Z 1
0
ρ
2n e
À2ρ=n dρ ¼
1
2
2nþ1
2n
ð Þ!n
2nþ1
ð
Þ
:
ð3:264Þ
Hence,
c n ¼
2
n
nþ
1
2 =
ffiffiffiffiffiffiffiffiffiffi
2n
ð Þ!
p
:
ð3:265Þ
To further normalize the other wave functions, we calculate the following inner
product:
ψ
n
ð Þ
l
D
ψ
n
ð Þ
l
E
¼ ψ
n
ð Þ
nÀ1 b
{
nÀ1 Á Á Áb
{
lþ2 b
{
lþ1
D
b lþ1 b lþ2 Á Á Áb nÀ1 ψ
n
ð Þ
nÀ1
E
:
ð3:266Þ
From (3.247) and (3.248), we have
b
{
l b l þ ε
lÀ1
ð
Þ
¼ b lþ1 b
{
lþ1 þ ε
l
ð Þ l ! 1
ð
Þ:
ð3:267Þ
Applying (3.267) to (3.266) repeatedly and considering (3.251), we reach the
following relationship:
ψ
n
ð Þ
l
D
ψ
n
ð Þ
l
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
l
ð Þ
h
i
ψ
n
ð Þ
nÀ1 ψ
n
ð Þ
nÀ1
D
E
:
ð3:268Þ
To show this, we use mathematical induction. We have already normalized ψ
n
ð Þ
nÀ1
in (3.261). Next, we calculate
D
ψ
n
ð Þ
nÀ2 jψ
n
ð Þ
nÀ2
E
such that
D
ψ
n
ð Þ
nÀ2 jψ
n
ð Þ
nÀ2
E
¼
D
ψ
n
ð Þ
nÀ1 b
{
nÀ1 jb nÀ1 ψ
n
ð Þ
nÀ1
E
¼ ψ
n
ð Þ
nÀ1 b
{
nÀ1 b nÀ1 ψ
n
ð Þ
nÀ1
D
E
¼ ψ
n
ð Þ
nÀ1 b n b
{
n þ ε
nÀ1
ð
Þ
À ε
nÀ2
ð
Þ
h
i
ψ
n
ð Þ
nÀ1
D
E
¼ ψ
n
ð Þ
nÀ1 b n b
{
n ψ
n
ð Þ
nÀ1
D
E
þ ε
nÀ1
ð
Þ
À ε
nÀ2
ð
Þ
h
i
ψ
n
ð Þ
nÀ1
D
ψ
n
ð Þ
nÀ1
E
¼ ε
nÀ1
ð
Þ
À ε
nÀ2
ð
Þ
h
i
ψ
n
ð Þ
nÀ1
D
ψ
n
ð Þ
nÀ1
E
:
ð3:269Þ
With the third equality, we used (3.267) with l ¼ n À 1; with the last equality we
used (3.251). Therefore, (3.268) holds with l ¼ n À 2. Then, it suffices to show that
assuming that (3.268) holds with
D
ψ
n
ð Þ
lþ1 ψ
n
ð Þ
lþ1
E
, it holds with
D
ψ
n
ð Þ
l ψ
n
ð Þ
l
E
as well.
Let us calculate
D
ψ
n
ð Þ
l ψ
n
ð Þ
l
E
, starting with (3.266) as below:
3.7 Radial Wave Functions of Hydrogen-Like Atoms
113