How Do You Solve the QHO with a Sinusoidal Perturbing Potential?

[taloc]

Homework Statement

I am tasked with solving the QHO with a sinusoidal perturbing potential of the form VoSIN(BX). I need to find the ground state energy as well as the ground state eigenket |g>.

Homework Equations

$$H_{o} = \frac{P^{2}}{2m} + \frac{1}{2}m \varpi^{2}$$
$$H = H_{o} + Asin(BX)$$
$$E^{(o)}_{n}=\hbar\varpi(n+\frac{1}{2})$$ Which is the unperturbed energy

The Attempt at a Solution

My first stab at this problem involved performing a Taylor expansion of the potential and the rewriting of the X operator in terms of the creation and annihilation operators:
$$X = \sqrt{\frac{\hbar}{2m\varpi}}(a^{\dagger}+a)$$

This process was not very rewarding. I found myself with no method for determining when to terminate the expansion.

An alternative approach would be to rewrite the potential as:

$$Asin(BX) = \frac{A}{2i}(e^{iBX}-e^{-iBX})$$

My concern with this method is that the |n> kets used in the QHO are not eigenkets of X and therefore do not play nicely with the exponentials. Do I need to perform a change of basis? Essentially creating a new set of kets composed of a linear combination of the |n> kets? Any advice would be greatly appreciated!

Cheers!

 

[nickjer]

Have you thought about using perturbation theory? I am guessing $V_0$ is small since you mention it is a perturbative potential. So are you supposed to find the ground state to first order in $V_0$? Or 2nd order?

 

[taloc]

Have you thought about using perturbation theory? I am guessing $V_0$ is small since you mention it is a perturbative potential. So are you supposed to find the ground state to first order in $V_0$? Or 2nd order?

nickjer,

Thank you for the reply. I do need to apply pertubation theory to solve this problem. The problem I am facing involves having the X operator of the perturbing potential "locked up" inside a trigometric function. If I were to expand the trig function in a Taylor Series I could then perform the analysis for first order corrections (which vanish) and then proceed to second order corrections. I asked my instructor about the Taylor expansion method and was told this was not the appropriate path to take. He confirmed that I must rewrite the Sin function in terms of the exponentials I gave in the initial post. I am completely lost as to how I should proceed from this point on.

Cheers,

 

[nickjer]

First order perturbation is just an integral over the perturbing potential:

$$E_n^{(1)}=\langle n^{(0)}|V|n^{(0)} \rangle$$

So just integrate over that potential with the ground state wavefunctions for a simple harmonic oscillator. No need for Taylor series.

 

[taloc]

First order perturbation is just an integral over the perturbing potential:

$$E_n^{(1)}=\langle n^{(0)}|V|n^{(0)} \rangle$$

So just integrate over that potential with the ground state wavefunctions for a simple harmonic oscillator. No need for Taylor series.

Wow! Is that all i need to do? That is far easier than i expected. i will attempt to generate the ground state energy tomorrow morning. I was attempting to evaluate the expression in a much more general case. I suspect that the integral will be zero and require a second order approximation, but here is to hoping.

Thank you very much for your help nickjer, i will get back to you in the morning with my results.

Ceers!