Part 1 showed that the problem of the fixation of the stationary states of a periodic system of several degrees of freedom, which is subject to the perturbing influence of a small external field, cannot be treated directly on the basis of the general principle of the mechanical transformability of the stationary states by considering the influence, which on ordinary mechanics a slow establishment of the external field would exert on the motion of some arbitrarily chosen stationary state of the undisturbed system (see Part I, page 41).

This is an immediate consequence of the fact, mentioned in the former section, that the stationary states of the perturbed system are characterised by a greater number of extra-mechanical conditions than the stationary states of the undisturbed system.

On the other hand, we were led to assume from the general formal relation between the quantum theory of line spectra and the ordinary theory of radiation, that it is possible to obtain information about the stationary states of the perturbed system from a direct consideration of the slow variations which the periodic orbit undergoes as a consequence of the mechanical effect of the external field on the motion.

Thus, if these variations are of periodic or conditionally periodic type, we may expect that, in the presence of the external field, the values for the additional energy of the system in the stationary states are related to the small frequency or frequencies of the perturbations, in a manner analogous to the relation between energy and frequency in the stationary states of an ordinary periodic or conditionally periodic system.

If the equations of motion for the perturbed system can be solved by means of separation of variables, it is easily seen that the relation in question is fulfilled if the stationary states are determined by the conditions (22).

Consider thus a system for which every orbit is periodic, and let us assume that in the presence of a given small external field a separation of variables is possible in a certain set of coordinates q1, . . . , qs.

For the undisturbed system we have then, according to equation (23), that the quantity I, defined by (5), is equal to κ1I1 + · · · + κsIs, where I1, . . . , Is are defined by (21) and calculated with respect to the set of coordinates just mentioned, and where the κ’s are a set of entire positive numbers without a common divisor.

For simplicity let us assume that at least one of the κ’s, say κs, is equal to one, and that consequently, as mentioned on page 40, the number n in (24), which characterises the stationary states of the undisturbed system, may take all positive values.

This condition will be fulfilled in case of all the applications to spectral problems discussed below; it will be seen, however, that the extension to problems where this condition is not fulfilled will only necessitate small modifications in the following considerations.

By use of (29) we get now for the difference in the total energy of two slightly different states of the perturbed system

Since for the undisturbed system ωk = κkωs, the differences ωk − κkωs appearing in the last term will, for the perturbed system, be small quantities which will just represent the frequencies of the slow variations which the orbit undergoes in the presence of the external field.

These quantities will in the following be denoted by vk. Consider now the multitude of states of the perturbed system for which Ps κkIk is equal to nh, where n is a given entire positive number.

This multitude will be seen to include all possible stationary states of the perturbed system, which satisfy (22), and the motion of which differs at any moment only slightly from some stationary motion of the undisturbed system, satisfying (24) for the given value of n.

Denoting the value of the energy of the undisturbed system in such a state by En, and the value of the energy of the perturbed system in a state belonging to the multitude under consideration by

for the energy difference between two neighbouring states of this multitude. Since this relation has the same form as (29), we see consequently that by putting I1, . . . , Is−1 equal to entire multipla of h, as claimed by the conditions (22), we obtain exactly the same relation between the additional energy E and the small frequencies vk, impressed on the system by the external field, as that which holds between the total energy and the fundamental frequencies in the stationary states of a conditionally periodic system of s − 1 degrees of freedom.

As a simple illustration of these calculations let us consider the system consisting of a particle moving in a plane and subject to an attraction from a fixed point, which varies proportional to the distance apart.

If undisturbed, the motion of this system will be periodic independent of the initial conditions, and the particle will describe an elliptical orbit with its centre at the fixed point.

Moreover the equations of motion of the undisturbed system may be solved by means of separation of variables in polar coordinates, as well as in any set of rectangular coordinates.

In the first case we have, taking for q1 the length of the radius vector from the fixed point to the particle and for q2 the angular distance of this radius vector from a fixed direction, κ1 = 2 and κ2 = 1, while in the second case we have κ1 = κ2 = 1.

In the presence of an external field the orbit will in general not remain periodic, but will in the course of time cover a continuous extension of the plane. If the external field is sufficiently small, however, the orbit will at any moment only differ little from a closed elliptical orbit, but in the course of time the lengths and directions of the principal axes of this ellipse will undergo slow variations.

In general the perturbed system will not allow of separation of variables, but two cases obviously present themselves in which such a separation is still possible; in the first case the external field is central with the fixed point as centre, and a separation is possible in polar coordinates; in the second case the external field of force is perpendicular to a given line and varies as some function of the distance from this line, and separation is possible in a set of rectangular coordinates with the axes parallel and perpendicular to the given line.

In the first case the perturbations will not affect the lengths of the principal axes of the elliptical orbit and will only produce a slow uniform rotation of the directions of these axes, while in the second case the lengths of the principal axes as well as their directions will perform slow oscillations.

It will consequently be seen that, by fixing the stationary states of the perturbed system by means of the conditions (22), the cycles of shapes and positions which the orbit of the particle will pass through in the stationary states will be entirely different in the two cases.

In both cases, however, it will be seen that the frequency v = ω1 −κ1ω2 will be equal to the frequency with which the orbit at regular intervals re-assumes its shape and position.

By fixing the stationary states by (22) we obtain therefore, as seen from (43), in both cases that the relation between this frequency and the additional energy of the system due to the presence of the field will be the same as the relation between energy and frequency in the stationary states of a system of one degree of freedom; and it will be seen that the above considerations offer a dynamical interpretation of the characteristic discontinuity involved in the application of the method of separation of variables to the fixation of the stationary states of perturbed periodic systems.

1. In general it will not be possible to solve the equations of motion of the perturbed system by means of separation of variables in a fixed set of positional coordinates, but we shall see that the problem of the fixation of the stationary states of the perturbed system may be attacked by a direct examination of the additional energy of the system and its relation to the slow variations of the orbit, on the basis of the usual theory of perturbations well known from celestial mechanics.

2. In this connection it may be of interest to note that the possibility of a rational interpretation of the discontinuity in question would seem to be essentially connected with the form of the principles of the quantum theory adopted in this paper. If for instance the quantum theory is taken in the form proposed by Planck in his second theory of temperature radiation, the consequent development to periodic systems of several degrees of freedom would seem to involve a serious difficulty as regards the question of the necessary stability of the temperature equilibrium among a great number of systems for small variations of the external conditions.

