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 line-spectra. Subsequently these conditions have been proved by Ehrenfest and especially by Burgers1) 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 E1 +· · ·+Es, where Ek contains qk and pk only.

An illustration of a system of this kind is presented by a particle moving in a field of force in which the 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:

…

where the integrals are taken over a complete period of the different q’s respectively, and where n1, . . . , ns 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 pk as a function of qk only. A simple system of this kind consists of a particle moving in a plane orbit in a central field of force.

Taking the length of the radius-vector from the centre of the field to the particle as q1, and the angular distance of this radius-vector from a fixed line in the plane of the orbit as q2, we get at once from (4), since E does not contain q2, the well known result that during the motion the angular momentum p2 is constant and that the radial motion, given by the variations of p1 and q1 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 non-periodic system Sommerfeld assumed therefore that the stationary states of the above system are given by two conditions of the form:

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 q2.

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 replaced by some function of this variable.

In fact, if instead of the angular distance of the radius-vector we take for q2 some continuous periodic function of this angle with period 2π, 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 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 multiple of h 2π .

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 follows 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 p2 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.1)

To this theory we shall come back in Part II.

As pointed out by Epstein 2) and Schwarzschild 3), the 1) In this connection it may be remarked that conditions of the same type as (16) were proposed independently by W. Wilson (Phil. Mag. XXIX p. 795 (1915) and XXXI p. 156 (1916)), 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

This theory, which has no direct relation to the problem of line-spectra discussed in the present paper, rests upon a profound analysis of the geometrical problem of dividing the multiple-dimensional phase 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).

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 so called conditionally periodic systems, to which we are led if the equations of motion are discussed by means of the Hamilton-Jacobi partial differential equation.1)

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:

obtained by putting this expression equal to an arbitrary constant α1. If then S = F(q1, . . . , qs, α1, . . . , αs) + C, where α2, . . . , αs, and C are arbitrary constants like α1, 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

where t is the time and β1, . . . , βs 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 α1, . . . , αs, β1, . . . , βs which may be determined for instance from the values of the q’s and ¨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

where Sk is a function of the s constants α1, . . . , αs and of qk only. In this case, in which the equation (17) allows of what is called “separation of variables”, we get from (19) that every p is a function of the α’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 α’s. Like in the case of a system of one degree of freedom, pk will become zero and change its sign whenever qk 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 q1, . . . , qs,

representing a point within a certain closed s-dimensional extension limited by s pairs of (s − 1)-dimensional surfaces 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 α’s and not on the β’s, which latter constants serve only to fix the exact configuration of the system at a given moment, when the α’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 α’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 q1, . . . , qs 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 central field, form a set of expressions of the type Ik = Z pk(qk, α1, . . . , αs) dqk, (k = 1, . . . , s) (21) where the integration is taken over a complete oscillation of qk. As, in general, the orbit will cover everywhere dense an s-dimensional extension limited in the characteristic way 35 mentioned above, it follows that, except in cases of degeneration, a separation of variables will not be possible for two different sets of coordinates q1, . . . , qs and q

), 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

Ik = nkh, (k = l, . . . , s) (22)

where n1, . . . , ns are positive entire numbers, we obtain therefore 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 α1, . . . , αs only and not on the β’s, the α’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 α1, 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 distances 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 ql and q2 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 q3 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 with period 2π.

A limiting case of this problem is afforded by an electron rotating round 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 f the electric force.

By applying the conditions (22) to this motion Epstein and Schwarzschild 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 in 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.

As mentioned above the conditions (22), first established from analogy with systems of one degree of freedom, have subsequently been proved generally to be mechanically invariant for any slow transformation for which the system remains conditionally periodic. The proof of this in variance has been given quite recently by Burgers2) by means of an interesting application of the theory of contacttransformations 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 § 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 a 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 coordinate-space can be made as small as we like.

This condition forms theref-ore 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 α’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 close 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 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 a 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,

where the integration is extended over one period of the motion, and where κ1, . . . , κs 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 satisfies (22), will be stationary. For any such motion we get from (23)

where n is a whole number which may take all positive values if, as in the applications mentioned below, at least one of the κ’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 satisfies 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 (10).

