XV. 2] 
THE ZEEMAN EFFECT 
209 
2. The Quantum Theory of the Zeeman Effect 
Difficulty is met with in applying the quantum theory even 
in the case of the simple Zeeman effect. The problem has been 
discussed by a number of workers, including Herzfeld,* * * § Bohr, f 
and later Debye | and Sommerfeld.§ Considerable progress has 
been made in the solution of the problem by the last two on 
this list. The difficulty of accounting for any effect at all had 
been emphasized by Mosharrafa,i| and more recently by Hicks, 
the former having shown how the extended form of the quantum 
restrictions put forward by W. Wilson leads to the simple Zeeman 
effect. Wilson himself has also discussed this problem. 
We shall, however, consider only the simple deduction of the 
Zeeman separation due to Sommerfeld. According to Bohr’s 
second postulate radiation is emitted only in the transition be 
tween two stationary states, the frequency v of the emitted 
spectral line being determined by the relation hv = H a — H e , 
where H a and H e are the initial and final energies of the system. 
The effect of the magnetic field is to produce a change in the 
energy in the initial state of amount AH a , and in the final state 
of amount AH e . 
Consequently there is a change in the frequency of the radia 
tion emitted determined by the equation 
hAv =AH a — AH e ... .15:4 
It is necessary then to determine the change in the energy 
of a stationary state due to some applied magnetic field of 
strength H. Sommerfeld supposes that orientation in space 
occurs as described in a previous chapter, but finds that there 
is no change in the form of the orbit, the change in energy being 
equal simply to the change in the kinetic energy, and the amount 
h 
of this change AH is equal to m—o where o is the angular velo- 
QTC 
city corresponding to the Larmor precession. When we sub 
stitute the value for o we obtain 
AH = nth— — 
m 0 47TC 
15:5 
In this expression m is the equatorial or magnetic quantum num 
ber which determines the resolved angular momentum about the 
* Herzfeld, Phys. Zeits., vol. 15, p. 193, 1914. 
t Bohr, Phil. Mag., vol. 27, p. 506, 1914. 
I Debye, Phys. Zeits., vol. 17, p. 507, 1916. 
§ Sommerfeld, Phys. Zeits., vol. 17, p. 491, 1916. 
II Mosharrafa, Proc. Roy. Soc. A, vol. 102, p. 529, 1923. 
TJ Hicks, Nature, vol. 115, p. 978, 1925.