252
22 Wigner- and Clebsch-Gordan Coefficients
[ ˆ
p i , ˆ
L j ] = ij k i ¯
h ˆ
p k
(22.1b)
[ ˆ
L i , ˆ
L j ] = ij k i ¯
h ˆ
L k ,
(22.1c)
which are sometimes summarized by the easy-to-memorize expressions
ˆ
r × ˆ
L = i ¯
hˆ r, · · · , ˆ
L × ˆ
L = i ¯
h ˆ
L.
(22.2)
For non-vanishing commutator equations we cannot simultaneously assign eigenvalues to all three components of the angular momentum operator. But we can
simultaneously measure the angular momentum square ˆ
L 2 = ˆ
L 2
1 + ˆ
L 2
2 + ˆ
L 2
3 with
any component of ˆ
L since these operators commute
[ ˆ
L 2 , ˆ
L i ] = 0 , i = 1, · · · , 3.
(22.3)
Thus the angular momentum eigenstates can be labeled by the eigenvalues of ˆ
L 2 and
one angular momentum component. It is customary to choose ˆ
L 2 and ˆ
L z . Denoting
the corresponding eigenvalues by l and m, we obtain
ˆ
L 2 |lm = l(l + 1) ¯
h
2
|lm l = 0, 1, 2, · · · and
ˆ
L z |lm = m ¯
h|lm m = −l, −l + 1, · · · , l − 1, l.
(22.4)
For practical purposes it is more comfortable to define the non-Hermitian ladder
operators
ˆ
L + = ˆ
L x + i ˆ
L y and ˆ
L − = ˆ
L x − i ˆ
L y ,
(22.5)
which fulfill the commutator relations
[ ˆ
L z , ˆ
L ± ] = ± ¯
h ˆ
L ± and [ ˆ
L + , ˆ
L − ] = 2 ¯
h ˆ
L z ,
(22.6)
and
ˆ
L ± |lm = ¯
h
l(l + 1) − m(m ± 1) |lm ± 1
(22.7)
In spherical coordinates the angular momentum operators hold
ˆ
L z =
¯
h
i
∂
∂φ
(22.8a)
ˆ
L ± = ¯
h exp(±iφ)
±
∂
∂θ
+ i cot θ
∂
∂φ
(22.8b)
ˆ
L 2 = − ¯
h
2
sin θ
∂
∂θ
sin θ
∂
∂θ
+
1
sin 2 θ
∂ 2
∂φ 2
,
(22.8c)