Perturbation theory / harmonic oscillator

[notist]

Homework Statement

An electron is confined by the potential of a linear harmonic oscillator V(x)=1/2kx² and subjected to a constant electric field E, parallel to the x-axis.
a) Determine the variation in the electron’s energy levels caused by the electric field E.
b) Show that the second order perturbation theory gives the exact value to the same variation.
c) The system’s electric dipole moment in the n-th state is defined by p_n=-e<x>_n. What is the electric dipole moment for the electron in the harmonic oscillator’s potential, in the absence and in the presence of the electric field E?Thanks in advance!

 

[kuruman]

Hi notist, welcome to PF. Please use the template for homework help. Note that we don't do people's homework for them. Please show the relevant equations and how far you got solving the problem before you seek help.

 

[notist]

Oh, sorry. My teacher didn’t explain this in class, so I’m supposed to learn it all by myself. That’s why I don’t really know how to do this. From some research in books I got:

a)
The perturbed hamiltonian is:

H=p²/2m+1⁄2 mw²-Ex, with Ex being the electric potencial energy.

Also, the eigenvalues of the harmonic oscilator are:

E_n=1⁄2 ℏw(n+1⁄2)

So with the perturbation it will be: E_n=1⁄2 ℏw(n+1⁄2)-1⁄2 E²/mw²

But I can't figure out how to get to the last term of this formula.

b)
The formula for the 2^nd order perturbation is

[PLAIN]http://img541.imageshack.us/img541/3295/lolq.jpg

but I don't know how to use it.


c) I have no idea how to do this one.

 

[bndnchrs]

Hi notist,


If you are able to write down the perturbed Hamiltonian, you should be able to run through these computations quite easily :).

The idea is that to first order perturbation, the energy shifts are essentially the same as the expectation value of the perturbing Hamiltonian. It seems like that's what you are trying to evaluate in the first problem. In the textbooks you have, it gives the simple formula for this.

You have the second order perturbation in that image you added. I think you should see its pretty straightforward, once you realize what that matrix element is. The H' you see there is your perturbing Hamiltonian (in your case this is just E(x)*x, and the psis are your regular old harmonic oscillator wavefunctions. Notice that the superscripts on all the perturbation theory things are indicating which order you will use. For energies, you never need to worry about the higher order wavefunctions.

For the last problem, you are going to need to get the wavefunction correction for higher orders. You can look this up in a book, its the same sort of idea. Because you note in (b) that the second order and higher will be as good as the first, you only need the first order. This is just an exercise in finding an expectation value, with the lowest order wavefunction, then again with the higher order wavefunction, and comparing them.


Hopefully when you don't have to worry about homework, you can really sit down and figure out how perturbation theory works. Its a very good tool!