84 THE QUANTUM [ VI . x 
electron. The radial quantum condition along OP is the same 
as in the two-dimensional problem. It can be shown, as is done 
in Sommerfeld’s book, Atomic Structure and Spectral Lines, 
p. 243, that the angular momentum in the plane of the orbit 
h 
pt is equal to k— } where f is the “ orbital azimuth ” in this 
plane, and k is the “ azimuthal ” quantum number. The third 
quantum condition may be expressed by saying that the 
generalized momentum corresponding to the resolved motion 
h 
of the electron in the equatorial plane is given by px = m— 
271, 
where m is a new quantum number which may be called the 
" equatorial ” quantum number. It must be noticed that this 
Fig. 12.—Quantization in Space. 
quantization is carried out for the system when the field tends 
to a limiting value of zero. Now let a denote the angle between 
the plane of the orbit and the equatorial plane, i.e. a is the 
angle at K in the spherical triangle KQP. Then we have 
■— — cos a. Substituting the values we have already found for 
p* and p+, we see that cos a = —. Reasons are given by Som- 
R 
merfeld for excluding the value m — o, which would mean that 
the plane of the orbit can be parallel to the applied field. Exclud 
ing this case we are left with k possible orientations in an external 
field, the possible orientations being given by 
cos a = 
k’ 
2 3 
k’ k 
k 
K