# Potential Barrier

1. Feb 19, 2009

### Feldoh

1. The problem statement, all variables and given/known data
Consider the diagram described by: http://filer.case.edu/pal25/well.jpg [Broken]

If a particle with $$E = \frac{-\hbar^2 q^2}{2m}$$ comes in from negative infinity with amplitude 1, what is the wave function for negative x?

Oh and V(x) < -a and > a = 0

2. Relevant equations

3. The attempt at a solution
I think the problem needs to be broken into two cases, depending on if E > V or E < V in region 2.

If E > V then...

Region I:
$$\frac{\partial ^2 \psi}{\partial{x}^2} = -q^2 \psi$$

Which gives the solution:

$$\psi (x) = exp(iqx) + A exp(-iqx)$$

So we have the wave equation be we still need to determine the coefficient A, so we have to solve for the other two regions...

Region II:
$$\frac{\partial ^2 \psi}{\partial{x}^2} = -\frac{2m}{\hbar ^2}(E-V_0) \psi$$

Which gives the solution:

$$\psi (x) = B exp(ikx) + C exp(-ikx)$$ where $$k^2 = \frac{2m}{\hbar ^2}(E-V_0)$$

Region III:

$$\psi (x) = D exp(iqx)$$

-----------------------

So I guess I need to do continuity conditions for these equations? I have a feeling this is going to be really tedious and that there is a different way?

Last edited by a moderator: May 4, 2017