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N.~ Bohr, D. Kgl. Danske Vidensk. Selsk. Skrifter,
Naturvidensk. og Mathem. Afd. 8. R\ae kke, IV.1, 1--3 \hfill {\large \bf 1918}\\
\vspace{2cm}
\begin{center}
{\large ON THE QUANTUM THEORY OF LINE--SPECTRA.}\\
\end{center}
\vspace{1cm}
\begin{flushleft}
{\small DEDICATED TO THE MEMORY OF MY VENERATED TEACHER\\
PROFESSOR C. CHRISTIANSEN}\\
October 9, 1843 November 28, 1917\\
\end{flushleft}
\vspace{1cm}
\begin{center}
N.~Bohr,\\
{\it Dr. phil. Copenhagen}\\
(Received 1918)\\
\end{center}
\section*{
{\it Introduction}}
\vspace{0.5cm}
In an attempt to develop certain outlines of a theory of line--spectra based on
a suitable application of the fundamental ideas introduced by Planck in his
theory of temperature-radiation to the theory of the nucleus atom of Sir Ernest
Rutherford, the writer has has shown that it is possible in this way to obtain a
sample interpretation of some of the main laws governing the line-spectra of the
elements, and especially to obtain a deduction of the well known Balmer formula
for the hydrogen spectrum\footnote{N.Bohr, Phil. Mag. {\bf 26} (1913) 1, 476,
857; {\bf 27} (1914) 506; {\bf 29} (1915) 332; {\bf 30} (1915) 394.}. The theory
in the form given allowed of a detailed discussion only in the case of periodic
systems, and obviously was not able to account in detail for the characteristic
difference between the hydrogen spectrum and the spectra of other elements, or
for the characteristic effects on the hydrogen spectrum of external electric and
magnetic fields. Recently, however, a way out of this difficulty has been opened
by Sommerfeld\footnote{A. Sommerfeld, Ber. Akad. M\"unchen. 1915, pp. 425, 459;
1916, p. 131; 1917, p. 83. Ann. d. Phys. {\bf 51} (1916) 1.} who, by introducing
a suitable
\begin{quotation}
{\small Editor's note. This paper is the Introduction and Part I (pp. 1--36) of
the paper {\it On the Quantum Theory of Line--spectra}, which was first
published (1918--1922) in the Memoires de l'Academie Royale des Sciences et des
Letters de Danemark, Copenhague, Section des Sciences, 8me serie t. IV,
n$^{\circ}$ 1, fasc. 1--3 (D. Kgl. Danske Vidensk. Selsk. Skrifter,
Naturvidensk. og Mathem. Afd. 8. R\ae kke, IV.1, 1--3). The Introduction was
signed `Copenhagen, November 1917'. The titles of Parts II and III, which are
not reproduced here, are respectively {\it on the hydrogen spectrum} and {\it On
the spectra elements of higher atomic number}.}
\end{quotation}
generalisation of the theory to a simple type of \mbox{non--periodic}
motions and by taking the small variation of the mass of the electron with its
velocity into account, obtained an explanation of the \mbox{line--structure} of
the hydrogen lines which was found to be in brilliant conformity with the
measurements. Already in his first paper on this subject, Sommerfeld pointed out
that his theory evidently offered a clue to the interpretation of the more
intricate structure of the spectra of other elements. Briefly afterwards
Epstein\footnote{P. Epstein, Phys. Zeitschr. {\bf 17} (1916) 148; Ann. d. Phys.
{\bf 50} (1916) 489; {\bf 51} (1916) 168.} and Schwarzschild\footnote{K.
Schwarzschild, Ber. Akad. Berlin (1916) 548.} independent of each other, by
adapting Sommerfeld's ideas to the treatment of a more extended class of \mbox{
non--periodic} systems obtained a detailed explanation of the characteristic
effect of an electric field on the hydrogen spectrum discovered by Stark,
Subsequently Sommerfeld\footnote{A. Sommerfeld, Phys. Zeitschr. {\bf 17} (1916)
491.} himself and Debye\footnote{P. Debye, Nachr. K. Ges. d. Wiss. G\"ottingen,
1916; Phys. Zeitschr. {\bf 17} (1916) 507.} have on the same lines indicated an
interpretation of the effect of a magnetic field on the hydrogen spectrum which,
although no complete explanation of the observations was obtained, undoubtedly
represents an important step towards a detailed understanding of this
phenomenon.
In spite of the great progress involved in these investigations many
difficulties of fundamental nature remained unsolved, not only as regards the
limited applicability of the methods used in calculating the frequencies of the
spectrum of a given system, but especially as regards the question of the
polarization and and intensity of the emitted spectral lines. These difficulties
are intimately connected with the radical departure from the ordinary ideas of
mechanics and electrodynamics involved in the main principles of the quantum
theory, and with the fact that it has not been possible hitherto to replace
these ideas by others forming an equally consistent and developed structure.
Also in this respect, however, great progress has recently been obtained by the
work of Einstein\footnote{A. Einstein, Verh. d. D. phys. Ges. {\bf 18} (1916)
318; Phys. Zeitschr. {\bf 18} (1917) 121.} and Ehrenfest\footnote{P. Ehrenfest,
Proc. Acad. Amsterdam {\bf 16} (1914) 591; Phys. Zeitschr. {\bf 15} (1914) 657;
Ann. d. Phys. {\bf 51} (1916) 327; Phil. Mag. {\bf 33} (1917) 500.}. On this
state of the theory it might therefore be of interest to make an attempt to
discuss the different applications from a uniform point of view, and especially
to consider the underlying assumptions in their relations to ordinary mechanics
and electrodynamics. Such an attempt has been made in the present paper, and it
will be shown that it seems possible to throw some light on the outstanding
difficulties by trying to trace the analogy between the quantum theory and the
ordinary theory of radiation as closely as possible.
The paper is divided into four parts.\\
\begin{minipage}[t]{2.5cm}
{\large Part I}\\
{\large Part II}\\
{\large Rart III}\\
{\large Part IV}\\
\end{minipage}
\begin{minipage}[t]{12.5cm}
contains a brief discussion of the general principles of the theory and deals
with the application of the general theory to periodic systems of one degree of
freedom and to the class of \mbox{non--periodic} systems referred to above.\\
contains a detailed discussion of the theory of the hydrogen spectrum in order
to illustrate the general considerations.\\
contains a discussion of the questions arising in connection with the
explanation of the spectra of other elements.\\
contains a general discussion of the theory of the constitution of atoms and
molecules based on the application of the quantum theory to the nucleus atom.
\end{minipage}
\vspace{0.3cm}
{\it Copenhagen, November 1917.}
\vspace{2cm}
\part*{
{\large PART I. ON THE GENERAL THEORY}}
\section*{{\bf 1. General principles}}
The quantum theory of \mbox{line--spectra} rests upon the following fundamental
assumptions:\\
\begin{itemize}
\item
{\it I. That an atomic system can, and can only, exist permanently
in a certain series of states corresponding to a discontinuous series of values
for its energy, and that consequently any charge of the energy of the system,
including emission and absorption of electromagnetic radiation, must take place
a complete transition between two such states. These states will be denoted as
the stationary states of the system.}
\item
{\it II. That the radiation absorbed or emitted during a transition between two
stationary states is `unifrequentic' and possesses a frequency $\nu$, given
by the relation}
\begin{equation}
E' = E'' = h \nu,
\end{equation}
{\it Where $h$ is Planck's constant and where $E'$ and $E''$ are the values of
the energy in the two states under consideration.}
\end{itemize}
As pointed out by the writer in the papers referred to in the introduction,
these assumptions offer an immediate interpretation of the fundamental {\it
principle of combination of spectral lines} deduced from the measurements of the
frequencies of the series spectra of the elements. according to the laws
discovered by Balmer, Rydberg and Ritz, the frequencies of the lines of the
series spectrum of an element can be expressed by a formula of the type:
\begin{equation}
\nu = f_{\tau''}(n'') - f_{\tau'}(n'),
\end{equation}
where $n'$ and $n''$ are whole numbers and $f_{\tau}(n)$ is one among a set of
functions of $n$, characteristic for the element under consideration. On the
above assumptions this formula may obviously be interpretated by assuming that
the stationary states of an atom of an element form a set of series, and that
the energy in the $n$th state of the $\tau$th series, omitting an arbitrary
constant, is given by
\begin{equation}
E_{\tau}(n) = - hf_{\tau}(n).
\end{equation}
We thus see that the values for the energy in the stationary states of an atom
may be obtained directly from the measurements of the spectrum by means of
relation (1). In order, however, to obtain a theoretical connection between
these values and the experimental evidence about the constitution of the atom
obtained from other sources, it is necessary to introduce further assumptions
about the laws which govern the stationary states of a given atomic system and
the transitions between these states.
Now on the basis of a vast amount of experimental evidence, we are forced to
assume that an atom or molecule consists of a number of electrified particles in
motion, and, since the above fundamental assumptions imply that no emission of
radiation takes place in the stationary states, we must consequently assume that
{\it the ordinary laws of electrodynamics cannot be applied} to these states
without radical alterations. In many cases, however, the effect of that part of
the electrodynamical forces which is connected with the emission of radiation
will at any moment be very small in comparison with the effect of the simple
electrostatic attractions or repulsions of the charged particles corresponding
to Coulomb's law. Even if the theory of radiation must be completely altered, it
is therefore a natural assumption that it is possible in such cases to obtain a
close approximation in the description of the motion in the stationary states,
by retaining only the latter forces. In the following we shall therefore, as in
all the papers mentioned in the introduction, for the present {\it calculate the
motions of the particles in the stationary states as the motions of \mbox{mass--
points} according to ordinary mechanics} including the modifications claimed by
the theory of relativity, and we shall later in the discussion of the special
applications come back to the question of the degree of approximation which may
be obtained in this way.
If next we consider a transition between two stationary states, it is obvious at
once from the essential discontinuity, involved in the assumptions I and II,
that in general it is impossible even approximately to describe this phenomenon
by means of ordinary mechanics or to calculate the frequency of the radiation
absorbed or emitted by such a process by means of ordinary electrodynamics. On
the other hand, from the fact that it has been possible by means of ordinary
mechanics and electrodynamics to account for the phenomenon of \mbox{
temperature--radiation} in the limiting region of slow vibrations, we may expect
that any theory capable to describing this phenomenon in accordance with
observations will form some sort of natural generalisation of the ordinary
theory of radiation. Now the theory of \mbox{temperature-radiation} in the form
originally given by Planck confessedly lacked internal consistency, since, in
the deduction of his radiation formula, assumptions of similar character as I
and II were used in connection with assumptions which were in obvious contrast
to them. Quite recently, however, Einstein\footnote{A. Einstein, loc. cit. [
[paper {\bf 1} and {\bf 1d}]} has succeeded, on the basis of the assumptions
I and
II, to give a consistent and instructive deduction of Planck's formula by
introducing certain supplementary assumptions about the {\it probability of
transition of a system between two stationary states} and about the manner in
which this probability depends on the density of radiation of the corresponding
frequency in the surrounding space, suggested from analogy with the ordinary
theory of radiation. Einstein compares the emission or absorption of radiation
of frequency $\nu$ corresponding to a transition between two stationary states
with the emission or absorption to be expected on ordinary electrodynamics for a
system consisting of a particle executing harmonic vibrations of this frequency.
In analogy with the fact that on the latter theory such a system will without
external excitation emit a radiation of frequency $\nu$, Einstein assumes in the
first place that on the quantum theory there will be a certain probability $A^{
n'}_{n''} dt$ that the system in the stationary state of greater energy,
characterised by the letter $n'$, in the time interval $dt$ will start {\it
spontaneously} to pass to the stationary state of smaller energy, characterised
by the letter $n''$. Moreover, on ordinary electrodynamics the harmonic vibrator
will, in addition to the above mentioned independent emission, in the presence
of a radiation of frequency $\nu$ in the surrounding space, and dependent on the
accidental \mbox{phase--defference} between this radiation and the vibrator,
emit or absorb \mbox{radiation--energy}. In analogy with this, Einstein assumes
secondly that in the presence of a radiation in the surrounding space, the
system will on the quantum theory, in addition to the above mentioned
probability of spontaneous transition from the state $n'$ to the state $n''$,
possess a certain probability, depending on this radiation, of passing in the
time $dt$ from the state $n'$ to the state $n''$, as well as from the state $
n''$ to the state $n'$. These latter probabilities are assumed to be
proportional to the intensity of the surrounding radiation and are denoted by $
\varrho_{\nu} B^{n'}_{n''} dt$ and $\varrho_{\nu} B^{n''}_{n'} dt$ respectively,
where $\varrho_{\nu} d \nu$ denotes the amount of radiation in unit volume of
the surrounding space distributed on frequencies between $\nu$ and $\nu + d
\nu$, while $B^{n'}_{n''}$ and $B^{n''}_{n'}$ are constants which, like
$A^{n'}_{n''}$, depend only on the stationary states under consideration.
