The time-independent Schrödinger equation can be written as
(1.1)
H
^
ψ
(
r
)
=
E
n
ψ
(
r
)
where the Hamiltonian operator for hydrogen-like atoms, in three different systems of units, is given in Table 1.1.
Table 1.1. The Hamiltonian operator for hydrogen-like atoms expressed in SI units, Gaussian units, and atomic units.
|
H
^
=
-
ℏ
2
2
μ
∇
2
+
e
2
4
πε
0
Z
r
| International System (SI) units | [Atkins, Levine, Brown, Struve, Haken, Lowe, Bernath] | |
H
^
=
-
ℏ
2
2
μ
∇
2
+
Ze
2
r
| Gaussian cgs units | [Schiff, Sakurai85, Liboff, Baym, Gasiorowicz, Saxon, Lefebvre-Brion, Shankar] | |
H
^
=
-
1
2
∇
2
+
Z
r
| Atomic units (au) | [Szabo, Leach, Thijssen, Levine, Koch, Parr89] | |
In Table 1.1, Z is the atomic number,
μ
is the reduced mass of the electron-nucleus system, and
ε
0
=
8.854
×
10
-
12
C
2
N
-
1
m
-
2
is the permittivity of free space. As can be seen from the references in Table 1.1, International System (SI) units units are typically used in the chemistry and spectroscopy literature, Gaussian cgs units are often used by physicists, and atomic units (au) are preferred by computational scientists. Table 1.2 shows some physical constants in the three systems of units.
Table 1.2. Physical constants expressed in atomic units (au), SI units, and Gaussian units.
|
| Electron mass | me | me ≡ 1 a.u. | 9.1093897 × 10-31 kg | 9.1093897 × 10-28 g |
| Electron charge magnitude | e | e ≡ 1 a.u. | 1.60217733 × 10-19 C | 4.8032068 × 10-10 statC |
| Rydberg energy |
R
∞
=
m
e
e
4
2
(
4
πε
0
)
2
ℏ
2
=
e
2
2
(
4
πε
0
)
a
0
| R∞ = 0.5 hartrees | 2.17987 × 10-18 J | 2.17987 × 10-11 erg |
| Speed of light | c | c = α-1 = 137.04 a.u. | ≡ 2.99792458 × 108 m s-1 | 2.99792458 × 1010 cm s-1 |
| Planck constant reduced | ℏ | ℏ ≡ 1 a.u. | 1.05457266 × 10-34 J s | 1.05457266 × 10-27 erg s |
| Bohr radius |
a
0
=
4
πε
0
ℏ
2
m
e
e
2
| a0 ≡ 1 bohr | 0.529177249 × 10-10 m | 0.529177249 × 10-8 cm |
1.2. Hydrogen-like atoms.
It will prove useful to write the Hamiltonian operator in spherical coordinates, in which case the contribution from the Laplacian operator is given by
(1.5)
∇
2
=
1
r
∂
2
∂
r
2
r
+
1
r
2
sin
2
θ
∂
2
∂
ϕ
2
+
1
r
2
sinθ
(
∂
∂
θ
sinθ
∂
∂
θ
)
.
The Laplacian can be simplified further if we note that the orbital angular momentum operator,
L
^
=
r
×
p
,
in the position-space representation, is given by the following differential operators
(1.6)
L
^
x
=
-
iℏ
(
y
∂
∂
z
-
z
∂
∂
y
)
=
iℏ
(
sinφ
∂
∂
θ
+
cotθ
cosφ
∂
∂
φ
)
(1.7)
L
^
y
=
-
iℏ
(
z
∂
∂
x
-
x
∂
∂
z
)
=
iℏ
(
-
cosφ
∂
∂
θ
+
cotθ
sinφ
∂
∂
φ
)
(1.8)
L
^
z
=
-
iℏ
(
x
∂
∂
y
-
y
∂
∂
x
)
=
-
iℏ
∂
∂
φ
(1.9)
L
^
2
=
L
^
x
2
+
L
^
y
2
+
L
^
z
2
=
-
ℏ
2
[
1
sin
2
θ
∂
2
∂
φ
2
+
1
sinθ
(
∂
∂
θ
sinθ
∂
∂
θ
)
]
.
In Eqs. (1.6) - (1.8), the Cartesian components of the orbital angular momentum operators are expressed in both Cartesian coordinates and spherical coordinates; As a reminder the two coordinate systems are related by
(1.10)
x
=
r
cosφ
sinθ
y
=
r
sinφ
sinθ
z
=
r
cosθ
.
