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Quantum Well & Barrier w/ Phase Shift

  1. Jan 25, 2009 #1
    1. The problem statement, all variables and given/known data

    V(x) = [tex]\inf[/tex] if x [tex]\leq[/tex] 0
    = -V if 0 [tex]<[/tex] x [tex]<[/tex] a
    = 0 if x [tex]>[/tex] a

    NOTE: E>V

    Find the wave functions in each region and the stated phase shift.

    2. Relevant equations

    Schrodinger Eq.

    Instructions note that wave functions are:
    [tex]\Psi_I[/tex] = [tex]A\sin{kx}[/tex]
    [tex]\Psi_{II}[/tex] = [tex]B\sin{k'x+\phi}[/tex]

    where,

    k = [tex]\sqrt{2m(V+E)}/\hbar[/tex]
    k' = [tex]\sqrt{2mE}/\hbar[/tex]

    Show [tex]2\phi = 2\left[\cot^{-1}\left(\frac{k}{k'}\cot\left({ka}\right)\right)-k'a\right][/tex]

    3. The attempt at a solution

    I can get Region I just fine (0:a well)

    [tex]\frac{-\hbar^2}{2m} \frac{d^2 \Psi}{dx^2} - V \Psi = E \Psi [/tex]
    [tex]-\frac{d^2 \Psi}{dx^2} = \left(E+V\right)\Psi\left(\frac{2m}{\hbar^2}[/tex]
    let [tex]k = \frac{\sqrt{2m\left(V+E\right)}}{\hbar}[/tex]
    [tex] \Psi_x\left(x\right) = A\sin{kx} [/tex]

    Region II

    [tex]\frac{-\hbar^2}{2m} \frac{d^2 \Psi}{dx^2} = E \Psi [/tex]
    [tex]-\frac{d^2\Psi}{dx^2} = \frac{2mE}{\hbar^2}\Psi [/tex]
    [tex] \Psi_{II}\left(x\right) = B\sin{k'x} [/tex]
    where [tex]k' = sqrt{2mE}{\hbar}[/tex]

    But my problem lies in the phase shift. How is it related to the wave functions and how can it be calculated?
     
  2. jcsd
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