Einstein does not introduce any detailed assumption as to the values of these
constants, no more than to the conditions by which the different stationary
states of a given system are determined or to the \mbox{`a--priory} probability'
of these states on which their relative occurrence in a distribution of
statistical equilibrium depends. He shows, however, how it is possible from the
above general assumptions of Boltzmann's principle on the relation between
entropy and probability and Wien's well known \mbox{displacement--law}, to
deduce a formula for the temperature radiation which apart from an undetermined
constant factor coincides with Planck's, if we only assume that the frequency
corresponding to the transition between the two states is determined by (1). If
will therefore be seen that by reversing the line of argument, Einstein's theory
may be considered as a very direct support of the latter relation.
In the following discussion of the application of the quantum theory to
determine the \mbox{line--spectrum} of a given system, it will, just as in the
theory of \mbox{temperature--radiation}, not be necessary to introduce detailed
assumptions as to the mechanism of transition between two stationary states. We
shall show, however, that the conditions which will be used to determine the
values of the energy in the stationary states are of such a type that the
frequencies calculated by (1), in the limit where the motions in successive
stationary states comparatively differ very little from each other, will tend to
coincide with the frequencies to be expected on the ordinary theory of radiation
from the motion of the system in the stationary states. In order to obtain the
necessary relation to the ordinary theory of radiation in the limit of slow
vibrations, we are therefore led directly to certain conclusions about the
probability of transition between two stationary states in this limit. This
leads again to certain general considerations about connection between the
probability of a transition between any two stationary states and the motion of
the system in these states, which will be shown to throw light on the question
of the polarisation and intensity of the different lines of the spectrum of a
given system.
In the above considerations we have by an atomic system tacitly understood a
number of electrified particles which move in a field of force which, with
approximation mentioned, possesses a potential depending only on the position of
the particles. This may more accurately be denoted as a system under constant
external conditions, and the question next arises about the variation in the
stationary states which may be expected to take place during a variation of the
external conditions, e.g. when exposing the atomic system to some variable
external field to force. Now, in general, we must obviously assume that this
variation cannot be calculated by ordinary mechanics, no more than the
transition between two different stationary states corresponding to constant
external conditions. If, however, the variation of the external conditions is
very slow, we may from the necessary stability of the stationary states expect
that the motion of the system at any given moment during the variation will
differ only very little from the motion in a stationary state corresponding to
the instantaneous external conditions. If now, moreover, the variation is
performed at a constant or very slowly changing rate, the forces to which the
particles of the system will be exposed will not differ at any moment from those
to which they would be exposed if we imagine that the external forces arise from
a number of slowly moving additional particles which together with the original
system form a system in a stationary state. From this point of view it seems
therefore natural to assume that, with the approximation mentioned, the motion
of an atomic system in the stationary states can be calculated direct
application of ordinary mechanics, not only under constant external conditions,
but in general also during a slow and uniform variation of these conditions.
This assumption, which may be denoted as the principle of the {\it `mechanical
transformability'} of the stationary states, has been introduced in the quantum
theory by Ehrenfest\footnote{P. Ehrenfest, loc.cit. In these papers the
principle in question is called the adiabatic hypothesis in accordance with the
line of argumentation followed by Ehrenfest in which considerations of
thermodynamical problems play an important part. From the point of view taken in
the present paper, however, the above notation might in a more direct way
indicate the content of the principle and the limits of its applicability.} and
is, as it will be seen in the following sections, of great importance in the
discussion of the conditions to be used to fix the stationary states of an
atomic system among the continuous multitude of mechanically possible motions.
In this connection it may be pointed out that the principle of the mechanical
transformability of the stationary states allows us to overcome a fundamental
difficulty which at sight would seem to be involved in the definition of the
energy difference between two stationary states which enters in relation (1). In
fact we have assumed that the direct transition between two such states cannot
be described by ordinary mechanics, while on the other hand we possess no means
of defining an energy difference between two states if there exists no
possibility for a continuous mechanical connection between them. It is clear,
however, that such a connection is just afforded by Ehrenfest's principle which
allows us to transform mechanically the stationary states of a given system into
those of another, because for the latter system we may take one in which the
forces which act on the particles are very small and where we may assume that
the values of the energy in all the stationary states will tend to coincide.
As regards the problem of the statistical distribution of the different
stationary states between a great number of atomic systems of the same kind in
temperature equilibrium, the number of systems present in the different states
may be deduced in the well known way from Boltzmann's fundamental relation
between entropy and probability, if we know the values of the energy in these
states and the {\it a-priori probability} to be ascribed to each state in the
calculation of the probability of the whole distribution. In contrast
considerations of ordinary statistical mechanics we possess on the quantum
theory no direct of determining these a-priori probabilities, because we have no
detailed information about the mechanism of transition between the different
stationary states. If the a-priori probabilities are known for the states of a
given atomic system, however, they may be deduced for any other system which can
be formed from this by a continuous transformation without passing through one
of the singular systems referred to below. In fact, in examining the necessary
conditions for the explanation of the second law of thermodynamics
Ehrenfest\footnote{ P. Ehresfest, Phys. Zeitschr. {\bf 15} (1914) 660. The above
interpretation of this relation is not states explicitly by Ehrenfest, but it
presents itself directly if the quantum theory is taken in the form
corresponding to the fundamental assumption I.} has deduced a certain general
condition as regards the variation of the a-priori probability corresponding to
a small change of the external conditions from which it follows, that the a-
priori probability of a given stationary state of an atomic system must remain
unaltered during a continuous transformation, except in special cases in which
the values of the energy in some of the stationary states will tend to coincide
during the transformation. In this result we possess, as we shall see, a
rational basis for the determination of the a-priori probability of the
different stationary states of a given atomic system.
\vspace{0.5cm}
\subsection*{
{\bf 2. System of one degree of freedom}}
\vspace{0.5cm}
As the simplest illustration of the principles discussed in the former section
we shall begin by considering systems of a single degree of freedom, in which
case it has been possible to establish a general theory of stationary states.
This is due to the fact, that {\it the motion will be simply periodic}, provided
the distance between the parts of the system will not increase infinitely with
the time, a case which for obvious reasons cannot represent a stationary state
in the sense defined above. On account of this, the discussion of the mechanical
transformability of the stationary states can, as pointed out by
Ehrenfest\footnote{P.Ehrenfest, loc.cit.
Proc. Acad. Amsterdam {\bf} (1914) 591.}
for systems of one degree of freedom be based on a mechanical theorem about
periodic systems due to Boltzmann and originally applied by this author in a
discussion of the bearing of mechanics on the explanation of the laws of
thermodynamics. For the sake of the considerations in the following sections it
will be convenient here to give the proof in a form which differs slightly from
that given by Ehrenfest, and which takes also regard to the modifications in the
ordinary laws of mechanics claimed by the theory of relativity.
Consider for the sake of generality a conservative mechanical system of $s$
degrees of freedom, the motion of which is governed by Hamilton's equations:
\begin{equation}
\frac{d p_k}{dt} = - \frac{\partial E}{\partial q_k},~~~~ \frac{d q_k}{dt} =
\frac{\partial E}{\partial p_k},~~~~ (k = 1, \ldots s)
\end{equation}
where $E$ is the total energy considered as a function of the generalised
positional coodinates $q_1, \ldots q_s$ and the corresponding canonically
conjugated momenta $p_1, \ldots p_s$. If the velocities are so small that the
variation in the mass of the particles due to their velocities can be neglected,
the $p$'s are defined in the usual way by
$$
p_k = \frac{\partial T}{\partial \dot q_k},~~~~ (k = 1, \ldots s)
$$
where $T$ is the kinetic energy of the system considered as a function of the
generalised velocities $\dot q_1, \ldots \dot q_s (\dot q_k = d q_k/dt)$ and of
$q_1, \ldots q_s$. If the relativity modifications are taken into account the $
p$'s are defined by a similar set of expressions in which the kinetic energy is
replaced by $T' = \Sigma m_0 c^2 \cdot (1 - \sqrt{1 - v^2/c^2)}$, where the
summation is to be extended over all the particles of the system, and $v$ is the
velocity of one of the particles and $m_0$ its mass for zero velocity, while $c$
is the velocity of light.
Let us now assume that the system performs a periodic motion with the period
$\sigma$, and let us form the expression
\begin{equation}
I = \int \limits^{\sigma}_0~ \sum^s_1 ~p_k \dot q_k ~ dt,
\end{equation}
which is easily seen to be independent of the special choice of coordinates $q_
1, \ldots q_s$ used to describe the motion of the system. In fact, if the
variation of the mass with the velocity is neglected we get
$$
I = 2 ~ \int \limits^{\sigma}_0~ T ~dt,
$$
and if the relativity modifications are included, we get a quite analogous in
which the kinetic energy is replaced by $T'' = \Sigma ~\frac{1}{2}
m_0 v^2/\sqrt{1 - v^2/c^2}.$
Consider next some new periodic motion of the system formed by a small variation
of the first motion, but which may need the presence of external forces in order
to be a mechanically possible motion. For the variation in $I$ we get then
$$
\delta I = \int \limits^{\sigma}_0~ \sum^s_1~(\dot q_k \cdot \delta \cdot
p_k + p_k ~ \delta \dot q_k) ~ dt + \mid \sum^s_1 ~p_k
\cdot \dot q_k \delta t \mid^{\sigma}_0,
$$
where the last term refers to the variation of the limit of the integral due to
the variation in the period $\sigma$. By partial integration of the second term
in the bracket under integral we get next
$$
\delta I = \int \limits^{\sigma}_0~ \sum^s_1~ \left(\dot q_k \cdot \delta
p_k - p_k \cdot \delta q_k \right) ~ dt + \mid \sum^s_1~ p_k
\cdot \left( \dot q_k \cdot \delta t + \delta q_k \right) \mid^{\sigma}_0,
$$
where the last term is seen to be zero, because the term in the bracket as well
as $p_k$ will be the same in both limits, since the varied motion as well as the
original motion is assumed to be periodic. By means of equations (4) we get
therefore
\begin{equation}
\delta I = \int \limits^{\sigma}_0 ~ \sum^s_1 ~ \left( \frac{\partial E}
{\partial p_k} \cdot \delta \cdot p_k + \frac{\partial E}{\partial q_k} \cdot
\delta \cdot q_k \right) ~ dt = \int \limits^{\sigma}_0~ \delta E ~ dt.
\end{equation}
Let us now assume that the small variation of the motion is produced by a small
external field established at a uniform rate during a time interval $\vartheta$,
long compared with $\sigma$, so that the comparative increase during a period is
very small. In this case $\delta E$ is at any moment equal to the total work
done by the external forces on the particles of the system since the beginning
of the establishment of the field. Let this moment be $t = - \vartheta$ and let
the potential of the external field at $t \geq 0$ be given by $\Omega$,
expressed as a function of the $q$'s. at any given moment $t > 0$ we have then
$$
\delta E = - \int \limits^0_{- \vartheta} ~ \frac{\vartheta + t}{\vartheta}
\cdot \sum^s_1~ \frac{\partial \Omega}{\partial q_k} \cdot \dot q_k
\cdot dt - \int \limits^t_0~ \sum^s_1 ~ \frac{\partial \Omega}{\partial q_k}
\cdot \dot q_k ~ dt,
$$
which gives by partial integration
$$
\delta E = \frac{1}{\vartheta} ~\int \limits^0_{- \vartheta} ~
\Omega ~ dt - \Omega_t,
$$
where the values for the $q$'s to be introduced in $\Omega$ in the first term
are those corresponding to the motion under the influence of the increasing
external field, and the values to be introduced in the second term are those
corresponding to the configuration at the time $t$. neglecting small quantities
of the same order as the square of the external force, however, we may in this
expression for $\delta E$ instead of the values for the $q$'s corresponding to
the perturbed motion take those corresponding to the original motion of the
system. With this approximation the first term is equal to the mean value of the
second taken over a period $\sigma$, and we have consequently
\begin{equation}
\int \limits^{\sigma}_0~ \delta E~ dt = 0.
