Y
0
3 θ, ϕ
ð
Þ¼ 3!
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
7 Á 3! Á 3!
4π
r
X 3
r¼0
À1
ð Þ
r cos
6À2r θ
2
sin
2r θ
2
r! 3 À r
ð
Þ! 3 À r
ð
Þ!r!
¼ 18
ffiffiffi
7
π
r cos
6 θ
2
0!3!3!0!
À
cos
4 θ
2
sin
2 θ
2
1!2!2!1!
þ
cos
2 θ
2
sin
4 θ
2
2!1!1!2!
À
sin
6 θ
2
3!0!0!3!
2
6
6
4
3
7
7
5
¼ 18
ffiffiffi
7
π
r
cos
6 θ
2
À sin
6 θ
2
36
þ
cos
2 θ
2
sin
2 θ
2
sin
2 θ
2
À cos
2 θ
2
!
4
8
> > <
> > :
9
> > =
> > ;
¼
ffiffiffi
7
π
r
5
4
cos
3
θ À
3
4
cos θ
,
where in the last equality we used formulae of elementary algebra and trigonometric
functions. At the same time, we get
Y
0
3 0, ϕ
ð
Þ ¼
ffiffiffiffiffi
7
4π
r
:
This is consistent with (3.147) in that Y
0
3 0, ϕ
ð
Þ is positive.
3.6.2 Orthogonality of Associated Legendre Functions
Orthogonality relation of functions is important. Here we deal with it, regarding the
associated Legendre functions.
Replacing m with (m À 1) in (3.174) and using the notation introduced before, we
have
1 À x
2
À
Á
d
mþ1 P l À 2mxd
m P l þ l þ m
ð
Þ l À m þ 1
ð
Þ d
mÀ1 P l ¼ 0:
ð3:219Þ
Multiplying both sides by (1 À x
2 )
m À 1 , we have
1 À x
2
À
Á m d
mþ1 P l À 2mx 1 À x
2
À
Á mÀ1 d
m P l
þ l þ m
ð
Þ l À m þ 1
ð
Þ1 À x
2
À
Á mÀ1 d
mÀ1 P l ¼ 0:
Rewriting the above equation, we get
3.6 Orbital Angular Momentum: Analytic Approach
103
0
3 θ, ϕ
ð
Þ¼ 3!
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
7 Á 3! Á 3!
4π
r
X 3
r¼0
À1
ð Þ
r cos
6À2r θ
2
sin
2r θ
2
r! 3 À r
ð
Þ! 3 À r
ð
Þ!r!
¼ 18
ffiffiffi
7
π
r cos
6 θ
2
0!3!3!0!
À
cos
4 θ
2
sin
2 θ
2
1!2!2!1!
þ
cos
2 θ
2
sin
4 θ
2
2!1!1!2!
À
sin
6 θ
2
3!0!0!3!
2
6
6
4
3
7
7
5
¼ 18
ffiffiffi
7
π
r
cos
6 θ
2
À sin
6 θ
2
36
þ
cos
2 θ
2
sin
2 θ
2
sin
2 θ
2
À cos
2 θ
2
!
4
8
> > <
> > :
9
> > =
> > ;
¼
ffiffiffi
7
π
r
5
4
cos
3
θ À
3
4
cos θ
,
where in the last equality we used formulae of elementary algebra and trigonometric
functions. At the same time, we get
Y
0
3 0, ϕ
ð
Þ ¼
ffiffiffiffiffi
7
4π
r
:
This is consistent with (3.147) in that Y
0
3 0, ϕ
ð
Þ is positive.
3.6.2 Orthogonality of Associated Legendre Functions
Orthogonality relation of functions is important. Here we deal with it, regarding the
associated Legendre functions.
Replacing m with (m À 1) in (3.174) and using the notation introduced before, we
have
1 À x
2
À
Á
d
mþ1 P l À 2mxd
m P l þ l þ m
ð
Þ l À m þ 1
ð
Þ d
mÀ1 P l ¼ 0:
ð3:219Þ
Multiplying both sides by (1 À x
2 )
m À 1 , we have
1 À x
2
À
Á m d
mþ1 P l À 2mx 1 À x
2
À
Á mÀ1 d
m P l
þ l þ m
ð
Þ l À m þ 1
ð
Þ1 À x
2
À
Á mÀ1 d
mÀ1 P l ¼ 0:
Rewriting the above equation, we get
3.6 Orbital Angular Momentum: Analytic Approach
103