditions

(1) J*=%A

can have a meaning attached to them.

Generalising our former considerations, we fix our attention on
mechanical systems x whose Hamiltonian functions H(j 1? p 1 . . . )
do not involve the time explicitly. We assume further that it is
possible to find new variables w k , 3 k derived from the q k , p k by means

1 The following conditions according to J. M Burgers, Hel Atoowmodel van
Rutherford-Bohr (Diss. Leyden), Haarlem, 1918, 10.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 87

of a canonical transformation with the generator S (q l9 Jj . . . q fy J f )
so that

=
o\ k ^?*

(2) as

in such a way as to fulfil the following conditions :

(A) The configuration of the system shall be periodic in the w k s
with the fundamental period 1. The q k s, which are uniquely deter-
mined by the configuration state of the system, shall be periodic
functions of the w k s with the fundamental period 1 ; if for u, given
configuration of the system q k is indeterminate to the extent of an
integral multiple of some constant (27r, say), it is only the residue
of q k to the modulus of this constant whi^h is periodic. In the
latter case there are also functions (e.g. sin q k ) which are periodic in
the w h9 in the strict sense of the word ( 13).

(B) The Hamiltonian function transforms into a function W,
depending only on the J fc 's.

It follows from this that the w k s are linear functions of the time,
and that the J fc 's are constant. The functions q k (w l . . . w f ) possess
a periodicity lattice in the w-space, the cells of the lattice being cubes
with sides 1.

Now it may be easily shown that the quantities J fc (apart from
being indeterminate to the extent of a linear integral transforma-
tion with the determinant 1) are not yet uniquely determined by
the two conditions (A) and (B).

A simple canonical transformation, which does not violate the
conditions (A) and (B), is as follows :

! J/)

where the c k s are constants. The arbitrariness in the choice of the
Cfc's prevents the application of the quantum conditions (1) for, if the
Jfc's are determined as integral multiples of h, this will not in general

be the case for the J fc 's. We must therefore do away with this remain-
ing arbitrariness in the choice of the variables. This may be accom-
plished by postulating generally a property of the w's and J's found
previously to hold in the case of separable systems,
(C) The function

s*=s-2>* J *

which is the generator of our transformation (qjcPk-+u>k3k) in the form

THE MECHANICS OF THE ATOM

(5)

shall be a periodic function of the w^'s with the period 1.

It is all the same here whether we regard S* as a function of q k and
w k or of J/, and w k9 since the g^'s are also periodic in the w k s.

If it be required in (C) that 1 shall be a fundamental period, (A)
will be superfluous. For, if the q k n are calculated as functions of the
w k & and JYs from the second system of equations, they will be periodic
in the w k s with the period 1 . Apart from this, it will be seen from the
first system of equations that the same is true of the p k 'a.

We must now prove that the conditions (A), (B), and (C) really
suffice for the logical applications of quantum conditions in the form
(1) ; we carry out the proof by finding the most general canonical
transformation

which satisfies the conditions (A), (B), and (C).

We seek the iirst group of the transformation equations, viz. those
for w k s in terms of the w's and J's. According to (A) the trans-
formation must transform into itself the system of lattice points
corresponding to the fundamental period 1. By (7), 13, the w k s
must transform as follows :

(6) Wfc

Here the system of integers r fcz has the determinant 1- The ^'s are
periodic in the w k s with the period 1, and, written as functions of the
w^s, periodic in these also ; they may therefore be expressed in the
form

The condition (B) introduces a fresh restriction. Considered as func-
tions of the time, the w k s, as well as the w k 's, must be linear ; from
(6) it follows that <^ fc 's are likewise linear functions of the time, but if
they vary at all with the time they must be multiply periodic, as has
just be shown ; they must therefore be constant. This means, how-
ever, that, in the exponent of the Fourier series, the only combina-
tions of the w k which can occur are such as make

.+(7,8,)

independent of t and therefore

PERIODIC AND MULTIPLY PERIODIC MOTIONS 89

(identically in the J k ). v k denotes here the derivative - .

&Jfc

The case where identical relations

exist between the frequencies will enter largely into our considera-
tions ; systems for which such relations exist wejjshall call degenerate
systems, while the others we shall refer to as non-degenerate.

We shall deal also with the case i?i which such relations exist only for certain
values of the J A , ; the mechanical system is then non- degenerate ; the particu-
lar motions in question, for which (TJ>)=(), we shall call accidentally degenerate,
whilst the motions of a degenerate system [(rv)=Q identically] are spoken of as
intrinsically degenerate.

We consider first non-degenerate systems. For these the trans-
formation (6) takes the form

(7) W Je =

/
In order now to find the second group of transformation equations

of a non-degenerate system (i.e. those for the J's in terms of the w'a
and J's), we write down the generator of the transformation (7), viz. :

V(w l9 J l . . . w f , J/) = 2 T J *^+^( J i J/)+W w,),

where X F has the partial derivatives i/j k . 1 The second group of the
transformation equations now becomes

(8) J fc = -^ -= 2TJ,+/( ?! . . . W f ).
In order to see if the transformation

(9)

actually leaves the conditions (A), (B), and (C) unaltered, or whether
we must still further restrict the number of permissible transforma-
tions, we resolve them into the three transformations

(10) w^toi+hdi . . . J/), J fc =I*

(12) &*=w to J*= *+/*(^i . . w f ).

1 It will be seen from this, that the 1/1% 's in (7) must fulfil certain differential
relations in order that the transformation may be canonical.

90 THE MECHANICS OF THE ATOM

All three are canonical ; to each may be assigned a generator in the
sense of 7.

The first transformation (10) does not conflict with (A) and (B).
That (C) likewise remains satisfied can be seen as follows : If S(#, J)
and %(q, $) are the generators of transformations of the form (2),
transforming the q, p into w, J and into to, 3f respectively, then

since the same variables are maintained constant during the differen-
tiation, it follows that S .$ is independent of q k . For S* j$* we
have then

k k

from which it is seen that (C) is fulfilled.

It will be seen at once that (11) leaves (A) and (B) unviolated ;
we test as follows for the condition (C). For $*(j, in) and J$*(g, 5)
we have on the one hand

since the same variables are kept constant during the differentiation
(the to's are transformed into the to's by a linear transformation
with a non-zero determinant) it follows that &* ^* docs not
depend on q. On the other hand

