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Editor: Professor E. N. DA C. ANDRADE, D$c., PH.D. 



Editor : 
Professor E. N. DA C. ANDRADE 

Uniform with this Volume 


By L. DUNOYER, Maitre de Conferences a la Sor- 
bonne, Paris. Translated by J. H. SMITH, 
M.Sc. 240pp. Illustrated, 
" . . . This book will be welcomed by all in- 
terested in the subject. The ground covered is 
of considerable extent, the treatment is lucid, and 
the volume abounds with practical information." 

Science Progress. 






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





Printed in Great Britain by 






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. 



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., 



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. 


G6TTINGEN, November 1924. 


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 


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. 


GETTING EN, January 1927. 


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 

D. R. H. 























































FIELD 229 



















INDEX 313 




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 



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 . 

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). 


This can clearly also be written 



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 


(%} 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 


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* . 


The summation of this geometric series gives 




1 e 
From this it follows that 


(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 


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


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. 

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 : 


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 

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. 


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 

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.) 


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 
(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 

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 


(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 


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 

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


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 

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 

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 

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. 


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


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 

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. 


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 


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 


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. 


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 

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 

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 

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 


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. 



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 

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- 

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 

17 2 


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 


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

d 0T*_8(-U)_ 

dt dJr k dx k 


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. 


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 


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 


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 

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. 


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 


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 


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 

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 


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

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. 


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 

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 


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). 


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

L=T* U= 
(10) p 


(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 


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), 



(&=2, 3 . . . n) 

it follows that 

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

The three integrals give therefore the principle of the conservation of 

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) 

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- 

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. 

. . .) 


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 

For motion under no forces (U~0). 

(1) H-T-1* 

<f) is therefore cyclic, and consequently 

p= constant, 

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

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 


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 


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 

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 



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 


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. 


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

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* = 


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. 


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 : 


(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 , 


(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 


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 


?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 : 


This function gives 



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 


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 


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 


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 

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


gives, by (1), 


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


it leads to 


y=V2p cos q 

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


(16) V = ~o>j a cotg 



(16') * 

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

mco 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 

and putting 


we get 


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 

and the equations of motion give 


a= constant 

The displacement q will therefore be given by 

sin (cot + ft), 

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 


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 

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) 








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. 


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



du du 

~dv ~dv 


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 


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 


(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 

satisfy the condition 



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 


the following special relations hold, 

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


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 


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 

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 

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 



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 

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 

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

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 


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 


-*.- /~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 

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. 


We find 

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

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



, 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 

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

, 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 



Since x and y are cyclic variables, we put 


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 

so that 

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




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

(13, ' frf<t 


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 



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 



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. 


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). 



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 



(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 


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. 


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 



(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 


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





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 

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 

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, 


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 


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 


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. 


the curves contract on the libration centre =0. For 

we have libration between the limits 


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

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. 





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 


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 


We obtain 

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



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 

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 


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 

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. 


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 

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- 

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 


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 


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 


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 

(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. 


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) 




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 



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 

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 


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 ) 


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 

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 

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. 


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, 


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 

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. 


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 


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. 


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


]. 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 


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, 


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 


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 



and the wave-length 


A=-=0-06a 2 cm. 

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). 

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 




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. 






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- 

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 : 

log 7=0-*, 


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. 


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 


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 


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 


and obtain the following expansion for J : 

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

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


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 : 


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 



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 


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 

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 


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 + 

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. 


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. 



* 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 

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. 


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 


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 . . .) 


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- 


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 ', 

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. 


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. 


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), ( .) : 


sin 2nw z P z = 


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 


(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, 


-|- ( 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 ( 


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) ' 


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 


(16) tf* 


Written as a function of t : 

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

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. 


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 

- 1 
(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 



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 

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 

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 

df k 

We thus have 


=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 

Since <f> is a cyclic co-ordinate, 


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 : 


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 

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 "' ,. , , . / . / * " 


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 

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 



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. 


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 



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 

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 

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 


independent of t and therefore 



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


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 


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. 


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 


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



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 = 


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 



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- 

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 

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 : 



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 



We now divide the transformation (16) in two parts : 

.> J > 

(7 / 


(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 

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 : 


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, : 


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 

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. 



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 


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 


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. 


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). 


