Phy5646/Another example

(Submitted by Team 1)

This example was taken from "Theory and Problems of Quantum Physics", SCHAUM'S OUTLINE SERIES, p. 190-192.

Problem: Consider a one dimensional harmonic oscillator embedded in a uniform electric field. The field can be considered as a small perturbation and depends on time according to

$\epsilon (t)={\frac {A}{{\sqrt {\pi }}\tau }}exp[-({\frac {t}{\tau }})^{2}]$

where A is constant. If the oscillator was in ground state until the field was turned on at t=0, compute in the first approximation, the probability of its excitation as a result of the action of the perturbation.

Solution: The harmonic oscillator eigenstates: $|n\rangle$, with eigen-energies $h\omega (n+1/2)$. Then

$\langle n|\psi (t)\rangle =\langle n|0\rangle +{\frac {qE}{i\hbar }}\int _{-\infty }^{t}dt'e^{{\frac {i}{\hbar }}(\epsilon _{n}-\epsilon _{0})t'}\langle n|x|0\rangle e^{-t'^{2}/\tau ^{2}}$