In fact, in connection with the development of his theory of the “physical structure of the phase space”, mentioned in Part I on page 31, in which conditions of the same type as (22) are established, Planck has deduced expressions for the total energy of a great number of systems in temperature equilibrium, which, if applied to systems of the same kind as those considered in the above example, show a dependency of this energy on the temperature which is different, according to whether polar coordinates or rectangular coordinates are used as basis for the structure of the phase space.

Consider a system for which every orbit, if undisturbed, is periodic independent of the initial conditions, and let us assume that the equations of motion for some set of coordinates q1, q2, . . . , qs are solved by means of the Hamilton-Jacobi partial differential equation, given by formula (17) in Part I.

The motion of the system is then determined by the equations (18), and the orbit is characterised by means of the constants α1, . . . , αs, β1, . . . , βs.

If now the system is subject to some small external field of force, the motion will no more be periodic, but, defining in the usual way the osculating orbit at a given moment as the periodic orbit which would result if the external forces vanished suddenly at this moment, we find that the constants α1, . . . , αs, β1, . . . , βs, characterising the osculating orbit, will vary slowly with the time.

Assuming for the present that the external forces possess a constant potential Ω given as a function of the q’s, we have according to the theory of perturbations that the rates of variation of the orbital constants of the osculating orbit will be given by

where Ω is considered as a function of α1, . . . , αs, β1, . . . , βs and t, obtained by introducing for the q’s their expressions as

functions of these quantities obtained by solving (18). The equations (44) allow to follow completely the perturbing effect of the external field on the motion of the system.

For the problem under consideration, however, a detailed examination of the perturbations is not necessary.

In fact, we shall not be concerned with the small deformation of the orbit characterised by the small oscillations of the orbital constants within a time interval of the same order of magnitude as the period of the osculating orbit, but only with the so called “secular perturbations” of the orbit, characterised by the total variation of these constants taken over a time interval long compared with the period of the osculating orbit.

As we shall see below, these variations may, with an approximation sufficient for our purpose, be obtained directly by taking mean values on both sides of the equations (44).

Before entering on these calculations, however, it may be observed that the part played by the constants α1 and β1 differs essentially from that played by the other orbital constants α2, . . . , αs, β2, . . . , βs.

Thus from the formulæ (17) and (18) on page 32, it follows that α1 is the total energy corresponding to the osculating orbit, while β1 will represent the moment in which the system would pass some distinguished point in this orbit. If for instance we consider the perturbations of a Keplerian motion, we may for β1 take the so called time of perihelium passage.

When discussing the secular perturbations of the shape and position of the orbit, we see therefore in the first place that the variations of β1 may be left out of consideration.

Further, it follows from the principle of conservation of energy, that α1 + Ω will remain constant during the motion, and that consequently during the perturbations α1 will change only by small quantities of the same order as λα1, where λ denotes a small constant of the same order of magnitude as the ratio between the external forces and the internal forces of the system.

Moreover, since the period σ of the undisturbed motion depends on α1 only, it follows that the period of the osculating orbit will remain constant during the perturbations, with neglect of small quantities of the same order as λσ. On the other hand it follows from (44) that, in a time interval of the same order as σ/λ, the constants α2, . . . , αs, β2, . . . , βs will in general undergo variations of the same order of magnitude as the values of these constants themselves.

As mentioned above, the total variations of the constants α2, . . . , αs, β2, . . . , βs, which characterise the secular perturbations of the shape and position of the orbit, may be obtained by taking mean values on both sides of the equations (44).

Introducing a function ψ of the α’s and β’s, equal to the mean value of the potential Ω taken over a period σ of the motion of the undisturbed system and defined by the formula

it is easily seen, since σ depends only on α1, that the mean values of the partial differential coefficients of Ω with respect to α2, . . . , αs, β2, . . . , βs, taken over an approximate period of the perturbed motion, may, if we look apart from small quantities proportional to λ , be replaced by the values of the corresponding partial differential coefficients of ψ at some moment within this period. With the approximation mentioned we get therefore

where the differential symbols on the left sides are written to indicate mean values of the rates of variation of the orbital constants during an approximate period of the perturbed motion.

From the definition of ψ it follows that this quantity in general will depend on α1 as well as on α2, . . . , αs, β2, . . . , βs, but that it will not depend upon β1.

From the above considerations it follows further that, with the approximation in question, α1 may be considered as a constant in the expressions on the right sides of (46), while for α2, . . . , αs, β2, . . . , βs we may take a set of values corresponding to some moment within the period to which the mean values on the left sides refer.

It will be seen that the equations (46) allow to follow the secular perturbations during a time interval sufficiently long for the external forces to produce a considerable change in the shape and position of the original orbit, if in the total variations of the orbital constants α2, . . . , αs, β2, . . . , βs we look apart from small quantities of the same order as the small oscillations of these constants within a single period.

As a consequence of the secular variations, the orbit will pass through a cycle of shapes and positions, which will depend on its original shape and position and on the character of the perturbing field, but not on the intensity of this field.

In fact, as seen from (46), the variations in the shape and position of the orbit will remain the same if ψ is multiplied by a constant factor, which will only influence the rate at which these variations are performed.

It will further be observed that the problem of determining the secular perturbations by means of (46) consists in solving a set of equations of the same type as the Hamiltonian equations of motion for a system of s−1 degrees of freedom.

In these equations the quantity ψ plays formally the same part as the total energy in the usual mechanical problem, and in analogy with the principle of conservation of energy it follows directly from (46) that, with neglect of small quantities proportional to λ 2, the value of ψ will remain constant during the perturbations, even if the external forces act through a time interval of the same order as σ/λ. In fact, with neglect of small quantities proportional to λ2 , we have

Since at any moment ψ will differ only by small quantities proportional to λ 2 from the mean value of the potential of the external forces taken over an approximate period of the perturbed motion, it follows from the above that, with neglect of small quantities of this order, also the mean value of the inner energy α1 of the perturbed system, taken over an approximate period, will remain constant during the perturbations, even if the perturbing forces act through a time interval long enough to produce a considerable change in the shape and position of the orbit. In the special case, where the perturbed system allows of separation of variables, this last result may be shown to follow directly from formula (28) in Part I.

Taking for the time interval ϑ in this formula the period σ of the undisturbed motion, we get Nk = κk, where κ1, . . . , κs are the numbers entering in formula (23).