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 )] 


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 


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, 



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, 


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 

2 This idea forms the starting-point of the new quantum mechanics. See W. 
Heisenberg, Zeit. f. Phya., vol. xxxiii, p. 879, 1925. 


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 

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. 


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 


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, 

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. 


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 

parallel to the fixed direction. On account of the constancy of , 


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 


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,). 


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- 

_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 


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 


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. 


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 

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. 


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 


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 


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, 

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. 



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 = 


" 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 

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 


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 

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. 


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. 


(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- 



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 


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^ 

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 

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 



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 : 

FD 2 ! cos 0dJ>=27rD 2 - ; 

T m 

if it encloses the positive polar axis : 

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). 


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 


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, 


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 : 




Dcos0= ; 


cos#= , 

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 : 


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 


(19) V=Ve 



(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. 


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 














i -^^^^ 




















... .j 

i | . 


i ~~ 



^L-^ 1 i 


"~ I I 


/i J^ 


Lx4rT ! 






II ! 





^7 1 

i i i 
i i i 







I I 


567 8 9 10 

// /^ 









/ (7 


I I 


O1 2 3 

5 6 


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 


(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 



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 




it ollows that 


FIG. 9. 

This is a generalisation of the formula for the energy of a simple 
rotator, which is obtained by putting 


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. 


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 

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 


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. 


(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 : 


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. 


p x =t*x. 

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 +^+| 



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 


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 

from this we obtain 

J 2 1 

1 47rV 


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 . 


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. 


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 

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 


(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, 


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 


FIG. 10. 


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. 



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. 



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. 


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, 


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 

the momenta : 

and use them to replace r, 9, <f>. The structure of the Hamiltonian 

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 : 


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 . 


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 

finite constant. 


(4) J 

J/) = 

With the help of the substitution 



a e 
the second integral takes the form 


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 


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 


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. 



r c 

*- -7 




2 "J 

J- a sin 2 fl 

The two integrals in dO may be evaluated. We have 


dO . cos 6 

;_~ = sin" 1 r+const. 

' T 2 . 

sin 2 (9 



sin 2 6 


= 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. 


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 



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 


dw UL 


Also since the angular momentum is J z /27r 

we eliminate dt and derive the differential equation of the orbit 




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 




27T J \2w. 


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), 

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 

. . - 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 


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 : 





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 =\ , 


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) 

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. 




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 


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, 

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 == 



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 

we find 

__ 1 

T ~ri 



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 ; 


and we find 


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 


so that 


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 

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, 


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 



(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. 

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 

2 f 7r 
B T = 1 sin [r(u sin u)] sin u du 


2 f * 

C T = I sin [r(u sin w)] sin w rfw 


1 2 f " 
D T = I cos [r(u sin w)l cos u du, 


A simple trigonometrical transformation leads to : 


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} 


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^) 


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. 


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= 



(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 


(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. 


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 


The value 13-53 volts is found for the energy of the hydrogen electron. 
The general transformation formula is 


(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 


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, 



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 


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. 


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 


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. 


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. 


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 



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 



(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. 



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 


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 



where this time 
(7) S=- 


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 

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- 

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* 

I Z* 




2o H 5 n 3 ife 7 


On introducing the Rydberg constant 

we get 

2a H hc 


l + f!+Ll + 

nk* nlc? 

- )c 4 

i 2 / 

Writing W in the form 

(9) W=-~ ~, 

v } n* 2 

we find, on neglecting products of the c/s, 


(11) "*=n+8 1 +??+ 


_ 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). 


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. 



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 

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. 


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 

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 

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 


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 : 




1-63 2-64 3-65 4-65 
2-12 3-13 4-14 

2-99 3-99 4-99 

4-00 5-00 




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 : 


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 


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 

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 



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, 


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. 









<5 a 













<5 2 

























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 


for ~~ for ~~ 

P 2 
d 3 
/ * 



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- 


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 


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 


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 


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. 


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 


and c 8 is determined by the condition that at the shells the potential 
varies continuously. It follows from this that 


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 


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. 


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 
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 




for the energy, we find for the effective quantum number 


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 : 


a is further related to the radius of the shell 


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 

n (0< _ 



(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 : 


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. 


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- 


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 

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. 


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 


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 

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. 



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 

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 


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). 


