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Time-Dependent Frequency Harmonic Oscillator

  1. Dec 24, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider an harmonic oscillator with time-dependent frequency as:
    [itex]\omega (t)=\omega_0 * \exp^{- \lambda t}[/itex]
    Find the time dependence of the ground state energy of this oscillator for [itex]\lambda << 1[/itex] situation.

    2. Relevant equations
    [itex]H=H_{0} + V(t)[/itex]
    [itex]H_{0} = \frac{p^2}{2m} + \frac{1}{2} m \omega_{0}^{2} x^{2}[/itex]
    and if we use the power series expansion for [itex]\lambda << 1[/itex] we get
    [itex]V(t) = - \frac{1}{2} m \omega_{0}^2 \lambda t x^{2}[/itex]

    3. The attempt at a solution
    I know that I should use the time-dependent perturbation theory, but I am not good at it. So I need some help to solve this problem.
     
    Last edited: Dec 24, 2011
  2. jcsd
  3. Dec 24, 2011 #2
    What are the relevant equations for time dependent perturbation theory?
     
  4. Dec 24, 2011 #3
    sorry I added them to my original post
     
  5. Dec 25, 2011 #4
    next, how do we express the energy eigenvalues in time-dependent perturbation theory? ;)
     
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