56 



queiicies of the secular perturbations of the orliil of the eleclron. 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 slate of the 

 perturbed system will correspond to 14 = 0, n, ^= n or to u^ = n, n.^ = n, according 

 to whether ß„ 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 trans- 

 formations of the external conditions (see Part I, page 9 and 27), it seems necessary 

 to conclude that no stationary state exists corresponding to n, = 0. For this value 

 of n^ 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 68). 



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 i 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 f will, besides on Cj, depend on a^ only, and from the equations (46) we get 

 therefor-e 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. For the frequency of this rotation we get from (46) " 



from which we get directly for the difference between the values of ¥ for two neigh- 

 bouring states of the perturbed system, for which the corresponding value of / is the 



^^™^' d¥ = 2 7toda^. (63) 



This relation, which corresponds to (56), is seen to coincide with (59), since in the 

 present case Oj = and S] = 2na^. From (63) it follows that the necessary rela- 

 tion between the additional energy of the atom and the frequency of the perturba- 



