# Time dependent perturbation theory of the harmonic oscilator

1. Mar 25, 2015

### CAF123

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. Mar 30, 2015