Time-Independent Perturbation Theory
Michael Fowler
2/16/06
Introduction
If an atom (not necessarily in its ground state) is placed in an external electric field, the energy
levels shift, and the wave functions are distorted.
This is called the
Stark effect
.
The new energy
levels and wave functions could in principle be found by writing down a complete Hamiltonian,
including the external field, and finding the eigenkets.
This actually can be done in one case: the
hydrogen atom, but even there, if the external field is small compared with the electric field
inside the atom (which is billions of volts per meter) it is easier to compute the changes in the
energy levels and wave functions with a scheme of successive corrections to the zero-field
values.
This method, termed
perturbation theory
, is the single most important method of solving
problems in quantum mechanics, and is widely used in atomic physics, condensed matter and
particle physics.
It should be noted that there
are
problems which cannot be solved using perturbation theory,
even when the perturbation is very weak, although such problems are the exception rather than
the rule.
One such case is the one-dimensional problem of free particles perturbed by a localized
potential of strength
λ
.
As we found earlier in the course, switching on an arbitrarily weak
attractive potential causes the
free particle wave function to drop below the continuum of
plane wave energies and become a localized bound state with binding energy of order
0
k
=
2
λ
.
However, changing the sign of
λ
to give a repulsive potential there is no bound state, the lowest
energy plane wave state stays at energy zero.
Therefore the energy shift on switching on the
perturbation cannot be represented as a power series in
λ
, the strength of the perturbation.
This
particular difficulty does not in general occur in three dimensions, where arbitrarily weak
potentials do not give bound states—except for certain many-body problems (like the Cooper
pair problem) where the exclusion principle reduces the effective dimensionality of the available
states.
The Perturbation Series
We begin with a Hamiltonian
having known eigenkets and eigenenergies:
0
H
0
0
0
0
.
n
H
n
E
n
=
The task is to find how these eigenkets and eigenenergies change if a small term
(an external
field, for example) is added to the Hamiltonian, so:
1
H
(
)
0
1
.
n
H
H
n
E
n
+
=
That is to say, on switching on
,
1
H
0
0
,
.
n
n
n
n
E
E
→
→

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