Section 2

# The stationary states of a perturbed periodic system

by Niels Bohr

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.

# Xs 2  ∂ψ ∂αk Dαk Dt + ∂ψ ∂βk Dβk Dt 

Xs 2  − ∂ψ ∂αk ∂ψ ∂βk + ∂ψ ∂βk ∂ψ ∂αk  = 0. 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 90 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, Z σ 0 δE dt = Xs 1 κk δIk, (47) 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 91 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 92 establishment of the perturbing field, and will therefore be given by ∆tα1 = − Z t 0 t ϑ Xs 1 ∂Ω ∂qk q˙k dt = 1 ϑ Z t 0 Ω dt − t ϑ Ωt , (48) 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 t = ϑ, ∆ϑα1 + Ωϑ = 1 ϑ Z ϑ 0 Ω dt = ψ, 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 93 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 transforma- 94 bility 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) 95 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 97 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) 98 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 I = Z s 0 α2 Dβ2 Dt dt = Z α2 Dβ2, (50)

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 two 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 com- 100 plex. 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, δψ = Xs−1 1 vk δIk, (52) where vk is the mean frequency of oscillation of βk+1 between its limits, and where the quantities Ik are defined by Ik = Z αk+1 Dβk+1, (k = 1, . . . , s − 1) (53) 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 101 small frequencies α β )

XCt1,…, ts−1 cos 2π  (t1v1 + . . . + ts−1vs−1)t +ct1,…, ts−1

# , (54) 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 s − 1 conditions Ik = nkh, (k = 1, . . . , s − l) (55) 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 102 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 two 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 degener- 103 ate, 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 I = Z v 0 Xs 2 αk Dβk Dt dt, (57) 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) 104 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 105 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 106 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 or- 107 bit 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 I1 = Z α2 Dβ2, I2 = Z 2π 0 α3 Dβ3 = 2πα3. (60) 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 conditions1 ) I1 = n1h, I2 = n2h, (61) 1 ) 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, 108 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 sysEpstein 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. 109 tem 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). 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

, ±y = XCτ sin 2π  (τω1 + ω2)t + cτ

, (73) where ω1 is the frequency of the radial motion and ω2 is the mean frequency of revolution, and where the summation is to be extended over all positive and negative entire values of τ . It will thus be seen that the motion may be considered as a 134 superposition of a number of circular harmonic vibrations, for which the direction of rotation is the same as, or the opposite of, that of the revolution of the electron round the nucleus, according as the expression τω1 + ω2 is positive or negative respectively. From the relation just mentioned between the quantum theory of line spectra and the ordinary theory of radiation, we shall therefore in the present case expect that, if the atom is not disturbed by external forces, only such transitions between stationary states will be possible, in which the plane of the orbit remains unaltered, and in which the number n2 in the conditions (16) decreases or increases by one unit; i. e. where the angular momentum of the electron round the nucleus decreases or increases by h/2π. From the relation under consideration, we shall further expect that there will be an intimate connection between the probability of a spontaneous transition of this type between two stationary states, for which n1 is equal to n 0 1 and n 00 1 respectively, and the intensity of the radiation of frequency (n 0 1 − n 00 1 )ω1 ± ω2, which on ordinary electrodynamics would be emitted by the atom in these states, and which would depend on the value Cτ of the amplitude of the harmonic rotation, corresponding to τ = ±(n 0 1 −n 00 1 ), which appears in the motion of the electron. Without entering upon a closer examination of the numerical values of these amplitudes, it will directly be seen that the amplitudes of the harmonic rotations, which have the same direction as the revolution of the electron, in general, are considerably larger than the amplitudes of the rotations in the opposite direction, and 135 we shall accordingly expect that the probability of spontaneous transition will in general be much larger for transitions, in which the angular momentum decreases, than for transitions in which it increases. This expectation is verified by Paschen’s observations of the fine structure of the helium lines, which show that, for a given line, the components corresponding to the transitions of the former kind are by far the strongest. On Paschen’s photographs, however, especially in the case of the application of a condensed discharge to the vacuum tube containing the gas, there appear, in addition to the main components corresponding to transitions for which the angular momentum changes by h/2π, a number of weaker components, corresponding to transitions for which the angular momentum remains unchanged or changes by higher multipla of h/2π. This fact obtains a simple interpretation on the considerations in Part I on page 64 about the influence of small external forces on the spectrum of a conditionally periodic system. Thus, in the presence of small perturbing forces, the motion will generally not remain in a plane, and in the trigonometric series representing the displacement of the electron in space, there will occur small terms corresponding to frequencies (τ1ω1+τ2ω2), where τ2 may be different from one. In the presence of such forces, we shall therefore expect that, in addition to the regular probabilities of the above mentioned main transitions, there will appear small probabilities for other transitions.1 ) 1 ) Note added during the proof. As remarked in Part I, this con- 136 A detailed discussion of these problems will be given in a later paper by Mr. H. A. Kramers, who on my proposal has kindly undertaken to examine the resolution of the motion of the electron in its constituent harmonic vibrations more closely, and who has deduced explicit expressions for the amplitudes of these vibrations, not only for the motion of the electron in the undisturbed atom, but also for the perturbed motion in the presence of a small external homogeneous electric field. As it will be shown by Kramers, these calculations allow to account in particulars for the observations of the relative intensities of the components of the fine structure of the hydrogen lines and the analogous helium lines, as well as for the characteristic way in which this phenomenon is influenced by the variation of the experimental conditions.

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