al*^_ _ = ^* = v

~~~ ~

and it follows from this that $* jj* is also independent of ffi and to.
In order that the complete transformation (9) may fulfil all three
conditions it is necessary and sufficient that this shall be the case
for (12).

For $*(<?, to) and S*((/, w) we have :

al* as*

_ - _ ~T)if
8q k 8q k

thus

^*-S*=R(W! . . . ibf).
Further, from (12)

thus

PERIODIC AND MULTIPLY PERIODIC MOTIONS 91

If (C) is to remain valid for the transformation (12), R must be
periodic in the w k , f k may therefore be represented by a Fourier
series without a constant term. If (B) is to remain valid,/*, must not
depend on the time. From these two conditions it follows that f k
must vanish. Hence if / fc =0, (A), (B), and (C) remain unviolated.

We have proved by this that the most general transformation for
the action variables is

(13) J*=2T Ifc J,.

If the jys are now determined as integral multiples of A, the same will

be true of the J k s and conversely.

Although we have been guided in our considerations by the idea
that Jfc/A must take integral values, we can state the mechanical
theorem proved in a form independent of any quantum theory :

Uniqueness theorem for non-degenerate systems : If we can intro-
duce in a mechanical system variables w k and J fc so that the conditions
(A), (B), and (C) are satisfied, and if between the quantities

no commensurability exists, then the J fc 's are determined uniquely,
apart from a homogeneous linear integral transformation with the
determinant 1.

We proceed now to the treatment of degenerate systems.

If between the v k s there exist (/ s) commensurability relations

(U) 2>*"*=0

we can arrange, by means of a canonical transformation satisfying

. 0W

the conditions (A), (B), and (C), that/ s of the frequencies i> k =

0J*

shall vanish and that between the remaining s no relation of the
type (14) shall exist. If we call the new variables w k and J k once
more, we have

e > a=l, 2 ... s,

and the Hamiltonian function has the form

W(JJ.

92 THE MECHANICS OF THE ATOM

The w a 's and J a 's we call proper angle and action variables, the w p 's
and J p 's improper or degenerate variables ; the //;/ remain constant
during the motion. The number s of the independent frequencies v a
is called the degree of periodicity of the system.

In the rase of accidental degeneration, the number of independent fre-
quencies is less for certain motions than for the number of the whole system.
We call the former number the degree of periodicity of the motion under con-
sideration.

We must now seek the most general transformation which violates
neither this division of the variables nor the conditions (A), (B), and
(C). The first group of transformation equations (i.e. that for the
w fc 'H) is now :

w fc =2 T *i'^+^*K+i ' f> J i J /)'

The generator is therefore

t .. J,)

'*/> J i
hi

where 1 F is periodic in the tf p . The second group of transformation
equations then becomes :

the derivative of X F is non-zero only if k is one of the numbers

* + l - ./

In order that the division into non-degenerate and degenerate vari-

ables may persist, the w p 's must not depend on the w a 's or the w p 's on
the w ft 's. This means, however, that the r pfl 's vanish. The transfor-
mation equations can then be written as follows :

',=2'

a, j8=l...

.p 9 (7=5 + 1 /

I =!...

where is put equal to <f> p . Since the r kl are whole numbers and
the r pa vanish, it follows, from the value of the determinant, that

also

PERIODIC AND MULTIPLY PERIODIC MOTIONS 93

We now divide the transformation (16) in two parts :

.> J >

(7 /

and

(18) te fc = fcf J*-3 fc +/ fc (),

and show that the first satisfies the condition (C) and that the second

does this only for/ a 0.

As before, let S(g, J) and ^(q, 3) be the generators of the trans-
formations q, p-+w, J and q, ^->itr, 31. We consider the function S ^
from tho point of view of its dependence on iu and J, i.e. we write

S=%(ta, J), J), 9 -= %(ut, J), 3(to, J))
and form

2*
r ^"*

from (17), the first two terms cancelling. We have therefore

(19) 8 _(S-$)^0, A(S_)^_V U , ^.

' ' v ' '

(B-9)-V - a?i -J - T^ ^'-T 8 ?i

We derive further :
(20)

-*-?-.i-*-?>.

It follows from (19) and (20) that

S-*=T(to^ J)-Z to A.

where T has the same meaning as in (15). We shall have therefore

k I

this denotes, however, that (0) remains valid.

The condition that (C) should be satisfied by the transformation
(18) is found as in the non-degenerate case, viz, :

94 THE MECHANICS OF THE ATOM

If (C) and (B) are satisfied, f k (w) is a periodic function of the form

in which only exponents containing w p alone may occur ; conse-
quently r a always. It follows from this, however, that

The most general permissible transformation of the non-degenerate
action variables is therefore

(21) J.=ZvV

The J 's, on the other hand, need not transform integrally. Since the
condition (C) does not forbid the occurrence here of w l in the trans-
formation equations of the J p 's, it follows that from a system J p , in

which all the J p 's are integral multiples of A, a system J p may always
be derived which does not possess this property (cf. examples of
14).

We can state the result of our investigations independently of the
quantum theory as follows :

If we can introduce in a mechanical system variables w k j k which
satisfy the conditions (A), (B), and (C), we can always arrange so
that certain of the partial derivatives

namely, the i> a 's (a=l . . . s), are incommensurable while the remain-
ing ones i/ p (p=s+l . . ./) vanish. The J t \s are then uniquely deter-
mined, apart from a homogeneous integral linear transformation
with the determinant ii. 1

We deduce still another consequence from the periodicity of S* as
a function of q and w or J and w ; the function

increases by J k when w k increases by 1 and the other w's and J's re-
main constant. We can write this in the form :

Jfc I dw k [ ) =1 dw k y*- -,

Jo \Wj Jo I fyi 3k

or :

1 J. M. Burgers, who refers to this theorem in his dissertation, does not give a
complete proof.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 95

(22)

This integral may be employed to ascertain if a given motion fulfils
the quantum conditions or not, since all that is necessary is a know-
ledge of the p and q as functions of the w u 's.

16. The Adiabatic Invariance of the Action Variables and the
Quantum Conditions for Several Degrees of Freedom

As in the case of one degree of freedom, the uniqueness of the
action variables is only one of the conditions necessary if the quan-
tum conditions in the form

J a =n a h

are to have a definite meaning attached to them.

As a second condition we must require that the J a 's shall be constant
not only for an isolated system but also in the case of a system subject
to slowly varying influences, in accordance with the principles of
classical mechanics.

In fact the following principle applies in this case also :

The action variables J a are adiabatically invariant so long as they
remain in a region free from new degenerations.

We carry out the proof (after J. M. Burgers) exactly as we did
in the case of one degree of freedom. 1 We imagine the canonical
transformation

as*

applied to the variables q k , p k , satisfying the canonical equations

. an . 8H
"-iff p *^;

so that, for constant a, the variables q k , p k are transformed into the
angle and action variables w k , J k . By (1), 7, H is transformed to

as*

H = H+ ir

Thus the transformed canonical equations can be written

1 The proof of the adiabatic invariance of the J's given here is not altogether
free from objection on account of difficulties due to the appearance of accidentally
degenerate motions in the course of the adiabatic change. A strict proof has been
given by M. v. Laue, Ann. d. Physik, vol. Ixxi, p. 619, 1925.

96 THE MECHANICS OF THE ATOM

8R 8 /8S*\

W= ~

___ s a

k ~~8w h ~dw k \df/'
Since H depends only on the J k '$, it follows that

T = _ -- i

k 8w k \ dt ) 8w k \ da I '

In the differentiation with respect to t and a, S* is to be regarded as
a function of q k , w k , and t or a. Now the variation of J fc in the time
interval (t^ t 2 ) is

and on account of the supposed slow alteration of a, independently of
the period of the system, a can be put before the integral sign. We
will now show that

a tk

is of the order of magnitude d(t 2 a ) (cf. 10).

f'S*

S* is a periodic function of the w k , so also is -- , and the inte-

grand of (1) is a Fourier series

without a constant term, so that the integral to be evaluated takes
the form

where A T , v, and S are functions of the J's and of a. We develop the
integrand in the neighbourhood of a certain value of t, which we
denote by =0 as in 10, and obtain

2 'A e'-'TiKT^M (r )]
TO

(2)

the notation being similar to that used in (4), 10. Consider
this expansion carried out at the beginning of the interval (^, t 2 )
arid the integral taken from t l to such a point that the integral

PERIODIC AND MULTIPLY PERIODIC MOTIONS 97

of the first term vanishes. This is always possible, since the indefinite
integral of the first term is a multiply periodic function, and in
intervals of the order of magnitude 1 /(TV O ) passes continually through
zero. The integral of the second term is of the order of magni-
tude dT or aT 2 . We imagine now a new expansion (2) carried out at
the beginning of the remaining interval and the integral again taken
so far that the first term vanishes. This process we suppose con-
tinued until finally there remains an interval over which the integral
of the first term has a non-zero value. It will be seen that, if none
of the (rv)'s vanish on the path of integration, the complete integral
is of the order of magnitude d(t 2 t t ).

In the case where an identical relation (i.e. a relation valid for all
J's) (VT) ^0 exists for a certain value of , the w's and J\s may be chosen
so that the i/ a 's are incommensurable and the v p 's equal to (cf. (14),
15). Constant exponents ((rv)=0) then appear in S*, but they in-
volve the w p 's only ; the terms in question disappear, therefore, on
differentiating with respect to w a , consequently the J a 's remain in-
variant at such places of degeneration ; this cannot be said generally
to hold for the J p 's.

In addition to those cases where (TV) is identically zero, it may
happen that (TV) is zero only for the particular values of 3 k under
consideration ; in the latter case we speak of " accidental degenera-
tion," and under such circumstances the J's need not be invariant
unless, in the expansion (2) of the integrand of (1) for the different
J fc 's, the term A r0 with the corresponding exponent (TW) occurs in S
with zero amplitude.

It follows then that if the J^'s are to be adiabatically invariant we
must exclude all cases for which an accidental commensurability
exists (i.e. one which holds only for the values of J under considera-
tion) between frequencies which occur conjointly in the form (TV) in
the exponent of a term of the Fourier series for S*.

As an example of the adiabatic invariance of an action variable we consider
the case where the mechanical system is invariant with respect to a rotation
about an axis fixed in space. If (/*, <, z) are cylindrical co-ordinates, the angle
of rotation <^ and the differences ^j. < x may be introduced as co-ordinates
instead of the individual <f> ; fa is then a cyclic variable, and (cf. 6) the
momentum conjugated with it is the angular momentum of the system about
the z-axis. The principle of the conservation of angular momentum about an
axis is also valid when the expression for the potential energy contains the time
explicitly, provided only that the invariance with respect to a rotation about
the axis persists identically in time. If the field of force of rotational symmetry
be strengthened or weakened, the angular momentum about the z-axis remains
invariant, and we have a special case of the principle of the adiabatic invariance
of the action variables,

98 THE MECHANICS OF THE ATOM

In order to see what may happen in the case of a passage of the system
through a degenerate state, we consider once again the spatial oscillator. We
suppose that the directions of the principal axes of the potential energy ellipsoid
as well as the magnitudes of the three frequencies are functions of a para-
meter a, which can be varied arbitrarily in time. If now for a certain value
of a no commensurability exists between the freq uencies, the J's will be adiabatic
invariants. If, however, for a certain value of a we have degeneration, e.g.
v v v y i this will no longer be the case, though certainly there are special varia-
tions for which the J's do remain invariant. If, for instance, the directions of
the principal axes are left unaltered and only the frequencies varied, the co-
ordinates behave as independent linear oscillators, and the J's are adiabatic
invariants for each individually. As an example of an adiabatic variation in
which the J's do not remain invariant in the ca>se of degeneration, we consider
the following. We allow the original potential energy ellipsoid with three
unequal axes to pass over into an ellipsoid of rotation, keeping the axes fixed ;
without varying the axis of rotation we now allow the axes of the ellipsoid
again to become all unequal, but with the other two axes turned through a
finite angle with respect to the original ones. In the instant of degeneration
the projection of the motion in a plane perpendicular to the axis of rotation is
an ellipse. The limiting values of the J's which are correlated with the J values
before and after the degeneration are determined by the amplitudes of this
elliptic motion in the directions of the principal axes of the potential energy
ellipsoid ; it will be seen at once that these values are different for different
directions of the axes.

The uniqueness of the J a 's (in the sense of 15), together with their
adiabatic invariance, strongly suggests the following generalisation
of the quantum condition for one degree of freedom :

In a mechanical system which satisfies the conditions (A), (B), and
(C)of 15,lettheit; A .'sandJ A: > sbechosensothatthey a J s(a=:l,2 . . .s)
are incommensurable and the v p 's (p^s+1 . . . /) are zero (it may be
that s=f). The stationary states of this system will be defined by

the conditions * T , . n rt x

J a =n a /i. (tt=l, 2 . . . s).

Since the Hamiltonian function depends only on the J a 's its value
is determined uniquely by the quantum numbers n a .

To this is added, as the second quantum principle, Bohr's frequency
condition W=sW -W.

1 The first generalisation of the quantum conditions for systems of more than
one degree of freedom was given by M. Planck ( Verh. d. Dtsch. Phys. Ges., vol. xvii,
pp. 407 and 438, 1915), W. Wilson (Phil. Mag., vol. xxix, p. 795, 1915), and A.
Sommerfeld (Sitzungsber. d. K. Bay. Akad. t p. 425, 1915). All three start out
by equating the action variables to integral multiples of h. The general case
of multiply periodic systems was dealt with by K. Schwa rzschild (Sitzungsber. d.
Preuss. Akad., p. 548, 1916), and tho conception of degeneration together with
the restriction of the quantum conditions to the non-degenerate J's was first made
clear by him. The unique determination of the J's through our conditions ( 15)
is given by J. M. Burgers, Het Atoommodel ran Rutherford- Bohr (Diss. Leyden,
1918).

PERIODIC AND MULTIPLY PERIODIC MOTIONS 99

We will now collect together once more the fundamental ideas
underlying the quantum mechanics as hitherto developed : the totality
of the motions (supposed multiply periodic) of a given model are to be
calculated according to the principles of classical mechanics (neglect-
ing the radiation damping) ; a discrete number of motions will be
selected from this continuum by means of the quantum conditions.
The energies of these selected states of motion will be the actual energy
values of the system, measurable by electron impact, and the energy
differences will be connected with the actual light frequencies emitted
by Bohr's frequency condition. Apart from the frequency the ob-
servable qualities of the emitted light comprise the intensity, phase,
and state of polarisation ; with regard to these the theory gives
approximate results only ( 17). This completes the properties of
the motion of atomic systems which are capable of observation. Our
calculation prescribes still other properties, however, namely, fre-
quencies of rotation and distances of separation ; in short, the progress
of the motion in time. It appears that these quantities are not in
general amenable to observation. 1 This leads us, then, to the con-
clusion that our method is, for the time being, only a formal scheme
of calculation, enabling us, in certain cases, to replace true quantum
principles, which are as yet unknown, by calculations on a classical
basis. We must require of these true principles that they shall
contain relations between observable quantities only, i.e. energies,
light frequencies, intensities, and phases. 2 So long as these principles
are unknown we must always be prepared for the failure of our
present quantum rules ; one of our main problems will be to deter-
mine, by comparison with observation, the limits within which these
rules are valid.

17. The Correspondence Principle for Several Degrees
of Freedom

As in 11, we must now investigate to what extent the classical
theory may be regarded as a limiting case of the quantum theory.
In this limiting case the discrete energy steps run together into the
continuum of the classical theory. We show further that a relation
similar to that holding in the case of one degree of freedom exists
between the classical and quantum frequencies.

When the classical radiation damping is neglected, the electric

1 Measurements of the radii of atoms and the like do not give a closer approxima-
tion to reality than, say, the agreement between rotation frequencies and light
frequencies.

2 This idea forms the starting-point of the new quantum mechanics. See W.
Heisenberg, Zeit. f. Phya., vol. xxxiii, p. 879, 1925.

100 THE MECHANICS OF THE ATOM

moment of the atomic system may be represented by a Fourier series
of the form

(i) D^^^^ZO^K"* KT5)j -

T T

The components of the vectors C T are complex quantities ; since the
components of p are real, the components of C T turn into the conju-
gate complex quantities when the signs of all the T fc 's are reversed.
By including in the constant the terms in w pj it may be arranged
that only the non-vanishing (and incommensurable) frequencies v a
occur in the exponent (see (14'), 15, for significance of suffixes a
and p).

Now the quantum frequency associated with a transition in which
the quantum numbers alter by r l . . . r s corresponds, in an analogous
way to the case of one degree of freedom, to the overtone of frequency

The relation between this classical frequency and the quantum fre-
quency is in this case also that between a differential coefficient and
a difference quotient.

We consider a fixed point J a in the J a -space and all the straight
lines

J.=J.-T a A,

going out from this point, the directions of which may be pictured
as lines joining J a with the angular points of a cubic lattice (of arbi-
trary mesh magnitude) surrounding this point. The classical fre-
quency may then be written in the form x

The quantum frequency may be written in the form

(3) --

In order to describe the relation between (2) and (3) we imagine the
above-defined grating chosen so that the side of the cube is equal to
h, v is then the decrease in the energy in going from the grating
point J a to the grating'point J a rji, expressed as a multiple of the
mesh magnitude h. The classical frequency is obtained when the mesh
magnitude h is made infinitely small.

The quantum frequency may also be looked upon as a mean value
of the classical frequency between the grating points J a and J a rji

1 The signs arc chosen so that emission occurs when all the r a 's are positive.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 101

for a finite h, i.e. as a certain mean value between the initial and final
orbits of the quantum transition, associated with radiation of that
frequency. We have in fact *

If the alterations r k of the quantum numbers are small in com-
parison with the numbers themselves, the expressions (3) and (2)
differ very little from one another.

As in the case of one degree of freedom, the correspondence prin-
ciple may be employed for the approximate determination of the
intensities and states of polarisation.

If the alterations r k of the quantum numbers are small in compari-
son with the numbers themselves, the Fourier coefficients C T for
the initial and final states differ by a relatively small amount. On
the basis of the correspondence principle we must now lay down the
following requirement : For large values and small variations of the
quantum numbers, the light wave corresponding to the quantum
transition T I . . . r a is approximately the same as that which would
be sent out by a classical radiator with electric moment

C^fr-o

This determines approximately the intensity and state of polarisa-
tion of the wave. The same quantities C r determine also the proba-
bilities of transitions between the stationary states.

If the alterations of the quantum numbers are of the same order
of magnitude as the numbers themselves, it seems likely that the
amplitudes arc determined by a mean value of C T between the initial
and final states. How this mean value is to be determined is still
an open question. 2 It can be answered only when certain components
of the classical C T are identically zero ; it may be assumed that the
corresponding oscillation is also absent in the quantum theory.

These considerations can be applied in practice to the determina-
tion of the polarisation only if, during the process, at least one
direction in space is kept fixed for all atoms by external conditions,
e.g. an external field. In other cases the orientations of the atoms
would be irregular and no polarisation could be established. If, for
example, a certain C T had the same direction for all atoms, then
to this would correspond a linearly polarised light wave with the

1 Comp. H. A. Kramers, Intensities of Spectral Lines (Diss. Ley den), Copenhagen,
1919.

2 This question is now answered by the new quantum mechanics founded by
Heisenberg (loc. cit.) and developed by Born, Jordan, Dirac, Schrodinger, and others.

102 THE MECHANICS OF THE ATOM

distribution of intensity given by the classical theory for different
directions in space.

Of special importance for the application of the quantum con-
ditions and of the correspondence principle is the case in which the
Hamiltonian function is not changed by the rotation as a whole of
an atomic system about a fixed direction in space. If we introduce
as co-ordinates the azimuth <f>=q f of one of the particles of the
system together with the differences of the azimuths of the other
particles from <j>, and other magnitudes depending only on the rela-
tive position of the particles of the system with respect to the fixed
direction in space, </> will be a cyclic variable and the momentum p^
conjugated to it is, by 6, the angular momentum of the system

flS
parallel to the fixed direction. On account of the constancy of ,

U</)

the function S, which transforms the q k and their momentum p k into
angle and action variables, has the form

It follows from this that

1 8F 0S

1 0F 88

If now qtfz . . . gy_ a be kept fixed and <f> allowed to increase by
2ir (i.e. if the whole system be rotated through 2-rr), the w k a must
change by whole numbers (for the q k s are periodic in the w k s with
the period 1) ; for this to be the case the derivatives of F must be
whole numbers and F has the form

F=r 1 J 1 + . . . +T/J/+C.

By means of a suitable integral transformation with the deter-
minant 1 this may always be brought to the form

so that

PERIODIC AND MULTIPLY PERIODIC MOTIONS 103

It follows from this that

"*=<&*(li - - - ?/-i> Ji - - J/-i J,) (*=1 - -/-I)

and by solving for the q k

f ?& TA^W! w f-ii Ji J/-i> Jj (& I . . ./I)

( 6 ) JL_^ -

so that we can write also

S=v(J,+c)+p(J,+c)+<FK . . . v_!, Ji . . . J/-L J,).

47T

Since S w^J must be periodic in w^ it follows that c=0 and so

(7) S= 1 ty+Sfo . . . 7/-i, Ji . . . J,_i, J,)-

The angular momentum in the direction of our fixed axis is conse-
quently

_as_ i

P *~^~~2i' *'
If there is no degeneration then we must put

In words : In every system for which the potential energy is in-
variant with respect to a rotation about an axis fixed in space, the
component of the angular momentum about the axis multiplied by
2rr is an action variable. If the energy depends essentially on this
quantity, it is to be quantised.

Since the functions <D/ C in (5) depend only on the relative positions
of the particles of the system with respect to one another and to
the fixed axis, these relative positions will be determined also by
w l . . . w/_i, while Wf fixes the absolute position of the system.
According to (6), 27rw/ can be regarded as the mean value of the
azimuth <f> of the arbitrarily selected particle of the system over the
motions of the " relative " angle variables w t . . . M/-I. The motions
can therefore be considered as a multiply periodic relative one on
which is superposed a uniform precession about the fixed axis. If
H, regarded as a function of the J^., does not depend on J , this pre-
cession is zero ; the system is then degenerate.

We consider first the case where the system moves under the action

104 THE MECHANICS OF THE ATOM

of internal forces only. Every fixed direction in space can then be
regarded as the axis of a cyclic azimuth. The energy does not depend
on the individual components of the resultant angular momentum,
but only on the sum of their squares, i.e. on the magnitude of the
resultant angular momentum. If the direction of the angular mo-
mentum be chosen as axis, the corresponding azimuth iff is cyclic
and w^ non-degenerate. The resultant angular momentum p is there-
fore determined by a quantum condition of the form

(8) 27rp=J^jA.

If we fix our attention on a second arbitrary axis fixed in space,
there will be a cyclic azimuth <j> about this ; the associated action
variable J =27rp^ does not occur, however, in the energy function in
addition to J^, because the energy of the system cannot depend on a
component of momentum in an arbitrary direction. The angle vari-
able w^ conjugated to J is therefore degenerate, and J may riot be
quantised. The significance of w^ will be recognised from the general
property of a cyclic angle variable, that it is equal to the mean value
of the azimuth of an arbitrary point of the system taken over the
motions relative to the axis, w^ is thus a constant angle which can
be chosen equal to the azimuth of the axis of the resultant angular
momentum about a plane through the fixed ^-axis.

We now consider the case where the mechanical system is sub-
jected to a homogeneous external (electric or magnetic) field. The
azimuth </> of a particle of the system about an axis parallel to the
field is then a cyclic variable ; in general H will depend on J^, and we
have the quantum condition

(9) 27rp,=J,=wA.

For an arbitrary external field, on the other hand, the resultant
angular momentum p is not in general an integral of the equations
of motion and cannot therefore be quantised, but it may happen,
in special cases, that p is constant and is an action variable. The
relations (8) and (9) will then be true at the same time ; but p^ is
the projection of p in the direction of the field and, if a denotes the
angle between the angular momentum and the direction of the field,
we have

(10) cosa= ^.

P J * 3

This angle is therefore not only constant (regular precession of the
resultant angular momentum about the direction of the field), but
is also restricted by the quantum condition to discrete values. One

PERIODIC AND MULTIPLY PERIODIC MOTIONS- 105

speaks, in this case, of " spatial quantisation." l Since by (10) m can
take only the values j, j+l,...j, it follows that for every j there
are in all 2j +1 possible orientations of the angular momentum. This
describes a cone of constant angle a about the direction of the field
with the processional velocity

This regular precession is, in general, possible only for certain
initial conditions. We shall show later (by the method of secular
perturbations, 18) that, for weak fields, the spatial quantisation
holds in general for every motion ; the only exceptions to this are
certain cases of double degeneration (e.g. hydrogen atom in an
electric field, cf. 35).

Certain predictions regarding the polarisation of the emitted light
and the transition possibilities may now be made with the help of
the correspondence principle.

If z is an axis of symmetry fixed in space, we combine the com-
ponents of the electric moment $ x , $ y , perpendicular to this, in the
form of a complex quantity and write :

* (4=1, 2, ... n).

Pi=2- e ***

If r k s are the distances from the axis and < fc 's the azimuths (one of
them being <f>), then

Now the bracketed expression (r k e l ^-^) 9 like the z k , depends only
on the #!,... y/_i ; substituting for these the values (6) we have

. . T/ i

The integer r can therefore assume only the value 1 in the x- and y-
components of the electric moment and the value in the case of the
z-component. 2 According to the correspondence principle, the corre-
sponding quantum number can alter only by 1 or 0. (This holds, of
course, only if J is to be quantised at all, i.e. provided there is no
degeneration.) The change of 1 corresponds to a right- or left-

1 A. fcommerfeld, Phys. Zeitschr., vol. xvii, p. 491, 1916 ; Ann. d. Physik, vol. li,
p. 1, 1916.

2 The sign of ris meaningless, since in the Fourier expansion T always occurs
as well as T.

106 THE MECHANICS OF THE ATOM

handed rotation of the electric moment about the axis of symmetry,
and, therefore, to right- or left-handed circularly polarised light.
Since the angular momentum of the system increases when the quan-
tum number changes by +1, that of the light therefore decreases, so
that for this transition of +1 in J^ the light is negatively circularly
polarised for emission and positively circularly polarised for absorp-
tion ; for the transition 1 in J (/> the reverse holds. 1 Corresponding
to the transition without change of angular momentum, we have
light polarised parallel to the axis of symmetry. 2 If the motion of
each point of the system is confined to a plane perpendicular to the
axis of symmetry then (except ioi r l = . . . r f _ 1 =0)

V -,-=

a transition without change of angular momentum does not then
occur.

We consider now the case of a system which is subject to internal
forces only. The above considerations are then applicable to the
axis of the resultant angular momentum, where, in place of <, the
angle denoted above by ifj appears and the quantum condition (8)
applies. The polarisation of the light cannot be observed, however,
since the atoms or molecules of a gas have all possible orientations.
The case mentioned above, where all the particles of the system move
in planes perpendicular to the axis, is of frequent occurrence, e.g. in
the case of the two-body problem (atom with one electron) and in
that of the rigid rotator (dumb-bell model of the molecule) ; the
transition j->j is then impossible.

We consider further the case in which the system is subject to the
action of an external homogeneous field and spatial quantisation
exists (which is approximately true for weak fields). The alterations
of m and the polarisation of the light are then subject to the rules
derived above. It is easy to see that the transition possibilities
Aj 1, 0, +1, which are valid for a free system, remain true for j.

We imagine a co-ordinate system f , 77, introduced so that the
-axis is in the direction of the angular momentum, and the ij-axis
perpendicular to the direction of the field. In this system of co-
ordinates the electric moment may be expressed in the form

1 Hubinowicz (Phyaikal. Zeitechr., vol. xix, pp. 441 and 456, 1918) used the
relation between polarisation and angular momentum (about the same time as the
general correspondence principle was given by Bohr) in order to arrive at the
selection principle for the alteration of quantum numbers.

2 In optics, such light would be said to be polarised perpendicular to the z-
direction, since the plane of polarisation is taken conventionally as the plane of
oscillation of the magnetic vector.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 107

in which only the angle variables of the relative motion w t . . . w/-i
(not w^ and w^) occur in the summations. The co-ordinates f , rj,
are connected with those of the fixed system x, y y z by the relations

x+iy=e 2irlw <l>(g cos a sin a+i^)
z= sin a-f cos a ;

which express the fact that the -axis makes a constant angle a with
the z-axis, and describes a regular precession w^=vj about it. The
same transformation formulae hold also for the components of the
vector p in the two co-ordinate systems. If the Fourier series (11) be
substituted for p f , p o , p^, it will be seen at once that the angle vari-
ables WQ and w^ occur only with the factors T =l ; r^ 0, J^l, in
the exponents of the Fourier series for PJ,. and p v , and in p z with the
factors r =0 ; r^=0, 1 only. The quantum number j can therefore
change only by or 1.

18. Method of Secular Perturbations

A multiply periodic degenerate system may frequently be changed
into a non-degenerate one by means of slight influences or variation
of the conditions. We shall consider, in particular, the simple case
where the Hamiltonian function involves a parameter A and the
system is degenerate for A=0. We imagine the energy function H
expanded in powers of A ; for sufficiently small values of A we can
break off this series after the term linear in A and write

(1) H^Ho+AH!.

To this approximation then each perturbation of the unperturbed
system, whose Hamiltonian function is H , may be taken account of
by the addition of an appropriate " perturbation function " AHj.
The effect of the perturbation function on the motion, when the
system whose Hamiltonian function is H is not degenerate, will be
examined later ; here we shall consider only the case where H is
degenerate. We suppose the problem of the unperturbed system
solved and angle and action variables w k Q , 3 k Q introduced by a canon-
ical substitution ; on account of the degeneration, H will depend
only on the proper action variables J a (a=l, 2 . . . s) (see (14'), 15,
for the significance of suffixes a, p). H x will be a function of all the
w k *'a and J fc 's, thus :

(2) H

108 THE MECHANICS OF THE ATOM

We obtain an approximate solution of the " perturbation problem "
by a method which will here be based on an intuitive line of argu-
ment ; it will be established mathematically later in a more general
context (Ch. 4, 43).

In the undisturbed motion the w p 's are constant and the w a 's vari-
able with the time. The effect of a small perturbation will be that
the w () 's will also be variable in time, but in such a manner that their
rates of change will be small, i.e. that they vanish with A. Since now
the co-ordinates q k , p k are periodic functions of all the w k 's with the
period 1, it follows that, during the time in which the w p 's vary by
a given amount, the system will have traversed a large number of
periods (rotations or libratioiis) of the w a 's. The coupling between the
motions of the w a 's and the w p 's may therefore be represented approxi-
mately by taking the mean value of the energy function over the
unperturbed motion of the w a ; (2) then becomes

(3) H-HO^+AH^ ; % <>, J P ).

In this expression the w a 's do not appear ; the J a 's are therefore
constant during the perturbed motion, and appear as parameters
only ; the only variables are the w p 's and J p 's. These satisfy the
canonical equations :

The only solutions which are of importance from the point of view of
the qiiantum theory are those of a multiply periodic nature. We
assume, therefore, that the perturbed motion has a principal function
of the form

(5)

where F is a periodic function of the w p 's with the fundamental period
1, and such that the canonical transformation with the generator S,
viz.

W.=. J = J a

(6) c)F 8V
W+*r J/^+aT,

P P

transforms the function H x into a function of the J fc 's alone :

(7) H^o; ^-VHW^J,,).
The portion of S depending on w> p , J p , viz.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 109

a=l

satisfies the Hamilton-Jacobi partial differential equation

The variations of w p J p are determined, therefore, from the mean of
the perturbation function just as the original co-ordinates of a system
are from the total energy function.

To this approximation the solution takes the form

J fi = -const. Wa
J p =const. w p =
where

_sn

" tt ~

We see, therefore, that the rates of variation of the w p \s arc in fact small
compared with those of the w a 's and vanish for A=0. In celestial
mechanics the name " secular perturbations " has been introduced
for such slow motions.

It will be seen from (6) that the original co-ordinates q and p of
the system are now periodic functions of the new angle variables w p as
well as of the old angle variables w a .

For the motions represented by equation (8) also, cases of libra-
tion. rotation, or limitation may occur. This problem is soluble
practically only if the differential equation (8) is separable in the
variables w? 9 or if it is possible to find other separation variables.
That is the case, for example, if all the variables w p , or all with the
exception of one, are cyclic ; the simplest case is that in which only
one variable w p appears, i.e. when the unperturbed system is simply
degenerate.

Further, it may happen that the problem defined by H is also
degenerate in respect to certain w p '&, in which case these w p s remain
constant during the motion. By the addition of a further perturba-
tion function these w p 9 s can, of course, become secularly variable.

The calculation of the mean value of the perturbation function H!
is frequently simplified by employing the original variables q, p in-
stead of the angle variables, and averaging with respect to the time.
The orbital constants of the unperturbed motion, which occur in the

110 THE MECHANICS OF THE ATOM

mean value H 19 have then to be replaced subsequently by the de-
generate angle variables w p and by the action variables J^. .

In the case of a system subject to internal forces only, the azimuth,
about any straight line fixed in space, of a plane passing through the
axis of the resultant angular momentum and this straight line is a
degenerate co-ordinate and is constant. If now a weak external
homogeneous field, having the direction of this straight line, acts on
this system, the mean value of the perturbation function AH^ cannot
depend on this azimuth. If now there is no other degenerate vari-
able of the unperturbed system which could be secularly varied by
means of the perturbation function (as is, for example, the case for a
hydrogen atom in an electric field, cf. 37), then the only secular
motion induced by the external field is a precession of the resultant
angular momentum about the direction of the field with the frequency

We have then an approximate realisation of the case of spatial quan-
tisation, dealt with in the foregoing paragraph. The exact motion
differs from that described by small superimposed oscillations ; it
is a " pseudo-regular precession."

19. Quantum Theory of the Top and Application
to Molecular Models

We have already examined (in 12) the motion of diatomic mole-
cules, which we considered as u rotators." We shall deal now with
the general case of molecules containing several atoms, regarded, to a
first approximation, as rigid bodies. The case of diatomic molecules,
mentioned above (and generally of molecules for which all the atoms
lie on a straight line), will then appear as a limiting case, and we shall
obtain, at the same time, a more rigorous foundation for our previous
results.

The conception of molecules as rigid bodies must, of course, be
founded on the electron theory ; for, actually, the molecule is a
complicated system made up of several nuclei and a large number
of electrons. It can in fact be shown * that the nuclei move, to a
close approximation, like a rigid system, but the resultant angular
momentum of the molecules will not be identical with the angular
momentum of the nuclear motion, because the electron system itself
possesses, relatively to the nuclei, an angular momentum of the same

1 M. Born and W. Heisenberg, Ann. d. Physik, vol. Ixxiv, p. 1, 1924.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 111

order of magnitude. We arrive, therefore, following Kramers and
Pauli, 1 at the conclusion that the adequate molecular model is not
simply a top, but a rigid body, in which is situated a fly-wheel with
fixed bearings. We shall consider then, in this paragraph, the theory
of this top provided with a fly-wheel.

Let the top, including the mass of the fly-wheel (which we take as
symmetrical about an axis, so that the mass distribution is not
altered by its rotation), have the principal moments of inertia A,, A tf ,
A a , the axes of which shall be, at the same time, the axes of a co-
ordinate system (x, ?/, z) fixed in the top ; let A be the moment of
inertia of the fly-wheel. Let a be the unit vector in the direction of
the axis of the fly-wheel, the angle of rotation of the fly-wheel
about its axis, and ^o> its angular velocity. As before, we note
by d the vector of the angular velocity of the whole top, and to
define the position of the top relative to axes fixed in space, we
again employ the Eulerian angles 6, ^, i/j (0 and i/t pole distance and
azimuth of the A s -axis, (f> the angle between nodal line and the A^-
axis). The relations between the derivatives of 0, <, iff and the com-
ponents of d have been given previously (in (2), 6). Let D be the
vector of the resultant angular momentum of the body.

The components of the total angular momentum are made up of
the components due to the top alone and those of the fly-wheel :

(1) D lf =A ir d lf +A .

The angular momentum of the fly-wheel about its axis is

(2) Q=A(oH-(da)).

The four equations of motion are obtained by applying the principle
of the conservation of angular momentum. In the first place, the
total angular momentum must remain constant in magnitude, and
in a direction fixed in space ; this gives the Eulerian equations

D=[D, d].

Secondly, the angular momentum of the fly-wheel can be changed
only through interaction with the body of the top resulting in a
change in direction of the axis ; its alteration is therefore perpen-
dicular to the axis, so that its component in the direction of the axis
is constant ; i.e.

1 H. A. Kramers, Zeitschr. f. Physik, vol. xiii, p. 343, 1923 ; H. A. Kramers and
W. Pauli, jr., Zeitschr. f. Physik, vol. xiii, p. 351, 1923.

112 THE MECHANICS OF THE ATOM

(3) S3=const.
The kinetic energy is

(4) T=H(dD)+coQ];

on substituting the expressions (1) this becomes

(5) T-|[AA 2 +AA 2 +AA 2 +Ao>(ad)+a>i}].

In order to obtain the energy as a function of the components of the
angular momentum we substitute in (5) the values of d*, d v , d z cal-
culated from (1) :

We calculate w by deducing a relation between aj and (da) by multi-

'A,

O O Q

plying the equations (1) by ~, -^, -^respectively, and adding; from

this and (2) we get

2 aJD,

_
A x . A v A~

and obtain therefore

U, r +S ni M^ W-v^,^ 2 "

1 ~~~

W T 2

A A 1 a 2 a 2 Q 2

k.,2 **-]/ -**-S J <*JR C*|/ <*2

x\. -^-a* ~^^y ~^z

Besides this integral, we have the principle of the conservation of
the angular momentum which gives :

(7 ) D 2 =J) X 2 +D y 2 +D, 2 -const.

The general character of the motion can be summarised as follows :
The components of D are the co-ordinates (relative to the axes (#, y, z)
fixed in the top) of the point in which the invariable axis of the
system (i.e. the axis of resultant angular momentum, which is fixed in
space) penetrates the sphere (7). This point traverses the curve of
intersection of the sphere with the ellipsoid (6), which is rigidly
connected to the top. In the fixed co-ordinate system, therefore,
the x, y, z system of axes, fixed in the top, executes a periodic nuta-
tion superposed on a precession about the axis of resultant angular
momentum. In the case where the sphere touches the ellipsoid the
motion becomes a rotation about a permanent axis.

In order to formulate the quantum conditions for the motion we

PERIODIC AND MULTIPLY PERIODIC MOTIONS 113

must return to the co-ordinates 9, ^, $ and calculate the correspond-
ing momenta. If we suppose the kinetic energy T expressed as a
function of 0, </), t/j and their derivatives, by means of the relations
(2), 6,

d >r -=0 cos (f)~\~i(f sin 6 sin <

& y -=9 sin (/) t/j sin 9 cos <

A z =--(j)-[-ifj cos 0,
we obtain :

_ar_aT ad. aT d& v aT ad 2

p '~dd~~d& t lrt ttiylri a< a

aT aT ad^ aT d& y aT ad,

<n T _ _ - _ _ ____ I __ .. I __ _ 7

a< ad, a</> ad y a</> ad 2 a<
ai aT ad x aT ad aT ad,

rn - ; _ - _ __ . I _ _ J _ _ *

* a</r ad x ai/r ad y dj> ad, a^
aT

Since, by (5), the derivatives of T with respect to d.r, d tf , d, are the
components D x , D^, D z of the angular momentum (1), it follows that

e x cos <
=rDa; sin sin <^ D y sin cos +-D~ cos

Since the constant angular momentum can have an arbitrary direc-
tion in space, the motion is degenerate and we can reduce the number
of degrees of freedom by I. We can, for example, without loss of
generality, choose the fixed polar axis 00 of the Kulerian co-ordinate
system in the direction of the resultant angular momentum D, in
which case we get :

D^Dsin 0sin0
(8) D v =-Dsin0cos</>

D,=D cos

(D= | D |), and the momenta become :

P ^= cos
p.=Q.

Cos is determined as a single valued function of 0, owing to the
fact that a curve on the ellipsoid (6) is prescribed for the end point

114 THE MECHANICS OF THE ATOM

of D, and that this curve will be traversed just once during one
revolution of <f). It will be seen then that the motion is separable
in the co-ordinates 9, $, <f>, , and leads to the action integrals

$p diff=-27rT) ; $jyfy=D$ cos
and to the quantum conditions : l

(10) D$ cos ed<f>=n*h

The second quantum condition admits of a simple interpretation.
The surface on the sphere (7), which the point of the vector D
passes round in a negative direction of rotation, is given by

F=-D a JJsin ed0d<f>=D*tfd(cos 0)<fy.

If we carry out the integration with respect to 0, we obtain, if the
boundary of the surface does not enclose the polar axis :

n*
FD 2 ! cos 0dJ>=27rD 2 - ;

T m

if it encloses the positive polar axis :

m
if it encloses the negative polar axis :

and if it encloses both ends of the polar axis :

F=D 2 $(2-cos 0)<ty=27rD 2 -- - .

In all cases the ratio to the hemisphere is

where n is a whole number, and the second quantum condition can
be formulated as follows : the ratio of the surface cut out from the
sphere (7) by the vector D to the hemisphere is equal to n/m ;
n can take the values 0, 1 ... 2m.

1 In the case of the top we do not denote the quantum number of the resultant
angular momentum by j, as in the general theory, but by w, because this letter is
used to denote the terms of a molecular rotation spectrum (see Rotator, 12).

PERIODIC AND MULTIPLY PERIODIC MOTIONS 115

We shall now apply our considerations to the case of an ordinary
top without an enclosed fly-wheel. 1 For the components of the angu-
lar momentum we obtain, in place of (1) :

the equation (5) for the energy becomes

T=HAA 2 +AA 2
On introducing the components of angular momentum,

m ,_a-w + v + vi

21 A x A y K z }

If in this case also we take the fixed polar axis in the direction of the
resultant angular momentum, the relations (8) are again valid and
we have

(13, T=

We get two quantum conditions :

(U \
v '

Dj> cos i

In the second condition we have to write cos 9 as a function of <f>
with the help of the energy W, which is equal to T since there are no
external forces. It follows from (13) that

OA\7 ' * 2 JL 2

^_ / L j 1

C080 = ~ ^ A * AV

1 (/) COS 2

A, \ A, ' A,
and the second quantum condition becomes

(15)

It leads to an elliptic integral, containing the energy W as para-
meter. The calculation of W as a function of the quantum numbers
m and n cannot be carried out explicitly, except in the case of rota-
tional symmetry (A^ A v ) which we have already dealt with ( 6).

1 See F. Reiche, Physikal. Zeitechr., vol. xix, p. 394, 1918; P. S. Epstein, Verh.
d. Dtsch. phys. Ge*\, vol. xviii, p. 398, 1916 ; Physikal. Zeitechr., vol. xx, p. 289,
1919.

116 THE MECHANICS OF THE ATOM

In this case, A^A^, the energy (13) becomes

tj> will likewise be a cyclic variable and 6 is constant. From (14), the
quantum conditions are :

27T

therefore

nh
Dcos0= ;

cos#= ,
m

i.e. we have a kind of space quantisation, for which the angular
momentum precesses not about an axis fixed in space but, relative
to axes fixed in the top, about the axis of figure. As a function of
the quantum numbers the energy becomes

If one considers how the co-ordinates of a point of the top are ex-
pressed in terms of the cyclic co-ordinates and (f> (by finite Fourier
series), it will be seen that, in the series for the electric moment,
the frequencies v^ and v^ occur in general with the coefficients and
1. The quantum numbers n and m can therefore change by and
1. When the electric moment has no component parallel to the
axis of figure the transition Aw is excluded.

An application of the energy equation (16) to multiply atomic
molecules would give several systems of rotation bands, displaced
from one another by fixed amounts, with the arrangement of
lines in any one band satisfying a formula of the simple Deslandres
type (c/. 12).

At this stage we raise the question how it is possible to derive
from the top formula (16), by a limiting process, the formula (1), 12,
for the rotator, and we shall show to what extent the application of
the rotator formula to a diatomic molecule is justified. If we have
the ideal case of a system consisting of two rigidly connected par-
ticles, then we have to put A a =0 in the top formula (16), and, in
order that the energy may remain finite, n can take the value only.
We obtain then for the energy the previous rotator formula (1), 12 :

PERIODIC AND MULTIPLY PERIODIC MOTIONS 117

Actually, however, in the case of diatomic molecules, we have to
deal with systems where, in addition to the nuclei which are practi-
cally points of large mass, a number of electrons are present, which
move around the nuclei and may, under certain circumstances,
possess angular momentum about the line joining the nuclei. This
system may be roughly compared to a top, whose moment of inertia
A z about the nuclear axis is small in comparison with the moment of
inertia A^ about a perpendicular direction. For an invariable electron
configuration, the quantum number n, and consequently the second
term in the energy (16), is a constant. For the dependence of the
energy on the state of rotation we have therefore

A 2
(17) W=W e

In general, in a quantum transition n, and consequently the contri-
bution W e to the energy from the motion of the electrons, varies, and
apart from this m varies by or 1. If we leave undetermined the
dependence of W, on the quantum numbers, since the conception of
the electrons as a rigid top is naturally very doubtful, we obtain for
the frequency radiated in a transition (neglecting the frequency v=v e
corresponding to Am 0)

W r , and so v e , is very large in comparison with the term originat-
ing from the rotation, on account of the smallness of A z in (16).
Since the rotation term alone gives rise, as already shown, to lines
hi the infra-red, the spectrum represented by (18) is displaced towards
higher frequencies, and so may lie in the visible or ultra-violet regions.
We have in this the simplest band formula which represents to the
roughest approximation the observed bands. From the observed
separation of the lines the moment of inertia A,,, of the molecule
may be calculated.

In passing from the energy equation (17) to the frequency equation
(18), the assumption is made that the moment of inertia A^ does not
vary with a change in the electron configuration. If this assumption
be dropped and we assume that A,,, changes from A a W to A B ( 8 ), we
get, for Am=l, the frequencies

118 THE MECHANICS OF THE ATOM

(19) V=Ve

where

=?+

h
(20) b=-

h I 1 1

n I 1

87r*\A,0) A X W/'

The frequencies (19) constitute the " positive and negative branches "
of the band. For Aw=0, the " null branch " is obtained :

1 1

It is absent if the electric moment of the molecule is perpendicular
to the axis of rotation.

We obtain the distribution of the lines in the three branches by
drawing the three parabolas (J9) (with + and signs) and (21), and
dropping perpendiculars on the i>-axis 1 from the points corresponding
to positive integral values of m (see fig. 8). One of the two branches
(19) covers part of the v-scale twice, the lines are concentrated (with
finite density) at the reversal point, the " band-head." The line in
which the positive and negative branch intersect (m 0) is called the
" null line." To calculate the moment of inertia from an observed
band, the constant b must be known, and for this one must know the
position of the null line of the band. If a null branch is present its
position serves to indicate the null line. If, however, the null branch
is absent, the properties of the band given here do not suffice. It
appears, however, that the intensities on the two sides of the null line
are symmetrically distributed, and the null line itself has the inten-
sity ; we shall return to this point again shortly.

Kramers and Pauli have endeavoured to treat the band spectra of
molecules whose electronic angular momentum has a direction fixed
in the molecule but is otherwise unrestricted, and to explain the
absence of the null line, by applying to molecules the model of the
top with an enclosed fly-wheel.

The top represents here the nuclear system (considered rigid) and
the fly-wheel represents the angular momentum of the electrons.
Since the dimensions of the electron orbits in a molecule are of the

1 Comp. A. Sommerfeld, Atomic Structure and Spectral Lines (Methuen), p. 427.

PERIODIC AND MULTIPLY PERIODIC MOTIONS

same order of magnitude as the nuclear separations, and the mass of
the electron is small in comparison with that of the nucleus, A is a
magnitude small in comparison with A^, A v , A z ; the quantum
conditions require, moreover, that the angular momentum 13 of the

6
5
9
