# 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!