324 



40 



The eccentricity ol the orbit and the position of its plane in space are undetermined 

 in the stationary states. 



In the deduction of the preceding formulae, the mass of the nucleus is 

 regarded as infinite compared with that ot the electron. If we take into account 

 that the mass of the nucleus is finite, the motion of the simplified iiydrogen 

 atom will still be periodic, the electron and the nucleus describing both a 

 closed Keplerian orbit with their common centre of gravity at one of the foci, 

 and separation of variables may again be obtained for any set of polar coordi- 

 nates as well as for any set of parabolic coordinates with this centre of gravity at the 

 centre. Performing the necessary calculations, it is easily found that the necessary 

 modifications to be introduced in the above formulae on account of the finite mass 

 of the nucleus are obtained by replacing, in the expressions for E and co, the quantity 

 m by = . "^ / > where M represents the mass of the nucleus. The expres- 



sion for the major axis of the orbit of the electron remains the same, while the 

 major axis of the orbit of the nucleus becomes equal to „ " , „ . For the energy 

 in the n"' stationary state of the simplified hydrogen atom we thus get 



In the calculations in § 2, § 3 and § 4, the correction for the finite mass of 

 the nucleus has not been taken into account, but since the motion of the electron 

 treated in these sections shows only small deviations from the periodic Keplerian 

 motion just considered, it is on account of the small value of "^Im obviously per- 

 mitted to neglect this correction in the calculation of these deviations and of their 

 effect on the total energy in the stationary states. 



From the above it is seen that the stationary states of a conditionally periodic 

 system are fixed by a number of conditions of the type = yj^./j. Calling this 

 number r, the total energy will be a function of »j, ... /?,., and according to (1) 

 the frequency u of the radiation emitted during a transition between two stationary 

 .states, which are characterised by /?, = /?',,.... n,. = n'j. and n, = n", .... n,. = n", 

 respectively, will be given by 



V = \ "r ) — ^ ("i- • • O ) • < 



The state of largest energy, ciiaracterised by n',, .... n',., will in the following be 

 denoted as the "initial state", the state of smallest energy, characterised by /i", ... /?'.', 

 as the "final state" of the transition in question. Formula (104) allows us to calculate 

 all possible values for the frequencies of the spectral lines which may be emitted 

 by the system. Thus, for the spectrum of the simplified hydrogen atom, we get from 

 (104) for the frequency y of the radiation emitted during a transition from an initial 

 state to a final state characterised by n' and n" respectively — such a transition 