J
2
1

^**^

^^^~\

.,^\

^x^"

^x^

yXi

i -^^^^

i^x*

i^^T

j^^^

... .j

i | .

i ~~

^L-^ 1 i

"~ I I

iVi

/i J^

Lx4rT !

}

Nll>^!

II !

^7 1

i i i
i i i

i
j

i
5

i
I

I I

positiue*

567 8 9 10

// /^

I
i
I

A?e0a/uel

rvx

V3J

/ (7

I I

Ai//A

O1 2 3

5 6

Band

FIG. 8.

electrons shall be of the same order of magnitude as the resultant
angular momentum D.

We now develop T in powers of A and break ofE the series after the
second term :

The first term of this expression is a constant (the energy of the
electron motion), the second term

120 THE MECHANICS OF THE ATOM

(22) E=

is the energy of the gyroscopic motion of the molecule.

The stationary motions are obtained when mlift-n is put for the
resultant angular momentum |D| and the values of E so chosen that
the ellipsoid represented by (22), whose centre is at the point 12a, cuts
from the sphere |D|=const. a surface whose ratio to that of the
hemisphere is n/m ; we shall return later to the consideration of the
significance of Q and the question whether this quantity is to be sub-
jected to a quantum condition.

In the case of diatomic molecules we take the z-axis in the line
joining the nuclei, and the #-axis in the plane determined by the
axis of the angular momentum of the electrons and the line joining
the nuclei. We then have a w =0, A z small in comparison with A^ and
A y (in the ratio of electron mass to nuclear mass), and (to the same
approximation) Aa.=A y . The ellipsoid represented by (22) degener-
ates into a flat circular disc, parallel to the (x, y)-plane, having Ua x ,
0, 2a 2 as the co-ordinates of its central point.

The curve of intersection of this degenerate ellipsoid with the
sphere encloses a surface the ratio of whose extension in the z direc-
tion to the radius of the sphere is VA Z /A X . For values of the result-
ant angular momentum D which are not too great, only the quantum
number n=0 is permissible. This signifies that the flat ellipsoid
touches the sphere. If E be allowed to increase from to oo , such a
contact occurs twice, irrespective of whether the centre point of the
ellipsoid lies inside or outside the sphere. Of the two corresponding
types of motion only that corresponding to the smaller value of E
is stable, since only in this case will the curve cut out from the sphere
for a small increase of E be closely confined to the region surround-
ing the point of contact, i.e. the motion remains in the immediate
proximity of the stationary motion.

The point of contact must lie in the plane passing through the
middle point of the ellipsoid and the nuclear axis ; from this it follows
that D v =0. We conclude from the relation

which implies that the normal to the sphere coincides with that of
the ellipsoid at the point of contact, that

Da-a^

PERIODIC AND MULTIPLY PERIODIC MOTIONS 121

is of the order of magnitude A^/Aa.. We can therefore neglect the
third term in the energy formula (22) and write

It will be seen from fig. 9 that for this we can write also

E =

If the quantum number m be
introduced together with the
quantities and , defined by

ZTT

Oa,

it ollows that
(23)

FIG. 9.

This is a generalisation of the formula for the energy of a simple
rotator, which is obtained by putting

==0.

If the angular momentum of the electrons is directed along the
nuclear axis (0), then

This formula agrees with that for the symmetrical top (16), if the
term there proportional to - (as electron energy) be removed and

put equal to n.

The general formula (23) has been used in different ways by
Kratzer, 1 and Kramers and Pauli, 2 to explain the observed
phenomena, that, in a system of equidistant band lines, one line
is missing.

Kratzer uses the formula (23) for the case where =0, i.e. the angu-
lar momentum of the electrons is perpendicular to the nuclear axis.
From 7 9

1 A. Kratzer, Sitz.-Ber. Bayr. Akad. Math.-phys. KL, p. 107, 3, 1922.

2 H. A. Kramers and W. Pauli, jr., Zeitschr.f. PhyM, vol. xiii, p. 351, 1923.

122 THE MECHANICS OF THE ATOM

he obtains for the frequency radiated in the transition
(keeping the electron configuration constant)

(24) ^+_*_( m _ +t ),

and for the frequency radiated in the transition m->m+l

(25) p = ?.__.( m

The positive and negative branches consist therefore of equidistant
lines, which begin in general at different places ; the positive branch
begins at \ , the negative at (). By forbidding the state
w=0 and putting ~\ Kratzer thus deduces a gap, of twice the
width of the ordinary separation of the lines, between the two
branches.

Kramers and Pauli show that this remains essentially valid if
does not vanish. In this case m must be > and the expansion of
E in terms of 1/m,

remains approximately valid even for small values of m (except for

2 f

m=0, which cannot occur if =)= 0)- If we neglect the term , we

obtain the same frequencies (24) and (25) as above, thus also the
correct size of the gap in the case f =J.

The value =% can arise by the angular momentum of the electrons
being h/fa and making an angle of 30 with the nuclear axis. This
assumption leads, however, to difficulties in connection with the
intensities given by the correspondence principle. 1 For this reason
Kramers and Pauli return to the assumption =, =0, in other
words to an electron momentum (with a " half " quantum number)
perpendicular to the nuclear axis.

20. Coupling of Rotation and Oscillation in the
Case of Diatomic Molecules

The bonds between the atoms which are combined to form a mole-
cule have hitherto been regarded as rigid ; this is only approximately

1 There are other difficulties, inasmuch as an electron angular momentum which
is not parallel to the nuclear axis is only possible for certain degenerations of the
electron motion (M. Born and W. Heisenberg, Ann. d. Physik, vol. Ixxiv, p. 1, 1924).
Prof. W. Pauli informs us that the rigorous treatment of these degenerations leads
to parallel and perpendicular orientations only for the angular momentum of the
electrons.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 123

true, however, for the atoms will in fact execute small oscilla-
tions with respect to one another. The problem now is to find what
influence these oscillations have on the energy and on the frequency
of the radiated or absorbed light.

The actual nature of the forces which bind the molecule together
will be determined in an extremely complicated manner by their
electronic and nuclear structure. Here we shall make the simplest
possible assumption, viz. that the atoms may be regarded as centres
of force which act on one another with a force depending only on the
distance ; it can be shown that the results so obtained represent a
correct approximation to the actual behaviour. 1

As regards the angular momentum of the electrons in diatomic
molecules, we have seen in the previous paragraph that it has no
influence on the rotational motion of the nuclei, and gives rise only
to an additive term in the energy if its axis is parallel to the line
joining the nuclei. The same must be true when the nuclei perform
oscillations in this direction ; we shall therefore restrict ourselves
here to this case.

We consider, therefore, a diatomic molecule, consisting of two
massive particles mi and w 2 , separated by a distance r, between which
there exists a potential energy U(r).

It may be shown quite generally, that such a two-body problem
may be reduced to a one-body problem. We choose the centre of
gravity of the two particles as the origin of co-ordinates and deter-
mine the direction of the line joining m 2 and m l by the polar co-
ordinates 0, (f>. If then 7 1 ! and r z are the distances of the particles
from 0, their polar co-ordinates will be r l9 0, <f> and r 2 , TT 0,
and further, r 1 +r 2 =r. The Hamiltonian function becomes

sin 2 9)+(' t ^+r+r^ sin 2 0) + U(r)

A A

sin 2 0)+U(r).
Since r x and r a are measured from the centre of gravity,

and therefore

If this be substituted in H, we get

1 M. Born and W. Heisenberg, Ann. d. Physik, vol. Ixxiv, p. 1, 1924.

124 THE MECHANICS OF THE ATOM

(1) H = (* 2 +r*fc +r*<f>* sin 2 6) + U(r),

on writing

Now the expression (1) is the Hamilton ian function of the motion
of a particle of mass /x under the action of a centre of force from
which it is separated by the distance r.

In the following chapter we shall investigate this problem quite
generally ; here we shall consider only the case where a position of
stable equilibrium exists, this being the only case of importance in
connection with molecules. 1 There will then be a distance r , for
which U(r) is a minimum, i.e.

(3) O '=0, U ">0,

where the index denotes here, and in what follows, the value of a

quantity at r r .

A possible state of motion of the system is a rotation with a con-
stant nuclear separation r and a uniform angular velocity < about
a fixed axis, passing through the centre of gravity of the masses
and perpendicular to the line joining them (nuclear axis). We take
the axis of rotation as the line 0=0, and have :

(4)

where the bar denotes here, and in the following, the value of a quality

We take this motion as the starting-point for an approximate
method of dealing with small oscillations. We suppose the separa-
tion /" increased by a small amount x so that r=r -\-x, and develop the
Hamiltonian function, regarded as a function of x, (f> and the corre-
sponding momenta, in powers of x. The Hamiltonian function is

The momentum

associated with (/> is constant, because <f> is cyclic ; moreover, p is the
angular momentum ; for x=Q, therefore

(5) P=/"Vo-

The momentum corresponding to x is

1 M. Born and E. Huckel, Phyaikal. Zeitechr., vol. xxiv, p. 1, 1923 ; see also A.
Kratzer, Zeitschr. /. Physik, vol. iii, pp. 289 and 460, 1920.

PERIODIC AND MULTIPLY PERIODIC MOTIONS 125

p x =t*x.
Consequently

On expanding in powers of x we get

The coefficient of x vanishes since by (4) and (5)

^,2 _

(7 > ^=u';

the Hamiltonian function has therefore the following form :

(8) H=W +^+|

where

(9)

This reduces the problem to that of the non-harmonic oscillator,
which we have discussed in 12.

If we now introduce angle and action variables we have to put

and then to introduce w x and 3 X in place of x and p x , in the manner
explained for the non-harmonic oscillator. If we take into considera-
tion the terms in x* in (8), we find (cf. (9), 12)

(10) H = W ( J) + JX J) + J>( J),

where for shortness we write

Similarly, if we take into account the term 6x 4 , H assumes to the same
approximation the same form, only a depends^also on b. The func-
tions W (J) and v(3) are found by calculating r as a function of p

126 THE MECHANICS OF THE ATOM

or J from (7) and substituting in (9). Actually in order to calculate
them the function U(r) must be known exactly. If we restrict our-
selves, however, to such small velocities of rotation that the deviation
r r =r 1 , caused by the centrifugal force, is small in comparison
with TQ, our objective may be attained by means of an expansion in
terms of r r Since U '=0, equation (7) may be written, to a first
approximation

from this we obtain

J 2 1

1 47rV
Further

therefore

a also can be expanded in the form

We have omitted here all terms of order higher than the first in J 2 .
The energy as a function of the action variables now becomes

where A=/u,r 2 is the moment of inertia in the rotationless state, and
v and a have the meaning assigned above.

If we neglect the terms in J x 2 and J^J 2 , and consequently the non-
harmonic character and the dependence of the v's on J, the energy is
resolved into a rotational component and an oscillation component
of the well-known form. As a nearer approximation we have a de-
pendence of the oscillation frequency on the rotation quantum
number and also the non-harmonic character of the oscillation.
Naturally our method admits of more accurate calculations of the
energy, involving higher powers of J and J c .

PERIODIC AND MULTIPLY PERIODIC MOTIONS 127

We shall apply the results obtained to the spectrum of diatomic
molecules. In the stationary states they have the energy

(12) W=U

where m is the rotation and n the oscillation quantum number.
The frequency corresponding to the transition

For fixed values of the initial and final oscillation quantum numbers
n 1 and n 2 and varying values of the rotational quantum number m,
this gives a band with the branches (to which a null branch may be
added) :

(14) v=abm+cm 2 ,

where a, 6, and c have a somewhat different meaning from that in
(20), 19.

The frequencies

(15) V^VQ^ n 2 )+ha Q (ni* n 2 2 ),

which can be ascribed to the change of oscillation quantum number
alone and so may be called " oscilktion frequencies," are displaced
from the null line of this band by

Thus we obtain a band system which is made up of individual bands
corresponding to the series of values of % and n 2 . The positions of
the individual bands in the system are given by (15), while (14) gives
the law of arrangement of the lines in the individual bands.

The infra-red spectra of the halogen hydrides are of the type de-
scribed here but without null branches. 1 These spectra consist of
individual " double bands," i.e. an approximately equidistant succes-
sion of lines, which are symmetrically situated with respect to a gap.
In this gap we have to imagine the null line mentioned in 19. A
doubling-back of the one branch is not observed in this case.

1 Measurements, especially by E. S. Imes, Astropkys. Journ., vol. 1, p. 251, 1919.
For the theory of A. Kratzer given hore, sec Zeitschr.f. Physik, vol. iii, p. 289, 1920.
See also H. BeU, Phil. Mag., vol. xlvii, p. 549, 1924.

128 THE MECHANICS OF THE ATOM

The oscillation frequencies in the case of HC1 are at v=2877 and
v=:5657 (in " wave numbers," i.e. number of waves per cm.). The
corresponding bands appear in the case of absorption at the ordinary
temperature. They correspond, therefore, to a change in the oscilla-
tion quantum number for which the initial state has so little energy
that it is present to a considerable degree at the ordinary tempera-
ture ; that, however, can only he the oscillation state n 2 --0. We
assign, therefore, to the two bands observed the two transitions

for absorption, or the values % 1 and n 1 =2 respectively, and n 2 =0
for emission. In accordance with the theoretical formula (15)

the second band is not situated exactly at twice the frequency of the
first.

An alteration of the rotation and oscillation quantum numbers
may be accompanied by a simultaneous alteration in the electron
configuration of the molecule. A frequency

corresponds to a transition between two stationary states with the
energies

where

(18) v l(tt ---

Altogether we obtain a band system whose individual bands exhibit
the structure described in 19, and are arranged according to the
formula (17). Written somewhat differently, it is

Since, in general, i> 01 and v 02 are of the same order of magnitude and
their difference is small compared with the values themselves, the
first term is the most important. It defines the position of a " band
group " in the band system ; a group contains, therefore, all bands
for which n changes by the same amount. The next term defines
the individual bands, inside the band group, in terms of their final
quantum numbers,

PERIODIC AND MULTIPLY PERIODIC MOTIONS 129

A beautiful example of a band system is provided by the violet
cyanogen bands. 1 Fig. 10 gives the positions of the null lines and

Offit
123VS

FIG. 10.

OiZ

their wave-lengths : the first row underneath the oscillation quantum
number in the initial state, the second row that in the final state. 2

1 Explained theoretically by A. Kratzer, Physikal. Zeitechr., vol. xxii, p. 552, 1921 ;
Ann. d. Physik, vol. Ixvii, p/127, 1922.

2 According to A. Kratzer, loc. cit.

THIRD CHAPTEK

SYSTEMS WITH ONE RADIATING ELECTRON

21. Motion in a Central Field of Force

THE applications of the principles of quantum mechanics, developed
in the second chapter, are at present considerably restricted, owing
to the fact that these principles are concerned only with multiply
periodic systems. The first example dealt with by Bohr, namely,
systems consisting of a nucleus and a single electron (the hydrogen
atom and the similar ions He + , Li ++ , etc.), satisfies this condition of
periodicity. In the case of other atoms the same difficulties underlie
an examination of the periodic properties as in the case of the many-
body problem of astronomy, and we can proceed only by a method
of approximation. Bohr realised that a large number of atomic
properties, especially those which exhibit themselves in the series
spectra, may be explained on the hypothesis that one electron, the
" radiating electron " or " series electron/' plays a special role in
the stationary states under consideration. The essential feature of
these states is that this one electron is in an orbit, which, at any
rate in part, is far removed from the rest of the atom, or " core," x
and exerts only a small reaction on the latter. We shall always
speak, therefore, of the stationary orbits of the radiating electron,
since we neglect the changes taking place in the core. The spectrum
of the atom corresponds then to transitions of the radiating electron
from one orbit to another.

This assumption implies that the motion of the outer electron is
multiply periodic, and that, in traversing the core, the electron
neither gives up energy to nor receives energy from it. Motions of
this kind are quite special cases according to classical mechanics, for
the motions of the core electrons must be such that their energy is
the same after every period of the outer electron, a condition which

1 German, Rumpf. The English equivalent of this word is not completely
standardised: the alternatives "body," "trunk," "kernel" have been used by
different writers.

130

SYSTEMS WITH ONE RADIATING ELECTRON 131

is evidently fulfilled only by strictly periodic solutions of the com-
plex many-body problem. Since, however, a large number of obser-
vations may be explained in a surprisingly simple way by such
stationary orbits of the radiating electron, it appears that we are
here dealing with some general process, which cannot easily be ex-
plained by such singular types of motion. We have here the same
failure of the classical mechanics as was brought to light by Franck's
researches on electron impact ; the exchange of energy between
electron and atom, or atom core, is restricted in a manner similar
to that familiar to us in the energy interchange between an atom
and radiation.

At present we cannot express this non-mechanical behaviour in
formulae. We endeavour to substitute for the atom a model which
possesses, in common with the actual atom, this characteristic pro-
perty of the absence of energy exchange between core and electron,
and to which the principles of the quantum theory, developed in the
second chapter, are applicable. The simplest assumption is that the
action of the core on the radiating electron can be represented by
a spherically symmetrical field of force. Further development has
shown that this simple hypothesis suffices to provide an explanation
of the main characteristics of the spectra of the first three divisions
of the periodic table and their sub-groups. The conception of a
single " radiating electron " is, however, no longer adequate to
explain the spectra of the remaining elements, but these considera-
tions are beyond the scope of this book. 1

For this reason we shall now deal with the motion of a particle in
a central field of force. The motion in a Coulomb field of force (such
as we have in the case of the hydrogen atom) will be found from this
as a special case.

So far as the calculation is concerned it is immaterial whether we
consider our problem as a one-body or as a two-body problem. In
the first case we have a fixed centre of force, and the potential of the
field of force is a function U(r) of the distance from the centre. In the
second case we have two masses, whose mutual potential energy U(r)
depends only on their distance apart ; they move about the common
centre of gravity. As we have shown generally in 20, the Hamil-
tonian function in polar co-ordinates is precisely the same for the
two cases, if, in the one-body problem, the mass p of the moving

1 For the development of the theory of complex spectra, see H. N. Russel and
F. A. Saunders, Astroph. Journ. t vol. Ixi, p. 38, 1925 ; W. Pauli, jr., Zeitsch. f.
Phys., vol. xxxi, p. 765, 1925 ; W. Heisenberg, Zeitschr. f. Phys., vol. xxxii, p. 841,
1925; F. Hund, Zeitsch. f. Phys., vol. xxxiii, p. 346; vol. xxxiv, p. 296, 1925.

132 THE MECHANICS OF THE ATOM

body and its distance r from the centre are used, and if in the two-
body problem /* is defined by the equation (2), 20,

-=-+-,

/LC m 1 m 2 '

and r is the distance between the two masses. The following equa-
tions admit then of both interpretations.

We work with polar co-ordinates r, 6, and <f>. Making use of the
canonical transformation (13), 7, which transforms rectangular into
polar co-ordinates, we obtain for the kinetic energy,

22*5

where p r , p e , p^ are the momenta conjugate with r, 0, ^ respectively.
We arrive, of course, at the same expression when we calculate from

sn 2
2

the momenta :

and use them to replace r, 9, <f>. The structure of the Hamiltonian
function

shows that r, 0, <f> are separation variables. If one puts
(2) S=S r (r)+S e (0)+S/<),

the Hamilton- Jacobi differential equation

splits up into three ordinary differential equations :

which can be solved for the derivatives of S :

SYSTEMS WITH ONE RADIATING ELECTRON 133

d$ r

Of the three integration constants W denotes the energy ;

==xr 2 sn

is the angular momentum about the polar axis (i.e. the line 6=0), and

is the magnitude of the resultant angular momentum. Since also
the direction of the angular momentum is constant (as in every
system subject to internal forces only), the orbit is plane and the
normal to the plane of the orbit is parallel to the vector representing
the angular momentum. The inclination i of the orbital plane to the
(r, <)-plane is given therefore by

a>4=a, cosi.

We consider next the general character of the motion and then
determine the energy as a function of the action variables for the
case of a periodic motion and, finally, we consider the progress of the
motion in time.

The co-ordinate (f> is cyclic and performs a rotational motion (cf.
9). The co-ordinate 6 performs a libration or limitation motion in
an interval, symmetrical about w/2, whose limits are given by the
zero points of the radicand in the expression of p e , i.e. by

sin 0=-_* = cos i, 0=--j-i.
a 2

Further, the character of the motion depends essentially on the be-
haviour of the radicand in the expression for p r ,

We investigate the various possible cases on the supposition that
U(r) is a monotonic function of r and that the zero of potential
energy is so chosen that it vanishes for r = oo .

134 THE MECHANICS OF THE ATOM

Case 1. In a repulsive central field of force U(r) is positive. In
order that positive values of F(r) shall occur at all, W must be posi-
tive. F(r) will then be positive for large values of r, decreasing con-
tinually with continuously decreasing r ; for small values of r, F(r)
is certainly negative ; F(r) has therefore exactly one root. The
motion takes place therefore between r=oo and a minimum value
of r.

Case 2. In an attractive central field, U(r) is negative, and W
may be positive or negative. The sign F(r) for large values of r is
determined by W. For positive W, F(r) is positive there, and there
are motions which extend to infinity. There are no such orbits for a
negative W. In the case of W=0 the variation of U with r, and, in
certain cases, the magnitude of a e , is the deciding factor. The sign
of F(r) for small values of r depends on the rate at which j U(r) | be-
comes infinite. If, for small r, it increases more rapidly than 1/r 2 , 1
F(r) will be positive there, and there will be orbits which approach
indefinitely close to the centre of force ; if | U(r) | becomes infinite
more slowly than J /r 2 there will be no such orbits ; if | U(r) j ap-
proaches infinity as 1/r 2 , the magnitude of a e is the deciding factor.
Further, there are cases where, in addition to paths extending to the
centre and to infinity, orbits exist which extend between two finite
and non-zero values of r, r mm and r max ; this is the case when r min
and r raax are consecutive zero points of F(r), between which F is
positive. In the case where | U(r) | becomes infinite more slowly than
1/r 2 it is certain that there are values of W for which such a libra-
tion sets in ; for negative W there are in fact, in this case, no other
motions but librations.

In many applications to atomic physics we are only concerned
with those motions in which the electron remains at a finite distance
from the centre and which are periodic. We consider therefore in the
following only the case of attraction and take for W values such that
F(r) is positive between two consecutive roots r mm and r max .

In this case we can apply the methods developed for periodic
motions. We obtain the action integrals

1 Mathematically expressed, this has the following significance : the order of
magnitude of | U(r) | is larger than that of 1/r 2 for small values of r. The order
of magnitude of a function f(x) (>0) is greater than the order of magnitude of the
function g(x) (>0), for small values of x, if

/(*) and g(x) have the same order of magnitude if the limiting value of -j-r- r is a

J\^f
finite constant.

SYSTEMS WITH ONE RADIATING ELECTRON 136

(4) J

J/) =

With the help of the substitution

1
i

a e
the second integral takes the form

Jn=

The evaluation of this integral (cf. (3) and (8), Appendix II) gives

J^M** %)
We can now express a e and a in terms of the action variables

In order to find the energy as a function of the J's we should have
to solve the equation

(6)

for W. This is impossible without a detailed knowledge of U(r) ; it
is seen, however, that W depends only on J r and the combination
J0+J . The two frequencies

are therefore equal and the system is degenerate. In accordance with
the fundamental principles developed in 15, we introduce new
variables w l9 w 2 , w& and J 1? J 2 , J 3 , so that w. 3 is constant. We
arrange at the same time that, in the case of the Coulomb field of
force, where v r =v & v^ that the variable w 2 shall also be constant.
We write, therefore, in accordance with (8), 7

136 THE MECHANICS OF THE ATOM

W l =W r Jl=Jr+J0+J

(7) W 2 =W e -W r J 2 =J0+J,

W3=w*We J3 =J ,-

The equation (6) contains then only J x and J 2 , and we derive the
energy W in the form

(8) W=W(J,, J 2 ).

For the stationary motions we have, provided there is no further
degeneration (e.g. no Coulomb field), the two quantum conditions :

n is called the principal quantum number and k the subsidiary quan-
tum number. 1

The action variables have the following physical significance : J 2
is 1/27T times the total angular momentum, J 3 is l/2ir times its com-
ponent in the direction of the polar axis.

It is obvious that J a cannot be zero. Also J 2 =0 would signify a
motion on a straight line through the centre of force, a " pendulum
motion " ; in physical applications, where the centre of force is the
atomic nucleus, this case must of course be excluded.

In order to find the physical significance of the angle variables,
we calculate them with the help of the transformation equations

If we introduce the J^'s in the equation (3) we obtain

47rV 2

T 2
v ' - 3

JrO o .

* sin 2 6

and for the angle variables I putting v l = =- ] :

1 k is also called the azimuthal quantum number. This term arises from the
fact that it can also be put in the form

where is the azimuth of the moving point in the orbital plane.

SYSTEMS WITH ONE RADIATING ELECTRON

137

r c

*- -7

(10)

2 "J

J- a sin 2 fl

The two integrals in dO may be evaluated. We have

(10/)

dO . cos 6

;_~ = sin" 1 r+const.

' T 2 .
Jo

sin 2 (9

sm^

and
(10")

sin 2 6

V'-.

= r :

Jsin 2

cos i dO

/ cos 2 !
V ""sin 2 !

J 2 2 sin 2 J V * sin 2

= sin* 1 (cot i cot 0) +const.

It will be seen from fig. 11 that, apart from the arbitrary constant
of integration, the integral (10') is
the angular distance if/ of the moving
point from the line of nodes,
measured on the orbital plane, and
the integral (10") is the projection
of this angular separation on the
(r, <)-plane. By subtraction of this
projection from <f> we get the longi-
tude of the line of nodes. The third
of our equations (10) states, there-
fore, that, apart from an arbitrary
additive constant, the longitude of the node is 27rt0 3 . According to
the second of the equations (10), 2irw z is the angular distance of

FIG. 11.

138 THE MECHANICS OF THE ATOM

the moving point from the node, measured on the orbital plane, in-
creased by a function of r :

, J 1? J 2 )

For given Jj and J 2 , F 2 is a single- valued function of r, for, during
a libration of r, Jp r rfr increases by Jj ; the partial derivative with
respect to J 2 assumes, therefore, its old value once more. Apart from
an additive constant, 27rw 2 is consequently the angular distance of a
point of the path with a given r from the line of nodes, measured on
the orbital plane, and therefore, apart from a constant, the angular
distance of the perihelion (r mm ) from the line of nodes. Finally, again
apart from an additive constant, 277^ is what astronomers call the
" mean anomaly," namely, the angular distance from perihelion of a
point imagined to rotate uniformly and to pass through perihelion
simultaneously with the actual moving point.

Since we have a system subjected only to internal forces, and the
motion takes place in a plane, the angle variable w 2 , associated with
the total angular momentum, occurs in the Fourier representation of
the electric moment with the factor 1 only (as was shown gener-
ally in 17). We can see this also directly, from the nature of the
expressions for the angle variables. These are :

^1= fi(r, Ji, J 2 )

w 2 - <A+/ 2 (r, J lf J 2 )

w 3 =const.,
or, if we solve for r, i/r,

r= ^(w l9 Ji, J 2 )

lfj = 2TTW 2 +(l> 2 (W v J 1? J 2 ).

If we transform to the rectangular co-ordinates , r], , where is
perpendicular to the orbital plane, we find for the components of the
electric moment p expressions of the form

(11')

P^=.

According, then, to the correspondence principle, the number k, of
the quantum numbers n and k introduced by (9), can alter by 1
only, while n can in general change by arbitrary amounts.

The orbit is best expressed in terms of the co-ordinates r and *//.
From the first equation (10) we get

SYSTEMS WITH ONE RADIATING ELECTRON 139

dw UL

-,dr.

Also since the angular momentum is J z /27r

we eliminate dt and derive the differential equation of the orbit

(12)

27T

dr~

477 V 2

Since the motion consists of a
libration of r, combined with a
uniform rotation of the perihelion,
the form of the orbit is that of a
rosette (cf. fig. 12).

FIG. 12.

22. The Kepler Motion

The simplest application of the results of 21 is to atoms consist-
ing of a nucleus with a charge Ze and only one electron. In this case
the motion concerned is that of two bodies under the influence of a
mutual attraction giving rise to the potential energy

(1) V(r) = ~-

This motion we shall now consider.

The action integral J r (6), 21, takes the form

(2)
where

(2')

A=2/*(-W)

27T J \2w.

140 THE MECHANICS OF THE ATOM

It will be seen that only when W is negative can the radicand have
two roots between r=0 and r=oo enclosing a region within which it
is positive ; consequently A, B, and C are all positive numbers. By
the method of complex integration we obtain (cf. (5), Appendix II) :

2n J J

ATT- - - ______ J fl J ..

V-2W ' *

We can now express the energy W in terms of the action variables,
the value we find being :

a\ w-

The motion is therefore doubly degenerate, since for a given value
of J the energy is independent of J 2 (the total angular momentum)
as well as of J^. Not only the longitude of the node, but also the
angular distance of the perihelion from the line of nodes, remains
unaltered. We have only one quantum condition,

and expressed in terms of this the energy is

(4) W -.

v ' h* n*

The motion has only one frequency different from zero ; from (3)
we find for this

_aW_477V 4 i

the period of revolution is therefore

We again express the orbit in terms of the co-ordinates r, ifj in the
orbital plane. As differential equation of the path, we get, by (12),
21:

d$ VC

dr~ I B C'

r\ /-A+2 f

V r r 2

where A, B, and C have the meanings (2'). Integration gives

SYSTEMS WITH ONE RADIATING ELECTRON 141
. . - C-Br

and, if we solve for r :

_ C

If, for shortness, we write

we obtain the well-known form for the equation of an ellipse, whoso
focal point coincides with the origin of the co-ordinates :

is the eccentricity and I the semi-latus rectum or parameter. If we
express these in terms of the angle variables we have

(8) --!-

These two quantities fix the form of the orbital ellipse. Since an
ellipse is usually determined by the semi-major axis a and the eccen-
tricity e or by means of the two semi-axes a and 6, let us express a
and b in terms of the action variables. We have

/ T'2

(10)

1-e 2 47T*
(11) 6=aVl^

Of these two quantities a alone is fixed by the quantum condition ;
e, and with it I and 6, can assume all values consistent with the corre-
sponding a. The relation between a and the quantities W and v lf
likewise fixed by the quantum condition, can be expressed as follows :

(12)

142

THE MECHANICS OF THE ATOM

The equation (13) expresses Kepler's third law. For the case of the
circular orbit, equation (12) states that the orbital energy is equal to
half the potential energy. As we shall see in a moment, it is in
general equal to half the time average of the potential energy.

We now consider the progress of the motion in time. By 10, 21,
we get for w l :

w 1 =v l t+8 1 =

If we resolve the radicand into its linear factors, we obtain

r prvidr

w 1 =\ ,

jVA.V[a(l+)-r][r-a(l-e)]

for a(l +e) and a(l e) are the libration limits of r. The substitution

(14) r =a(l cos u)
transforms the integral to

(15) Wl

In order to make clear the geometrical significance of u, we introduce
rectangular co-ordinates , 77 such that the -axis is the major axis
of the orbit and the origin is the centre of force, Z (fig. 13), thus :

We obtain then from (7) and (14)
(16)

qr aq
= - a cos u =a (cos u e)

fy / b <" V A c A A wo W J

77 =a Vl e 2 sin u.

In fig. 13 ON-a, ZQ-f =a [cos (ZON)-e] and QM=>q=Vl- 2 .
QN=aVl 2 sin (ZON). The angleZON is there-
fore just the auxiliary quantity u. On account of
fl its significance u is called the eccentric anomaly.
Now that we have found expressions for all the
principal magnitudes of the Kepler motion we
shall write them down once more in collected
f orjtn. The energy of the motion is

FIG. 13.

(3)

SYSTEMS WITH ONE RADIATING ELECTRON 143

the motion is confined to an ellipse with the semi-axes

(10) =; T ' 2

(11) b=^

the parameter of this ellipse is
(9) Z=

the eccentricity

(8a)

and the inclination i of the normal to the plane of the orbit to the
polar axis of the r, 6, $ co-ordinate system is given by

The progress of the motion is given by

(14) r=a(l ecosu)

(16) | a(cosu e)

(17) i^aVl-c 88 "^.
Here w is defined by

(150) 27Tv l t=u sin w,
where

i being reckoned from the instant at which the perihelion is traversed.
A knowledge of the progress of the motion in time enables us to
calculate certain mean values. Later we shall often require the mean
values of certain powers of 1/r which we now proceed to evaluate.
We have

f- n = l-= = 9

yn l/yn-2

Now the areal velocity r^ifi is equal to 2v x times the area of the ellipse,
from which it follows that

Vl dt dj

if n ==

and

144 THE MECHANICS OF THE ATOM

For n~2 we can quickly find, in this way, the mean value sought ; if
we take 1/r from the ellipse-equation (7)

1 = 1
Til

we find

__ 1

T ~ri

(19)

a 4 (l-e 2 )'

a 5 (l-e

The mean values r~ l , r, r 2 . . . are more easily calculated by means
of the eccentric anomaly. Using (14) and (15)

- If

r n ~fr n vidta n I (1 cos u) n + l du ;

2tTTj

and we find

(20)

Mean values of the form r" cos m 0(m>0) are best calculated for
n^ 2 by the ellipse equation (7'), for n^m 1 by the eccentric
anomaly ; with the help of (18) we obtain

/* cos m

and from (14), (15), and (16)

_ 1 p

r" cos w 0=ai n I (1 cos w) n ~ m+1 (cos u c) m du 9

&7TJ

so that

SYSTEMS WITH ONE RADIATING ELECTRON 145

COS =

r~ 2 cos iff=(
(22) r-*cos0=-

Mean values of the form r" cos w i/j sin' iff vanish for odd values of /.
For even values of J, sin 2 may be replaced by 1 cos 2 and the mean
value reduced to the form just considered. In particular

/OO\ 1 " 9 I

(23) r -3 sln 2^,

2o 3

We can now find the time average of the potential energy, it is

and is thus twice the orbital energy. The mean kinetic energy
becomes

This theorem that the mean kinetic energy is equal to half the mean
potential energy is valid generally for a system of electric charges
which act on one another with forces obeying Coulomb's law.

Further, the co-ordinates of the electrical centre of gravity for a
charge revolving in a Kepler ellipse are the time averages of the
actual co-ordinates and 77, thus

(23') f=-30

and, by symmetry,

f,=0.

The electrical centre of gravity is therefore situated on the major
axis half-way between the middle point of the ellipse and that focus
not occupied by the centre of force.

In the case of the Kepler motions the Fourier .series for the rect-
angular co-ordinates , rj and for the distance r are comparatively
easy to find. Noting that r/a and g/a are even functions, and rj/a an
uneven function of u, and therefore also of w l9 we can put

146 THE MECHANICS OF THE ATOM

(24) - = JCo+Z C r cos

a *

sn >i

For the coefficients we obtain the integrals :

ri r

B T =4 - cos (27TW 1 r)dw 1
J a

(25) C T =4 1 - cos (2rrw 1 r)dw 1

D T =4 I - J= sin (27TW l r)dw 1 .
JoaVl e 2

By partial integration we get from these

2 f* /V\

B T = -- sin (27TW l r)d( -

7TTj W

C -- (*si

2 r* r)

D T -+ COS (Zrrwrfdl ' __ I.
^rJ

Now by (16) and (17) we have

d( - =sin?/ du

= cos u du.

If now we introduce u as an integration variable from (15), we
obtain

2 f 7r
B T = 1 sin [r(u sin u)] sin u du

7TTJ

2 f *

C T = I sin [r(u sin w)] sin w rfw

7TTJ

1 2 f "
D T = I cos [r(u sin w)l cos u du,

7rrJ

A simple trigonometrical transformation leads to :

SYSTEMS WITH ONE RADIATING ELECTRON 147

elf" (" }

B T = {I cos [(r+l)u resin u]du\ cos [(r l)u resin u}du\

1 f f " - 7 f * 1

C T = i ~ 1 cos[(r+l)w TSinw|w+ 1 cos[(r 1)?^ rSinw]d!w}

TTTl J Q J J

D T = I I cos [(r-\-l)u re sin w"]dM+ I cos [(rl)uresinu]du\.

7TTlJ J J

The integrals appearing here are Bessel functions x defined by
J T (X) = - 1 cos (TH x sin u)du.

7T JQ

We have therefore

Since these formute fail for. r=0 we must calculate B , C , D from
(25). We find:

f*r 2 f 7 "

B -4 -^=- (l-

J a 7rJ

f*l 2

C =4l -^ 1 =-

Ja 7r

D =0.

If, finally, we substitute the calculated values of the coefficients in
(24) we derive :

j = l + ^+2 ^iW-JI^Tc)] COS (27^)
(26)

28. Spectra of the Hydrogen Type

The calculations given in 22 provide us now with a basis for the
explanation of certain line spectra. According to the conception of

1 The Bessel functions are here indicated by Gothic J's t to avoid confusion with
the action variables.

148 THE MECHANICS OF THE ATOM

atomic structure described in the Introduction, the hydrogen atom
in the uncharged (neutral) state consists of a nucleus of charge +e
and considerable mass M and an electron of charge e and small
mass m. Of similar structure are the singly ionised helium atom
(He+) and the doubly ionised lithium atom (Li+ + ), only the nuclear
charge is 2e and 3e respectively in the two cases. In all of these
atoms, therefore, we have a Z-fold charged nucleus and one electron ;
their mechanics is consequently included in the theory given in 22.
The energy in the stationary states is, by (4), 22,

(1) W=

7fr~

where

(2) R=_^!!.

R is known as the Rydberg Constant, because Rydberg was the
first to notice that it occurred in the representations of numerous
spectra. Since

_ wM _ 1

(3) **~m+M.~ m ~ m 9

R depends on the ratio of the electron mass m to the nuclear mass
M. The limiting value for infinitely heavy nuclei is

For other atoms

(5) m

1+ M

The correction factor is here very nearly 1, since even for hydrogen
w/M=l/1830 ; in the majority of cases, therefore, R may, to a suffi-
cient approximation, be replaced by R^.

Spectroscopists prefer to specify spectral lines not in frequencies
but in wave numbers, i.e. number of waves per cm. We will follow
the usual notation and write v for the wave number of a line or term
in 23-29. This should not be confused with the earlier use of v
for the mechanical frequency in an orbit.

The wave numbers of the spectral lines corresponding to the terms
(l)are

SYSTEMS WITH ONE RADIATING ELECTRON 149

(6) v = I (WW - WW) =RZ 2 / -L - -I

Ac v ' \n a * 2

According to the correspondence principle all transitions between
the stationary states occur, since in the Fourier series for the motion
(26), 22, the coefficients of all the harmonics differ from zero.

For Z=l the spectrum of the hydrogen atom is obtained from
equation (6), and, for n 2 = 2, in particular, the long-familiar Balmer
series :

The strongest support of the Bohr theory consists in the agreement
of the quantity R n , determined from the spectroscopic measure-
ments of this series, with that expressed by (4) and (5), in terms of
atomic constants (the difference between R H and R x is smaller than
the relative errors of measurement of the atomic constants).
According to deviation experiments on cathode rays

m gm.

by Millikan's measurement of the smallest charge on a drop
e-4-77 . 10- 10 E.S.U.,

according to heat radiation measurements and determinations of the
limit of the continuous X-ray spectrum (see later)

h ==6-54 . 10~ 27 erg sec ;

with these numerical values one finds from (4)
cR=3-28 . 10 15 sec- 1 ,
R^l-Og.lOScm- 1 ;

the value deduced directly from the observed spectrum is
R H =109678 cm- 1 .

The agreement of the two numbers lies within the limits of accuracy
in the value calculated from (4) using the observed values of e, e/m,
and h.

This gives for the work done in separating the electron when in
the one-quantum orbit

W 1 =-cRA=2-15 . 10- 11 erg.

This value can also be expressed in kilocalories per gram molecule
by multiplying by Avogadro's number N=6-0^ . 10 23 and dividing
by the mechanical equivalent of heat 4-18 X 10 10 ergs per kcal. The
result is 312 kcal. Finally, as a measure of the energy, use is often.

150 THE MECHANICS OF THE ATOM

made of the potential V in volts through which an electron must pass
in order to gain the energy under consideration ; we have

w- eV

300'

The value 13-53 volts is found for the energy of the hydrogen electron.
The general transformation formula is

kcal.

(7) 1 volt=23-0 -- - =-1-59 . 10- 12 erg=8-ll . 10 3 cm- 1 .
gm. mol.

It is the potential V which is directly measured in the method of
electron impact (see Introduction, 3).

The formula (6) contains, in addition to the Balmer series, the
following hydrogen series :

1. The ultra-violet Lyman series,

Since the constant term in this series formula corresponds to the
normal state of the atom, the series occurs in " non-excited " atomic
hydrogen as an absorption series.
2. The infra-red Paschen series,

For Z 2 we obtain the spectrum of ionised helium (the " spark
spectrum " of helium). In this spectrum the lines which correspond
to even quantum numbers (w=2N),

2 ~ N?/'

are situated in close proximity to the hydrogen lines,

This similarity between the spark spectrum of helium and the spec-
trum of hydrogen was responsible for the fact that the former used
to be written in the form

.. 2,

and the lines, observed in certain stars and nebulae, which fitted
this formula were ascribed to hydrogen. Bohr made the situation

SYSTEMS WITH ONE RADIATING ELECTRON 151

clear and showed that the difference between the two Rydberg con-
stants R u and R Ho was due to the differences in the nuclear masses
M in (3).

The hitherto unobserved spectrum of doubly ionised lithium
(Li ++ ) is given by putting Z=3.

In addition to the quantitative agreement of the spectra the
orders of magnitude are also in favour of Bohr's model of the atom.
For the radius of the normal orbit of the hydrogen atom, considered
as a circle, we have by (10), 22, for fj,=m

(8) a H = -4 =0-532 . 10~ 8 cm ;

* ' ** A 2//i/ ,o2

this falls within the order of magnitude of estimates deduced from
the kinetic theory of gases and other atomic theories. For the semi-
major axis of the excited hydrogen ellipses we have by (10), 22,

(9) a=a H .n 2 ;

the radii of the corresponding orbits of He + and Li++ are smaller
in the ratio 1 : 2 and 1 : 3 respectively.

24. The Series Arrangement of Lines in Spectra not of
the Hydrogen Type

We proceed now to those spectra not of the hydrogen type. As
we have already mentioned in 21 we endeavour, following Bohr,
to ascribe the production of these spectra to transitions between
stationary states of the atom, each of these stationary states being
characterised essentially by the motion of a single " radiating " or
" series " electron in an orbit under the influence of the core, which
is represented approximately by a central field of force. This con-
ception explains some of the most important regularities of the
series of spectra, namely, the existence of several series, each of
which is more or less similar to the hydrogen type, and the possi-
bility of combinations between these.

In a (non-Coulomb) central field of force the motion depends,
according to 21, on the subsidiary quantum number k in addition
to the principal quantum number n. k has a simple mechanical
significance, being in fact the total angular momentum of the electron
measured in units of A/27T.

The Bohr relation between frequencies of radiation and energy
differences of the radiating system,

152 THE MECHANICS OF THE ATOM

corresponds to the general observation that the regularities which
occur in observed spectra can be expressed by writing the wave
number of a line as the difference of two terms, the number of terms
being less than the number of lines ordered by means of them. In
our simple atomic model the terms depend on two integers n and k
and can therefore be denoted by the symbol n k . We found, by
applying the correspondence principle, that only such terms may
combine with one another as have values of k differing by 1
(sec (11'), 21).

With this theoretically predicted spectrum we compare that
actually observed. The empirical set of terms of any one spectrum
is arranged by spectroscopists in a number of term series ; l an indi-
vidual term is denoted by its number in the term series and by the
name of this series. The usual designation of these term series is
derived from the historical designation of the corresponding line
series : s (sharp or second subordinate series), p (principal series),
d (diffuse or first subordinate series), / (fundamental series, often
called also 6, Bcrgmann series), g (called sometimes /' or /*), etc.
There is therefore a series of s-terms, one ol p- 9 d- 9 f- . . . terms ;
further, each of these may be multiple, but this possibility we shall
disregard for the time being. 2

With the usual spectroscopic numbering of the terms in the series
we derive the following scheme :

Is 25 3s 4s 5s 6s ...

3d 4d 5d 60! ...
4/ 5/ 6/ . . .

In each of these series the terms with increasing order number de-
crease towards zero.

In order to see how our numbers n and k are related to these letters,
we refer to the following observations respecting the combination
of the terms. Under normal conditions (i.e. when the atoms are in
direct interaction with the radiation without being disturbed by
external influences) the following rules hold : 3

1 Thc^ word " sequence " is sometimes used for a term series, and the word
"series " is then restricted to mean a scries of lines in the spectrum.

2 The multiplicity of the terms cannot be explained on the assumption of a point
electron and a central field of force. It was first ascribed to a space quantisation
of the orbit of the radiating electron with respect to an axis in the core, and later to
a spin of the electron itself (</. p. 155).

3 Thoy are obeyed strictly in the more simply constructed types of spectra, e.g.
those of the alkalies and of Cu and Ag. In the other spectra also they are for the

SYSTEMS WITH ONE RADIATING ELECTRON 153

1. Two terms of the same term series never combine.

2. The only combinations are s- with y-terms, p- with s- and d-
terms, d- with p- and /-terms, etc.

From this it is clear that the separate term series differ in the quan-
tum number k and that taking the term series in the order s, p, d,
/ . . . the number k increases or decreases by 1 from one to the
next. Since s represents the end of the series of combinations, pre-
sumably in the term series s-, p-, d-,f- . . ., k is to be put equal to
1, 2, 3, 4 respectively. 1

We shall now see what can be said regarding the magnitudes of
the terms.

The field of force of the core of an atom is, at a sufficiently great
distance, a Coulomb, field of force. In the case of the neutral atom
it corresponds to the " effective " nuclear charge Z=l, in the case
of the 1-, 2- ... fold ionised atom Z=2, 3 ... respectively. The
orbits of the radiating electron at a large distance are therefore
similar to those in the case of hydrogen. They differ from the Kepler
ellipses only by the fact that the perihelion executes a slow rotation
in the plane of the orbit. The semi-axes and parameter of the
ellipses are, by (9), (10), and (11) of 22,

The perihelion radius vector is :

for a fixed value of k this distance lies between 1/2 and I, the exact
value depending on the value of n. The larger the value of k the
more of the orbit is situated in the Coulomb part of the field of
force ; for large values of k the terms are consequently similar to
those of hydrogen. This confirms the adopted numbering of the
series by the values of k, for observation shows that the terms

most part valid ; the exceptions point to a deficiency of our model (they may depend
on quantum transitions of the core electrons, or to interactions of the scries electron
with the core which cannot bo represented by a central field).

1 A. Sommcrfeld, tiitz.-Ber. d. Bay. Akad. d. Wiss., Math. Phys. C7., p. 131, 1916,
and A. Sommerfeld and W. Kossel, Verh. d. Dtsch. Phys. Ges. 9 vol. xxi, p. 240, 1919.
This co-ordination is possible only in those spectra where one electron can be
singled out as the radiating electron. In the case of more complicated spectra
the designations s- t p- t d- terms must be associated with the resultant angular
momentum of all the external electrons.

154 THE MECHANICS OF THE ATOM

approximate more and more nearly to those of hydrogen the further
we proceed in the order s, p,d,f....

From the term series the line series are obtained by keeping one
term fixed and allowing the other to traverse a term series. The
most commonly observed series by far, and those which have given
their names to the terms, are the following :

Principal series (H.-S.) .... v=ls mp

Diffuse (1st subordinate) series (I. N.-S.) . v=2p md

Sharp (2nd subordinate) series (II. N.-S.) . v2pms

Fundamental (Bergmann) series (F.-S.) . . v=3dmf.

In addition to these the following combinations occur :

Second principal series .... v=2s mp
Second diffuse series ..... v=Sp md

v=Sdmp
v4fmd.

Not only these term differences, but also the terms themselves,
have a physical significance. Thanks to our hypothesis regarding
the potential energy, which we have supposed to vanish at infinity,
the magnitude | W | of the energy constant denotes the work which
is necessary to remove an electron from its stationary orbit to infinity
and to bring it to rest there (relatively to the nucleus). If the station-
ary orbit of the electron is that of the normal state, then this work is
the work of ionisation.

Also the energies W converge to zero, with increasing n, 1 as in the
case of hydrogen, and further the empirical terms of a single term
series likewise converge to zero, so the energy values ascribed theoreti-
cally are in agreement with the empirical terms ; the wave number of
a term multiplied by he is therefore a measure of the work required
to remove the electron from the orbit to a state of rest at infinity.

The largest existing term corresponds to the orbit of the electron
in the normal state and gives a measure of the ionisation potential.
If this term is an s-term, as is the case for several of the simpler
spectra, the ionisation potential is the frequency of the limit (w=oo )
of the principal series multiplied by h ; if the largest term is a p-teim,
the ionisation potential is the frequency of the common limit of the
two subordinate series multiplied by h. Simple spectra are also
known for which a d-term corresponds to the normal state (e.g. Sc++).

All that we can expect of our simple atomic model, by means of

t .

1 This result arises from the behaviour of the integral (6), 21, for negative values
of W tending to zero, when U(r) 1/r for r large.

SYSTEMS WITH ONE RADIATING ELECTRON 155

which we replace the non-mechanical motion of the radiating electron
by a mechanical one based on the assumption of a spherically sym-
metrical field of force for the core, is that it shall give a rough indica-
tion of the general characteristics of the line spectra. As a matter of
fact it makes comprehensible the series arrangement of the lines and
terms as well as the increasing similarity of the higher series to those
of hydrogen. Of the most important remaining unexplained facts we
mention once more the multiplicity of the terms. In all of the alkali
spectra the p-, d- . . . terms are double, in the alkaline earths there
are also triple p-, d- . . . terms. Other elements, e.g. Sc, Ti, Va, Cr,
Mn, Fe exhibit still higher multiplicities. We mention further the
fact that many elements have term systems of the structure described
here, e.g. the alkaline earths have a complete system of single terms
as well as a system with single s-terms and triple p-, d- . . . terms.
Finally, exceptions occur to the above-mentioned rule for the change
of k in quantum transitions.

The multiplicity may be accounted for in principle by assuming
deviations from the central symmetry of the core. If these deviations
are small, they produce a secular precession of the angular momen-
tum vector of the radiating electron and core about the axis of the
resultant angular momentum of the system. Space quantisation
