1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Probability of perturbed harmonic oscillator

  1. Mar 10, 2010 #1
    An experimenter has carefully prepared a particle of mass m in the first excited state of a one dimensional harmonic oscillator. Suddenly he coughs and knocks the center of the potential a small distance, a, to one side. It takes him a time T to recover and when he has done so he immediately puts the center back where it was.
    A) Find, to lowest order in a, the probability that the oscillator will now be in its ground state.
    B) Find, to lowest order in a, the probability that the oscillator will now be in its second excited state.

    I know that there are two sudden approximations that can be made, the first after he coughs, and the second after he moves the center back. I was thinking Hi = -hbar^2/2m * d^2 phi/dx^2 + 1/2 m w^2 x^2. For the first sudden approximation, the x^2 would be replaced by (x-a)^2, and then it would return to x^2 for the second approximation. I think I'm not sure where the a fits into the calculation, and how to find the answer to the lowest order in a.
  2. jcsd
  3. Mar 10, 2010 #2
    First step, you will need to find the first order expansion using perturbation theory. Use the perturbed potential:

    [tex]V = \frac{1}{2}m\omega^2 (x-a)^2- \frac{1}{2}m\omega^2x^2[/tex]

    And expand the the wavefunction to first order in 'a' by finding,

    [tex]|n'> = |n>+\sum_{k\ne n}|k>\frac{<k|V|n>}{E_n - E_k}[/tex]

    Once you have that you will have to make use of the fact that the change in the potential was sudden.
    Last edited: Mar 11, 2010
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook