Page 142 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

5.2 Generalized angular momentum 133
^ ^ ^
[J x , J y ] ˆ i"J z (5:13a)
^ ^ ^
[J y , J z ] ˆ i"J x (5:13b)
^ ^ ^
[J z , J x ] ˆ i"J y (5:13c)
The square of the angular-momentum operator is de®ned by
^
^
^
^ : ^
2
2
2
J ˆ J J ˆ J ‡ J ‡ J ^ 2 (5:14)
x y z
^
^
^
^ 2
and is hermitian since J x , J y , and J z are hermitian. The operator J commutes
^
^
^
with each of the three operators J x , J y , J z . We ®rst evaluate the commutator
^ 2 ^
[J , J z ]
^
^
^
^
^
^
^
^
2
2
2
2
[J , J z ] ˆ [J , J z ] ‡ [J , J z ] ‡ [J , J z ]
x y z
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
ˆ J x [J x , J z ] ‡ [J x , J z ]J x ‡ J y [J y , J z ] ‡ [J y , J z ]J y
^ ^ ^ ^ ^ ^ ^ ^
ˆÿi"J x J y ÿ i"J y J x ‡ i"J y J x ‡ i"J x J y
ˆ 0 (5:15a)
^
where the fact that J z commutes with itself and equations (3.4b) and (5.13)
have been used. By similar expansions, we may also show that
^
^
2
[J , J x ] ˆ 0 (5:15b)
^
^
2
[J , J y ] ˆ 0 (5:15c)
^
^
^
^ 2
Since the operator J commutes with each of the components J x , J y , J z of
^
J, but the three components do not commute with each other, we can obtain
^ 2
simultaneous eigenfunctions of J and one, but only one, of the three compo-
^
^
nents of J. Following the usual convention, we arbitrarily select J z and seek the
^
^ 2
simultaneous eigenfunctions of J and J z . Since angular momentum has the
2
same dimensions as ", we represent the eigenvalues of J ^ 2 by ë" and the
^
eigenvalues of J z by m", where ë and m are dimensionless and are real
^
^ 2
because J and J z are hermitian. If the corresponding orthonormal eigenfunc-
tions are denoted in Dirac notation by jëmi, then we have
^ 2 2
J jëmiˆ ë" jëmi (5:16a)
^ 2
J z jëmiˆ m" jëmi (5:16b)
We implicitly assume that these eigenfunctions are uniquely determined by
only the two parameters ë and m.
^ 2
^ 2
The expectation values of J and J are, according to (3.46), and (5.16)
z
^
^
2
2
hJ iˆhëmjJ jëmiˆ ë" 2
2 2
2
^
2
^
hJ iˆhëmjJ jëmiˆ m "
z
z