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.
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.