PHY 4604 Fall 2009 – Homework 4
Due by 5:00 p.m. on Monday, October 19. Please turn in your homework in class
before the deadline, or else bring it to NPB 2162. (You may push it under the
door if NPB 2162 is unoccupied.) No credit will be available for homework submitted
after the start of class on Friday, October 24.
Answer all three questions. Please write neatly and include your name on the front page of
your answers. You must also clearly identify all your collaborators on this assignment. To
gain maximum credit you should explain your reasoning and show all working.
You may find useful the following:
cos2 x + sin2 x = 1
2 sin x cos x = sin 2x
cos2 x − sin2 x = cos 2x
sin(x ± y) = sin x cos y ± cos x sin y
1. Consider a piecewise constant potential V (x).
(a) Show that in any region where V (x) = Vj = constant, the general stationary state
ψ(x) = Aj exp(ikj x) + Bj exp(−ikj x) can be rewritten in the equivalent form
ψ(x) = Cj cos kj x + Dj sin kj x.
(1)
Provide expressions for Cj and Dj in terms of Aj and Bj .
(b) Writing C = |C|eiγ and D = |D|eiδ where γ and δ are real, find an expression
for the probability density ρ(x) in the stationary state specified in Eq. (1). You
should find that ρ(x) varies sinusoidally with x, reaching a maximum value
o
p
1n 2
ρmax =
|C| + |D|2 + |C|4 + |D|4 + 2|C|2 |D|2 cos[2(γ − δ)] .
2
(c) Using wave functions of the form of Eq. (1), write down the boundary conditions
on the wave function at x = 0, where V (x) undergoes a finite jump from V1
to V2 < E. Assume that V1 < V2 < E, and hence the wave numbers satisfy
k1 > k2 . Express your results both (i) as relations between the (generally) complex
amplitudes C1 , D1 , C2 , and D2 , and (ii) as relations between the real quantitities
|C1 |, γ1 , |D1 |, δ1 , |C2 |, γ2 , |D2 |, and δ2 .
(d) Show that for the situation considered in part (c), the maximum probability density ρmax,2 in the region where V (x) = V2 is greater than or equal to the maximum
probability density ρmax,1 in the region where V (x) = V1 . (Assume that each region of constant potential is wide enough for the probability density to reach the
maximum of its sinusoidal oscillation.) Under what special circumstances does
ρmax,2 = ρmax,1 ?
Note: The result ρmax,2 > ρmax,1 that holds generally (i.e., when the special circumstances above do not apply) agrees with the conclusion that one draws by
considering the relative probability densities for finding a classical particle in each
region. This agreement justifies use of the classical argument even for situations
that are far from the classical limit.
2. A particle of mass m moves in the one-dimensional potential
V (x) = −V0 aδ(x),
where V0 > 0 is an energy and a > 0 is a length scale. The bound state wave function
for this problem was discussed in class; see also Eq. (2.129) of Griffiths (where α ≡ V0 a).
(a) Find hV i, the expectation value of the potential in the bound state.
(b) Find hT i, the expectation value of the kinetic energy in the bound state, where
T̂ = −(~2 /2m)∂ 2 /∂x2 . Verify that hT i + hV i = E, the energy of the bound state.
Hint: In order to obtain the correct value for hT i, you must take into account the
fact that the slope of the wave function undergoes a jump at x = 0.
3. A particle of mass m moves in the one-dimensional potential
V (x) = V0 Θ(x) − V1 aδ(x),
where V0 and V1 are positive energy scales, a > 0 is a length scale, and Θ(x) is the
Heaviside or step function:
0
for x < 0,
Θ(x) =
1
for x > 0.
This potential describes a delta-function well located right at a jump in the background
potential.
(a) Write down the form of wave function for a bound state of energy E < 0 in the
regions x < 0 and x > 0. You may leave in your answer any unknown amplitude
that may have a nonzero value in a physically acceptable state. You should define
any other symbol that doesn’t appear in the statement of the problem.
(b) By applying the appropriate boundary conditions, obtain an equation relating the
bound-state energy E to other quantities defined above. Express your answer in
the form f (E) = constant.
(c) Use your answer to (b) to determine the range of V1 over which a bound state
exists.
(d) Find the bound-state energy E in closed form. You should be able to eliminate
all square roots from the equation you found in (b) by squaring both sides, then
carrying some terms from one side to the other side before squaring again. As
a check, for V0 = 0, the energy should reduce to E = E1 = −ma2 V12 /(2~2 ), the
standard result for a pure delta-function potential V (x) = −aV1 δ(x).
(e) Now consider stationary state wave functions of energy E > V0 . Apply the appropriate boundary conditions to construct the wave function describing a rightward
moving particle incident from the far left, plus any reflected and transmitted
products.
(f) Find the transmission probability T (E) for E > V0 .