Subatomic Physics

5.1. Mass and Spin. Fermions and Bosons 81
Figure 5.1: Vector diagram for an angular momentum with quantum number l =2,m =1. The
other possible orientations are indicated by dashed lines.
The first equation states that the magnitude of the angular momentum is quantized
and restricted to values [l(l+1)] 1/2�. The second equation states that the component
of the angular momentum in a given direction, called z by general agreement, can
assume only values m�. The quantum numbers l and m must be integers, andfor
a given value of l, m can assume the 2l +1 valuesfrom−lto +l. The spatial
quantization is expressed in a vector diagram, shown in Fig. 5.1 for l =2. The
component along the arbitrarily chosen z direction can assume only the values
shown.
We repeat again that the quantization of the orbital angular momentum Eq. (5.2)
leads to integral values of l and hence to odd values of 2l +1, thenumberof possible orientations. It was therefore a surprise when the alkali spectra showed
unmistakable doublets. Two orientations demand 2l +1 = 2 or l = 1
2
. Many
attempts were made before 1924 to explain this half-integer number. The first half
of the correct solution was found by Pauli in 1924; he suggested that the electron
possesses a classically nondescribable two-valuedness, but he did not associate a
physical picture with this property. The second half of the solution was provided by
Uhlenbeck and Goudsmit, who postulated a spinning electron. The two-valuedness
then arises from the two different directions of rotation.
Of course, a way has to be found to incorporate the value 1
2
into quantum
mechanics. It is easy to see that the quantum mechanical operators that correspond
to L, Eq. (5.2), satisfy the commutation relations
LxLy − LyLx = i�Lz
LyLz − LzLy = i�Lx
LzLx − LxLz = i�Ly.
(5.5)
It is postulated that the commutation relations, Eq. (5.5), are more fundamental