\end{equation}
From (6) and (7) it follows that $I$ will remain constant during the slow
establishment of the small external field, if the motion corresponding to a
constant value of the field is periodic. If next the external field
corresponding to $\Omega$ is considered as an inherent part of the system, it
will be seen in the same way that $I$ will remain unaltered during the
establishment of a new small external field, and so on. Consequently {\it I will
be invariant for any finite transformation of the system which is sufficiently
slowly performed,} provided the motion at any moment during the process is
periodic and the effect of the variation is calculated on ordinary mechanics.
Before we proceed to the applications of this result we shall mention a simple
consequence of (6) for systems for which every orbit is periodic independent of
the initial conditions. In that case we may for the varied motion take an
undisturbed motion of the system corresponding to slightly different initial
conditions. This gives $\delta E$ constant, and from (6) we get therefore
\begin{equation}
\delta E = \omega \cdot \delta \cdot I,
\end{equation}
where $\omega = 1/\sigma$ is the frequency of the motion. This equation forms a
simple relation between the variations in $E$ and $I$ for periodic systems,
which will be often used in the following.
Returning now to systems of one degree of freedom, we shall take our starting
point from Planck's original theory of a {\it linear harmonic vibrator}.
according to this theory the stationary states of a system, consisting of a
particle executing linear harmonic vibrations with a constant frequency $\omega_
0$ independent of the energy, are given by the well known relation
\begin{equation}
E = n \cdot h \cdot \omega_0,
\end{equation}
where $n$ is a positive entire number, $h$ Planck's constant, and $E$ the total
energy which is supposed to be zero if the particle is at rest.
From (8) it follows at once, that (9) is equivalent to
\begin{equation}
I = \int \limits^{\sigma}_0~ p \dot q~ dt = \int~ p~dq = nh,
\end{equation}
where the latter integral is to be taken over a complete oscillation of $q$
between its limits. On the principle of the mechanical transformability of the
stationary states we shall therefore assume, following Ehrenfest, that (10)
holds not only for a Planck's vibrator but for {\it any periodic system of one
degree of freedom} which can be formed in a continuous manner from a linear
harmonic vibrator by a gradual variation of the field of force in which the
particle moves. This condition is immediately seen to be fulfilled by all such
systems in which the motion is of oscillating type i.e. where the moving
particle during a period passes twice through any point of its orbit once in
each direction. If, however, we confine ourselves to systems of one degree of
freedom, it will be seen that system in which the motion is of rotating type, i.
e. where the particle during a period passes only through every point of its
orbit, cannot be formed in a continuous manner from a linear harmonic vibrator
without passing through singular states in which the period becomes infinite
long and the result becomes ambiguous . We shall not here enter more closely on
this difficulty which has been pointed out by Ehrenfest, because it disappears
when we consider systems of several degrees of freedom, where we shall see that
a simple generalisation of holds for any system for which every motion is
periodic.
As regards the application of (9) to statistical problems it was assumed in
Planck's theory that the different states of the vibrator corresponding to
different values of $n$ are {\it a-priori equally probable,} and this assumption
was strongly supported by the agreement obtained on this basis with the
measurements of the specific heat of solids at low temperatures. Now it follows
from the considerations of Ehrenfest, mentioned in the former section, that the
a-priori probability of a given stationary state is not changed by a continuous
transformation, and we shall therefore expect that for any system of one degree
of freedom the different corresponding to different entire values of $n$ in (
10) are a-priori equally probable.
As pointed out by Planck in connection with the application of (9), it is simply
seen that statistical considerations, based on the assumption of equal
probability for the different states given by (10), will show the necessary
relation to considerations of ordinary statistical mechanics in the limit where
the latter theory has been found to give results in agreement with experiments.
Let the configuration and motion of a mechanical system be characterised by $s$
independent variables $q_1, \ldots q_s$ and corresponding momenta $p_1, \ldots
p_s$, and let the state of the system be represent in a $2s$--dimensional
\mbox{phase--space} by a point with coordinates $q_1, \ldots q_s,~ p_1, \ldots
p_s$. Then, according to ordinary statistical mechanics, the probability for
this point to lie within a small element in the \mbox{phase-space} is
independent of the position and shape of this element and simply proportional to
its volume, defined in the usual way by
\begin{equation}
\delta W = \int~ dq_1 \ldots dq_s~ dp_1 \ldots dp_s.
\end{equation}
In the quantum theory, however, these considerations cannot be directly applied,
since the point representing the state of a system cannot be displaced
continuously in the $2s$--dimensional \mbox{phase--space}, but can lie only on
certain surfaces of lower dimensions in this space. For systems of one degree of
freedom the \mbox{phase--space} is a two dimensional surface, and the points
representing the states of some system given by (10) will be situated on closed
curves on this surface. Now, in general, the motion will differ considerably for
any two states corresponding to successive entire values of $n$ in (10), and a
simple general connection between the quantum theory and ordinary statistical
mechanics is therefore out of question. In the limit, however, where $n$ is
large, the motions in successive states will only differ very little from each
other, and it would therefore make little difference whether the points
representing the systems are distributed continuously on the
\mbox{phase--surface} or situated only on the curves corresponding to (10),
provided the number of systems which in the first case are situated between two
such curves is equal to the number which in the second case lies on one of these
curves. But it will be seen that this condition is just fulfilled in consequence
of the above hypothesis of equal a-priori probability of the different
stationary states, because the element of \mbox{phase--surface} limited by two
successive curves corresponding to (10) is equal to
\begin{equation}
\delta W = \int dp~ dq = [\int p ~ dq]_n - [\int p~ dq]_{n-1} = I_n - I_{n-
1} = h,
\end{equation}
so that on ordinary statistical mechanics the probabilities for the point to lie
within any two such elements is the same. We see consequently that the
hypothesis of equal probability of the different states given by (10) gives the
same result as ordinary statistical mechanics in all such applications in which
the states of the great majority of the systems correspond to large values of $
n$. Considerations of this kind have led
Debye\footnote{P. Debye, Wolfskehl--Vortrag. G\"ottengen (1913).}
to point out that condition (10) might have a
general validity for systems of one degree of freedom, already before Ehresfest,
on the basis of his theory of the mechanical transformability of the stationary
states, had shown that this condition forms the only rational generalisation of
Planck's condition (9).
We shall now discuss the relation between the theory of {\it spectra of atomic
systems of one degree of freedom,} based on (1) and (10), and the ordinary
theory of radiation, and we shall see that this relation in several respects
shows a close analogy to the relation, just considered, between the statistical
applications of (10) and considerations based on ordinary statistical mechanics.
Since the values for the frequency $\omega$ in two states corresponding to
different values of $n$ in (10) in general are different, we see at once that we
cannot expect a simple connection between the frequency by (1) of the radiation
corresponding to a transition between two stationary states and the motions of
the system in these states, except in the limit where $n$ is very large, and
where the ratio between the frequencies of the motion in successive stationary
states differs very little from unity. Consider now a transition between the
state corresponding to $n = n'$ and the state corresponding to $n = n''$, and
let us assume that $n'$ and $n''$ are large numbers and that $n'- n''$ is small
compared with $n'$ and $n''$. In that case we may in (8) for $\delta E$ put
$E'- E''$ and for $\delta I$ put $I' - I''$, we get therefore from (1) and (
10) for the frequency of the radiation emitted or absorbed during the transition
between the two states
\begin{equation}
\nu = \frac{1}{h} \cdot (E' - E'') = \frac{\omega}{h} \cdot (I' - I'') =
(n' - n'') \cdot \omega.
\end{equation}
Now in a stationary state of a periodic system the displacement of the particles
in any given direction may always be expressed by means of a
\mbox{Fourier--series} as a sum of harmonic vibrations:
\begin{equation}
\xi = \Sigma ~C_{\tau} ~ \cos~ 2 \pi \cdot (\tau \cdot \omega \cdot t +
c_{\tau}),
\end{equation}
where the $C$'s and $c$'s are constants and the summation is to be extended over
all positive entire values of $\tau$. On the ordinary theory of radiation we
should therefore expect the system to emit a spectrum consisting of a series of
lines of frequencies equal to $\tau \omega$, but as it is seen, this is just
equal to the series of frequencies which we obtain from (13) by introducing
different values for $n'- n''$. As far as the frequencies are concerned we see
therefore that in the limit where $n$ is large there exists a close relation
between the ordinary theory of radiation and the theory spectra based on (1) and
(10). It may be noticed, however, that, while on the first theory radiations of
the different frequencies $\tau \omega$ corresponding to different values of
$\tau$ are emitted or absorbed at the same time, these frequencies will on the
present theory, based on the fundamental assumption I and II, be connected with
entirely different processes of emission or absorption corresponding to the
transition of the system from a given state to different neighbouring stationary
states.
In order to obtain the necessary connection, mentioned in the former section, to
the ordinary theory of radiation in the limit of slow vibrations, we must
further claim that a relation, as that just proved for the frequencies, will, in
the limit of large $n$, hold also for the intensities of the different lines in
the spectrum. Since now on ordinary electrodynamics the intensities of the
radiations corresponding to different values of $\tau$ are directly determined
from the coefficients $C_{\tau}$ in (14), we must therefore expect that for
large values of $n$ these coefficients will on the quantum theory determine the
{\it probability of spontaneous transition} from a given stationary state for
which $n = n'$ to a neighbouring state for which $n = n''= n' - \tau$. Now, this
connection between the amplitudes of the different harmonic vibrations into
which the motion can be resolved, characterised by different values of $\tau$,
and the probabilities of transition from a given stationary state to the
different neighbouring stationary states, characterized by different values of $
n' - n''$, may clearly be expected to be of a general nature. Although, of
course, we cannot without a detailed theory of the mechanism of transition
obtain an exact calculation of the latter probabilities, unless $n$ is large, we
may expect that also for small values of $n$ the amplitude of the harmonic
vibrations corresponding to a given value of $\tau$ will in some way give a
measure for the probability of a transition between two states for which $n' -
n''$ is equal to $\tau$. Thus in general there will be a certain probability of
an atomic system in a stationary state to pass spontaneously to any other state
of smaller energy, but if for all motions of a given system the coefficients $C$
in (14) are zero for certain values of $\tau$, we are led to expect that no
transition will be possible, for which $n' - n''$ is equal to one of these
values.
A simple illustration of these considerations is offered by the linear harmonic
vibrator mentioned above in connection with Planck's theory. Since in this case
$C_{\tau}$ is equal to zero for any $\tau$ different from 1, we shall expect
that for this system only such transitions are possible in which $n$ alters by
one unit. From (1) and (9) we obtain therefore the simple result that the
frequency of any radiation emitted or absorbed by a linear harmonic is equal to
the constant frequency $\omega_0$. This result seems to be supported by
observations on the \mbox{absorption--spectra} of diatomic gases, showing that
certain strong \mbox{absorption--lines}, which according to general evidence may
be ascribed to vibrations of the two atoms in the molecule relative to each
other, are not accompanied by lines of the same order of intensity and
corresponding to entire multipla of the frequency, such as it should be
expected from (1) if the system had any considerable tendency to pass between
\mbox{non--successive} states. In this connection it may be noted that the fact,
that in the absorption spectra of some diatomic gases faint lines occur
corresponding to the double frequency of the main lines\footnote{See E.C.
Kemble, Phys. Rev. {\bf 8} (1916) 701.} obtains a natural explanation by
assuming that for finite amplitudes the vibrations are not exactly harmonic and
that therefore the molecules possess a small probability of passing also between
\mbox{non--successive}states.
