Now, we know that m takes (2l + 1) different values that correspond to each l.
This implies that the operators can be expressed with (2l + 1, 2l + 1) matrices. As
implied in (3.151), M
(À) takes the following form:
M
À
ð Þ
¼
0
ffiffiffiffiffiffiffiffiffi ffi
2l Á 1
p
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2l À 1
ð
ÞÁ 2
p
0
⋱
⋱
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2l À k þ 1
ð
ÞÁk
p
0
⋱
⋱
0
ffiffiffiffiffiffiffiffiffi ffi
1 Á 2l
p
0
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
,
ð3:152Þ
where diagonal elements are zero and a (k, k + 1) element is
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2l À k þ 1
ð
ÞÁk
p
. That
is, nonzero elements are positioned just above the zero diagonal elements. Correspondingly, we have
M
þ
ð Þ
¼
0
ffiffiffiffiffiffiffiffiffi
2l Á 1
p
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2l À 1
ð
ÞÁ2
p
0
⋱
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2l À k þ 1
ð
ÞÁk
p
0
⋱
⋱
0
⋱
0
ffiffiffiffiffiffiffiffiffi
1 Á 2l
p
0
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
,
ð3:153Þ
3.5 Orbital Angular Momentum: Operator Approach
87