83 



of the electron round the axis of the field will he c<|ual to an entire multiple 

 of /I/2-. 



As regards the expression for the total energy of the hydrogen atom in the 

 presence of the magnetic field, formula (80) is in agreement with the formulae ob- 

 tained by Sommerfeld and Debye on the basis of the conditions (22), holding for con- 

 ditionally periodic systems which allow of separation of variables. As shown by these 

 authors, a system, which consists of an electron moving under the influence of 

 the attraction from a fixed nucleus and of a homogeneous magnetic field, allows of 

 separation of variables in polar coordinates, if the polar axis is chosen parallel to 

 the magnetic field. Looking apart from the effect of the relativity modifications, 

 and choosing for q^, q^ and q^ the length of the radius vector from the nucleus to 

 the electron, the angle between this radius vector and the axis of the system, and 

 the angle which the plane through the electron and this axis makes with a 

 fixed plane through the axis respectively, they obtain the following expression for 

 the total energy: ') 



/i^(;ji + 7J2 + "3) 4;rmc ' ^ ^ 



where n^, n^ and n^ are the integers which appear as factors to Planck's constant 

 on the right side of the conditions (22). As mentioned this formula gives the same 

 result as (80); in fact, if we put n = n^ -{-n^-^-n^ and if we look apart from the 

 small correction due to the finite mass of the nucleus, the first term in (81) is seen 

 to coincide with the expression for — W„ given by (41), while the last term in (81) 

 coincides with the last term in (80), if we put | ii , = n.^. It will be observed, however, that, 

 while in the theories of Sommerfeld and Debye the stationary states are charac- 

 terised by three conditions, only two conditions were necessary on the above con- 

 siderations in order to secure the right relation between the energy and frequencies 

 of the system in the stationary states. Thus, besides the conditions which prescribe 

 the length of the major axis of the rotating orbit and the value of the angular 

 momentum of the system round the axis of the field, the theories of the mentioned 

 authors involve the further condition, that the value of the total angular momentum 

 of the electron round the nucleus must be equal to an entire multiple of ''/2-; and 

 that consequently the minor axis of the orbit has the same values as in a hydrogen 

 atom perturbed by a small external central field (compare page -57). This is due to 

 the circumstance, that the perturbed atom forms a degenerate system if we 

 look apart from the effect of the relativity modifications, because the secular per- 



') A. Sommerfeld, Phys. Zeitschr. XVil, p. 491 (1916) and P. Debye, Phys. Zeitschr. XVII, p. 507 

 (1916). While Debye proceeds directly by tlie application of tlie conditions (22) in a fixed set of posi 

 tional polar coordinates, Sommerfeld determines tlie stationary states by applying these conditions to 

 the motion of the system relative to a set of coordinates wliich rotates uniformlj' round the polar axis 

 with tlie frequency o,,; a procedure which in the special case under consideration is simply shown to 

 give the same result as the direct application of (221 to lixed pohir coordinates. 



