# Homework Help: Harmonic oscillator perturbation

1. Dec 9, 2015

### Santiago Paz

1. The problem statement, all variables and given/known data
Consider the one-dimensional harmonic oscillator of frequency ω0:
H0 = 1/2m p2 + m/2 ω02 x2

Let the oscillator be in its ground state at t = 0, and be subject to the perturbation
Vˆ = 1/2 mω22 cos( ωt )at t > 0.

(a) Identify the single excited eigenstate of H0 for which the transition amplitude is nonzero in first-order time-dependent perturbation theory. Calculate this amplitude explicitly.
(b) Calculate the dominant term of the first-order transition probability to the state identified in (a) for ω = 2ω0. Give a condition for which this result becomes meaningless.

2. Relevant equations
H = H0 + V

3. The attempt at a solution
Used the above equation to start my solution
ended up with something in the form of

H = p2/2m + m/2 (ω02 + ω2cos( ωt )) x2

I am unsure as to how to keep going and find the eigenstates of H0. I thought it was more intuitive to solve for the eigenstates of H not H0. And I'm also not quite sure how to find the transition amplitude. Any help is greatly appreciated.

2. Dec 10, 2015

### Orodruin

Staff Emeritus
But you are not asked to find the eigenstates of H, you are asked to find out things in terms of the eigenstates of H0. In addition, the eigenstates of H are time dependent, which leads you to further complications. You only need to apply simple first order perturbation theory using the properties of the harmonic oscillator here.