Comparing a given perturbed motion of the system with some undisturbed motion of which it may be regarded as a small variation, we get therefore from (28), with neglect of small quantities proportional to the square of the intensity of the external forces,

where the I’s are calculated with respect to a set of coordinates in which a separation can be obtained for the perturbed motion, and where δE is the difference between the total energy of the undisturbed motion and the energy which the system would possess in its perturbed state, if the external forces vanished suddenly at the moment under consideration, and which in the above calculations was denoted by α1.

Now the energy E of the undisturbed motion is determined completely by the value of I = PκkIk.

If therefore the perturbed motion is all the time compared with a neighbouring undisturbed motion of given constant energy, it follows directly from (47), that, with neglect of small quantities of the same order as the square of the external forces, the integral on the left side, taken over an approximate period of the perturbed motion, will remain unaltered during the perturbations through any time interval, however long.

Before proceeding with the applications of the equations (46) which apply to the case of a constant perturbing field, it will be necessary to consider the effect of a slow and uniform establishment of the external field. Let us assume that, within the interval 0 < t < ϑ where ϑ denotes a quantity of the same order as σ/λ, the intensity of the external field increases uniformly from zero to the value corresponding to the potential Ω.

Since the variation in the perturbing field during a single period will only be a small quantity of the same order as λ 2, we see in the first place that the secular variations of the constants α2, . . . , αs, β2, . . . , βs, with the same approximation as for a constant field, will be given by a set of equations of the same form as (46), with the only difference that ψ is replaced by t ϑ ψ.

Moreover it may be shown that in these equations the quantity α1 may be considered as constant, just as in the equations which hold for a constant perturbing field.

In fact the total variation in α1 at any moment t will be equal to the total work performed by the external forces since the beginning of the establishment of the perturbing field, and will therefore be given by

where the expression on the right side is obtained by partial integration; but, since both terms in this expression are of the same order of magnitude as λα1, we see that the total variation in α1 within the interval in question will, just as in case of a constant perturbing field, be only a small quantity of this order.

We get therefore the result, that, for the same shape and position of the original orbit, the cycle of shapes and positions passed through by the orbit during the increase of the external field will be the same as that which would appear for a constant perturbing field, and that, with neglect of small quantities proportional to λ 2, the value of the function ψ will consequently remain constant during the establishment of the field. With this approximation we get therefore from (48), putting

which shows that the change in the total energy of the system, due to the slow and uniform establishment of the external field, is just equal to the value of the function ψ, and consequently equal to the mean value of the potential of the external forces taken over an approximate period of the perturbed motion.

This result may also be expressed by stating, that, with neglect of small quantities proportional to the square of the external forces, the mean value of the inner energy taken over an approximate period of the perturbed motion will be equal to the energy possessed by the system before the establishment of the perturbing field.

Returning now to the problem of the fixation of the stationary states of a periodic system subject to the influence of a small external field of constant potential, we shall base our considerations on the fundamental assumption that these states are distinguished between the continuous multitude of mechanically possible states by a relation between the additional energy of the system due to the presence of the external field and the frequencies of the slow variations of the orbit produced by this field, which is analogous to the relation discussed on page 80 in the special case in which the perturbed system allows of separation of variables in a fixed set of coordinates.

On this assumption we shall expect in the first place that, apart from small quantities proportional to λ, the cycles of shapes and positions of the orbit belonging to the stationary states of the perturbed system will depend only on the character of the external field, but not on its intensity.

Since now, as shown above, such a cycle will remain unaltered during a slow and uniform increase of the intensity of the external field if the effect of the external forces is calculated by means of ordinary mechanics, we are therefore, with reference to the principle of the mechanical transformability of the stationary states, led to the conclusion that it is possible by direct application of ordinary mechanics, not only to follow the secular perturbations of the orbit in the stationary states corresponding to a constant external field, but also to calculate the variation in the energy of the system in the stationary states which results from a slow and uniform change in the intensity of this field.

If we denote the energy in the stationary states of the perturbed system by En + E, where En is the value of the energy in the stationary state of the undisturbed system characterised by a given entire value of n in the condition I = nh, we may therefore conclude from the above that the additional energy E in the stationary states of the perturbed system will be equal to the value in these states of the function ψ defined by (45), if we look apart from small quantities proportional to the square of the intensity of the external forces.

It will be seen that this result is equivalent to the statement, that the mean value of the inner energy taken over an approximate period of the perturbed motion will be equal to the value En of the energy in the corresponding stationary state of the undisturbed system.

In case of the perturbed system allowing of separation of variables in a fixed set of coordinates, this result may be simply shown to be a direct consequence of the fixation of the stationary states by means of the conditions (22).

In fact, if we assume that the undisturbed motion, considered in (47), corresponds to some stationary state, satisfying (24) for a given value of n, and that the perturbed motion is also stationary and satisfies (22), we see that the right side of (47) will be zero, and we get the result that the mean value of the inner energy in the stationary states of the system, with the approximation mentioned, will not be altered in the presence of the external field.

Due to the above result that the additional energy E in the stationary states of the perturbed system, with neglect of small quantities proportional to λ 2, may be taken equal to the value in these states of the function ψ entering in the equations (46) which determine the secular perturbations of the orbits, we are now able to draw further conclusions from the fact, mentioned above, that these equations are of the same type as the Hamiltonian equations of motion for a mechanical system of s − 1 degrees of freedom.

In fact, we see that the fixation of the stationary states of the perturbed system is reduced to a problem which is formally analogous to the fixation of these states for a mechanical system of less degrees of freedom.

As it will appear from the following applications this problem may, quite independent of the possibility of separation of variables for the perturbed system, be treated directly on the basis of the fundamental relation between energy and frequency in the stationary states of periodic or conditionally periodic systems, discussed in Part I, if only the solution of the equations (46) is of a periodic or conditionally periodic character.

In this connection it may once more be emphasised that these equations, according to the manner in which they were deduced, allow to follow the secular perturbations only through a time interval of the same order of magnitude as that sufficient for the external forces to produce a finite alteration in the shape and position of the orbit.

With reference to the necessary stability of the stationary states of an atomic system, it seems justified, however, to conclude that any possible small discrepancy between the motion to be expected from a rigorous application of ordinary mechanics and that determined by a calculation of the secular perturbations, based on the equations (46), cannot cause a material change in the character of the stationary states as fixed by a consideration of the periodicity properties of these perturbations.

