(adsbygoogle = window.adsbygoogle || []).push({}); 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 [tex]E = \frac{-\hbar^2 q^2}{2m}[/tex] 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:

[tex]\frac{\partial ^2 \psi}{\partial{x}^2} = -q^2 \psi[/tex]

Which gives the solution:

[tex]\psi (x) = exp(iqx) + A exp(-iqx)[/tex]

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:

[tex]\frac{\partial ^2 \psi}{\partial{x}^2} = -\frac{2m}{\hbar ^2}(E-V_0) \psi[/tex]

Which gives the solution:

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

Region III:

[tex]\psi (x) = D exp(iqx)[/tex]

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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?

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