FIG. 16. 


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. 


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 

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. 


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. 


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. 


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. 



n* of the 


Rydberg Correction 
large n of the 












1 H 














+ 0-01 








- 30 

- 0-07 



3 Li 
















10 Ne 1 







11 Na 









12 Mg 

/I -33 








13 Al 









16 3 

/I -97 





19 K 

















20 Ca 



1 95 






24 (!r 







25 Mn 







29 Cu 









30 Xn 

/I -20 








31 CJa 







37 Kb 









38 8r 










42 Mo 

1 -36 





47 Ag 









48 Cd 

/I -23 








49 Tn 







55 Cs 









K(l "liu 

/I -62 








t)\J -L>d 









79 Au 












80 Hg 









81 Tl 









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. 


$-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. 


however, must be somewhat smaller on account of the higher nuclear 

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 

calculated from 

From the 
Kinetic Theory 
of Gases 

From Grating 

3 Li+ 



11 Na+ 





19 K+ 



20 Ca+ 


29 Cu+ 




37 Rb+ 



38 Sr+ 



0-9-2-2 3 





56 Ba+ 


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. 


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). 



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 



3 Li 

- 0-00 


11 Na 



19 K 



29 Cu 


37 Rb 



47 Ag 


55 Cs 





(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 



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 


Terms of each Series 










*1 ^2 3 3 

4 4 

2 He 






1 9 Q 
O ^2 *8 

3 Li 





2i 2 2 3 3 

4 4 





3i 2 2 3 3 

4 4 

10 Ne 




3O 1 Q 
1 "2 f) 3 

4 4 

11 Na 






4 4 

12 Mg 





3l 3 2 a 3' 

4 4 

13 Al 





4j 3 2 3 3 

4 4 

19 K 





4! 4 2 3 3 

4 4 

20 Ca 





gj 4 2 3 3 

4 4 






4i 4 2 3 3 

4 4 

30 Zn 




gj 4 2 3 3 

4 4 

31 <!a 



4 2 3 3 

37 Kb 





5 t 5 2 3 3 

4 4 

38 Sr 

\ 3-37 




5i r \ 
p O 2 4 3 

4 4 

47 Ag 





5i 5 2 3 3 


48 Cd 




\\ 5 2 3, 


49 In 



61 5 2 3 3 

55 Cs 





6i 6 2 3 3 


56 Ba 






4 4 

79 Au 




61 6 2 3 3 


80 Hg 




?J 6 a 3 8 

81 Tl 




7 X 6 2 3 3 

4 4 

1 Normal orbit of last electron added. See footnote, p. 185. 


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 

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. 



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 


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 ). 


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. 



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 

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 

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. 


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 


coloured ions 
incomplete inter- 
mediate shells with 
90 loosely bound 

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. 


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. 


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 


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. 




3i 3 3 3 3 

*i 2 4 3 4 4 

1 H 







2 1 

2 (2) 


12 M 
13 A] 
14 Si 







2 1 


20 Ca 
21 Sc 
22 Ti 

29 CA 
30 Zn 
31 Ga 

36 Kr 





8 1 
8 2 






2 1 





37 Rb 
38 Sr 
39 Y 
40 Zr 

47 Ag 
48 Cd 
49 In 

54 X 









8 1 
8 2 






2 1 





55 Cs 
56 Ba 
57 La 
58 Ce 
59 Pr 








18 1 
18 2 

S 1 
8 1 
8 1 




71 Cp 
72 Hf 




8 1 
8 2 


2 1 

79 Au 
80 Hg 
81 Tl 

86 Nt 













88 Ra 
89 Ac 
90 Th 








8 1 
8 2 










Horn, Mechanics of the Atom, I. To face p, 198. 


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 


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. 


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 




FIG. 19. 






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 


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 



Therefore, by (2), 
(4) H 

1 A. Sommerfeld, Ann. d. Phynik, vol. li, p. 1, 1916. 


If we calculate from this the sum of the squares of the momenta, we 

(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 = 


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), 



/ - ^ 

( . Y - A aZ 
v ' V i 




If the equation be solved for 1 -\ -- we find 



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 

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 


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 

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 : 


1 2 

FIG. 20. 

3 f 
123 123* 


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 . 

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 


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 

-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). 


