Phy5646/Simpe Example of Time Dep Pert

This example taken from "Quantum Physics" 3rd ed., Stephen Gasiorowicz, p. 237-238.

Problem: A particle of charge $q$ in a one-dimensional harmonic oscillator of characteristic frequency $\omega$ is placed in an electric field that is turned on and off so that the potential energy is

$\lambda V(t)=qExe^{-t^{2}/\tau ^{2}}$

If the particle is initially in the ground state, what is the probability that after time $t$, such that $t>>\tau$, the particle is found in the first excited state of the harmonic oscillator? What is the probability that it is found in the second excited state?