e b l Φ
n
ð Þ
l
ρ
ð Þ
¼
2
n
lþ
3
2 n þ l
2l
nl n À l
ð
Þ!
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
n þ l
ð
Þ n À l
ð
Þ
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1
2n n þ l
ð
Þ! n À l À 1
ð
Þ !
r
e
Àρ=n
ρ
l L
2lÀ1
nÀl
2ρ
n
¼
2
n
lþ
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n À l
ð
Þ!
2n n þ l À 1
ð
Þ !
s
e
Àρ=n
ρ
l L
2lÀ1
nÀl
2ρ
n
¼
2
n
lÀ1
ð
Þþ
3
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n À l À 1
ð
ÞÀ1
½
!
2n n þ l À 1
ð
Þ
½
!
s
e
À
ρ
n ρ
lÀ1
ð
Þþ1 L
2 lÀ1
ð
Þþ1
nÀ lÀ1
ð
ÞÀ1
2ρ
n
Φ
n
ð Þ
lÀ1 ρ
ð Þ:
ð3:291Þ
Thus, we find out that Φ
n
ð Þ
l
ρ
ð Þ behaves exactly like e
ψ
n
ð Þ
l . Moreover, if we replace
l in (3.279) with n – 1, we find
Φ
n
ð Þ
nÀ1 ρ
ð Þ ¼ e
ψ
n
ð Þ
nÀ1 :
ð3:292Þ
Operating e b nÀ1 on both sides of (3.292),
Φ
n
ð Þ
nÀ2 ρ
ð Þ ¼ e
ψ
n
ð Þ
nÀ2 :
ð3:293Þ
Likewise successively operating e b l (1 l n À 1),
Φ
n
ð Þ
l
ρ
ð Þ ¼ e
ψ
n
ð Þ
l
ρ
ð Þ,
ð3:294Þ
with all allowed numbers of l (i.e., 0 l n À 1). This permits us to identify
Φ
n
ð Þ
l
ρ
ð Þ e
ψ
n
ð Þ
l
ρ
ð Þ:
ð3:295Þ
Consequently, it is clear that the parameter n introduced in (3.249) is identical to a
principal quantum number and that the parameter l (0 l n À 1) is an orbital
angular momentum quantum number (or azimuthal quantum number). The functions
Φ
n
ð Þ
l
ρ
ð Þ and e
ψ
n
ð Þ
l
ρ
ð Þ are identical up to the constant c n expressed in (3.265). Note,
however, that a complex constant with an absolute number of 1 (phase factor)
remains undetermined, as is always the case with the eigenvalue problem.
The radial wave functions are derived from the following relationship as
described earlier:
R
n
ð Þ
l
r
ð Þ ¼ e
ψ
n
ð Þ
l =ρ:
ð3:296Þ
To normalize R
n
ð Þ
l
r
ð Þ, we have to calculate the following integral:
120
3 Hydrogen-Like Atoms
n
ð Þ
l
ρ
ð Þ
¼
2
n
lþ
3
2 n þ l
2l
nl n À l
ð
Þ!
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
n þ l
ð
Þ n À l
ð
Þ
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1
2n n þ l
ð
Þ! n À l À 1
ð
Þ !
r
e
Àρ=n
ρ
l L
2lÀ1
nÀl
2ρ
n
¼
2
n
lþ
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n À l
ð
Þ!
2n n þ l À 1
ð
Þ !
s
e
Àρ=n
ρ
l L
2lÀ1
nÀl
2ρ
n
¼
2
n
lÀ1
ð
Þþ
3
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n À l À 1
ð
ÞÀ1
½
!
2n n þ l À 1
ð
Þ
½
!
s
e
À
ρ
n ρ
lÀ1
ð
Þþ1 L
2 lÀ1
ð
Þþ1
nÀ lÀ1
ð
ÞÀ1
2ρ
n
Φ
n
ð Þ
lÀ1 ρ
ð Þ:
ð3:291Þ
Thus, we find out that Φ
n
ð Þ
l
ρ
ð Þ behaves exactly like e
ψ
n
ð Þ
l . Moreover, if we replace
l in (3.279) with n – 1, we find
Φ
n
ð Þ
nÀ1 ρ
ð Þ ¼ e
ψ
n
ð Þ
nÀ1 :
ð3:292Þ
Operating e b nÀ1 on both sides of (3.292),
Φ
n
ð Þ
nÀ2 ρ
ð Þ ¼ e
ψ
n
ð Þ
nÀ2 :
ð3:293Þ
Likewise successively operating e b l (1 l n À 1),
Φ
n
ð Þ
l
ρ
ð Þ ¼ e
ψ
n
ð Þ
l
ρ
ð Þ,
ð3:294Þ
with all allowed numbers of l (i.e., 0 l n À 1). This permits us to identify
Φ
n
ð Þ
l
ρ
ð Þ e
ψ
n
ð Þ
l
ρ
ð Þ:
ð3:295Þ
Consequently, it is clear that the parameter n introduced in (3.249) is identical to a
principal quantum number and that the parameter l (0 l n À 1) is an orbital
angular momentum quantum number (or azimuthal quantum number). The functions
Φ
n
ð Þ
l
ρ
ð Þ and e
ψ
n
ð Þ
l
ρ
ð Þ are identical up to the constant c n expressed in (3.265). Note,
however, that a complex constant with an absolute number of 1 (phase factor)
remains undetermined, as is always the case with the eigenvalue problem.
The radial wave functions are derived from the following relationship as
described earlier:
R
n
ð Þ
l
r
ð Þ ¼ e
ψ
n
ð Þ
l =ρ:
ð3:296Þ
To normalize R
n
ð Þ
l
r
ð Þ, we have to calculate the following integral:
120
3 Hydrogen-Like Atoms