Solve two eigenfunctins for a Finite Square Well

[speedofdark8]

Homework Statement

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.

Homework Equations

See image

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]

First you must do those integrals to give yourself the final condition.

Which lines should be integrated? The ones for the 3 regions or the 4 conditions? I am 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.

 

[paranormal]

.I would first start by identifying the unknowns and the conditions that need to be satisfied. From the given information, it looks like the unknowns are the coefficients A, B, C, D, and E. The conditions that need to be satisfied are the continuity of the wavefunction and its derivative at x = ±a/2, as well as the normalization condition.

To solve for the unknowns, we can start by considering the continuity condition at x = ±a/2. This means that the wavefunction and its derivative must be equal on both sides of the potential well. This gives us two equations:

A + B = C + D

and

A - B = C - D

Next, we can use the normalization condition to solve for the remaining unknown, E. This condition states that the integral of the square of the wavefunction over all space must equal 1. This gives us the equation:

∫ψ(x)^2dx = 1

Substituting in the expressions for ψ(x) and solving for E, we get:

E = √(2V0/3a)

Now, we can use the two equations from the continuity condition to solve for the remaining unknowns, A, B, C, and D. This will involve some algebraic manipulation and substitution of the expression for E.

Once we have solved for all the unknowns, we can plug them back into the original expressions for ψ(x) to get the first two eigenfunctions for the finite square well.