Page 213 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

204 Spin
^ : ^
^ 2
^ 2
constants of motion. However, the operators L and S do commute with L S,
2
2
which follows from equations (5.15), so that the quantities L and S are still
constants of motion.
We now introduce the total angular momentum J, which is the sum of L
and S
J ˆ L ‡ S (7:36)

^
^
^ 2
The operators J and J commute with H 0 . The addition of equations (7.34)
and (7.35) gives
^ ^ : ^
^ ^ : ^
^ ^ : ^
[J z , L S] ˆ [L z , L S] ‡ [S z , L S] ˆ 0
The addition of similar relations for the x- and y-components of these angular
^ : ^
^ ^ : ^
^
momentum vectors leads to the result that [J, L S] ˆ 0, so that J and L S
^ : ^
commute. Furthermore, we may easily show that J ^ 2 commutes with L S
^ : ^
^ : ^
^
^ 2
^ 2
^ 2
because each term in J ˆ L ‡ S ‡ 2L S commutes with L S. Thus, J
^
^ 2
2
and J commute with H in equation (7.33) and J and J are constants of
motion.
2
2
2
That the quantities L , S , J , and J are constants of motion, but L and S are
not, is illustrated in Figure 7.1. The spin magnetic moment M s , which is
antiparallel to S, exerts a torque on the orbital magnetic moment M, which is
antiparallel to L, and alters its direction, but not its magnitude. Thus, the orbital
angular momentum vector L precesses about J and L is not a constant of
2
motion. However, since the magnitude of L does not change, the quantity L is
a constant of motion. Likewise, the orbital magnetic moment M exerts a torque
on M s , causing S to precess about J. The vector S is, then, not a constant of
S

J
L

Figure 7.1 Precession of the orbital angular momentum vector L and the spin angular
momentum vector S about their vector sum J.