Homework 5
(due Wednesday, September, 26)
This is the homework assignment dealing with piecewise constant potentials. There is
only one problem on it because of the amount of algebra required. It is recommended that
you start this problem well ahead of the due date.
1. Consider the following potential in one dimensions: V (x) = 0 for x < −L, V (x) =
Vo < 0 for −L < x < L, and V (x) = 0 for x > L. Assume that the the energy of an
incoming electron, E, is greater than zero.
The general form of a the solution to the Schrodinger equation for this problem is
φ(x < −L) = A1 eik1 x + A′1 e−ik1 x
φ(−L < x < L) = A2 eik2 x + A′2 e−ik2 x
φ(L < x) = A3 eik3 x + A′3 e−ik3 x .
(a) What are k1 , k2 and k3 ?
(b) For the above wave function what is the probability current in each of the three
regions? Note that the A’s are in general complex.
(c) If a wave is incident from the left, which coefficient is zero? Also, in this case
what are the reflection and transmission probabilities. From the results of part
(b) show that the sum of the reflection and transmission probabilities is one.
(d) If a wave is incident from the right, which coefficient is zero? Also, in this case
what are the reflection and transmission probabilities. From the results of part
(b) show that the sum of the reflection and transmission probabilities is one.
(e) Write down the boundary conditions at x = −L.
(f) Write down the boundary conditions at x = L.
(g) Specialize to the case of a wave incident from the left and solve for the reflection
and transmission probabilities.
(h) Check your result by showing explicitly that the sum of the reflection and transmission probabilities is one.
(i) Plot the transmission probability as a function of k2 L? Are there any values of
k2 L for which T = 1? If so, can you offer a qualitative explanation?