67 



due to the attraction from Ihe nucleus.') Now it is e:isily shown thai this lalio 



will be a small quantity of the same order of magnitude as A'- ( ^ ) , and it would 



therefore beforehand seem justified in the expression for the total energy in the 

 stationary states to retain small terms of the same order as the second term in 

 (11), while at the same time it might appear highly questionable, whether, in the 

 complete expression for the total energy in the stationary stales deduced by Sommer- 

 feld and Debye on the basis of the conditions (16), it has a physical meaning to 

 retain terms of higher order than those retained in formula (68); unless A' is a large 

 number, as in the theory of the Röntgenspectra to be discussed in Part III. 



While the preceding considerations, which deal with the determination of the 

 energy in the stationary states of the hydrogen atom, allow to determine the fre- 

 quency of the radiation which would be emitted during a transition between 

 two such states, they leave quite untouched the problem of the actual occurrence 

 of these transitions in the luminous gas, and therefore give no direct information 

 about the number and relative intensities of the components into which 

 the hydrogen lines may be expected to split up as a consequence of the relativity 

 modifications. This"problem has recently been discussed by Sommerfeld-), who in 

 this connection emphasises the importance of the different a-priori probabilities of 

 the stationary states, characterised by different sets of values of the ;!'s in the con- 

 ditions (16). Thus Sommerfeld attempts to obtain a measure for the relative in- 

 tensities of the components of the fine structure of a given line, by comparing the 

 intensities observed with the products of the values of the a-priori probabilities of 

 tlie two states, involved in the emission of the components under consideration; and he 

 tries in this connection to test different expressions for these a-priori probabilities 

 (See Part I, pag. 26). In this way, however, it was not found possible to account 

 in a satisfactory manner for the observations; and the difficulty in obtaining an ex- 

 planation of the intensities on this basis was also strikingly brought out by the 

 fact, that the number and relative intensities of the components observed varied in 

 a remarkable way with the experimental conditions under which the lines were 



') Compare Part I, p. (i. It may in this connection be noted tliat the degree of appi'o.ximation, uivulved 

 in the determination of tlie frequencies of an atomic system by means of relation il if in the fi.xation 

 of the stationary states we look apart from small forces of the same order of magnitude as the radia- 

 tion forces in ordinarj' electrodynamics, would appear to be intimately connected with the limit of 

 sharpness of the spectral lines, which depends on the total number of waves contained in the 

 radiation emitted during the transition between two stationary states. In fact, from a consideration 

 based on the general connection between the quantum theory and the ordinary theory of radiation, it 

 seems natural to assume that the rate, at which radiation is emitted during a transition between two 

 stationary states, is of the same order of magnitude as the rate, at which radiation would be emitted 

 from the system in these states according to ordinary electrodynamics. But this will be seen to imply 

 that the total number of waves in question will just be of the same order as the ratio between the 

 main forces acting on the particles of the system and the reaction from the radiation in ordinary 

 electrodynamics. 



-) A. Sommerfeld, Ber. Akad. München, 1917. p. 83. 