\vspace{0.5cm}
\subsection*{
{\bf 3. Conditionally periodic systems}}
\vspace{0.5cm}
If we consider systems of several degrees of freedom the motion will be periodic
only in singular cases and the general conditions which determine the stationary
states cannot therefore be derived by means of the same simple kind of
considerations as in the former section. As mentioned in the introduction,
however, Sommerfeld and others have recently succeeded, by means of a suitable
generalisation of (10), to obtain conditions for an important class of systems
of several degrees of freedom, which, in connection with (1), have been found to
give results in convincing agreement with experimental results about \mbox{
line--spectra}. Subsequently these conditions have been proved by Ehrenfest and
especially Burgers\footnote{J.M. Burgers, Versl. Akad. Amsterdam {\bf 25} (1917)
849, 918, 1055; Ann. d. Phys. {\bf 52} (1917) 195; Phil. Mag. {\bf 33} (1917)
514.} to be invariant for slow mechanical transformations.
To the generalisation under consideration we are naturally led; if we first
consider such systems for which the motions corresponding to the different
degrees of freedom are dynamically independent of each other. This occurs if the
expression for the total energy $E$ in Hamilton's equations (4) for a system of
$s$ degrees of freedom can be written as a sum $E_1 + \ldots E_s$, where $E_k$
contains $q_k$ and $p_k$ only. An illustration of a system of this kind
is presented by a particle moving in a field of force in which the
\mbox{force--components} normal to three mutually
perpendicular fixed planes are functions of
the distances from these planes respectively. Since in such a case the motion
corresponding to each degree of freedom in general will be periodic, just as for
a system of one degree of freedom, we may obviously expect that the condition (
10) is here replaced by a set of $s$ conditions:
\begin{equation}
I_k = \int~ p_k \cdot d q_k = n_k \cdot h, ~~~ (k = 1, \ldots s)
\end{equation}
where the integrals are taken over a complete period of the different $q$'s
respectively, and where $n_1, \ldots n_s$ are entire numbers. It will be seen at
once that these conditions are invariant for any slow transformation of the
system for which the independency of the motions corresponding to the different
coordinates is maintained.
A more general class of systems for which a similar analogy with systems of a
single degree of freedom exists and where conditions of the same type as (15)
present themselves is obtained in the case where, although the motions
corresponding to the different degrees of freedom are not independent of each
other, it is possible nevertheless by a suitable choice of coordinates to
express each of the momenta $p_k$ as a function of $q_k$ only. A simple of this
kind consists of a particle moving in a plane orbit in a central field of force.
Taking the length of the \mbox{radius--vector} from the centre of the field to
the particle as $q_1$, and the angular distance of this \mbox{radius--vector}
from the centre of the field to the particle as $q_1$, and the angular distance
of this \mbox{radius--vector} from a fixed line in the plane of the orbit as $q_
2$, we get at once from (4), since $E$ does not constant $q_2$, the well known
result that during the motion the angular momentum $p_2$ is constant and that
the radial motion, given by the variation of $p_1$ and $q_1$ with the time, will
be exactly the same as for a system of one degree of freedom. In his fundamental
application of the quantum theory to the spectrum of a {\it non--periodic
system} Sommerfeld assumed therefore that the stationary states of the above
system are given by two conditions of the form:
\begin{equation}
I_1 = \int~ p_1 ~ d q_1 = n_1 \cdot h, ~~~ I_2 = \int~ p_2~ dq_2 = n_2 \cdot h.
\end{equation}
While the first integral obviously must be taken over a period of the radial
motion, there might at first sight seem to be a difficulty in fixing the limits
of integration of $q_2$. This disappears, however, if we notice that an integral
of the type under consideration will not be altered by a change of coordinates
in which $q$ is replaces by some function of this variable. In fact, if instead
of the angular distance of the \mbox{radius--vector} we take for $q_2$ some
continuous periodic function of this angle with period $2 \pi$, every point in
the plane of the orbit will correspond to one set of coordinates only and the
relation between $p$ and $q$ will be exactly of the same type as for a periodic
system of one degree of freedom for which the motion is of oscillating type. It
follows therefore that the integration in the second of the conditions (16) has
to be taken over a complete revolution of the \mbox{radius--vector}, and that
consequently this condition is equivalent with the simple condition that the
angular momentum of the particle round the centre of the field is equal to an
entire multiplum of $h/2 \pi$. As pointed out by Ehrenfest, the conditions (16)
are invariant for such special transformations of the system for which the
central symmetry is maintained. This following immediately from the fact that
the angular momentum in transformations of this type remains invariant, and that
the equations of motion for the radial coordinate as long as $p_2$ remains
constant are the same as for a system of one degree of freedom. On the basis of
(16), Sommerfeld has, as mentioned in the introduction, obtained a brilliant
explanation of the fine structure of the lines in the hydrogen spectrum, due to
the change of the mass of the electron with its velocity.\footnote{In this
connection it may be remarked that conditions of the type as (16) were proposed
independently by W. Wilson [Phil. Mag. {\bf 29} (1915) 795 and {\bf 31} (1916)
156] but by him applied only to the simple Keplerian motion described by the
electron in the hydrogen atom if the relativity modifications are neglected. Due
to the singular position of periodic systems in the quantum theory of systems of
several degrees of freedom this application, however, involves, as it will
appear from the following discussion, an ambiguity which deprives the result of
an immediate physical interpretation. Conditions analogous to (16) have also
been established by Planck in his interesting theory of the physical structure
of the phase space of systems of several degrees of freedom [Verh. d. D. Phys.
Ges. {\bf 17} (1915) 407 and 438; Ann. d. Phys. {\bf 50} (1916) 385]. This
theory, which has no direct relation to the problem of line--spectra discussion
in the present paper, rests upon a profound analysis of the geometrical problem
of dividing the multiple--dimensional space corresponding to a system of several
degrees of freedom into `cells' in a way analogous to the division of the phase
surface of a system of one degree of freedom by the curves given by (10).} To
this theory we shall come back in Part II. As pointed by Epstein\footnote{P.
Epstein, loc. cit.} and Schwarzschild\footnote{K. Schwarzschild, loc. cit.} the
central systems considered by Sommerfeld form a special case of a more general
class of systems for which conditions of the same type as (15) may be applied.
These are the socalled {\it conditionally periodic systems,} to which we are led
if the equations of motion are discussed by means of the Hamilton--Jacobi
partial differential equation.\footnote{See f. inst. C.V.L. Charlier,
Die Mechanik
des Himmels, Bd. I, Abt. 2.} In the expression for the total energy $E$ as a
function of the $q$'s and the $p$'s, let the latter quantities be replaced by
the partial differential coefficients of some function $S$ with respect to the
corresponding $q$'s respectively, and consider the partial differential
equation:
\begin{equation}
E \cdot \left( q_1, \ldots q_s,~ \frac{\partial S}{\partial q_1}, \ldots
\frac{\partial S}{\partial q_s} \right) = \alpha_1,
\end{equation}
obtained by putting this expression equal to an arbitrary constant $\alpha_1$.
If then
$$
S = F \cdot (q_1, \ldots q_s,~ \alpha_1, \ldots \alpha_s) + C,
$$
where $\alpha_1, \ldots \alpha_s$ and $C$ are arbitrary constants like $\alpha_
1$, is a total integral of (17), we get, as shown by Hamilton and Jacobi, the
general solution of the equations of motion (4) by putting
\begin{equation}
\frac{\partial S}{\partial \alpha_1} = t + \beta_1,~~~ \frac{\partial S}
{\partial \alpha_k} = \beta_k,~~~ ~~~ (k = 2, \ldots s)
\end{equation}
and
\begin{equation}
\frac{\partial S}{\partial q_k} = p_k,~~~ (k = 1, \ldots s)
\end{equation}
where $t$ is the time and $\beta_1, \ldots \beta_s$ a new set of arbitrary
constants. By means of (18) the $q$'s are given as functions of the time $t$ and
the $2s$ constants $\alpha_1, \ldots \alpha_s,~ \beta_1, \ldots \beta_s$ which
may be determined for instance from the values of the $q$'s and $\dot q$'s at a
given moment.
Now the class of systems, referred to, is that for which, for a suitable choice
of orthogonal coordinates, it is possible to find a total integral of (17) of
the form
\begin{equation}
S = \sum^s_1~~ S_k \cdot (q_k, ~ \alpha_1, \ldots \alpha_s),
\end{equation}
where $S_k$ is a function of the $s$ constants $\alpha_1, \ldots \alpha_s$ and
of $q_k$ only. In this case, in which the equation (17) allows of which is
called `separation of variables', we get from (19) that every $p$ is a function
of the $\alpha$'s and of the corresponding $q$ only. If during the motion the
coordinates do not become infinite in the course of time or converge to fixed
limits, every $q$ will, just as for systems of one degree of freedom, oscillate
between two fixed values, different for the different $q$'s and depending on
the $\alpha$'s. Like in the case of a system of one degree of freedom, $p_k$
will become zero and change its sign whenever $q_k$ passes through one of these
limits. Apart from special cases, the system will during the motion never pass
twice through a configuration corresponding to the same set of values for the $
q$'s and $p$'s, but it will in the course of time pass within any given, however
small, distance from any configuration corresponding to a given set of values $
q_1, \ldots q_s$, representing a point within a certain closed $s$--dimensional
extension limits by $s$ pairs of
$(s - 1)$ -- dimensional surface corresponding to constant
values of the $q$'s equal to the above mentioned limits of oscillation. A motion
of this kind is called `conditionally periodic'. It will be seen that the
character of the motion will depend only on the $\alpha$'s and not on the
$\beta$'s, which latter constants serve only to fix the exact configuration of
the system at a given moment, when the $\alpha$'s are known. For special systems
it may occur that the orbit will not cover the above mentioned $s$--dimensional
extension everywhere dense, but will, for all values of the $\alpha$'s, be
confined to an extension of less dimensions. Such a case we will refer to in the
following as a case of `degeneration'.
Since for a conditionally periodic system which allows of separation
in the variables $q_1, \ldots q_s$ the $p$'s are functions of the corresponding
$q$'s only, we may, just as in the case of independent degrees of freedom or in
the case of quasiperiodic motion in a certain field, from a set of expressions
of the type
\begin{equation}
I_k = \int~ p_k \cdot (q_k, \alpha_1, \ldots \alpha_s) \cdot d q_k, ~~~ (k = 1,
\ldots s)
\end{equation}
where the integration is taken over a complete oscillation of $q_k$. As, in
general, the orbit will cover everywhere dense an $s$--dimensional extension
limited in the characteristic way mentioned above, it follows that, except in
cases of degeneration, a separation of variables will not be possible for two
different sets of coordinates $q_1, \ldots q_s$ and $q'_1, \ldots q'_s$, unless
$q_1 = f_1 (q'_1), \ldots q_s = f_s (q'_s)$, and since a change of coordinates
of this type will not affect the values of the expressions (21), it will be seen
that the values of the $I$'s are completely determined for a given motion of the
system. By putting
\begin{equation}
I_k = n_k \cdot h, ~~~ (k = 1, \ldots s)
\end{equation}
where $n_1, \ldots n_s$ are positive entire numbers, we obtain therefore {\it a
set of conditions which form a natural generalisation of condition} (10) holding
for a system of one degree of freedom.
Since the $I$'s, as given by (21), depend on the constants $\alpha_1, \ldots
\alpha_s$ only and not on the $\beta$'s, the $\alpha$'s may, in general,
inversely be determined from the values of the $I$'s. The character of the
motion will therefore, in general, be completely determined by the conditions (
22), and especially the value for the total energy, which according to (17) is
equal to $\alpha_1$, will be fixed by them. In the cases of degeneration
referred to above, however, the conditions (22) involve an ambiguity, since in
general for such systems there will exist an infinite number of different sets
of coordinates which allow of a separation of variables, and which will lead to
different motions in the stationary states, when these conditions are applied.
As we shall see below, this ambiguity will not influence the fixation of the
total energy in the stationary states, which is the essential factor in the
theory of spectra based on (1) and in the applications of the quantum theory to
statistical problems.
