L z ¼ Àiħ
∂
∂ϕ
:
ð3:24Þ
In a similar manner, we have
∂
∂z
¼
∂r
∂z
∂
∂r
þ
∂θ
∂z
∂
∂θ
¼ cos θ
∂
∂r
À
sin θ
r
∂
∂θ
:
Combining this relation with either (3.23) or (3.22), we get
L x ¼ iħ sin ϕ
∂
∂θ
þ cot θ cos ϕ
∂
∂ϕ
,
ð3:25Þ
L y ¼ Àiħ cos ϕ
∂
∂θ
À cot θ sin ϕ
∂
∂ϕ
:
ð3:26Þ
Now, we introduce following operators:
L
þ
ð Þ
L x þ iL y
and
L
À
ð Þ
L x À iL y :
ð3:27Þ
Then, we have
L
þ
ð Þ
¼ ħe
iϕ ∂
∂θ
þ i cot θ
∂
∂ϕ
and L
À
ð Þ
¼ ħe
Àiϕ
À
∂
∂θ
þ i cot θ
∂
∂ϕ
: ð3:28Þ
Thus, we get
L
þ
ð Þ L
À
ð Þ
¼ ħ
2 e
iϕ ∂
∂θ
þ i cot θ
∂
∂ϕ
e
Àiϕ
À
∂
∂θ
þ i cot θ
∂
∂ϕ
¼ ħ
2 e
iϕ e
Àiϕ
À
∂
2
∂θ
2
þ i À
1
sin
2
θ
∂
∂ϕ
þ i cot θ
∂
2
∂θ∂ϕ
!
&
þe
Àiϕ cot θ À
∂
∂θ
þ i cot θ
∂
∂ϕ
þ ie
Àiϕ cot θ À
∂
2
∂ϕ∂θ
þ i cot θ
∂
2
∂ϕ
2
'
¼ Àħ
2 ∂
2
∂θ
2
þ cot θ
∂
∂θ
þ i
∂
∂ϕ
þ cot
2
θ
∂
2
∂ϕ
2
ð3:29Þ
In the above calculation procedure, we used differentiation of a product function.
For instance, we have
64
3 Hydrogen-Like Atoms