H
l
ð Þ
À
d
2
dρ 2 þ
l l þ 1
ð
Þ
ρ 2 À
2
ρ
!
:
Then, from (3.243) and (3.244) as well as (3.245) and (3.246) we have
H
l
ð Þ
¼ b lþ1 b
{
lþ1 þ ε
l
ð Þ l ! 0
ð
Þ,
ð3:247Þ
where ε
l
ð Þ
À
1
lþ1
ð
Þ
2 . Alternatively,
H
l
ð Þ
¼ b
{
l b l þ ε
lÀ1
ð
Þ l ! 1
ð
Þ:
ð3:248Þ
If we put l ¼ n À 1 in (3.247) with n being a fixed given integer larger than l, we
obtain
H
nÀ1
ð
Þ
¼ b n b
{
n þ ε
nÀ1
ð
Þ
:
ð3:249Þ
We evaluate the following inner product of both sides of (3.249):
χ H
nÀ1
ð
Þ
χ
D
E
¼ χ
h jb n b
{
n χ
j i þ ε
nÀ1
ð
Þ
h χ χ
j i
¼ b
{
n χ
b
{
n χi þ ε
nÀ1
ð
Þ
h χ χ
j i
¼ b
{
n χ
j i
2 þ ε
nÀ1
ð
Þ
h χ χ
j i
! ε
nÀ1
ð
Þ
:
ð3:250Þ
Here we assume that χ is normalized. On the basis of the variational principle [9],
the above expected value must take a minimum ε
(n À 1) so that χ can be an
eigenfunction. To satisfy this condition, we have
jb
{
n χi ¼ 0:
ð3:251Þ
In fact, if (3.251) holds, from (3.249) we have
H
nÀ1
ð
Þ
χ ¼ ε
nÀ1
ð
Þ
χ:
ð3:252Þ
We define such a function as below
ψ
n
ð Þ
nÀ1 χ:
ð3:253Þ
From (3.247) and (3.248), we have the following relationship:
110
3 Hydrogen-Like Atoms