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Harmonic oscillator perturbation

  1. Dec 9, 2015 #1
    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. jcsd
  3. Dec 10, 2015 #2

    Orodruin

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    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.
     
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