(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I showed earlier this semester that in the presence of a "constant force", [itex]F_{o}[/itex], i.e. [itex]V=-Fx[/itex], that the eigenvalues for the Harmonic oscillator are shifted by

[itex]\frac{F^{2}}{2m\omega^{2}}[/itex]

from the "unperturbed" case. It was also discussed that [itex]x\rightarrow x-\frac{F}{m\omega^{2}}[/itex].

An oscillator is initially in its ground state (n=0). At t=0, a perturbation V is suddenly applied. What is the probability of finding the system in its (new) ground state for t>0, i.e. find [itex]|a_{o}|^{2}[/itex].

2. Relevant equations

For this [itex]|a_{n}|^{2}=|\int \Phi^{*}_{n}(x)\Psi_{o}(x)dx|^{2}[/itex] over all space.

3. The attempt at a solution

For t>0, the state of the system is [itex]\Psi(x,t)=\sum a_{n}exp(-i(\frac{E_{n}}{\hbar})t)\Phi_{n}(x)[/itex]. Here [itex]\Phi_{n}(x)[/itex] is an eigenvector of H. And the coefficients [itex]a_{n}[/itex] are obtained by expanding [itex]\Psi_{o}(x)[/itex], the ground state of [itex]H_{o}[/itex], in terms of [itex]\Phi_{n}(x)[/itex].

I also know that the basis states [itex]\Phi_{n}(x)[/itex] as well as [itex]\Psi_{o}(x)[/itex] are Hermite polynomials.

With that in mind my assumption would simply be to integrate the following:

[itex]|a_{o}|^{2}=|\int \Phi_{o}(x) \Psi_{o}(x) dx|^{2} =|\int 1*1 dx|^{2}[/itex]

If I integrate this over all space I end up with a probability that goes to infinity. . . .Maybe I am missing something as far as Hermite polynomials go. . . or maybe I have the wrong idea about this problem. Any suggestions would be helpful.

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# Perturbed harmonic oscillator

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