On the other hand, from the point of view of the general formal relation between the quantum theory and the ordinary theory of radiation, we must be prepared to find that the motion and the energy in the stationary states of a perturbed periodic system, for which we only know that the secular perturbations as determined by (46) are of conditionally periodic type, will not be as sharply defined as the motion and the energy in the stationary states of a conditionally periodic system for which the equations of motion allow of a rigorous solution by means of the method of separation of variables.

Thus, if we consider a large number of similar atomic systems of the type in question, we may be prepared to find that the values of the additional energy in a given stationary state will for the different systems deviate from each other by small quantities; but it must be expected that the values of the additional energy for the large majority of systems will differ from the value of ψ, as determined by the method indicated above, only by small quantities proportional to λ 2, and that only for a small fraction (at most of the same order as λ 2) of the systems the values of the additional energy will show deviations from this value of ψ, which are of the same order as λ.

As to the application of the preceding considerations to special problems, it will be seen in the first place that in case of a perturbed periodic system possessing two degrees of freedom, as for instance that considered in the example on page 82, the problem of the fixation of the stationary states of the perturbed system in the presence of a small external field allows of a general solution on the basis of the method developed above, because in this case the secular perturbations will in general be simply periodic. In fact, in this case the shape and position of the orbit are characterised by two constants α2 and β2, and from the equations (46), which will be analogous to the equations of motion of a system of one degree of freedom, it follows directly that during the perturbations α2 will be a function of β2 and that in general these quantities will be periodic functions of the time with a period s which, besides on α1, will depend on the value of ψ only.

Considering two slightly different states of the perturbed system for which the corresponding states of the undisturbed system (i. e. the states which would appear if the external forces vanished at a slow and uniform rate) possess the same energy and consequently the same value for the quantity I defined by (5), we get therefore by a calculation completely analogous to that leading to relation (8) in Part I, which was deduced directly from the Hamiltonian equations, for the difference in the values of the function ψ for these two states δψ = v δI, (49)

where v = 1 s is the frequency of the secular perturbations, and where the quantity I is defined by

…

where the latter integral is taken over a complete oscillation of β2.

In order to fix the stationary states, it will now be seen in the first place that, among the multitude of states of the perturbed system for which the value of I in the corresponding states of the undisturbed system is equal to nh where n is a given positive integer, the state for which I = 0 must beforehand be expected to be a stationary state.

In fact, for this value of I, the shape and position of the orbit will not undergo secular perturbations but will remain unaltered for a constant external field as well as during a slow and uniform establishment of this field.

In contrast to what in general will take place during a slow establishment of the external field, we may therefore expect that, for this special shape and position of the orbit, a direct application of ordinary mechanics will be legitimate in calculating the effect of the establishment of the field, since there will in this case obviously be nothing to cause the coming into play of some non-mechanical process, connected with the mechanism of a transition between two stationary states accompanied by the emission or absorption of a radiation of small frequency.

With reference to relation (49) we see therefore that, by fixing the stationary states of the perturbed system by means of the condition

I = nh, (51)

where n is an entire number, we obtain a relation between the additional energy E = ψ of the system in the presence of the field and the frequency v of the secular perturbations, which is exactly of the same type as that which holds between the energy and frequency in the stationary states of a system of one degree of freedom, and which is expressed by (8) and (10).

By means of (51) it is possible, with neglect of small quantities proportional to the square of the perturbing forces, directly to determine the value of the additional energy in the stationary states of a periodic system of 2 degrees of freedom subject to an arbitrarily given small external field of force, and consequently with this approximation, by use of the fundamental relation (1), to determine the effect of this field on the frequencies of the spectrum of the undisturbed periodic system.

In general this effect will consist in a splitting up of each of the spectral lines into a number of components which are displaced from the original position of the line by small quantities proportional to the intensity of the external forces.

When we pass to perturbed periodic systems of more than two degrees of freedom, the general problem is more complex.

For a given external field, however, it may be possible to choose a set of orbital constants α2, . . . , αs, β2, . . . , βs in such a way, that during the motion every of the β’s will depend on the corresponding β only, while every of the β’s will oscillate between two fixed limits.

From analogy with the theory of ordinary conditionally periodic systems which allow of separation of variables, the perturbations may in such a case be said to be conditionally periodic, and, from a calculation quite analogous to that leading to equation (29) in Part I which is based entirely on the use of the Hamiltonian equations, we get for the difference in ψ for two slightly different states of the perturbed system, for which the value of I in the corresponding states of the undisturbed system is the same,

where vk is the mean frequency of oscillation of βk+1 between its limits, and where the quantities Ik are defined by:

where the integral is taken over a complete oscillation of βk+1.

In analogy with the expression (31) for the displacements of the particles of an ordinary conditionally periodic system which allows of separation of variables, we get further in the present case that every of the α’s and β’s may be expressed as a function of the time by a sum of harmonic vibrations of small frequencies

where the C’s and c’s are constants, the former of which, besides on I, depend on the I’s only, and where the summation is to be extended over all positive and negative entire values of the t’s.

If therefore the secular perturbations are conditionally periodic, we may conclude that the stationary states of the perturbed system, corresponding to a given stationary state of the undisturbed system, will be characterised by the

where n1, . . . , ns−1 form a set of entire numbers. In fact, as seen from (52), we obtain in this way a relation between the additional energy and the frequencies of the secular perturbations of exactly the same type as that holding for the energy and frequencies of ordinary conditionally periodic systems and expressed by (22) and (29); moreover we may conclude beforehand that the state in which every of the quantities Ik, defined by (53), is equal to zero must belong to the stationary states of the perturbed system, because in this case the orbit will not undergo secular perturbations for a constant external field, nor during a slow and uniform establishment of this field. Since the conditions (55), with neglect of small quantities proportional to the square of the intensities of the external forces, allow to determine the additional energy of the system due to the presence of the external field, we see therefore that the effect of this field on the spectrum of the undisturbed system, if the secular perturbations are conditionally periodic, will consist in a splitting up of each spectral line in a number of components, in analogy with the effect of a perturbing field on the spectrum of a periodic system of two degrees of freedom. In general, however, the perturbations, which a periodic system of more than 2 degrees of freedom undergoes in the presence of a given external field, cannot be expected to be conditionally periodic and to exhibit periodicity properties of the type expressed by formula (54).

