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7.4 Spin±orbit interaction 205
2
motion, but S is. Since J is ®xed in direction and magnitude, both J and J 2
are constants of motion.
^
^
If we form the cross product J 3 J and substitute equations (7.36), (5.11),
and (7.3), we obtain
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
J 3 J (L S) 3 (L S) (L 3 L) (S 3 S) i"L i"S i"J
^
^
^
^
where the cross terms (L 3 S) and (S 3 L) cancel each other. Thus, the
^
operator J obeys equation (5.12) and the quantum-mechanical treatment of
^
^ ^
Section 5.2 applies to the total angular momentum. Since J x , J y , and J z each
^
^ 2
commute with J but do not commute with one another, we select J z and seek
the simultaneous eigenfunctions jnlsjm j i of the set of mutually commuting
^ 2 2 2 ^
operators H, L , S , J , and J z
^
Hjnlsjm j i E n jnlsjm j i (7:37a)
^ 2 2 (7:37b)
L jnlsjm j i l(l 1)" jnlsjm j i
^ 2 2 (7:37c)
S jnlsjm j i s(s 1)" jnlsjm j i
^ 2 2
J jnlsjm j i j(j 1)" jnlsjm j i (7:37d)
^
J z jnlsjm j i m j "jnlsjm j i, m j ÿj, ÿj 1, ... , j ÿ 1, j (7:37e)
From the expression
^ ^ ^
J z jnlsjm j i (L z S z )jnlsjm j i (m m s )"jnlsjm j i
obtained from (7.36), (5.28b), and (7.5), we see that
m j m m s (7:38)
The quantum number j takes on the values
l s, l s ÿ 1, l s ÿ 2, ... , jl ÿ sj
The argument leading to this conclusion is somewhat complicated and may be
3
found elsewhere. In the application being considered here, the spin s equals 1
2
and the quantum number j can have only two values
1
j l (7:39)
2
The resulting vectors J are shown in Figure 7.2.
^ : ^
The scalar product L S in equation (7.33) may be expressed in terms of
^
operators that commute with H by
^ : ^ 1 ^ ^ : ^ ^ 1^ : ^ 1^ : ^ 1 ^ 2 ^ 2 ^ 2 (7:40)
L S (L S) (L S) ÿ L L ÿ S S (J ÿ L ÿ S )
2 2 2 2
3 B. H. Brandsen and C. J. Joachain (1989) Introduction to Quantum Mechanics (Addison Wesley Longman,
Harlow, Essex), pp. 299, 301; R. N. Zare (1988) Angular Momentum (John Wiley & Sons, New York), pp.
45±8.
