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Time dependent perturbation theory of the harmonic oscilator

  1. Mar 25, 2015 #1

    CAF123

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    Gold Member

    1. The problem statement, all variables and given/known data
    A 1-d harmonic oscillator of charge ##q## is acted upon by a uniform electric field which may be considered to be a perturbation and which has time dependence of the form ##E(t) = \frac{K }{\sqrt{\pi} \tau} \exp (−(t/\tau)^2) ##. Assuming that when ##t = -\infty##, the oscillator is in its ground state, it can be shown that the probability that it is in its first excited state at ##t = +\infty## using time-dependent perturbation theory is $$P_{t} = \frac{q^2 K^2}{2 \hbar m \omega} e^{-\frac{\tau^2 \omega^2}{4}}$$

    Discuss the behaviour of the transition probability and the applicability of the perturbation theory result when a) ##\tau \ll 1/\omega## and b) ##\tau \gg 1/\omega##.

    2. Relevant equations

    (Given in question)

    3. The attempt at a solution

    The given conditions imply that either ##\tau^2 \omega^2 \ll 1## or ##\tau^2 \omega^2 \gg 1##.In the former case, I can expand the exp factor for small argument and for the latter case, the exp term dies away to zero quickly giving a transition prob of zero. I am just wondering what further comments can be made, in particular regarding the applicability of the perturbation theory result? Thanks!
     
  2. jcsd
  3. Mar 30, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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