ψ
n
ð Þ
lþ1
D
ψ
n
ð Þ
lþ1
E
¼ ε
nÀ1
ð
Þ
À ε
nÀ2
ð
Þ
h
i
ε
nÀ1
ð
Þ
À ε
nÀ3
ð
Þ
h
i
Á Á Á ε
nÀ1
ð
Þ
À ε
lþ1
ð
Þ
h
i
ψ
n
ð Þ
nÀ1
D
ψ
n
ð Þ
nÀ1
E
:
Inserting this equation into (3.272), we arrive at (3.268). In other words, we have
shown that if (3.268) holds with l ¼ l + 1, (3.268) holds with l ¼ l as well. This
completes the proof to show that (3.268) is true of ψ
n
ð Þ
l
D
ψ
n
ð Þ
l
E
with l down to 0.
The normalized wave functions e
ψ
n
ð Þ
l are expressed from (3.255) as
e
ψ
n
ð Þ
l ¼ κ n, l
ð Þ
À
1
2 b lþ1 b lþ2 Á Á Áb nÀ1 e
ψ
n
ð Þ
nÀ1 ,
ð3:273Þ
where κ(n, l ) is defined such that
κ n, l
ð Þ ε
nÀ1
ð
Þ
À ε
nÀ2
ð
Þ
h
i
Á ε
nÀ1
ð
Þ
À ε
nÀ3
ð
Þ
h
i
Á Á Á Á ε
nÀ1
ð
Þ
À ε
l
ð Þ
h
i
,
ð3:274Þ
with l n À 2. More explicitly, we get
κ n, l
ð Þ ¼
2n À 1
ð
Þ! n À l À 1
ð
Þ ! l!
ð Þ
2
n þ l
ð
Þ! n!
ð Þ
2 n nÀlÀ2
ð
Þ
2
:
ð3:275Þ
In particular, from (3.265) we have
e
ψ
n
ð Þ
nÀ1 ¼
2
n
nþ
1
2
1
ffiffiffiffiffiffiffiffiffiffi
2n
ð Þ!
p
ρ
n e
Àρ=n
:
ð3:276Þ
From (3.272), we define the following operator:
e b l ε
nÀ1
ð
Þ
À ε
lÀ1
ð
Þ
h
i À
1
2 b l :
ð3:277Þ
Then (3.273) becomes
e
ψ
n
ð Þ
l ¼ e b lþ1 e b lþ2 Á Á Á e b nÀ1 e
ψ
n
ð Þ
nÀ1 :
ð3:278Þ
3.7 Radial Wave Functions of Hydrogen-Like Atoms
115