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Homework Help: Reflection Coeff - Quantum Mech

  1. Apr 8, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the reflection coefficient for electrons traveling toward a potential change from [tex]V[/tex] to [tex]V_0[/tex] with a total energy [tex]E > V_0[/tex].
    The potential diagram is just a unit step function. It goes from [tex]V = 0[/tex] to [tex]V = V_0[/tex] at [tex]x=0[/tex]. In piecewise notation:
    [tex]
    \begin{displaymath}
    V(x) = \left\{
    \begin{array}{lr}
    0 & : x < 0 \\
    V_0 & : x \ge 0
    \end{array}
    \right.
    \end{displaymath}
    [/tex]
    The piecewise notation does not account for the [tex]V(x)[/tex] being continuous at [tex]x=0[/tex].



    2. Relevant equations
    a) Probability flux:
    [tex]S\left( x,t \right)=-\frac{i\hbar}{2m}\left[ \Psi^*\left( x,t \right) \frac{\partial \Psi\left( x,t \right)}{\partial x} - \Psi\left( x,t \right) \frac{\partial \Psi^*\left( x,t \right)}{\partial x}\left][/tex]

    b) Reflection coefficient:
    [tex]R=\frac{S_{I}^{-x}\left( x,t \right)}{S_{I}^{+x}\left( x,t \right)}[/tex]

    3. The attempt at a solution
    I've solved for the wave equations at [tex]x > 0[/tex] and [tex]x < 0[/tex]. I'm stuck as far as where to go from there.
     
  2. jcsd
  3. Apr 8, 2010 #2
    Are you asking for help on (a)? Can you also show us your final wave function? And have you tried plugging that wavefunction into (a)?
     
  4. Apr 8, 2010 #3

    vela

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    Require continuity of the wavefunction and its derivative at x=0. That will allow you to solve for most of the constants.
     
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