43 



327 



Although the radiation process in the quantum llieory is so much unlike the 

 radiation process in or(hnary electrodynamics, it was shown by Bohu that there 

 exists an intimate formal connection between these two theories. This connec- 

 tion refers in the first place to the frequencies of the radiation which on the 

 quantum theory will be emitted by the atom and the frequencies which on the 

 ordinary theory of electrodynamics would be emitted. Consider thus a transition 

 between two stationary states of a non-degenerate conditionally periodic .system of 

 the type described in Part I, the initial state and final state of which are characteri- 

 sed by ;j, -= n\, .... n, and i\ = n',', . . n, - n" respectively, where 

 />,, .... n, are the integers appearing in the conditions (99), and consider the 

 multitude of mechanically possible states of the system lying "between" the initial 

 state and the final stale, for which the quantities I^, .... 1^ are equal to 4 = 

 {n'k Å(n', — n',')) h. ik 1, .... .si, where / assumes all values between <\nd 1. 

 Then it is easily proved that the frequency i/ of the radiation emitted during the 

 transition under consideration is equal to the mean value, taken over all states 

 from / = to ^ 1, of the frequency (n', — .... (/j.' — ng')w, which ap- 

 pears in the motion of the electron when this motion according to (12) is resolved 

 in its constituent harmonic components. In fact, from (11) it follows that the 

 difference in the total energy for two neighbouring mechanically possible slates, 

 characterised by /, , . . /, and 4- dl^, .... /. - fil, respectively, may be expressed 

 by the formula 



so that we get from ( 1 



âE ^ w.âl, — (o,àI,, (106) 



Syl = 1 /»>} = 1 - .1 



(107) 



Especially in the region of stationary states where the a s are so large, that 

 for small values of the numbers nl- — n',' the motion in the initial and in the final 

 state differ relatively little from each other, the w's may be considered as constant 

 when / varies from to 1, so that the frequency of the emitted radiation 

 approaches asymptotically to the frequency (/?' — /i',')w, 4- • • ih'](o,, 

 present in the motion of the system.') 



From this remarkable connection in the limit of large n's between the fre- 

 quencies of the spectral lines to be expected on the quantum theory and the fre- 

 quencies rjWi + .... -,(0., of the harmonic vibrations in which, according to (12). 

 the motion of a conditionally periodic system may be resolved, and which there- 

 fore according to ordinary electrodynamics would occur in Ihe electromagnetic 



' See BoHH, loc. cit. Part 1. p. .U. ('.ompaic .1. .\I Bliigeks Met atooinmoclel van Hutlierford-liolir, 

 Haarlem, li)18\ wlio recently has also called attention to this asymptotical relation in the region of large 

 HS, without entering, however, on the hearing of this relation on the prohlcm of the intensity and 

 polarisation of sjiectral lines. 



42* 



