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Potential Barrier

  1. Feb 19, 2009 #1
    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]

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

    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
  2. jcsd
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