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 systems 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 42 field at a rate so slow that the comparative change of its intensity during a period 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 process 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, 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 43 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 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 14). 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 κ1n1+· · ·+κsns 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 κ’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). 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 ini- 44 tial 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 36, 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 also 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 (21), and let us assume that by the above means the orbit is originally so adjusted that in plane polar coordinates it will correspond to n1 = m and n2 = 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 κ1 and κ2 in (22) which for the first state are equal to κ1 = κ2 = 1, will be for the second state κ1 = 2 and κ2 = 1. Since during the transformation n1 and n2 will keep their values, we get therefore in the final state I = h

2m + (n − m)
= h(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 rectangula 45 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 > l, will also satisfy (22). By an infinite small change of the force components in the directions of the axes, in such a way that the motions in 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 given 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 n1 = 0 and n2 = 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(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 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 § 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 degener- 46 ation. 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 the 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 any system 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 q1, . . . , qs, p1, . . . , ps which represent states for which the value of Ik given by (21) lies between Ik and Ik + δIk, is seen at once to be equal to1 ) δW = δI1 δI2 . . . δIs, (25) if the coordinates are so chosen that the motion corresponding to every degree of freedom is of oscillating type. The volume of the 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 1 ) Comp. A. Sommerfeld, Ber. Akad. M¨unchen, 1917, p. 83. 47 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 n1, . . . , ns in (22), as would be obtained by application of ordinary statistical mechanics. The fact that the last considerations hold for every nondegenerate conditionally periodic system suggests the assumption that in general the 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 characterised 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 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 mov- 48 ing in a 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 characterised by different combinations of n1 and n2 in (16) cannot be the same. Thus the volume of the phase-space, corresponding to states for which I1 lies between I1 and I1 + δI1 and for which I2 lies between I2 and I2 + δI2, is found by a simple calculation1 ) to be equal to δW = 2I1 δI1 δI2, if the motion is described by ordinary polar coordinates. For large values of n1 and n2, we must therefore expect that the apriori probability of a stationary state corresponding to a given combination (n1, n2) is proportional to n2. 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 phase-space, which might be considered as associated with the states characterised by different combinations (n1, n2), Sommerfeld proposes several different expressions for the 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 a-priori probability of states corre1 ) See A. Sommerfeld, loc. cit. 49 sponding to small values of n1 and n2. 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 n1, n2 and n3, 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 systems 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 0 h 2π , where n 0 is a positive entire number equal to or less than n2, and which for the system considered in the spectral problems must be assumed 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 n1 and n2 in the presence of the field can be established in 2n2 different ways. The a-priori probability of the different states of the system may consequently for all combinations of n1 and n2 be assumed to be proportional to n2. The assumption just mentioned that the angular momentum round the axis of the field cannot be equal to zero is deduced from considerations of systems for which the mo- 50 tion 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 (n1, n2, . . . , ns) under consideration, and on the above principle of the invariance of the a-priori probability for continuous transformations we shall accordingly expect that the 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 proceed to consider the spectrum of a conditionally periodic system, calculated from the values of the energy in the stationary states by means of relation (1). If E(n1, . . . , ns) is the total energy of a stationary state determined by (22) and if ν is the frequency of the line corresponding to the transition between two stationary states characterised by nk = n 0 k and nk = n 00 k respectively, we have ν = 1 h

E(n 0 1 , . . . , n0 s ) − E(n 00 1 , . . . , n00 s )

. (26)

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 § 2 51 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 supported by observations. In order to discuss this question we shall first deduce a general expression for the energy difference between two neighbouring states of a conditionally periodic system, which can be simply obtained by a calculation analogous to that used in § 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 q1, . . . , qs, and let us assume that at the time t = ϑ the configuration of the system will to a close approximation be the same as at the time t = 0. By taking ϑ 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 0 1 , . . . , q 0 s which may differ slightly from the set q1, . . . , qs, we get by means of Hamilton’s equations (4), using the coordinates q 0 1 , . . . , q 0 s , Z ϑ 0 δE dt = Z ϑ 0 Xs 1  ∂E ∂p0 k δp0 k + ∂E ∂q0 k δq0 k  dt