A well known characteristic example of a conditionally periodic system is
afforded by a particle moving under the influence of the attractions from two
fixed centres varying as the inverse squares of the distance apart, if the
relativity modifications are neglected. As shown by Jacobi this problem can be
solved by a separation of variables if so called elliptical coordinates are
used, i.e. if for $q_1$ and $q_2$ we take two parameters characterising
respectively an ellipsoid and a hyperboloid of revolution with the centres as
foci and passing through the instantaneous position of the moving particle, and
for $q_3$ we take the angle between the plane through the particle and the
centres and a fixed plane through the latter points, or, in closer conformity
with the above general description, some continuous periodic function of this
angle period $2 \pi$. A limiting case of this problem is afforded by an
electron rotating a positive nucleus and subject to the effect of an additional
homogeneous electric field, because this field may be considered as arising from
a second nucleus at infinite distance apart from the first. The motion in this
case will therefore be conditionally periodic and allow a separation of
variables in parabolic coordinates, if the nucleus is taken as focus for both
sets of paraboloids of revolution, and their axes are taken parallel to the
direction of the electric force. By applying the conditions (22) to this motion
Epstein and Schwarschild have, as mentioned in the introduction, independent of
each other, obtained an explanation of the effect of an external electric field
on the lines of the hydrogen spectrum, which was found to be convincing
agreement with Stark's measurements. To the results of these calculations we
shall return in Part II.
In the above way of representing the general theory we have followed the same
procedure as used by Epstein. By introducing the so called `angle variables'
well known from the astronomical theory of perturbations, Schwarzschild has
given the theory a very elegant form in which the analogy with systems of one
degree of freedom presents itself in a somewhat different manner. The connection
between this treatment and that given above has been discussed in detail by
Epstein.\footnote{P. Epstein, Ann. d. Phys. {\bf 51} (1916) 168. See also Note
on page 33 of the present paper.}
As mentioned above the conditions (22), first established from analogy with
systems of one degree of freedom, have subsequently been proved generally to be
{\it mechanically invariant for any slow transformation for which the system
remains conditionally periodic.} The proof of this invariance has been given
quite recently by Burgers\footnote{J.M. Burgers, loc. cit. Versl. Akad.
Amsterdam {\bf 25} (1917) 1055.} by means of an interesting application of the
theory of \mbox{contact--transformations} based on Schwarzschild's introduction
of angle variables. We shall not enter here on these calculations but shall only
consider some points in connection with the problem of the mechanical
transformability of the stationary states which are of importance for the
logical consistency of the general theory and for the later applications. In
$\S$~2 we saw that in the proof of the mechanical invariance of relation (10)
for a periodic system of one degree of freedom, it was essential that the
comparative variation of the external conditions during the time of one period
could be made small. This may be regarded as an immediate consequence of the
nature of the fixation of the stationary states in the quantum theory. In fact
the answer to the question whether a given state of a system is stationary, will
not depend only on the motion of the particles at a given moment or on the field
of force in the immediate neighbourhood of their instantaneous positions, but
cannot be given before the particles have passed through a complete cycle of
states, and so to speak have got to know the entire field of force of influence
on the motion. If thus, in the case of a periodic system of one degree of
freedom, the field of force is varied by a given amount, and if its comparative
variation within the time of a single period was not small, the particle would
obviously have no means to get to know the nature of the variation of the field
and to adjust its stationary motion to it, before the new field was already
established. For exactly the same reasons it is necessary condition for the
mechanical invariance of the stationary states of a conditionally periodic
system, that the alteration of the external conditions during an interval in
which the system has passed approximately through all possible configurations
within the above mentioned $s$--dimensional extension in the \mbox{coordinate--
space} can be made as small as we like. This condition forms therefore also an
essential point in Burgers' proof of the invariance of the conditions (22) for
mechanical transformations. Due to this we meet with a characteristic difficulty
when during the transformation of the system we pass one of the cases of
degeneration mentioned above, where, for every set of values for the $\alpha$'s,
the orbit will not cover the $s$--dimensional extension everywhere dense, but
will be confined to an extension of less dimensions. It is clear that, when by a
slow transformation of a conditionally periodic system we approach a degenerate
system of this kind, the time--interval which the orbit takes to pass to any
possible configuration will tend to be very long and will become infinite when
the degenerate system is reached. As a consequence of this {\it the conditions (
22) will generally not remain mechanically invariant when we pass a degenerate
system,} what has intimate connection with the above mentioned ambiguity in the
determination of the stationary states of such systems by means of (22).
A typical case of a degenerate system, which may serve as an illustration of
this point, is formed by system of several degrees of freedom for which every
motion is simply periodic, independent of the initial conditions. In this case,
which is of great importance in the physical applications, we have from (5) and
(21), for any set of coordinates in which a separation of variables is
possible,
\begin{equation}
I = \int \limits_0^{\sigma} ~ (p_1 \dot q_1 + \ldots + p_s \dot q_s)~ dt
= \kappa I_1 + \ldots + \kappa_s I_s,
\end{equation}
where the integration is extended over one period of the motion, and where
$\kappa_1, \ldots \kappa_s$ are a set of positive entire numbers without a
common divisor. Now we shall expect that every motion, for which it is possible
to find a set of coordinates in which it satisfied (22), will be stationary.
For any such motion we get from (23)
\begin{equation}
I = (\kappa_1 \cdot n_1 + \ldots + \kappa_s \cdot n_s) \cdot h = nh,
\end{equation}
where $n$ is a whole number which may take all positive values if, as in the
applications mentioned below, at least one of the $\kappa$'s is equal to one.
Inversely, if the system under consideration allows of separation of variables
in an infinite continuous multitude of sets of coordinates, we must conclude
that generally every motion which satisfies (24) will be stationary, because in
general it will be possible for any such motion to find a set of coordinates in
which it satisfied also (22). It will thus be seen that, for a periodic system
of several degrees of freedom, condition (24) forms a simple generalisation of
condition (1)). From relation (8), which holds for two neighbouring motions of
any periodic system, it follows further that the energy of the system will be
completely determined by the value of $I$, just as for systems of one degree of
freedom.
Consider now a periodic system in some stationary state satisfying (24), and let
us assume that an external field is slowly established at a continuous rate and
that the motion at any moment during this process allows of a separation of
variables in a certain set of coordinates. If we would assume that the effect of
the field on the motion of the system at any moment could be calculated directly
by means of ordinary mechanics, we would find that the values of the $I$'s with
respect to the latter coordinates would remain constant during the process, but
this would involve that the values of the $n$'s in (22) would in general not be
entire numbers, but would depend entirely on the accidental motion, satisfying (
24), originally possessed by the system. That mechanics, however, cannot
generally be applied directly to determine the motion of a periodic system under
influence of an increasing external field, is just what we should expect
according to the singular position of degenerate system as regards mechanical
transformations. In fact, in the presence of a small external field, the motion
of a periodic system will undergo slow variations as regards the shape and
position of the orbit, and if the perturbed motion is conditionally periodic
these variations will be of a periodic nature. Formally, we may therefore
compare a periodic system exposed to an external field with a simple mechanical
system of one degree of freedom in which the particle performs a slow
oscillating motion. Now the frequency of a slow variation of the orbit will be
seen to be proportional to the intensity of the external field, and it is
therefore obviously impossible to establish the external field at a rate so slow
that the comparative change of its intensity during a periodic of this variation
is small. The process which takes place during the increase of the field will
thus be analogous to that which takes place if an oscillating particle is
subject to the effect of external forces which change considerably during a
period. Just as the latter generally will give rise to emission or absorption of
radiation and cannot be described by means of ordinary mechanics, we must expect
that the motion of a periodic system of several degrees of freedom under the
establishment of the external field cannot be determined by ordinary mechanics,
but that the field will give rise to effects of the same kind as those which
occur during a transition between two stationary states accompanied by emission
or absorption of radiation. Consequently we shall expect that, during the
establishment of the field, {\it the system will in general adjust itself in
some unmechanical way} until a stationary state is reached in which the
frequency (or frequencies) of the above mentioned slow variation of the orbit
has a relation to the additional energy of the system due to the presence of the
external field, which is of the same kind as the relation, expressed by (8) and
(10), between the energy and frequency of a periodic system of one degree of
freedom. As it will be shown in Part II in connection with the physical
applications, this condition is just secured if the stationary states in the
presence of the field are determined by the conditions (22), and it will be seen
that these considerations offer a means of fixing the stationary states of a
perturbed periodic system also in cases where no separation of variables can be
obtained.
In consequence of the singular position of the degenerate systems in the general
theory of stationary states of conditionally periodic systems, we obtain a means
of {\it connecting mechanically two different stationary states of a given
system} through a continuous series of stationary states without passing through
systems in which the forces are very small and the energies in all the
stationary states tend to coincide (comp. page 9). In fact, if we consider a
given conditionally periodic system which can be transformed in a continuous way
into a system for which every orbit is periodic and for which every state
satisfying (24) will also satisfy (22) for a suitable choice of coordinates, it
is clear in the first place that it is possible to pass in a mechanical way
through a continuous series of stationary states from a state corresponding to a
given set of values of the $n$'s in (22) to any other such state for which
$\kappa_1 n_1 + \ldots + \kappa_s n_s$ possesses the same value. If, moreover,
there exists a second periodic system of the same character to which the first
periodic system can be transformed continuously, but for which the set of
$\kappa$'s is different, it will be possible in general by a suitable cyclic
transformation to pass in a mechanical way between any two stationary states of
the given conditionally periodic system satisfying (22).
{\small To obtain an example of such a cyclic transformation let us take the
system consisting of an electron which moves round a fixed positive nucleus
exerting an attraction varying as the inverse square of the distance. If we
neglect the small relativity corrections, every orbit will be periodic
independent of the initial conditions and the system will allow of separation of
variables in polar coordinates as well as in any set of elliptical coordinates,
of the kind mentioned on page 23, if the nucleus is taken as one of the foci. It
is simply seen that any orbit which satisfies (24) for a value of $n > 1$, will
satisfy (22) for a suitable choice of elliptical coordinates. By imagining
another nucleus of infinite small charge placed at the other focus, the orbit
may further be transformed into another which satisfies (24) for the same value
of $n$ but which may have any given value for the eccentricity. Consider now a
state of the system satisfying (24), and let us assume that by the above means
the orbit is originally so adjusted that in plane polar coordinates it will
corresponding to $n_1 = m$ and $n_2 = n - m$ in (16). Let then the system
undergo a slow continuous transformation during which the field of force acting
on the electron remains central, but by which the law of attraction is slowly
varied until the force is directly proportional to the distance apart. In the
final state, as well as in the original state, the orbit of the electron will be
closed, but during the transformation the orbit will not be closed, and the
ratio between the mean period of revolution and the period of the radial motion,
which in the original motion was equal to one, will during the transformation
increase continuously until in the final state it is equal to two. This means
that, using polar coordinates, the values of $\kappa_1$ and $\kappa_2$ in (22)
which for the first state are equal to $\kappa_1 - \kappa_2 = 1$, will be for
the second state $\kappa_1 = 2$ and $\kappa_1 = 1$. Since during the
transformation $n_1$ and $n_2$ will keep their values, we get therefore in the
final state $I = h \cdot (2m + (n - m)) = h \cdot (n + m)$. Now in the latter
state, the system allows a separation of variables not only in polar coordinates
but also in any system of rectangular Cartesian coordinates, and by suitable
choice of the direction of the axes, we can obtain that any orbit, satisfying (
24) for a value of $n > 1$, will also satisfy (22). By an infinite small change
of the force components in the axes, in such a way that the motions of these
directions remain independent of each other but possess slightly different
periods, it will further be possible to transform the elliptical orbit
mechanically into one corresponding to any ratio between the axes. Let us now
assume that in this way the orbit of the electron is transformed into a circular
one, so that, returning to plane polar coordinates, we have $n_1 = 0$ and $n_2 =
n + m$, and let then by a slow transformation the law of attraction be varied
until again it is that of the inverse square. It will be seen that when this
state is reached the motion will again satisfy (24), but this time we will have
$I = h \cdot (n + m)$ instead of $I = nh$ as in the original state. By repeating
a cyclic process of this kind we may pass from any stationary state of the
system in question which satisfies (24) for a value of $n > 1$ to any other such
state without leaving at any moment the region of stationary states.}\\
The theory of the mechanical transformability of the stationary states gives us
a means to discuss the question of the {\it a-priori probability} of the
different states of a conditionally periodic system, characterised by different
sets of values for the $n$'s in (22). In fact from the considerations, mentioned
in $\S$~1, it follows that, if the a--priori probability of the stationary
states of a given system is known, it is possible at once to deduce the
probabilities for the stationary states of any other system to which the first
system can be transformed continuously without passing through a system of
degeneration. Now from the analogy with systems of one degree of freedom it
seems necessary to assume that, for a system of several degrees of freedom for
which the motions corresponding to the different coordinates are dynamically
independent of each other, the a--priori probability is the same for all states
corresponding to different sets of $n$'s in (15). According to the above we
shall therefore assume that the a--priori probability is the same for all
states, given by (22), of a by which can be formed in a continuous way from a
system of this kind without passing through systems of degeneration. It will be
observed that on this assumption we obtain exactly the same relation to the
ordinary theory of statistical mechanics in the limit of large $n$'s as obtained
in the case of systems of one degree of freedom. Thus, for a conditionally
periodic system, the volume
given by (11) of the element of phase--space, including all points
$q_1, \ldots q_s,~ p_1, \ldots p_s$ which represent states for which the value
of $I_k$ given by (21) lies between $I_k$ and $I_k + \delta I_k$ it seen at once
to be equal to\footnote{Comp. A. Sommerfeld, Ber. Akad. M\"unchen, 1917, p. 83.}
\begin{equation}
\delta W = \delta I_1 \delta I_2 \ldots \delta I_s,
\end{equation}
if the coordinates are so chosen that the motion corresponding to every degree
of freedom is of oscillating type. The volume of the \mbox{phase--space} limited
by $s$ pairs of surfaces, corresponding to successive values for the $n$'s in
the conditions (22), will therefore be equal to $h^s$ and consequently be the
same for every combination of the $n$'s. In the limit where the $n$'s are large
numbers and the stationary states corresponding to successive values for the $
n$'s differ only very little from each other, we thus obtain the same result on
the assumption of equal a--priori probability of all the stationary states,
corresponding to different sets of values of $n_1, n_2, \ldots n_s$ in (22), as
would be obtained by application of ordinary statistical mechanics.
