Thus, we get a recurrence relation with respect to I (3.132) such that
I ¼
2l
2l þ 1
Z π
0
sin
2lÀ1
θdθ:
ð3:134Þ
Repeating the above process, we get
I ¼
2l
2l þ 1
Á
2l À 2
2l À 1
Á Á Á
2
3
Z π
0
sin θdθ ¼
2
2lþ1 l!
ð Þ
2
2l þ 1
ð
Þ!
:
ð3:135Þ
Then,
κ l
j j ¼
1
2
l l!
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2l þ 1
ð
Þ!
4π
r
or κ l ¼
e
iχ
2
l l!
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2l þ 1
ð
Þ!
4π
r
χ : real
ð
Þ,
ð3:136Þ
where e
iχ is an undetermined constant (phase factor) that is to be determined below.
Thus we get
Y
l
l θ, ϕ
ð
Þ ¼
e
iχ
2
l l!
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2l þ 1
ð
Þ!
4π
r
sin
l
θe
ilϕ
:
ð3:137Þ
Meanwhile, in the second equation of (3.101) replacing J
(À) , j, and μ in (3.101)
with M
(À) , l, and m, respectively, and using Y
m
l θ, ϕ
ð
Þ instead of jζ, μi, we get
Y
mÀ1
l
θ, ϕ
ð
Þ ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l À m þ 1
ð
Þl þ m
ð
Þ
p
M
À
ð Þ Y
m
l θ, ϕ
ð
Þ:
ð3:138Þ
Replacing m with l in (3.138), we have
Y
lÀ1
l
θ, ϕ
ð
Þ ¼
1
ffiffiffiffi
2l
p M
À
ð Þ Y
l
l θ, ϕ
ð
Þ:
ð3:139Þ
Operating M
(À) (l À m) times in total on Y
l
l θ, ϕ
ð
Þ of (3.139), we have
Y
m
l θ, ϕ
ð
Þ ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2l 2l À 1
ð
ÞÁÁÁ l þ m þ 1
ð
Þ
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 Á 2Á Á Á l À m
ð
Þ
p
M
À
ð Þ
h
i lÀm
Y
l
l θ, ϕ
ð
Þ
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
l þ m
ð
Þ!
2l
ð Þ! l À m
ð
Þ!
s
M
À
ð Þ
h
i lÀm
Y
l
l θ, ϕ
ð
Þ:
ð3:140Þ
Meanwhile, putting m ¼ l, n ¼ l À m, and j ¼ l in (3.117), we have
84
3 Hydrogen-Like Atoms
I ¼
2l
2l þ 1
Z π
0
sin
2lÀ1
θdθ:
ð3:134Þ
Repeating the above process, we get
I ¼
2l
2l þ 1
Á
2l À 2
2l À 1
Á Á Á
2
3
Z π
0
sin θdθ ¼
2
2lþ1 l!
ð Þ
2
2l þ 1
ð
Þ!
:
ð3:135Þ
Then,
κ l
j j ¼
1
2
l l!
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2l þ 1
ð
Þ!
4π
r
or κ l ¼
e
iχ
2
l l!
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2l þ 1
ð
Þ!
4π
r
χ : real
ð
Þ,
ð3:136Þ
where e
iχ is an undetermined constant (phase factor) that is to be determined below.
Thus we get
Y
l
l θ, ϕ
ð
Þ ¼
e
iχ
2
l l!
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2l þ 1
ð
Þ!
4π
r
sin
l
θe
ilϕ
:
ð3:137Þ
Meanwhile, in the second equation of (3.101) replacing J
(À) , j, and μ in (3.101)
with M
(À) , l, and m, respectively, and using Y
m
l θ, ϕ
ð
Þ instead of jζ, μi, we get
Y
mÀ1
l
θ, ϕ
ð
Þ ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l À m þ 1
ð
Þl þ m
ð
Þ
p
M
À
ð Þ Y
m
l θ, ϕ
ð
Þ:
ð3:138Þ
Replacing m with l in (3.138), we have
Y
lÀ1
l
θ, ϕ
ð
Þ ¼
1
ffiffiffiffi
2l
p M
À
ð Þ Y
l
l θ, ϕ
ð
Þ:
ð3:139Þ
Operating M
(À) (l À m) times in total on Y
l
l θ, ϕ
ð
Þ of (3.139), we have
Y
m
l θ, ϕ
ð
Þ ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2l 2l À 1
ð
ÞÁÁÁ l þ m þ 1
ð
Þ
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 Á 2Á Á Á l À m
ð
Þ
p
M
À
ð Þ
h
i lÀm
Y
l
l θ, ϕ
ð
Þ
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
l þ m
ð
Þ!
2l
ð Þ! l À m
ð
Þ!
s
M
À
ð Þ
h
i lÀm
Y
l
l θ, ϕ
ð
Þ:
ð3:140Þ
Meanwhile, putting m ¼ l, n ¼ l À m, and j ¼ l in (3.117), we have
84
3 Hydrogen-Like Atoms