In such cases it seems impossible to define stationary states in a way which leads to a complete fixation of the total energy in these states, and we are therefore led to the conclusion, that the effect of the external field on the spectrum will not consist in the splitting up of the spectral lines of the original system into a number of sharp components, but in a diffusion of these lines over spectral intervals of a width proportional to the intensity of the external forces.

In special cases in which the secular perturbations of a perturbed periodic system of more than two degrees of freedom are of conditionally periodic type, it may occur that these perturbations are characterised by a number of fundamental frequencies, which is less than s − 1.

In such cases, in which the perturbed periodic system from analogy with the terminology used in Part I may be said to be degenerate, the necessary relation between the additional energy and the frequencies of the secular perturbations is secured by a number of conditions less than that given by (55), and the stationary states are consequently characterised by a number of conditions less than s. With a typical example of such systems we meet if, for a perturbed periodic system of more than two degrees of freedom, the secular perturbations are simply periodic independent of the initial shape and position of the orbit.

In direct analogy to what holds for perturbed periodic systems of two degrees of freedom, the difference between the values of ψ in two slightly different states of the perturbed system, corresponding to the same value of I, will in the present case be given by δψ = v δI, (56) where v is the frequency of the secular perturbations, and where I is defined by

where s = 1/v is the period of the perturbations. We may therefore conclude that the stationary states of the perturbed system, corresponding to a given stationary state of the undisturbed system, will be characterised by the single condition:

I = nh, (58)

in which n is an entire number, and which will be seen to be completely analogous to the condition which fixes the stationary states of ordinary periodic systems of several degrees of freedom.

In the following sections we shall apply the preceding considerations to the problem of the fixation of the stationary states of the hydrogen atom, when the relativity modifications are taken into account, and when the atom is exposed to small external fields.

In this discussion we shall for the sake of simplicity consider the mass of the nucleus as infinite in the calculations of the perturbations of the orbit of the electron.

This involves, in the expression for the additional energy of the system, the neglect of small terms of the same order as the product of the intensity of the external forces with the ratio between the mass of the electron and the mass of the nucleus, but due to the smallness of the latter ratio the error introduced by this simplification will be of no importance in the comparison of the results with the measurements. Since in the case under consideration the system possesses three degrees of freedom, the equations which determine the secular perturbations of the orbit of the electron will correspond to the equations of motion of a system of two degrees of freedom, and it will therefore not be possible to give a general treatment of the problem of the stationary states.

Thus, for any given external field, we meet with the question whether the perturbations are conditionally periodic and, if so, in what set of orbital constants this periodicity may be conveniently expressed. Now, in many spectral problems, the external field possesses axial symmetry round an axis through the nucleus, and in this case it is easily shown that the problem of the fixation of the stationary states allows of a general solution.

A choice of orbital constants which is suitable for the discussion of this problem, and which is well known from the astronomical theory of planetary perturbations, is obtained by choosing for α2 the total angular momentum of the electron round the nucleus and for α3 the component of this angular momentum round the axis of the field. For the set of β’s, corresponding to this set of α’s, we may take β2 equal to the angle, which the major axis makes with the line in which the plane of the orbit cuts the plane through the nucleus perpendicular to the axis of the field, and β3 equal to the angle between this line and a fixed direction in the latter plane.

For the problem under consideration it will be seen that, with this choice of constants, the mean value ψ of the potential of the perturbing field will, besides on α1, generally depend on α2 and β2 as well as on α3, but due to the symmetry round the axis it will obviously not depend on β3.

In consequence of this, the equations (46), which determine the secular perturbations, will possess the same form as the Hamiltonian equations of motion for a particle moving in a plane and subject to a central field of force.

Thus corresponding to the conservation of angular momentum for central systems, we get in the first place from (46) that α3 will remain unaltered during the perturbations.

Next corresponding to the simple periodicity of the radial motion in central systems, we see from (46), if α3 as well as α1 is considered as a constant, that during the perturbations α2 will be a function of β2 and vary in a simple periodic way with the time. The perturbations of the orbit of the electron produced by an external field which possesses axial symmetry will therefore always be of conditionally periodic type, quite independent of the possibility of separation of variables for the perturbed system.

As regards the form of the conditions which fix the stationary states, it may be noted, however, that with the choice of orbital constants under consideration the β’s will not, as it was assumed for the sake of simplicity in the general discussion on page 100, oscillate between fixed limits, but it will be seen that β2 during the perturbations may either oscillate between two such limits or increase (or decrease) continuously, while β3 will always vary in the latter manner.

This constitutes, however, only a formal difficulty of the same kind as that mentioned in Part I in connection with the discussion of the conditions (16), which fix the stationary states of a system consisting of a particle moving in a central field of force.

Thus from a simple consideration it will be seen that, in complete analogy to the relations (52) and (53), we get in the present case for the difference between the energy of two slightly different states of the perturbed system, which correspond to the same value of I, δψ = v1 δI1 + v2 δI2, (59)

where v1 is the frequency with which the shape of the orbit and its position relative to the axis of the field repeats itself at regular intervals and which is characterised by the variation of α2 and β2, while v2 is the mean frequency of rotation of the plane of the orbit round this axis characterised by the variation of β3, and where I1 and I2 are defined by the equations

In case β2 varies in an oscillating manner with the time, the first integral must be taken over a complete oscillation of this orbital constant, while, if β2 during the perturbations increases or decreases continuously, the integral in the expression for I1 must be taken over an interval of 2π, just as the integral in the expression for I2.

By fixing the stationary states of the perturbed system by means of the two conditions

Quite apart from the problem of perturbed periodic systems, the second of these conditions would also follow directly from certain interesting considerations of Epstein (Ber. d. D. Phys. Ges. XIX, p. 116 (1917)) about the stationary states of systems which allow of what may be called “partial separation of variables”.

In this case it is possible to choose a set of positional coordinates q1, . . . , qs in such a way that, for some of the coordinates, the conjugated momenta may be considered as functions of the corresponding q’s only, so that, for these coordinates, quantities I may be defined by (21) in the same way as for systems for which a complete separation of variables can be obtained.

From analogy with the theory of the stationary states of the latter systems, where n1 and n2 are entire numbers, it will therefore be seen that we obtain the right relation between the additional energy E = ψ of the perturbed atom and the frequencies of the secular perturbations of the orbit of the electron.