The fact that the last considerations hold for every \mbox{non--degenerate}
conditionally periodic system suggests the assumption that in general {\it a--
priori probability will be the same for all the states determined by (22)}, even
if it should not be possible to transform the given system into a system of
independent degrees of freedom without passing through degenerate systems. This
assumption will be shown to be supported by the consideration of the intensities
of the different components of the Stark--effect of the hydrogen lines,
mentioned in the next Part. When we consider a degenerate system, however, we
cannot assume that the different stationary states are a--priori equally
probable. In such a case the stationary states will be characterized by a number
of conditions less than the number of degrees of freedom, and the probability of
a given state must be determined from the number of different stationary states
of some \mbox{non--degenerate} system which will coincide in the given state, if
the latter system is continuously transformed into the degenerate system under
consideration.
In order to illustrate this, let us take the simple case of a degenerate system
formed by an electrified particle moving in the plane orbit in a central field,
the stationary states of which are given by the two conditions (16). In this
case the plane of the orbit is undetermined, and it follows already from a
comparison with ordinary statistical mechanics, that the a--priori probability
of the states characterized by different combinations of $n_1$ and $n_2$ in (16)
cannot be the same. Thus the volume of the \mbox{phase--space}, corresponding to
states for which $I_1$ lies between and $I_1$ and $I_1 + \delta I_1$ and for
which $I_2$ lies between $I_2$ and $I_2 + \delta I_2$ , is found by a simple
calculation\footnote{See A. Sommerfeld, loc. cit.} to be equal to $\delta W = 2
I_2 \delta I_1 \delta I_2$, if the motion is described by ordinary polar
coordinates. For large values of $n_1$ and $n_2$, we must therefore expect that
the a--priory probability of a stationary state corresponding to a given
combination $(n_1, n_2)$ is proportional to $n_2$. The question of the a--priori
probability of states corresponding to small values of the $n$'s has been
discussed by Sommerfeld in connection with the problem of the intensities of the
different components in the fine structure of the hydrogen lines (see Part II).
From considerations about the volume of the extensions in the \mbox{phase--
space}, which might be considered as associated with the states characterised by
different combinations $(n_1, n_2)$, Sommerfeld proposes several different
expressions for the \mbox{a--priori} probability of such states. Due to the
necessary arbitrariness involved in the choice of these extensions, however. we
cannot in this way obtain a rational determination of the \mbox{a--priori}
probability of states corresponding to small values of $n_1$ and $n_2$. On the
other hand, this probability may be deduced by regarding the motion of the
system under consideration as the degeneration of a motion characterised by
three numbers $n_1, n_2$ and $n_3$, as in the general applications of the
conditions (22) to a system of three degrees of freedom. Such a motion may be
obtained for instance by imagining the system placed in a small homogeneous
magnetic field. In certain respects this case falls outside the general theory
of conditionally periodic system discussed in this section, but, as we shall see
in Part II, it can be simply shown that the presence of the magnetic field
imposes the further condition on the motion in the stationary states that the
angular momentum round the axis of the field is equal to $n' h/2 \pi$, where, $
n'$ is a positive entire number equal to or less than $n_2$, and which for the
system considered in the spectral problems must be assume to be different from
zero. when regard is taken to the two opposite directions in which the particle
may rotate round the axis of the field, we see therefore that for this system a
state corresponding to a given combination of $n_1$ and $n_2$ in the presence of
the field can be established in $2 n_2$ different ways. The \mbox{a--priori}
probability of the different states of the system may consequently for all
combinations of $n_1$ and $n_2$ be assumed to be proportional to $n_2$.
The assumption just mentioned that the angular momentum round the axis of the
field cannot be equal to zero is deduced from considerations of system for which
the motion corresponding to special combinations of the $n$'s in (22) would
become physically impossible due to some singularity in its character. In such
cases we must assume that no stationary states exist corresponding to the
combinations $(n_1, n_2, \ldots n_s)$ under consideration, and on the above
principle of the invariance of the \mbox{a--priori} probability for continuous
transformations we shall accordingly expect that the \mbox{a--priori}
probability of any other state, which can be transformed continuously into one
of these states without passing through cases of degeneration, will also be
equal to zero.
Let us now produced to consider the {\it spectrum of a conditionally periodic
system}, calculated from the values of the energy in the stationary states by
means of relation (1). If $E \cdot (n_1, \ldots n_s)$ is the total energy of a
stationary state determined by (22) and if $\nu$ is the frequency of the line
corresponding to the transition between two stationary states characterised by $
n_k = n'_k$ and $n_k = n''_k$ respectively, we have
\begin{equation}
\nu = \frac{1}{h} \cdot \left[ E \cdot \left(n'_1, \ldots n'_s \right) - E \cdot
\left(n''_1, \ldots n''_s\right) \right].
\end{equation}
In general, this spectrum will be entirely different from the spectrum to be
expected on the ordinary theory of electrodynamics from the motion of the
system. Just as for a system of one degree of freedom we shall see, however,
that in the limit where the motions in neighbouring stationary states differ
very little from each other, there exists a close relation between the spectrum
calculated on the quantum theory and that to be expected on ordinary
electrodynamics. As in $\S$~2 we shall further see, that this connection leads
to certain general considerations about the probability of transition between
any two stationary states and about the nature of the accompanying radiation
which are found to be supposed by observations. In order to discuss this
question we shall first deduce a general expression for the energy difference
between neighbouring of a conditionally periodic system, which can be simple
obtained by a calculation analogous to that used in $\S$~2 in the deduction of
the relation (8).
Consider some motion of a conditionally periodic system which allows of
separation of variables in a certain set of coordinates $q_1, \ldots q_s$ and
let us assume that at the time $t = \vartheta$ the configuration of the system
will to a close approximation be the same as at the time $t = 0$. By taking
$\vartheta$ large enough we can make this approximation as close as we like. If
next we consider some conditionally periodic motion, obtained by a small
variation of the first motion, and which allows of separation of variables in a
set of coordinates $q'_1, \ldots q'_s$ which may differ slightly from the set $
q_1, \ldots q_s$, we get by means of Hamilton's equations (4), using the
coordinated $q'_1, \ldots q'_s$,
$$
\int \limits^{\vartheta}_0~ \delta E~dt = \int \limits^{\vartheta}_0~ \sum^s_
1~ \left( \frac{\partial E}{\partial p'_k} \cdot \delta p'_k + \frac{\partial
E}{\partial q'_k} \cdot \delta q'_k \right)~ dt =
$$
$$
= \int \limits^{\vartheta}_0~
\sum^s_1~ \left(\dot q_k \cdot \delta p'_k - p'_k \cdot \delta
q'_k \right) ~ dt.
$$
By partial integration of the second term in the bracket this gives:
\begin{equation}
\int \limits^{\vartheta}_0~ \delta E ~ dt = \int \limits^{\vartheta}_{0}~
\sum^s_1~ \delta ~ \left( p'_k \cdot \dot q'_k \right) \cdot dt - \mid
\sum^s_1~ p'_k \cdot \delta q'_k \mid^{t=\vartheta}_{t=0}.
\end{equation}
Now we have for the unvaried motion
$$
\int \limits^{\vartheta}_0 ~ \sum^s_1~ p'_k \cdot \dot q_k dt = \int
\limits^{\vartheta}_0~ \sum^s_1~ p_k \dot q_k ~ dt = \sum^s_1~ N_k \cdot I_
k,
$$
where $I_k$ is defined by (21) and where $N_k$ is the number of oscillations
performed by $q_k$ in the time interval $\vartheta$. For the varied motion we
have on the other hand:
$$
\int \limits^{\vartheta}_0~ \sum^s_1~ p'_k \dot q'_k ~ dt = \int
\limits^{t = \vartheta}_{t = 0} ~ \sum^s_1~ p'_k \cdot d q'_k = \sum^s_1 ~ N_k
\cdot I'_k + \mid \sum^s_1~ p'_k \cdot \delta q'_k \mid^{t = \vartheta}_{t
= 0},
$$
where the $I$'s correspond to the conditionally periodic motion in the
coordinates $q'_1, \ldots q'_s$, and the $\delta q'$'s which enter in the last
term are the same as those in (27). writing $I'_k - I_k = \delta \cdot I_k$, we
get therefore from the latter equation
\begin{equation}
\int \limits^{\vartheta}_0~ \delta E ~ dt = \sum^s_1~ N_k \cdot \delta
I_k.
\end{equation}
In the special case where the varied motion is an undisturbed motion belonging
to the same system as the unvaried motion we get, since $\delta \cdot E$ will be
constant,
\begin{equation}
\delta E = \sum^s_1~ \omega_k \cdot \delta I_k,
\end{equation}
where $\omega_k = N_k/\vartheta$ is the mean frequency of oscillation of $q_k$
between its limits, taken over a long time interval of the same order of
magnitude as $\vartheta$. This equation forms a simple generalisation of (8),
and in the general case in which a separation of variables will be possible only
for one system of coordinates leading to a complete definition of the $I$'s it
might have been deduced directly from the analytical theory of the periodicity
properties of the motion of a conditionally periodic system, based on the
introduction of angle variables.\footnote{See Charlier, die Mechanik des
Himmels, bd. I abt. 2, and especially P. Epstein, Ann. d. Phys. {\bf 51} (1916)
178. By means of the well known theorem of Jacobi about the change of variables
in the canonical equations of Hamilton, the connection between the notion of
\mbox{angle--variables} and the quantities $I$, discussed by Epstein in the
latter paper, may be briefly exposed in the following elegant manner which has
been kindly pointed out to me by Mr. H.A. Kramers. Consider the function $S (q_
1, \ldots q_s,~ I_1, \ldots I_s)$ obtained from (20) by introducing for the
$\alpha$'s their expressions in terms of the $I$'s given by the equations (21).
