Z 1
0
R
n
ð Þ
l
r
ð Þ
2
r
2 dr ¼
Z 1
0
1
ρ 2 e
ψ
n
ð Þ
l
2 a
Z
ρ
2 a
Z
dρ ¼
a
Z
3 Z 1
0
e
ψ
n
ð Þ
l
2
dρ
¼
a
Z
3 :
ð3:297Þ
Accordingly, we choose the following functions e
R
n
ð Þ
l
r
ð Þ for the normalized radial
wave functions:
e
R
n
ð Þ
l
r
ð Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
Z=a
ð
Þ
3
q
R
n
ð Þ
l
r
ð Þ:
ð3:298Þ
Substituting (3.296) into (3.298) and taking account of (3.279) and (3.280), we
obtain
e
R
n
ð Þ
l
r
ð Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2Z
an
3
Á
n À l À 1
ð
Þ !
2n n þ l
ð
Þ!
s
2Zr
an
l
exp À
Zr
an
h
i
L
2lþ1
nÀlÀ1
2Zr
an
: ð3:299Þ
Equation (3.298) is exactly the same as the normalized radial wave functions that
can be obtained as the solution of (3.241) through the power series expansion. All
these functions belong to the same eigenenergy E n ¼ À
ħ
2
2μ
Z
a
À Á 2 1
n 2 ; see (3.258).
Returning back to the quantum states sharing the same eigenenergy, we have such
n states that belong to the principal quantum number n with varying azimuthal
quantum number l (0
l
n À 1). Meanwhile, from (3.158), (3.159), and
(3.171), 2l + 1 quantum states possess an eigenvalue of l(l + 1) with the quantity
M
2 . As already mentioned in Sects. 3.5 and 3.6, the integer m takes the 2l + 1
different values of
m ¼ l, l À 1, l À 2, Á Á Á1, 0, À 1, Á Á Á À l þ 1, À l:
The quantum states of a hydrogen-like atoms are characterized by a set of integers
(m, l, n). On the basis of the above discussion, the number of those sharing the same
energy (i.e., the principal quantum number n) is
X nÀ1
l¼0
2l þ 1
ð
Þ¼n
2
:
Regarding this situation, we say that the quantum states belonging to the principal
quantum number n, or the energy eigenvalue E n ¼ À
ħ
2
2μ
Z
a
À Á 2 1
n 2 , are degenerate n
2 -
fold. Note that we ignored the freedom of spin state.
In summary of this section, we have developed the operator formalism in dealing
with radial wave functions of hydrogen-like atoms and seen how the operator
formalism features the radial wave functions. The essential point rests upon that
3.7 Radial Wave Functions of Hydrogen-Like Atoms
121
0
R
n
ð Þ
l
r
ð Þ
2
r
2 dr ¼
Z 1
0
1
ρ 2 e
ψ
n
ð Þ
l
2 a
Z
ρ
2 a
Z
dρ ¼
a
Z
3 Z 1
0
e
ψ
n
ð Þ
l
2
dρ
¼
a
Z
3 :
ð3:297Þ
Accordingly, we choose the following functions e
R
n
ð Þ
l
r
ð Þ for the normalized radial
wave functions:
e
R
n
ð Þ
l
r
ð Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
Z=a
ð
Þ
3
q
R
n
ð Þ
l
r
ð Þ:
ð3:298Þ
Substituting (3.296) into (3.298) and taking account of (3.279) and (3.280), we
obtain
e
R
n
ð Þ
l
r
ð Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2Z
an
3
Á
n À l À 1
ð
Þ !
2n n þ l
ð
Þ!
s
2Zr
an
l
exp À
Zr
an
h
i
L
2lþ1
nÀlÀ1
2Zr
an
: ð3:299Þ
Equation (3.298) is exactly the same as the normalized radial wave functions that
can be obtained as the solution of (3.241) through the power series expansion. All
these functions belong to the same eigenenergy E n ¼ À
ħ
2
2μ
Z
a
À Á 2 1
n 2 ; see (3.258).
Returning back to the quantum states sharing the same eigenenergy, we have such
n states that belong to the principal quantum number n with varying azimuthal
quantum number l (0
l
n À 1). Meanwhile, from (3.158), (3.159), and
(3.171), 2l + 1 quantum states possess an eigenvalue of l(l + 1) with the quantity
M
2 . As already mentioned in Sects. 3.5 and 3.6, the integer m takes the 2l + 1
different values of
m ¼ l, l À 1, l À 2, Á Á Á1, 0, À 1, Á Á Á À l þ 1, À l:
The quantum states of a hydrogen-like atoms are characterized by a set of integers
(m, l, n). On the basis of the above discussion, the number of those sharing the same
energy (i.e., the principal quantum number n) is
X nÀ1
l¼0
2l þ 1
ð
Þ¼n
2
:
Regarding this situation, we say that the quantum states belonging to the principal
quantum number n, or the energy eigenvalue E n ¼ À
ħ
2
2μ
Z
a
À Á 2 1
n 2 , are degenerate n
2 -
fold. Note that we ignored the freedom of spin state.
In summary of this section, we have developed the operator formalism in dealing
with radial wave functions of hydrogen-like atoms and seen how the operator
formalism features the radial wave functions. The essential point rests upon that
3.7 Radial Wave Functions of Hydrogen-Like Atoms
121