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!

 

[paranormal]

I would like to address the content provided in this question regarding perturbation theory and the harmonic oscillator potential.

Firstly, perturbation theory is a mathematical tool used to approximate solutions to complex systems by breaking them down into simpler, solvable parts. In this case, we are considering the effect of a constant electric field on an electron confined in a harmonic oscillator potential.

a) To determine the variation in the electron's energy levels caused by the electric field E, we can use the first order perturbation theory. This theory states that the energy shift of a system due to a perturbation is equal to the expectation value of the perturbation operator in the unperturbed state. In this case, the perturbation operator is the electric field E and the unperturbed state is the harmonic oscillator potential. Therefore, the variation in the electron's energy levels can be calculated by taking the expectation value of the electric field operator in the harmonic oscillator potential.

b) The second order perturbation theory provides a more accurate approximation by taking into account higher order corrections. In this case, we can use the second order perturbation theory to calculate the exact value of the variation in the electron's energy levels caused by the electric field E. This can be done by considering the second order correction to the energy, which is given by the sum of all possible intermediate states. By taking into account all possible intermediate states, we can obtain the exact value for the variation in the electron's energy levels.

c) The electric dipole moment of the electron in the n-th state is defined as pn=-e<x>n, where e is the charge of the electron and <x>n is the expectation value of the position operator in the n-th state. In the absence of the electric field E, the electric dipole moment will be zero as the electron is confined in a symmetric potential and its position is equally likely to be on either side of the origin. However, in the presence of the electric field E, the potential becomes asymmetric and the electron will experience a shift in its position, resulting in a non-zero electric dipole moment.

In conclusion, the use of perturbation theory in the context of the harmonic oscillator potential and a constant electric field allows us to accurately calculate the variation in the electron's energy levels and the resulting electric dipole moment. This is an important tool in understanding the behavior of electrons in complex systems and has various applications in the field of quantum mechanics.