This function will be a many valued function of the $q$'s which increases by $I_
k$ if $q_k$ described one oscillation between its limits and comes back to its
original value while the other $q$'s remain constant. If we therefore introduce
a new set of variables $w_1, \ldots w_s$ defined by
$$
w_k = \frac{\partial S}{\partial I_k}, ~~~~ (k = 1, \ldots s)
\eqno(1^{\ast})
$$
it will be seen that $w_k$ increases by one unit while the other $w$'s will come
back to their original values if $q_k$ described one oscillation between its
limits and the other $q$'s remain constant. Inversely it will therefore be seen
that the $q$'s, and also the $p$'s which were given by
$$
p_k = \frac{\partial S}{\partial q_k}, ~~~~ (k = 1, \ldots s),
\eqno(2^{\ast})
$$
when considered as functions of the $I$'s and $w$'s will be periodic functions
of every of the $w$'s with period. according to Fourier's theorem any of the $
q$'s may therefore be represented by an $s$--double trigonometric series of the
form
$$
q = \sum ~ A_{\tau_1, \ldots \tau_s} ~ \cos ~ 2 \pi \cdot (\tau_1 \cdot w_1 +
\ldots \tau_s \cdot w_s = \alpha_{\tau_1 \ldots \tau_s}),
\eqno(3^{\ast})
$$
where the $A$'s and $\alpha$'s are constants depending on the $I$'s and where
the summation is to be extended over all entire values of $\tau_1, \ldots \tau_
s$. On account of this property of the $w$'s, the quantities $2 \pi w_1, \ldots
2 \pi w_s$ are denoted as `angle variables'. Now from (1$^{\ast}$)
and (2$^{\ast}$) it follows
according to the above mentioned theorem of Jacobi (see for instance Jacobi,
Vorlesungen \"uber Dynamik $\S$~37) that the variations with the time of the $
I$'s and $w$'s will be given by
$$
\frac{d I_k}{dt} = - \frac{\partial E}{\partial w_k}, ~~~~ \frac{d w_k}{dt} =
\frac{\partial E}{\partial I_k}, ~~~~ (k = 1, \ldots s)
\eqno(4^{\ast})
$$
where the energy $E$ is considered as a function of the $I$'s and $w$'s. Since $
E$, however, is determined by the $I$'s only we get from (4$^{\ast}$),
besides the evident result that the $I$'s are constant during the motion, that
the $\omega$'s will vary linearly with the time and can be represented by
$$
\omega_k = \omega_k t + \delta_k, ~~~ \omega_k = \frac{\partial E}{\partial I_
k}, ~~~~ (k = 1, \ldots s) \eqno(5^{\ast})
$$
where $\delta_k$ is a constant, and where $w_k$ is easily seen to be
equal to the mean frequency of oscillation of $q_k$.
From (5$^{\ast}$) eq. (28) follows
at once, and it will further be seen that by introducing
(5$^{\ast}$) in (3$^{\ast}$) we get
the result that every of the $q$'s, and consequently also any \mbox{one--valued}
function of the $q$'s, can be represented by an expression of the type (31).
In this connection it may be mentioned that the method of Schwarzschild of
fixing the stationary states of a conditionally periodic system, mentioned on
page 117, consists in seeking for a given system a set of canonically conjugated
variables $Q_1, \ldots Q_s,~ P_1, \ldots P_s$ in such a way that the positional
coordinates of the system $q_1, \ldots q_s$ and their conjugated momenta $p_1,
\ldots p_s$, when considered as functions of the $Q$'s and $P$'s, are periodic
in every of the $Q$'s with period $2 \pi$, while the energy of the system
depends only on the $P$'s. In analogy with the condition which fixes the angular
momentum in Sommerfeld's theory of central systems Schwarzschild next puts every
of the $P$'s equal to an entire multiplum of $h/2 \pi$. In contrast to the
theory of stationary states of conditionally periodic systems based on the
possibility of separation of variables and the fixation of the $I$'s by (22),
this method does not lead to an absolute fixation of the stationary states,
because, as pointed out by Schwarzschild himself, the above definition of the $
P$'s leaves an arbitrary constant undermined in every of these quantities. In
many cases, however, these constants may be simple determined from
considerations of mechanical transformability of the stationary states, and as
pointed out by Burgers [loc. cit. Versl. Akad. Amsterdam 25 (1917) 1055]
Schwaszschild method possesses on the other hand the essential advantage of
being applicable to certain classes of system in which the displacements of the
particles may be represented by trigonometric series of the type (31), but for
which the equations of motion cannot be solved by separation of variables in any
fixed set of coordinates. An interesting application of this to the spectrum of
rotating molecules, given by Burgers, will be mentioned in Part IV.}
From (29)
it follows moreover that, if the system allows of a separation of
variables in an
infinite continuous of sets of coordinates, the total energy will be the same
for all motions corresponding to the same values of the $I$'s, independent of
the special set of coordinates used to calculate these quantities. as mentioned
above and as we have already shown in the case of purely periodic systems by
means of (8), the total energy is therefore also in cases of degeneration
completely determined by the conditions (22).
Consider now a transition between two stationary states determined by (22) by
putting $n_k = n'_k$ and $n_k = n''_k$ respectively, and let us assume that $n'_
1, \ldots n'_s,~ n''_1, \ldots n''_s$ are large numbers, and that the
differences $n'_k - n''_k$ are small compared with these numbers.
Since the motions of the system in
these states will differ relatively very little from each other we may calculate
the difference of the energy by means of (29), and we get therefore, by means of
(1), for the frequency of the radiation corresponding to the transition between
the two states
\begin{equation}
\nu = \frac{1}{h} \cdot \left( E' - E'' \right) = \frac{1}{h}~ \sum^s_1~ \omega_
k \cdot \left( I'_k - I''_k \right) = \sum^s_1 ~ \omega_k \cdot
\left( n'_k - n''_k \right),
\end{equation}
which is seen to be a direct generalisation of the expression (13) in $\S$~2.
Now, in complete analogy to what is the case for periodic systems of one degree
of freedom, it is proved in the analytical theory of the motion of conditionally
periodic mentioned above that for the latter systems the coordinates $q_1,
\ldots q_s$, and consequently also the displacements of the particles in any
given direction, may be expressed as a function of the time by an $s$--double
infinite Fourier series of the form:
\begin{equation}
\xi = \sum ~ C_{\tau_1, \ldots \tau_s} \cdot \cos 2 \pi \left \{ \left( \tau_1
\cdot \omega_1 + \ldots \tau_s \cdot \omega_s \right) \cdot t + c_{\tau_1,
\ldots \tau_s} \right \},
\end{equation}
where the summation is to be extended over all positive and negative entire
values of the $\tau$'s, and where the $\omega$'s are the above mentioned mean
frequencies of oscillation for the different $q$'s. The constants $C_{\tau_1,
\ldots \tau_s}$ depend only on the $\alpha$'s in the equations (18) or, what is
the same, on the $I$'s, while the constants $c_{\tau_1, \ldots \tau_s}$ depend
on the $\alpha$'s as well as on the $\beta$'s. In general the quantities $\tau_1
\omega_1 + \ldots \tau_s \omega_s$ will be different for any two different sets
of values for the $\tau$'s, and in the course of time the orbit will cover
everywhere dense a certain $s$--dimensional extension. In a case of
degeneration, however, where the orbit will be confined to an extension of less
dimensions, there will exist for all values of the $\alpha$'s one or more
relations of the type $m_1 \omega_1 + \ldots m_s \omega_s = 0$ where the $m$'s
are entire numbers and by the introduction of which the expression (31) can be
reduced to a Fourier series which is less than $s$--double infinite. Thus in the
special case of a system of which every orbit is periodic we have $\omega_1/
\kappa_1 = \ldots = \omega_s/\kappa_s = \omega$, where the $\kappa$'s are the
numbers which enter in eq. (23), and the Fourier series for the displacements in
the different directions will in this case consist only of terms of the simple
form $C_{\tau} \cdot \cos 2 \pi \cdot \left\{ \tau \cdot \omega \cdot t +
c_{\tau} \right\}$, just as for a system of one degree of freedom.
On the ordinary theory of radiation, we should expect from (31) that the
spectrum emitted by the system in a given state would consist of an $s$--double
infinite series of lines of frequencies equal to $\tau_1 \omega_1 + \ldots \tau_
s \omega_s$. In general, this spectrum would be completely different from that
given by (26). This follows already from the fact that the $\omega$'s will
depend on the values for the constants $\alpha_1, \ldots \alpha_s$ and will vary
in a continuous way for the continuous multitude of mechanically possible states
corresponding to different sets of values for these constants. Thus in general
the $\omega$'s will be quite different for two different stationary states
corresponding to different sets of $n$'s in (22), and we cannot expect any close
relation between the spectrum calculated on the quantum theory and that to be
expected on the ordinary theory of mechanics and electrodynamics. In the limit,
however, where the $n$'s in (22) are large numbers, the ratio between the
$\omega$'s for two stationary states, corresponding to $n_k = n'_k$ and $n_k =
n''_k$ respectively, will tend to unity if the differences $n'_k = n''_k$ are
small compared with the $n$'s, and as seen from (30) the spectrum calculated by
(1) and (22) will in this limit just tend to coincide with that to be expected
on the ordinary theory of radiation from the motion of the system.
As far as the frequencies are concerned, we thus see that for conditionally
periodic systems there exists a connection between the quantum theory and the
ordinary theory of radiation of exactly the same character as that shown in $\S$
~2 to exist in the simple case of periodic systems of one degree of freedom. Now
on ordinary electrodynamics the coefficients $C_{\tau_1, \ldots \tau_s}$ in the
expression (31) for the displacements of the particles in the different
directions would in the well known determine the intensity and polarisation of
the emitted radiation of the corresponding frequency $\tau_1 \omega_1 + \ldots
\tau_s \omega_s$. As for systems of one degree of freedom we must therefore
conclude that, in the limit of large values for the $n$'s, the probability of
spontaneous transition between two stationary states of a conditionally periodic
system, as well as the polarisation of the accompanying radiation, can be
determined directly from the values of the coefficient $C_{\tau_1, \ldots \tau_
s}$ in (31) corresponding to a set of $\tau$'s given by $\tau_k = n'_k - n''_
k$, if $n'_1, \ldots n'_s$ and $n''_1, \ldots n''_s$ are the numbers which
characterise the two stationary states.
Without a detailed theory of the mechanism of transition between the stationary
states we cannot, of course, in general obtain an exact determination of the
{\it probability of spontaneous transition} between two such states, unless the
$n$'s are large numbers. Just as in the case of systems of one degree of
freedom, however, we are naturally led from the above considerations to assume
that, also for values of the $n$'s which are not large, there must exist an
intimate connection between the probability of a given transition and the values
of the corresponding Fourier coefficient in the expressions for the
displacements of the particles in the two stationary states. This allows us at
once to draw certain important conclusions. Thus, from the fact that in general
negative as well as positive values for the $\tau$'s appear in (31), it follows
that we must expect that in general not only such transitions will be possible
in which all the $n$'s decrease, but that also transitions will be possible for
which some of the $n$'s increase while others decrease. This conclusion, which
is supported by observations on the fine structure of the hydrogen lines as well
as on the Stark effect, is contrary to the suggestion, put forward by
Sommerfeld with reference to the essential positive character of the $I$'s, that
every of the $n$'s must remain constant or decrease under a transition. Another
direct consequence of the above considerations is obtained if we consider a
system for which, for all values of the constants $\alpha_1, \ldots \alpha_s$
the coefficient $C_{\tau_1, \ldots \tau_s}$ corresponding to a certain set
$\tau^0_1, \ldots \tau^0_s$, of values for the $\tau$'s is equal to zero in the
expressions for the displacements of the particles in every direction. In this
case we shall naturally expect that no transition will be possible for which the
relation $n'_k - n''_k = \tau^0_k$ is satisfied for every $k$. In the case where
$C_{\tau^0_1, \ldots \tau^0_s}$, is equal to zero in the expressions for the
displacement to a certain direction only, we shall expect that all transitions,
for which $n'_k - n''_k \tau^0_k$ for every $k$, will be accompanied by a
radiation which is polarized in a plane perpendicular to this direction.
A simple illustration of the last considerations is afforded by the system
mentioned in the beginning of this section, and which consists of a particle
executing motions in three perpendicular directions which are independent of
each other. In this case all the fourier coefficients in the expressions for the
displacements in any direction will disappear if more than one of the $\tau$'s
are different from zero. Consequently we must assume that only such transitions
are possible for which only one of the $n$'s varies at the same time, and that
the radiation corresponding to such a transition will be linearly polarized in
the direction of the displacement of the corresponding coordinate. In the
special case where the motions in the three directions are simply harmonic, we
shall moreover conclude that none of the $n$'s can vary by more than a single
unit, in analogy with the considerations in the former section about a linear
harmonic vibrator.