It will moreover be seen that a state in which the electron moves in a circular orbit perpendicular to the axis of the field, and which beforehand must be expected to belong to the stationary states of the perturbed atom since this orbit will not undergo secular perturbations during a uniform establishment of the external field, will be included among the states determined by (61).

In fact, if n is the number which characterises the corresponding stationary state of the undisturbed system, this state of the perturbed system will correspond to n1 = 0, n2 = n or to n1 = n, n2 = n, according to whether β2 during the perturbations oscillates between fixed limits, or increases (or decreases) continuously.

As regards the application of the conditions (61) it is of importance to point out that, from considerations of the invariance of the a-priori probability of the stationary states of an atomic system during continuous transformations of the external conditions (see Part I, page 14 and page 49), it seems necessary to conclude that no stationary state exists corresponding to n2 = 0.

For this value of n2 the motion of the electron would take place in a plane through the axis, but for certain external fields such motions cannot be regarded as physically realisable stationary states of the atom, since in the course of the perturbations the electron would collide with the nucleus (compare page 134).

Epstein proposes therefore the assumption, that some of the conditions to be fulfilled in the stationary states of the systems in question may be obtained by putting the I’s thus defined equal to entire multipla of h. It will be seen that, in case of systems possessing an axis of symmetry, this leads to the second of the conditions (61), which expresses the condition that in the stationary states the total angular momentum round the axis must be equal to an entire multiple of h/2π.

As pointed out in Part I on page 64, this condition would also seem to obtain an independent support from considerations of conservation of angular momentum during a transition between two stationary states.

A special case of an external field possessing axial symmetry, in which the secular perturbations are very simple, presents itself if the external forces form a central field with the nucleus at the centre. In this case the solution of the problem of the fixation of the stationary states is given by Sommerfeld’s general theory of central systems, discussed in Part I, which rests upon the fact that these systems allow of separation of variables in polar coordinates.

In connection with the above considerations it may be of interest, however, to consider the problem in question directly from the point of view of perturbed periodic systems, because it presents a characteristic example of a degenerate perturbed system.

In the present case ψ will, besides on α1, depend on α2 only, and from the equations (46) we get therefore the well known result, that the angular momentum of the electron and the plane of its orbit will not vary during the perturbations, and that the only secular effect of the perturbing field will consist in a slow uniform rotation of the direction of the major axis. 110 For the frequency of this rotation we get from (46) v = 1 2π Dβ2 Dt

1 2π δψ δα2 , (62) from which we get directly for the difference between the values of ψ for two neighbouring states of the perturbed system, for which the corresponding value of I is the same, δψ = 2πv δα2. (63) This relation, which corresponds to (56), is seen to coincide with (59), since in the present case v2 = 0 and I1 = 2πα2. From (63) it follows that the necessary relation between the additional energy of the atom and the frequency of the perturbations is secured if the stationary states in the presence of a small external central field are characterised by the condition I = 2πα2 = nh, (64) where n is an entire number. This condition, which is equivalent with the second of Sommerfeld’s conditions (16), corresponds to (58) and is seen to coincide with the first of the conditions (61), while the second of the latter conditions in the special case under consideration loses its validity corresponding to the fact that the orientation of the plane of the orbit in space is obviously arbitrary. Since, for a Keplerian motion, the major axis of the orbit depends on the total energy only while the minor axis is proportional to the angular momentum, it will be seen from (64) that the presence of 111 a small external field imposes the restriction on the motion of the atom in the stationary states, that the minor axis of the orbit of the electron must be equal to an entire multiple of the n th part of the major axis, which was given by 2αn in (41). This result has been pointed out by Sommerfeld as a consequence of the application of the conditions (16). In the preceding it has been shown how it is possible to attack the problem of the stationary states of a perturbed periodic system by an examination of the secular perturbations of the shape and position of the orbit, and to fix these states if the perturbations are of periodic or conditionally periodic type. While these considerations allow to determine the possible values for the total energy of the perturbed system and thereby the frequencies of the components into which the lines of the spectrum of the undisturbed system are split up in the presence of the external field, it is necessary, however, for the discussion of the intensities and polarisations of these components to consider more closely the motion of the particles in the perturbed system and the relation of the total energy of this system to the fundamental frequencies which characterise the motion. In the first place it will be seen that, if the secular perturbations as determined by the equations (46) are of conditionally periodic type, the displacements of the particles of the system in any given direction may, with neglect of small quantities proportional to the intensity of the external forces, be represented, within a time interval sufficiently large for these forces to produce a 112 considerable change in the shape and position of the orbit, as a sum of harmonic vibrations by expressions of the type: ξ = XCτ,t1,…, ts−1 cos 2π  (τωP + t1v1 + · · ·

