Solve two eigenfunctins for a Finite Square Well

speedofdark8

1. The problem statement, all variables and given/known data

Solve Explicitly the first two eigenfunctions ψ(x) for the finite square wave potential V=V₀ for x<a/2 or x>a/2, and V=0 for -a/2<x<a/2, with 0<E<V₀.

2. Relevant equations

See image

3. The attempt at a solution

See image. After modeling an in class example, my classmates and i were stuck here. We have 5 unknowns, and 4 conditions. We know we have integrate the square of each region (as shown) and add to normalize and solve this, but we don't know how to handle/solve for the unknowns

jfy4

First you must do those integrals to give yourself the final condition. Then you will have five equations, two from each matching point, and the normalization condition. After, its a matter of solving 5 equations and 5 unknowns. If you feel comfortable, use mathematica, if not, do it by hand. Methods of substitution or elimination can get it done.

speedofdark8

Which lines should be integrated? The ones for the 3 regions or the 4 conditions? Im unclear on how i will be getting 5 equations.

And I presume that you are saying to integrate first, and then solve for the unknowns?

vela

The five equations are the two you wrote down for x=a/2, the two you wrote down for x=-a/2, and the normalization condition.

You should recheck your work. There are a few errors in your equations.

Add the first and third equations together, and add the second and fourth equations together. Then divide one result by the other. You should get something like
$$\beta \tan (\beta a/2) = \alpha$$ (or maybe cot instead of tan). You should be able to show that for this to hold, either C or D has to vanish.