By substituting Eq. (1.9) into Eq. (1.5), one obtains the following expression for the Laplacian
(1.11)
∇
2
=
1
r
∂
2
∂
r
2
r
-
1
ℏ
2
L
^
2
r
2
and the Hamiltonian operator in SI units becomes
(1.12)
H
^
=
-
ℏ
2
2
μ
1
r
∂
2
∂
r
2
r
+
1
2
μ
L
^
2
r
2
+
e
2
4
πε
0
Z
r
.
Next, we will use the fact that the spherical harmonics,
Y
l
m
l
(
θ
,
φ
)
,
are the simultaneous eigenfunctions of
L
^
2
and
L
^
z
;
it is not difficult to show that
(1.13)
L
^
2
Y
l
m
l
(
θ
,
φ
)
=
ℏ
2
l
(
l
+
1
)
Y
l
m
l
(
θ
,
φ
)
and
(1.14)
L
^
z
Y
l
m
l
(
θ
,
φ
)
=
ℏ
m
l
Y
l
m
l
(
θ
,
φ
)
.
The form of Equations (1.12) and (1.13) tells us that the separation of variables method can be used to write the solution to the Schrödinger equation as
(1.15)
ψ
(
r
)
=
ψ
nlm
l
(
r
,
θ
,
ϕ
)
=
R
nl
(
r
)
Y
l
m
l
(
θ
,
ϕ
)
where n is the principle quantum number, and l and ml are the orbital angular momentum quantum numbers.
The Schrödinger equation becomes
(1.16)
[
-
ℏ
2
2
μ
1
r
∂
2
∂
r
2
r
+
1
2
μ
L
^
2
r
2
+
e
2
4
πε
0
Z
r
]
R
nl
(
r
)
Y
l
m
l
(
θ
,
ϕ
)
=
E
n
R
nl
(
r
)
Y
l
m
l
(
θ
,
ϕ
)
.
Then, by using the orbital angular momentum eigenvalue equation [Eq. (1.13)], one can obtain the following radial differential equation
(1.17)
[
-
ℏ
2
2
μ
1
r
∂
2
∂
r
2
r
+
ℏ
2
2
μ
l
(
l
+
1
)
r
2
+
e
2
4
πε
0
Z
r
]
R
nl
(
r
)
=
E
n
R
nl
(
r
)
.
The energy eigenvalues are
(1.18)
E
n
=
-
μ
(
Ze
2
)
2
2
(
4
πε
0
)
2
ℏ
2
n
2
=
-
Z
2
e
2
2
(
4
πε
0
)
n
2
a
with
(1.19)
ρ
=
(
8
μ
|
E
n
|
ℏ
2
)
r
=
2
Zr
na
(1.20)
a
=
4
πε
0
ℏ
2
μe
2
.
For an infinitely massive nucleus
(1.21)
lim
M
→
∞
μ
=
lim
M
→
∞
M
m
e
M
+
m
e
=
m
e
and Eq. (1.20) is equal to the Bohr radius
(1.22)
lim
M
→
∞
a
=
a
∞
=
a
0
=
4
πε
0
ℏ
2
m
e
e
2
.
In the case of the hydrogen atom itself
(1.23)
μ
=
M
m
e
M
+
m
e
=
(
1.6726231
×
10
-
27
kg
)
(
9.1093897
×
10
-
31
kg
)
(
1.6726231
×
10
-
27
kg
)
+
(
9.1093897
×
10
-
31
kg
)
=
0.9994557
m
e
and throughout the text we will often make the approximation
μ
≈
m
e
.
The solutions to the radial Schrödinger equation [Eq. (1.17)] are
(1.24)
R
nl
(
r
)
=
-
{
(
2
Z
na
)
3
(
n
-
l
-
1
)
!
2
n
[
(
n
+
l
)
!
]
3
}
1
2
ρ
l
e
-
ρ
/
2
L
n
+
l
2
l
+
1
(
ρ
)
where the associated Laguerre polynomials are
(1.25)
L
n
+
l
2
l
+
1
(
ρ
)
=
∑
k
=
0
n
-
l
-
1
(
-
1
)
k
+
2
l
+
1
[
(
n
+
l
)
!]
2
ρ
k
(
n
-
l
-
1
-
k
)
!
(
2
l
+
1
+
k
)
!
k
!
(Note that some authors use the notation
L
n
-
l
-
1
2
l
+
1
(
ρ
)
for the associated Laguerre polynomials [9].). Equations (1.24) and (1.25) were used to generate the first several radial wavefunctions shown in Table 1.3.
