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

I have a charged particle in a 1D harmonic oscillator, on which an externally applied electric field acts, such that the Hamiltonian can be written:

[tex] \frac{p^2}{2m}+\frac{kx^2}{2}-qEx[/tex]

The problem asks to first find the first (trivial) and second order corrections to the energy levels via perturbation theory, which I have done. Now the problem asks to verify that the second order correction agrees with the exact solution of the Schrodinger equation. The hint is to use a change of variable

[tex]x'=x-(\frac{qE}{m \omega ^2})[/tex]

3. The attempt at a solution

Using the suggested change of variable, it is straight-forward to show that the Hamiltonian can be written as that of the conventional, unperturbed oscillator, only in [tex]x'[/tex] rather than [tex]x[/tex]. My question is, where do I go now? What I essentially want to do is calculate the inner product

[tex]\left\langle\Psi(x)\right|H^{new}\left|\Psi(x)\right\rangle[/tex]

But I want to do this for a general case of all [tex]\Psi(x)[/tex], and I have an operator, [tex]H^{new}[/tex] that acts on x', and not on x. My only guess is that I can rewrite the known solutions of the 1D SHO, (The Hermite Polynomials) in terms of x' and integrate directly, but this gets very messy very quickly, even for the ground state! It strikes me I'm missing some obvious trick.

Can anyone suggest an approach here?

Aside: This is a (reworded) question out of a very common quantum text. I've refrained from specifying which in the interest of avoiding this thread popping up in an internet search for the question number. Hopefully my rewording makes sense!

Thanks in advance!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: 1D SHO in external electric field

**Physics Forums | Science Articles, Homework Help, Discussion**