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 probability of a transition from the state n to the state k is given by

$P_{nk}={\frac {1}{\hbar ^{2}}}|\int _{-\infty }^{\infty }<k|V|n>e^{it(E_{k}^{(0)}-E_{n}^{(0)})/\hbar }dt|$

Let e, m and w denote the charge, mass and natural frequency of the oscillator, respectively, where x denotes its deviation from its equilibrium position. In the case of an uniform field, the perturbation is given by

$V(x,t)=-ex\epsilon (t)-x$

The oscillator is in the ground state (n=0), so the non vanishing elements of the perturbation matrix are

$V_{01}=V_{10}=-{\frac {p}{{\sqrt {\pi }}\tau }}{\sqrt {\frac {\hbar }{2mw}}}exp[-({\frac {t}{\tau }})^{2}]$

In the first approximation a uniform field can produce a transition of the oscillator only to the first excited state, then:

$P_{01}={\frac {p^{2}}{2\pi \tau ^{2}mtw}}|\int _{-\infty }^{\infty }exp[iwt-({\frac {t}{\tau }})^{2}]dt|$

Using the identity

$\int _{-\infty }^{\infty }e^{i\beta x-\alpha x^{2}}dt={\sqrt {\frac {\pi }{\alpha }}}e^{\frac {-\beta ^{4}}{4\alpha }}$

we have:

$P_{01}={\frac {p^{2}}{2m\hbar w}}exp[-{\frac {1}{2}}(wt)^{2}]$

Note that for $\tau >>{\frac {1}{w}}$ the probability for the excitation is extremely small. This is the case of a so-called adiabatic perturbation. On the contrary, for a rapid perturbation $\tau <<{\frac {1}{w}}$ the probability of excitation is constant.