Solving Schrodinger Equation for Finite Potential Barrier

[irmanbearpig]

Homework Statement

I am trying to find the coefficients in a Schrödinger equation approaching a finite potential.
https://www.physicsforums.com/showthread.php?t=203385
It is a problem similar to this, except a little easier. In my case, though, there is no V₁ as shown in the picture at the thread, the potential is at x = 0 and doesn't drop back down, just continues going in the positive x direction. So at x = 0 it just goes straight up vertically, then levels off and keeps going, creating a sort of wall. (hope that is a good enough explanation)

Homework Equations

Schrodingers Equation

The Attempt at a Solution

I have defined the regions as:
Region 1: V(x) = 0 for x < 0
Region 2: V(x) = V₀ for x > 0

I have used Schrödinger's Equation to get the wave function for each region:
Region 1:
Psi(x) = Ae^ik₁x + Be^-ik₁x
With A being the incident wave and B being the reflected wave off of the potential barrier.
I also have K₁ = (2mE/h²)^1/2

Region 2:
Psi(x) = Ce^ik₂x
With C being for the continuing wave, there is no reflected wave here because region 2 contains the barrier. I also think C would be considered T (transmitted).
I also have K2 = (2m(E-V₀)/h²)^1/2

So I can say that A + B = C at x = 0 (I think). I can also say that their derivatives are equal at that point, so ik₁A - ik₂B = ik₂C

I also know that 1 = P(R) + P(T)
and R = abs(B²/A²)
and I have written T = 1 - R

I guess I'm supposed to solve B and C in terms of A, I'm almost positive all of the information thus far is correct, unless I just typed something incorrectly.

It's something that I haven't ever seen/done before so I'm absolutely stumped... I even looked at some old physics books from the library trying to get some ideas but am lost. I'm sure it's something really simple... I just need to know where to go from this point in order to solve for A B and C.

Thanks!

 

[lanedance]

so you have 2 equations linking A,B & C due to the boundary conditions (continuous wave function & derivative)

you should be able to solve for B & C in terms of A...

 

[quasar_4]

Well, it's not too bad. You can relate everything using your two equations at the boundary, and finding an expression for C/A and B/A. These expressions should end up containing k1 and k2.

You can then find a way to relate C/A and B/A to transmission and reflection coefficients.

I'm pretty sure most quantum texts discuss this - try ch. 7 of Liboff's book, or ch. 5 of Shankar's. This would be best classified as the single step potential.