Z ϑ 0 Xs 1 ( ˙q 0 k δp0 k − p˙ 0 k δq0 k ) dt. By partial integration of the second term in the bracket this 52 gives: Z ϑ 0 δE dt = Z ϑ 0 Xs 1 δ(p 0 k q˙ 0 k ) dt −      Xs 1 p 0 k δq0 k      t=ϑ t=0 . (27) Now we have for the unvaried motion Z ϑ 0 Xs 1 p 0 k q˙ 0 k dt = Z ϑ 0 Xs 1 pkq˙k dt = Xs 1 NkIk, where Ik is defined by (21) and where Nk is the number of oscillations performed by qk in the time interval ϑ. For the varied motion we have on the other hand: Z ϑ 0 Xs 1 p 0 k q˙ 0 k dt = Z t=ϑ t=0 Xs 1 p 0 kdq0 k = Xs 1 NkI 0 k+      Xs 1 p 0 k δq0 k      t=ϑ t=0 , where the I’s correspond to the conditionally periodic motion in the coordinates q 0 1 , . . . , q 0 s , and the δq’s which enter in the last term are the same as those in (27). Writing I 0 k − Ik = δIk, we get therefore from the latter equation Z ϑ 0 δE dt = Xs 1 Nk δIk. (28) In the special case where the varied motion is an undisturbed motion belonging to the same system as the unvaried motion we get, since δE will be constant, δE = Xs 1 ωk δIk, (29) 53 where ωk = Nk ϑ is the mean frequency of oscillation of qk between its limits, taken over a long time interval of the same order of magnitude as ϑ. 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.1 ) From (29) it follows moreover that, if the system allows of a separation of variables in an infinite continuous multitude 1 ) See Charlier, Die Mechanik des Himmels, Bd. I Abt. 2, and especially P. Epstein, Ann. d. Phys. LI p. 178 (1916). 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 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(q1, . . . , qs, I1, . . . , Is) obtained from (20) by introducing for the α’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 Ik if qk describes 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 w1, . . . , ws defined by wk = ∂S ∂Ik , (k = 1, . . . , s) (1∗ ) it will be seen that wk increases by one unit while the other w’s will come back to their original values if qk describes one oscillation between its limits and the other q’s remain constant. Inversely it will therefore 54 be seen that the q’s, and also the p’s which were given by pk = ∂S ∂qk , (k = i, . . . , s) (2∗ ) when considered as functions of the I’s and w’s will be periodic functions of every of the w’s with period 1. According to Fourier’s theorem any of the q’s may therefore be represented by an s-double trigonometric series of the form q = XAτ1,…, τs cos 2π(τ1w1 + . . . + τsws + ατ1,…, τs ), (3∗ ) where the A’s and α’s are constants depending on the I’s and where the summation is to be extended over all entire values of τ1, . . . , τs. On account of this property of the w’s, the quantities 2πw1, . . . , 2πws are denoted as “angle variables”. Now from (1∗ ) and (2∗ ) it follows according to the above mentioned theorem of Jacobi (see for instance Jacobi, Vorlesungen ¨uber Dynamik § 37) that the variations with the time of the I’s and w’s will be given by dIk dt = − ∂E ∂wk , dwk dt = ∂E ∂Ik , (k = 1, . . . , s) (4∗ ) 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∗ ), besides the evident result that the I’s are constant during the motion, that the w’s will vary linearly with the time and can be represented by wk = ωkt + δk, ωk = ∂E ∂Ik , (k = 1, . . . , s) (5∗ ) where δk is a constant, and where ωk is easily seen to be equal to the mean frequency of oscillation of qk. From (5∗ ) equation (28) follows at once, and it will further be seen that by introducing (5∗ ) in (3∗ ) we get the result that every of the q’s, and consequently also any one-valued function of the q’s, can be represented by an expression of the type (31). 55 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 alIn this connection it may be mentioned that the method of Schwarzschild of fixing the stationary states of a conditionally periodic system, mentioned on page 36, consists in seeking for a given system a set of canonically conjugated variables Q1, . . . , Qs, P1, . . . , Ps in such a way that the positional coordinates of the system q1, . . . , qs, and their conjugated momenta p1, . . . , ps, when considered as functions of the Q’s and P’s, are periodic in every of the Q’s with period 2π, 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 multiple of h 2π . 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 undetermined in every of these quantities. In many cases, however, these constants may be simply determined from considerations of mechanical transformability of the stationary states, and as pointed out by Burgers (loc. cit. Versl. Akad. Amsterdam XXV p. 1055 (1917)). Schwarzschild’s method possesses on the other hand the essential advantage of being applicable to certain classes of systems 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. 56 ready 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 nk = n 0 k and nk = n 00 k respectively, and let us assume that n 0 1 , . . . , n 0 s , n 00 1 , . . . , n 00 s are large numbers, and that the differences n 0 k−n 00 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 ν = 1 h (E 0−E 00) = 1 h Xs 1 ωk(I 0 k−I 00 k ) = Xs 1 ωk(n 0 k−n 00 k ), (30) which is seen to be a direct generalisation of the expression (13) in § 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 systems mentioned above that for the latter systems the coordinates q1, . . . , qs, 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: ξ = XCτ1,…, τs cos 2π  (τ1ω1+. . .+τsωs)t+cτ1,…, τs

