Biomedical Engineering Reference
In-Depth Information
14.4 Perturbation Theory
There are very few physically interesting problems that we can solve exactly by the standard
methods of quantum theory. The great majority of problems, including those of atomic and
molecular structure, must therefore be tackled by approximate methods.
Suppose that our problem is to solve, for example, the heliumatomelectronic Schrödinger
equation
H
i
Ψ
i
(
r
1
,
r
2
)
=
ε
i
Ψ
i
(
r
1
,
r
2
)
(14.7)
We might suspect that this problem is similar to that of two superimposed hydrogen atoms,
for which we can find exact solutions to the
zero-order
Schrödinger equation
H
(0)
Ψ
(0)
i
ε
(0
i
Ψ
(0)
=
(
r
1
,
r
2
)
(
r
1
,
r
2
)
(14.8)
i
The aim of perturbation theory is to relate the solutions of problem (14.7) to the exact
zero-order solutions of (14.8). To simplify the notation, I will drop all references to two
electrons; perturbation theory is a general technique, not one that is specific to helium.
It is also general in that it can be applied to every solution not just the lowest energy
one. There are two technical points. First, I am going to assume that the state of interest
is not degenerate. There is a special version of perturbation theory that is applicable to
degenerate states, and I refer the interested reader to the classic texts such as Eyring
et al.
(1944). Second, I am going to assume that the wavefunctions are real rather than complex.
It makes the equations look a bit easier on the eye.
We proceed as follows. First we write the Hamiltonian as
H
=
H
(0)
λ
H
(1)
+
(14.9)
where λ is called the
perturbation parameter
. The second term is called the
perturbation
.
We assume that the energies and wavefunctions for our problem can be expanded in terms
of the zero-order problem as
Ψ
(0)
i
λΨ
(1)
i
λ
2
Ψ
(2)
i
Ψ
i
=
+
+
+···
(14.10)
ε
(0)
i
λε
(1)
i
λ
2
ε
(2)
i
ε
i
=
+
+
+···
The superscript '(
k
)' refers to the order of perturbation theory, and the equation should
demonstrate why the perturbation parameter λ is added; it is a formal device used to keep
track of the 'orders'of the perturbation. It might physically correspond to an applied electric
field (as in the Stark effect) or an applied magnetic induction (as in the Zeeman effect, in
which case we need to use complex wavefunctions). If we substitute the expansions into
our problem we find
λ
λ
2
H
(0)
Ψ
(0)
i
H
(1)
Ψ
(0)
i
+
H
(0)
Ψ
(1)
i
H
(1)
Ψ
(1)
i
+
H
(0)
Ψ
(2)
i
+
+
+···
(14.11)
λ
ε
(1
i
Ψ
(0)
+
λ
2
ε
(2
i
Ψ
(0)
+···
=
ε
(0
i
Ψ
(0)
+
+
ε
(0
i
Ψ
(1)
+
ε
(1
i
Ψ
(1)
+
ε
(0
i
Ψ
(2)
i
i
i
i
i
i