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 


The Lagrangian function is by (8), 4 : 

where the sum is to be taken over all the electrons. From this we 
calculate the momenta. For one electron they are : 

dL e 


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. 


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 

(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. 



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, 


the angle a between the directions of the angular momentum and 
the magnetic field is given by 



cos a= ; 

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 ) 


FIG. 21. 


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 


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 


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> : 


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 

Since p^d^ is never negative, J^.^.0 always. We find further : 


(6) \ 
/ 2 (,)=2mWr ? H2a a -- 


(7) a 1 +a,=2me 2 Z. 


The action integrals J^ and J^ are consequently : 




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 


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 


and obtain 

J , = : J 


The motion has two frequencies : 
(14) v 



We have two quantum conditions : 

If we introduce them into the expression (13) for the energy, we have 


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 



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 


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 


, 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, 



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 


-6 -3 03 6 

-12 -0 - O 8 12 

-20 -1 


-/<? -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 


02 V 6 8101211 

Fio. 25. 

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, 


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 


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). 


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. 


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 : 


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 \/ 




/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 


(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 



a' i n 
L> e - 
i) 2 

Now by (7) : 


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 : 


(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 

this is the well-known expression for the Bessel function x J n (/o). 
Using the relations 2 


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. 


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/ 



d x 

~ g 

(a 2 +6 2 )(a 2 +6 2 cos x ) 
and consequently, substituting for (x+iy) from (15) and (4), 


From this we can deduce at once the B (we now write : 


TT- - 

* 4:77 


[ 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 


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. 


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) : 


for the (z+iy)-components (r^= 
(24) R i 


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 



T^ r^ Tj, 

Ra a 

R* 2 


Const, x 
(Ra 2 + R) 







102 -> 002 






201 -> 101 







201 -> Oil 



, , 

111 -> 002 

1 1 -1 



1 2-6 


J_ J 102 -> 101 







102 -> Oil 


1 -1 1 

. . 

. . 

[ 201 ~> 002 


2 0-1 


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 


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. 



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 ? 

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 


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 . 


(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 


nucleus as origin and the major axis as the -axis, we have (fig. 31) 



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 



(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 : 


we have then 




2 =m 

'-lA 1 -; 

If we calculate r- 2 - from the last relation we find 

/ 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) : 


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 



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), 

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), 



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. 


is a maximum, J^ and J 3 , and thus 

being constant. Now the function 



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 


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 

J 1 <J e <J 1 


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 

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. 


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 


The magnetic contribution can likewise be simply expressed, if we 
introduce the angular momentum P by means of the well-known 
vector relation 


and remember that the time average of 

is zero. We find in this way 

2[r[Hr]J = -[HP] 
(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 


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) 

and from (8), 22 (p. 141), 


where J is the non-degenerate action variable of the motion in the 
absence of a field. Elimination of leads to 


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 


(2) and (8) now become 


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]]. 


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 


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), 


(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", 

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 ) 

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 



This implies a somewhat modified type of space quantisation, since 
by (19) : 


cos(fi, w e +w w )=2- 


cosfc, w e -w m )=2-. 

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 


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" 

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 



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 


1 V 

or, in elliptic co-ordinates : 

w u= "o(^ 

The kinetic energy is 


and by the relations (cf. (3)) 


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 

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. 




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 


(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. 


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. 


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. 


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. 



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 



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- 


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. 


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 

(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 

(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, 


(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 



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. : 


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), 


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 

(15) ^=^o +A i 


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 


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, 


where O n again denotes the " periodic component " of the function 


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 ' 



/ 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. 


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., 

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. 


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 + . . ., 

"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 



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 




We now find W X (J) and - from equation (9), 41 ; this gives 

(4) W^H^O, 



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 


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 



\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) " 


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 +..., 


/ v 

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 

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 


(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. 


we find : 
(12) H 1 == 



If this be arranged as a Fourier series 

(13) H 1= 2X sin (^)= 

(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 : 

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 : 


aj, ' 33, 


**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 

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. 


(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 : 


(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 : 

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- 

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 



_ 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 


( } 

We can now write 

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 


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. 


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. 


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^ 


(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, 



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 


167T 2 A 

If we put 

'(I) Vl -V si 



(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 


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 


sufficiently small values of A, the motion of w' Q is clearly a rotation, 
and for E(&) we can make use of the expansion 


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 ; 



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 

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 

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 


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 


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 

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*. 


