378 



94 



atom M, this does not constitute a difficulty for the theory but it just what should be 

 expected according to the above considerations of the ellecl of perturbing fields on 

 the fine structure. 



§ 8. The effect of a magnetic field on the fine structure 

 of the hydrogen Hnes. 



In this chapter we shall briefly consider certain points which present them- 

 selves in connection with the application of the quantum theory to the problem of 

 the effect of a magnetic field on the fine structure of the hydrogen 

 lines, and from wdiich it is possible to draw conclusions which are of interest in 

 connection with the problems discussed in the preceding chapters. 



The problem of the influence of a homogeneous magnetic field on the hydrogen 

 atom may be treated in a similar way as the influence of an electric field on the 

 simplified hydrogen atom, since the equations of motion of the electron also in the 

 presence of the magnetic field may be written in the canonical form and since, if 

 we look apart from small quantities proportional to the square of the intensity of 

 the magnetic force, a solution of these equations may be obtained by separation of 

 variables in the Hamilton-Jacobi partial differential equation if polar coordinates 

 are introduced.-) The motion in the stationary states will then be fixed by three 

 conditions of the type (99). The results obtained in this way may be very simply 

 interpreted. In fact, as mentioned in § 2, the mechanical motion of the electron 

 in the hydrogen atom in the presence of a homogeneous magnetic field differs from 

 a mechanical motion in the absence of this field only by a slow- and uniform super- 

 posed rotation round an axis through the nucleus parallel to the magnetic force, the 

 frequency Oa of which is given by (40), and it is simply shown that the stationary 

 states of the system in the presence of the field are obtained by superposing a 

 rotation of this kind on a stationary motion of the atom without field which, be- 

 sides satisfying the conditions (117) characterising the stationary states of the un- 

 disturbed atom, satisfies the further condition that the value of the angular momen- 

 tum of the electron round the axis is equal to an entire multiple of ''/27r 0- Denoting 

 this value by "''/'iTr, the stationary states may, in analogy with the notation used in 

 the preceding chapter, be characterised by the symbol (n^, n^; u). Further the different 

 possible stationary states corresponding to different combinations of Hj, n^, n {n.^ ■> n) 

 will again be a-priori equally probable, but just as in the case of the perturbed 

 system treated in the preceding chapter, we must assume that neither n., nor ii can 

 assume the value zero. 



') See T. R. Merton and .J. \V. Nicholson. Trnns. Roy. Soc. A .'S5.') (1918). 



■-■) Compare A. Sommerfklu, Fhys. Zeitsclir. XVII, p. 491 (191(;), and es|)ecially P. Di- nyi:. ibid. p. .")07. 

 ■ J ('.oni])arc Rohh, loc. c it. Part. II, p. S2. 



