Homework 5
(due Friday, September, 27)
This is the last homework assignment before the exam. For this problem you need to
solve for the transmission and reflection coefficients for a piecewise constant potential. There
is only one problem because of the amount of algebra involved - do not wait until the last
minute to start this assignment. This problem is a variation on problems covered in lecture
and in previous years’ homework assignments. You can get a good idea from these old
problems of how to proceed, but you will need to do this from scratch to get the right
answer.
1. Consider the following potential in one dimension: V (x) = 0 for x < −L, V (x) = Vo >
0 for −L < x < L, and V (x) = Vo /2 for x > L. Assume that the the energy of an
incoming electron, E, in the range Vo > E > Vo /2.
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 eρ2 x + A′2 e−ρ2 x
φ(L < x) = A3 eik3 x + A′3 e−ik3 x .
(a) What are k1 , ρ2 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 E/Vo , which is between 1/2 and
1.