Another example which has more direct physical importance, since it includes all
the special applications of the quantum theory to spectral problems mentioned in
the introduction, is formed by a conditionally periodic system possessing an
axis of symmetry. In all these applications a separation of variables is
obtained in a set of three coordinates $q_1, q_2$ and $q_3$, of which the first
two serve to fix the position of the particle in a plane through the axis of the
system, while the last is equal to the angular distance between this plane and a
fixed through the same axis. Due to the symmetry, the expression for the total
energy in Hamilton's equations will not contain the angular distance $q_3$ but
only the angular momentum $p_3$ round the axis. The latter quantity will
consequently remain constant during the motion, and the vibrations of $q_1$ and
$q_2$ will be exactly the same as in a conditionally periodic system of two
degrees of freedom only. If the position of the particle is described in a set
of cylindrical coordinates $z, \varrho, \vartheta$, where $z$ is the
displacement in the direction of the axis, $\varrho$ the distance of the
particle from this axis and $\vartheta$ is equal to the angular distance $q_3$,
we have therefore
$$
z = \sum ~ C_{\tau_1, \tau_2}~ \cos 2 \pi \cdot \left\{ \left( \tau_1
\cdot \omega_1 + \tau_2 \cdot \omega_2 \right) \cdot t + c_{\tau_1, \tau_2}
\right\}
$$
and
\begin{equation}
\varrho = \sum ~ C'_{\tau_1, \tau_2}~ \cos 2 \pi \cdot \left\{ \left(
\tau_1 \cdot \omega_1 + \tau_2 \cdot \omega_2 \right) \cdot t + c'_{\tau_1,
\tau_2} \right\},
\end{equation}
where the summation is to be extended over all positive and negative entire
values of $\tau_1$ and $\tau_2$, and where $\omega_1$ and $\omega_2$ are the
mean frequencies of oscillation of the coordinates $q_1$ and $q_2$. For the rate
of variation of $\vartheta$ with the time we have further
$$
\frac{d \vartheta}{dt} = \dot q_3 = \frac{\partial E}{\partial p_3} = f \cdot
\left( q_1, q_2, p_1, p_2, p_3 \right) =
$$
$$
= \pm \sum~ C''_{\tau_1, \tau_2} \cos 2
\pi ~ \left \{ \left( \tau_1 \cdot \omega_1 + \tau_2 \cdot \omega_2 \right)
\cdot t + C''_{\tau_1, \tau_2} \right\},
$$
where the two signs correspond to a rotation of the particle in the direction of
increasing and decreasing $q_3$ respectively, and are introduced to separate the
two types of symmetrical motions corresponding to these directions. This gives
\begin{equation}
\pm \vartheta = 2 \pi \cdot \omega_3 \cdot t + \sum ~ C'''_{\tau_1, \tau_2}
\cdot \cos ~ 2 \pi ~ \left\{ \left( \tau_1 \cdot \omega_1 + \tau_2 \cdot
\omega_2 \right) \cdot t + c'''_{\tau_1, \tau_2} \right\},
\end{equation}
where the positive constant $\omega_3 = C''_{0,0}/2 \pi$ is the mean frequency
of rotation round the axis of symmetry of the system. Considering now the
displacement of the particle in rectangular coordinates $x, y$ and $z$, and
taking as above the axis of symmetry as $z$-axis, we get from (32) and (33)
after a simple contraction of terms
$$
x = \varrho \cos \vartheta = \sum~ D_{\tau_1, \tau_2} \cdot \cos 2 \pi \cdot
\left\{ \left( \tau_1 \cdot \omega_1 + \tau_2 \cdot \omega_2 + \omega_3 \right)
\cdot t + d_{\tau_1, \tau_2} \right\}
$$
and
\begin{equation}
y = \varrho \cos \vartheta = \pm \sum~ D_{\tau_1, \tau_2} \cdot \sin 2 \pi
\cdot \left\{ \left( \tau_1 \cdot \omega_1 + \tau_2 \cdot \omega_2 + \omega_3
\right) \cdot t + d_{\tau_1, \tau_2} \right\},
\end{equation}
where the $D$'s and $d$'s are new constants, and the summation is again to be
extended over all positive and negative values of $\tau_1$ and $\tau_2$.
From (32) and (34) we see that the motion in the present case may be considered
as composed of a number of linear harmonic vibrations parallel to the axis of
symmetry and of frequencies equal to the absolute values of $(\tau_1 \cdot
\omega_1 + \tau_2 \cdot \omega_2)$, together with a number of circular harmonic
motions round this axis equal to the absolute values of $( \tau_1 \cdot \omega_1
+ \tau_2 \cdot \omega_2 + \omega_3)$ and possessing the same direction of
rotation as that of the moving particle or the opposite if the latter expression
is positive or negative respectively. According to ordinary electrodynamics the
radiation from the system would therefore consist of a number of components of
frequency $\mid \tau_1 \cdot \omega_1 + \tau_2 \cdot \omega_2 \mid$ polarised
parallel to the axis of symmetry, and a number of components of frequencies
$\mid \tau_1 \cdot \omega_1 + \tau_2 \cdot \omega_2 + \omega_3 \mid$ and of
circular polarisation round this axis (when viewed in the direction of the
axis). On the present theory we shall consequently expect that in this case only
kinds of transition between the stationary states given by (22) will be
possible. In both of these $n_1$ and $n_2$ may vary by an arbitrary number of
units, but in the first kind of transition, which will give rise to a radiation
polarised parallel to the axis of the system, $n_3$ will remain unchanged, while
in the second kind of transition $n_3$ will decrease or increase by one unit and
the emitted radiation will be circularly polarised round the axis in the same
direction as or the opposite of that of the rotation of the particle
respectively.
In the next Part we shall see that these conclusions are supported in an
instructive manner by the experiments on the effects of electric and magnetic
field on the hydrogen spectrum. In connection with the discussion of the general
theory, however, it may be of interest to show that the formal analogy between
the ordinary theory of radiation and the theory based on (1) and (22), in case
of systems possessing an axis of symmetry, can be traced not only with respect
to frequency relations but also by considerations of {\it conservation of
angular momentum}. For a conditionally periodic system possessing an axis of
symmetry the angular momentum round this axis is, with above choice of
coordinates, according to (22) equal to $I_3/2 \pi = n_3 h/ 2 \pi$. If
therefore, as assumed above for a transition corresponding to an emission of
linearly polarised light, $n_3$ is unaltered, it means that the angular momentum
of the system remains unchanged, while if $n_3$ alters by one unit, as assumed
for a transition corresponding to an emission of circularly polarised light, the
angular momentum will be altered by $h/2 \pi$. Now it is easily seen that the
ratio between this amount of angular momentum and the amount of energy $h \nu$
emitted during the transition is just equal to the ratio between the amount of
angular momentum and energy possessed by the radiation which according to
ordinary electrodynamics would be emitted by an electron rotating in a circular
orbit in a central field of force. In fact, if $\alpha$ is the radius of the
orbit, $\nu$ the frequency of revolution and $F$ the force of reaction due to
the electromagnetic field of the radiation, the amount of energy and of angular
momentum round an axis through the centre of the field perpendicular to the
plane of the orbit, lost by the electron in unit of time as a consequence of the
radiation, would be equal to $2 \pi \nu \alpha F$ and $\alpha F$ respectively.
Due to the principles of conservation of energy and of angular momentum holding
in ordinary electrodynamics, we should therefore expect that the ratio between
the energy and the angular momentum of the emitted radiation would be $2 \pi
\nu$, \footnote{Comp. K. Schaposchnikow, Phys. Zeitschr. {\bf 15} (1914) 454.}
but this seen to be equal to the ratio between the energy $h \nu$ and the
angular momentum $h/2 \pi$ lost by the system considered above during a
transition for which we have assumed that the radiation is circularly polarised.
This agreement would seem not only to support the validity of the above
considerations but also to offer a direct support, independent of the equations
(22), of the assumption that, {\it for a atomic system possessing an axis of
symmetry, the total angular momentum round this axis is equal to an entire
multiple of $h/2 \pi$.}
A further illustration of the above considerations of the relation between the
quantum theory and the ordinary theory of radiation is obtained if we consider a
conditionally periodic system subject to the {\it influence of a small
perturbing field of force}. Let us assume that the original system allows of
separation of variables in a certain set of coordinates $q_1, \ldots q_s$, so
that the stationary states are determined by (22),. From the necessary stability
of the stationary states we must conclude that the perturbed system will possess
a set of stationary states which only differ slightly from those of the original
system. In general, however, it will not be possible for the perturbed system to
obtain a separation of variables in any set of coordinates, but if the
perturbing force is sufficiently small the perturbed motion will again be of
conditionally periodic type and may be regarded as a superposition of a number
of harmonic vibrations just as the original motion. The displacements of the
particles in the stationary states of the perturbed system will therefore be
given by an expression of the same type as (31) where the fundamental
frequencies $\omega_k$ and the amplitudes $C_{\tau_1, \ldots \tau_s}$ may differ
from those corresponding to the stationary states of the original system by
small quantities proportional to the intensity of the perturbing forces. If now
for the original motion the coefficients $C_{\tau_1, \ldots \tau_s}$
corresponding to certain combinations of the $\tau$'s are equal to zero for all
values of the constants $\alpha_1, \ldots \alpha_s$, these coefficients will
therefore for the perturbed motion, in general, possess small values
proportional to the perturbing forces. From the above considerations we shall
therefore expect that, in addition to the main probabilities of such transitions
between stationary states which are possible for the original system, there will
for the perturbed system exist small probabilities of new transitions
corresponding to the above mentioned combinations of the $\tau$'s. Consequently
we shall expect that the effect of the perturbing field on the spectrum of the
system will consist partly in a small displacement of the original lines, partly
in the appearance of new lines of small intensity.
A simple example of this afforded by a system consisting of a particle moving in
a plane and executing harmonic vibrations in two perpendicular directions with
frequencies $\omega_1$ and $\omega_2$. If the system is undisturbed all
coefficients $C_{\tau_1, \ldots \tau_2}$ will be zero, except $C_{1, 0}$ and $
C_{0, 1}$. When, however, the system is perturbed, for instance by an arbitrary
small central force, there will in the Fourier expressions for the displacements
of the particle, in addition to the main terms corresponding to the fundamental
frequencies $\omega_1$ and $\omega_2$, appear a number of small terms
corresponding to frequencies given by $\tau_1 \omega_1 + \tau_2 \omega_2$ where
$\tau_1$ and $\tau_2$ are entire numbers which may be positive as well as
negative. On the present theory we shall therefore expect that in the presence
of the perturbing force there will appear small probabilities for new
transitions which will give rise to radiations analogous to the socalled
harmonics and combination tones in acoustics, just as it should be expected on
the ordinary theory of radiation where a direct connection between the emitted
radiation and the motion of the system is assumed. Another example of more
direct physical application is afforded by the effect of an external homogeneous
electric field in producing new spectral lines. In this case the potential of
the perturbing force is a linear function of the coordinates of the particles
and, whatever is the nature of the original system, it follows directly from the
general theory of perturbations that the frequency of any additional term in the
expression for the perturbed motion, which is of the same order of magnitude as
the external force, must correspond to the sum or difference of two frequencies
of the harmonic vibrations into which the original motion can be resolved. With
applications of these considerations we will meet in Part II in connection with
the discussion of Sommerfeld's theory of the fine structure of the hydrogen
lines and in Part III in connection with the problem of the appearance of new
series in the spectra of other elements under the influence of intense external
electric field.
As mentioned we cannot without more detailed theory of the mechanism of
transition between stationary states obtain quantitative information as regards
the general question of the intensities of the different lines of the spectrum
of a conditionally periodic system given by (26), except in the limit where the
$n$'s are large numbers, or in such special cases where for all values of the
constants $\alpha_1, \ldots \alpha_s$ certain coefficient $C_{\tau_1, \ldots
\tau_s}$ in (31) are equal to zero. From considerations of analogy, however, we
must expect that it will be possible also in the general case to obtain an {
estimate of the intensities} of the different lines in the spectrum by comparing
the intensity of a given line, corresponding to a transition between two
stationary states characterised by the numbers $n'_1, \ldots n'_s$ and $n''_1,
\ldots n''_s$ respectively, with the intensities of the radiations of
frequencies $\omega_1 \cdot (n'_1 - n''_1) + \ldots + \omega_s \cdot (n'_s -
n''_s)$ to be expected on ordinary electrodynamics from the motions in these
states; although of course this estimate becomes more uncertain the smaller the
values for the $n$'s are. as it will be seen from the applications mentioned in
the following Parts this is supposed in a general way by comparison with the
observations.
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