occurs, a somewhat different energy value becoming associated with
each orientation. But this argument leads to multiplicities which
do not correspond exactly to those observed. 1 Pauli 2 has shown
that these could be explained by ascribing four quantum numbers to
each electron instead of three ; and to account for the fourth quan-
tum number Uhlenbeck and Goudsmit 3 suggested that the electron
had a quantised spin about an axis. This hypothesis has been very
fruitful for the understanding of spectra with multiple terms, but it
will not be considered in this volume.

25. Estimation of the Energy Values of Outer Orbits in
Spectra not of the Hydrogen Type

We found that the orbit of the radiating electron was hydrogen-
like for large values of k, since it is situated in an approximately
Coulomb field of force. For smaller values of k the orbit approaches

1 For multiplicities and Zeeman effects cf. E. Back and A. Lande, Zeetrtanejfekt
und MuUipkttetruktur der Spektrallinien, Berlin, Julius Springer, 1925, vol. i of
the German series, Rtruktur der Matene ; and K Hurid, Linienspektra t vol. iv of the
same series.

2 W. Pauli, Zeitschr. f. Physik, vol. xxxi, p. 765, 1925. '

3 G. E. Uhlenbeck and S. Goudsmit, Natunvissenschaften, vol. xiii, p. 953, 1925;
Nature, vol. cxvii, p. 264, 1926.

156 THE MECHANICS OF THE ATOM

the region of the core electrons. As long as it does not penetrate
this region it will be permissible, to a first approximation, to expand
the potential energy of the central field of force in powers of 1/r when
calculating the value of a term. 1 We write

where a denotes a length which may conveniently be put equal to
a n (see (8), 23). The radial action integral is then, by (4), 21 :

B C D -11

where

D- +-2we 2 Za I[ 2 c 2 .

We assume now that in the, expansion for U(r ) the term quadratic in
a jr is small in comparison with the linear term, and calculate as a
first approximation the influence of the subsidiary term c x a/r in the
potential energy on the value of the term. This calculation may be
carried out rigorously for all values of c v The phase integral has the
same form as in 22, and we obtain by complex integration (cf. (5),
Appendix II) :

and from this

If we substitute for B and C their values and introduce the Rydberg
constant R from (2), 23, we get

(2)
(2)

where

(using (8), 23). If the deviation from the Coulomb field is small
only, we can write *

1 See A. Sommerfeld, Atomic Structure and Spectral Lines (Methuen), p. 596.

SYSTEMS WITH ONE RADIATING ELECTRON 157

(3,

The influence of the additional term in the potential energy on the
value of the term may be expressed as follows : If the energy be

-pr ^2

written in the form -- - , the " effective quantum number " w*
w* 2

differs from the integral value n, which it has for hydrogen, by a
small amount 8. The difference depends on k, but not on n, and its
amount will be smaller the larger the value of k. The deviation
from the Coulomb field, caused by the core electrons, will consist
mainly of a more rapid variation of the potential with r, since as
r decreases the attractive action of the highly charged nucleus will
be less and less weakened by the core electrons. Assuming that the
first term of the expansion is the determining factor, this means that
in our expansion (1) c t is positive. 8 is then negative, so that we
should expect the magnitude n*, the effective quantum number, to
be smaller than n.

The form of the orbit is, as in every multiply-periodic central
motion, a rosette. Its equation is easily found. In order to derive
it we again introduce the co-ordinates r, if; in the orbital plane.
By (12), 21, we obtain then for the differential equation of the orbit :

dr y I 2B C

A2 / A _j

^y if ^2

The equation has almost the same form as in the case of the Kepler
motion ; A and B have the same meaning as there :

A=2 M (-W), B
C is somewhat different :

and y has the value

The integration of the equation (4) is carried out in precisely the

158 THE MECHANICS OF THE ATOM

same way as in the case of the Kepler motion, and leads to (cf. 22)

9*= -

B+ VfiS-
If we introduce here the abbreviations (cf. (6), 22)

we get

(6) r = .

l+ COS y(*A ^o)

The equation of the path differs from that of an ellipse with the
parameter I and eccentricity e by the factor y. While r goes through
one libration, the true anomaly increases by 2?r/y. The path ap-
proaches more nearly to an ellipse the smaller the coefficient c x of
the additional term in the potential energy, and for c 1 =0 it becomes
an ellipse. For small values of c l we can regard the path as an
ellipse, whose perihelion slowly rotates with the angular velocity

co 1 is here the mean motion of the point on the ellipse.

We now take into account the term c 2 (a/r) 2 in (1), but only in the
case where its influence is small. We find then by complex integra-
tion (cf. (10), Appendix II) :

and from this

_
A=-2mW

BD
and

where this time
(7) S=-

SYSTEMS WITH ONE RADIATING ELECTRON 159

The following term c 3 (a/r) 8 may be taken into account in a similar
way and would lead to a dependence of the quantity 8 on n in the
form

However, we shall not carry out the calculation in this way ;
instead, we will again calculate the influence of the additional terms
in the potential energy, this time with the help of the method of
secular perturbations, 18. The result will be of less generality
only inasmuch as we must suppose the quantity c t to be small as
well as c a , c 3 . We write

where H is the Hamiltonian function of the Kepler motion, con-
sequently

H-W-
HO o-

and we regard

as the perturbation function. The unperturbed motion is doubly de-
generate ; the perturbation makes it singly degenerate. We obtain
the secular motion of the angle variables now no longer degenerate,
and the influence of the perturbation on the energy, by averaging H l
over the unperturbed motion. In this way we find

The mean values are by (19), 22 :

_ 1 _ Z*
""06"

I Z*

2o H 5 n 3 ife 7

160 THE MECHANICS OF THE ATOM

On introducing the Rydberg constant

we get

2a H hc

W=-

l + f!+Ll +

nk* nlc?

- )c 4

i 2 /

nF
Writing W in the form

(9) W=-~ ~,

v } n* 2

we find, on neglecting products of the c/s,

(11) "*=n+8 1 +??+

where

_ Zcj Z 2 c 2 3Z 3 c 3 5Z 4 c 4
1= ""^"""

We now compare these theoretical formulae with observation.
The terms derived from observations of spectra of the non-hydrogen
type may in fact be written in the form

RZ 2

where, in general, 8 depends very little on n. Rydberg 1 was the
first to suggest this form and verified it by measurements of numerous
spectra. We shall therefore denote the quantity 8 as the Rydberg

correction. The remaining deviations have been represented by

- < -

1 J. R. Rydberg, K. Svenska Akad. HandL, vol. xxiii, 1889 : an expansion in

1/w 2 equivalent to the Rydberg formula has been given independently by H. Kayser

and C. Runge (Berlin. Akad., 1889 to 1892).

SYSTEMS WITH ONE RADIATING ELECTRON 161

Ritz, 1 who gave a series expansion for the difference between n* and
the whole number

(12) S=S 1 +8 8 I+...
Eitz used also the implicit formula

(13) = -??L .

26. The Rydberg-Ritz Formula

The Rydbcrg-Ritz formula can be established empirically not
only for the terms of the outer orbits, but also for orbits which pene-
trate the core and which we shall call " penetrating orbits." It may
in fact be derived theoretically for very general cases.

We show next that for an arbitrary central field the formula

m RX2

(1) = _ .

(n l-^+S^) 2

corresponds to a reasonable series expansion. 2

The connection between the quantum numbers and the wave
number v of tho term is given by the equation (cf. (4), 21 )

(U(r) is negative, sec 21.) We compare this with the expression

-2| hv- e ~

which, for the same v, corresponds to a Coulomb field of force.
For this n* is of course not an integer, but has the value given by

_RZ 2
n* 2

The difference of the two integrals is a function of v and k alone. If
we imagine it expanded in terms of v and put equal to

we obtain

1 W. Ritz, Ann. d. Physik, vol. xii, p. 264, 1903 ; Physikal. Zeitschr., vol. ix,
p. 621, 1908 ; see also Ges. Werke, Paris, 1911.
8 G. Wentzel, Zeitechr. /. Physik, vol. xix, p. 53, 1923.

162 THE MECHANICS OF THE ATOM

and

RZ 2

Since for larger values of n the term v rapidly approaches zero,
we can conclude from this consideration that the correction Sj+Sg^
rapidly converges to a fixed limiting value for increasing n.

The following argument due to Bohr, 1 goes much further towards
providing a theoretical basis for the Rydberg-Ritz formula (1), and
gives this formula greater physical significance.

The real object of the introduction of the central field was to
describe, by means of a simple model, the (certainly non-mechanical)
interaction between core and radiating electron, for which no ex-
change of energy between core and electron occurs. Now this
assumption regarding the constancy of the energy of the radiating
electron is alone enough to enable us to deduce the series formula,
without special assumptions regarding the field of force ; this
derivation is, in consequence, not only valid for any atom whose
spectrum can be ascribed to a single series electron, but even for
molecules. Certainly molecules do not emit line but band spectra ;
these, however, are also produced chiefly by transitions of a radiating
electron, on which are superimposed the quantum transitions of the
molecule as a whole from one state of rotational or oscillatory
motion to another.

Further, this derivation is altogether independent of whether an
exchange of angular momentum between core and electron takes
place or not, i.e. whether or not an azimuthal quantum number k
can be defined in a manner analogous to that in the case of central
motion.

The only assumption which we make is that the core (which
includes one nucleus in the case of one atom and several in the case
of a molecule) is small in comparison with the dimensions of the
path of the radiating electron. The field will then closely resemble
a Coulomb field over most of the path outside the core ; the distance
of the aphelion from the centre point of the core will be determined
only by the potential energy in the aphelion, it is therefore equal
for all loops of the path independently of whether these loops are
similar to one another (as for a central field) or not. Accordingly
an effective quantum number n* may be so defined that the relation

1 We are indebted to Professor Bohr for kindly communicating the ideas on
which the following paragraphs are based.

SYSTEMS WITH ONE RADIATING ELECTRON 163

holds which is valid in the Coulomb field, between n* and aphelion
distance and energy respectively :

dRAZ 2

We assume, on account of the periodicity of the electron motion,
that it has a principal quantum number n ; W is then a function of
J=wA, and for the radial period r of the motion (i.e. the time from
aphelion to aphelion) we have

( ' r dJ h 8n'

The radial period T* of the motion in the Kepler ellipse with the
same energy (2) is

(4)
v '

In a single term series we consider the variation of energy with
the principal action integral J or principal quantum number n, for
constant values of the other quantum numbers ; for such a variation
we may invert the derivatives (3) and (4), and find for a term series

h '

Now the Rydberg correction 8 is equal to n* n (compare (2) with
(2), 25), and W is he times the wave number v of the corresponding
term in the spectrum, so that for a term series

(5) =**->

The radial motion in the two orbits is only different over that
part of the actual orbit where the field of the core is appreciable ;
the proportion of a radial period spent in this part of the orbit is
small, if the core is small compared to the dimensions of the orbit
of the series electron as assumed, so r* T is small compared to r.
If it can be taken to have a constant value

T *_ T= 8 2 /c

over the range of v covered by the term series, (5) integrates directly
to

8 = 8 1+ 8 2 v,

whence the Ritz formula (1) follows at once. t

If T* T cannot be taken as constant, it seems probable that it
will be expansible in a power series in W or v (this can certainly be

164 THE MECHANICS OF THE ATOM

done if the field is central) ; integration of (5) then gives 8 as a
power series in v the " extended Kitz formula."

In order to provide a survey of the validity of this formula we
give the values of the effective quantum number n* for the terms
of two typical spectra, those of Na and Al :

Na.

P
d

1-63 2-64 3-65 4-65
2-12 3-13 4-14

2-99 3-99 4-99

4-00 5-00

P
d

219 3-22 4-23 5-23 6-23

1-51 2-67 3-70 4-71 5-72

2-63 3-42 4-26 5-16 6-11 7-08 8-07

3-97 4-96 5-96

The Na-spectrum and the s- 9 p-, and /-scries of the Al-spectrum
show the behaviour which we find for almost all term series, namely,
very little dependence of the Kydberg correction n*n on the term
number n. The d-series of aluminium and a few other known series
form the exception, inasmuch as the limiting value of the correction
is reached only for comparatively high term number.

Since, for the time being, we do not know the quantum number
n only the fractional part of 8 can yet be found, the integer is un-
determined. If we choose the integers here so that the magnitudes
of 8 decrease with increasing k and at the same time are as small as
possible, we obtain as limiting values for large n :

N.a
Al

1-35 -0-80 -0-01 0-00

1-77 -1-28 - 0-93 -004

Now if the analysis of 25 were applicable, |S| would increase as
l/k or 1/F or I/A: 5 (cf. (JO), 25), as k decreased and the orbit at
perihelion came closer to the nucleus ; it will be seen from these
examples, and from all other series spectra, that there comes a stage
at which the increase of |8| with decreasing k is very much more
rapid than that given by any of these inverse powers. The large
values of 8 show us, moreover, that we can no longer regard it as a
small correction of n.

The large deviations of the term values from the hydrogen terms
may be explained if we consider that the orbit of the series electron
is not always situated entirely outside the core, even in the excited

SYSTEMS WITH ONE RADIATING ELECTRON 165

states, but penetrates into it. Such a penetrating orbit (Tauchbahn)
is in its innermost parts much more strongly subject to the influence
of the nucleus ; it traverses, therefore, a field of force similar to a
Coulomb field of force with a higher nuclear charge. Under such
conditions use of (1), 25, for the potential energy will not be
justified.

In the case of Na a noticeable irregularity is present in the course
of the 8-values between the d- and y-terms ; this suggests that the
d-orbits are situated entirely outside the core and that the s- and
y-orbits penetrate into the core.

27. The Rydberg Corrections of the Outer Orbits and the
Polarisation of the Atomic Core

We now consider in greater detail the physical influences which
cause a departure of the field of force outside the core from a Coulomb
field of force. 1 First we can determine approximately which power
of a/r is especially important in the potential. We write the orbital
energy in the form

w _ cRAZ 2

'^+5

2*7

An additional term . c l in the potential energy gives by

(10), 25, a " Rydberg correction "

and a " Ritz correction

e 2 Z a 2

An additional term -- . c gives
r r 2

~\ S 2 =0;

e 2 Z a 3
an additional term -- . c 3 -^- gives

1 M. Born and W. Heisenberg, Zeitschr. f Physik, vol. xxiii, p. 388, 1924 ; the
numerical values of the following tables are taken from this work. For further
work on this subject, see D. R. Hartree, Proc. Roy. ti<9b. t vol. cvi, p. 552, 1924;
E. Schrodinger, Ann. d. Physik, vol. Ixxvii, p. 43, 1925 ; A. Unsold, Zeitschr. f.
Phyaik, vol. xxxvi, p. 92, 1926 ; B. Swirles, Proc. Camb. Phil Soc., vol. xxiii, p. 403,
1926*

166

THE MECHANICS OF THE ATOM
a % s _ Z3c 3 S 2 __P

-, o 2 ;rr^> ~ i?9

and an additional term . C A gives

r *r 4

5 Z 4 c 4

5J *

1- 2 A; 7 '

8 2 =

3Zc 4

The following table gives the values of the Rydberg and Ritz
corrections and their ratio, determined from the spectra of the alkali
metals, whose structure is especially simple.

-<5i

0-049

<5 a

0-031

-A0i

0-63

-A

0-016

0-25

0-35

<5 2

0-036

0-80

0-99

-<5iA5i

2-4

3-2

2-8

-A

0-0020

0009

0-36

0-032

0-0064

0-035

0-35

0-16

-<V<5i

3-2

3-9

9-8

5-0

The letter T in the table denotes that the Rydberg correction is too
large so that an expansion of the potential in powers of l/r does not
appear justifiable.

The large value of Sg/Sj^ shows that the higher powers of l/r are
present in the potential to an appreciable extent. For the terms with
c 3 /r 4 and cjr 5 we obtain theoretically the values

"~~do/d|

for ~~ for ~~

P 2
d 3
/ *

1-33
3-0
5-33

2-4
5-4
9-6

From this it appears that the term containing c 3 /r 4 is the essential
additional term.

Now such an additional term in the potential energy has in fact a
theoretical significance. For if the core of the atom, instead of being
regarded as absolutely rigid, is considered to be capable of deforma-
tion, it will acquire an electric moment in the field of the series elec-
tron. If the electron is at a sufficient distance from the core, the
field | E | =e/r 2 produced by it in the vicinity of the core may be con-
sidered as homogeneous. The induced moment of the core is pro-

SYSTEMS WITH ONE RADIATING ELECTRON 167

portional to this field: p = aefr 2 . The moment of such a doublet
produces an electric field in its neighbourhood ; if it be considered
to arise from the approach of two charges p/l at a distance I apart it
will be seen that the force exerted on the radiating electron, in the
direction of its axis, will be

pel 1 "_ At 1\ 2pe_2ae*

Its potential is ae 2 /2r 4 . If the other deviations from the Coulomb
field be neglected, we have

and

3 Z 2 a

2Za [t 3

Our assumption, that the departure of the field of force from a
Coulomb field is due essentially to the induced doublet in the core,
may be tested by calculating the " polarisability " a, from the
empirical values of 8 l and 8 2 . It must be assumed that the cores of
the alkalies Li, Na, K, Kb, Cs are vsimilar in structure to the neiitral
atoms of the inert gases (containing the same number of electrons)
He, Ne, A, Kr, X (see further, 30). The values of a for these atoms
may be determined from the dielectric constants ; between them and
the a- values of the alkali cores a simple relation should exist.

From the empirical 8 ^ values of the alkalies we get

I Li+ Na+ K+ Rb+ O+

a-10 24 - | 0-314 0-405 1-68 .. 6-48

For this the /-terms are used with the exception of Li, the j9-term of
which serves for the calculations ; Rb is omitted on account of its
somewhat anomalous Rydberg and Ritz correction. The polaris-
abilities of the inert gases are related to the dielectric constants
or with the refractive indices n for infinitely long waves by the
Lorentz-Lorenz formula

3 -l 3 n 2 -!

where N is the number of atoms per unit volume. If the optically
measured refractive indices be extrapolated for infinitely long waves,
one finds .

| He Ne A Kr X

a-10 24 = I 0-20 0-39 1-63 2-46 4-00

168 THE MECHANICS OF THE ATOM

The a- values of the alkali ions must be somewhat smaller since the
volumes of the ions must be less than those of the preceding inert
gas atoms on account of the higher nuclear charge.

We find consequently that the a- values calculated from the spec-
trum have the right order of magnitude, but that they are all rather
too large. One might be inclined to account for the difference by
assuming that, in addition to the induced moment, still another
deviation from the Coulomb law of force is present, likewise corre-
sponding to an auxiliary term of the approximate form c 3 /r 4 . We
cannot at this point prove whether such an assumption is admissible.
It should, however, be mentioned that our knowledge of the structure
of the ions of the inert gas type hardly admits of such a possibility.

If the explanation given here of the Rydberg correction as being
due to polarisation of the core be retained, then a contradiction re-
mains which, from the standpoint of our quantum rules, cannot be
removed. We have, however, already referred to the fact that the
explanation of the finer details of the spectra (the multiplets and the
closely allied anomalous Zeeinan effect) does not appear possible
within the range of a quantum theory of multiply-periodic systems.
One is led by the theory of these phenomena to the formal remedy of
giving to the quantum number k half integral values, i.e. to give it
the values J, , fj, etc. It is to be expected that in the further
development of the theory the real quantum numbers will remain
integral as before and that the quantity k, occurring in our approxi-
mate theory, is not itself such a quantum magnitude, but is built
up indirectly out of them. We shall not go into these questions in
the present book ; we shall content ourselves with seeing what values
are obtained for a when we choose half values for k in our formula.
We find, then, from the spectroscopic values of S, the following
a- values :

| Li+ Na< K+ Rb+ Cs+

a -1C 24 - | 0-075 0-21 0-87 .. 3-36

These numbers are related in the right sense to the a-values of the
inert gases. This connection can be traced still further by considering
the a-values of other (multiple- valued) ions of inert gas type, which
may be determined partly from the Rydberg corrections of spectra
of the ionised element (spark spectra), partly from the refractive
indices of solid salts, (ionic lattice). In this way further support is
obtained for the view that the Rydberg correction of the terms of
the outer orbits in the spectra under consideration is due to the

SYSTEMS WITH ONE RADIATING ELECTRON 169

polarisation of the atomic core and that the quantum number Jc is
to be given half values. 1

The investigations dealt with in this volume are otherwise inde-
pendent of a decision for whole or half values for k.

28. The Penetrating Orbits

In 26 we have ascribed the large values of the Rydberg correc-
tions to the fact that the electron penetrates deeply into the atomic
core, and is thus subjected to an increased nuclear influence.

An estimate of the orders of magnitude to be expected for the 8-
values for such " penetrating orbits " may be obtained by a procedure
due to E. Schrodinger. 2 He considers the core of the atom replaced by
a spherical shell uniformly charged with negative electricity, external
to which there is then a Coulomb
field of force, corresponding to the
nuclear charge Z (a) (1 for a neutral,
2 for a singly ionised atom), and in
the interior of which there is like
wise a Coulomb field, but corre-
sponding to a higher nuclear charge
Z (l) . As soon as the perihelion
distance of a quantum orbit, cal-
culated as an ellipse in the field of force with the nuclear charge
Z (rt) , becomes smaller than the radius of this spherical shell, the orbit
penetrates into the interior ; it consists then of two elliptic arcs which
join smoothly at the intersection with the spherical shell (fig. 14).

For given quantum numbers n and fc, given shell radius, and given
charges of the shell and nucleus, the effective quantum number n*
or the correction 8 may be calculated.

We shall not repeat Schrodinger's calculations here ; we shall
show only that by means of such an atomic model, which may even
consist of several concentric shells with surface charges, the relation
between quantum numbers and energy may be expressed in terms of

1 This evidence is by no moans conclusive, since values of the polarisability a
deduced from terms of spectra corresponding to external orbits depend to a con-
sideiable extent on the term series from which they are deduced, so conclusions
drawn from comparison of values of a deduced from a single series with values
deduced from other phenomena must be regarded with caution. We may also
mention here that the polarising field on the core due to an electron in an orbit
radius 9 H (the radius of the 3-quantum circular orbit of hydrogen) is about 10 8
times the field in strong sunlight, and the displacement of the electrons in the core
polarised by this field may be an appreciable fraction of the core radius.

2 E. Schrodinger, Zeitschr. f. Physik, vol. iv, p. 347, 1921.

170 THE MECHANICS OF THE ATOM

elementary functions. 1 Let the shell radii be p l9 p 2 > arranged in
decreasing order of magnitude, and their charges ztf, z 2 e ....
The potential energy in the space between the shells p s and p s+1 is

where

and c 8 is determined by the condition that at the shells the potential
varies continuously. It follows from this that

^iPcr

Since we now know the potential energy as a function of r, we can
calculate the perihelion distance r min and state within which shells
p l9 p 2 . . p p it lies. The radial action integral has, according to
(4), 21, the form

Bn C

where

All the integrals may be expressed in terms of elementary functions ;
in this way we obtain J r and hence (w k) as a function of W and k,
and finally W as a function of n and k.

Following van Urk 2 we shall make use of Schrodinger's conception
of the charged shells to estimate the 8-values for the penetrating
orbits. It will be seen that the larger the radius of the spherical
shell the larger will be the radial action integral, for a given external
ellipse ; for the larger this radius, the longer will the electron move

1 Cf. also G. Wentzel, Zeitschr. f. Physik, vol. xix, p. 53, 1923, especially p. 55.

2 A. Th. van Urk, Zeitechr. f. Physik, vol. xiii, p. 268, 1923.

SYSTEMS WITH ONE RAMATING ELECTRON 171

under the influence of the full nuclear charge. One obtains, there-
fore, from the Schrodinger model, on the assumption that an orbit
is of the penetrating kind, a lower limit for the magnitude of 8
by choosing the radius of the shell so that it touches the external
ellipse. If we wish to find the value to which 8 tends for large values
of n (the dependence on n is extremely small in the case of the
Schrodinger model), we can take as the perihelion distance of the
external ellipse that of the parabola ; in the case of the s-orbit, there-
fore, 5^7-)%, we shall write generally ^f\i- Since we choose the

radius of the sphere equally large, the total orbit of the radiating
electron will be given to a close approximation by the two complete
ellipses.
We get for the radial action integral

J __j ()_|_J (0
u r r \ u r >

affix (a) indicating the contribution from the part of the orbit
outside the core, and affix (i) the contribution from the part inside.
Now the spectrum term is proportional to the work of separation
of the outer electron, and consequently equal to the energy of the
outer ellipse

where J^ is 2?r times the common angular momentum for the two
ellipses. If we compare this with the form

W=-

for the energy, we find for the effective quantum number

But

J r +J t =n
so that

T < l)
(1) 8 = M *-n=-^ = -

n \ n

where J (t) is the sum of the action integrals for the inner ellipse.
J (0 is determined by the semi-major axis a of the inner ellipse :

172 THE MECHANICS OF THE ATOM

a is further related to the radius of the shell

where

If a and e be eliminated from these three equations we find

J (f)

and from this, by solving for and substituting in (1) :

(2) 8= , +k.

' J 7- 2

) T~m~f \ '^

The equation (1) is also approximately valid if the outer ellipse
cuts the shell at a small angle instead of touching it, so long as the
shell radius is small in comparison with the major axis of the outer
ellipse (which is certainly the case for large values of the principal
quantum number) and if TP^ is considerably greater than Z (a) . The
error which is then made in replacing the action integral over the
outer portion of the orbit by that over the complete outer ellipse is
then small ; likewise the error made in replacing the inner portion
by the complete inner ellipse ; the aphelion of the inner ellipse is
situated only slightly outside the shell (on account of the rapid
decrease of the potential energy in the field with the nuclear charge
Z (l) ). The sum J (t) of the action integrals of the inner ellipse is
determined uniquely by the major axis of this ellipse, and is con-
sequently almost independent of n.

In the approximation given by formula (1) 8 is not dependent on
n. This approximation is better the larger the major axis of the outer
ellipse ; since that is rapidly attained by increasing n, we see how 8
very soon assumes a constant value with increasing n.

If there are quantum paths which are contained completely in
the interior of the shell, and if n is the principal quantum number of
the largest of them, then

J<
n (0< _

h
and

SYSTEMS WITH ONE RADIATING ELECTRON 173

(3) S = -(w (t) +e-fc) 0<<1.

This formula is essentially independent of the Schrodinger model
of the charged spherical shells, and depends only on the fact that the
aphelion distance of the outer orbit is large in comparison with the
core radius, and that the electron penetrating the core soon comes
into the region of higher effective nuclear charges. Bohr * derived
it, before v. Urk, in the following way :

The radial action integral J r =h(n k) of the orbit is composed of
the outer portion of the orbit and of the inner loop :

J/ a) is only slightly smaller than the radial action integral A(n* k)
of the complete external ellipse :

J^^n*-*--*!),

and J r (t) differs but little from the radial action integral h(n^k) of
the largest orbit completely contained in the core :

It is not necessary here that n^ should be integral, but it is the sum
of the action integrals of the largest possible mechanical (not quan-
tum) orbit divided by h. One obtains consequently

(4) 8=n*-n=-(nW-/j- 1 +,)

and the result may be formulated as follows :

The Rydberg correction for penetrating orbits is not very different 2
from the radial action integral of the largest orbit, completely con-
tained in the core, divided by h.

The question as to the accuracy with which all optical (and X-ray)
terms may be consistently represented by a suitable central field
has been examined by E. Fues ; 3 he arrived at eminently satis-
factory results in the case of the arc spectrum of Na and the analogous
spark spectra of Mg + and Al ++ .

29. The X-ray Spectra

The optical series spectra of the elements provide one of the
principal means of obtaining information regarding the structure of

1 Bohr, N., Lectures at Gottingen, June 1922 (unpublished).

2 In practice, e l e a i s no * always small compared to 1 ; comparison with observed
spectra shows it may be greater than J.

3 E. Fues, Zeitschr. f. Physik, vol. xi, p. 364 ; vol. xii, pp. 1, 314 ; vol. xiii, p. 211,
1923 ; vol. xxi, p. 265, 1924. See also W. Thomas, Zeitschr. f. Physik, vol. xxiv,
p. 169, 1924. For further work on penetrating orbits, especially the relations
between corresponding terms of different atoms of the same electronic structure,
see E. Fues, Ann. d. Physik, vol. Ixxvi, p. 299, 1924 ; D. R. Hartree, toe. cit.,
and Proc. Camb. Phil. Soc., vol. xxiii, p. 304, 1926.

174 THE MECHANICS OF THE ATOM

atoms. In as far as they can be comprehended on the basis of our
theoretical conceptions we can draw conclusions regarding the pro-
cesses taking place in the exterior portions of atoms only ; they
afford us little or no information about those occurring in the
inner regions. The most important means of investigating the
internal structure of the atom is the study of the X-ray spectra.
Our theory of the motion of an electron in a central field of force is
applicable also to these, since it may be inferred from the observa-
tions that we are here concerned with quantum transitions of the
atom in which one electron (corresponding to the series electron
in the optical spectra) changes its position in the interior of the
atom while the rest of the atom remains approximately a structure
possessing central symmetry.

Before we follow out these ideas in detail, we shall give a brief
summary of some of the results of observations on X-ray spectra.
Since the discovery of v. Laue, the natural gratings of crystals have
been available for the analysis of these spectra. Each X-ray spec-
trum consists of a continuous band and a series of lines.

The continuous spectrum has a short-wave limit, whose frequency
^inax * s rp ' a ted to the kinetic energy of the generating cathode rays
by the equation

._ m
*"= 2 V -

This result can be looked upon as a kind of converse to the photo-
electric effect, on the assumption that the incident cathode rays are
retarded in the anti-cathode and that their energy is transformed
into radiation according to the Einstein law ( 2) ; the highest fre-
quency emitted corresponds then to the total loss of kinetic energy
of the incident electrons.

The line spectrum is characteristic of the radiating matter, and is
called, therefore, " characteristic radiation." The most important
fact relating to it is that every element exhibits the same arrange-
ment of lines, and that with increasing atomic number the lines
shift towards the shorter wave-lengths. This line spectrum contains
various groups of lines : a short-wave group (called K-radiation) has
already been found in the case of the light elements (from elements
in the neighbourhood Na and onwards). These become continually
shorter for the heavier elements, and are followed by a group of
longer waves (L-radiation) ; behind this group follows, in the case
of still heavier elements, a group of still longer wave-lengths (M-
radiation).

SYSTEMS WITH ONE RADIATING ELECTRON 175

If these spectral lines are to be related to the motions of the
electrons in the atom in accordance with the principles of the quan-
tum theory, the X-ray frequencies must be given in terms of the
energies of two stationary electron configurations by the equation

The large values of v (about 1000 times as great as in the visible
spectrum) indicate that we have to do with variations in the orbits
of the inner electrons where, on account of the high nuclear charge, a
large amount of work must be expended in the displacement of an
electron.

The fact that the X-ray lines are arranged in simple series, and
may be characterised by small integers, forms the ground for the
assumption that, as in the case of the simpler optical spectra, we
are here concerned principally with the motion of a single " radiating
electron." Although we arc compelled to assume that this electron
moves in the interior of the atom we shall replace the action of the
nucleus and remaining electrons, for reasons analogous to those
holding in the case of the visible spectra, by a central symmetrical
field of force. By so doing we express once again the fact that no
exchange of energy takes place between the radiating electron and
the remainder of the atom ; the existence of quantum numbers
for the radiating electron points to its motion being periodic, and
assuming, therefore, the same energy after each revolution.

There is, however, a fundamental difference between the optical
spectra and the X-ray spectra. Whereas the lines of the optical
spectra can occur also in absorp-
tion, the X-ray lines are never
observed as absorption lines.
The absorption coefficient for
Rontgen rays exhibits, in fact,
no maxima of the kind which
produces absorption lines ; it
shows rather a continuous vari-
ation, broken only at certain Fia. 15.

places by the so-called " ab-

sorption edges," at which a sudden increase in absorption coefficient
occurs if the frequency is increased through them (fig. 15).

An explanation of this phenomenon has been given by Kossel. 1
According to him, the absorption spectra are concerned with the

1 W. Kossel, Verhandl d. Ditch, physikal. Ges., vol. xvi, pp. 899 and 953, 1914,
and vol. xviii, p. 339, 1916.

176 THE MECHANICS OF THE ATOM

ionisation of the atom in such a way that an inner electron is removed.
The frequency condition gives for this process

where v is the velocity of the electron after separation and W is
the work of separation. It follows then that all frequencies will
be absorbed which are greater than the limiting frequency

which will thus be the frequency of the absorption edge. The hy-
pothesis that in the atom there are electrons with various different
binding energies W leads then to a variation of the absorption
with frequency in qualitative agreement with observation.

According to Kossel the emission lines are caused by an electron
falling in from a higher quantum orbit to replace the ejected electron,
whereby the energy of the atom decreases. Further, an electron from
a still higher quantum orbit can fall into the vacated place until
finally the last gap will be filled by a free electron.

The emission spectra of the X-rays arise then from the re-establish-
ment of a stable state of the atom after its disturbance through the
ejection of an inner electron.

We can express this hypothesis of KossePs, which has been com-
pletely verified, as follows : For every system of quantum numbers,
corresponding to inner orbits, there corresponds a maximum number
of electrons. This is reached in the stable state. An exchange of
place occurs, however, when an electron is removed from its inner
orbit. All electrons which possess the same quantum numbers are
considered as together forming a shell ; we shall be led subsequently,
by altogether different considerations, principally from the domain
of chemistry, to the same conception of a shell-like structure of atoms
( 30). We shall now endeavour to establish the truth of these con-
ceptions from the quantitative standpoint.

Our model, in which the electron under consideration moves in a
central field, gives rosettes for the electron paths, and these are
determined by two quantum numbers n and k. Orbits with different
values of n must in fact occur in the interior of the atom. The
behaviour of the Rydberg corrections shows that, for almost all
elements, the p-orbit^ penetrate ; in order that this may be possible
the core must at least contain orbits with n=2. Of the orbits in
the core those with n\ (&=1) are nearest to the nucleus, then

SYSTEMS WITH ONtl RAM v r ' ENO ELECTBOTST 177

follow those with n=2 (k=l 9 2), and then perhaps come orbits with
n=3 (k=l, 2, 3).

In the elements of high atomic number the innermost orbits are
subject mainly to the attractive force of the nucleus, while the influ-
ence of the remaining electrons is comparatively small. The energy
of the innermost electron orbit is then given approximately by

with n 1 and Z equal to the atomic number ; as we proceed out-
wards the energy decreases rapidly, partly on account of the decrease
of n and also on account of the shielding of the nuclear charge by
the remaining electrons. The wave number of the first line to be
expected is

(1) ,=

approximately. The formula requires that Vv shall increase
linearly with the nuclear charge. Moseley, 1 who was the first to
study the X-ray spectra systematically, found that for the K-series
Vv is actually very nearly a linear function of the atomic number ;
by atomic number is understood the number expressing the position
of an atom in the series order of the periodic system (1 H, 2 He,
3 Li . . .), thus practically in the order of the atomic weights ; the
gaps required by chemistry (e.g. that of the element 43 homologous
to manganese) are to be taken into account as well as the reversals
required by chemical behaviour [e.g. 18 A (at. wt. 39-88), and 19 K
(39-10)].

This provides an excellent verification of the already long-inferred
principle first put forward by van den Brock (cf. 3, p. 13), that
the atomic number is equal to the number of the nuclear charges. 2
This enables us also to determine uniquely the atomic numbers of
elements with very high atomic weights, among which occur long series
of elements differing very little chemically from one another (e.g.
the rare earths), and also to determine accurately the existing gaps.

In order to show the accuracy with which the law (1) holds, we

give the values of . / - for some elements.
o ri

1 H. 0. J. Mofleley, Phil Mag., vol. xxvi, p. 1024, 1913 ; vol. xxvii, p. 703, 1914.

2 Strictly speaking, Moseley's law only confirms tha*; the difference between
the atomic number and nuclear charge is the same for all elements whose X-ray
spectrum has been observed ; it does not show that this constant difference is
necessarily zero.

178 THE MECHANICS OF THE ATOM

For Na(Z=ll) the value is 10-1, for Rb(Z=37) it is 36-3, and for
W(Z=74) it is 76-5. We associate therefore the first K-line with
the transition of an electron from a two-quantum to a one-quantum
orbit. This suggests associating the remaining K-lines with transi-
tions from higher quantiim orbits to a one-quantum orbit. The
K-lines have actually the theoretically required limit

RZ 2

T 2 '

Situated at the same place is one of the above-mentioned absorption
edges.

The principle of linear increase of Vv is valid also for the L-lines.
We attempt to identify these lines as transitions to a two-quantum
orbit (ft =2), and obtain for one of the L-lines the approximate
wave number

This formula does not hold so well as it does for the K-series ; this
we can understand since here we are at a greater distance from the
nucleus. We can take account of this quantitatively, 1 by writing

(3)

the empirical values are then in agreement with a value for s which,
for medium values of Z, lies approximately between 6 or 7.
Here again the series limit coincides with an absorption edge. The
M-lines correspond finally to transitions to a three-quantum orbit.
We obtain a clearer survey of the stationary orbits of the electrons in
the atom if from the system of the X-ray lines we proceed to that of
the X-ray terms. The end term of the K-lines we call the K-term,
it corresponds to the K absorption edge, and corresponding to it
(in our model) are the quantum numbers w 1, it 1. In order to
account for the multiplicity of the L-lines we must assume three
end terms (L-terms) for which w=2 and Jc=l or 2. The fact that
three terms exist instead of two implies that the quantum numbers
n and k are not sufficient to define them ; we are confronted here
by a phenomenon very closely allied to that of the multiplicity of
the optical terms. On the basis of our model we cannot give an
explanation of this phenomenon. 2 Again, investigations of the

1 A. Sommerfeld, Anil. d. Physik, vol. li, p. 125, 1916.

2 A satisfactory interpretation in terms of the " spinning electron " can be given,
as for the multiplets of optical spectra (cf. p. 155 and footnote 2, p. 152).

SYSTEMS WITH ONE RADIATING ELECTRON 179

FIG. 16.

180 THE MECHANICS OF THE ATOM

X-ray lines give M-terms with w=3 (fc=l, 2, 3), and seven N-terms
with n=4 ; some 0-terms have also been established.

To provide a survey of the occurrence of these different terms
we reproduce here a graphical representation of the terms, taken
from the work of Bohr and Coster * (fig. 16). We find there the K-
and one L-term (w=l, n=2) even for the lightest elements ; 2 an M-
tenn (n=3) appears about the atomic number 21, an N-term (n=4)
about 39, and an 0-term (n=5) at about 51. With regard to the
number of the terms corresponding to each principal quantum num-
ber, the resolution into 3, 5, and 7 terms mentioned above is readily
noticeable ; this resolution occurs in two stages ; we find first
two L-, three M-, and four N-terms, all of which, with the exception
of the first of each, again split up into two terms. If we disregard this
further splitting up, which occurs only for higher atomic numbers,
we have just as many terms as there are values which the subsidiary
quantum number can assume. The rule in accordance with which
the terms combine corresponds exactly to the selection principle for

We refer finally to the departures of the square roots of the term
values from a linear variation with the atomic number. These
are clearly shown in fig. 1 6, given by Bohr and Coster. The general
curvature of the graphs (especially of that for the K-term) is attri-
buted by Sommerfeld 3 to the " relativity correction " ( 33, p. 201).
The small kinks, e.g. at Z=56 and Z 74, are connected, according
to Bohr and Coster, with the building up of the inner electron groups,
to a consideration of which we shall shortly return ( 32, p. 191).

30. Atomic Structure and Chemical Properties

The final aim of a theory of atomic structure must be to construct
the whole periodic system of the elements from an atom model. Bohr
had already made attempts in this direction in his earlier works.
He made use of " ring models," in which the individual electrons were
situated at the corners of concentric regular polygons (the " rings ").
A considerable amount of work has been expended on the calcula-
tions of such ring systems by Bohr, 4 Sommerfeld, 6 Debye, 8 Kroo, 7

1 N. Bohr and D. Coster, Zeitschr. f. Physik, vol. xii, p. 342, 1923.

* A state of the atom giving another L-term presumably exists for the lighter
elements, but has not been experimentally determined as it is not involved in any
line in the K spectrum, which is the only one of their X-ray spectra yet observed.

3 A. Sommerfeld, Ann. d. Physik, vol. li, p. 125, 1916.

4 N. Bohr, Phil Mcttj., vol. xxvi, p. 476, 1913.

c A. Sommerfeld, Physical. Zeitschr., vol. xix, p. 297, 1918.

6 P. Debye, ibid., vol. xviii, p. 276, 1917.

7 J. Kroo, ibid., vol. xix, p. 307, 1918.

SYSTEMS WITH ONE RADIATING ELECTRON 181

Smekal, 1 and others, particularly with reference to the explanation
of the X-ray spectra ; the results were, however, altogether unsatis-
factory. The most important mechanical result arising out of this
was Sommerfeld's observation that such an^electron polygon can not
only rotate about the nucleus, but that it can^execute a motion in
which the electrons traverse congruent Kepler ellipses (family of
ellipses). Sommerfeld dealt also with the mutual perturbations of
such rings for the case in which they are coplanar as well as for that
in which they lie in different planes. Models of this kind have
indeed a spatial structure just like the real atoms, but they do not
show the symmetry of the latter as exhibited chemically (e.g. carbon
tetrahedra) as well as crystallographically. Lande 2 therefore en-
deavoured to construct models with spatial symmetry such that,
in common with Sommerfeld's family of ellipses, the electrons tra-
verse congruent paths in exact phase relations (e.g. simultaneous
passage through the perihelion). But these models also failed when
it came to quantitative investigations.

Bohr realised that, by purely theoretical considerations and the
construction of models, the desired object of explaining the regu-
larities in the structures of atoms (periodic system of the elements)
would be very difficult to attain ; he therefore adopted a procedure
by means of which, half theoretically and half empirically, making
use of all the evidence provided by physics and chemistry, and,
especially, by a thorough application of the data derived from the
series spectra, there was evolved a picture of the building up of
atoms.

The chemical results which are to be taken into account in such
an investigation have been expressed in a suitable form by Kossel. 3
He takes as a starting-point the fact that the periods of the system
of elements begin with an inert gas, the atoms of which are char-
acterised by the fact that they enter into no combinations and can
be ionised only with extreme difficulty. The atoms of the inert gases
are, therefore, particularly stable configurations which, perhaps as
a result of the high degree of symmetry, are surrounded only by
small fields of force and, on account of this great stability, neither
take up electrons easily nor part with them. The atoms preceding
the inert gases are the halogens (F, 01, Br, I) which occur readily

1 A. Smekal, Zeitschr. f. Physik, vol. v, p. 91, 1921.

2 A. Lande, Verhandl. d. Dtsch, physikal. Ges., vol. xxi, pp. 2, 644, 653, 1919 ;
Zeitschr. f. Physik, vol. ii, pp. 83, 380, 1920. *

8 W, Kossel, Ann. d. Physik, vol. xlix, p. 229, 1916; see also G. N. Lewis,
Journ. Amer. Chem. Soc. t vol. xxxviii, p. 762, 1919, and J. Langmuir, ibid., vol. xli,
p. 868, 1919.

182 THE MECHANICS OF THE ATOM

as singly charged negative ions ; this, according to Kossel, is due
to the fact that their electron systems lack one electron to make up
the stable inert gas configurations and that they endeavour, with
loss of energy, to take up the missing electron. Conversely the
atoms following the inert gases, the alkalies (Li, Na, K, Rb, Cs),
occur always as singly charged positive ions, and so must easily give
up an electron ; in their case consequently it may be assumed that
an easily removable electron revolves outside a stable core of the
inert gas type. The positive or negative electro valency of the remain-
ing atoms may be accounted for in a similar manner ; the former is
due to the presence of easily separable electrons, after the removal of
which the inert gas-like core remains ; the latter is due to the en-
deavour on the part of " incomplete " electron structures to form
complete inert gas configurations by taking up electrons.

The application of this principle to the periodic system leads to
the conception of the shell structure of atoms (see also 29, p. 176).
The first period, consisting of the elements H and He, represents the
structure of the innermost shells. The system of two electrons of
the inert gas He must therefore be a very stable arrangement.

The second period commences with Li. This element will have a
core of the character of the He atom, external to which a third
electron is loosely bound. In the next element, Be, a further outer
electron is added, and so on, until at the tenth element, Ne, the
second shell has become a stable inert gas configuration with 8
electrons. This completes the second shell.

The first element of the third period, Na, has again the loosely
bound outer electron, which represents the commencement of the
third shell ; this closes with the inert gas A, and, since this has the
atomic number 18, the complete third shell is again made up of
8 electrons.

The process is continued in a similar way, the periods, however,
becoming longer (they contain first 18, afterwards 32 elements).
Among them occur the elements Cu, Ag, Au, which have a certain
resemblance to the alkalies ; they will thus be characterised by an
easily separable electron and a relatively stable core.

By means of these qualitative considerations, Kossel was able to
make a considerable part of inorganic chemistry comprehensible
from the physical standpoint ; this theory proved particularly fruit-
ful in the domain of the so-called complex combinations, i.e. com-
binations in which ipolecules arise by the superposition of atomic
complexes, which, from the standpoint of the simple valency
theory, are completely saturated.

SYSTEMS WITH ONE RADIATING ELECTRON 183

Langmuir and Lewis * have (independently of Kossel) added to
the theory by imagining that the stable configuration of 8 electrons,
which we met with in the case of Ne, A, and the ions of the neigh-
bouring elements, is a cube (octet theory), at the corners of which
these 8 electrons remain in equilibrium. According, then, to these
American investigators, we have to do with static models, a hypothesis
which does not agree with our ideas of atomic mechanics, and which
will therefore not be considered any further here.

The manner in which Bohr arrives at the building up of the atoms
step by step in the order of their atomic number is as follows.

He considers the capture of the most loosely bound electron by
the remainder of the atom. This process takes place by transitions
of this electron between the stationary orbits, regarding which
information is obtained from the arc spectrum of the element.
During this process the atom can be thought of as resolved into a
core and a radiating electron. The core has the same number of
electrons as that of the foregoing atom and a nuclear charge one
unit greater. The first question arising is whether the electrons
in the core have the same arrangement as in the foregoing neutral
atom ? Information is obtained on this point in many cases from
the spark spectrum. The second question is, in what orbit does
the newly captured electron finally move ? It either takes a place
as one of a group of outer electrons already existing in the core, or
it traverses an orbit not yet occurring in the core. In the former
case it adds further to an already existing shell, in the latter case
it commences a new shell. In order to answer these questions
we must know the quantum numbers of the orbits in the atom.
The answer to the first question is sometimes Yes and sometimes
No ; in the latter case the same two questions have to be asked
about the last electron but one captured, and so on.

The idea underlying this procedure is called by Bohr the " Aufbau-
prinzip " (atom building).

31. The Actual Quantum Numbers of the Optical Terms

Our next problem will be the more exact determination of the
number of electrons occupying the individual electron orbits and
the values of n and k associated with them. Two methods are avail-
able for the solution : the examination of the optical spectra and of
the X-ray spectra.

If one goes through the series of the elements, and considers in

1 Loc. cit. t see p. 181.

184 THE MECHANICS OF THE ATOM

each case the scheme of the spectral terms, the great similarity
between the spectra of homologous elements will be recognised.
Each alkali spectrum exhibits the same characteristics, likewise each
spectrum of the alkaline earths. We attribute this to the equal
numbers of outer electrons (cf. Kossel, 30).

We turn to the term values themselves. We imagine them written
in the form

vv *r*
n* 2

The spectrum of an element can then be expressed by the system
of n*-values. In order to give a survey of the dependence of the
spectrum on the atomic number, we give here the effective quantum
numbers n* of the lowest term of each series, for the arc spectra so
far analysed, together with the decimal places of the absolute magni-
tude of the Rydberg correction taken as the limiting value for large
values of n. 1

The table shows that for neutral atoms of almost all elements the
/-terms are still of hydrogen type. The Rydberg corrections are
smallest here for Cu and Ag, apart from the light elements ; they
are largest for the alkaline earths, and in this case increase in the
order of the atomic numbers. The rf-tcrms arc of hydrogen type
in the case of the lightest elements (i.e. not heavier than Na) ; it
seems probable also that for Cu, Ag, and perhaps for Or, Mn, the
correction is nearly zero (not approximately equal to another whole
number). The Rydberg correction is still relatively small for the
alkalies, but increases definitely with the atomic number ; in the
