130_notes.dvi

22 Time Independent Perturbation Theory

Perturbation Theory is developed to deal withsmall corrections to problems which we have
solved exactly, like the harmonic oscillator and the hydrogen atom. We will make a series expansion
of the energies and eigenstates for cases where there is only a small correction to the exactly soluble
problem.

First order perturbation theory will give quite accurate answers ifthe energy shifts calculated are
(nonzero and) much smaller than the zeroth order energy differences between eigenstates. If the
first order correction is zero, we will go to second order. If the eigenstates are (nearly) degenerate
to zeroth order, we will diagonalize the full Hamiltonian using only the (nearly) degenerate states.

Cases in which the Hamiltonian is time dependent will be handled later.

This material is covered inGasiorowicz Chapter 16,inCohen-Tannoudji et al. Chapter XI,
and in Griffiths Chapters 6 and 7.

22.1 The Perturbation Series

Assume that the energy eigenvalue problem for the HamiltonianH 0 can besolved exactly

H 0 φn=En(0)φn

but that the true Hamiltonian has a small additional term orperturbationH 1.

H=H 0 +H 1

The Schr ̈odinger equation for thefull problemis

(H 0 +H 1 )ψn=Enψn

Presumably this full problem, like most problems, cannot be solved exactly. To solve it using a
perturbation series,we will expand both our energy eigenvalues and eigenstates in powers of the
small perturbation.

En = En(0)+E(1)n +En(2)+...

ψn = N



φn+

∑

k 6 =n

cnkφk





cnk = c(1)nk+c(2)nk+...

where the superscript (0), (1), (2) are the zeroth, first, and second order terms in the series.Nis
there to keep the wave function normalized but will not play an important role in our results.

By solving the Schr ̈odinger equation at each order of the perturbation series, wecompute the
corrections to the energies and eigenfunctions.(see section 22.4.1)

En(1) = 〈φn|H 1 |φn〉