• ts−1vs−1)t + cτ,t1,…, ts−1

, (65) where the summation is to be extended over all positive and negative entire values of τ , t1, . . . , ts−1, and where the C’s and c’s are two sets of constants, the former of which depend only on the values of the quantities I1, . . . , Is−1 defined by (53) and on the value of the quantity I, which characterises the corresponding state of the undisturbed system which would appear if the external field vanished at a slow and uniform rate. While the quantities v1, . . . , vs−1 are the same as those which appear in the formula (54), and represent the small frequencies of the secular perturbations of the shape and position of the orbit, the quantity ωP may be considered as representing the mean frequency of revolution of the particles in their approximately periodic orbit. As regards the total energy of the perturbed system, it may next be proved that, looking apart from small quantities proportional to the square of the intensity of the external forces, the difference in the total energy in two slightly different states of the perturbed system, for which the values of I, I1, . . . , Is−1 differ by δI, δI1, . . . , δIs−1 respectively, is given by the relation1 ) δE = ωP δI + Xs−1 1 vk δIk, (66) 113 which coincides with (52) if δI = 0, and which will be seen to 1 ) From a comparison with formula (8), holding for the energy difference between two neighbouring states of the undisturbed system, and with formula (52), it will be seen that (66) implies the condition ωP = ω + ∂ψ/∂I, where ω is the frequency of revolution in the corresponding state of the undisturbed system characterised by the given value of I, and where, in the partial differential coefficient, ψ is considered as a function of I and I1, . . . , Is−1. This relation can be verified by means of a consideration based on the perturbation equations (44), which takes into account the simple relation between α1 and I for the undisturbed system, as well as the relation between the mean rate of variation of β1 with the time and the difference between ωP and ω. We shall not enter, however, on the details of the rather intricate calculations involved in such a consideration, since the problems in question allow of a more elegant treatment by means of another analytical method. Thus it will be shown by Mr. H. A. Kramers, in the paper mentioned in the end of § 4, that, quite independent of the possibility of separation of variables for the perturbed system in a fixed set of positional coordinates, the theory of secular perturbations exposed in this section offers—if these perturbations as determined by (46) are of conditionally periodic type—a means of disclosing a set of angle variables, which may be used to describe the motion of the perturbed system with the same degree of approximation as that involved in the preceding calculations. According to the definition of angle variables, mentioned in the Note on page 53 in Part I, this means that it is possible, in stead of the positional coordinates q1, . . . , qs of the perturbed system and their conjugated momenta p1, . . . , ps to introduce a new set of s variables in such a way, that the q’s and p’s are periodic in every of the new variables with period 1, when they are considered as functions of these variables and of their canonically conjugated momenta. These momenta will just coincide with the quantities denoted above by I, I1, . . . , Is−1, and the corresponding angle variables may 114 be completely analogous with formula (29) in Part I, holding for an ordinary conditionally periodic system which allows of separation of variables in a fixed set of positional coordinates; just as (65) is analogous to formula (31) representing the displacements of the particles for such a system. Since moreover, in complete analogy to the conditions (22), the stationary states of the perturbed system are characterised by I = nh, Ik = nkh, (k = 1, . . . , s − 1) (67) we see consequently that, for sufficiently small intensity of the external forces, we obtain in the region of large values of n and of the n’s a connection between the frequencies of the components of the spectral lines, determined on the quantum conveniently be denoted by w, w1, . . . , ws−1 respectively. Introducing the new variables, the total energy of the perturbed system will be a function of I, I1, . . . , Is−1 only, if we look apart from small quantities proportional to λ 2 . With this approximation we get consequently by a calculation, analogous to that given in the Note referred to, that the angle variables w, w1, . . . , ws−1 may be represented as linear functions of the time within an interval of the same order as σ/λ. Denoting the rates of variation of w, w1, . . . , ws−1 by ω, v1, . . . , vs−1 respectively, the formulæ (65) and (66) are therefore directly obtained, just as the corresponding formulæ (31) and (29) in Part I. In this connection it will be observed that, due to the possibility of introduction of angle variables, the conditions (67) appear in the same form as that in which the conditions, which fix the stationary states of ordinary conditionally periodic systems which allow of separation of variables, have been formulated by Schwarzschild, and which, as mentioned in the Note in Part I, has already been applied by Burgers to certain systems for which such a separation cannot be obtained. 115 theory by means of relation (1), and those to be expected on ordinary electrodynamics, which is of exactly the same type as the analogous connection, discussed in Part I, in case of ordinary conditionally periodic systems which allow of separation of variables. In perfect analogy with the general considerations in Part I, we are therefore led directly to certain simple conclusions as regards the intensities and polarisations of the components into which the lines of the spectrum of the undisturbed periodic system are split up in the presence of the external field. Thus we shall expect that there will exist an intimate connection between the probability of spontaneous transition between two stationary states of the perturbed system, for which n = n 0 , nk = n 0 k and n = n 00 , nk = n 00 k respectively, and the values in these states of the coefficient Cτ,t1,…, ts−1 in the expressions for the displacements of the particles, for which τ = n 0 − n 00 and tk = n 0 k − n 00 k . If for instance, for a certain set of values of τ and t1, . . . , ts−1, the coefficient Cτ,t1,…, ts−1 in the expressions for the displacements in every direction will be equal to zero for all motions of the perturbed system, we shall expect that the corresponding transitions between two stationary states will be impossible in the presence of the given external field; and if this coefficient is zero for the displacements of the particles in a certain direction only, we shall expect that the corresponding transitions will give rise to the emission of a radiation which is polarised in a plane perpendicular to this direction. With a characteristic example of these considerations we 116 meet in the case of the spectrum of a hydrogen atom exposed to an external field of force which possesses axial symmetry round an axis through the nucleus. In analogy with the resolution of the motion of an ordinary conditionally periodic system which possesses an axis of symmetry in its constituent harmonic vibrations, discussed in Part I on page 61, it follows from the discussion of the general character of the secular perturbations on page 104 that the motion of the electron in the perturbed atom in this case can be resolved in a number of linear harmonic vibrations parallel to the axis with frequencies τωP + t1v1 and in a number of circular harmonic rotations perpendicular to the axis with frequencies τωP +t1v1+v2. In complete analogy with the considerations in Part I, we are therefore led to conclude that in the present case only two types of transitions between the stationary states of the perturbed atom are possible. In the transitions of the first type n2 will remain unaltered and the emitted radiation will give rise to components of the hydrogen lines which will show linear polarisation parallel to the axis. In the transitions of the second type n2 will change by one unit and the emitted radiation will show circular polarisation when viewed in the direction of the axis. Remembering that, according to the conditions (61), the angular momentum of the system round the axis in the stationary states is equal to n2 h 2π , it will be seen moreover that, also in the present case, these conclusions obtain an independent support from a consideration of conservation of angular momentum during 117 the transitions (Compare Part I page 64).1 ) In the following we will meet with applications of these considerations when discussing the effect of electric and magnetic fields on the hydrogen lines. In the latter case, however, the preceding considerations need some modifications due to the fact, that the external forces acting on the electron cannot be derived from a potential expressed as a function of its positional coordinates; to this point we shall come back in § 5. Before leaving the general theory of perturbed periodic systems we shall still consider the problem of the effect on the spectrum of a periodic system, undergoing secular perturbations of conditionally periodic type under the influence of a given small external field, if this system is further subject to the influence of a second external field which is small compared with the first field, but the perturbing effect of 1 ) Note added during the proof. In an interesting paper by A. Rubinowicz (Phys. Zeitschr. XIX, p. 441 and p. 465 (1918)) which has just been published, a similar consideration of conservation of angular momentum has been used to draw conclusions, as regards the possibility of transitions between the stationary states of a conditionally periodic system possessing an axis of symmetry, and as regards the character of the polarisation of the radiation accompanying these transitions. In this way Rubinowicz has arrived at several of the results discussed in the present paper; in this connection, however, it may be remarked that, from a consideration of conservation of angular momentum, it is not possible, even for systems possessing axial symmetry, to obtain as complete information, as regards the number and polarisation of the possible components, as from a consideration based on the resolution of the motion of the electron in harmonic vibrations. 118 which is yet large compared with the small effects on the motion, proportional to the square of the intensity of the first perturbing field, which were neglected in the preceding calculations. This problem is closely analogous to the problem, briefly discussed in Part I, of the effect of a small perturbing field on the spectrum of an ordinary conditionally periodic system which allows of separation of variables. As mentioned on page 64, we have in this case, quite independent of the possibility of separation of variables for the perturbed system, that in general the motion under the influence of the external field may still be represented as a sum of harmonic vibrations by a formula of the type (31), if we look apart from small terms proportional to the square of the perturbing forces. Corresponding to this we have in the case under consideration that, independent of the nature of the second external field, the resultant secular perturbations may in general be expressed as a sum of harmonic vibrations of small frequencies of the type (54), if we look apart from small terms of the same order as the product of the secular perturbations produced by the first external field with the square of the ratio between the intensities of the forces due to the first and those due to the second external field. Let us denote this ratio by µ and let, as above, λ represent a small constant of the same order as the ratio between the external forces due to the first field and the internal forces of the system. On the basis of the general relation between energy and frequency in the stationary states, we may then expect that it is possible to fix the motion in these states 119 for the perturbed periodic system in the presence of both external fields with neglect of small terms of the same order as the largest of the quantities µ 2 and λ, and to fix the corresponding values for the energy with neglect of small terms of the same order as the largest of the quantities λµ2 and λ 2 . 1 ) In general, however, the effect on the spectrum of the perturbed system, produced by the second external field, may be calculated without considering the perturbing effect of this field in detail. In fact, it is in general possible, by means of the principle of the mechanical transformability of the stationary states, with the approximation mentioned to determine the alteration of the energy of the system, due to the presence of the second external field, directly from the character of the secular perturbations produced by the first external field only. Thus let us assume that the second field is slowly established at a uniform rate within a time interval of the same order of magnitude as that in which the system will pass approximately through any state belonging to the cycle of shapes and positions, which the orbit passes through in the stationary states in the presence of the first external field only. Denoting a time interval of this order by ϑ and 1 ) In analogy with the considerations on page 97 it may be expected, however, that these limits for the definition of the energy in the stationary states will hold only for the great majority among a large number of atomic systems. Thus in the present case we must be prepared to find that for a small fraction of the systems of the same order as µ 2 (if µ 2 > λ) the energy will differ from that fixed by the method under consideration by small quantities of the same order as µλ. 120 the potential of the first perturbing field by Ω and that of the second by ∆Ω, we get then, by a calculation quite analogous to that given in Part I on page 17 for the alteration in the mean value of the energy of a periodic system during a slow establishment of a small external field, that the alteration in the mean value of α1 + Ω taken over a time interval of the same order as ϑ, due to the establishment of the second external field, will be a small quantity of the same order of magnitude as ϑ(∆Ω)2 ; but with the notation used above this means, in general, a small quantity of the same order as λµ2 . It follows consequently that, with this approximation, the alteration in the energy in a given stationary state, due to the presence of the second perturbing field, is equal to the mean value of the potential of this field taken over the cycle of shapes and positions, which the orbit would pass through in the corresponding stationary state of the perturbed system under the influence of the first external field only. In general, the effect on the spectrum will therefore consist in a small displacement of the original components proportional to the intensity of the forces due to the second perturbing field; and as regards the degree of approximation with which these displacements are defined, it will be seen from the above that, if µ is smaller than √ λ, the fixation of the energy in the stationary states in the presence of the second external field, and therefore also the determination of the frequencies of the spectral lines by means of (1), allow of the same degree of approximation as the fixation of the energy in the stationary states of the original perturbed periodic system. If µ is larger 121 than √ λ, however, the stationary states will in general not be as well defined as for the original system, and from relation (1) we may therefore expect that the components will be diffuse, although, as long as µ remains small compared with unity, the width of the components will remain small compared with the displacements from their positions in the presence of the first external field alone. Only when µ becomes of the same order as unity, the simultaneous effect of both perturbing fields may be expected to consist in a diffusion of the lines of the undisturbed periodic system; unless of course the secular perturbations due to the simultaneous presence of both fields are still of conditionally periodic type, as it may happen in special problems. In certain cases the second external field will not only give rise to small displacements of the original components but also to the appearance of new components of small intensities proportional to µ 2 . This occurs if for the original perturbed periodic system, due to some peculiarity of the motion, some of the coefficients Cτ,t1,…, ts−1 in the expressions (65) for the displacements of the particles as a sum of harmonic vibrations, corresponding to certain combinations of the numbers τ , t1, . . . , ts−1, are equal to zero, while in the presence of the second external field these coefficients are small quantities proportional to µ (compare Part I, page 64).1) In the preceding con1 ) As regards the degree of definition with which the positions of the new components will be determined, we must be prepared to find that the frequencies of these components are only defined with neglect of small quantities proportional to λµ. Compare the detailed discussion siderations it has been assumed that the perturbed system in the presence of the first external field is non-degenerate.