case of the alkaline earths it is considerably larger. Finally the
p- and s-terms depart considerably from the values in the case of
hydrogen. It appears, consequently, that /-orbits are in general
situated outside the core, that the d-orbits in many neutral atoms
approach the core too closely to remain hydrogen-like, and in several
cases many actually penetrate into the core, and that the p- and

1 The numbers arc mostly calculated from the data in Paschen-Gotze (Serien-
gesetze der Linicnspektren, 1922). In the case of doublets or triplets the mean
value of n* is given ; for the alkaline earths the values in the first and second row
correspond to the singlet and triplet terms respectively, for O, S they correspond
to the triplet and quintet terms, and for He to the singlet and " doublet " (which
are possibly really triplet) terms ; the figures for Cr, Mo refer to septet terms and
those for Mn to octet terms. Except for those of the alkali metals and He, most
of these spectra, especially those of Cr, Mn, Mo, include terms which cannot be
explained on the assumption of a single radiating electron ; only those terms are
included in the table which can be so explained. From O onwards only the decimal
places of d are given i the columns for the Rydberg corrections. In those
places where the known terms permit of no extrapolation for n=oo, the Rydberg
correction of the last known is given in brackets.

SYSTEMS WITH ONE RADIATING ELECTRON

185

n* of the

first

Rydberg Correction
large n of the

for

P-

Terms

1 H

1-00

2-00

3-00

4-00

0-00

/0-74

2-01

3-00

4-00

-0-14

+ 0-01

0-00

\l-69

1-94

2-99

4-00

- 30

- 0-07

0-00

3 Li

1-59

1-96

300

4-00

-0-40

-0-05

0-00

/1-82
\l-74

1-00
2-17

2-98
2-97

14
23

70
78

02
04

10 Ne 1

1-67

2-15

2-99

11 Na

l-3

2-12

2-99

4-00

12 Mg

/I -33
\2-31

203
1-66

268
2-83

3-96

52
63

04
12

56
17

13 Al

2-19

1-51

2-63

397

16 3

/I -97
\l-88

1-15
234

2-65

(20)
(06)

19 K

1-77

2-23

2-85

3-99

fl-49

2-07

2-00

3-97

20 Ca

\2-49

1-79

1 95

3-92

24 (!r

1-42

1-88

2-99

(12)

(01)

25 Mn

2-31

1-63

2-89

(37)

29 Cu

1-33

1-86

2-98

4-00

(09)

30 Xn

/I -20
\2-34

1-94
1-60

2-87
2-90

3-98

62
72

09
20

20
08

31 CJa

2-16

1-52

2-84

37 Kb

1-80

2-27

2-77

3-99

38 8r

/1-54
\2-55

2-13
1-87

206
1-99

4-14
3-91

(26)
37

(59)
85

10
12

42 Mo

1 -36

1-81

2-74

(65)

(01)

47 Ag

1-34

190

2-98

3-99

(05)

48 Cd

/I -23
\2-28

195
1-62

2-87
2-89

3-97

57
67

05
14

21
07

49 Tn

2-21

1-55

2-82

55 Cs

1-87

2-35

2-55

3-98

K(l "liu

/I -62

2-14

1-89

2-85

(73)

(92)

t)\J -L>d

\2-63

1-94

1-82

3-84

79 Au

1-21

1-77

2-97

/M4

1-91

2-92

80 Hg

\2-24

1-59

2-93

3-97

81 Tl

2-19

1-56

2-90

3-97

1 The neon spectrum is known to have two systems of terms which converge to
different limits. In calculating n* the term under consideration has to be counted
from the limit of the system to which it belongs. The |)-term given is the lowest
which can be assigned to a definite series. A deep-lying term (n*=0-79) is known
from measurements on electron impact (G. Hertz, Zeitschr. f. Physik, vol. xviii,
p. 307, 1923), and from the spectrum in the extreme ultra-violet (G. Hertz, Zeitschr. f.
Physik, vol. xxxii, p. 933, 1925 ; T. Lyman and F. A. Sauniers, Proc. Nat. Acad. Sci.,
vol. xii, p. 192, 1926) ; it corresponds to the normal state, but cannot be explained
on the assumption of a single radiating electron.

186 THE MECHANICS OF THE ATOM

$-orbits are always penetrating orbits, except in the case of the
very lightest elements. (These conclusions, as far as they are based
on this table, refer to neutral atoms only ; it is not necessarily true
that the orbits of the series electron of an ionised atom will be of
the same character as those with the same Jc in a neutral atom of
the same atomic structure.)

In order to substantiate this view we consider the radii of the
atomic cores. The sizes of the cores in the case of the arc spectra
of the alkaline earths, or, what comes to the same thing, the sizes
of the singly charged ions of the alkaline earths, can be derived
from the spark spectra. These ions possess only one external
electron ; the aphelion of its orbit is situated in a region where
the field of force of the atom has approximately the character of a
Coulomb field, and the aphelion distance depends in the same way
on the energy, and consequently on n*, as in the case of hydrogen :

Since the first s-orbit is the normal orbit of the series electron
of the ions of the alkaline earths, we take from the spark spectra
of the alkaline earths the n* values corresponding to the first s-terms,
and regard the aphelion distances calculated from them as the core
radii of the alkaline earths. In the same way we may draw con-
clusions regarding the cores of the elements Zn and Cd, which are
similar to the alkaline earths, since we must assume also of their
ions that they have only one external electron. We obtain an
upper limit for the radii of the alkali ions and the ions of Cu and Ag
from their distances of separation in the crystal gratings of their
salts ; the separation of the Na+ and Cl~ ions in the rock-salt grating
must, for example, be larger than the sum of the ionic radii. By
means of such considerations all radii of all monovalent ions are
determined, apart from an additive constant which is additive for
positive and subtractive for negative ions. This constant can be
determined approximately by putting the two ions K> and Cl~,
both of which are similar to the A-atom, equal to one another ; this
gives upper limits for the radii of the positive ions, since K+ must
be smaller than Cl~ on account of the difference of nuclear charge. 1
A second upper limit for the ionic radii of the alkali metals is given
by the known radii of the atoms of the preceding inert gases, deduced
from the kinetic theory of gases ; the alkali ions we must regard
as being similar in structure to the inert gases their dimensions,

1 Cf. W. L. Bragg, Phil. Mag., ser. 7, vol. ii, p. 258, 1926.

SYSTEMS WITH ONE RADIATING ELECTRON 187

however, must be somewhat smaller on account of the higher nuclear
charge.

The ionic radii calculated in this way are collected in the following
table. They are expressed in terms of the hydrogen radius a 1L as
unit. 1

The table shows the growth in the core radii of homologous ele-
ments with atomic number as well as the fact that the radii of the
alkaline earth cores are relatively large, while those of Cu and Ag
are smaller.

An /-orbit has, in a strict Coulomb field, a perihelion distance
which is larger than 8a n (cf. 24). Owing to the departures from a
Coulomb field of force in the neighbourhood of the atomic cores it
will be decreased. We shall not carry out this calculation, how-
ever, 2 since for our purpose (the determination of the real quantum

Upper Limit of the Radius

Radius
calculated from
n*

From the
Kinetic Theory
of Gases

From Grating
Separation

3 Li+

1-8

1-7

11 Na+

2-2

2-4

12Mg+

3-3

19 K+

2-6

2-9

20 Ca+

4-3

29 Cu+

1-2-14

30Zn+

2-5

37 Rb+

3-0

3-2

38 Sr+

4-7

47Ag+
48Cd+

0-9-2-2 3

8W
2-6

56Cs+

3-3

3-7

56 Ba+

5-2

numbers) a qualitative consideration suffices. We see that an /-orbit
can most easily approach the core in the case of the heavy alkaline
earths ; we understand the large Rydberg correction in the case
of Ba and the relatively large ones in the case of Sr and Ca ; we
find generally a complete correspondence between the core radii

1 There are still other methods of determining the radii of the alkali cores, which
we shall not enter into here. The results are in agreement with the upper limits
given here. Cf. the summary by K. F. Herzfeld, Jahrb. d. Radioakt. u. Ekk-
tronik, vol. xix, p. 259, 1922. j

2 The calculations have been carried out by F. Hund, Zeitschr. f. Physik, vol. xxii,
p. 405, 1924.

8 The values obtained from different Ag salts are widely different.

188 THE MECHANICS OF THE ATOM

and the Bydberg corrections. This connection enables us to draw
conclusions regarding the ionic radius also in the case of the few
other elements the Rydberg corrections of which are known ; we
conclude in this way that it is rather smaller for Al than for Mg, and
that in the case of Hg and Tl it is of the same order of magnitude
as for Zn and Cd.

The d-orbits in the hydrogen atom have a perihelion distance of
more than 4r-5a, r (the circular orbit n= 3 has radius 9a H ) ; in the field
exterior to the cores, which deviates appreciably from a Coulomb
field, they are smaller. The very small Rydberg corrections in
the case of Cu and Ag we ascribe to the fact that in these cases
the d- or bits are situated at a considerable distance from the core.
The small values for the alkalies and for Zn, Cd, and Hg show that
in these cases also the d-orbits are still external paths ; in the case
of Rb and Cs, they must approach very close to the boundary of
the core. In the case of the heavier alkaline earths, Ca, Sr, Ba,
we must assume that penetration occurs. In this connection it is
striking, that in spite of the increase in the core radius from Ca
to Ba, the n* values (for large n) increase ; this leads to the assump-
tion that the Rydberg corrections in the table are to be altered by
whole numbers and are, for Ca, 0-95, 0-92 respectively ; for Sr,
1-75, 1-80 respectively ; for Ba, 245, 2-77 respectively. In
the case of Ca the lowest d-teim would still correspond to a 3 3 -orbit,
in the case of Sr to a 4 3 -, and in the case of Ba to a 5 3 -orbit. The
cases are worthy of note in which the Rydberg corrections of the
/- and d-orbits do not go hand in hand. Thus in the case of Zn the
magnitude of the /-correction is larger, that of the ^-correction
smaller than for K ; Cd and Hg have considerably smaller d-correc-
tions than Rb and Cs, whereas the /-corrections are about the same.
The explanation of this is the high degree of symmetry of the alkali
ions ; this causes the potential in the vicinity of their boundaries to
vary in accordance with a high power of r } while, in the case of the less
symmetrical cores of Zn, Cd, and Hg, it varies more slowly with r.

In the case of the very light elements, the j?-orbits are still external ;
the smallness of the Rydberg corrections, and small core radii,
suggest that this may also be the case for Cu, Ag, and Au, but the
magnitude of the doublet separations and the variation of the value
of the correction for different atoms of the same atomic structure
(Cu, Zn+, Ga++, Ge+ ++ , etc. 1 ) seem to show conclusively that the
p-orbits penetrate. t The apparently small Rydberg corrections for

1 See J. A. Carroll, Phil. Trans. Roy. Soc., vol. ccxxv, p. 357 (1926).

SYSTEMS WITH ONE RADIATING ELECTRON

189

Mg (-0-04 and -0-12), Zn (-0-09 and -0-20), Cd (-0-05 and
0-14), as well as Hg (0-00 and 0-10), have certainly to be in-
creased by a whole number ; their magnitudes would otherwise be
no larger than those of the ^-corrections. If we again note that the
n*-values in the series of the alkalies increase with increasing core
radius, we must assume that the real n-values are 3 for Na, 4 for K,
5 for Rb, 6 for Cs, and that the Rydberg corrections are 0-85,
1-70, 2-66, and 3-57 respectively. Their magnitudes for the
alkaline earths must be somewhat larger ; we assume, therefore,
1-04, 1-12 respectively for Mg; 1-93, 1-95 respectively forCa;
2-59, 2-85 respectively for Sr ; 3-73, 3-67 respectively for Ba.

The s-orbits penetrate from Li onward. In order that the magni-
tudes of the Rydberg corrections may increase with increasing atomic
radius, we must take 8 = 1-34 for Na (0-34 would be smaller in
amount than the ^-correction) ; 2-17 for K ; 3-13 for Rb, and
4-05 for Cs. The somewhat larger values for the alkaline earths
may likewise be found uniquely from the table. For Al we assume
1-76 ; for Cr to Ga values ranging from 2 to 3 ; for Ag, Cd
from 3 to 4 ; for Hg and Tl values between 4 to 5 are very
probable. According to the estimate (4), 28, of the Rydberg cor-
rection, the essential factor is the principal quantum number of the
largest s-orbit confined to the interior of the core, and this is clearly,
in the case of Cu, Zn, Ga, the same as for Rb, and in the case of Ag,
Cd, In, the same as for Cs ; the values in the sixth period can be
inferred by analogy.

We supplement this consideration by another rough estimation
of the 8-values for the s-terms, namely, that given by van Urk.
We replace the electron structure of the atomic cores by charged
spherical shells, the radii of which are somewhat larger than ^a H

'Vorr

3 Li

- 0-00

-0-40

11 Na

074

1-35

19 K

-1-24

-2-18

29 Cu

-2-59

37 Rb

-2-08

-3-14

47 Ag

-3-54

55 Cs

-2-74

(-4-06)

-3-69

(they must be as large as this for the s-orbits to be penetrating
orbits), and imagine the full charge of the nuclei (equal to the order
in the periodic system) to be operative in thfe interior of the shells.
Since the ^-orbits under consideration have the same angular

190

THE MECHANICS OF THE ATOM

momentum as the innermost orbits of the core but smaller amounts
of energy, and since the field of the core again resembles a Coulomb
one in the vicinity of the nucleus, it follows that the inner loops of
these s-orbits have the same parameter as the core orbits next to
the nucleus ; they are therefore subjected to the undiminished
nuclear charge. Application of the equation (2), 28, leads to the
8-values (8 cal ) given in the following table. Together with these the
only S-values (S corr ) which can be in agreement with these lower
limits and the empirical terms are given.
As a consequence of this we can regard the actual principal

Negative Rydberg Corrections
/ ,x\

Quantum Numbers of the

First

Terms of each Series

,5?

0-00

*1 ^2 3 3

4 4

2 He

ro-14

\0-30

-0-01
0-07

0-00
0-00

1 9 Q
O ^2 *8

3 Li

0-40

0-05

000

0-00

2i 2 2 3 3

4 4

/M4
\l-23

0-70
0-78

0-02
004

3i 2 2 3 3

4 4

10 Ne

J-30

0-83

0-02

3O 1 Q
1 "2 f) 3

4 4

11 Na

1-34

0-85

0-01

0-00

333

4 4

12 Mg

fl-52
U-63

1-04
1-12

0-56
0-17

0-06

3l 3 2 a 3'

4 4

13 Al

1-76

1-28

0-93

0-05

4j 3 2 3 3

4 4

19 K

2-17

1-70

0-25

0-01

4! 4 2 3 3

4 4

20 Ca

/233
\2-44

1-93
1-95

0-95
0-92

0-09
0-10

gj 4 2 3 3

4 4

29Cu

2-58

(2-09)

0-02

0-00

4i 4 2 3 3

4 4

30 Zn

/2-02
\2-72

0-20
0-08

0-04

gj 4 2 3 3

4 4

31 <!a

2-78

0-24

4 2 3 3

37 Kb

3-13

2-66

0-35

0-03

5 t 5 2 3 3

4 4

38 Sr

/(3-20)
\ 3-37

(2-59)
2-85

1-75
1-80

0-10
0-12

5i r \
p O 2 4 3

4 4

47 Ag

3-52

(2-05)

001

5i 5 2 3 3

48 Cd

/3-57
\3-67

0-21
0-07

0-03

\\ 5 2 3,

49 In

3-73

0-29

61 5 2 3 3

55 Cs

4-05

3-57

0-45

0-04

6i 6 2 3 3

56 Ba

/4-43
\4-28

(3-73)
3-67

2-45

2-77

(0-92)
0-12

4 4

79 Au

4-60

(3-2)

3-00

61 6 2 3 3

80 Hg

/4-63
\4-71

0-08
0-05

0-03

?J 6 a 3 8

81 Tl

4-74

0-10

0-03

7 X 6 2 3 3

4 4

1 Normal orbit of last electron added. See footnote, p. 185.

SYSTEMS WITH ONE RADIATING ELECTRON 191

quantum numbers and the actual Kydberg corrections of the empiri-
cally known terms as determined, with a few exceptions. To sum-
marise these results we now give a table (p. 190) of the negative
values 8 of the true Kydberg corrections (for large n) and the
quantum numbers of the first terms of each series. The normal state
is denoted by heavy type ; it is distinguished by the fact that the
lines for which it is the initial state occur in absorption at ordinary
temperatures. It must be emphasised that this table only refers
to neutral atoms, and it must not be assumed that the relative
magnitudes of the terms, or the quantum numbers of the first term
in each term series, are necessarily the same for all ions containing
the same number of electrons.

32. The Building Up of the Periodic System of the
Elements

We are now in a position to deal with the building up of the
periodic system step by step, for which purpose we have now at
our disposal all of the data hitherto collected, namely, the properties
of the X-ray spectra ( 29), the chemical behaviour ( 30), and the
characteristics of the optical spectra collected in the table on p. 190
and similar data for many ions.

As a reminder of the order of the elements in the periodic system
we give the scheme (fig. 17) often used by Bohr and dating back to
J. Thomsen.

In the normal state, hydrogen (1 H) has an electron in an orbit
with the principal quantum number 1. As long as the orbit is re-
garded as an exact Kepler ellipse the subsidiary quantum number
is undetermined. We shall see, however, on taking into account the
relativity theory in 33, that the total angular momentum is also
to be fixed by a quantum condition, without thereby appreciably
altering the energy. The normal orbit of the electron is thus a
l r orbit.

For helium (2 He) in the excited states the core will correspond,
according to Bohr's principle, with that of the hydrogen atom in
the normal state, the only difference being the higher nuclear charge.
Now the orbit of maximum energy, or normal orbit, of the series
electron is likewise a li-orbit, so that helium in the normal state
would have two (presumably equivalent) ^-electron orbits. This
system will be considered in greater detail later ( 48). According
to Kossel, a special stability must be ascribed to such a system of
two l r orbits, such as is the case with all inert gases ; in X-ray
terminology this structure comprises the K-shell.

192

THE MECHANICS OF THE ATOM

The question why there are two systems of terms a singlet system
(parhelium), to which belongs the normal state, and a doublet system
(orthohelium) and why these cannot combine with one another,
cannot be dealt with from the standpoint of our book.

The configuration of two Ij-orbits occurs again in the core of the
excited lithium atom (3 Li). According to the spectroscopic evidence,
the normal state in this case is not a l x - but a 2 r orbit. We must
conclude from this that, according to the principles which limit the

^Hc

Fio. 17.

number of electrons in orbits with the same n k , 1 a system of three
li-orbits under the influence of a nuclear charge 3 is not possible.
The ions Be+, B++, C+++ . . . have a structure similar to that of
the lithium atom. Millikan and Bowen 2 were able to confirm experi-
mentally the fact that the spectra of the " stripped atoms " Be + to
+5 are similar to that of the Li-atom.

The spectra of the two following elements, beryllium (4 Be) and
boron (5 B), are not sufficiently well known for us to be able to draw
conclusions regarding the electronic orbits. We can conclude only,

1 These principles have been formulated by W. Pauli (Zeitschr. f. Physik, vol.
xxxi, p. 765, 1925) but will not be explained in this book.

2 R. A. Millikan and I. S. Bowen, Proc. Nat. Acad. Sci., vol. x, p. 199, 1924
(B++) ; Nature, vol. exiv, p. 380, 1925 (Be+ and C+++) ; Phys. Pev., vol. xxviii, p. 256,
1920 (Be+); Phys. Rev., vol. xxvii, p. 144, 1926 (O+ 5 ).

SYSTEMS WITH ONE RADIATING ELECTRON 193

from the bivalent character of beryllium and trivalent character
of boron, that the newly added electron occupies orbits with the
principal quantum number 2, and that the number of l^orbits
remains equal to two ; the K-shell is therefore closed with the He
configuration. The spectra of B+ and C++ are known, 1 at least
in part ; they are presumably similar to the spectrum of neutral
Be, and their lowest terms indicate that the normal orbit of the
series electron is a 2 1 -orbit. Also the spark spectrum of carbon
is known ; 2 the lowest term occurring in it is the 2 2 -term. Since
the boron atom is most probably similar in structure to the single-
charged carbon ion, we may assume that, in addition to the K-shell,
one 2 2 - and two 2 1 -orbits exist in boron. We arrive here at the
same result as for lithium, that not more than two equivalent
electrons with Jc=l exist.

A further electron is added in carbon (6 C) and occupies, in all
probability, a 2 2 -orbit. Such a system of two 2 r and two 2 2 -orbits
does not necessarily possess the tetrahedral symmetry with which one
is familiar from the chemical and physical properties (e.g. diamond
lattice) of the carbon atom. Since, however, nothing is known
regarding the complicated motions in the atom, this does not neces-
sarily imply a contradiction.

Too little is known spectroscopically regarding the next elements
(7 N, 8 0, 9 F). The chemical evidence affirms that N, 0, F have
an affinity for three, two, and one electrons, and the spectrum of
shows that the normal orbit of the last electron is a 2 2 -orbit. The
eight-shell required by Kossel's theory must be reached in the case
of the inert gas neon (10 Ne) ; we can assume, therefore, that the
eight electrons added since Li are bound in orbits with the principal
quantum number 2. The question as to how they are distributed
among the 2 r and 2 2 -orbits we leave unanswered. 3

The conception of the fully occupied eight-shell is confirmed by
the well-known spectrum of sodium (11 Na). The normal orbit of
the series electron is a 3 1 -orbit, the p-oibit of maximum energy
being a 3 2 -orbit. Outside the core, then, no more orbits occur with
w=2. We conclude from this, that the series of electrons for which
n=2 is completed by the number 8 reached in the case of neon.
Using the terminology of the X-ray spectra we call this structure

1 R. A. Millikan and I. S. Bowen, Phys. Rev., vol. xxvi, p. 310, 1925.

2 A. Fowler, Proc. Roy. Soc. (A), vol. cv, p. 299, 1924.

8 Later investigations by E. C. Stoner (Phil. Mag., voL^xlviii, p. 719, 1924) and
W. Pauli, jr. (loc. cit.), have shown that two of the electrons traverse 2 r orbits and
six 2 a -orbits. Here and in the following, however, we shall not enter further into
these details.

194 THE MECHANICS OF THE ATOM

the L-shell. The construction of this L-shell is therefore completed
in the second period of the system of elements, while the K-shell is
built up in the first period.

Since in the case of magnesium (12 Mg) the normal orbit of the
series electron is again a Sj-orbit, we assume, in accordance with the
double valency, that the magnesium atom in the normal state has
two equivalent Sj-electrons in addition to the K- and L-shells.

In aluminium (13 Al), a 3 2 -orbit appears as the normal orbit. We
see, therefore, that a system of three S^orbits cannot be formed as
the outside shell. In the case of Li and O we arrived at a similar
conclusion, namely, the impossibility of the existence of three l r or
2 r orbits.

In the case of silicon (14 Si) we meet with an instance in which
the spectrum can no longer be accounted for with the help of one
" radiating electron." 1 We conclude from the tetravalent character
that the L-ring is surrounded by four orbits with n=3.

With regard to the following elements (15 P, 16 S, 17 Cl) the only
relevant evidence at present available is the affinities for one, two,
and three electrons and the spectrum of S which indicates that the
normal orbit of the last bound electron is a 3 2 -orbit. The final
element of the period is the inert gas argon (18 A), in which, again,
a closed shell of 8 electrons must exist. The detailed construction
of this shell is best considered from the standpoint of the follow-
ing element potassium (19 K), the core of which must have this
structure.

The potassium spectrum indicates a 4 r orbit as the normal orbit
of the radiating electron, and a 4 2 -orbit as the p-oibit with maximum
energy ; the series of 3 X - and 3 2 -orbits is therefore completed on the
attainment of the eight-shell of argon. The 3 3 -orbit of potassium is
more loosely bound than the 4 r and even the 4 2 -orbits ; it has, in
fact, a larger effective quantum number (2-85 in comparison with
2'23 for the 4 2 - and 1-77 for the 4 1 -orbit). The closed shells in argon
do not therefore contain all orbits with the principal quantum
number 3, but only the 3 r and 3 2 -orbits.

In the case of the divalent calcium (20 Ca), chemical and
spectroscopic results both point to a second electron occupying a
^-orbit.

The elements now following exhibit very complicated spectra, for

1 Experimental determinations by J. C. McLennan and W. W. Shaver, Trans.
Roy. Soc. Canada, vol. xviii, p. 1, 1924, and A. Fowler, Phil. Trans. Roy. Soc.
London (A), vol. ccxxv, p. 1, 1925. Theoretical interpretation by F. Hund, Zeitschr.
f. Physik, vol. xxxiii, p. 345, 1925 ; vol. xxxiv, p. 296, 1925.

SYSTEMS WITH ONE RADIATING ELECTRON 195

whose resolution into series very few data are at present available. 1
Their terms have a very high multiplicity, e.g. the terms of Mn and
others are octets ; further, the elements have each several systems
of terms, so that, for example, an element can have several p- or rf-
terms of the same multiplicity, which do not belong to a series ;
the normal state is not always, as hitherto, an s- or y-state ; d- and
/-terms also occur as normal terms, but the spectroscopic character

30 40 50
Atomic number

FIG. 18.

60 70

paramagnetic
coloured ions
incomplete inter-
mediate shells with
90 loosely bound
electrons.

of these terms is not determined by one electron only. Here too for
the first time we meet ions having electron arrangements different
from those of the neutral atoms with the same number of electrons ;
for some of these ions the normal orbit of the series electron is a
rf-orbit (see p. 200).

The elements from scandium to nickel also form, chemically,
a special group. With regard to their chemical valency, they do
not form a continuation of the series K, Ca ; rather they exhibit
multiple valencies which vary irregularly though their maximum

1 For the connection between these spectra and the periodic system, see F. Hund,
ZeitscJir. f. Physik, kc. cit.

196 THE MECHANICS OF THE ATOM

values correspond in general to their position in the general scheme
of the periodic system (Ti 4-, V 5-, Cr 6-, M 7-valent) ; the minimum
valency may be as low as 2. At this juncture the well-known curve
(fig. 18) of atomic volumes, after Lothar Meyer, can be used as an
example of the relations between different elements (atomic weights
and densities in the solid state). The alkali elements form sharply
defined maxima on this curve, which, according to our ideas, arises
from the fact that they have one outer electron in an elliptic orbit.
The fact which concerns us here is that the elements Ti to Ni are
all situated in the neighbourhood of the third minimum of the
curve, and have only slightly different atomic volumes. A further
difference between these elements and the preceding ones arises
from their magnetic behaviour and the colouration of the heteropolar
compounds in which the elements occur as ions.

According to Ladenburg, 1 these compounds are paramagnetic for
the group Ti to Cu (the latter only in divalent form) and exhibit
characteristic colouration (cf. fig. 18), i.e. electron jumps exist with
such a small energy difference that they absorb visible light. Pre-
vious to Bohr's system of quantum numbers Ladenburg attributed
this behaviour to the formation of an " intermediate shell " in the
group of elements from Sc to Ni. The newly added electrons are not
to take up positions externally but internally, while the two outer
electrons of Ca remain.

Bohr has made this conception more precise by assuming that,
in the group Sc to Ni, the series of the 3^ and 3 2 -orbits are com-
pleted by 3. r orbits. We shall consider later how such a completion
of inner groups can occur ; for the present it may be mentioned that
the interpretation of the complex spectra of Sc to Ni 2 fully confirm
this assumption. The appearance of the last M-term in the X-ray
spectra of Cu (cf. fig. 16, p. 179) shows that 3 3 -orbits are actually
present in the interior of the atoms of the following elements. The
3 3 -orbits in the core do not prevent the existence of excited 3 3 -orbits
in the exterior, as the table on p. 190 shows for Cu, Zn, Ga, Rb.

The elements copper (29 Cu) and zinc (30 Zn) resemble the alkalies
and alkaline earths respectively in some series of their spectra. In
Cu we have to assume an outer electron confined to a 4 r orbit, and in
Zn two such ^-electrons. Corresponding to Al, the radiating electron
in gallium (31 Ga) is situated on a 4 2 -orbit. In the eighth place
after Ni we have the inert gas krypton (36 Kr), so that the group

1 R. Ladonburg, Zeitschr. /. Elektrochem., vol. xxvi, p. 262, 1920. Fig. 18 is
taken from this paper.
F. Hund, loc. cit.

SYSTEMS WITH ONE RADIATING ELECTRON 197

Cu to Kr resembles very closely the second and third periods. We
assume, therefore, that in this group the eight four-quantum electron
orbits (4 X - and 4 2 -orbits) are added to the complete three-quantum
shell completed in the case of Ni.

The fact that in Kr the N-ring (w=4) is closed is shown by the
spectra of rubidium (37 Kb) and strontium (38 Sr) ; they prove, in
conjunction with the chemical behaviour of these elements, that, in
the normal state, we have one and two outer electrons respectively
in S^orbits. The following elements, yttrium (39 Y) to palladium
(46 Pd) (like the group Sc to Ni), do not form a simple continuation
of the series, but exhibit multiple and rapidly changing valency.
This suggests that in these elements the 4 3 -orbits, hitherto absent,
appear for the first time ; in the case of silver (47 Ag) we actually
observe a corresponding X-ray term. The occurrence of 4 3 -orbits
in the core, again, does not prohibit electrons in an excited state
from moving in a 3 3 -orbit outside the core, as is the case in Ag, Od,
and In.

The elements silver (47 Ag), cadmium (48 Cd), and indium (49 In)
correspond in their spectra and chemical behaviour to the elements
Cu, Zn, Ga. In their case one 5 2 - and two 5 x -orbits are superposed
on the four-quantum shell (4 t -, 4 2 -, 4 3 -orbits). In xenon (54 X) we
must, for the time being, regard the 5^ and 5 2 -groups as closed.

The sixth period begins with caesium (55 Cs) and barium (56 Ba)
in analogy with the fifth ; the normal orbits of the radiating electrons
are 6 1 -orbits. Lanthanum (57 La) and the elements immediately
preceding platinum (78 Pt) resemble the group Y to Pd. We may
assume there the building up of the 5 3 -group ; a 5 3 -X-ray term oc-
curs, in fact, soon after platinum. In this group is included a further
group of elements which all have very much the same chemical be-
haviour, the rare earths ; we may ascribe them to the formation of
the 4 4 -orbits, which have not occurred hitherto ; a 4 4 -X-ray term
occurs in the case of tantalum (73 Ta). The elements gold (79 Au) to
niton (86 Nt) correspond to the elements Ag to X, and involve the
initial formation of the 6 X - and 6 2 -orbits. The last period involves
the superposition of 7 1 -orbits.

We would expect that in the seventh period the addition of
5 4 -orbits would give a group of elements of very similar chemical
properties, analogous to the rare earths. The heaviest known
elements do not appear to belong to such a group, so the addition of
5 4 -orbits must begin later in the seventh period than the addition
of 4 4 -orbits in the sixth period ; this different is probably to be
ascribed to the greater eccentricity and consequent looser binding

198 THE MECHANICS OF THE ATOM

of the 5 4 -orbits of an element of the seventh period, as compared
with the 4 4 -orbits of the corresponding element of the sixth period. 1

If we again cast a glance over the periodic system and omit for
the moment those groups (framed in fig. 17) having special chemical
and spectroscopic behaviour, we see that, in the first period, two
Ij-electrons are added and in each following period altogether eight
n t - and w 2 -electrons. The iron group (Sc to Ni) introduces ten
further electrons in three-quantum orbits, so that altogether we
obtain 18 three-quantum orbits. The palladium group (Y to Pd)
introduces 10, and the group of the rare earths 14 further four-
quantum orbits, the number of which is thereby raised to 32.

A corroboration of this conception of the building up of inner
electron groups is found (according to Bohr and Coster) in the dia-
gram of the X-ray terms (fig. 16, p. 179), where marked kinks occur
in the curves for the values of Z concerned in this process.

For convenience we give a table of the numbers of electrons occupy-
ing the various shells. 2

In order to be able to derive deductively the construction of the
periodic table, one must be able to deduce theoretically the maximum
number of electrons which can occupy orbits of the same n k . This
can now be done ; consideration of physical 3 and chemical 4 experi-
mental evidence first suggested the rule that the maximum number
of electrons which can occupy equivalent %-orbits in a single atom
is 2(2& 1) ; a theoretical explanation of this rule can now be given, 5
but it lies outside the scope of this book.

If these maximum numbers of occupation be regarded simply as
given, then the order of addition of the quantum orbits becomes, to
a certain degree, comprehensible. We must suppose that the addi-
tion of a fresh electron to an already existing configuration takes
place in such a way that the electron finally enters that quantum
orbit in which it has the least energy (in which it is most firmly
bound), and that it remains in this orbit during the capture of sub-
sequent electrons. And here it must be borne in mind that an atom

1 Calculations by Y. ISugiura and H. C. Urey (Det. Kongel. Danske Vidensk.
tielskab., vol. vii, No. 13, 192(5) suggest that in the seventh period the group
analogous to the rare earths should begin with the element atomic number 95.

2 The table gives the numbers of occupying electrons only in as far as they are
determined to a fair degree of certainty trom our considerations. Later investiga-
tions permit of these numbers being given with a fair degree of probability also
in the case of the remaining elements. See Jb\ Hund, Zeitschr. f. Physik, loc. cit.

1 E. C. Stoner, Phil. Mag., vol. xlviii, p. 719, 1924.

* J. I). Main Smith, Journ. Soc. Chem. Ind., vol. xliii, p. 323, 1924; vol. xliv, p, 944,
1925 ; H. G. Grimm anS A. Sommerfeid, Zeitschr. f. Phys., vol. xxxvi, p. 36, 1926.

6 This explanation depends on the work of Pauli (loc. cit.) and the concept of the
spinning electron.

DISTRIBUTION OF ELECTRONS AMONG THE ^.-ORBITS

3i 3 3 3 3

*i 2 4 3 4 4

1 H

1
2

3Li
4B
5B
6C

ION

2
2

1
2
2 1

2 (2)

11X
12 M
13 A]
14 Si

ISA

2
2
2
2

8
8
8
8

2 1

19K
20 Ca
21 Sc
22 Ti

29 CA
30 Zn
31 Ga

36 Kr

2
2
2
2

2
2
2

8
8
8
8

8
8
8

8
8
8 1
8 2

18
18

1
o

(2)
(2)

1
2

2 1

37 Rb
38 Sr
39 Y
40 Zr

47 Ag
48 Cd
49 In

54 X

2
2
2
2

2
2
2

8
8
8
8

8
8
8

18
18
18
18

18
18
18

8 1
8 2

18
18
18

1
o

(2)

1
2

2 1

55 Cs
56 Ba
57 La
58 Ce
59 Pr

2
2
2

2
2

8
8

18
18
18
18
18

18
18
18
18 1
18 2

8
8
S 1
8 1
8 1

1
2

(2)

71 Cp
72 Hf

8
8

18
18

32
32

8 1
8 2

(2)
(2)

1
2
2 1

79 Au
80 Hg
81 Tl

86 Nt

2
2
2

8
8
8

18
18
18

32
32
32

18
18

87
88 Ra
89 Ac
90 Th

118

2
>
>

8
8
8
8

18
18
18
18

32
32
32
32

18
18
18

8
8
8 1
8 2

1
2

2)
2)

'
8

Horn, Mechanics of the Atom, I. To face p, 198.

SYSTEMS WITH ONE RADIATING ELECTION 199

is not produced from the preceding one by the addition of an electron,
but from its own positive ion ; this certainly has the same number
of electrons as the preceding atom, but a somewhat higher nuclear
charge. That this nuclear, charge may on occasion be an important
factor in deciding which orbit of the added electron is most firmly
bound is shown by the following arguments.

We assume that an ion contains a number of fully occupied quan-
tum orbits, and we inquire now which of those not occupied is the
most strongly bound. We can give an answer to this in two limiting
cases. If the nuclear charge is much greater than the number of
electrons, the field of force in the ion and its surroundings is nearly
a Coulomb field and the energies of the orbits are in the same order
as in the case of hydrogen, only the p-, rf-, etc., orbits with a given
n are slightly less firmly bound than the s-orbits with the same n ;
the order is, therefore : l x , 2 lf 2 2 , 3 lf 3 2 , 3 3 , 4 X . . . .

If now we imagine, say, the uranium atom to be produced by a
nucleus of charge 92 collecting 92 electrons in turn, it will first capture
two l r elcctrons, then altogether eight 2 r and 2 2 -electrons, eighteen
3r, 3 2 -, 3 3 -electrons, etc. Since now the number of electrons gradually
becomes comparable with the nuclear charge, the order of capture is
no longer quite certain. The Bohr-Coster diagram of the X-ray terms
(fig. 16, p. 179) shows us, however, that the energies of the orbits,
at any rate in completed atoms, are in the order 4 1? 4 2 , 4 3 , 4 4 , 5 X . . . .

If the number of electrons is only one less than the nuclear charge,
that is, if we have to do with the addition of the last electron and the
consequent formation of the neutral atom, we can fall back on the
rough estimation of the effective quantum number given by (4),
28, as soon as we have to do with orbits of the penetrating type.
For s-orbits we have

n*=n-(n>-l -!+,).

Since the aphelia of the s-orbits of the core determine its magnitude,
it follows that n is the real quantum number of the largest s-orbit
in the core, and therefore w w =w 1. We shall then have approxi-
mately w*=2.

In the case of the p-orbits n will be somewhat larger than the
quantum number of the p-orbit completely contained in the core,
so that we get 2 < n* < 3.

These values agree in a certain measure with the empirical values
(first table of 31). 1 In general, the dimensions of the d-orbits of

1 For half integral values of k one finds n* = l-5 for s- terms, n*==l-6 to 2-6 for
p-texim.

200 THE MECHANICS OF THE ATOM

neutral atoms are such that they do not penetrate into the core, or
penetrate to such a small extent that the equation (4), 28, does not
seem to be applicable ; the 3 3 -orbit is then the most firmly bound
d-orbit and its n* will be somewhat less than 3. Only in the case of
Sr and Ba do the d-orbits appear to penetrate more deeply. The
estimate would lead to

the empirical value is approximately 2, but is still higher than for
the s-orbits. In either case, then, the first s-term of a neutral atom
(whatever the value of n) is likely to be more firmly bound than the
first rf-orbit.

This estimation affords an explanation of the fact that after the
completion of an n x - and n 2 -group, an outer electron of a neutral
atom will become bound in an (n+l^-orbit, and that, in conse-
quence, after the closing of the 3 X - and 3 2 -groups in A or K + , the next
electron in K traverses a 4 r (not a 3 3 -) orbit, or after completion of
the 4 r and 4 2 -groups in Kr or Rb+, Rb begins a 5 r (not a 4 3 - or 4 4 -)
group. Whilst in the successive capture of electrons by a slightly
ionised atom the 3 2 -orbit is followed by a 4 1 -orbit, for atoms of high
atomic number with the same number of electrons, which are highly
ionised, the 3 2 -orbit is succeeded by a 3 3 -orbit. Consequently, if we
traverse the series of potassium-like ions K, Ca + , Sc ++ , Ti +++ , V< 4) . . .
TJ( 73 ) we must, sooner or later, arrive at a point where the outermost
electron is confined to a 3 3 -orbit. Actually, in the spectrum of K, the
3 3 -orbit (n* 2-85) is less strongly bound than the 4 r orbit (w*=l-77),
for Ca + the difference is much less (n*=2-31 ; 2-14) ; in Sc ++ the n*
of the s-term will be still larger than in the case of Ca+ (in accordance
with the general behaviour of the penetrating orbits), so that the
d-orbit could be more strongly bound than the s-orbit. 1 It has
recently been confirmed by experiment 2 that for Sc++, Ti+++, V+ 4
the lowest rf-term (3 3 -orbit) is lower than the lowest s-term (ij-orbit),
and similarly 3 in the next row of the periodic table the lowest rf-term
for Yt++, Zr+++ is lower than the lowest s-term. It may therefore be
assumed that, in the building up of the Sc-atom from the argon-like
configuration of Sc++ + , a 3 3 -orbit is added and subsequently two 4 X -
orbits, and in the case of Ti from Ti ++++ , two 3 3 - and then two 4 X -
orbits. 4

1 ISce N. Bohr, Zcibchr. f. Physik, vol. ix, p. ], 1922.

2 R. E. Gibbs and H. E. White, Proc. Nat. Acad. Sri., vol. xii, p. 598, 192(5.

3 R. A. Millikan and T. S. Bo wen, Phys. Rev., vol. xxviii, p. 923, 1926.

4 This hypothesis is siif)poitcd by the investigations of these spectra and their
theoretical significance (F. Kund, loc. cit.), even though the conception of a single
** radiating electron " is no longer adequate.

SYSTEMS WITH ONE RADIATING ELECTRON 201

In order to represent the numbers of electrons occupying the
quantum orbits with different n, a two-dimensional diagram must be
employed, for in order to include all the elements together with all
their ions down to the bare nucleus the values of n must be shown as a
function of both the atomic number Z and of the numbers of electrons
z. An illustration of the ideas in question is provided by fig. 19, in

Pt-Group

Fe-Gioup

FIG. 19.

90Z

which only the group is represented (by shading) which is in process
of completion, i.e. the quantum orbit of the last electron added. The
regions where this quantum orbit is not uniquely determined are
doubly shaded.

33. The Relativistic Kepler Motion

In our investigations of the periodic system we found the non-
relativistic mechanics adequate. The more rigorous treatment of

202 THE MECHANICS OF THE ATOM

the orbits in the case of hydrogen requires, however, that the rela-
tivity theory should be taken into account.

A simple calculation shows in fact that already in the one-quantum
circular orbit of the hydrogen atom the velocity of the electron attains
a value whose ratio to the velocity of light c is not negligible for all
purposes. This velocity is

p h

v u = = ;

ma n 27rwa 1[

if for % we substitute the value (8), 23,

A 2

a n

we find for the ratio a
(1) a=^=-^=7-29.10-3.

For observations which attain this order of accuracy, the ordinary
mechanics will therefore no longer suffice. Consequently we must in-
vestigate the motion of an electron in a Coulomb field of force arising
from a nucleus of charge Z, taking the relativity theory into considera-
tion ; in this investigation we follow Sommerfeld. 1

Here also the Hamiltonian function is identical with the total
energy (cf. (11), 5). We have

where fiv/c. The components of the momentum are by (10), 5,

which, on squaring and adding, gives

/ 1

- - *- -- 1 p " yi__p2

and

vT=/

Therefore, by (2),
(4) H

1 A. Sommerfeld, Ann. d. Phynik, vol. li, p. 1, 1916.

SYSTEMS WITH ONE RADIATING ELECTRON 203

If we calculate from this the sum of the squares of the momenta, we
find:

(5) L (p .. +ft . +ft -)

This equation differs from the corresponding one in the non-relativ-
istic Kepler motion by the term

only. Since this term depends on r only, the present problem is like-
wise separable in polar co-ordinates.

Now, however, we have single degeneration only. Following the
notation introduced in 21 for the central motion, we write

The action integrals J and J e are the same as before, in particular

is 27T times the angular momentum. J r takes the same form (2), 22,
as before

but here A, B, and C have a somewhat different meaning :

W 2 I

=2ro (-W) -- -=>Vc 2 1
c* L

B =

I-V+

We 2 Z ../ W

m c 2

a 2 Z 2

a being given by (1). The evaluation of the integral gives as before
(c/. (5), Appendix II),

VA/
therefore

/ - ^

( . Y - A aZ
v ' V i

204 THE MECHANICS OF THE ATOM

If the equation be solved for 1 -\ -- we find

(6)

/
V + ^

2 Z 2

We have here the exact expression for the energy. As in the case
of every multiply- periodic central motion, we know that the orbit is
a rosette.

Only the case in which a is very small is of interest to us. The
first few terms of the expansion in a are therefore sufficient. We find

W a 2 Z 2 a 4

If the value (1) be substituted for a and the Rydberg constant R
be introduced, in accordance with (2), 23, we obtain

(7) w=

( }

Before we enter into a fuller discussion of this equation, we shall
give another deduction of it, this time using the theory of secular
perturbations.

We take as our starting-point the expression (4) for the Hamil-
tonian function. In this the second term under the root is of the
order of magnitude /J 2 ; if we expand in terms of this we get

If we put

H = (PS+PS+P.

H is the Hamiltonian function of the non-relativistic Kepler motion,
which we regard as the unperturbed motion, and H! is a perturba-
tion function. In order to find the influence of this perturbation on
the Kepler motion, we have to average H! over the unperturbed
motion. If we express the sum of the squares of the momenta occur-
ring in Hj with the help of the equation for W , we obtain

This additional term in the energy corresponds to the additional

SYSTEMS WITH ONE RADIATING ELECTRON 205

term in (5), only in this case W is replaced by W , in accordance with
our degree of approximation. We have already calculated the mean
values of 1/r and 1/r 2 for the Kepler motion in (19) and (20), 22 :

_ l ~~ 2__ l
a ab

so that

Remembering that

a n
b k 9

we get for the relativity contribution to the energy

or, if we again introduce a and R,
(8) W^

n 3
l~~l

in agreement with (7).

The smaller the principal quantum number the larger is the
relativity correction (8), and it is therefore greatest for the l^orbit.
For the same value of n it is greater the greater the eccentricity of
the orbit. The frequency of rotation of the perihelion will be

8W 1 dWi cRZ 2 a 2 Z 2 a 2 Z 2

Ut) 2 fl ufa W /C ifC

where v is the frequency of revolution of the electron in its ellipse.

The terms of the spectrum (H, He+, Li ++ ) represented by (7) do
not form a singly ordered set, as do the terms obtained by the non-
relativistic calculation, but a doubly ordered one.

Since the influence of k on the magnitude of the term is small in
comparison with that of n, we can regard the modification brought
about by the relativity correction as a splitting up of the non-
relativistic terms. The arrangement of the terms (with considerable
magnification of the relativistic " fine structure ") is as follows :

Jt-f

1 2

FIG. 20.

3 f
123 123*

206 THE MECHANICS OF THE ATOM

In the absence of external disturbances, only those terms combine,
according to the correspondence principle ( 17), for which the sub-
sidiary quantum number k differs by 1. The line series whose
limiting term is n=l (in H the Lyman series) consists of single lines ;
the line series having the limiting term n= < 2 (in H the Balmer series)
consists of triplets, the lines of the remaining series show a still
more complex character.

As a measure of the relativistic fine structure we take, following
Sommerfeld, that of the limiting term (n=2) of the Balmer series
of hydrogen. This has the theoretical value

Ea 2

Ay ir = - =0-365 cm- 1 .
16

The vahie for the corresponding term in the case of a general value
of Zis

e.g. for He + it is 16A^ H . The quantity Av u will be the approximate
amount by which all terms of the Balmer series are split up, for the
separations of the variable terms (n=3, 4 . . .) will be very small.

With regard to the verification of this theory by observation,
measurements on hydrogen and helium have actually disclosed' the
expected components. Regarding the magnitude of the effect, how-
ever, the experimental results are not in agreement with one another,
measurements on H rt , H^ . . ., for instance, for which theoretically
Ai> IT must be 0-365 cm- 1 , vary between 0*29 and 0-39. 1 In the
case of He + the fine structure may be observed in the series

Paschen has made measurements with direct currents as well as with
alternating currents ; in the latter case many more lines appear,
since, on account of the rapidly changing field strengths, disturbances
arise as a result of which the selection rule based on the correspond-
ence principle breaks down. The numbers of components, as well as
the relative magnitudes of the separations, are in agreement with
the theory. 2

1 Compare the comprehensive report by E. Lau in Physikal. Zeitechr., vol. xxv,
p. 60, 1924 ; Lau considers the value 0-29 to 0-30 as the most probable. The new
measurements by J. C. McLennan and (I. M. Shrum (Proc. Roy. Soc., vol. cv,
p. 259, 1924) give, however, again 0-33 to 0-37. Measurements by G. Hansen
(Dis8. t Jena, 1924) also support the theory.

2 In the report by Lau, referred to above, matters are represented as if in the
case of He also the measurements by Paschen gave values smaller than those

Sr STEMS WITH ONE BADIATING ELECTRON 207

Sommerf eld 1 has used the relativity correction to explain the
multiplicity of the X-ray terms and the departures from Mosley's
law, (1), (2), and (3), 29. The numerical agreement is surprisingly
good throughout the whole periodic system ; the foundations of the
theory are, however, too uncertain to justify its treatment in this
volume.

-84. The Zeeman Effect

Hitherto we have considered atoms as isolated systems ; we now
proceed to investigate the action of constant external influences on
them, commencing with that of a constant external magnetic field,
the Zeeman effect.

We can start out from a very general atomic model with a station-
ary nucleus and any number of revolving electrons. We assume
that the energy of the undisturbed system (without magnetic field) is
a function of certain action variables J l9 J 2 . . . .

Wo^, J 2 . . .)

If now a homogeneous magnetic field exists, the potential energy
of the system is invariant with respect to a rotation about the
direction of the field. The azimuth <f> of an arbitrary point of the
system is then a cyclic variable, as proved in 6 and 17, and the
corresponding conjugated momentum p^ is the angular momentum
of the system about the direction of the field.

The principal function

defines angle variables w t w z . . . w^ ; w^ is the mean azimuth about
the direction of the field.

In the absence of a magnetic field J^ does not appear in the
Hamiltonian function, the motion is degenerate and w, is constant.

If now we wish to find the influence of the magnetic field on the
energy, we meet with the case, mentioned in 4, where the forces
which act on the various particles of the system depend on the
velocities. Owing to the magnetic field H (supposed for the moment

required by the theory. This is due to the fact that Lau bases his observations
only on the direct current measurements of Paschen, whereas Paschen includes also
the alternating current measurements.

1 A. Sommerfeld, Ann. d. Physik, vol. li, p. 126, 1916. A. Land6 (Zeitschr. f.
Physik, vol. xxv, p. 46, 1924) has shown that even certain optical doublets, in the
case of terms not of the hydrogen type, follow the appropriate relativity formula.
Millikan and Bowen (Phys. Rev., vol. xxiii, p. 1, 1924, and vol. xxv, p. 295, 1925)
have brought forward much empirical evidence in support of this. These effects
are now ascribed in part to a spin of the electron (cf. footnote, p. 152).

208 THE MECHANICS OF THE A.TOM

to depend arbitrarily on x, y, z), an electron of charge e is subjected
to the so-called Lorentz force l

(1) K=-%H].

According to 4 we have to determine a function M such that

d dM M__

dt~8x~~dx~ x '
The function

has this property ; A is the vector potential of the magnetic field,
defined by

H --curl A.
We have :

d d

y 8A, r

c[ \ 'Ox tiy i \ tiz
e,.

The Lagrangian function is by (8), 4 :
(2)

where the sum is to be taken over all the electrons. From this we
calculate the momenta. For one electron they are :

dL e

p^r^WJ'-^

d.f c

dL e

(S) ^_,^-.A v

dL e

Pz=-zr=*M A Z -

dz c

The Hamiltonian function becomes, by (3), 5 :

+ypv +*PZ) L

1 See, for example, M. Abraham, Theorie der Ehktrizitat, vol. ii, third edition,
Leipzig, 1914, 4, p. 20, or H. A. Lorentz, Theory of Electrons, p. 15.

SYSTEMS WITH ONE RADIATING ELECTRON 209

It is, therefore, equal to the total energy in this case also. No addi-
tional term occurs in the energy, corresponding to the magnetic field,

since the magnetic forces do no work ; the force [vH] is always

perpendicular to v. If in H we express the velocity components
in terms of the momenta we get

We restrict ourselves in the following to the case where the field is
so weak that we can neglect the squares of A^, A y , A z . We can then
write

(5) H=

so that the Hamiltonian function differs only by the term

from its value for no field.

We now examine the effect of a homogeneous magnetic field H
on the motion of the electrons. The vector potential of such a field is

where r is the position vector from an arbitrary origin, which we take
as the nucleus. In the additional term we have therefore

where p is the resultant angular momentum of the system of elec-
trons, and PQ, as above, its component in the direction of the field.
Apart from terms proportional to H, p^ is the momentum conjugate
to an absolute azimuth. If we pass over to the angle and action
variables w lt w 2 . . . w^ J t , J 2 . . . J^ of the motion in the absence
of a field, (5) takes the form l

(6) H

From this we can deduce at once the influence of the magnetic

1 The double sign is due to the fact that p. can be positive or negative, whereas,
by definition, J^ is only positive.

1/1

210 THE MECHANICS OF THE ATOM

field H on the motion of the electrons. The angle and action vari-
ables of the motion in the absence of a field remain angle and action
variables in the presence of a magnetic field, since the total energy
depends only on the J fc 's. The angle variable w^ is, however, no
longer constant, but has the frequency ^=v w , where

corresponding to a wave number

=l e _El

27T 2mc'

while the frequencies of all the remaining angle variables are ex-
pressed in terms of the J fc , in just the same way as with no field
acting. The sole effect of the magnetic field H is then to superpose
on the motion occurring in the absence of a field a uniform precession
of the whole system with the frequency v m (the Larmor precession).

The motion of an electron may then be resolved into oscillations
parallel to the field with frequencies (vr)=^ 1 T 1 +^2 T 2+- inde-
pendent of the field, and into oscillations perpendicular to the field
with the frequencies (vr)+v m and (vr) v m . This, on the classical
theory, would give rise to radiation of frequency (TV) polarised parallel
to the field and to radiation with the frequencies (vr) v m circularly
polarised about the direction of the field.

We shall see that the quantum theory leads to the same resolution
of a line into three components.

Since J 1? J 2 . . . are adiabatic invariants (cf. 16) they remain con-
stant in a magnetic field slowly generated, so on switching on the
field the only change in the motion of the electrons is the super-
position on the already existing motion of a uniform precession of
frequency v^

To the quantum conditions of the unperturbed system

J k =n k h

there is now added a new condition
(8) J,=mA;

it states that the angular momentum of the electron system in the
