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Grad 1D scattering from step function

  1. Oct 14, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider the 1D potential V(x) such that V(x)=0 for x<0 and V(x) = V for x>0 and assume that a wave packet with energy E0=p20/2m<V is incident on the barrier from the left. Calculate in terms of E0 and V the difference in time between the arrival of the incident packet at the step and the departure of the reflected packet from the step. Relate this time delay to a “distance of travel” within the step.

    2. Relevant equations

    time independent Schrödinger equation

    3. The attempt at a solution

    I've calculated the time independent wavefunction to be
    [tex]
    \Psi(x) = \begin{cases}
    A_Ie^{ik_0x} + B_Ie^{-ik_0x} & x<0\\
    A_{II}e^{kx} & x>0 \end{cases}
    \\
    k = \sqrt{\frac{2mE_0}{\hbar^2}}
    \\
    k = \sqrt{\frac{2m(V-E_0)}{\hbar^2}}
    [/tex]
    with the relations
    [tex]
    \frac{B_I}{A_I} = \frac{ik_0+k}{ik0-k}
    \\
    \frac{A_{II}}{A_I} = \frac{2ik_0}{ik_0-k}.
    [/tex]
    From that, I was thinking about writing the wavefunction as
    [tex]
    \Psi(x) = \begin{cases}
    e^{ik_0x} + e^{-i(k_0x-\phi)} & x<0\\
    ce^{kx+i\theta} & x>0 \end{cases}
    [/tex]
    where I've removed the normalization constant and use the two phases [itex]\phi,\theta[/itex] given by
    [tex]
    \tan(\phi) = \frac{2kk_0}{k^2-k_0^2}
    \\
    c = \frac{2k_0}{k'}
    \\
    \tan(\theta) = \frac{k}{k_0}
    \\
    k'^2=k_0^2+k^2=\frac{2mV}{\hbar^2}
    [/tex]

    I'm stuck with what the question means by calculate the time difference.
    One of my classmates said to apply the time evolution operator [itex]e^{-i\hat{H}t/\hbar}[/itex], which because we're acting on one eigenfunction, becomes [itex]e^{-iE_0t/\hbar}[/itex] and the wavefunction is
    [tex]
    \Psi(x,t) = \begin{cases}
    e^{i(k_0x-E_0t/\hbar)} + e^{-i(k_0x-\phi+E_0t/\hbar)} & x<0\\
    ce^{kx+i(\theta-E_0t/\hbar)} & x>0 \end{cases}
    [/tex]

    Where would I go from here?
     
  2. jcsd
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