# Homework Help: Grad 1D scattering from step function

1. Oct 14, 2012

### frogjg2003

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

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

Where would I go from here?