# Time dependent perturbation theory of the harmonic oscilator

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## Homework Statement

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##.

## Homework Equations

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(Given in question)

## The Attempt at a Solution

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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!

## Answers and Replies

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?