362 



78 



With reference to the general relation which, according to Bohr, must be ex- 

 pected to exist between the additional energy of a degenerate system due to the 

 presence of small external forces and the small frequency (or frequencies) impres- 



the mean value of the potential energy of the sj'stem with regard to these forces, taken over a long 

 time interval for the "corresponding " stationary motion of the undisturbed system, i. e. the motion in 

 the state which would appear if the perturbing field decreased to zero infinitely slowly and at a uni- 

 form rate. This theorem follows directly from the principle of the mechanical transformability of the 

 stationary states, since it may be sliown that during such a slow change of the perturbing field the 

 external forces will, with this approximation, not perform work on the particles of the system (com- 

 pare Bohr, loc. cit. Part II). In order to apply the theorem in the present case we have to calculate 

 the mean value of the potential energy of the electron with respect to a homogeneous electric field of 

 force, taken over the motion which tliis electron performs in tlie stationary states of the undisturbed 

 hydrogen atom, but owing to the symmetry of the latter motion round the nucleus this mean value is 

 always equal to zero. In fact, with the notation of § 4, the perturbing potential is equal to Fez, and 

 from (90) it is seen that the trigonometric series representing z as a function of the time for the 

 undisturbed system does not contain a constant term, so that the mean value of z is equal to zero. 

 (Compare also .1. M. Burgers, Het atoommode! van Rutlierford-Bohr, Haarlem 1918, p. 128.) 



In the later paper, referred to above, which deals with the transmutation of the fine structure 

 into the Stark effect, it will be proved that, for small values of the intensity of the electric field, 

 the additional energy in the stationary states may be represented by a series of terms of the form 

 I , if we disregard small terms the ratio of which to one of these terms is 

 of the order F, F^, or o, o^. . The largest term in the expression for the additional energy is 

 thus seen to be a small quantity of the order ^ , and it is of interest that this term may be calcu- 

 lated already from the formulae deduced in § 4. Thus it is seen from (94) that the mean position of the 

 electron taken over a large time interval, which for the undisturbed orbit coincided with the nucleus, 

 under the influence of the electric field is displaced in the direction of the positive z-axis by an amount 



equal to '^^^ ^ • ^ ' = ^'^ ""^^ imagine that the electric force increases slowly and 



uniformly from zero, it will be seen that the work performed hy the external force on the atom during 

 this process will be equal to eFdisF} = - IseF^. Since further the mean value of the potential 

 energy of the perturbed atom with respect to the electric field is equal to sF-eF = seF-, it is seen, 

 with reference to the principle of the mechanical transformability of the stationary states, that we 

 may conclude that the additional energy in the stationary states of the system under consideration 



will be given by seF- — } seF- = \ seF^. Introducing /" = nh, s' = n^Jn. ,/ = n o = 



(2jj^g2.2 J 2 2 ' /}• 



He I iv'n' ''^"'"P^"'^ formula 186;), and denoting the additional energy by J/t, we have tiuis 



2 



-^^=-4(2^) N^e^n^^'^^^^''"^^^^- 



This formula allows in first approximation to calculate tlie displacements of the components of the fine 

 structure under the influence of an external electric field. In fact, the ])resence of the perturbing forces 

 will cause that the frequency of the radiation corresponding to a transition ; 11' — n", /)"; n"), 



which gives rise to one of the components into which the fine structure component (n', n' ^ n'\ n") 

 is split up, will differ from the value of v given by (119) by an amount 



JE' - JE" 9 /j'c-'F- r ,. , , , , „, nul 



h 4 |27:)*'N'' e"'m L 2 2 ' ' 2 2 J 



For the sake of orientation it maj' be of interest here to note that for increasing intensity of the elec- 

 tric field a state of the system, for which /',' = n.h, I'i = (;i., ii)/i, /!] = nil, will be continuously 