, (31) 57 where the summation is to be extended over all positive and negative entire values of the τ ’s, and where the ω’s are the above mentioned mean frequencies of oscillation for the different q’s. The constants Cτ1,…, τs depend only on the α’s in the equations (18) or, what is the same, on the I’s, while the constants cτ1,…, τs depend on the α’s as well as on the β’s. In general the quantities τ1ω1 + . . . + τsωs will be different for any two different sets of values for the τ ’s, and in the course of time the orbit will cover everywhere dense a certain sdimensional 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 α’s one or more relations of the type m1ω1+. . .+msωs = 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 ω1 κ1 = · · · = ωs κs = ω, where the κ’s are the numbers which enter in equation (23), and the Fourier series for the displacements in the different directions will in this case consist only of terms of the simple form Cτ cos 2π{τωt + cτ}, 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 τ1ω1 + · · · + τsωs. In general, this spectrum would be completely different from that given 58 by (26). This follows already from the fact that the ω’s will depend on the values for the constants α1, . . . , αs 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 ω’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 ω’s for two stationary states, corresponding to nk = n 0 k and nk = n 00 k respectively, will tend to unity if the differences n 0 k − n 00 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 § 2 to exist in the simple case of periodic systems of one degree of freedom. Now on ordinary electrodynamics the coefficients Cτ1,…,τs in the expression (31) for the displacements of the particles in the different directions would in the well known way determine the intensity and polarisation of the emitted radiation of the corresponding frequency τ1ω1 + . . . + τsωs. As for systems of one degree of freedom we must therefore 59 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 Cτ1,…, τs in (31) corresponding to a set of τ ’s given by τk = n 0 k − n 00 k , if n 0 1 , . . . , n 0 s and n 00 1 , . . . , n 00 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 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 τ ’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, 60 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 α1, . . . , αs, the coefficient Cτ1,…, τs corresponding to a certain set τ 0 1 , . . . , τ 0 s of values for the τ ’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 0 k−n 00 k = τ 0 k is satisfied for every k. In the case where Cτ 0 1 ,…, τ 0 s is equal to zero in the expressions for the displacement in a certain direction only, we shall expect that all transitions, for which n 0 k −n 00 k = τ 0 k for every k, will be accompanied by a radiation which is polarised 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 that case all the Fourier coefficients in the expressions for the displacements in any direction will disappear if more than one of the τ ’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 polarised 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 61 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 q1, q2 and q3, 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 plane through the same axis. Due to the symmetry, the expression for the total energy in Hamilton’s equations will not contain the angular distance q3 but only the angular momentum p3 round the axis. The latter quantity will consequently remain constant during the motion, and the variations of q1 and q2 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, ρ, ϑ, where z is the displacement in the direction of the axis, ρ the distance of the particle from this axis and ϑ is equal to the angular distance q3, we have therefore z = XCτ1,τ2 cos 2π  (τ1ω1 + τ2ω2)t + cτ1,τ2

