How to Determine the Transmission Coefficient for a Rectangular Barrier?

[maggicmike660]

Homework Statement

Determine the transmission coefficient for a rectangular barrier. Treat separately the three cases E<V_o, E>V_o, and E=V_o.

Homework Equations

V(x)= +V_o if -a<xa
V(x)= 0 otherwise

Transmission coefficient=(amplitude of transmited wave)²/(amplitude of incoming wave)²

I am also using the Time independent Schrödinger equation

The Attempt at a Solution

What I did for each case was to solve the Schrödinger equation in the following regions:

a) x<-a
b) -a<x<a
c) x>a

After doing this I exploited the fact that my solutions had to be continuous at the boundaries. The derivatives also had to be continuous at the boundaries.

ie)$$\Psi$$(a)^-=$$\Psi$$(a)⁺ and,

d/dx($$\Psi$$(a)^-)=d/dx($$\Psi$$(a)⁺

After doing this I get a bunch of equations and my goal is to isolate the amplitude of transmiited wave function and the amplitude of the initial wave function. I think that after a bunch of tedious algebra I will be able to isolate for the desired things.

Is there an easier way to isolate my amplitudes? My professor mentioned something about treating the even and odd solutions of the wavefunctions separately, but I do not know what he means.

If any of this is unclear please let me know!

 

[buffordboy23]

I think that after a bunch of tedious algebra I will be able to isolate for the desired things.

This method will work. I solved this problem before, and if I recall correctly, it did require very tedious algebra.

This problem is a common one in introductory QM. Many texts and websites give the final result without derivation, so you can consult those for guidance toward the final answer should get stuck.

 

[maggicmike660]

Thanks for the tip. I just got home after working on it and yes it does just require tedious algebra. Not difficult otherwise.