M
À
ð Þ
h
i lÀm
Y
l
l θ, ϕ
ð
Þ ¼ e
Ài lÀm
ð
Þϕ
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À ξ
2
q
Àm ∂
lÀm
∂ξ
lÀm
Â
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À ξ
2
q
l
Y
l
l θ, ϕ
ð
Þ
"
#
:
ð3:141Þ
Further replacing M
À
ð Þ
Â
à lÀm Y
l
l θ, ϕ
ð
Þ in (3.140) with that of (3.141), we get
Y
m
l θ, ϕ
ð
Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
l þ m
ð
Þ!
2l
ð Þ! l À m
ð
Þ!
s
e
Ài lÀm
ð
Þϕ
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À ξ
2
q
Àm ∂
lÀm
∂ξ
lÀm
Â
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À ξ
2
q
l
Y
l
l θ, ϕ
ð
Þ
"
#
:
ð3:142Þ
Finally, replacing Y
l
l θ, ϕ
ð
Þ in (3.142) with that of (3.137) and converting θ to ξ,
we arrive at the following equation:
Y
m
l θ, ϕ
ð
Þ ¼
e
iχ
2
l l!
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2l þ 1
ð
Þ l þ m
ð
Þ!
4π l À m
ð
Þ!
s
e
imϕ 1 À ξ
2
À
Á Àm=2 ∂
lÀm
∂ξ
lÀm
1 À ξ
2
À
Á l
h
i
: ð3:143Þ
Now, let us decide e
iχ . Putting m ¼ 0 in (3.143), we have
Y
0
l θ, ϕ
ð
Þ ¼
e
iχ
À1
ð Þ
l
2
l l! À1
ð Þ
l
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2l þ 1
ð
Þ
4π
r
∂
l
∂ξ
l
1 À ξ
2
À
Á l
h
i
,
ð3:144Þ
where we put (À1)
l on both the numerator and denominator. In RHS of (3.144),
À1
ð Þ
l
2
l l!
∂
l
∂ξ
l
1 À ξ
2
Þ
l
À
à P l ξ
ð Þ:
Â
ð3:145Þ
Equation (3.145) is well known as Rodrigues formula of Legendre polynomials.
We mention characteristics of Legendre polynomials in the next section. Thus,
Y
0
l θ, ϕ
ð
Þ ¼
e
iχ
À1
ð Þ
l
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2l þ 1
ð
Þ
4π
r
P l ξ
ð Þ:
ð3:146Þ
According to the custom [2], we require Y
0
l 0, ϕ
ð
Þto be positive. Noting that θ ¼ 0
corresponds to ξ ¼ 1, we have
3.5 Orbital Angular Momentum: Operator Approach
85