Phy5646/Simpe Example of Transition Probability Calculation

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?

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

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

Since $\tau >>T$,we may take the upper and lower limits in the time integral to be $\pm \infty$. We therefore get

${\begin{aligned}\langle n|\psi (t)\rangle &={\frac {qE}{i\hbar }}\langle n|x|0\rangle \int _{-\infty }^{\infty }dt'e^{in\omega }e^{-t'^{2}/\tau ^{2}}\\&={\sqrt {\pi }}{\frac {qE\tau }{i\hbar }}\langle n|x|0\rangle e^{-n^{2}\omega ^{2}\tau ^{2}/4}\end{aligned}}$

Therefore,

$P_{n}=\pi {\frac {q^{2}E^{2}\tau ^{2}}{\hbar ^{2}}}|\langle n|x|0\rangle |^{2}e^{-n^{2}\omega ^{2}\tau ^{2}/2}$

To calculate $\langle n|x|0\rangle$ we use

$x={\sqrt {\frac {\hbar }{2m\omega }}}(a+a^{\dagger })$

and recall that $a|0\rangle =0$ and $a^{\dagger }|0\rangle =|1\rangle$. Thus, $|\langle n|x|0\rangle |^{2}={\frac {\hbar }{2m\omega }}\delta _{n1}$ and therefore

$P_{1}={\frac {\pi }{2}}{\frac {q^{2}E^{2}\tau ^{2}}{m\hbar \omega }}e^{-\omega ^{2}\tau ^{2}/2}$

We also see that $P_{n}=0$ for $n$ = 2, 3, ...

Note the following

1. As $\tau \to \infty P_{1}\to 0$. When the electric field is turned on very slowly, then the transition probability goes to zero; that is, the system adjusts adiabatically to the presence of the field, without being "jolted" into making a transition.

2. The potential, through its $x$-dependence, will allow only certain transitions to take place. In other words, we see that there is a selection rule that operates here. If the potential were proportional to $x^{2}$, for example, then transitions to $n$ = 2 would be allowed.