狄拉克方程求解氢原子(含详细推导过程)

1.1 Stationary-state Dirac equation
Hˆ ψ =
cαˆ
·
pˆ
+
mc2βˆ
−
e21 r
ψ = Eψ
(1.1)
For matrices we add a hat, while for 3D Euclidean vectors we add an arrow. The e1 is the effective charge as is easier to use in SI units:
1.3 Total angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Spin-angular functions . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Contents
1 Assumptions
1
1.1 Stationary-state Dirac equation . . . . . . . . . . . . . . . . . . . . . 1
1.2 Pauli matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.5.1
First look at the commutation rules between αˆ, βˆ and other operators. We know that αˆ, βˆ emerge in Dirac equation, whose validity does not depend on space-time coordinates. Because of the homogeneity of space-time, although αˆ and βˆ are matrices, they do not depend on space-time coordinates. Hence,
1.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
(1.5)
{σˆi, σˆj} = 2δij
(1.6)
Here {· · · } is the anti-commutation operator and δ here is the Kronecker delta func-
tion.
[σˆi, σˆj] = 2i
ijkσˆk
(1.7)
k
Here [· · · ] is the commutation operator and here is the Levi-Civita symbol.
4 Solving the Dirac equation
21
4.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 The radial wave function . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3 Energy level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
We already know that, as far as Dirac hydrogen atom is concerned, Jˆ2 and Jˆz commute with the Hamiltonian. The total angular momentum operator is:
Jˆ = Lˆ + Sˆ = Lˆ + 2 Σˆ = Lˆ + 2
Sˆ2
=
3 4
2
(1.13)
All over this work we shall use the Dirac-Pauli representation and hence the explicit forms of αˆ and βˆ are:
αˆ =
0 σˆ σˆ 0
(1.14)
βˆ =
Iˆ 0 0 −Iˆ
e1
=
√e 4π
0
(1.2)
In addition, we denote m as the rest mass for an electron (∼ 9.11E − 31 kg), and it will never denote the relativistic mass in the context.
1.5 Commutation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Lemmas
12
2.1 Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Σˆ 2i = 1
{Σˆ i, Σˆ j} = 0 f or i = j
(1.9) (1.10)
[Σˆ i, Σˆ j] = 2i
ijkΣˆ k
k
(1.11)
From (1.8) and (1.9) we obviously get these properties:
Σˆ · Σˆ = 3
(1.12)
In the Dirac problem, we write the electron spin operator as Sˆ, different from sˆ.
And we introduce Σˆ , as below:
Sˆ = 2 Σˆ
(1.8)
There are several properties of Σˆ:
1.6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
mj
=
m
Fra Baidu bibliotek
+
1 2
(1.22)
Finally, we know that φA and φB are opposite in parity.
1.5 Commutation rules
We shall do lots of commutation calculations in this work. Let us obtain some results for future use.
3 Complete set of commuting observables
16
3.1 The set of commuting observables . . . . . . . . . . . . . . . . . . . . 16
3.2 The mutual eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Jˆ2
=
Lˆ2
+
3 4
2+
Σˆ
·
Lˆ
=
Lˆ2
+
3 4
2+
σˆ · Lˆ 0 0 σˆ · Lˆ
(1.18)
1.4 Spin-angular functions
For spin 1/2 particles, we already know that the spin-angular functions can be written as (Sakurai 3.7.64):
1.5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Pauli matrices
σˆx =
01 10
;
σˆy =
0 −i i0
;
σˆz =
10 0 −1
sˆ = 2 σˆ
(1.3) (1.4)
1
The properties of Pauli matrices are demonstrated below:
σˆx2 = σˆy2 = σˆz2 = 1
Jˆ2 = j(j + 1) 2
j
=
l
+
1 2
(1.21)
3
Note
that
we
cannot
have
j
=
l
−
1 2
because
we
used
l+1
for
φB ,
rather
than
l − 1.
If the “l” for φB is denoted as l , then l = l + 1.
Jˆz = mj
1.5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Other formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
(1.15)
where Iˆ is the 2 × 2 unit matrix:
Iˆ =
10 01
2
In Dirac-Pauli representation, Σˆ is written explicitly as:
Σˆ =
σˆ 0 0 σˆ
(1.16)
1.3 Total angular momentum
1.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
σˆ 0 0 σˆ
(1.17)
Jˆ2 = Lˆ + Σˆ 2 = Lˆ2 + 2 Σˆ 2 + (Lˆ · Σˆ + Σˆ · Lˆ)
2
4
2
Since the orbital angular momentum and the spin are operating on totally different spaces, they must commute. Also, applying (1.12) we get:
5 Non-relativistic limit
31
6 The wave function
33
Chapter 1 Assumptions
In doing the Dirac hydrogen atom problem, we must be aware of which equations or results we can in fact use, while many of the results we cannot. Let us enumerate all what we have already known on this problem, as the assumptions for this problem.
√

φAjmj
=
√1 2l +
1

l + m + 1 Yl,m
√

l − m Yl,m+1
(1.19)
√
φBjmj
=
√1
2l + 3
− l − m + 1 Yl+1,m √
l + m + 2 Yl+1,m+1

(1.20)
They are both simultaneous eigenstates of (Jˆ2, Jˆz), with: