H
l
ð Þ b lþ1 ¼ b lþ1 H
lþ1
ð
Þ l ! 0
ð
Þ:
ð3:254Þ
Meanwhile we define the functions as shown below
ψ
n
ð Þ
nÀs b nÀsþ1 b nÀsþ2 ∙ ∙ ∙ ∙ b nÀ1 ψ
n
ð Þ
nÀ1 2 s n
ð
Þ :
ð3:255Þ
With these functions (s – 1) operators have been operated on ψ
n
ð Þ
nÀ1 . Note that if
s took 1, no operation of b l would take place. Thus, we find that b l functions upon the
l-state to produce the (l À 1)-state. That is, b l acts as an annihilation operator. For the
sake of convenience we express
H
n,s
ð Þ
H
nÀs
ð
Þ
:
ð3:256Þ
Using this notation and (3.254), we have
H
n,s
ð Þ
ψ
n
ð Þ
nÀs ¼ H
n,s
ð Þ b nÀsþ1 b nÀsþ2 Á Á Áb nÀ1 ψ
n
ð Þ
nÀ1
¼ b nÀsþ1 H
n,sÀ1
ð
Þ b nÀsþ2 Á Á Áb nÀ1 ψ
n
ð Þ
nÀ1
¼ b nÀsþ1 b nÀsþ2 H
n,sÀ2
ð
Þ
Á Á Áb nÀ1 ψ
n
ð Þ
nÀ1
Á Á ÁÁ Á Á
¼ b nÀsþ1 b nÀsþ2 Á Á ÁH
n,2
ð Þ b nÀ1 ψ
n
ð Þ
nÀ1
¼ b nÀsþ1 b nÀsþ2 Á Á Áb nÀ1 H
n,1
ð Þ
ψ
n
ð Þ
nÀ1
¼ b nÀsþ1 b nÀsþ2 Á Á Áb nÀ1 ε
nÀ1
ð
Þ
ψ
n
ð Þ
nÀ1
¼ ε
nÀ1
ð
Þ b nÀsþ1 b nÀsþ2 Á Á Áb nÀ1 ψ
n
ð Þ
nÀ1
¼ ε
nÀ1
ð
Þ
ψ
n
ð Þ
nÀs :
ð3:257Þ
Thus, total n functions ψ
n
ð Þ
nÀs 1 s n
ð
Þbelong to the same eigenvalue ε
(n À 1) .
Notice that the eigenenergy E n corresponding to ε
(n À 1) is given by
E n ¼ À
ħ
2
2μ
Z
a
2 1
n 2 :
ð3:258Þ
If we define l n À s and take account of (3.252), total n functions ψ
n
ð Þ
l (l ¼ 0, 1, 2,
∙ ∙ ∙ ∙, n À 1) belong to the same eigenvalue ε
(n À 1)
.
The quantum state ψ
n
ð Þ
l is associated with the operators H
(l ) . Thus, the solution of
(3.242) has been given by functions ψ
n
ð Þ
l parametrized with n and l on condition that
(3.251) holds. As explicitly indicated in (3.255) and (3.257), b l lowers the parameter
3.7 Radial Wave Functions of Hydrogen-Like Atoms
111