Adding a small extra potential to a simple harmonic oscillator

[kramleigh]

Hi,

I've been scouring through many textbooks to try find some kind of solution to a question I have been asked for a problem sheet and was wondering if any1 would be able to help. The question is as follows;

The simple harmonic oscillator with hamiltonian H = (p^2/2m) + (1/2(mw^2x^2) is modified by adding a small extra potential V(x) = (1/2(ma^2x^2)). Use the first order perturbation theory to calculate the change in the ground state energy. How does this estimate differ from the exact result?

Unfortunately, I don't think we've covered it clearly enough in our lectures and I can't quite find anything specific enough in textbooks to answer it, so any help would be much appreciated.

Cheers

 

[Physics Monkey]

The obvious observation here is that the perturbed potential is still just a harmonic oscillator potential but with a shifted frequency. That should immediately tell you what the exact energy levels are. Now the remaining question is do you know how to do first order perturbation theory. I'm sure the details are in your book, just look up perturbation theory. The answer is very simple: the first order energy shift is just the expectation value of the perturbation.