PHY 6646 - Quantum Mechanics II - Spring 2013
Homework set #8, due March 1
1. Problem 16.2.7 in Shankar’s book.
2. A particle of mass µ is in linear motion in the potential
V (x) = 5kx
= −kx
for x > 0
for x < 0
.
(0.1)
with k positive. Use the WKB approximation to estimate the energies of the states of the
particle.
3. A particle of mass µ is in linear motion in a potential V (x) with the following
properties: V (x) is continuous everywhere except x = 0, goes to zero when x → −∞, jumps
from V− at x → 0− to V+ at x → 0+ , and goes to V∞ when x → +∞. The particle is
incident on the potential from the left with energy E larger than the maximum value of
V (x). Assume the WKB approximation is valid everywhere except at x = 0.
a. Draw a graph of such a potential.
b. In the neighborhood of x = 0, the wavefunction has the form
Ψ(x) = Aeik− x + Be−ik− x
= Ceik+ x
for x < 0
for x > 0
(0.2)
to leading order in an expansion in powers of h̄. Justify this form of the wavefunction near
x = 0 and give expressions for k− and k+ . What jump conditions must Ψ(x) satisfy at
x = 0? What relations between A, B and C are implied by the jump conditions?
c. What are the reflection and transmission coefficients of the potential barrier V (x) taken
as a whole? (Hint: The reflection and transmission coefficients do not depend on V∞ .)
4. Problems 17.2.2 and 17.2.4 in Shankar’s book.
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