334 



50 



where /j, /., and are the quantities defined by (45), and where n,, /i, and n.^ are 

 positive integers. The different stationary states, characterised by different combina- 

 tions of the »'s, will in the following be denoted by the symbol (n^, n^, /J.,); they 

 are, as mentioned, a-priori equally probable, but it must be kept in mind that, 

 while and may assume the values 0, 1, 2, 3 ... ., n,j can only assume one of 

 the values 1, 2, 3, 4 ... . In fact, it was pointed out by Bohr that states correspond- 

 ing to »3 = cannot represent possible stationary states of the atom because there 

 is an essential singularity involved in the motion in these states'). 



The value of the total energy in the stationary states will be obtained by 

 introducing (110) in the expression (46) for the total energy of the system. The 

 frequency v of the radiation emitted during a transition between an initial state 

 (n'j, /?'„, n'3) and a final state {n", n", n") — such a transition will in the following 

 be denoted by the symbol (n[, n\, n[ n", n", /j'J) — will then, according to (1), 

 be given by 



where 



and where 



n' = n'^ + n'g -f- «3 , n" = n" n" -\- n'^. 



(Ill) 



The expression for v,, coincides with formula (105) holding for the frequencies of 

 the spectral lines emitted by the simplified hydrogen atom, when the mass of the 

 nucleus is considered as infinite. The additional term in the expression for v is pro- 

 portional to the intensity F of the electric force and allows us to calculate the magni- 

 ludes of the displacements from the position of the original line of the different 

 components in which this line splits up under the influence of the electric force ^). 

 As shown by Epstein and by Schwarzschild, formula (111) is in excellent agree- 

 ment with the frequencies of the different components of the hydrogen lines ob- 



') Bohr, loc. cit. Part II, p. 75. In this connection it may be observed that in states for which 

 «3=0 the motion of the electron would take place in a plane, and that, if the relativity modifications are 

 neglected, the angular momentum of the electron round the nucleus would in the course of the motion 

 become equal to zero at regular intervals and change its sign, so that in the course of time the elec- 

 tron would in general collide with the nucleus. On the other hand, if the relativity modifications are 

 taken into account, the perturbing influence of these modifications would become very large and of the 

 same order of magnitude as the influence of the electric field when the angular momentum approaches 

 to zero. As will be shown in the paper mentioned in the beginning of § 4, the value of this angular 

 momentum will never pass through zero and the motion of the electron would in the states in ques- 

 tion be essentially different from that in the non-relativity case. It was pointed out by Bohb, however, 

 that this circumstance does not, from the point of view of the quantum theory, remove the singular 

 cliaracter of these slates, which compels us to exclude them from the ensemble of possible stationary states. 



'0 See P. Epstein, Ann. d. Phys. L., p. 489 (1916), K. Schwarzschild. Berl. Ber. p. 548 (1916). The 

 correction for the finite mass of the nucleus in the expression for will, according to what has been 

 said in § 5 on page 40, be taken into account by simpU' replacing the above expression for v„ by that 

 which is given in formula (105). 