In case, however, this system is degenerate, it is obviously impossible, by a direct application of the principle of the mechanical transformability of the stationary states, to determine the alteration in the energy in the stationary states of the system, which will be due to the presence of a second external field small compared with the first field; because, as mentioned, the stationary states of the system, in the presence of this field only, will be determined by a number of conditions which is less than the number s of degrees of freedom, and that consequently the cycles of shapes and positions, which the orbit will pass through in these states, will not be completely determined. For the calculation of the energy in the stationary states it will therefore be necessary to consider the secular perturbing effect of the second external field on these cycles. In the special case where the secular perturbations due to the first field are simply periodic, it wall in this way be seen that the problem of the fixation of the stationary states in the presence of the second external field, by means of the method exposed in this section, may be reduced to the problem of the fixation of the stationary states of a system of s − 2 degrees of freedom. If, as in the applications considered below, s is equal to 3, this problem allows of a general solution, and we must therefore expect that in this case the effect on the spectrum of the perturbed of the example in § 5 on page 192.

system produced by an arbitrary second external field, which is small compared with the first, will consist in the splitting up of every component into a number of separate components, just as the effect of an arbitrary small external field on the lines of the spectrum of a simple periodic system of 2 degrees of freedom. We will meet with applications of the above considerations when considering the effect on the hydrogen spectrum of the combined action of different external fields and when considering the effect of an external field on the spectra of other elements, which latter problem will be discussed in Part III.