direction of the magnetic field can have only certain values. For
a weak magnetic field we have here an example of spatial quantisa-
tion, which we have dealt with generally in 17. If the angular
momentum J/27T, where J is one of the quantities J lf J 2 . . ., is fixed
by the quantum number j,

SYSTEMS WITH ONE RADIATING ELECTRON 211

the angle a between the directions of the angular momentum and
the magnetic field is given by

(9)

cos a= ;
3

The axis of the angular momentum can therefore be orientated
only in 2j+l different directions (m=j, j 1 . . . j) with respect to
the axis of the field.

The additional magnetic energy is, by (6), (7), and (8),
(10) W m =hv m .m;

each term will, in consequence, be split up into 2j+l equidistant
terms separated by a distance v m .

According to the correspondence principle, the quantum number
m can change by 1, 0, 1 where, for the transition m-+m, the light
radiated is polarised parallel to the direction of the field and for
transition m^\-*m it is circularly polarised about the direction
the field. A decrease in m corresponds to a Larmor precessiodnl
the positive sense in the classical theory, and therefore to positive
circularly polarised radiation ; an increase of m corresponds to
negative circularly polarised radiation.

The frequency radiated in the transition m-+m is the same as
the frequency V Q radiated in the absence of a magnetic field for the
same variations of the remaining quantum numbers. The frequency
radiated in the transition m-^l^m is

One finds consequently for longitudinal observation, as in the
classical theory, a doublet of circularly polarised spectral lines,
situated symmetrically with ^-^ ^--^

respect to v . The line with ( j ( )

the greater frequency cor- ^ >
responds to the transition
m+I-*m ; it is therefore
positively circularly polarised
with respect to the field, i.e.
left-handed to an observer
looking in the opposite direc-
tion to that of the field. ~"
For transverse observation
a triplet is observed, the
centre line of which is situated at V Q and is polarised parallel to the
lines of force, the outer lines being separated*from V Q by v m and
polarised in a perpendicular direction (fig. 21).

longitudinal (axis of
lieid directed to front )

transversal

FIG. 21.

212 THE MECHANICS OF THE ATOM

This result is the same as in the classical theory of H. A. Lorentz.
It is verified experimentally for such lines of the other elements as
are simple (singlets). This simple theory (which is analogous to the
classical theory of Lorentz), does not suffice for the explanation of
the complicated Zeeman effects which occur in the case of multi-
plets. The theory of these " anomalous Zeeman effects " lies outside
the scope of this book. 1

36. The Stark Effect for the Hydrogen Atom

The next example of the action of an external field which we shall
consider is that of the Stark effect for the hydrogen atom, i.e. the
influence of a homogeneous electric field E on the motion in the
hydrogen atom (more generally in an atom with only one electron).
We shall treat this problem in considerable detail, in order to illus-
trate the various methods employed for its solution.

The first method to which we resort is that of the introduction of
separation variables ; 2 afterwards we shall calculate the secular
perturbations by two different methods. The result will, of course,
be the same in every case.

If we choose the z-axis of a rectangular co-ordinate system as the
direction of the field, the energy function becomes

m e z 7i

(1) H=-(x 2 +2/ 2 +z 2 )- +Ez, E=|B|.

It is easy to see that the Hamilton- Jacobi differential equation is
separable neither in rectangular nor in polar co-ordinates. It may,
however, be made separable by introducing parabolic co-ordinates.
We put

x=grj cos <f>

(2) y=r?sin<

The surfaces =const. and ry=const. are then paraboloids of rota-
tion about the z-axis ; they intersect the (x, z)-plane in the curves

X*=2*(--Z

1 Cf. E. Back and A. Land6, Zeemaneffekt und MuUiplettstruktur der Spek-
trallinien, vol. i of the German series, Struktur der Materie (Springer).

1 First worked out by P. S. Epstein, Ann. d. Physik, vol. 1, p. 489, 1916 ; vol. Iviii,
p. 553, 1919; and K. Schwarzschild, Sitzungsber. d. Berl Akad., 1916, p. 548 t

SYSTEMS WITH ONE RADIATING ELECTRON 213

i.e. in parabolas with their focus at the origin, and having the para-
meters 2 and rj 2 ; </> is the azimuth about the direction of the field.
In the new co-ordinates the kinetic energy is

(3) T=

which gives for the momenta conjugated to , 77, <f> :

(4)

If we substitute these in T and add the potential energy

we get

If this be equated to W and the resulting equation multiplied by
it becomes separable. We have first :

^ = ^

since <f> is a cyclic co-ordinate, and

J*=#lVfy
Since p^d^ is never negative, J^.^.0 always. We find further :

where

(6) \
/ 2 (,)=2mWr ? H2a a --

and

(7) a 1 +a,=2me 2 Z.

214 THE MECHANICS OF THE ATOM

The action integrals J^ and J^ are consequently :

(8)

where

A=2w(-W),

In order that the integrals (8) shall remain real also for zero external
field, a i and a 2 must be positive. If the field strength is small, the
terms involving D x and 1) 2 arc small in comparison with the remain-
ing ones and the integrals may be evaluated approximately by com-
plex integration. We find (cf. (11), in Appendix II), if we take the
roots in (8) so that the integrals are positive :

j i
*"2m 2(-2mW) 3

= ~

a x and a a are to be eliminated from the three equations (7) and (9)
and W evaluated. To a first approximation the term proportional
to E in (9) can be omitted and, afterwards, the values of a and a 2 ,
calculated to this first approximation, can be substituted in this
correction term. In this way one finds
tt 2J +J meti

V-2mW 2ir 87rV(-2mW) 3
and then, iising (7),

This gives to a first approximation (omitting the term proportional
to E) the energy of the motion in the absence of a field

SYSTEMS WITH ONE RADIATING ELECTRON 215

and, if we substitute this value of W in the correction term, to a
second approximation :

'> T-

To our approximation, then, the energy depends only on two linear
combinations of the action variables, i.e. we have to do with a case
of single degeneration. This would no longer be the case if we cal-
culated higher terms in E in the expression for the energy. In
accordance with our general considerations ( 15) we now introduce,
in place of J^, J^, J^, new action variables, derived from these by an
integral transformation with the determinant 1, and so chosen
that the energy (11) depends on only two of the new action vari-
ables, and the energy (10) of the unperturbed motion (corresponding
to the double degeneration) on only one of the action variables.

We write therefore

(12)

and obtain

J , = : J

The motion has two frequencies :
(14) v

and

We have two quantum conditions :

If we introduce them into the expression (13) for the energy, we have

SEA*

where R is again the Rydberg constant (see (2), 23). A more
accurate calculation gives higher terms which depend also on a
third quantum number ri.

J, has the same meaning as the correspondingly denoted magni-
tude of the Kepler ellipse in the absence of a t field ; it can assume
values between and J only. The sum of the positive quantities

216

THE MECHANICS OF THE ATOM

Jj and J^ lies by (12) similarly between and J and their difference
J e between J and +J. The quantum number n e can therefore
have only the values n, (w 1) . . .+n. As will be seen from a
study of the orbits, the values n are also to be excluded.

The parabolic co-ordinates and rj execute librations between the
zero points of /^f) and/ 2 (ry) in (6). We will consider the character
of the motion first for the case in which J , and consequently C,
does not vanish. Here the region in which /i() and/ 2 (^) are positive
does not extend to the positions f=0 and 77=0; the zero points
f mm an( l ^min arc different from 0. The third co-ordinate in this
case performs a rotation. The path is confined to the interior of a
ring having the direction of the field as axis of symmetry and the

Fia. 22.

cross-section bounded by the parabolas f =f mln> = max , ^
and 7?=^ ax (cf. fig. 22). In particular, if J^==J n =0, min and max
and likewise rj mm and 7y max coincide and the path is a circle. Since
f mm 4 s ^imu ^ s pboi& does not pass through the nucleus ; it is displaced
in the opposite direction to that of the external field B, as will be seen
by a consideration of the equilibrium between the positive nucleus and
the orbit of the electron in the field, or by calculating the double
roots. If J^=0 and J,>0, the orbit lies on the paraboloid 1=^,^
= max , between the circles of intersection with the paraboloids TJ ^^
and ^=i7 max . Finally, in the general case, for J f >0 and J,,>0, it
lies in a three-dimensional ring. If we disregard the motion of <f>,
the (, 77) co-ordinates in general fill completely the curvilinear
quadrilateral contained between the parabolas for the extreme
values of f and 77, since the frequencies associated with J^ and J^ are
different and their ratio is rational only for certain values of E.
Proceeding to the case J^=0, <f> remains constant, the motion

SYSTEMS WITH ONE RADIATING ELECTRON 217

takes place in a meridional plane parallel to the direction of the field.
The region in which / x (f ) and/ a (ry) are positive comprises the values
f =0 and ^=fO, since in this case/^) and / 2 (^) remain positive
there (cf. (6)), i.e. the path completely fills the two-dimensional
region bounded by |=f max and i?=i7 max . The orbit approaches
therefore indefinitely close to the nucleus.

The case in which the electron approaches infinitely close to the
nucleus is to be excluded on principle, just as in the case of central
motions (21). This excludes at the same time the case n e = n,
since in this case J^ or J^ would be equal to nh=J and J^=0.

The stationary state represented by the quantum number n in
the absence of a field splits up, on application of the field, into
2w 1 states of different energy with the quantum numbers

n,=-(-l). -(-2) . . . +(n-l).

We now consider the radiation from such an atom. The radiated
frequencies and the possible changes of n and n e depend on the terms
of the Fourier expansion of the electric moment or of the co-ordinates
of the electron. To the action variables J f , J^, J^ correspond angle
variables w^ w^ w^. With the help of these the Fourier expansion
of the co-ordinates may be written in the form

Since w^ and <f> are proportional to one another and <f> performs a
uniform rotation about the direction of the field, the values of r^ for
the components of the electric moment perpendicular to the field
are 1 only, and for the component in the direction of the field
the value is 0. The co-efficients r f and r^ on the other hand, do
not appear to be restricted (see 36).

Passing over now to the angle variables which correspond to the
action variables J, J e , J', we have to write (by 7) :

and since only J and J e appear in the energy (see (13)), w* is con-
stant. The Fourier series becomes

where

, T f .

w is the angle variable for the motion in the Absence of a field and
corresponds to the revolution of the electron in the elliptic orbit,

218

THE MECHANICS OF THE ATOM

r may therefore be any integer ; r e is also unrestricted, since r^ and
Ty are so. This means that w and n e can change by any amount
consistent with their values, and that frequencies corresponding to
all these transitions will be radiated.

The polarisation is derived as follows : If r+r e , which is equal to
2^+r^, is an even number, r^ can only be zero. Such a Fourier
term represents consequently a motion in the direction of the field ;
a light-wave polarised parallel to the field corresponds then to a
transition for which Aw+An e is even. If An+An e is odd, r^ = l ;
the wave corresponding to such a transition is polarised perpen-
dicularly to the field.

We illustrate the above remarks by considering the resolution of
the Balmer lines H a , H^ ... of hydrogen. The terms which combine
to give these lines are split up in the following way (the numbers

. , , . , ,.

give the energy change as a multiple of

-202

-6 -3 03 6

-12 -0 - O 8 12

-20 -1

MINI h

-/<? -5 O 5 fO f5 2O

FIG. 23.
We obtain from this for the line H a (n=3->n=2) the lines ;

01234 56 8

ForH,,:

02 V 6 8101211

Fio. 25.

SYSTEMS WITH ONE RADIATING ELECTRON 219
For H y :

I I i i I I i I II I

O2 35 78 10 1Z 73 75 7778 20 22

FIG. 26.

The calculation of the Stark effect by parabolic co-ordinates
allows us to illustrate by an example some previous considera-
tions regarding the restriction of the quantum conditions to non-
degenerate action variables.

For | E | =0 the motion of the Stark effect passes over into the
simple Kepler motion. This is separable in polar co-ordinates as
well as in parabolic co-ordinates. From the separation in polar
co-ordinates ( 22) we obtain the action variables J r , J e , J^, and the
quantum condition

J f +J0+J^=ni.

J0+J^ is now 2?r times the total angular momentum, and J^ is 2?r
times its component in the direction of the polar axis. The motion
remains separable in these co-ordinates if the field is no longer a
Coulomb field, but is still spherically symmetrical ; in the latter
case, however, a second quantum condition,

J*+J*=**,

is to be added. To make J^ an integral multiple of h would have no
significance, since the direction of the polar axis of the co-ordinate
system is altogether arbitrary and the integral value of J^/h would
be destroyed by a rotation of the co-ordinate system. The restric-
tion J +3^=kh 9 on the other hand, would lead to no impossibility in
the case of the simple Kepler motion.

If now we calculate the Kepler motion in parabolic co-ordinates,
we have only to put E=0 in the above calculations. We obtain the
action variables J^, J n , and J^ (the last has the same significance as
in polar co-ordinates) and the quantum condition

The second quantum condition

Jf-J^A

which we had in the electric field, must now be dropped, since this
combination of the J's no longer appears in the energy. It has a
meaning only if an electric field is present (though this need only be
a weak one).

220 THE MECHANICS OF THE ATOM

The stationary motions in a weak electric field are, however,
essentially different from those in a spherically symmetrical field
differing only slightly from a Coulomb field. In the latter (for which
the separation variables are polar co-ordinates) the path is plane ; it
is an ellipse with a slow rotation of the perihelion. In the former
(separable in parabolic co-ordinates) it is likewise approximately an
ellipse, but this ellipse performs a complicated motion in space. If
then, in the limiting case of a pure Coulomb field, k or n e be intro-
duced as second quantum number, altogether different motions would
be obtained in the two cases. The degenerate action variable has
therefore no significance for the quantisation.

Our considerations lead to yet another result ; the calculation of
the Stark effect and the quantising of J e can have a meaning only
if the influence of the relativity theory, or of a departure of the atomic
field of force from a Coulomb field, is small in comparison with that
of the electric field. Further, our former calculation of the relativ-
istic fine structure is valid only if the influence of the electric fields,
which are always present, is small compared with the relativity
perturbation. 1

36. The Intensity of the Lines in the Stark Effect
of Hydrogen 2

The correspondence principle, which, by its nature, allows of only
approximate calculations of intensities, leads to relatively accurate
results when we are concerned with ratios of intensities of the lines
within a fine structure, e.g. in the Stark effect.

Following Kramers, 3 we shall deduce in the following the Fourier
expansion for the orbit of an electron which moves round the nucleus
under the action of an external field E and compare the classical
intensity ratios with those observed. In the Fourier coefficients we
shall omit all terms proportional to E, E 2 , etc., since they lead only
to unimportant corrections.

From 35 we obtain for the principal function S :

1 Kramers has succeeded in dealing with the simultaneous action of the relativity
variation of mass and a homogeneous field for the case in which the corresponding
changes in the energy are of the same order of magnitude (H. A. Kramers, Zeitechr.
/. Physik, vol. iii, p. 199, 1920).

* In this section we have given the calculations more shortly than previously in
this book.

* H. A. Kramers, Intensities of Spectral Lines, Copenhagen, 1919.

SYSTEMS WITH ONE RADIATING ELECTRON 221

If the values of a x and a a be taken from (9), 35, and the value of
W from (10), 35, both of them for E=0, we find :

(1)

where for shortness we write :

For the angle variables w^, w , w^ conjugate to J f , J^, J^ we find
from (1) the equations :

V -
1 rdr,

(3) * J J V-

jcJ 2 J 17 \/

2^=2.

/cJ 2 J I V

I fdr,

Since the calculation of the w'a as functions of and 77 from
the above formulae would obviously be very laborious, it is advisable
to write the squares of the variables, 2 and -q 2 , which oscillate between
two fixed limits (cf. 35), in the form

222 THE MECHANICS OF THE ATOM

(4) ^ 2 =a 1 +6 1 cos^, r)*=a 2 +b 2 coa x ,

just as before ( 22) we found it convenient to introduce the mean
and eccentric anomalies. In order that the new variables if/ and ^
may increase by 2?r during one libration of or 77, we must put :

This gives :

and for ^, w n , w^ we find :

sin ij+b 2 sin

sn + 2 sn +

2 \ J o^l+^l COS ^ J 0^2+^2 COS

In these expressions we have chosen the still arbitrary constants
of integration in such a way that the final result takes the simplest
possible form.

Introducing the abbreviations

we get

27TW=or 1 sn i

^ ' 27^=0"! sin 0+^2 s i n X+X+ 77 -

The similarity between these equations and (15), 22, shows clearly

the analogy between iff, #, and the eccentric anomaly.

We can now write down without difficulty the Fourier series for
the co-ordinates z and x-\-iy. By (2), 35, z=(f 2 rj 2 ). Since
z does not depend on <f>, it is also independent of w . ; we write
therefore :

(8) Z

where

SYSTEMS WITH ONE RADIATING ELECTRON 223

a' i n
L> e -
i) 2

Now by (7) :

COS

Again since, by (4) and (5),

a l a 2 &! cos 6 2 cos

^ J,,)+/<:J 2 (C71 COS /f (7 2 COS x),

we have :

1 J-STT^TT

(11) A 00 = J J^ <tyd x [*J(J f -J,)+*JVi COB </r-(7 2 cos x)]

. (l+CTj. COS ^+<7 2 COS X )=(icJ(j J^).

For the remaining values A T T , for which not both the T'S are zero,

the constant term /cJ(J f J n ) in z can be omitted at once, since by
(9) it will disappear on averaging. In this way we find (T+T I) =T)

K 3*(iyr 2 "( 2
(12) A =

1 4:7T JO JO

COS xe --r < r l -r 1tK -r<r t mnx t

If in this equation cos ^r and cos^ be replaced by ^(e^+er^) and
\(e ix +e~ lx ) respectively, it will be seen that the integral on the right
may be split up into a sum of products, each factor of which is of the
form

this is the well-known expression for the Bessel function x J n (/o).
Using the relations 2

and

1 E. Jahnko and F. Emde, Funktionentafeln (Leipzig, 1909), p. 169; or see, for
example,. G. N. Watson, Theory of Bessel Functions (Cambridge, 1923), p. 20. The
Bessel functions are here indicated by Gothic letters to avoid confusion with the
action integrals.

2 Jahnke-Emde, op. cit. t p. 165, or Watson, op. cit. t p. 17.

224 THE MECHANICS OF THE ATOM

we find in this way from (12) :

(13) \, = ^{

Finally we get for z :
(U) ^(J^

(The dash on the summation sign signifies that ^=^=0 is to
be excluded from the summation.) For r=0 the expression (13) is
indeterminate. It follows, however, directly from (12), that the
corresponding A T r (^+^=0, r^ + O) vanish.

In order to calculate the Fourier expansion for x+iy we take from
(2), 35:

(15) x+iy^fye*.

We can conclude at once from (15) and (3) or (6), that
(x-\-iy) . e~" 2lrtl 4 depends only on w^ and w n . The most convenient
method of procedure is to expand (x+iy)e 2iri ( w n" w ^ in a Fourier
series :

(16) (x+iy)W>i- w 4t =Z, B r T ^M^+( T n+ 1 KL

In order to write the left side of (16) as a function Ox and x we
deduce from (6)

On putting

we have :

2 \oa 1 +fe 1 cos0^oa 2 +6 2 cosx/

cos

d x

~ g

(a 2 +6 2 )(a 2 +6 2 cos x )
and consequently, substituting for (x+iy) from (15) and (4),
(18)

SYSTEMS WITH ONE RADIATING ELECTRON 225

From this we can deduce at once the B (we now write :

TT- -

* 4:77

O JO

[ Jf . C . 0\/ y . C y

.cos - +* - sm cos-4^ - - sin
\ 2^ a^ 2 A 2 a 2 +6 2 2

We can express the quantities cos ^r, cos ^, cos -, etc., in terms of

exponential functions exactly as in equation (12) and so represent
B T T as a sum of products of Bessel functions. We obtain finally,

in the same way as in the case of the magnitudes A T T (r=l +r f +r) 9

(20) B T =-V(J f +^

For r=0 we can calculate B T T directly from (19). It is found
that B T T =0 for r=0, with the exception of the values

(21) B_ 1>0 =

Finally, as the Fourier series for x+iy, we find

(22)

Now that we have calculated the Fourier coefficients we can pro-
ceed to an approximate estimation of the intensities on the basis
of the correspondence principle. We assume that the simple degen-
eration of the variables J^, J^, J , which still existed in (11), 35,
has now been removed, either by including in the energy terms
quadratic in E, or by taking account of relativity.

226 THE MECHANICS OF THE ATOM

In accordance with the fundamental principles of the quantum
theory we must then write

J^ =nji ; J^ =n^ ; J == nji.

According to the correspondence principle we find the approximate
intensity of a line corresponding to a transition in which n^ changes
by An f , n^ by An^, and n^ by An^, if we examine the intensity of
the harmonic r ( =An^, r^An^, r^==An^ in the classical spectrum
represented by (14) or (22). This leaves open the question whether
the classical spectrum shall be taken to correspond with the initial
orbit, the final orbit, or an intermediate one. In the following we
shall investigate only the relative intensities within a fine struc-

A B
ture. Consequently we shall introduce the magnitudes - --

as " relative amplitudes " R, and then compare the simple arithmetic
mean of the relative intensities R 2 corresponding to the initial and
final orbits with the observed relative intensities. One result of intro-
ducing the " relative amplitudes " is that in forming the average the
initial and final orbits have the same weight attached to them as
regards the intensity relations. It may be conjectured that this latter
assumption involves a fundamental aspect of the quantum calculation
of the intensities ; in the case of the Zeeman effect, for example, it
implies that the relative intensities in the Zeeman fine structures
shall be independent of the principal quantum number, a result that
must certainly be expected to hold from analogy with the classical
theory and which has always been verified empirically.

From (13) and (20) we find for the z components of the relative
amplitudes (r^=0, T^+T )? =T) :

(23)

for the (z+iy)-components (r^=
(24) R i

SYSTEMS WITH ONE RADIATING ELECTRON 227

The amplitudes of the ^-components correspond to the lines
polarised parallel to the field ; those of the (+iy)-components to
the lines polarised perpendicular to the field.

H a 6562-8 A

Transition

T^ r^ Tj,

Ra a

R* 2

Obser.
Intens.

Const, x
(Ra 2 + R)

!111->011

0-21

1-0

0-35

102 -> 002

0-26

0-43

201 -> 101

0-38

0-33

1-2

1-16

201 -> Oil

2-1

, ,

(003->002
111 -> 002

1
1 1 -1

1-00
0-07

1-00

1 2-6

),4

J_ J 102 -> 101

0-56

0-39

1-0

1-56

102 -> Oil

1 -1 1

. .

[ 201 ~> 002

2 0-1

0-00

The table gives a comparison between theory and observation for
H a (n =3->n =2) 6562-8 A. The transitions, characterised by the quan-
tum numbers in the initial and final states (nf, nf, nf-

, n

are given in the first column. The second column gives the dis-
placement A of the corresponding lines from their positions for

3EA

E=0 in multiples of the smallest displacement - - (in wave
r 2 v

numbers), as calculated by (11), 35. The third column contains the
values of r^ r , r^ corresponding to these transitions ; in the fourth
and fifth columns the quantities R 2 and R,, 2 , for the initial and final
orbits respectively, are given as a measure of the relative intensities.
The sixth column contains the intensities observed by Stark. The
seventh gives the values of (R a 2 +R/) ; in order to make a com-
parison with Stark's values possible a constant factor is introduced,
so that the total intensity of the theoretical and observed groups
of lines are the same.

We see from the table that the sum of the calculated intensities
of the parallel components (1*9), differs considerably from the sum
of the intensities of the perpendicular components (5-0), while ob-
servation gives for each sum nearly the same value (3'3 and 3*6).

The figs. 27 to 30 x represent the comparison between theory and
observation in the case of H a and the other hydrogen lines examined
by Stark. An essential point for the agreement between theory and
observation is the absence, required by 35, of the case J^=n^=0.

To sum up, we can conclude, from the calculations of the previous

After H. A. Kramers, foe. ctf., figs. 1 to 4.

228

THE MECHANICS OF THE ATOM

paragraphs, that the correspondence principle, combined with the
method of averaging applied here (taking the arithmetic average of

. I J . "

H a calculated

HQ calculated

FIG. 27.

FIG. 28.

H n observed

it it Mli

? I I ?
. 1 1 III in

tip observed

? I I I . . If I . I I I ?

.lit,
y~, calculated

,1 1,11,,

H^ observed

Fid. 29.

. " ! I , I I M , I I I , I J

..II ,,,

//* calculated

FIG. 30.

; .1 1, ii , , h , i hi

A/5 observed

the relative intensities of initial and final orbits) approximates
closely to the quantum law of intensities. Among other things, the

SYSTEMS WITH ONE RADIATING ELECTRON 229

fact that these calculations do not give exactly the true quantum
intensities is to be expected, since, according to the above calcula-
tions, the resolved lines should exhibit a polarisation as a whole
(mentioned above for H a ), the existence of which seems highly im-
probable both for theoretical reasons and on account of the experi-
mental results.

37. The Secular Motions of the Hydrogen Atom in
an Electric Field

The method so far employed for the treatment of the Stark effect
depends on the special circumstance, which might almost be con-
sidered accidental, that separation co-ordinates exist having a
simple geometrical significance. We shall now show how we may
attain our object, without making use of this peculiarity, by a
systematic application of the theory of secular perturbations. We
shall adopt two different methods of procedure, beginning with one
which investigates the secular motions of those angle and action
variables which occur as degenerate variables in the investigation
of the Kepler motion by polar co-ordinates ; the second method,
which is more suited to the geometrical aspects of the perturba-
tion, has the advantage of being capable of extension to a more
general case (crossed electric and magnetic fields).

We write the Hamiltonian function in the form

(i) H=H O +AH I .

Here

(2) H =

is the energy of the Kepler motion in the absence of a field, and

denotes the perturbation function. (The field strength E may be
considered as the small parameter A.) According to the rules given
in 18, we have to express the mean value over a period of the un-
perturbed motion,
(3) AH^eEz

in terms of the degenerate angle variables and the action variables
of the unperturbed motion (see 22), which we now denote by
w 2 Q , w 3 , and JfJfJj*.

If in the unperturbed motion and rj are the rectangular co-
ordinates of the electron in the plane of the orbit referred to the

230 THE MECHANICS OF THE ATOM

nucleus as origin and the major axis as the -axis, we have (fig. 31)

cos

since apart from a constant of integration which can be taken

as zero, 27rw 2 is the angular dis-
tance of the perihelion from the
nodal line, measured in the plane
of the orbit ( 21, p. 137) ; and
averaging over a period of the un-
perturbed motion

FIG. 31.

z=sn sn 7rw 2 +rj cos

In 22 we found for the mean
values (see (21), (23'), 22)

they are the co-ordinates of the
electrical " centre of gravity " of the

moving electron. If we express sin i and in terms of J 1 J a J3 we

get

and

(4) W^-AH^-sin 2^ 2 . . etil-

The angle variables w 2 and w 3 vary ; w 3 Q varies in a cyclic
manner and J 3 remains an action variable of the perturbed motion.
w 2 is consequently the only non-cyclic co-ordinate in the averaged
perturbation function, and we obtain as the only new action variable

It may be found as a function of W^ 3 1 Jj , and J 3 =J 3 from equa-
tion (4). On evaluating the integral, W : and hence W is found as a
function of the action variables.
We write for brevity :

and

we have then

SYSTEMS WITH ONE RADIATING ELECTRON 231

-
dx

where

2 =m
J

'-lA 1 -;

If we calculate r- 2 - from the last relation we find
dx

/ r R

L =dl~-

dw 2 sin 3 27TW 2 /B 1

da? 47rC cos 2nw 2 Q \x 2 Ay

^ 2 \/AC.(o; 2 -AB)

thus our integral becomes

Since the integrand is a rational function of x and the root of an
expression quadratic in x, the integral may be evaluated by the
method of complex integration. We find (c/. (9), Appendix II) :

thus

T / T TO TO

U o 1 1/1 t/ o tl 1

O \ ^ "

W x may be calculated from this ; we find (on putting J 1 =J 1 ,

and, if we express a in terms of J A by (10), 22,

2 3EA 2 _ _

This equation becomes equation (13), 35, if we put

232

THE MECHANICS OF THE ATOM

We show subsequently that our present considerations lead to the
same range of values for J e as found previously. Once again we have
the quantum conditions (15), 35, and the energy equation (16),
35.

We now examine the secular motions caused by the electric field.
The perihelion of the orbital ellipse alters its position relatively to
the line of nodes, and the latter itself moves uniformly about the
axis of the field. It follows from (5) that two periods of the peri-
helion motion occur during one revolution of the line of nodes.

This motion of the perihelion, together with its accompanying
phenomena, can best be studied by referring to the curve representing
the motion in the (w 2 , J 2 )-plane (fig. 32). Its equation is, by (4),

(7)

v 1 (j>

for shortness, we write here

3eEa ll (J 1 ) 2

(using (9), 23, o n (Ji) a /A 2 has been substituted for a in (4)). It is
symmetrical with respect to the straight lines w 2 Q =% or w 2 =f . If
W x 0, either w 2 Q is or J, or J 2 has one of the values J^ or J 3 .

For W 1 <0, w 2 can no longer have
the values or ^, or J 2 the values
Ji or J 3 ; the curve is confined to
the interior of a rectangle bounded

When | Wj | is sufficiently small, w 2
can only depart from the immediate
vicinity of or when J 2 lies close
to J^ or J 3 . The representative
w/ curve lies close to the rectangle
mentioned and passes over into the
circumference of the rectangle for
Wi0. The curve becomes less extended for larger values of | W x |,
until w 2 Q can assume only such values as lie in the neighbourhood
of J (sin 27rw 2 Q =l) 9 and, finally, only this value itself; the curve
contracts in this case to a point (cf. fig. 32). The same holds for
W!>G, only the limiting point is at w 2 =f . For a given value of
W 1? the reversal points for w 2 are situated at those places where
sin 27TW 2 is a minimum, or where

FIG. 32.

SYSTEMS WITH ONE RADIATING ELECTRON 233

is a maximum, J^ and J 3 , and thus

Ji
being constant. Now the function

(1-sXl-y),

where

xy const.,

will be a maximum if x=y. Thus w 2 Q reverses when

(8) (J 2 ) 2 -JiJ3 ,

or when J 2 is the geometric mean of J^ and J 3 .

The secular motions of the orbit under the influence of the electric
field are thus as follows : while the line of nodes revolves once, the
perihelion of the orbital ellipse performs two oscillations about the
meridian plane perpendicular to the line of nodes. For a transit
through this meridian plane in one direction, the total momentum
J 2 /27r is a maximum and consequently the eccentricity is a mini-
mum ; for a transit in the other direction the eccentricity is a
maximum. Since the component J 3 /27r of the angular momentum
in the direction of the field remains constant, the inclination of the
orbital plane oscillates with the same frequency as the eccentri-
city. It has its maximum or minimum value when the perihelion
passes through the equilibrium position, and it assumes both its
maximum and minimum value twice during one revolution of the
line of nodes. The major axis remains constant during this oscilla-
tion of orbital plane and perihelion (since J^ remains constant) ;
the eccentricity varies in such a way that the electrical centre of
gravity always remains in the plane

In this plane it describes a curve about the direction of the field ;
since the inclination and rotation of the line of nodes have the
frequency ratio 2:1, the curve is closed and, in the course of one
revolution, the electrical centre of gravity attains its maximum
distance from the axis twice and also its minimum distance twice.
We shall show later ( 38) that the electric centre of gravity executes
an harmonic oscillation about the axis of the field.
We still have the two limiting cases of the perihelion motion to

234 THE MECHANICS OF THE ATOM

consider. If the representative curve in the (w 2 , J 2 )-plane has con-
tracted to a point (the libration centre), then J 2 =0 and J 3 =Ji+J
is an integral multiple of h. The orbital ellipse has a constant eccen-
tricity, and constant inclination, and is spatially quantised. The
major axis is perpendicular to the line of nodes (since w 2 = J), and
the latter revolves uniformly about the direction of the field. To our
approximation it is not a special state of motion singled out by the
quantum theory, since J 2 is not fixed by a quantum condition. The
necessity for fixing J 2 would be arrived at only by a closer approxi-
mation in calculating the energy.

In the other limiting case, W^O or J 2 =J(Ji J 3 ), where the
curve in the (J 2 , w 2 )-plane coincides with the perimeter of the
rectangle, the motion is rather complicated. The line of nodes
revolves uniformly. In a certain phase of the motion the orbit is
a circle (J 2 =Ji), whose configuration is determined by J 3 and J x .
This circle changes gradually into an ellipse, whose perihelion lies in
the line of nodes ; the orbital plane is orientated perpendicular to
the field during this process. Certainly in this configuration the
direction of the line of nodes is indeterminate ; but if we define it
by continuing the uniform motion which it had previously, the
perihelion lags behind the line of nodes until the separation is TT.
At this stage the orbital plane changes its orientation once more
and the orbit gradually becomes a circle again. When it is a circle,
the position of the perihelion is indeterminate. We can deduce,
however, from the representative curve that it lies once again in the
line of nodes when the eccentricity again increases and the path
once more becomes orientated. During one revolution of the line
of nodes the orbit twice becomes a circle.

The range of values of J e or n e is found by the following considera-
tion. J 3 =J 3 is positive and at the most equal to J x . J 3 can never
become zero, for otherwise J 2 would execute a libration between
J! and Ji, as can be seen from (4) ; this would give a limiting
case in which the orbital ellipse would have to traverse a straight
line (Pendelbahn, cf. 21 and 35) and, on account of the incommen-
surability of the periods of revolution in the ellipse and of the
libration, would approach indefinitely close to the nucleus. From

and the relation

evident from fig. 3^, since 3 2 =$3 2 dw 2 by (4'), which is at most
equal to the area of the rectangle in the figure, we find for J e

SYSTEMS WITH ONE RADIATING ELECTRON 235
J 1 <J e <J 1

and

In place of the single quantum state characterised by a single n,
as in the case of the Kepler motion in the absence of a field, we have
the 2n 1 states already mentioned in 35.

38. The Motion of a Hydrogen Atom in Crossed
Electric and Magnetic Fields

Bohr has given another and more illuminating method of cal-
culating the secular motions of the hydrogen atom in an electric
field. 1 Using a similar method, Lenz and Klein 2 succeeded in de-
ducing the effect of the simultaneous influence of a magnetic field
and of an electric field arbitrarily orientated with respect to it.

We reproduce the calculation for the case of an electric field E
and a magnetic field H. The unperturbed motion (E=H=0) has
six independent integration constants ; as such constants we first
choose the components of the angular momentum vector P and of
the position vector f of the electric centre of gravity of the orbit.
Since P and f are always perpendicular to one another, this provides
only five independent quantities ; as the sixth we can choose a
magnitude which determines the phase of the motion ; for this
problem, however, it is unimportant. P and f suffer variations under
the influence of the fields E and H, and we commence by writing
down the differential equations for P and f .

Both the electric field and the magnetic field exert couples on the
electron orbit, and these determine the time-rate of variation of
the angular momentum P. On multiplying the equation of motion
of the electron, viz. :

(1) mi =Ze 2 grad -L -eE+ Jffi]

vectorially by r we get the time-rate of variation of the angular
momentum

p =m[rr] =e[Er] + >[Hr

1 N. Bohr, Quantum Theory of Line Spectra (Copenhagen, 1918), p. 72.

2 The problem was first solved by P. Epstein, Physical. Rev., vol. xxii, p. 202,
1923 ; O. Halpern gave another solution, Zeitschr. /. fhysik, vol. xviii, p. 287,
1923. The method given here was originally given by W. Lenz (Lecture in
Brunswick, 1924, and in more detailed form, Zeitschr. f. Physik, vol. xxiv, p. 197,
1924), and 0. Klein, ibid., vol. xxii, p. 109, 1924.

236 THE MECHANICS OF THE ATOM

The secular component of this motion is found by taking the mean
value over a period of the undisturbed motion ; the electric contri-
bution is

e[Ef].

The magnetic contribution can likewise be simply expressed, if we
introduce the angular momentum P by means of the well-known
vector relation

[r[Hf]]=[H[rr]]+[f[Hr]J

=i[HP]+[rtHr]],
and remember that the time average of

[r[Hr]]+[r[Hr]J=|[r[Hr]j
is zero. We find in this way

2[r[Hr]J = -[HP]
and
(2) P^[ES]+[HP].

The first term represents the couple due to the electric field acting
on an electron situated at the centre of gravity of the orbit ; the
second term corresponds to Larmor's theorem, and signifies an
additional rotation of the vector P about H with the angular velocity

e|H[
2mc'

In addition to the three equations included in (2), we will now
find three others. In the first place the mean value of the per-
turbation energy, taken over a period of the undisturbed motion, is
a constant

Secondly, P and f are perpendicular, so that

(4) Pf=0,

and, thirdly, P and f are connected through the eccentricity. We

have from (23'), 22 (p. 145)

||=fac
and from (8), 22 (p. 141),

SYSTEMS WITH ONE RADIATING ELECTRON 237

where J is the non-degenerate action variable of the motion in the
absence of a field. Elimination of leads to

(5)
where

From (3), (4), and (5) it is possible, with the help of (2), to derive
an equation for r of the same form as (2). If (3), (4), and (5) be
differentiated with respect to the time and the value of P substituted
from (2), we obtain

0=ff+eK 2 P[Ef]=f(f+eK 2 [PE]).
This implies, however, that the scalar products of the vector

(7) f+eK'CPEH^CrH]

with E, P, and f vanish. Since in general these three vectors do
not all vanish nor do they all lie in one plane, the vector (7) must
itself be zero. Consequently

(8) f=eK'[EP] + ^-[Hf].

Our problem is solved when we can solve the system of equations
(2), (8). This is best accomplished by introducing the new vectors

m fx=r+KP

(9) f,=i-KP,

instead of the unknowns P and f. Since f and KP are perpendicular
to one another, the two vectors (9) have the same magnitude which,
by (5), is _

(10) |f 1 |=|f 2 l-V^+K^"==|a.

Further, in terms of f and P the variables f x and f 2 are given by
the equations

238 THE MECHANICS OF THE ATOM

(2) and (8) now become

(12)

Writing for shortness

(13) ^c E ^'

the system of equations becomes

ra=[-w,+w m , f J.
This denotes simply that the vectors f x and f 2 rotate uniformly
about the axes defined by w e +w m =(H/2mc)+KE and w e + w m
(H/2wc) KE respectively with the respective angular velocities
| w m +w e | and | w m W 6 1. At each instant the separation of the end
points of the two vectors is proportional (by (11)) to the angular
momentum of the motion, and half their sum gives the radius vector
of the electrical centre of gravity.

We consider first the case in which only an electric field E acts.
f i and f 2 both rotate with the same velocity about the direction of
the field, but in opposite directions. In the course of a complete
rotation of each of the vectors they come twice into a configuration
in which E is coplanar with them and they both lie on the same side
of E. In this position their difference, and therefore the resultant
angular momentum P, is a minimum, the eccentricity attains its
maximum and the plane of the orbit deviates least from the equa-
torial plane of the field. Between these positions there are two
others where f^ f 2 , and E likewise lie in a plane, but with f l and
f 2 on opposite sides of E. P is then a maximum and the eccentricity
a minimum, while the plane of the orbit has its greatest inclination
with the equatorial plane. While the magnitude of P goes through
two librations during such a revolution, the direction P completes
only one rotation, i.e. the line of nodes of the orbital plane completes
one revolution.

If the motion of the electrical centre of gravity be alone considered,
it may be found directly from the equations (2) and (8) (for H=0).
If (8) be differentiated with respect to the time and P substituted
from (2), we get .

if=e 2 K*[E[Ef]].

SYSTEMS WITH ONE RADIATING ELECTRON 239

This expresses the fact that f is directed perpendicularly to the

direction of the field and that | r | is proportional to the distance of
the electric centre of gravity from the axis of the field, | [fE]| / |E|.
The electric centre of gravity performs, in other words, an harmonic
oscillation about the axis of the field (cf. 37, p. 233).

If only a magnetic field is acting, f a and r 2 rotate in the same sense
about the axis of the field with the same velocity

i.e. the whole system performs a uniform precession (the Larmor
precession) about the axis of the field.

When both fields are acting, the rotations of f x and f 2 occur about
different axes. Thus the simple phase relation, which we had in the
case of an electric field only, between the rotation of the line of
nodes on the one hand and the orbital eccentricity and inclination
on the other, will be destroyed and a much more complex motion
sets in. Special difficulties arise when the two cones described by
the vectors i l and f 2 intersect. If the rotation frequencies are
incommensurable, the vectors TI and f 2 will then approach indefinitely
close to one another, and, therefore, the angular momentum becomes
indefinitely small. If now the frequency of rotation in the ellipse
is incommensurable with the other two frequencies, the electron
approaches indefinitely close to the nucleus. On the basis of the
fundamental principles we have previously used, we should have to
exclude such motions. We shall see later, however, when fixing the
quantum conditions, that such orbits may be transformed adia-
batically into those of the pure Stark or Zeeman effect which we
must allow.

We turn now to the energy of the perturbed motion and the
fixing of the stationary states.

Under the influence of the two fields E and H, an additional
term W x is added to the energy W of the unperturbed motion,
where (cf. (3))

(15) W

2mc

If we express here r and P in terms of f x and f 2 by (11), we get

and, introducing the vectors w and w m from (13),

240 THE MECHANICS OF THE ATOM

(16) W

If we define the frequencies v' and v" by

" / =5-|w.+w m

(17) 2 ;

I/' _ I m \n I
V O I e W l

the energy can be expressed in the form

(18) W^i/J'+i/'J",
where

1 2?r
J/== 2' K' fl l COs(flj

1 2?r
J"= 2 ' 1; ' f 2 ' COS (f 2 ' w ~ w )'

By (6) and (10) we can write this

J' =iJ COS (f 1} W e +W m )

J^iJcoBft.w.-wJ.
Since i/ and i^" in equation (18) are constant, it follows from the form
of this equation that J' and J" are the action variables conjugated
to the angle variables

The periodicity conditions of 15 are all satisfied. The quantities
J' and J" are therefore to be determined by the quantum conditions

J'-rc'A

J'W'A.

This implies a somewhat modified type of space quantisation, since
by (19) :

cos(fi, w e +w w )=2-
n

cosfc, w e -w m )=2-.
n

The quantum numbers n' and n" are thus restricted in this case to

I n n\

the range (-5,5).
\ ^ z/

If the magnetic field H vanishes we have a case of degeneration,
for then f

SYSTEMS WITH ONE RADIATING ELECTRON 241

The old action variable 3 e =3' + J" is then to be introduced in place
of J' and J" and we get

in agreement with the previous results. In a similar way we have,
for a pure magnetic field,

J m =J' J"
and

If we have only a weak magnetic field in addition to a finite
electric field, the axes of rotation of the vectors f x and f 2 have almost
opposite directions. Since the cones generated by these vectors
may not coincide in the case of a vanishing magnetic field (for
this would give P=0 in the Stark effect), they do not intersect in
the case of a weak magnetic field. If, however, we allow H to in-
crease adiabatically, the angles of the cones remain constant and,
finally, a point is reached where the cones meet. A similar thing
takes place when we start with a weak electric field and a finite
magnetic field. The axes of rotation have then very nearly the same
direction, and the cones do not intersect. Nevertheless, by an adia-
batic increase of E, a point is again reached when the cones meet,

It is possible, therefore, to transform orbits which we have hitherto
permitted, and which have been confirmed empirically, into orbits
in which the electron approaches indefinitely close to the nucleus.
At present no explanation of this difficulty can be given. There is a
possibility that the J's need not be strictly invariant for the adiabatic
changes considered in this connection, since states are continually
traversed where (non-identical) commensurabilities exist between
the frequencies (" accidental degeneration," see 15, p. 89, and 16,
p. 97).

39. Problem of Two Centres

The parabolic co-ordinates used in the separation method to
determine the motion of an electron in the hydrogen atom under the
influence of an electric field are a special case of elliptic co-ordinates.
The latter are the appropriate separation variables for the more
general problem of the motion of a particle attracted to two fixed
centres of force by forces obeying Coulomb's law. If one centre of
force be displaced to an infinite distance, with an appropriate simul-
taneous increase in the intensity of its field, we get the case of the
Stark effect ; at the same time the elliptic co-ordinates become
parabolic.

242 THE MECHANICS OF THE ATOM

If the distance apart of the fixed points F x and F a is 2c, the
elliptic co-ordinates of a point f , 77, distant r l and r 2 from the fixed
points, are given by the equations

It is evident from these equations that
(2) >1,

and, moreover, that the surfaces const, are ellipsoids of revolution
with semi-major axis c and foci Pj and F 2 , whilst the surfaces
77=const. are hyperboloids of revolution of two sheets with a dis-
tance 2crj between their vertices, and the same focal points. To
determine a point uniquely a third co-ordinate is required, e.g. the
azimuth <f> about the line FjF 2 .

Taking cylindrical co-ordinates (r, <, z) with F^Fg as z-axis, and
its mid-point as origin, we can write the equations of these surfaces
of revolution

* , ^ _2

+ ~ c

If l-7f

These give the equations of transformation

We shall show that the " problem of two centres " referred to above
is separable in the co-ordinates , 77, <f>. The potential energy of an
electric charge e attracted by two positively charged points is

u=-^+fn

1 V

or, in elliptic co-ordinates :

w u= "o(^

The kinetic energy is

SYSTEMS WITH ONE RADIATING ELECTRON 243

and by the relations (cf. (3))

r=rf

this takes the form
(5) T=

This gives for the momenta conjugate to , 77, <,

. 2 -n 2

P( =mc *t-j^

(6) , . ?n*

^ }

If we express T in terms of the co-ordinates and momenta, and add
the potential energy, we obtain the Hamiltonian function

It will be seen at once that our problem may be solved by separa-
tion of the variables. The three momenta are found to be

where C is an arbitrary constant and
A-C+1=--

We may now proceed to investigate the possible types of orbit,
leaving out of consideration individual limiting cases, and restricting
ourselves to the case of a negative W. We shall A*ot give the method
of proof in detail.

244

THE MECHANICS OF THE ATOM

I. ORBITS WHICH ARE COPLANAR WITH THE CENTRES. 1

In this case p(\ therefore A C-|-l=0 and f=l, r) = l are

roots of the expressions
under the square root sign
(radicand) in (8). We
distinguish the following
cases :

1. The radicand of p$ is
positive for f >1 ; then
performs a libration be-
tween =1 and a value

max*

(a) The radicand of p
FIG. 34. * s positive throughout the

whole interval ~1<^<1.
The orbit lies within the ellipse =~f max (fig. 33).

(b) The radicand of p n has a root in the interval l<rj<l in ad-
dition to the roots 77 1. The orbit is then
contained within a region bounded by the ellipse
~ max an( l a hyperbola 7?^const. (fig. 34). The
case in which two roots occur in the interval
I<rj<l does not arise.

2. The radicand of p^ is negative for > 1 and later
assumes positive values in the interval (f nun , f lnox ) ;
then performs a libration in this interval. In this
case, the radicand of p^ must be positive throughout
the whole interval 1<^<1. The curve is then
confined between the two ellipses = min and f = f maac
(fig. 35).

FIG. 33.

Fio. 35.

II. ORBITS WHICH ARE NOT COPLANAR WITH THE CENTRES. 2

The radicand of p^ is at most positive in an interval ( min , inax ),
which does not extend to 1 : the radicand of p^ is likewise
negative for 17 = 1 and can have two or four roots in the interval
1<^<1. Finally, p^ is not zero and < performs a rotation about
the line of centres. In all cases where motions are possible at all, they
are confined to a ring bounded by two hyperboloids of rotation and

1 For a detailed discussion of these orbits, see C. L. Charlier, Die Mechanik des
Himmels, vol. i, Leipzig, 1902, iii, 1 (p. 122).

* Detailed discussion by W. Pauli, jr., Ann. d. Physik, vol. Ixviii, p. 177, 1922,
ii, 6, and K. F. Niessen, Zur Quantentheorie des Wasserstoffmolokul-Iona (Diss.,
Utrecht, 1922), section 1.

SYSTEMS WITH ONE RADIATING ELECTRON 245

two ellipsoids of rotation, whose axes pass through the centres (figs.
36 and 37). In the case of double roots two of the ellipsoids or
hyperboloids can coincide ; limitation motions can also occur.

The regions mentioned here will be completely filled if the motion
is not strictly periodic. In the two cases I, 1 (a) and (6), this would
involve an infinitely close approach of the moving point to the centres
of force.

Pauli l and Niessen 2 have endeavoured to treat the quantum
theory of the problem of two centres, and to apply it to the hydrogen
molecule positive ion, which consists of two nuclei with charges +e
(i.e. Z l =Z 2 =l) y and one electron. To a first approximation, the
motion of the nuclei can be neglected on account of their large mass.
The first step is to calculate the motion of the electrons when the

FIG. 36.

FIG. 37.

nuclei are an arbitrary distance apart ; the nuclear separation has
then to be determined so that the nuclei are in stable equilibrium for
definite values of the action variables of the electron motion. It has
been found in this way that a configuration of minimum energy (the
normal state) is uniquely determined by these conditions (it is of
the type in fig. 36, the figure being symmetrical for nuclei with equal
charges). Not only can the value of the energy be found in this case,
but the small oscillations of the nuclei, which are brought about by
small perturbations, can also be calculated.

It has been found, however, that the numbers obtained in this
way do not agree with experimental determinations of the ionisation
and excitation potentials. On this account we shall refrain from
discussing more fully this model for H 2 + . At present the reason for
the failure of the theory is by no means clear. We shall see later that
the treatment of atomic problems with the help of classical mechanics

1 W. Pauli, toe. tit.

2 K. F. Niessen, toe. ctt.

,246 THE MECHANICS OF THE ATOM

leads to false results immediately several electrons arc present ; in
other words, whenever we have to deal with a problem involving three
or more bodies. The artificial reduction of a multiple-body problem
to a one-body problem, on the basis of the small ratio of electron to
nuclear mass, is, perhaps, not permissible.

FOURTH CHAPTER

THEORY OF PERTURBATIONS

40. The Significance of the Theory of Perturbations
for the Mechanics of the Atom

IF we glance back at the atomic models dealt with in the previous
chapter, we see that they are all characterised by the fact that the
motion of only one electron is taken into consideration. The results
tend to show that, in such cases, our method of procedure is legiti-
mate or, in other words, that we are justified in first calculating
the motions in accordance with classical mechanics, and subse-
quently singling out certain stationary states by means of quantum
conditions. The problem now arises of how to treat atoms with
several electrons.

At first glance a similar method would appear applicable to this
case, the mechanical many-body problem being first solved, and
the quantum conditions introduced subsequently. It is well known,
however, what difficulties arise even in the three-body problem of
astronomy ; and in the present case things are still more unfavour-
able, the reason being that whereas the perturbing forces which two
planets exert on one another in the problems of celestial mechanics
are extremely small in comparison with the attraction of the sun for
either of them, the repulsive force between two electrons in an atom
