l by one from l to l – 1, when it operates on ψ
n
ð Þ
l . The operator b 0 cannot be defined as
indicated in (3.243), and so the lowest number of l should be zero. Operators such as
b l are known as a ladder operator (lowering operator or annihilation operator in the
present case). The implication is that the successive operations of b l on ψ
n
ð Þ
nÀ1 produce
various parameters l as a subscript down to zero, while retaining the same integer
parameter n as a superscript.
3.7.2 Normalization of Radial Wave Functions
Next we seek normalized eigenfunctions. Coordinate representation of (3.251) takes
À
dψ
n
ð Þ
nÀ1
dρ
þ
n
ρ
À
1
n
ψ
n
ð Þ
nÀ1 ¼ 0:
ð3:259Þ
The solution can be obtained as
ψ
n
ð Þ
nÀ1 ¼ c n ρ
n e
Àρ=n ,
ð3:260Þ
where c n is a normalization constant. This can be determined as follows:
Z 1
0
ψ
n
ð Þ
nÀ1
2
dρ ¼ 1:
ð3:261Þ
Namely,
c n
j j
2
Z 1
0
ρ
2n e
À2ρ=n dρ ¼ 1:
ð3:262Þ
Consider the following definite integral:
Z 1
0
e
À2ρξ dρ ¼
1
2ξ
:
Differentiating the above integral 2n times with respect to ξ gives
Z 1
0
ρ
2n e
À2ρξ dρ ¼
1
2
2nþ1
2n
ð Þ!ξ
À 2nþ1
ð
Þ
:
ð3:263Þ
Substituting 1/n into ξ, we obtain
112
3 Hydrogen-Like Atoms