PHY 6645 Fall 2001 – Homework 6
Due at the start of class on Thursday, October 25. No credit will be available
for solutions submitted after 4 p.m. on Friday, October 26.
Answer all questions. To gain full credit you should explain your reasoning and show all
working. Please write neatly and include your name on the front page of your answers.
1. Starting from
*
+
d
∂Ω
i
,
hΩi = h[H, Ω]i +
dt
h̄
∂t
and defining the operator
J = {X, P } − 2hXihP i,
where {X, P } is the anticommutator of X and P , show that the following equations
are valid for the time evolution of any state of a quantum harmonic oscillator:
d
hXi
dt
d
hP i
dt
d
hX 2 i
dt
d 2
hP i
dt
=
1
hP i,
m
= −mω 2 hXi,
=
1
h{X, P }i,
m
= −mω 2 h{X, P }i,
d
1
(∆X)2 =
hJi,
dt
m
d
(∆P )2 = −mω 2 hJi,
dt
d
2
hJi =
(∆P )2 − 2mω 2 (∆X)2 .
dt
m
Note that the first two properties correspond to the classical equations of motion for
the dynamical variables x and p.
2. Consider a harmonic oscillator wave function that at time t = 0 has the form of a
Gaussian wave packet,
"
eiθ
ip0 (x − x0 )
x − x0
ψ(x, 0) =
exp
−
2
1/4
(2πw )
h̄
2w
2 #
,
where w, θ, x0 , and p0 are all real parameters.
(a) Evaluate hXi, hP i, ∆X, and ∆P in the state ψ(x, 0), and confirm that it is a
minimum-uncertainty state.
(b) Evaluate dn (∆X)2 /dtn and dn (∆P )2 /dtn in the state ψ(x, 0), where n is a positive
integer. Hint: Apply some of the equations that you proved in the previous
problem.
(c) Use your results from part (b) to show that the state ψ(x, t) q
that evolves from
ψ(x, 0) is also a minimum uncertainty state provided that w = h̄/2mω.
3. Consider a coherent state of the harmonic oscillator,
|zi = exp(za† − 12 |z|2 )|0i,
where |0i is the ground state, a† is the creation operator, and z is any complex number.
Coherent states have many remarkable properties, but this question focuses on their
position and momentum uncertainties.
(a) Express |zi as a sum over all eigenstates |ni of the harmonic oscillator, and confirm
that |zi is normalized to unity.
(b) Suppose that a harmonic oscillator is initially in state |ψ(0)i = |zi. Using the
representation of |zi derived in part (a), find |ψ(t)i. Simplify your answer as much
as possible.
(c) Evaluate hXi, hP i, ∆X, and ∆P in the state |zi. Use the number basis (rather
than the position basis) to obtain your answers.
(d) Demonstrate that |zi is an eigenket of the annihilation operator a. Use this fact
to derive the wave function ψz (x) = hx|zi. Show that your result is a special case
of the initial wave function of the previous problem.
4. Shankar Exercise 9.4.4.
5. Consider the operators Lx , Ly , and Lz defined on Shankar Exercise 4.2.1.
(a) Show that these operators satisfy the commutation relations
[Li , Lj ] = iεijk Lk ,
where εijk is the Levi-Civita symbol (defined, e.g., on page 319 of Shankar).
(b) Calculate ∆Lx , ∆Ly , and ∆Lz in the states


1
|ψ1 i =  0  ,
0


0
|ψ2 i =  1  .
0
(c) Calculate ∆Lx ∆Ly in the states |ψ1 i and |ψ2 i defined above, and check that the
appropriate uncertainty relation ∆Ω ∆Λ ≥ 12 |h[Ω, Λ]i| is obeyed in each case.