is of the same order of magnitude as the force of attraction between
each and the nucleus. Moreover, in astronomical problems it suffices
to calculate the motions in advance for periods of a few hundred
or thousand years ; in atomic theory, on the other hand, only those
multiply-periodic motions can be employed whose course can be repre-
sented for all time by one and the same Fourier series. It appears,
then, that all progress in this direction is barred by insurmountable
analytical difficulties, and so it might be concluded that it is im-
possible from a purely theoretical basis to arrive at an explanation of
the structures of the atoms right up to uranium.

The object of the investigations of this chapter is to show that this

247

248 THE MECHANICS OP THE ATOM

is not the decisive difficulty. It would, indeed, be remarkable if
Nature fortified herself against further advance in knowledge behind
the analytical difficulties of the many-body problem. Atomic
mechanics overcomes the above-mentioned difficulties arising from
the like order of magnitude of all the forces acting, by precisely those
characteristics which distinguish it from celestial mechanics, namely,
the quantum restrictions on the possible types of motion. We shall
show, by a systematic development of the perturbation theory, that
it is only the simplest types of orbits which are of importance in the
quantum theory, and in astronomy these occur only as exceptional
cases and so receive no attention. These quantum orbits admit of
relatively simple analytical description. One might, therefore, pro-
ceed in this way to compute the atoms of the periodic system one
after another.

An attempt has actually been made to subject to the theory of
perturbations the second simplest atom, that of helium, with its one
nucleus and two electrons. The result, however, was entirely nega-
tive ; the discrepancies between theory and observation were much
too large to be accounted for by the inaccuracy of the calculations.
This indicates that there is some basic error in the principles of our
atomic mechanics.

When we set forth these fundamental principles ( 16) we called
attention to their provisional nature ; this is shown in particular
by the fact that the theory introduces magnitudes such as frequen-
cies of rotation, distances of separation, etc., which, in all probability,
are by nature incapable of being observed. Again, the phenomena of
dispersion show that the system is not in resonance with an external
alternating electric field of the frequency (TV) calculated by classical
mechanics, but of the quantum frequencies v which are associated
with the quantum transitions. Finally, in the course of our investi-
gations we have come across several cases where the failure of our
hypothesis has been indubitably established by experiment, e.g.
the appearance of " half " quantum numbers, the multiplets and
anomalous Zeeman effects, etc. The presentation of atomic me-
chanics given here must therefore be regarded as only a first step
towards a final theory, which can be approached only by gradually
eliminating all false trails.

In order to set about this thoroughly, it is necessary to follow
through the method suggested, and to examine the consequences
to which we are led by the application of classical mechanics in con-
junction with the quantum restrictions. We shall therefore give in
this chapter a detailed account of the theory of perturbations, in-

THEORY OF PERTURBATIONS 249

eluding all cases permitted by the quantum theory ; finally, we shall
demonstrate the failure of this theory in the case of helium.

We are of opinion that this will not be labour spent in vain, but
that this broad development of the theory of perturbations will,
together with the negative results, form the foundation for the true
quantum theory of the interaction of several electrons. 1

41. Perturbations of a Non-degenerate System

Even the throe-body problem, to say nothing of those involving
more bodies, belongs to that class of mechanical problems which have
not been solved by the method of separation of the variables, and,
indeed, are hardly likely to be. In all such cases one is compelled to
fall back on methods which give the motion to successive degrees
of approximation. These methods are applicable if a parameter A
can be introduced into the Hamiltonian function in such a way that
for A=0 it degenerates into the Hamiltonian function H of a problem
soluble by the method of separation, provided also that it may be
expanded in a series

(1) H=H +AH 1 +A 2 H 2 + . . .,

which converges for a sufficiently large range of values of the co-
ordinates and momenta.

Problems of this kind are dealt with in celestial mechanics, and
the various methods adopted for their solution are referred to under
the heading " Theory of Perturbations." The additional terms
AH 1 +A 2 H 2 + ... are in fact regarded as a " perturbation " of the
" unperturbed " motion characterised by H .

It is only the multiply-periodic solutions which are of importance
for the quantum theory. The methods which we shall employ for
their deduction in what follows are essentially the same as those
which Poincare has treated in detail in his M&thodes nouvelles de la
Mecanique celeste. 2 By a solution we mean, as usual, the discovery
of a principal function S which generates a canonical transformation,

as as

Pk=^r> "^ST"'

v<lk <Mk

as a result of which the original co-ordinates and momenta are trans-
formed into angle and action variables.

1 The first applications of the theory of perturbations to atomic mechanics will
be found in the following works : N. Bohr, Quantum Theory of Line Spectra, parts i,
ii, iii, Copenhagen, 1918 and 1922 ; M. Born and E. Brody, Zcitschr. f. Physik,
vol. vi, p. 140, 1921 ; P. S. Epstein, Zeitschr. f. Physik y voJ, viii, pp. 211, 305, 1922 j
vol. ix, p. 92, 1922.

2 Three vols., Paris, 1892-99.

250 THE MECHANICS OF THE ATOM

Let us suppose that the unperturbed motion is already known and
assume, for the time being, that this motion is non-degenerate. In
other words, we suppose that there exists no integral relation of the
form

(2) (r^)=r 1 V+... +r,iv =0

between the frequencies v k of the unperturbed motion, either identi-
cally in the action variables J fc or for the special values of the
J fc 's which characterise the initial state of motion.

We now introduce the angle and action variables w k 9 J k of the
undisturbed motion, and consider the Hamiltonian function of the
perturbed motion defined in terms of them. They are still canonical
co-ordinates, but, in general, they are no longer angle and action
variables ; in fact, it is evident from the canonical equations

an an

that J fc depends on time and that w k is no longer a linear function
of time. For A=0, H becomes the Hamiltonian function H of the
unperturbed system, which depends only on the J fc 's :

H (J 1 , J 2 . . .)

Similarly, the angle and action variables of the perturbed system
become those of the unperturbed system for A=0.

To find them, we have to look for the generator S(w, J) of a
canonical transformation

as dS

< 3 > J *%v. "'-af?

which transforms the variables W Q , J into fresh variables w, J, in
such a way as to satisfy the following three conditions (c/. 15) :

(A) The position co-ordinates of the system are periodic functions
of the w k s with the fundamental period 1.

(B) H is transformed into a function W depending only on the
J*'s.

(0) S*=S^w k J k is periodic in the w k s with the period 1.

The rectangular co-ordinates of the system are thus periodic func-
tions of the w k Q '8, as well as of the w k s : in other words, a periodi-
city parallelepiped in the w A -space will be transformed into another
in the w^-space. Apart from an arbitrary integral linear transforma-
tion of the w k s among themselves with the determinant 1, we
have, therefore,

THEOBY OF PERTURBATIONS 251

(4) w k =w k *+ a periodic function of the w k Q 's (period 1).

From this and from (C) we conclude that S ^w k Q J k is also peri-

odic in the w k 's with period 1 . Or conversely, taking S 2 w k* Jfc to

be periodic in the w k Q 's with the fundamental period 1, equation
(4), and with it the periodicity of S*, follows from the relation

"nsr;

and further, since from the beginning we have assumed that the
position co-ordinates are periodic functions of the w k 's, they must
also be periodic functions of the w k '&. The conditions (A) and (C)
are thus satisfied.

The function S which we require is now supposed to be capable of
expansion as a power series in A, of the form

(5) S=S +AS 1 +A 2 S 2 + .-..

S is here the generator of the identical transformation and has
therefore (cf. 7, p. 31) the form

(6) S

and Sx, S 2 . . . are periodic in the w fc 's. Conversely, every function
S possessing these properties leads to variables which satisfy the
conditions (A) and (C).

We now substitute the series (5) for S in the Hamiltonian-Jacobi
equation for the perturbed motion

and expand W in turn in powers of A :

W-W (J)+AW 1 (J)+A 2 W 2 (J)+- .

A number of differential equations then result on equating the co-
efficients of like powers of A.
First of all we have

(8) H (JHW (J),

i.e. W is found by replacing J*. by J k in the energy of the unper-
turbed motion. We shall refer to W as the zero approximation to
the energy.

We find the equation for the first approximation by equating the
coefficients of A, viz. :

252 THE MECHANICS OP THE ATOM

in which H (J) and H^w , J) mean that in H (J) and H^w , J)
the J's are simply replaced by the J's, the form of the function
remaining unaltered. The two unknown functions W x and S x may
be determined by means of this equation. Since S x is to be periodic
in the w k Q 's, the mean value of the sum in (9), taken over the unit
cube of the w-space, or over the time variation of the unperturbed
motion, is zero. It follows, then, from (9) that

(10) W 1 (J)=H^T),

where Hj is likewise to be averaged over the time variation of the
unperturbed motion. Hence we obtain for AV^ the same expression
as in the calculation of the secular perturbations, although in this
case we have started out from the totally different hypothesis that
the unperturbed motion is not degenerate. Here again we have the
theorem :

The energy of the perturbed motion is, to a first approximation,
equal to the energy of the unperturbed motion increased by the time
average of the first term of the perturbation function taken over the
unperturbed motion. Apart, then, from the determination of the
unperturbed motion, no new integration is involved in the calculation
of the energy to this degree of approximation.

After calculation of W^J), we have for S x the equation

_
"

where the sign ~ over H! denotes the difference of the function
from its mean value :

We may conveniently refer to H x as the " periodic component " of
Hj. It may be represented as a Fourier series

without a constant term (this being denoted by the accent on the
summation sign). If we imagine S x expressed as a Fourier series

the unknown coefficients B r (J) may be expressed in terms of the
known A T (J) with the* help of (11). It is found that
2V)B T (J)=A T (J),

THEORY OP PERTURBATIONS 253

if we write

(12) Ir^W

dJ k

so that i>fc( J) can be derived from the frequencies v k ( J) of the un-
perturbed motion by replacing J fc by J fc . In this way we find as
a solution of (11)

In addition to this there can occur an arbitrary function which de-
pends only on the J fc 's. We are now in a position to calculate the
influence of the perturbation on the motion to a first approximation.
To this degree of approximation we have for the angle variables
of the motion

from which the w k 's are given as functions of the time. Superposed
on the unperturbed motion are small periodic oscillations, the ampli-
tudes of which are of the order of magnitude A, and are therefore
proportional to the perturbing forces, while the frequencies

8R,
(15) ^=^o +A i

vJk

deviate but slightly from those of the unperturbed motion.
For the J fc 's we have

,,_.

which implies that the J^'s, which in the unperturbed motion are
constant, are likewise subject to small oscillations with amplitudes
of the order of A. So-called secular perturbations do not occur, i.e.
quantities constant in the unperturbed motion do not undergo
changes of their own order of magnitude, such as occur in the case
of a degenerate unperturbed motion (cf. 18).

The necessity for the hypothesis of the non-degenerate character
of the unperturbed motion is evident from (13), since, if this were
not the case, the expression (13) would be meaningless, owing to
certain of the denominators vanishing. We see further, however,
that, even if such degeneration be absent, the denominators can be
made arbitrarily small by a suitable choice of the numbers r l . . . r f ,
and, moreover, this may happen for an infinite-ciumber of terms if the
vary from oo to +00 . In view of this, the convergence of the

254 THE MECHANICS OP THE ATOM

Fourier series (13) appears questionable. We shall return to this
at the end of the paragraph and meanwhile continue the formal
development of the method of approximation.

By comparison of the coefficients, more differential equations may
be deduced from (7), the second (coefficients of A 2 ) and n ih (co-
efficients of A n ) of which we give below :

gH gs 2 i a*H as, aSt

_,Hi 8S X

+ ft "a
2L^ + yJ.

J d3 k dw k Q 7^2!

oo QC!
,_ C?O (7O

Qgv *.-*

aH x as n _!

All the equations have the form

where O n is a function, involving only the results of previous stages
of approximation, and so known at the stage to which (19) refers,
and periodic in the w's, and S n and W n are the required functions.
By forming the time average over the unperturbed motion we find,
in exactly the same manner as in the first stage of the process,

(20)
and

where O n again denotes the " periodic component " of the function

THEORY OF PERTURBATIONS 255

If now we again express the right-hand side in the form of a
Fourier series

in which no constant term appears, integration of (21) gives

(22) S n = y AT e **W).

1 ' n ^2ni(^f

This is a formal solution of the proposed problem.

As an illustration of the method of procedure, we shall carry out
the calculation as far as the expression for W 2 in terms of the Fourier
coefficients of the perturbation function. By (13)

where the A T 's are the Fourier coefficients of H 19 and the term for
which T 1 =r 2 = . . . =T/=0 is absent. The equation (17) for W 2 we
now rewrite as

V 0^1 +V I ^ V ' V '

a,.')

/ 2!

W z is obtained by averaging

T , T *_ T -

* ~ + *

This can be written

or (what comes to the same thing, the case (TI>)=() being excluded)

We shall now consider briefly the question of the convergence of
the series so obtained. The point to be decided is whether the small
values of the denominators (TI>) which must continually recur in
the higher terms of the series, will prevent the series being con-
vergent, or whether the convergence can be maintained by cor-
respondingly small values of the numerators. Bruns l has shown

1 H. Bruns, Aatr. Nachr., vol. cix, p. 215, 1884 ; C. . Charlier, Mechanik dea
Himmda, vol. ii, p. 307, Leipzig, 1907.

256 THE MECHANICS OF THE ATOM

that this depends entirely on the character of the frequency ratios
VI Q : J> 2 ' " *v. He deduced the following theorem : Those values
of the periods v k Q for which the series are absolutely convergent
and those for which even the individual terms of the series do not
converge to zero, lie indefinitely close to one another. Since the
i> fc J s are functions of the J^'s, it follows that the function S, de-
rived according to the above procedure, is not a continuous func-
tion of the jys. Since, on the other hand, this continuity must be
presumed, in order that the Hamiltonian equations should be satis-
fied on the basis of (3) and the equations

Ja=const., w k =t+ const.,
dJk

it follows that our series do not necessarily represent the motion to
any required degree of accuracy, even when they happen to converge.

These results of Bruns have been supplemented by PoincarS's
investigations ; l these lead to the following conditions : Apart from
special cases, it is not possible to represent strictly the motion of
the perturbed system by means of convergent /-fold Fourier series in
the time and magnitudes J k constant in time, which could serve
for the fixation of the quantum states. For this reason it has
hitherto been impossible to carry out the long-sought-for proof of
the stability of the planetary system, i.e. to prove that the distances
of the planets from one another and from the sun remain always
within definite finite limits, even in the course of infinitely long
periods of time.

Although the method of approximation under consideration is not,
in the strict sense of the word, convergent, it has proved very useful
in celestial mechanics. It may in fact be shown that the series
possess a kind of semi-convergence. 2 If they are discontinued at
certain points they give a very accurate representation of the motion
of the perturbed system, not indeed for arbitrarily long periods of
time, but still over what are for practical purposes long intervals.
This shows that the absolute stability of atoms cannot be estab-
lished purely theoretically in- this way. We may, however, ignore
these fundamental difficulties for the time being and carry out the
calculation of the energy, in order to see if our results are in agree-
ment with observation, as is the case in celestial mechanics.

1 H. PoincarS, Method^ nouvelles de la Mecanique celeste, Paris, 1892-99, vol. i,
chap. v.

8 H. Poincare, Joe. cit., vol. ii, chap. viii.

THEORY OF PERTURBATIONS 257

42. Application to the Non-harmonic Oscillator

In the case of one degree of freedom the motion may always be
found by a quadrature (cf. 9) ; the desired result, however, is
often obtained more simply by adopting the method of approxima-
tion described in 41.

Let us take as an example a linear oscillator whose motion is
slightly non-harmonic, a case already treated by a simple method
(12). Here we will consider an oscillator for which the potential
energy contains a small term proportional to the cube of the dis-
placement y, and a term in j 4 which is of the second order of small
quantities. The Hamiltonian function has the form (cf. (3), 12) :

(1) H=H,+AH 1 +A 2 H 2 + . . .,
where

"I fry*

Ho=^ 2 +^K)Y

(2) 2m 2

( '

The angle and action variables of the unperturbed motion, in this
case that of the harmonic oscillator, are given by the canonical trans-
formation with the generator (cf. (16), 7)

) =cD Q q 2 cot
or by

sin

/~J~~

q= * I ^ si

* V 77o>m

cos 27rt0.

If we express H in terms of W Q and J we obtain
H =vJ,

/ J \l

H!=a( ) sin 3

(3) / TO \ 2

H,=6(-i-)sin:

\7ra>m/

as,

We now find W X (J) and - from equation (9), 41 ; this gives

(4) W^H^O,

258 THE MECHANICS OF THE ATOM

From (4) it follows that in this case the deviation from a linear
restoring force does not give rise to terms in the energy which are
proportional to the deviation. On the other hand, to this approxi-
mation, the motion does contain an additional term, which arises
from S x .

In order to find an additional term in the expression for energy,
we must make a second approximation. From equation (17), 41,
we deduce

and

The calculation gives

15 J 2 3 J 2

(6) Wl

4 (27r) 6 (i/) 4 m 3
The term proportional to a 2 is in agreement with our previous result

(9), 12.

Finally, we can deduce from (5) the effect on the oscillation of the
deviation from a linear restoring force. We find that

and

\a 1 2 J \t

(sin 3 2<Trw Q cos 27rt0-f 2 cos
/

By solving the first equation for w and substituting the values of
w, J in

the result (11) of 12 is arrived at by a simple calculation :

(3+C 8 4WW) "

THEORY OF PERTURBATIONS 259

As an example of a more complicated case, we may indicate the
method of calculation applicable to a spatial non-harmonic oscillator
consisting of any number / of coupled linear non-harmonic oscil-
lators. 1 Its Hamiltonian function is

(9) H=H +AH 1 +A 2 H 2 +...,

where

/ v
(10)

kjl kjlm

here we make the convention that different suffixes j, k, I . . . in the
same product always signify different numbers of the set 1, 2, . . ./.
The coefficients have, of course, the same symmetrical properties as
the products of the q's which they multiply.

We shall assume that the v k 's are incommensurable. Introducing
the angle and action variables w, J of the unperturbed motion, we
have

and in TL 19 H 2 we have to substitute

^ . , (* rw~ , \

qk=Qk sm <fc I Qfc= /v / 5-, 0*=27r?V ).

\ V Trco^m ^ /

Since H x is a polynomial of odd degree in the q k s 9 it follows at once

that

(11) W^H^O.

To calculate W 2 we have only to find the Fourier coefficients A T of Hj.
In order to obtain H x in the form of a Fourier series we make use
of the identity

4 sin a sin ]8 sin y= sin (a+j3+y)+sin ( a+]8+y)

+sin (a j8+y)+sin (a+j8 y),

1 M. Born and E. Brody, Zeitachr. /. Physik, vol. vi, p. 140, 1921.

260 THE MECHANICS OF THE ATOM

we find :
(12) H 1 ==

If this be arranged as a Fourier series

(13) H 1= 2X sin (^)=
where

(14) A T =I(B T -B_ r )

the following values are found for the coefficients :

W* fo=i al l othcr T ' 8 zcr ).

B _=.

(Tfc"3, all other T'S zero),
(^^2, Tj-^1, all other r's zero),
(T*=2, Tj= l, all other T'S zero),
(T*=T^=T,=!, all other T'S zero),
(T*=^f=ljT|= 1, all other T'S zero),
(in all other cases).

The terms with like combinations of the T'S (e.g. ^=^=^=1 for
(k, j, J)=(l, 2, 3) and (1, 3, 2) and (2, 1, 3), etc.), are already grouped
together here.

From |A T | 2 =A T A_ T =4(B T B_ T ) 2 it follows that :
(15)

all other T'S zero),
(|T*|=S f
all other T'S zero),

all other T'S zero),

,Q l (|T t |=|T,|=|T,|=l,

all other T'S zero),
(in all other cases).

By (23), 41, \rohave :

THEORY OF PERTURBATIONS 261

aj, ' 33,

+v?

**w-*i '

The quantities Q fc 2 are of the first order in the J's, the quantities A
are of the third order, and so W 2 is quadratic in the J fc 's. The total
energy may therefore be written

(17) W=2 **J*+i 2 **J J*

k kj

The v k , Q may be calculated from (16).

It will be seen that the method fails even to this degree of approxi-
mation if one of the following commensurabilities occur :

that is, if one frequency of the unperturbed system is twice one of
the others, or is equal to the sum of two others.

The formula (17) finds an application in the theory of the thermal
expansion of solid bodies x and in the theory of the band spectra
of polyatomic molecules. 2

43. Perturbations of an Intrinsically Degenerate System

As we have seen, certain denominators in the terms of the series
of 41 will be zero if an integral linear relation exists between the
frequencies V Q of the unperturbed system, and so the method is not
applicable.

We consider next the case of " intrinsic " degeneration, i.e. we
assume that a relation

between frequencies v of the unperturbed motion is true identically
in the J's. In this case the angle and action variables w k Q , J k Q can
be transformed in such a way that they can be separated into non-
degenerate w 's and J a 's, and degenerate w p 's and J p 's (v /) =0)

1 For literature on this subject, see M. Born, Atomtheorie des fasten Zustandes,
Leipzig, 1923 ; also Encykl d. math. Wiss., v, 25, 29f.

a M. Born and E. Huckel, Physikal Zeitschr., vol. xxiv, p. 1, 1923; M. Born and
W. Heisenberg, Ann. d. Physik, vol. Ixxiv, p. 1, 1924.

262 THE MECHANICS OF THE ATOM

(a=l, 2 ... s i p=s+l . . . /). H depends then only on the J a 's
(15, p. 91).
We might now try

On substituting in the Hamilton- Jacobi equation (7), 41, equation
(9) would again result ; but in averaging subsequently over the
unperturbed motion, H 1 (^, J) would remain dependent on w*.
We cannot therefore apply the method without further considera-
tion. The deeper physical reason for this is that the variables W Q ,
J, with which the angle and action variables w 9 J of the perturbed
motion are correlated, are not determined by the unperturbed
motion ; on account of its degenerate character, other degenerate
action variables, connected with the J p 's by linear non-integral
relations, could be introduced in place of the J p 0> s, by a suitable
choice of co-ordinates.

Our first problem will therefore be to find the proper variables

w p , J p in place of the w p 's, J p 's, to serve as the limiting values in
an approximation to w p , J p . For this purpose we make use of the
method of secular perturbations already discussed (cf. 18). It

consists in finding a transformation w J ->wJ such that the first
term of the perturbation function, when averaged over the unper-

turbed motion, depends only on the J's. We assume at the start
that Hj is not identically zero ; we shall return later to the case
where it vanishes identically. We have now, as before, to solve a
Hamiltonian- Jacobi equation

(1) H^j.o; V,J p )=W 1 (JO).

We have considered this problem in detail in 18. If the equation
(1) is soluble by separation of the variables, we obtain new angle and

action variables w k g , J 7c . If the generator of the transformation is

we have

,.=}.;

We now introduce' w k J k into the Hamiltonian function of the
motion :

THEORY OF PERTURBATIONS 263

(2) H=H (J )+AH 1 (w A! ) J^+A'H^', j.)+. . .

and, as in 41, try to find the generator S(w k Q , 3 k )
S=S +AS 1 +A 2 S 2 +. . .

of a canonical transformation, which transforms the w fc 's and J fc 's
into angle and action variables w k , 3 k of the perturbed motion.
This again leads to the equations (9), (17), and generally (18) of

41, if, instead of w k Q , J fc we again write w k , J fc .
The solution takes a somewhat different form, since the quantities

- vanish. If we solve equation (11) of 41 :
0J,

where H 1 =H 1 Hj is the periodic component of H l5 there remains
in Sj an indeterminate additive function Rj which depends on the
jys and also on the w p 's but not on the w a 's. We shall determine
this in the course of the next approximation. Sj now takes the form

where 8^ can be found by solving (3).

If this be substituted in equation (17), 41, for the next approxi-
mation

y a y

( ' *dJ a 8w tt + ~2 dJ k dJi 8w k <>

+H 2 =W 2 (J)

all the terms containing Sx can be taken as known ; the terms in
Kj are not yet known, so that (17), 41, takes the form

O being a known function. It should be noticed that the coefficients
of the quadratic terms in the differential equation differ

from zero only if both J fc and J, belong to the J a 's.

From equation (6), W 2 (J), I^and a part S 2 of S 2 may be deter-
mined. Indicating mean values over a unit cube of the w a -space
by a single bar as before, and mean values over a unit cube of the
whole w^-space by a double bar, we have

further

264 THE MECHANICS OF THE ATOM

_ P f

where O = O O. This equation is of the same type as (3) and may
be solved in an analogous manner. Finally we have also

(9)

( }

We can now write
(10)

and determine S 2 as a function of w k Q , J fc from (9) ; R 2 is a function
of w p , Jfc, which so far remains undetermined.

The process may be continued ; the next step determines W 3 (J),
R^Wp , Jfc) and a part S 3 of S 3 , etc. The final result is again a
series for the energy

(11) W=W (J a )+AW 1 (J /c )+A 2 W 2 (J fc )+' .

These considerations provide a justification for our previous
method of determining the secular perturbations ( 18) by regarding
them as first approximations in a method of successive approxima-
tions. The higher approximations lead to periodic variations of the
w k Q 's and J fc 's, whose amplitudes are at most of the order of magni-
tude of A. Secular motions of w a Q , J a do not occur ; also in addition
to the secular motions of w p , J p which we recalculated in the first
stage of the process, only periodic variations occur having frequencies
of the same order of magnitude and amplitudes proportional to A.

We see further that the terms fL 1 =H 1 H 1 merely contribute to
the energy an amount of the second order in A, although they pro-
duce effects of the first order in the motion of the system.

The method hitherto discussed fails if

H^O

identically (in the w p 's, Jjfc 0> s), a case which very frequently occurs.
A more rigorous investigation shows that the secular motion of
w p , J p and the additional energy W 2 follow from the Hamilton-
Jacobi equation, if we substitute in (5) the expression for Sj given
by (3), and average the equation over the unperturbed motion.
The procedure can be continued, the main object being to eliminate
H! altogether from the perturbation function by means of a suitable
canonical substitution. 1
Further special cases can occur, e.g. when the secular motion de-

1 See M. Born and W. Heisenberg, Ann. d. Physik, vol. Ixxiv, p. 1, 1924.

THEOEY OF PERTURBATIONS 265

termined by (1) is itself degenerate, inasmuch as commensurabilities

exist between the quantities -=1. The secular motions of the vari-

ables which are still degenerate to a first approximation would then

have to be found from the second approximation.

44. An Example of Accidental Degeneration

The method of approximation described in 41 can also fail
when the unperturbed system is not intrinsically degenerate, if there
exist relations of the form

(i) 2>*v fc o =o

for the unperturbed motion with those values of the J fc 's which
are fixed by quantum conditions. In such cases we speak of acci-
dental degeneration. The w 7 .'s may then be chosen so that for those
particular values of J*. the frequencies v p vanish (ps+l . . . f)
and the frequencies i> a (a--l, 2 ... s) are incommensurable. In the
unperturbed motion, however, the J p 's are also to be determined by
quantum conditions, as already mentioned. Accidentally degenerate
degrees of freedom are therefore subject to quantum conditions,
intrinsically degenerate are not.

Accidental degeneration is a rare and remarkable exception in
astronomy ; the odds against (1) being exactly fulfilled are infinite.
A close approach to it is found in the case of perturbations of some
minor planets (Achilles, Patroclus, Hector, Nestor) which have very
nearly the same period of revolution as Jupiter. In atomic theory,
on the other hand, where the J /{; 's can only have discrete values,
accidental degeneration is very common.

We may illustrate the most important properties of accidentally
degenerate systems by a simple example. 1

Consider two similar rotating bodies of moment of inertia A, with
a common axis, their positions being defined by the angles fa and
<f> 2 . As long as they do not interact they rotate uniformly. The
angle and action variables are given by

1 M. Born and W. Hcisenborg, Zcitschr. f. Physik, vol. xiv, p. 44, 1923.

266 THE MECHANICS OF THE ATOM

where p l9 p% are the angular momenta. The energy is

If we fix J^ and J 2 by means of quantum conditions, the two fre-
quencies of rotation are always commensurable ; in particular, they
are equal when J^^ J 2 .

Let us now suppose the motion to be perturbed by an interaction
between the two rotators, consisting of a couple proportional to
sin (<i ^ 2 ) > ^ e ener gy is then

(3) H^

where

(4) H^l-cos

and A measures the strength of coupling. In this case we can give

a rigorous solution of the problem of the perturbed motion. If we

carry out the canonical transformation

=< J^ f J, =J,

then

(J0)2_|_/J'0\2

and this expression involves only one co-ordinate w'. w is cyclic,
and consequently J is constant ; suppose its value is J. Since the
determinant of the transformation (5) of the J fc 's is not 1, it follows
that J and J' are not action variables of the unperturbed system.
J can therefore only be fixed by quantum conditions in such a way
that, in passing over to the unperturbed system, J+J' is an integral
multiple of h. Instead of J' we have, in the case of the perturbed
motion, the action integral

(7)
where

167T 2 A

If we put

'(I) Vl -V si

THEORY OF PERTURBATIONS 267

then

(9) J'

In order to obtain the energy as a function of the action variables,
the equation (9) must be solved for k and the solution substituted in
the equation

derived from (8). For k>I, w' executes a motion of libration
within the libration limits

T,
k

and the integral E(&) has to be evaluated over a complete period
between limits sin $=l/k. For k<l, w' Q performs a rotational
motion ; the limits of the integral are and 2?r, and K(k) denotes
the complete elliptic integral of the second kind.

For the purposes of further calculation we have to distinguish
between two different cases :

I. J 1 4 S J 2 ; J' + 0; the unperturbed motion has two unequal

frequencies. W is not zero, and k vanishes with A. For

lOTT A

sufficiently small values of A, the motion of w' Q is clearly a rotation,
and for E(&) we can make use of the expansion

(11)

We find then from (9) :

Jf 1677 Z A

and from (10) :

II. Jf=3f, J'=0, i.e. the frequencies of the unperturbed

motion are equal. We shall then have W =0, the denomi-

nator in equation (8) will be of the same order as A, and for finite
values of Wj, k 2 is of the order of magnitude 1. Both libration and
rotation of w' can occur, and the expansion 11) is no longer valid.
For the larger values of W lf we have k<l, and therefore a rotation ;

268

THE MECHANICS OP THE ATOM

for the smaller values of W x we have i>l, and hence libration
(cf. fig. 38). The libration limits approach one another as W x
diminishes ; for W 1 =0 the curve representing the motion in the
(w' y J')-plane contracts to the libration centre w'=0, J'=0, or
w'=, J'=0; negative values of W x do not occur since, by (7),
J' would then be imaginary. Disregarding the limitations imposed

FIG. 38.

by quantum conditions, all these motions are possible, since W x can
assume a continuous set of values.

The quantum theory requires, however, that J' should be an
integral multiple of h ; moreover, J' is proportional to V\ (by (7))
and must, therefore, be capable of becoming arbitrarily small for
small values of A. These two conditions are fulfilled only by the
value

In the case of a rotation of w' Q this is not possible, and for a libration
it can hold only in the limiting cases w'=0, J'=0, and w'=,
J'=0. Hence in the perturbed motion the two rotating bodies are
exactly in phase. We have only one frequency, but two quantum
conditions.

If all that is required is that the equations of motions shall be
satisfied without the state necessarily being stable, the cases M>' =,
J'-0, and w'=f , J'=0 are also possible.

In any neighbourhood of each of the motions defined by w'=J
and f there are, however, motions of rotation and libration for
which w' takes values widely different from J or |-. For w'=%
or f the motion with a definite phase relation is therefore unstable,
in the mechanical sense of the word. In this case the motions w'= J
and f are also energetically unstable, inasmuch as H is then a maxi-
mum. We shall also meet with cases, however, where the mechani-
cally stable motion is energetically unstable.

These special motions can be very simply characterised by the fact

THEORY OF PERTURBATIONS 269

that they are the only solutions of the equations of motion

du/_8H L ^___ 2H
( ' ~dt W* ~dt ~~^7

for which w' Q is constant and hence for which the bodies rotate with
a constant difference of phase. It then follows from the conserva-
tion of energy

H(J, J',w')=W,

that since J is constant, J' must likewise be constant ; consequently

According to (6) this equation has the solutions

w'=0, 1 i f,
Putting (6) into the first of equations (13) it then follows that

J'=0.

This is our first example of a case in which the selection of a
particularly simple motion as a stationary state, from the mass of
complex mechanical motions, is due entirely to the quantum condi-
tions. We shall see quite generally that the simple motions with
phase relations have a special significance.

45. Phase Relations in the Case of Bohr Atoms and
Molecules

As already mentioned, the accidental degeneration of the un-
perturbed system is a very exceptional case in astronomy. In
atomic physics, on the other hand, it plays an important role,
for firstly, according to Bohr's ideas, a whole set of equivalent orbits
occur in the higher atoms ; and again according to the quantum
theory the periods of rotation of the Kepler motions with different
principal quantum numbers are always commensurable, since they
vary as the cubes of whole numbers.

After the discussion of the example in the foregoing paragraph,
we should expect quite generally, in such cases of accidental de-
generation, that the quantum conditions would enforce exact phase
relations, and consequently particularly simple types of motion.
Since the proof of this for any degree of approximation is somewhat
complicated, and since the necessary mathematical method can only
be given later, we shall indicate here a simpler method by means of
which the phase relations can be found to a first approximation only.

In this section we shall therefore neglect all expressions involving
higher power of A than the first, even, for example, A*.

270 THE MECHANICS OF THE ATOM

If for the moment we disregard the presence of intrinsic degenera-
tions, but assume the existence of several accidental degenerations,
we can choose the angle and action variables w k , J k (i=l, 2 . . ./)
of the unperturbed system so that the j> a 's (a=l, 2 . . . s) differ
from zero and are incommensurable, while v p (p=s+l . . ./) vanish
for the particular values which the J fc 's have in the case of the un-
perturbed motion. We assume therefore that an (/ s)-fold acci-
dental degeneration exists.

We may write (with an alteration of suffixes from those used
previously) the Hamiltonian function in the form

(1) H=H (J fc )+AH 2 K ,J. )

and endeavour to represent the energy constant as a series of the
form

(2) W=W (J*)+AW 2 (J fc ).

If, as before, we made the assumption

we should obtain for S 2 expressions in which denominators occur

which vanish for A 0, i.e. S is no longer an analytic function of

A at A=0. Now Bohlin 1 has shown that a series in increasing

powers of A/A of the form

(3) 8=S +VAJS 1 +A8 a + . . .

is what is required. Here again (cf. 41)

and Sj, S 2 are periodic in the w k 's (period 1). If - - be substituted

vW k

for J fc in the Hamiltonian function (1), we obtain an expression of
the form (2) if the equations

(4o) H (J)=W (J)

(4l)

<*) 2.

are satisfied.

aS 2 1

1 K. Bohlin, "t)ber eine neue Annaherungsmethode in der Storungstheorie/*
Bihang till K. Svenska Vet. Akad. Handl., vol. xiv, Afd. i, Nr. 5, 1888 ; see also,
for example, H. Poincare, Methodes nouvelles, vol. ii, chap, xix, and C. L. Charlier,
Mechanik des Himmels, voj. ii, p. 446. The application to the quantum theory is
due to L. Nordheim, Zeitschr. f. Physik, vol. xvii, p. 316, 1923 ; vol. xxi, p. 242,
1924.

THEORY OF PERTURBATIONS 271

W is found from (4 ). Since Sj is to be a periodic function of
the w k Q 's, it follows from (4^ that

as,

the quantities - - remain, however, indeterminate. By averaging
Q

over the unperturbed motion (that is, over the w a Q9 s only) we obtain
from (4) :

(5) M-*. fc .-,+.... A

(Suffixes p and a both refer to accidentally degenerate variables.)
This is a partial differential equation of the Hamilton- Jacobi type.
It does not admit of integration in all cases, and the method fails,
therefore, for the determination of the motion for arbitrary values
of the JVs. We can show, however, as in the example of 44, that
the motions for which the w p 's are constant to zero approximation,
and remain constant also to a first approximation, are stationary
motions in the sense of quantum theory.

We shall now demonstrate this for one accidentally degenerate
degree of freedom, the last (/). In this case equation (5) has the
form

This differential equation of the Hamilton- Jacobi type for one degree
of freedom can always be solved by the method of quadratures and
we find

2! 8J/
The constant of integration must satisfy the condition that

is an integral multiple of h. From this it follows, according to
whether wf performs a rotation (fdw f *=l) or a libration (fdw f *=0) 9

272 THE MECHANICS OF THE ATOM

The integrand ^ is never negative along the path of integra-

OWf

tion. Hence in the case of rotation we must have

for all values of w f 9 i.e. H 2 is totally independent of w f . It follows,
of course, that with this approximation, nothing is known about
t0/. In the case of libration, J/ must decrease to zero with VX,
but since on the quantum theory J/ must be an integral multiple of
h, it follows that J/=0, i.e. the integral is to be taken over an
infinitely short section of the (w f , J/J-plane ; the libration con-
tracts therefore to a point. Since w f Q is now constant during the
motion, the perturbed motion has only/ 1 frequencies, and has
therefore no higher degree of periodicity than the unperturbed
motion.

The value which wf has for the motion must be a double root of
W 2 H 2 (w/) ; it must therefore satisfy the equations

(9) ' W 2 =H 2 )

and _

(io) m,

3w f Q

The fact that wf can only have certain definite values, namely,
the roots of (10), signifies a phase relation in the motion of the
system.

If the motion determined in this way is to be actually the limiting
case of a libration and only if this is the case will it be stable the
radicand of (6) must be negative in the neighbourhood of the root
w f , i.e.

2! 0J/

must have a minimum. If the latter condition is not fulfilled the
equations of motion

will still be satisfied, but in any immediate neighbourhood of the

THEORY OF PERTURBATIONS 273

solution with constant values of w f Q and J r there will be solutions
of the equations of motion for which the co-ordinates differ widely
from these constant values. The motions determined by (9) and
(10) are thus mechanically unstable.

In the case where - is positive (as in the example of the two
dJf

rotators, 44), the mechanically stable motion has the smallest value

of H 2 . If, however, is negative (this case occurs in atomic

U<J f

mechanics), the mechanically stable motion has the largest value
of H 2 , and the mechanically unstable the smallest. As yet we
are unable to decide whether only the mechanically stable motions
are permissible for stationary states. If only the stable motions
are permitted it can so happen that the perturbation energy H 2 is a
maximum, as opposed to static models where the energy is always
a minimum. If mechanically unstable motions be also allowed
(on the quantum theory their neighbouring motions are not allowed
as they do not satisfy quantum conditions) it may happen that the
normal state (state of minimum energy) is included among them.

In order to illustrate this behaviour, consider two electrons re-
volving in circular Kepler orbits (it is immaterial whether they
revolve about the same nucleus or about different nuclei) and at
the same time exercising small perturbations on one another. Sup-
pose the position and form of the orbits are fixed, and let us consider
only the variation of the phase of the motion under the influence of
the perturbing forces. The energy of the unperturbed motion is

the unperturbed frequencies are

_ 2 A _ 2A

They are therefore commensurable for each quantum state
(J 1 =w 1 A; J 2 H 2 /i), since T 1 ^ 1 -fr 2 v 2 =0 if r l =n 1 ? 9 r 2 = n 2 3 , and
these T 1? r 2 are both integral. If now, by means of a canonical sub-
stitution, we separate the angle and action variables into those which
are degenerate and those which are not, we have to put
_ _1

*"l rtV'!""! ''&""&!* ~ 1 o

_ 1

274 THE MECHANICS OF THE ATOM

we find

J 2 is the degenerate action variable. If we now evaluate

it will be seen that this expression is negative for all values of J.
Hence in this case the minimum of the perturbation energy H 2
corresponds to the unstable motion.

It will be seen that this result is due to the fact that

where H denotes the energy of the unperturbed Kepler motion. It
will therefore be true generally when electronic orbits in atoms or
molecules exert a mutual influence on one another.

Our considerations show that in the case of one degree of freedom
the motions for which phase relations hold are the only ones possible
according to the quantum theory. The same is true if the equation
(5) is soluble by separation of the variables or can be made so by a
transformation of the w p 's. Equations of the form (6) are then
obtained for the individual terms of S l3 and all conclusions which
follow from this equation can be arrived at in the same manner.

In the general case, it is true, the necessity for phase relations
cannot be proved ; it can, however, be shown that there are per-
turbed motions with the same degree of periodicity s as the unper-
turbed, for which phase relations exist and which are of significance
from the point of view of the quantum theory.

The differential equation (5) is equivalent to a system of canonical
equations

in which K is the expression obtained by replacing the w p 's in the

38
left-hand side of (5) by " co-ordinates " q p , and the - 's by the con-

GWp

jugate " momenta " p p , i.e. :
(11) K

the quantities v pff = -> - being treated as constants. The mechani-
p ff

THEORY OF PERTURBATIONS 275

cal system defined by (11) has, in general, several equilibrium
configurations : for if the values of q p =q p Q be determined from

q p =q p , p p =Q will be solutions of the canonical equations. Also

(12) H) =o ' s i= c nst -

is a particular integral of the differential equation (5), if the constant
value of w Q be calculated from the equations

(13) - Q =0

and

This method fails only if the system of equations (13) is not soluble
for the Wp Q '& 9 i.e. if the " Hessian determinant "

3*5,

vanishes.

The motion of the perturbed system found in this way has the
same degree of periodicity s as the unperturbed motion. The fact
that the constants w p Q can have only certain definite values indicates
the existence of phase relations in the perturbed motion.

The motion is stable only if the auxiliary variables q p of equation
(11) have a stable equilibrium for q p =q p . The neighbouring motions
then consist of small oscillations about the particular motion under
consideration.

The fact that the motions found here satisfy the quantum condi-
tions can be seen as follows. J p is constant and equal to the value
which it has in the case of the unperturbed motion ; in addition,

and so, by (12),

so that J p is also quantised.

V-J, ;

46. Limiting Degeneration

A common characteristic of the two cases of Regeneration which
have been considered is the fact that the trajectory occupies a region

276 THE MECHANICS OF THE ATOM

of less than / dimensions in the co-ordinate space. A third possi-
bility, characterised by the same property, occurs in the case of
multiply periodic systems ; it arises in a number of atomic pro-
blems and leads to typical difficulties in the application of the quan-
tum theory. It is therefore advisable to generalise somewhat the
conception of degeneration and to regard a multiply periodic
motion as degenerate whenever the trajectory occupies a region of
less dimensions than the number of degrees of freedom.

Generalising our previous terminology ( 15, p. 92), we shall refer
to the number of dimensions of the region of the y-space filled by
the trajectory of the motion as the degree of periodicity of the
motion. A motion is thus always degenerate when its degree of
periodicity is less than/.

We shall consider a system whose motion may be found by the
method of separation of the variables when unperturbed. As we
have seen ( 14), in separable systems the trajectory in the j-space
is bounded by a series of surfaces, each of the separation co-ordinates
oscillating backwards and forwards between two surfaces of such a
series. In certain cases these surfaces may coincide. The number of
dimensions of the region filled by the path is then decreased by 1.

This coincidence of two libration limits characterises the third
and, it appears, last possibility of a degeneration.

An example will at once make clear what is meant. Let us take the
relativistic Kepler motion, or, in other words, motion in an ellipse with
a perihelion rotation. In general, the path fills a circular ring and,
therefore, a two-dimensional region, densely everywhere. The bound-
aries for the libration of the radius vector are here concentric circles.

If now we suppose the eccentricity of the initial orbit to decrease,
the two limiting circles approach one another until finally they
coalesce and the orbit becomes a one-dimensional circular orbit.
This does not involve any degeneration in the previous sense of the
word. Actually, however, one angle variable (in this case the longi-
tude of the perihelion) will be indeterminate owing to its geometrical
definition, whilst one of the action variables assumes a limiting value
consistent with being real. For the relativistic Kepler motion, for
instance, we always have J 2 ^ J l9 while here J 2 = J x . We may there-
fore call this appropriately " limiting degeneration."

Other examples are provided by an orbit perpendicular to the
direction of the field in the case of the Zeeman effect and, in the case
of the problem of two centres ( 39), by one which is confined to the
surface of an ellipsoid of rotation, etc. For the purpose of illustra-
tion we shall continue to speak of circular orbits, eccentricities,

THEORY OF PERTURBATIONS 277

etc., although our considerations will have a much more general
significance.

Let the degree of freedom subject to limiting degeneration be
denoted by the separation co-ordinate q f , whose libration limits
coincide. The action variable corresponding to it,

has, obviously, the value 0. If we allow perturbing forces to act
on such a motion with J/=0, the degree of freedom q f will in
general be excited (quite apart from the quantum theory) and the
phase integral J f will differ from zero (in our example the path would
not remain circular).

According to the principles of the quantum theory, J f must be an
integral multiple of h ; since it must be equivalent to J/ for a vanish-
ingly small perturbation, it can have only the value zero. We shall
see that the only solution which satisfies this condition is that for
which J f also remains zero during the perturbed motion. The per-
turbed motion has therefore (as in the case of accidental degeneration)
the same degree of periodicity as the unperturbed motion.

The problem of finding this solution involves a mathematical
difficulty. Eeturning to our example, the perturbation function
contains in general terms which are linear in the eccentricity, that
is in terms in A/J/. 1 Now this can occur quite generally if the un-
perturbed system has limiting degeneration. Terms in l/Vj f then
occur in

dt 0J/'

i.e. in passing over in the limit to the unperturbed motion, the co-
ordinate w f (perihelion longitude) will vary very rapidly and will
have no finite limiting value. The expansions of 41 are now no
longer applicable.

The behaviour of the variables 3 f Q w f Q resembles that of polar co-
ordinates : wf is indeterminate when J/^0. We can, as a matter
of fact, overcome the difficulty which has been mentioned by re-
placing them by the Poincar6 " rectangular " canonical co-ordinates : a

In our previous notation the eccentricity is

V-S

the degree ot treedom subject to limiting degeneration corresponds to the radial
action integral

Jr=Ji Jj.

It is seen at once that, for small J/-, the eccentricity is proportional to V J r .
2 Cf. H. Poincare, Mithodes nouvelles, vol. ii, chap. xii.

278 THE MECHANICS OF THE ATOM

(1)

(the generator of the transformation is %(r) Q ) 2 tan 27rw f ). w f Q can
then be varied in the neighbourhood of J/^0 without and 77
being at the same time subject to rapid variations.

Since in the perturbed motion J/ can deviate but slightly from
the corresponding action variable J/=0, we can consider and 77
to be small. If we substitute the new variables in the Hamiltonian
function, we can expand this in terms of and 77 in such a way that
each coefficient of the powers of A will itself be a series in increasing
powers of and 77.

On account of (1) the expansion of H , and therefore of the energy
function of the unperturbed motion, proceeds in even powers of and
77 only, since it depends only on J f and not on w f . In the perturba-
tion function, on the other hand, linear terms will also occur. The
difficulty previously mentioned may now be formulated analytically.

The circular orbit =0, 77 =0 is indeed an exact solution of the
equations of motion for the unperturbed system, since

d_dH __ ^77 H

~di~~drf =o,v^-o~~ ' ~dt~~~~d^

but it is no longer so in the case of the perturbed motion, since the
perturbation function contains in general terms which are linear in
the 's and r7's.

This consideration indicates a method of solution. If, by a suit-
able transformation, variables , 77 can be introduced, such that all
linear terms in the development of the Hamiltonian function are
absent, we have in f =0, 77=0 a rigorous solution of the equations of
motion for the perturbed system as well. This transformation may
be found by means of a recurrence method, the integration of the
remaining equations of motion being accomplished at the same time.

We postulate then a mechanical problem with the Hamiltonian
function

(2) H=H 1 +AH 1 +AH a +...

WV+ -

WV+ ' ' '
The H WO , a n , b n . . .'s (w=l, 2 . . .) are here periodic functions of the

THEORY OF PERTURBATIONS

w a Q 'a (period 1). When transformed, the expression (2) must take the
form

(3) H=W +AW 1 +AW,+ . . .,
where

(4) W B =V n (J a )+E B

and the R n 's denote power series in , 77 commencing with quadratic
terms.
We assume for the generating function of the transformation

(5) S=2 J a^+T+^+B^-A7 7 ,

i
where

T=AT 1 +A 2 T 2 +. . .

(6) A.=XA 1 +X*A 2 + . . .
B=AB 1 +A 2 B 2 +...

are to be power series in A, whose coefficients T n , A n , B n are periodic
functions of the quantities w^ . . . w/-i-

We find in this way for the transformation formulae for and 77 :

(7)

1 ?=5
and employing these in turn :

The new variables difEer therefore from the old only by terms of the
order of A, so that for A^O we have once again the unperturbed
circular orbits g Q =r)=Q.

If now we carry out the transformation and expand everything
in powers of A, then, to each approximation, there are three, and
only three, functions available T n , A n , B n which are so far unde-
termined and can be chosen so as to satisfy our conditions. Com-
parison of the coefficients of A in (2) and (3) gives

. . . =V 1 +R 1 .

280 THE MECHANICS OF THE ATOM

On making the coefficients of and 77 zero, equations for AJ and 'B l
are obtained, viz. :

Equations of the same type occur very frequently in the theory of
