Harmonic oscillator perturbation

[Santiago Paz]

Homework Statement

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 mω²xˆ² 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.

Homework Equations

H = H₀ + V

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.

 

[Orodruin]

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.