Table 1.3. The first several radial wavefunctions.
| n | l | Spectroscopic notation | Radial wavefunction |
|---|
| 1 | 0 | 1s |
R
10
(
r
)
=
(
Z
a
)
3
2
2
e
-
ρ
/
2
| |
| 2 | 0 | 2s |
R
20
(
r
)
=
(
Z
2
a
)
3
2
(
2
-
ρ
)
e
-
ρ
/
2
R
21
(
r
)
=
(
Z
2
a
)
3
2
ρ
3
e
-
ρ
/
2
| |
| 2 | 1 | 2p |
| 3 | 0 | 3s |
R
30
(
r
)
=
(
Z
3
a
)
3
2
(
1
3
)
(
6
-
6
ρ
+
ρ
2
)
e
-
ρ
/
2
R
31
(
r
)
=
(
Z
3
a
)
3
2
(
1
3
2
)
(
4
-
ρ
)
ρe
-
ρ
/
2
R
32
(
r
)
=
(
Z
3
a
)
3
2
(
1
3
10
)
ρ
2
e
-
ρ
/
2
| |
| 3 | 1 | 3p |
| 3 | 2 | 3d |
| 4 | 0 | 4s |
R
40
(
r
)
=
(
Z
4
a
)
3
2
(
1
12
)
(
24
-
36
ρ
+
12
ρ
2
-
ρ
3
)
e
-
ρ
/
2
R
41
(
r
)
=
(
Z
4
a
)
3
2
(
1
4
15
)
(
20
-
10
ρ
+
ρ
2
)
ρ
e
-
ρ
/
2
R
42
(
r
)
=
(
Z
4
a
)
3
2
(
1
12
5
)
(
6
-
ρ
)
ρ
2
e
-
ρ
/
2
R
43
(
r
)
=
(
Z
4
a
)
3
2
(
1
12
35
)
ρ
3
e
-
ρ
/
2
| |
| 4 | 1 | 4p |
| 4 | 2 | 4d |
| 4 | 3 | 4f |
To completely specify the spatial part of the hydrogen-like wavefunction, the radial wavefunction needs to be combined with a spherical harmonic with the same orbital angular momentum quantum number, l. The spherical harmonics can be written as
(1.30)
Y
l
m
(
θ
,
ϕ
)
=
2
l
+
1
4
π
(
l
-
m
)
!
(
l
+
m
)
!
P
l
m
(
cos
θ
)
e
i
m
ϕ
for l ≥ 0
where
P
l
m
(
cosθ
)
are the associated Legendre functions. The first several spherical harmonics are shown in Table 1.4.
Table 1.4. The first several spherical harmonics.
| l | ml |
Y
l
m
(
θ
,
ϕ
)
|
|---|
| 0 | 0 | (1.31)
Y
0
0
(
θ
,
ϕ
)
=
1
2
π
|
| 1 | 0 |
(1.32)
Y
1
0
(
θ
,
ϕ
)
=
1
2
3
π
cosθ
Y
1
±
1
(
θ
,
ϕ
)
=
∓
1
2
e
±
i
ϕ
3
2
π
sinθ
|
| 1 | ± 1 |
| 2 | 0 |
Y
2
0
(
θ
,
ϕ
)
=
1
4
5
π
(
-
1
+
3
cos
2
θ
)
(1.33)
Y
2
±
1
(
θ
,
ϕ
)
=
∓
1
2
e
±
i
ϕ
15
2
π
cosθ
sinθ
Y
2
±
2
(
θ
,
ϕ
)
=
±
1
4
e
±
2
i
ϕ
15
2
π
sin
2
θ
|
| 2 | ± 1 |
| 2 | ± 2 |
| 3 | 0 |
Y
3
0
(
θ
,
ϕ
)
=
1
4
7
π
(
-
3
cosθ
+
5
cos
3
θ
)
(1.34)
Y
3
±
1
(
θ
,
ϕ
)
=
∓
1
8
e
±
i
ϕ
21
π
(
-
1
+
5
cos
2
θ
)
sinθ
Y
3
±
2
(
θ
,
ϕ
)
=
∓
1
4
e
±
2
i
ϕ
105
2
π
cosθ
sin
2
θ
Y
3
±
3
(
θ
,
ϕ
)
=
∓
1
8
e
±
3
i
ϕ
35
π
sin
3
θ
|
| 3 | ± 1 |
| 3 | ± 2 |
| 3 | ± 3 |
| 4 | 0 |
Y
4
0
(
θ
,
ϕ
)
=
3
16
π
(
3
-
30
cos
2
θ
+
35
cos
4
θ
)
Y
4
±
1
(
θ
,
ϕ
)
=
∓
3
8
e
±
i
ϕ
5
π
cosθ
(
-
3
+
7
cos
2
θ
)
sinθ
(1.35)
Y
4
±
2
(
θ
,
ϕ
)
=
±
3
8
e
±
2
i
ϕ
5
2
π
(
-
1
+
7
cos
2
θ
)
sin
2
θ
Y
4
±
3
(
θ
,
ϕ
)
=
∓
3
8
e
±
3
i
ϕ
35
π
cosθ
sin
3
θ
Y
4
±
4
(
θ
,
ϕ
)
=
±
3
16
e
±
4
i
ϕ
35
2
π
sin
4
θ
|
| 4 | ± 1 |
| 4 | ± 2 |
| 4 | ± 3 |
| 4 | ± 4 |
In Table 1.4 we used the fact that
Y
l
-
m
=
(
-
1
)
m
(
Y
l
m
)
*
.