perturbations. To integrate them, A and B are each separated into
a constant part, depending only on the J's, and a purely periodic
component :

The former is found from the equations which result on averaging
(10), viz :

and the latter is then found directly from (10), as in the case of equa-
tion (11), 41. As usual, Vj. and Tj may be calculated as functions
of the J a 's and w a 's, from the terms in (9) independent of and rj.

The higher approximations can be obtained in exactly the same
way. Since in the case of even the second approximation the for-
mula) are already very involved, we shall not write them down.
Finally, it should be noticed that to the first approximation no new
terms occur in the energy W 1? but that this is again obtained by
simply averaging H 10 over w l . . . w f ^ t ; in the second approxima-
tion, however, a whole series of new terms appears.

The final result is an expression for the Hamiltonian function in
the form

(12) H=V(J a )+c(J a )f +d(J a )^+e(J )^+ .

It is the Hamiltonian function of a system in which all co-ordinates
but one are cyclic. The motion may be found in the usual way by
solving a Hamilton-Jacobi differential equation for one degree of
freedom. Since and ^ (like and ^) must vanish with A, we
need only consider small motions, that is, those belonging to a
system whose Hamiltonian function is

(13) cp+jhf+efr.

By means of a suitable homogeneous linear transformation from f , v\
to new variables X, Y it takes the form

(14) CX 2 +DY 2 .

THEORY OF PERTURBATIONS 281

If the quadratic form (13) is " definite," i.e. C and D have the same
sign in (14), the motions in the neighbourhood of X=Y=0 or
f =77=0 are small oscillations of X and Y about this point. The
only motion compatible with the quantum condition

is one in which and T? remain zero. The energy of this particular
state is a minimum, if the quadratic form is positive definite ; it is
a maximum if the form is negative definite.

If the quadratic form (13) is indefinite there are motions in each
neighbourhood of f ==^=0, for which and rj do not remain small.
The only values which satisfy the equations of motion and the
quantum condition are again rj=Q: the motion is, however,
mechanically unstable.

In every case the perturbed motion has the same degree of periodi-
city/!, whilst its energy is

(15) W=V(J a ).

The restriction to simple limiting degeneration is not necessary.
The corresponding considerations and calculations are also valid for
limiting degeneration of arbitrary multiplicity. The appropriate
expression for the generator S is

(16) S=iX'

The result of the transformation is an expression for H in the form

(17) H=v(j a

to which must be added terms of the third and higher orders in ,,
T? P . The Hamilton-Jacobi equation to which this function leads is
not, in general, separable for finite values of f p , 7j p . We need ex-
amine, however, only those motions for which g p and t] p remain
small. By means of a suitable homogeneous linear transformation,
the quadratic terms in (17) may be written in the form

(18) H=V(J.)+2(C,X,+D,Y,).

H is now separable. The only motions compatible with the
quantum conditions are those for which X p , Y p and consequently
p , rj p are always zero.

The conditions for stability are analogous to those in the case of
one degree of freedom. The particular motion ^ p =^ p =0 is stable
when, and only when, the quadratic form i (17) is definite. The
energy is a minimum if it is positive definite.

282 THE MECHANICS OF THE ATOM

To summarise, we may state : For an initial motion possessing
limiting degeneration, the perturbed motion, selected in accordance
with the quantum theory, has the same degree of periodicity 8 as
the unperturbed motion. Its energy is

(19) W=V(JJ.

47. Phase Relations to any Degree of Approximation

In 45 we had to leave unanswered the question whether, in the
case of an accidentally degenerate initial motion, the motions singled
out by the quantum theory have the same degree of periodicity as
the initial one, when the work is carried to any degree of approxima-
tion. The method developed for limiting degeneration now enables
us to answer this question. At the same time the restriction on the
w p Q 's given in 45 will be established by an independent method.

Let us again state the problem : we wish to study those motions
of the mechanical system with the Hamiltonian function

(1) H=H (J jfc o)+AH 1 (J^ ^)+. . . (*=1 . . ./)

which are connected with the accidentally degenerate motions of the
unperturbed system (A=0), i.e. those for which, as a result of the
choice of integration constants, certain frequencies vanish :

OTT

(2) V=aj^= (/>=+!.../).

The path fills a region of only s dimensions (s<f) in the case of
the unperturbed system, since the w p 's are constant.

Let us assume that the perturbed motion is connected with a
certain unperturbed motion for which

It follows from the assumption of accidental degeneration that the
J p 's must have perfectly definite values in the initial motion. The
J p *'s may be determined if equation (2) be solved for the J p 0> s ; they
appear as functions of the J a 's. That w p must necessarily have
definite discrete values in the initial motion is certainly an assump-
tion ; it is also conceivable that the perturbed motion could be
associated with every system of values w p Q of a continuum, but our
argument cannot be applied to this case.

If we assume, therefore, that only certain initial motions are
possible, the J p *'s and w p *'s are perfectly definite functions of the
J a 's ; so far we do net know w p *(J a ), but this will be found in the
course of the investigations. We now introduce new variables

THEORY OF PERTURBATIONS 283

(3) f/Wp'-W. )' V=V-V( J .')-

This may be accomplished by means of a canonical transformation
with the generator

(4) 2. J.'+2[VJp*+fp (V-/)] 5

a p

the transformation equations are

The new J a 's will be equal to the original J a 's, while the w a 's will
differ from the w a 's only by quantities which are constant in the
unperturbed motion ; they retain their character of action and angle
variables respectively. The p 's and 7y p 's tend to zero with vanish-
ing perturbation.

We can now develop the Hamiltonian function with respect to
p> ^p ? thus obtaining

(6) H=H '+AH 1 '+A 2 H 2 '+
where (omitting the bar in w a )

H '=H 00 (J a , J

(7) H/=H 10 (w a > w p *, J (

From (5)

C P ^2!

while the expressions H 00 , H 10 . . . are obtained from H , H! ... in
(1) simply by writing J p *, w p * instead of Jp, w p Q . (6) has now a form
analogous to that of (2) in 46, and may in consequence be dealt
with, to any degree of approximation, by >he method employed
there.

284 THE MECHANICS OF THE ATOM

There is the one difference, that the ^'s do not appear at all in
H '. If, therefore, we make the transformation given by (16), 46,
the equations for determining the A/'s and B/'s (</. (10), 46)
become :

It follows from the second of these equations that the mean value

bjf vanishes.
Finally, the Hamiltonian function is obtained in the form

(9) H=V(J B )+R(J., ,, ,,),

where the expansion of K in terms of p , rj p commences with quad-
ratic terms. For small values of g p , TJ P , which are all that we need
consider, H is separable and gives, as the only solution satisfying the
quantum conditions,

The perturbed motion has therefore the same degree of periodicity
as the unperturbed motion. It is stable (in the ordinary mechanical
sense) when, and only when, the quadratic form in p , rj p in (9) is
definite.
The condition

(10) &7^<>

implies a determination of the w p *'s. For since the mean values of

the 3 %, which are pure periodic functions without a constant

VWgf

term, vanish, it follows from (8) that

This equation implies, however, phase relations for the w p *'s.
It is, in fact, equation (13), 45, since H 10 in the present notation
is identical with H 2 in 45.

In 45 we considered in great detail the case of one accidentally
degenerate degree of freedom ; we may show finally how it fits in
with our general considerations of stability. Equation (5'), 45 (H 2
there is equivalent to H 10 here),

THEORY OF PERTURBATIONS 285

is, to a first approximation, equivalent to

2! dJ, 2
for motions in the neighbourhood of solutions of the equation

, _

If -? is positive, we have in the neighbourhood of the stable
dJ f 2

solution (H 2 is a minimum) a positive definite quadratic form,
whilst in the neighbourhood of the unstable solution (H 2 a maxi-

# 2 H

mum) the form is indefinite. If ? is negative, the form is negative

dJf

definite (H 2 is a maximum) in the neighbourhood of the stable
solution, indefinite (H 2 a minimum) in the neighbourhood of the
unstable solution.

It remains to consider the cases of combinations of different kinds
of degeneration. It has been shown that accidental degeneration and
limiting degeneration can be treated in the same way, and so it is
obvious that they do not interfere with one another. In this case
the number of the , 77 variables is simply increased. In addition
the sole remaining possibility, a combination of intrinsic degenera-
tion with limiting degeneration, does not, as a rule, involve any
difficulty. In such a case the secular motions of the intrinsically
degenerate variables are first of all calculated and then the procedure
of 46 adopted. 1

Special cases, in which, for example, by averaging over the non-
degenerate variables, their dependence on the degenerate variables
disappear (e.g. H 1 =0), must of course be examined separately.

We have now justified the statement made in 40, that the
stationary states are to be found chiefly among the particularly
simple types of motion, which can be calculated by comparatively
easy approximate methods.

With this mathematical tool at hand we shall now proceed to the
calculation of the next simplest atom to hydrogen, that of helium.
We shall show (as mentioned in 40) that the results are not in
agreement with observation ; but quite apart from this, we consider

1 The case in which the degrees of freedom exhibiting limiting degeneration
are at the same time intrinsically degenerate is cfealt with by L. Nordheim,
Zeitechr.f. Physik, vol. xvii, p. 316, 1923.

286 THE MECHANICS OF THE ATOM

that working out this example is a necessary preliminary to any
attempt to discover the true principles of quantum mechanics.

48. The Normal State of the Helium Atom

According to 32, two one-quantum electron orbits are present in
helium in its normal state. Our problem is to investigate their
possible arrangements in the atom.

We shall take the unperturbed motion to be one in which the
electrons are only subject to the action of the nucleus, of charge
Ze. Let the angle and action variables of the first electron be w l9
w 2 , w 3 , Jj, J 2 , J 3 , and let us distinguish by a dash the correspond-
ing quantities for the second electron. The energy of the unper-
turbed motion is then

where

The perturbation function is the mutual potential energy of the
electrons

(2) AH 1= =: = ,

ft V(x-x')*+(y-y')*+(z-sf)*

where R denotes the distance between the electrons and (x, y, z),
(x f , y', z'), their respective cartesian co-ordinates in any co-ordinate
system with the nucleus as origin.

The expansions of the cartesian co-ordinates as functions of the
angle variables (to be calculated from (26), 22) must now be intro-
duced, to provide a starting-point for the calculation of the perturba-
tions. In this connection, however, there is one point to be borne
in mind. In the unperturbed Kepler motion (without taking account
of the variation in mass) only Jj is fixed by the quantum theory,
whilst J 2 , i.e. the eccentricity, remains arbitrary ; in the relativistic
Kepler motion, J 2 is also to be quantised and, for a one-quantum
orbit, J 2 =J 1 =h. We shall not take account quantitatively of the
relativistic variation of mass, but we shall assume that the initial
orbit of each electron is circular with limiting degeneration Ji=A,
J 2 =h.

The unperturbed system consists therefore of two circular orbits
of the same size. In addition to the double limiting degeneration
due to the circular orbfys, we have also a double intrinsic degenera-
tion, arising from the fact that the planes of the two orbits are

THEORY OF PERTURBATIONS 287

fixed, and in addition we have an accidental degeneration, since the
rotation frequencies of the two electrons are equal.

By the principle of conservation of angular momentum, the inter-
action of the two electrons must leave still one intrinsic degeneration
(the difference of the longitudes of the nodes of the two orbits on
the invariable plane remains zero). The line of nodes, however,
processes uniformly about the axis of the resultant angular momen-
tum ; as long as we confine our attention to secular perturbations,
the latter makes the same angle with the angular momentum
vectors of the two electron orbits. Limiting degeneration also per-
sists in the perturbed motion (by the argument of 46). The
same is true ( 47) of the accidental degeneration. The per-
turbed motion will, however, only be related to those unperturbed
motions for which the two electrons have some quite definite phase
relations.

In this special state the mutual energy of the electrons will have
a stationary value. It is evident, on visualising the motion, that this
will be the case only if the electrons are as far apart as possible at
every instant, that is, if they are always in the same meridional
plane passing through the axis of the angular momentum.

This almost self-evident result may be arrived at analytically.
In this connection we must first of all choose the variables of the
unperturbed motion, so that they can be separated into those which
are degenerate and those which are non-degenerate.

The limiting degeneration

j.-j^o, jY--j 2 '=o

necessitates the transformation (which we shall only give for the
first electron)

COS

In what follows we shall again omit the bars over w and J x :

is then the angular distance of the electron in its orbit from the

line of nodes ; and 77 are zero in the unperturbed motion.

The accidental degeneration requires the following canonical
transformation :

t^tax-toi', Ji'=i(Ii-3Ii'),
or, solved for the new variables,

288
(3')

THE MECHANICS OF THE ATOM
Ii=Ji+J/,

The geometrical significance of w 39 w 3 ' 9 J 3 , and J 3 ' depends on the
position of the co-ordinate system. If we take the (x 9 y)- and (x' 9 y')-
planes in the invariable plane of the system (elimination of the lines
of nodes), J 3 +J 3 ' is the total angular momentum and w 3 w 3 '=$.
Since the energy of the perturbed motion can depend only on the
combination J 3 +J 3 ', we may write

(4)

so that his' J.

In order to calculate the phase
relations in the initial motion we
have to express the perturbation
function (2) in terms of the vari-
ables toi, to/, iti 3 , <t lt |/, | 3 .
A simple geometrical treatment
gives (fig. 39)

FIG. 39.

(5)

X~XQ cos
?/=^ sin

sn
cos

cos
cos i

where X Q and y are the rectangular co-ordinates of the electron in
its orbit (the nodal line is the # -axis) and i is the inclination of the
orbital plane to the (z^)-plane. We have

X =a COS 277^!

y =a sin 27710!,

(6)

For x , y 9 , we have

(7)

The perturbation function is now
(8) AH 1=

where

II 2

THEORY OF PERTURBATIONS

289

(9) = COS 277(111! +W) COS !

-f-sin 27r(itr 1 +ii / ) sin !
= (1p 2 ) cos 477^1^ cos

faj 8 does not appear ; it is a cyclic variable, and Jf 3 , the resultant
angular momentum, is constant.

We must now average the perturbation function over the unper-
turbed motion :

(10) XfL 1 = -^ f 1 tol

and determine the constant value which to/ has in the case of the
unperturbed motion from

This equation takes the form

and is satisfied only if p0, or if for/ =|(^i w^) has one of the values
or | (0 and are equivalent, as they give the same configuration).
p~Q would lead to J 3 =0 ; the two electrons would revolve in the
same circle in opposite directions, and this case must be excluded.
In the case taj'^J the electrons will collide on the nodal line each
period. The only remaining possibility is iti/rr^O, for which the two
electrons pass simultaneously through their ascending nodes. They
then lie at each instant in the same meridian
plane through the axis of angular momentum.

Let us now introduce the quantum conditions.
In the perturbed motion J' remains zero ; f^
is to be put equal to 2A, and for J[ 3 we have
the values 2A, A, or ; correspondingly, p will
be equal to 1, J, or 0. As already mentioned,
the case y=0 can be rejected ; ^=1 gives a
plane model of the helium atom ; p=% gives
a spatial model, in which the normals to the
electron orbits are inclined to one another at
an angle of 120 (fig. 40 shows this case). FIG. 40.

The plane model is the He-model first proposed by Bohr. 1 The

1 K. Bohr, Phil Mag., vol. xxvi, p. 476, 1913.

290 THE MECHANICS OF THE ATOM

two electrons are situated at the extremities of a diameter of the
orbit. The problem reduces to a one-body problem ; each electron
moves in a field of force with potential

r 4r r

It describes a Kepler motion of energy

so that the energy of the whole atom becomes

(11) W=-2cKA(Z-i) 2 .
In the special case of helium (Z=2)

(12) W=-VdM.

This enables the energy to be calculated which is necessary for re-
moval of the first electron, since after its separation the atom must
have the normal state of ionised helium with energy

The energy difference

(13) W taL =

gives the work done in separating the first electron, or the ionisation
energy of the neutral helium atom.

To calculate the ionisation potential 13-53 volts has to be sub-
stituted for the energy cRh of the hydrogen atom ; it follows that

V ton> =28-75 volts.

This value is not in agreement with observation, the method of
electron impact giving the value

(14) V lon> =24-6 volts. 1

Although the motion so found satisfies the equations of motion
and the quantum conditions, yet it is not the limiting case of a
libration and is therefore not stable. Applying the result obtained
in 45 for an accidentally degenerate degree of freedom, the motion
with phase relations is only stable if

has a maximum for it. Here Hj has obviously a minimum and
hence the numerator a maximum, whilst the denominator (as we
have shown in 45) is negative.

_ _ c _ ______ _ _____

1 J. Franck, Zeitachr. f. Phyaik, vol. xi, p. 155, 1922.

THEORY OF PERTURBATIONS 291

This last difficulty alone would not definitely point to the in-
correctness of our model, since it is not known if the ordinary
stability conditions are valid in the quantum theory. The dis-
crepancy between the calculated and observed values of the ionisa-
tion potential shows, however, that the model is not correct.

The spatial model was likewise proposed by Bohr and investigated
in detail by Kramers. 1 Here we shall merely calculate the energy to
a first approximation. The energy of the unperturbed motion is

where R is the Rydberg frequency. The first approximation to the
perturbation energy is, by (10),

l =AH 1= ^ * f
a V2Jo

,
JoV(l+;> 2 ) +(l-

a 47rJ VI sin 2 i sin 2 \jj a *
where K is the complete elliptic integral of the first kind :

o V 1 sin 2 i sin 2 iff

In our case i=- and K=2-157. 2 It follows that
o

W^O-687 - -=l-373cR/*Z.
a

and to this approximation the total energy is given by

W cR/K2Z 2 -l-373Z) :
for Z=2,

(15) W=-5-254cRA.

We cannot expect this first approximation to be very accurate,
since at times the perturbing force attains half the value of the force
due to the nucleus. Kramers has carried out the calculation with
greater accuracy and finds

(16) W=-5-525cRA.

Energy equivalent to l525cRA must therefore be expended to liberate
the electron, and the ionisation potential is 2O63 volts. This is
almost 4 volts too small.

1 H. A. Kramers, Zeitachr. /. Physik, vol. xiii, p. 312, 1923 ; also J. H. van Vleck,
Phys. Rev., vol. xxi, p. 372, 1923.
8 Jahnke-Emde, Funktionentafeln, p. 57, Leipzig and Berlin, 1909.

292 THE MECHANICS OF THE ATOM

In addition, the motion of this molecule is unstable, as may be
shown in the same way as for the plane model.

We find, then, that a systematic application of the theory of per-
turbations does not lead to a satisfactory model of the normal
helium atom. It might be supposed that the failure of our method
was due to the fact that we are dealing here with the normal state,
where several electrons move in equivalent orbits, and that a better
result would be anticipated in the case of the excited states, where
the main characteristics of the spectra are reproduced by the quan-
tum theory in the form used here. We shall now show that this
again is not the case.

49. The Excited Helium Atom

Before proceeding to calculate the excited states of the helium
atom we may mention a few facts about the helium spectrum.
The terms consist of two partial systems which do not combine with
one another. Both are approximately hydrogen-like ; one consists
of singlets, and gives rise to the so-called parhelium spectrum ;
this also includes the normal state. The other component system
yields the orthohelium spectrum, and consists (apart from the simple
s-terms) of very close doublets. The lowest orthohelium term is
(according to its effective quantum number) a 2 1 -term. Since the
corresponding state cannot pass into the normal state with emission
of radiation, it has a particularly long life, or, to use Pranck's expres-
sion, it is metastable. The transition from the normal atom to this
metastable state can be brought about by electron impact. 1

We shall now investigate the highly excited orbits of the helium
atom on the basis of the theory of perturbations, by which we mean
the external orbits which can be occupied by an electron when
added to a helium ion. We shall assume that the orbit of the
first electron in the ion is circular. Our problem is to investigate
those types of orbits for which the inner electron, if unperturbed,
would move in a one-quantum circle.

In this connection it is convenient to choose the reciprocal radius
of the outer electron, or some quantity connected with it, as the

1 J. Franck and F. Reiche, Zeitschr. f. Physik, vol. i, p. 154, 1920. According
to measurements of H. Schuler, Natururissenschaflen, vol. xii, p. 579, 1924, the
spectrum of Li+ likewise shows the two corresponding systems of terms (see further
Y. Sugiura, Jour, de Physique, Ser. 6, vol. vi, p. 323, 1925; S. Werner, Nature,
vol. cxv, p. 191; vol. cxvi, f. 574, 1925; vol. cxviii, p. 154, 1926; H. Schuler,
Zeitschr. f. Physik, vol. xxxvii, p. 568, 1926). Moreover, M. Morand (Comptes
Rendus, vol. clxxviii, p. 1897, 1925) has found a new spectrum of neutral Li which
he ascribes to the metastable state of the Li+ core (corresponding to the lowest
level of orthohelium).

THEORY OF PERTURBATIONS 293

small " parameter " A in calculating the perturbation, for the farther
away the " outer " electron, the more will the motion of the inner
electron resemble the " unperturbed motion.'* We shall take into
account the relativistic variation of mass.

If we denote the polar co-ordinates of the outer electron by r, 6, <f>,
those of the inner electron by r', 0', <', and the conjugated momenta
by Pr JV> the Hamiltonian function of the three-body problem
of the helium type has the form

-
^r* sin 2 02m\ r ' r' 2 r' 2 sin 2 0'

6 2 Z6 2 Z

VV 2 +r' 2 2rr'[cos cos 0'+sin sin 6' cos (< 0'
+relativity terms.

Let us resolve this function into H and H 1} where H is the
Hamiltonian function of the (non-relativistic) Kepler motion of the
inner electron and H! the remaining part of the above expression.

After calculating the unperturbed motion of the inner electron,
we can find the secular motions of the remaining variables by intro-
ducing a new Hamiltonian function, the mean value of H x taken over
the unperturbed motion of the inner electron. The integration of
the corresponding Hamilton- Jacobi equation is again performed by
the methods of the theory of perturbations.

We can decrease the number of degrees of freedom in the problem
by an application of the theorem of the conservation of angular
momentum (elimination of the nodes).

If the p-jlar axis of the co-ordinate system be taken jui the direc-
tion of the resultant angular momentum P=3f 3 /27r, the angular
separation of the line of nodes from a fixed line in the invariable
plane is a cyclic variable conjugate to P. For the other co-ordinates
let us take the radius vector r of the outer electron and the conju-
gate momentum p r , together with the angular separation t/r of the
outer electron from the line of nodes and the conjugate momentum

finally we also require the variables w^, w z ', J x ', J 2 ' of the inner
electron, where (as before) w^, J/ correspond to the principal quan-
tum number, w 2 ', J a ' to the subsidiary quantum number.

294 THE MECHANICS OB 1 THE ATOM

Since the initial motion of the inner electron exhibits limiting
degeneration, it is convenient to replace the variables tu/, w 2 ', J/,
J 2 ' by other variables. We therefore perform the canonical trans-
formation

(2)

) = /

j / _ j f

cos 2nw 2 ' 9

and then omit the bars once again.

We shall now calculate the mean value of H x in these new variables.
At the same time we shall develop H! in terms of spherical har-
monics, i.e. in powers of 1/r, and powers of and 77. We shall stop
after terms in 1/r 3 : it appears that this approximation is equivalent
to taking into account terms linear in and 77.

We have now

(3) W =

and

W,=H

+relativity terms,

where a H stands for the hydrogen radius, and evaluation gives for

and A 2 :

We have neglected terms of degree higher than the first in and 77.
The partial differential equation H^const. is not separable.
Since, however, it may be resolved into terms of different orders of
magnitude, it can be /iealt with by the methods of the theory of
perturbations. Let us put

THEORY OF PERTURBATIONS 295*

(6) H^

where

H 2 = A 2 ~- +relativity terms.
r 3

It is easy to see that the relativistic terms are small compared with
1^2, so that our expansion is legitimate.

We must now introduce into H! the angle and action variables
w l9 w 2 , J l9 J 2 of the unperturbed Kepler motion of the outer electron,
represented by the term U . We shall, however, replace w^ by the
true anomaly ^ which is connected with w l by the equation

J (XCD-i

XQV x7/O \ 2 i I

^ ' * l'"~~" T a 7i ^x-v~ j \a*

(cf. (18), (7'), and (8)-(ll), 22 ; fr here=7r+^ of (18), 22) ; let us
also put ^ 2 =27rM> 2 . If we take Ji=h, which is the only case of
interest, we obtain :

ti-

cos (+-f sin

(Z-l)cEA

a is the Sommerf eld fine structure constant a= -r (c/. 33) ; the

terms proportional to a 2 contain the relativitjr correction for the inner
and outer electrons.

296 THE MECHANICS OF THE ATOM

In order to solve our problem we have to apply the method dis-
cussed in 46.
Let us therefore try to find a function

(10) S^J^+fY+Brf-AxY,

which introduces variables fo^, $ 19 X, Y, such that H! has no linear
terms in X and Y, and is quite independent of to x . The terms T 1?
T| . . . ; A 8 , A, . . . ; B 8 , B, . . . of (10) are omitted, since we do
not require them to this degree of approximation. The transforma-
tion generated by (10) is

The reason why we do not need the function T 1 is that J^ l has no
term independent of and 77.
Writing for shortness

the method leads to the following equations :
(12) o=W

(14) A.+.A-W

We have neglected the terms in (14) which involve and 77. It
follows from (13) that

1 A '

991** *'' W,^ 1
and from this and from (14) by averaging over w a for =?j=0,

(16) &~

Hence we do not need to calculate Aj. The mean values may easily
be found with the help of (8) (cf. 22). We obtain from (9) and (15) :

THEOBY OF PERTURBATIONS 297

whence

3 V^AVAIA |,a-J t t- A t

1-. cos , srn

It follows that

and finally

__.
2 4Z 2 J 1 8

We notice that in averaging over w 1 in ^ 2 and ^2 the dependence
on w 2 has of itself vanished : w 2 is cyclic to this approximation and
J 2 remains an action variable.

The quantum conditions are therefore

t =nh, J 2 =| 2 =M, J,=jA.

The relativistic terms are of no practical importance (we have
taken them into consideration throughout only to show that they
give rise to no difficulties). If we omit them, the energy W 1 =H 1
may be expressed as a Kydberg series formula. It is found that

(20) W-

(20) Wl

where

Writing j~k+p, and expanding in powers of -, the result is

298 THE MECHANICS OF THE ATOM

The total energy of the excited helium atom becomes :

(23) W=

(+8) 2

with Z=2. This solves our problem. 1

The formula (20) must lead to the spectrum of helium. Since p
can have the values 1,0, 1, it must give three systems of terms.
Their Rydberg corrections would be (for Z=2) :

(24)

P= 0: 8 =S7IS'

p=-l: S=

The following table gives the values of 8 for k =2, 3, 4, and below
them the empirical values of 8 :

k=2

fc=3

fc=4

(P= 1

-0-063

-0-029

-0-017

Theoretical 1 p=

+0-014

+0-004

+ 0-002

U=-l

+0-078

+0-034

+0-019

T? i /Orthohelium
Empmcal ^p arhelilim

-0-069
+0-011

-0-003
-0-002

-0-001
-0-001

Comparison of the two shows clearly that the theoretical values
do not agree with the empirical values.

We may therefore conclude that the systematic application of the
principles of the quantum theory proposed in the second chapter,
namely, the calculation of the motion according to the principles
of classical mechanics, and the selection of the stationary states
from these by determining the action variables as integral multiples
of Planck's constant, gives results in agreement with experiment
only in those cases where the motion of a single electron is con-
sidered ; it fails even in the treatment of the motion of the two
electrons in the helium atom.

This is not surprising, for the principles used are not really con-
sistent ; on the one hand the classical differential relation is replaced
by a difference relation, in the shape of the Bohr frequency condition,

1 The general solution pi this problem without restriction to circular orbits
of the inner electron has Seen obtained by M. Born and W. Heisenberg, Zeitschr.
/. Physik, vol. xvi, p. 229, 1923.

THEORY OF PERTURBATIONS 299

in describing the interaction of an atom with radiation, while on
the other hand the classical differential relations have hitherto been
employed in dealing with the interaction of several electrons. A
complete systematic transformation of the classical mechanics into
a discontinuous mechanics of the atom is the goal towards which
the quantum theory strives.

APPENDIX
I. Two Theorems in the Theory of Numbers

(a) THEOREM. If A is an irrational number, two integers r and r'
differing from zero can be chosen so that (r+r'X) is arbitrarily small.

Proof. On the unit distance OE, imagine the distances OT? l9
OP 2 . . . measured out from 0, their lengths being A [A], 2A
[2 A] ...([#] denotes here the greatest integer which is not greater
than x). It follows from the irrationality of A that none of the points
0, P 1? P 2 . . . coincide. Further, since they are all situated on the
unit length they must have a point of concentration P, in the neigh-
bourhood of which there are points P a and P^ T f of the series, be-
tween which the distance is smaller than a given quantity 8. This
separation, however, is given by

and is smaller than an integer by r'A. Let this whole number be
r ; then

|T+r'A|<S.

(6) The trajectory in the space of the angle variables w is a
straight line. Without loss of generality we can choose a point on
the trajectory as origin ; it will then be seen that the direction
cosines of the trajectory are proportional to the frequencies v l9
v% . . . v f . We have then the

THEOREM. If no degeneration is present, then for any given
point in the w-space it is always possible to find an equivalent
point to which the trajectory approaches indefinitely close.

If we confine the trajectory to a single cube, by replacing each
point of the trajectory by the equivalent point in the unit cube, we
can state the theorem in the following form :

THEOREM. The trajectory approaches infinitely close to every
point of the unit cube.

This corresponds ^o the following theorem in the theory of

numbers :

300

APPENDIX 301

If n irrational numbers a l . . . a n and any number b are given, n
integers r l . . . r n can always be found so that

()-6=r 1 a 1 + . +r n a n -b

differs from an integer by an arbitrarily small amount.

We can prove the theorem for the trajectory in the following
way : 1

Let be the origin and OE^ OE 2 . . . OE, unit lengths along the
axes of the (w l9 w 2 . . . ^-co-ordinate system. Let P , PI, P a -
be the points of intersection of the
path, confined to the unit cube, with
the (/ l)-dimensional surfaces bound-
ing the unit cube, which intersect in
OE 1? OE 2 . . . OE f . Let P and be
identical. Since the direction cosines
are incommensurable none of these
points P w coincide ; they have at least
one limit point in each of the bounding
surfaces perpendicular to the axes. In
each of these (/ l)-dimensional sur-
faces, there is therefore an infinite number of vectors P m P m+n ,
whose magnitudes are less than a given number S.

We must be quite clear as to the distribution of the points of
intersection on the bounding surfaces, each of which is perpendicular
to one of the axes OEj .... For this purpose, let us consider any
one of the surfaces, say that which is perpendicular to OE/. Of the
series of points of intersection Pj, P 2 . . ., let P^. be the first which
falls in this bounding surface (a is a finite number, since otherwise
we should have degeneration). We may suppose that the vectors
PwPw+n * n fche bounding surface are drawn from f ff) and so we arrive
at new points of our series, Qi, Q 2

We have now to show that these do not all lie in one (/ 2)-
dimensional space passing through P^. We shall prove this in-
directly, by first assuming it to be true, and showing that this leads
to a contradiction.

The point P^ has the co-ordinates

FIG. 41.

(*=1 ./-I)
in the bounding surface under consideration. For/ 1 other points

1 Appended to the proof by F. Lettenmeyer (Proc. London Math. Soc. (2), vol.
xxi, p. 306, 1923) of this theorem in the theory of numbers.

302

THE MECHANICS OP THE ATOM

P^, Pa. . . . P 9M of the Q-series we have, if P, and the (/I) other
points all lie on a surface of / 2 dimensions,

L v f

vi_r vi~|

"" Vf L ~ Vfj "

or, after a simple rearrangement,

r v /-i"i

- ^!-^
L ^/ J

=0.

Vf __Vf

=0.

Since no integral relation

Tl~+T 2 -+ . . . +T / _ 1 '^+T / -0
Vf V f V f

may exist, apart from the case when all the r's are zero, the co-
efficient of -i must vanish :

If we divide the first row by
we obtain

! 1 and proceed to the limit

=0.

APPENDIX

303

The coefficient of in this expression must vanish. If we divide

"/
first row by x% 1 , and allow x 2 to tend to oo , it will be seen that we

must have

V f

=0.

may continue this process until we arrive at the relation

r~ i

=0.

This contradicts, however, the irrationality of - .

If the points of the Q-series do not all lie in one linear (/ 2)-
dimensional space passing through P^, we can pick out/ 1 of the
vectors F^Q, which form an (/ l)-dimensional (/ l)-edge. If
we again attach all the/ 1 vectors to the end point of each of these
vectors, and continue this process, we can cover the whole (/!)-
dimensional surface of the unit cube perpendicular to OE, with a
net of cells, the sides of which are smaller than 8. Evidently the
same is true for those boundaries perpendicular to the other OE,-.
This shows, however, that the points of intersection of the trajec-
tory fill the bounding surfaces completely, and hence the trajectory
approaches infinitely close to every point of the unit cube.

n. Elementary and Complex Integration

Integrals of the form

, V

where R is a rational function of the given argument, are of frequent
occurrence in our problems. We have to deal with the definite
integral, taken over a libration of x, in calculating the energy as a
function of the J's, and with the indefinite form in calculating, for
example, the angle variables.

The indefinite integration may be performed by elementary
means : if e l and e a ( e i >e 2) denote the roots of the expression under

304 THE MECHANICS OF THE ATOM

the square root sign, this expression takes the form (neglecting the
factor A)

2
on making the substitution

p f

dx 1 2 cos

2
The integral then becomes

sm "" cos -~ cos

which is the integral of a rational function of sin i/t and cos if/, which
in every case may be reduced to the integral of a rational function of
u by the substitution w=tan |0, or alternatively, if the integrand is
an even function of its argument, by the substitution w=tan iff. Let
us consider the following examples :

1. jVa 2 x*dx.

The substitution xa sin i// gives

(1) a 2 j

I cos 2 fidtft= |(l+cos 2iff)d2iff== a 2 -H sin !
4J [_2 4

If x 1

= - a 2 sin- 1 -+va 2 x 2 .
2L a J

The definite integral taken over one libration of x is

(2) $Va 2 x 2 dx=a 2 \ cos 2 if/diff=7ra 2 .

r
-dx.

By the substitutions #=sin 0, w=tan 0, we obtain

f JL ^0=1

J 1 a sin 2 $ J 1 +w 2 (l a) 1 +u 2 '

The integrand can be resolved into partial fractions
11 1 1

a 1+u* a 1

1 a

Hence the indefinite integral is
fl

APPENDIX 305

- tan" 1 uqp tan" 1 ( uVl a) for a <: 1,

a a

1 x. i , Va 1 luVa 1

- tan" 1 w log , for aj> 1,

1 x. i , Va 1 littVa 1

- tan" 1 w-t log r= for a^l,

^ a l^puVal ~~

where, if Vl x 2 be positive, the value H == is to be substituted

Vlx*
j?__. ..

case when a<l the integral over a libration of x is :

JT Vl x 2 f 2tr cos 2 t/r 2?r /

fc di/j= (1 VI a).

1 ax z J o 1 a sin 2 /f

for u.
In the

- C * ,, 7T , -

<Pl - r*= i - rY-d0= (1-Vl-a).
j lax z Jo 1 a sin 2 iff a

If it is only necessary to find the values of the definite integral
J=#R(z, V-Ax*+2Rx-C)dx,

the method of complex integration is usually the most convenient.

If x be represented in the complex plane, the function R can be
pictured on a Riemann surface of two sheets with branch points at
the roots e l and e 2 (e 1 >e 2 ) of the radicand. The path of integration
encloses the line joining the two roots. If it goes from e 2 to e x
(dx>0) in that sheet of the surface where the root is positive, it goes
from 6 X to e 2 (dx<0) in the sheet with the negative root (see, for
example, fig. 42).

The simplest way of evaluating the integral is to distort the path
of integration and separate it into individual contours, each of which
encloses one pole of the function. With the direction of rotation
indicated in fig. 42, J is then equal to the negative sum of the
residues of the integrand in these poles (the residue is 2ni times the
coefficient of l/(xa) in the Laurent expansion in the neighbourhood
of the pole a ; we will use the symbol Res for the residue at the
pole a) :

Let us consider a few types of integrals.
Group 1.

J =

The constants A, B, C are supposed positive. If real roots exist

tive

306 THE MECHANICS OF THE ATOM

real axis. The only possible poles of the integrand are at x=0 and
3=00 . We have therefore

J= -

The diagrams of the original and deformed paths of integration in
this instance are clearly shown in fig. 42, in which the pole 05=00 is
represented as if it were at a finite distance. Outside the range e ly
e 2 on the real axis, the root is purely imaginary, and has the sign
+i from e l to oo , and i from oo to e 2 .

FIG. 42.

We calculate Res> as the Res of the integrand of the integral
arising from the substitution y=l/x ; since in the representation of
the x-surface on the y-surface, the direction in which the path of
integration is traversed remains unaltered, we have

Resoo [x*(V-Ax*+2Bx-Cy]

= -Res |

The root has sign i from l/e 2 to y=oo , and +i from oo to l/e r
(a) a=-l, =+1:

Taking account of the above determination of sign, the expansions
of the integrand necessary for the calculation of the residues at o?=0
and y=0 are

and

respectively. It follows therefore that

Res ao = 27T =

APPENDIX 307

Ji= (j>- V-
(6)

For aj=oo the integral is regular. For 35=0, the expansion of the
integrand is

I/ 1 B

that is

=
CVC

and

xy.2

(6)

B C\- B

CVC
(c) a=-

The integral is regular at #=0. The expansion of the correspond-
ing integrand for y=l/x=0 is

1 T 1 B 1 / B 2 C \ 1

-o =H =VH .(3 = n)v 2 + . ,

y*LiVA. iAVA 2t\ A 2 VA AVA/ J

that is

B 2 C

consequently

T
*~~ V-Ax*+2Ex-CJ I B

(7)

-
VA\A 2 A/

Group 2 :

(

r Vla; 2
a) CD -r ~ dx. We can distinguish two'possible cases.

308 THE MECHANICS OF THE ATOM

1. a<l. The poles of the integrand given by the roots of 1 ax 2
lie outside the path of integration surrounding the zero points 1
of the roots (branch points of the integrand) ; they lie on the real
axis for 0<a<l, and on the imaginary axis for a<0. The integral
is composed of the residues at

x=. I and #=oo .
V a

The root is positive and imaginary on the positive real axis, and
negative and imaginary on the negative real axis ; it is positive and
real on the negative imaginary axis, and negative and real on the
positive imaginary axis. Taking these signs into account, the expan-

sion of the integrand at its poles . commences with

. /

\l &

The residues in both poles are the same, viz. :

The contribution of the contour about x =00 works out to be

Since the root is positive and imaginary for positive real values in
the neighbourhood of zero, the expansion of the function starts with

277

This gives for the required contribution and finally

277., ,- v

2. a>l. The poles ./ -fall in the interval ( 1, +1) of the real

axis and lie therefore inside the path of integration. The integrand
does not remain integrable at them, so that this case must be
excluded.

APPENDIX 309

(&)

with /(s)=(A-*)(s-B), F(aO=/(aO-ACte.

Let A, B, C be positive and real, A>B, and C chosen so that F(x) can
assume positive values. The roots a, j8 of F(x) are then real and lie
between A and B.

The integrand possesses simple branch points at a and ]8 : it
becomes infinite there, but remains integrable. Simple poles lie

FIG. 43.

at A and B. Again, a circulation about x =00 will contribute to the
integral. The signs of the roots are given in fig. 43. In the neigh-
bourhood of A the expansion of the integrand commences with

*Vc X

in the vicinity of B with

The residues axe therefore

Res A =--,

Using the substitution yl/xwQ find

__ p 1-ABy 2
OB* es ~

__ p 1-ABy 2 _ 1 _ "I
OB* es o|_~ (A2/ _ 1)(1 _ By) ' v^-l)(l-By)-ACtJ'

where the root for positive real values of y in the neighbourhood of
zero has the sign +i. The expansion commences therefore with

, and

Hence
(9)

f(x)V$(x)

310 THE MECHANICS OF THE ATOM

In conclusion we will consider one or two other integrals of the
form

V - Ax* +2Bz-C +Xf(x))dx,

where \f(x) represents a correction term. Under these circumstances
the positions of the branch points are not essentially different from
those in the integrals of group 1, and the previous figures and deter-
minations of sign and paths of integration remain the same.

In order to carry out the integration we have to expand the
integrand in powers of the factor A of the correction term, and in this
connection it should be noticed that the expansion must be valid
for the whole path of integration, so that in this case the path of
integration must first of all be suitably deformed. Should new
branch points be added on account of the correction term, they must
be avoided by the deformed path of integration.

The integration may then be carried out by the same process as
before, since for the individual terms only the branch points e l and
e 2 , and the poles ce 0, #=oo occur.

(a) Jt

For sufficiently small values of D the expansion for D=0 holds for
the whole path of integration. Let us restrict ourselves to terms of
the first order in D :

Hence

B 1 BD

(6) J 7 =

=(L / -A+2? - 2. +Dx dx.

JV * *

APPENDIX 311

The expansion of the square root in powers of D yields

Confining ourselves to terms of the first order in D, this leads to

/ E ,-\
(11) J 7 -5

INDEX

(The numbers refer to pages}

a-rays, 13.

Accidental degeneration, 89, 97,
265 et seq.

Action variable, 45.

Adiabatic hypothesis, 54.

Adiabatic invariance (for one
degree of freedom), 52 et
seq. ; (for several degrees
of freedom), 95 et seq.

Alkali metals, spectra, 166 et seq.

Aluminium atom, 194.

Aluminium spectrum, 164.

Amplitudes, correspondence of
the, 60, 101.

Angle variables, 45 ; (for central
orbit), 137.

Angular momentum, 23, 25 ;
(quantisation of the), 104 et
seq. ; (of an electron), 152,
155 ; (of the electrons in
molecule), 110, 118 et seq.

Anomalous Zeeman effect, 168,
212.

Anomaly (eccentric), 142 ; (mean),
138; (true), 295.

A priori probability, 9.

Argon atom, 194.

Arrangement of spectra in series,
151 et seq.

Atomic core, 130, 151, 183, 187.

Atomic number, 13, 177.

Atomic structure, 12 et seq.

Atomic volumes, 196 et seq.

Aufbauprinzip, 183.

0-rays, 12.

Balmer series, 149, 206.

Band groups, 128.

Band-head, 118.

Band spectra, 116 et seq. ; (rota-
tion), 63 et seq.

Band system, 128.

Barium atom, 197.

Bergmanii series, 152.

Beryllium atom, 192.

Bessel functions, 147, 223.

Bohr's frequency condition, 7, 11,
14, 59, 98.

Bohr's fundamental hypothesis,
14.

Boron atom, 192.

Branch (of a band), 118.

Cadmium atom, 197.

Caesium atom, 197.

Calcium atom, 194.

Canal rays, 12.

Canonical equations, 20 et seq.

Canonical transformations, 28 et

seq.

Carbon atom, 193.
Cathode rays, 13.
Celestial mechanics, 247.
Central field, 130 et seq., 151 et seq.
Centre of gravity (electrical), 145.
Characteristic radiation, 174 et

seq.
Chemistry and atomic structure,

180 et seq.
Commensurability of frequencies,

77, 81, 83 et seq., 91 et seq., 97.
Complex integration, 305 et seq.
Conditionally periodic systems,

81.
Conjugate variables, 21.

313

314

THE MECHANICS OP THE ATOM

Continuous X-ray spectrum, 174.

Convergency of perturbation cal-
culations, 253, 255.

Copper (atom), 196 ; (spectrum),
184, 188.

Core of the atom, 130, 151, 183,
187.

Core radii, 187.

Correspondence principle, 8, 9 ;
(for one degree of freedom),
60 et seq. ; (for several de-
grees of freedom), 99 et seq.}
(in the Stark effect), 220 et seq.

Coulomb field, 131, 139 et seq.

Coupling of oscillation and rota-
tion in molecules, 122 et seq.

Coupling of several electrons, 247
et seq.

Crossed electric and magnetic
fields, 235 et seq.

Cyanogen bands, 129.

Cyclic variables, 24 et seq., 102
et seq.

Deformation of the atomic core,

166.
Degenerate systems, 89 et seq., 97

et seq. ; (perturbations of),

107, 261 et seq.
Degenerate variables, 92.
Degeneration (intrinsic), 89, 261

et seq. ; (accidental), 89, 97,

265 et seq.\ (limiting), 275

et seq.

Degree of freedom, 20.
Deslandre's band formula, 64,

116.

Difference quotient, 61, 100.
Double bands, 127.
Dulong and Petit's law, 5, 71.
Dumb-bell (molecular model), 63

et seq.

Eccentric anomaly, 142.
Effective nuclear charge, 153.
Effective quantum number, 157,

164, 184.
Ehrenfest's adiabatic hypothesis,

54.

Einstein's law (for the photo-
electric effect), 7 et seq., 174.

Electric moment of an atomic
system, 60, 100.

Electron, 12.

Electron bombardment, 15, 59.

Elliptic co-ordinates, 241 et seq.

Energy, conservation of (for radia-
tion), 6 ; (in mechanics), 22
et seq.

Energy quanta, 4.

Equations of motion, 17.

External orbits of series electron,
155 et seq.

Failure of classical mechanics,

131.

Fall, free, 40.

Flywheel in top, III et seq.
Foundations of atomic mechanics,

1 et seq.

Frequency, 2, 47 ; (correspond-
ence for the), 60 et seq., 90

et seq.
Frequency condition, 7, 11, 14,

59, 98.
Fundamental series, 152, 154,

166.
Fundamental system of periods,

73.
Fundamental vibration, 61.

Gallium atom, 196.

Generator (of a transformation),

31.
Gold atom, 197.

h (Planck's Constant), 3.
Half -integral k, 168.
Halogen hydrides, 127.
Hamiltonian function, 21.
Hamilton - Jacobi differential

equation, 36 et seq.
Hamilton- Jacobi theory, 17 et seq.
Hamilton's Principle, 17.
Harmonic oscillator, 2, 34, 50, 62 ;

(spatial), 77 et seq., 80, 84.
Heat radiation, 1 et seq.

INDEX

315

Helium atom, 191 ; (normal
state), 286 et seq. ; (excited
states), 292 et seq.

Helium spark spectrum, 150.

Hydrogen atom, 148, 191 ; (in
electric field), 212 et seq. ; (in
crossed electric and magnetic
fields), 235 et seq.

Hydrogen molecular ion, 245.

Hydrogen spectrum, 15, 149.

Hydrogen-type spectra, 147 et seq.

Indium atom, 197.
Integral invariants, 35.
Integration (elementary), 303 et

seq. ; (complex), 305 et seq.
Intensities (correspondence of),

62, 101 ; (in the Stark effect),

220 et seq.
Interaction (between matter and

radiation), 2, 6, 9 ; (of

several electrons), 247.
Intermediate shell, 196.
Intrinsic degeneration, 89, 261 et

seq.

Ionic radii, 186, 187.
lonisation, work of, 154.
Ions, 13.

Jacobi's integral, 23.

Kepler motion, 139 et seq. ; (rela-
tivistic), 201 et seq. ; (in
parabolic co-ordinates), 219.

Kepler's law (third), 142.

Krypton atom, 196.

Lagrange's equations, 18, 20.
Lanthanum atom, 197.
Larmor precession, 210, 236.
Legendre transformation, 20.
Libration centre, 49.
Libration limits, 49 ; (coincidence

of the), 276.
Libration, motion of, 45, 48 et

seq.
Limitation, motion of, 49.

Limiting degeneration, 275 et seq.
Lissajous figure, 77, 81.
Lithium atom, 192.
Lyman series, 150, 206.

Magnesium atom, 194.

Many-body problem, 130, 247.

Mean values for the Kepler
motion, 143 et seq.

Molecular heats (solids), 5, 71 ;
(diatomic gases), 65 et seq.

Molecular models, 63, 110, 117 et
seq.

Molecule, diatomic, 63 et seq., 110,
117 et seq.

Momentum, 17, 20 ; (conserva-
tion of), 25 ; (angular), 23,
25.

Moseley's law, 14, 177.

Multiplicity of spectral terms,
152, 155.

Multiply periodic functions, 71
et seq.

Multiply periodic systems (separ-
able), 76 et seq. ; (general), 86
et seq.

Negative branch (of a band), 118.
Neon (atom), 193 ; (spectrum),

185.

Nickel atom, 196.
Niton atom, 197.
Nodes, line of, 137, 232 et seq., 238

et seq., 287.
Non-harmonic oscillator, 66 et seq. y

125, 257 et seq. ; (spatial), 259

et seq.
Normal state, 154 ; (of the

helium atom), 286 et seq.
Nucleus, 13.

Null branch (of a band), 118, 127.
Null line (of a band), 118, 127,

129.

Occupation numbers, 198.
One-body problem, 123, 131 et

seq. '
Orthohelium, 192, 292, 298.

316

THE MECHANICS OF THE ATOM

Oscillations of molecules, 123 et
seq.

Oscillator (harmonic), 2, 34, 50,
62 ; (non-harmonic), 66 et
seq., 125, 257 et seq. ; (spatial
harmonic), 77 et seq., 80, 84 ;
(spatial non-harmonic), 259
et seq.

Overtones, 61, 100.

Oxygen atom, 193.

Palladium atom, 197.
Parabolic co-ordinates, 212.
Parhelium, 192, 292, 298.
Partition function, 3.
Paschen series, 150.
Pendulum, 40, 49 et seq., 52, 55.
Pendulum orbit (Pendelbahn),

136, 234.

Penetrating orbits, 161, 169 et seq.
Perihelion, rotation of, 139, 158,

205.
Periodic functions (multiply), 71

et seq.
Periodic motions, 45 et seq. ;

(multiply), 76 et seq.
Periodic system of the elements,

13, 181 et seq., 191 et seq.
Periodicity, degree of, 92.
Perturbation function, 107.
Perturbation problem, 107.
Perturbations, theory of, 247 et

seq.
Perturbations of an intrinsically

degenerate system, 107 et

seq., 261 et seq.
Perturbations of a non-degenerate

system, 249 et seq.
Perturbations, secular, 107 et seq.,

261 et seq.
Phase relations, 268 et seq., 282

et seq.

Phase space, 2.
Photo-electric effect, 7.
Planck's Constant, 3.
Planck's law of radiation, 3.
Platinum atom, 197.
Polarisability (of atom core), 167.
Polarisation (of radiation), 62,

101, 106.

Positive branch of a band, 118.

Positive rays, 12.

Potassium atom, 194, 200.

Principal quantum number, 136,
151 ; (actual values of, for
terms of optical spectra),
183 et seq.

Principal series, 152, 154.

Probability, a priori, 9.

Projection (under gravity), 40, 41.

Pyroelectricity, 70.

Quantum conditions, 4, 14 (for
one degree of freedom), 56,
59 ; (for several degrees of
freedom), 98.

Quantum number, 16 ; (effec-
tive), 157, 164, 184; (prin-
cipal), 136, 151 ; (actual
values of, for terms of
optical spectra), 183 et seq.

Quantum transition, 6, 54.

Radiating electron, 131, 151, 175.
Radiation from an atomic system,

60, 100.

Radiation, law of, 3.
Radiation, theory of, 1 et seq.
Rare earths, 197.
Rayleigh- Jeans formula, 3, 11.
Relativistic Kepler motion, 201

et seq.
Relativistic resolution of the H

and He + lines, 205 et seq.
Relativity theory, 17, 19, 24.
Relativity, X-ray and optical

doublets, 207.
Resonator, 2, 62.
Ring models, 180.
Ritz correction, 165.
Ritz spectral formula, 161 et seq.
Rosette orbit, 139, 158.
Rotational motion, 45, 48 et seq.
Rotation bands, 64.
Rotation, heats of, 65 et seq.
Rotator, 26, 63 et seq.
Rotators, two coupled (accidental

degeneration), 265 et seq.

INDEX

317

Kubidium atom, 197, 200.
Rydberg Constant, 148, 151.
Rydberg correction, 160, 165 et

seq., 173, 184 et seq.
Rydberg-Ritz spectral formula,

161 et seq.
Rydberg spectral formula, 160.

Scandium atom, 195, 196, 200.

Secular perturbations, 107 et seq.,
261 et seq.

Selection principle, 138, 152, 180.

Separable systems, 76 et seq.

Separation of the variables, 44, 76
et seq.

Serie> electron, 130, 151.

Series spectra, 131, 151 et seq.

Shell structure of atoms, 176, 182,
191 et seq.

Silicon atom, 194.

Silver (atom), 197 ; (spectrum),
184, 188.

Sodium (atom), 193 ; (spectrum),
164.

Somrnerfeld's theory of fine struc-
ture, 202 et seq.

Space quantisation, 105, 110, 155,
211, 240.

Spatial oscillator (harmonic), 77
et seq., 80, 84 ; (non-har-
monic), 259 et seq.

Specific heats (solids), 5, 71 ; (dia-
tomic gases), 65.

Spectra (of molecules), 63 et seq.,
Ill et seq., 127 et seq. ; (of
the halogen hydrides, infra-
red), 127 et seq. ; (of atoms,
hydrogen type), 147 et seq. ;
(of atoms, non - hydrogen
type), 151 et seq.

Stark effect for hydrogen, 212 et
seq.

Stationary [states, 6, 11, 14, 52,
130.

Strontium atom, 197.

Subordinate series, 152, 154.
Subsidiary quantum number,
136, 151.

Term sequence, 152.

Terms, 152 et seq.

Theory of numbers, theorems,

300.

Titanium atom, 196, 200.
Top (free symmetrical), 26 et seq. ;

(symmetrical, in field of

force), 42 et seq. ; (free un-

symmctrical), 115 et seq. ;

(with flywheel), 111 et seq.
Transformability, mechanical, 54.
Transformation (canonical), 28 et

seq. ; (Legendre), 20.
Transition, probability of, 9, 62.
Two bodies, problem of, 123, 131.
Two centres, problem of, 241 et

seq.

Uniqueness of the J's (in separ-
able systems), 83 et seq. ; (in
general), 86 et seq.

Variation principle (Hamilton's),
17.

Wave number, 128, 148.
Wien's displacement law, 4, 11.
Wien's law of radiation, 3.

Xenon atom, 197.

X-ray spectra, 13, 15, 173 et seq.

X-ray terms, 178 et seq.

Yttrium atom, 197, 200.

Zeeman effect, 168, 207 et seq.
Zinc atom, 196.
Zustandsintegral, 3.

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