PHY 6645 Fall 2002 – Final Exam
Instructions: Attempt all four questions, each of which is worth 25 points. The maximum
score for each part part of each question is shown in square brackets. To gain full credit you
should explain your reasoning and show all working. Please write neatly and remember to
include your name on the front page of your answers.
Please read carefully: During this exam, you may use Shankar’s Principles of Quantum
Mechanics and lecture notes from this course. You are not permitted to consult any other
books, notes, or papers, or to communicate with anyone other than the proctor. In accordance with the UF Honor Code, by turning in this exam to be graded, you affirm the
following pledge: On my honor, I have neither given nor received unauthorized aid in doing
this assignment.
1. Consider a particle of mass m moving in the one-dimensional potential


 +∞
V (x) =
0


+∞
x ≤ 0,
0 < x < a,
x ≥ a.
Suppose that the particle is in the n’th stationary state (where n = 1 is the ground
state and En+1 > En ).
(a) [12 points] Calculate the position uncertainty ∆X in this state.
(b) [8 points] Calculate the momentum uncertainty ∆P in this state.
(c) [5 points] Find ∆X∆P , and verify that it obeys the necessary bound.
You may find the following useful:
Z
dx xj sink x =
xj−1 sink−1 x
(j sin x − kx cos x)
k2
k−1Z
j(j − 1) Z
+
dx xj sink−2 x −
dx xj−2 sink x.
k
k2
2. This question concerns a one-dimensional harmonic oscillator described by the Hamiltonian H = P 2 /2m+mω 2 X 2 /2, where X is the position operator and P is the momentum
operator.
The two parts of this question are independent of one another.
(a) [10 points] Show that the Heisenberg equation of motion for X 2 can be written
dX 2
{X, P }
=
,
dt
m
where {Ω, Λ} is the anticommutator of Ω and Λ.
(b) [15 points] Suppose that at time t = 0, the harmonic oscillator is in the state
1
|ψ(0)i = √ (|0i − |2i) ,
2
where |ni is the n’th eigenket of the number operator. Calculate hX 2 i in the
time-evolved state |ψ(t)i.
3. A particle of mass m is subject to a spherically symmetry potential V = −V0 r0 δ(r−r0 ),
where V0 and r0 are positive quantities.
Consider a bound state of angular momentum l = 0, described by a wave function
ψE00 (r, θ, φ) = (4π)−1/2 UE0 /r with E < 0.
(a) [4 points] Write down the general form of UE0 (r) in the region 0 ≤ r ≤ r0 .
Hint: Here and in (b), don’t bother to use solutions of the spherical Bessel equation. Express your answer directly in terms of elementary functions.
(b) [4 points] Write down the general form of UE0 (r) in the region r ≥ r0 .
(c) [10 points] Apply the appropriate boundary conditions at r = r0 to obtain an
equation that determines the allowed value(s) of E.
(d) [7 points] Show that the equation found in part (c) has at most one physically
acceptable solution, and find the minimum value of V0 for which a bound state
exists.
4. A simple model of glassy behavior in an amorphous solid focuses on “tunneling centers,”
each of which consists of an atom that can tunnel quantum mechanically between two
sites, labeled 1 and 2. Suppose that occupancy of site j is described by a ket |ji,
where |1i and |2i form an orthonormal basis for a two-dimensional vector space. The
Hamiltonian for a single tunneling center can be written in the basis {|1i, |2i} as
−ε −0.1iε
0.1iε
ε
H=
!
,
where ε is an energy and i is the square-root of −1. Let us also define two other
operators in the same basis:
N1 =
1 0
0 0
!
,
and P =
0 ed
ed 0
!
,
which represent, respectively, the occupation number operator for site 1 and the electric
dipole moment operator. Here, e is the charge of an electron and d is a (real) length.
A certain sample contains millions of tunneling centers. You may assume that all the
tunneling centers have identical parameters ε and d, and that they are subject to the
same preparation procedure. A series of experiments reveal that the prepared state
has the following expectation values: hHi = 0.86ε, hN1 i = 0.1, and hP i = 0.
(a) [20 points] Use the information provided above to find the density matrix for the
ensemble of prepared states. Express the density matrix in the basis {|1i, |2i}.
(b) [5 points] Deduce whether the prepared state is pure or mixed.