The radial wavefunctions from Table 1.3 can be combined with the spherical harmonics in Table 1.4 to form the hydrogen-like wavefunctions, as shown in Table 1.5.
Table 1.5. The first several hydrogen-like wavefunctions.
| n | l | ml | Spectroscopic notation | Wavefunction |
|---|
| 1 | 0 | 0 | 1s |
ψ
(
r
)
=
ψ
100
(
r
,
θ
,
ϕ
)
=
(
Z
a
)
3
2
2
e
-
ρ
/
2
Y
0
0
(
θ
,
ϕ
)
|
ρ
=
2
Zr
a
| |
| 2 | 0 | 0 | 2s |
ψ
(
r
)
=
ψ
200
(
r
,
θ
,
ϕ
)
=
(
Z
2
a
)
3
2
(
2
-
ρ
)
e
-
ρ
/
2
Y
0
0
(
θ
,
ϕ
)
|
ρ
=
Zr
a
| |
| 2 | 1 | 1 | 2p |
ψ
(
r
)
=
{
ψ
111
(
r
,
θ
,
ϕ
)
ψ
110
(
r
,
θ
,
ϕ
)
ψ
11
-
1
(
r
,
θ
,
ϕ
)
=
(
Z
2
a
)
3
2
(
2
-
ρ
)
e
-
ρ
/
2
{
Y
1
1
(
θ
,
ϕ
)
Y
1
0
(
θ
,
ϕ
)
Y
1
-
1
(
θ
,
ϕ
)
|
| 2 | 1 | 0 |
| 2 | 1 | -1 |
| 3 | 0 | 0 | 3s |
ψ
(
r
)
=
ψ
200
(
r
,
θ
,
ϕ
)
=
(
Z
2
a
)
3
2
(
2
-
ρ
)
e
-
ρ
/
2
Y
0
0
(
θ
,
ϕ
)
|
ρ
=
2
Zr
3
a
| |
| 3 | 1 | 1 | 3p |
ψ
(
r
)
=
{
ψ
311
(
r
,
θ
,
ϕ
)
ψ
310
(
r
,
θ
,
ϕ
)
ψ
31
-
1
(
r
,
θ
,
ϕ
)
=
(
Z
3
a
)
3
2
(
1
3
2
)
(
4
-
ρ
)
ρ
e
-
ρ
/
2
{
Y
1
1
(
θ
,
ϕ
)
Y
1
0
(
θ
,
ϕ
)
Y
1
-
1
(
θ
,
ϕ
)
|
| 3 | 1 | 0 |
| 3 | 1 | -1 |
| 3 | 2 | 2 | 3d |
ψ
(
r
)
=
{
ψ
322
(
r
,
θ
,
ϕ
)
ψ
321
(
r
,
θ
,
ϕ
)
ψ
320
(
r
,
θ
,
ϕ
)
ψ
32
-
1
(
r
,
θ
,
ϕ
)
ψ
32
-
2
(
r
,
θ
,
ϕ
)
=
(
Z
3
a
)
3
2
(
1
3
10
)
ρ
2
e
-
ρ
/
2
{
Y
2
2
(
θ
,
ϕ
)
Y
2
1
(
θ
,
ϕ
)
Y
2
0
(
θ
,
ϕ
)
Y
2
-
1
(
θ
,
ϕ
)
Y
2
-
2
(
θ
,
ϕ
)
|
| 3 | 2 | 1 |
| 3 | 2 | 0 |
| 3 | 2 | -1 |
| 3 | 2 | -2 |
The energy eigenvalues, En, depend on the value of principle
quantum number, n, but not on the orbital angular momentum quantum numbers, l and ml. The range of possible values of n, l, and ml are
Using the sum rule
(1.40)
∑
i
=
0
k
i
=
k
(
k
+
1
)
2
it is not difficult to see that the degeneracy of the energy levels, En, is
(1.41)
∑
l
=
0
n
-
1
(
2
l
+
1
)
=
n
2
.
The degeneracy is clearly shown in Table 1.5, where there is the one ground state wavefunction for n = 1, the four wavefunctions for n = 2, and the nine for n = 3. When the spin degrees of freedom are included, the degeneracy is 2n2.
![[Note]](images/note.png)