How Does a Perturbation Affect Energy Levels in an 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?

 

[Galileo]

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.

 

[Gokul43201]

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).

 

[Gokul43201]

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.