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The basic idea of perturbation theory is very simple: we split the
Hamiltonian into a piece we know how to solve (the ``reference'' or
``unperturbed'' Hamiltonian) and a piece we don't know
how to solve (the ``perturbation''). As long as the perburbation is
small compared to the unperturbed Hamiltonian, perturbation theory
tells us how to correct the solutions to the unperturbed problem to
approximately account for the influence of the perturbation. For
example, perturbation theory can be used to approximately solve an
anharmonic oscillator problem with the Hamiltonian

(132) 
Here, since we know how to solve the harmonic oscillator problem (see
5.2), we make that part the unperturbed Hamiltonian
(denoted
), and
the new, anharmonic term is the perturbation (denoted
):
Perturbation theory solves such a problem in two steps. First, obtain
the eigenfunctions and eigenvalues of the unperturbed Hamiltonian,
:

(135) 
Second, correct these eigenvalues and/or eigenfunctions to account for
the perturbation's influence. Perturbation theory gives these
corrections as an infinite series of terms, which become smaller and
smaller for wellbehaved systems:
Quite frequently, the corrections are only taken through first or
second order (i.e., superscripts (1) or (2)). According to
perturbation theory, the firstorder correction to the energy is

(138) 
and the secondorder correction is

(139) 
One can see that the firstorder correction to the wavefunction,
, seems to be needed to compute the secondorder
energy correction. However, it turns out that the correction
can be written in terms of the zerothorder
wavefunction as

(140) 
Substituting this in the expression for , we obtain

(141) 
Going back to the anharmonic oscillator example, the ground state
wavefunction for the unperturbed problem is just (from section
5.2)
The firstorder correction to the ground state energy would be

(145) 
It turns out in this case that , since the integrand is
odd. Does this mean that the anharmonic energy levels are the same as
for the harmonic oscillator? No, because there are higherorder
corrections such as which are not necessarily zero.
Next: The Variational Method
Up: Approximate Methods
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David Sherrill
20060815