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## Homework Statement

Consider the one-dimensional harmonic oscillator of frequency ω0:

H

_{0}= 1/2m p

^{2}+ m/2 ω

_{0}

^{2}x

^{2}

Let the oscillator be in its ground state at t = 0, and be subject to the perturbation

Vˆ = 1/2 mω

^{2}xˆ

^{2}cos( ωt )at t > 0.

(a) Identify the single excited eigenstate of H

_{0}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 = 2ω

_{0}. Give a condition for which this result becomes meaningless.

## Homework Equations

H = H

_{0}+ V

## The Attempt at a Solution

Used the above equation to start my solution

ended up with something in the form of

H = p

^{2}/2m + m/2 (ω

_{0}

^{2}+ ω

^{2}cos( ωt )) x

^{2}

I am unsure as to how to keep going and find the eigenstates of H

_{0}. I thought it was more intuitive to solve for the eigenstates of H not H

_{0}. And I'm also not quite sure how to find the transition amplitude. Any help is greatly appreciated.