# Infinite Potential Well

cyberdeathreaper
Sorry for all the questions - I tend to save them till I'm done with assignments:

Here's the question:
Consider a particle of mass 'm' in a one-dimensional infinite potential well of width 'a'
$$V (x) = \left\{\begin{array}{c} 0 \ \ \ if \ \ \ 0 \leq x \leq a \\ \infty \ \ \ otherwise$$
The particle is subject to a perturbation of the form:
$$\omega (x) = a \omega_0 \delta \left(x - \frac{a}{2} \right)$$
Where 'a' is a real constant with dimension of energy. Calculate the changes in the energy level of the particle in the first order of $\omega_0$

I just need some help starting off at this point. Can anyone suggest how to begin?

## Answers and Replies

Science Advisor
Homework Helper
What does first-order pertubation theory say?

cyberdeathreaper
Okay, I think I've got it. Does this look correct?
ANS:
I'm looking for first-order correction to the nth eigenvalue - so I need to solve this:
$$E_n^1 = \left< \psi_n^0 | H' | \psi_n^0 \right>$$
Where
$$\psi_n^0 (x) = \sqrt{ \frac{2}{a} } sin \left( \frac{n \pi x}{a} \right)$$
and
$$H' = a \omega_0 \ \delta \left( x - \frac{a}{2} \right)$$
Substituting and solving gives:
$$E_n^1 = \left< \psi_n^0 | H' | \psi_n^0 \right> = 2 \omega_0 \int_0^a sin^2 \left( \frac{n \pi x}{a} \right) \delta \left( x - \frac{a}{2} \right) dx$$
$$= 2 \omega_0 sin^2 \left( \frac{n \pi}{2} \right)$$
$$= \left\{\begin{array}{c} 2 \omega_0 \ \ \ if \ \ \ n = "odd" \\ 0 \ \ \ if \ \ \ n = "even"$$
Do I "choose" only non-zero answers then, or is the array the complete answer? Thanks.

Staff Emeritus
Science Advisor
Gold Member
Looks right ! And no, you don't throw away the zero-terms. In fact, you should be asking yourself if it makes sense for the perturbation to have no effect on half the spectrum (the eigenvalues for even n).

Staff Emeritus
Science Advisor
Gold Member
Found this java applet from ZapperZ's link :

http://www.quantum-physics.polytechnique.fr/en/pages/p0204.html

It shows you the solutions to the SE for a double well. You can play with the potentials to essentially mimic your problem. Make a (infinitesimally) thin, high wall in the middle, and see what happens when you just barely increase its width : only the odd eigenvalues move, as predicted.