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 

(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) 



<*) 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, 


W is found from (4 ). Since Sj is to be a periodic function of 
the w k Q 's, it follows from (4^ that 


the quantities - - remain, however, indeterminate. By averaging 

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 

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 



The integrand ^ is never negative along the path of integra- 


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 

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 

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 


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 

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 



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 

in which K is the expression obtained by replacing the w p 's in the 

left-hand side of (5) by " co-ordinates " q p , and the - 's by the con- 


jugate " momenta " p p , i.e. : 
(11) K 


the quantities v pff = -> - being treated as constants. The mechani- 
p ff 


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 


This method fails only if the system of equations (13) is not soluble 
for the Wp Q '& 9 i.e. if the " Hessian determinant " 



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 

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 


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, 


etc., although our considerations will have a much more general 

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 


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. 



(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 

(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 


w a Q 'a (period 1). When transformed, the expression (2) must take the 

(3) H=W +AW 1 +AW,+ . . ., 

(4) W B =V n (J a )+E B 

and the R n 's denote power series in , 77 commencing with quadratic 
We assume for the generating function of the transformation 

(5) S=2 J a^+T+^+B^-A7 7 , 


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 : 


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 . 


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 . 


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. 


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 : 


(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 


(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 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 
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 


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), 



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 


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. 


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 


The perturbation function is the mutual potential energy of the 

(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 


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 

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) 


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, 



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 


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. 


X~XQ cos 
?/=^ sin 


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!, 


For x , y 9 , we have 


The perturbation function is now 
(8) AH 1= 


II 2 



(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. 



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. 


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 

W^O-687 - -=l-373cR/*Z. 

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. 


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). 


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. 


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- 



) = / 

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 = 



+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 


(6) H^ 


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 : 


cos (+-f sin 


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. 


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) : 




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 



Writing j~k+p, and expanding in powers of -, the result is 



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) : 


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 : 




(P= 1 




Theoretical 1 p= 



+ 0-002 





T? i /Orthohelium 
Empmcal ^p arhelilim 




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. 


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. 

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 


(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 : 



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. 



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 


Vf __Vf 


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 





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 

V f 



may continue this process until we arrive at the relation 

r~ i 



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 


the square root sign, this expression takes the form (neglecting the 
factor A) 

on making the substitution 

p f 

dx 1 2 cos 

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 . 


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 


- 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 

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 



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 




respectively. It follows therefore that 

Res ao = 27T = 



Ji= (j>- V- 

For aj=oo the integral is regular. For 35=0, the expansion of the 
integrand is 

I/ 1 B 


that is 






B C\- B 

(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 


*~~ V-Ax*+2Ex-CJ I B 


VA\A 2 A/ 

Group 2 : 


r Vla; 2 
a) CD -r ~ dx. We can distinguish two'possible cases. 


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 



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 



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 





In conclusion we will consider one or two other integrals of the 

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 : 




B 1 BD 

(6) J 7 = 

=(L / -A+2? - 2. +Dx dx. 

JV * * 


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 


(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, 

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, 

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 


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 

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, 

Conjugate variables, 21. 




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, 

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, 

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, 


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, 


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, 


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, 

Fundamental system of periods, 

Fundamental vibration, 61. 

Gallium atom, 196. 

Generator (of a transformation), 

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. 



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 


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 

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 

Molecule, diatomic, 63 et seq., 110, 
117 et seq. 

Momentum, 17, 20 ; (conserva- 
tion of), 25 ; (angular), 23, 

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), 


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, 


Occupation numbers, 198. 
One-body problem, 123, 131 et 

seq. ' 
Orthohelium, 192, 292, 298. 



Oscillations of molecules, 123 et 

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, 

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 

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. 



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), 

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 

Stationary [states, 6, 11, 14, 52, 

Strontium atom, 197. 

Subordinate series, 152, 154. 
Subsidiary quantum number, 
136, 151. 

Term sequence, 152. 

Terms, 152 et seq. 

Theory of numbers, theorems, 


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 


Uniqueness of the J's (in separ- 
able systems), 83 et seq. ; (in 
general), 86 et seq. 

Variation principle (Hamilton's), 

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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