and ρ = XC 0 τ1,τ2 cos 2π  (τ1ω1 + τ2ω2)t + c 0 τ1,τ2

, (32) 62 where the summation is to be extended over all positive and negative entire values of τ1 and τ2, and where ω1 and ω2 are the mean frequencies of oscillation of the coordinates q1 and q2. For the rate of variation of ϑ with the time we have further dϑ dt = ˙q3 = ∂E ∂p3 = f(q1, q2, p1, p2, p3) = ± XC 00 τ1,τ2 cos 2π  (τ1ω1 + τ2ω2)t + c 00 τ1,τ2

, where the two signs correspond to a rotation of the particle in the direction of increasing and decreasing q3 respectively, and are introduced to separate the two types of symmetrical motions corresponding to these directions. This gives ±ϑ = 2πω3t+ XC 000 τ1,τ2 cos 2π  (τ1ω1+τ2ω2)t+c 000 τ1,τ2

, (33) where the positive constant ω3 = 1 2π C 00 0,0 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 = ρ cos ϑ

XDτ1,τ2 cos 2π  (τ1ω1 + τ2ω2 + ω3)t + dτ1,τ2

and y = ρ sin ϑ = ± XDτ1,τ2 sin 2π  (τ1ω1 + τ2ω2 + ω3)t+dτ1,τ2

, (34) 63 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 τ1 and τ2. 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 (τ1ω1 + τ2ω2), together with a number of circular harmonic motions round this axis of frequencies equal to the absolute values of (τ1ω1+ τ2ω2 + ω3), 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 τ1ω1 + τ2ω2 polarised parallel to the axis of symmetry, and a number of components of frequencies τ1ω1 + τ2ω2 + ω3 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 two kinds of transitions between the stationary states given by (22) will be possible. In both of these n1 and n2 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, n3 will remain unchanged, while in the second kind of transition n3 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. 64 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 fields 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 conservation of angular momentum. For a conditionally periodic system possessing an axis of symmetry the angular momentum round this axis is, with the above choice of coordinates, according to (22) equal to I3 2π = n3 h 2π . If therefore, as assumed above for a transition corresponding to an emission of linearly polarised light, n3 is unaltered, it means that the angular momentum of the system remains unchanged, while if n3 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 2π . Now it is easily seen that the ratio between this amount of angular momentum and the amount of energy hν 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 a is the radius of the orbit, ν the frequency of revolu- 65 tion 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 2πνaF and aF 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 2πν, 1 ) but this is seen to be equal to the ratio between the energy hν and the angular momentum h 2π 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, for an atomic system possessing an axis of symmetry, the total angular momentum round this axis is equal to an entire multiple of h 2π . 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 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 coordi1 ) Comp. K. Schaposchnikow, Phys. Zeitschr. XV, p. 454 (1914). 66 nates q1, . . . , qs, 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 ωk and the amplitudes Cτ1,…, τs 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 Cτ1,…, τs corresponding to certain combinations of the τ ’s are equal to zero for all values of the constants α1, . . . , αs, 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 τ ’s. Consequently we shall expect that the effect of the perturbing field on the 67 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 is afforded by a system consisting of a particle moving in a plane and executing harmonic vibrations in two perpendicular directions with frequencies ω1 and ω2. If the system is undisturbed all coefficients Cτ1,τ2 will be zero, except C1,0 and C0,1. 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 ω1 and ω2, appear a number of small terms corresponding to frequencies given by τ1ω1 + τ2ω2 where τ1 and τ2 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 so called 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 68 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 fields. As mentioned we cannot without a 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 α1, . . . , αs certain coefficients Cτ1,…, τs 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 0 1 , . . . , n 0 s and n 00 1 , . . . , n 00 s respectively, with the intensities of the radiations of frequencies ω1(n 0 1−n 00 1 )+· · ·+ωs(n 0 s−n 00 s ) to be expected on ordinary electrodynamics from the motions in these states; although of course this estimate becomes 69 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 supported in a general way by comparison with the observations.