Thermodynamics, Statistical Physics, and Quantum Mechanics

The result from first-order perturbation theory is obtained by taking the

integral of the perturbation with the ground state wave function

The ground state energy is The first term in has odd

parity and integrates to zero in the above expression. The second term in

has even parity and gives a nonzero contribution. In this problem it is

easiest to keep the eigenfunctions in the separatebasis of rather than

to combine them into In one dimension the average of so

we have

where This is probably the simplest way to leave the answer.
This completes the discussion of first-orderperturbation theory.
The otherterm in contributes an energy of insecond-
order perturbation theory. The excited state must have the symmetry of
which means it is the state This has three
quanta excited, so it has an energy

Now we combine the results from first- and second-order perturbation the-
ory:

5.43 Uand Perturbation(Princeton)

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