66 



a small repulsion on the electron, proportional to the inverse cube of the distance 

 and equal and opposite lo tlie attraction just mentioned, we would therefore ob- 

 tain a system for which, with neglect of small quantities of higher order than 

 the square of the ratio between the velocity of the electron and the velocity of 

 light, every orbit would be periodic independent of the initial conditions, and for 

 which consequently the stationary states would be fixed by the single condition 

 / = nil. Now the actual hydrogen atom may obviously be considered as a per- 

 turbed system, formed by this periodic system, when it is exposed to a small 

 central field for which the value of '/' is given by (70). With the approximation 

 mentioned, we get therefore for the total energy in the stationary states of the atom 



8 JT* A" e* m 1 

 E = K- - !) . 4: , (71) 



where E' is the energy in the stationary states of the periodic system just mentioned, 

 and where the last term is obtained by introducing in (70) the value of u^ given 

 by (64) and the value of wa given by (41), neglecting the small correction due lo 

 the finite mass of the nucleus. Remembering that in our notation n^ — ;;., = n 

 and n., = n, it will be seen that, as regards the small differences in the energy of 

 the different stationary states corresponding to the same value of n, formula (71) 

 gives the same result as Sommerfeld's formula (68). In fact, comparing (68) and 

 (71), we get 



h-n- 



c-h-n- j 



which is seen to be a function of /! only. This expression might also have been deduced 

 directly from the condition / = nh by considering, for instance, a circular orbit, 

 in which case the calculation can be very simph' performed. 



In connection with the above calculations, it may be remembered that the fixation 

 of the stationary states, leading to the formulae (68) or (71), is based on the assump- 

 tion, that the motion of the electron can be determined as that of a mass point 

 which moves in a conservative field of force, according to the laws of ordinary 

 relativistic mechanics, and that we have looked apart from all such forces which, 

 according to the ordinary theory of electrodynamics, would act on an accellerated 

 charged particle, and which constitute the reaction from the radiation which on this 

 theory would accompany the motion of the electron. Some procedure of this kind, 

 which means a radical departure from the ordinary theory of electrodynamics, is 

 obviously necessary in the quantum theory in order to avoid dissipation of energy 

 in the stationary states. Since we are entirely ignorant as regards the mechanism 

 of radiation, we must be prepared, however, to find that the above treatment will 

 allow to determine the motion in the stationary states, only with an approximation 

 which looks apart from small quantities of the same order as the ratio between 

 the radiation forces in ordinarv electrodvnamics and the main forces on the electron 



