Take
A
to be real with

A


E
0

.
(a) (25 points) Find the exact eigenvalues of
H
. What are the corresponding normalized
eigenstates in terms of the up state
(
1
0
)
and the down state
(
0
1
)
?
(b) (25 points) Next suppose that the molecule is placed in an electric field that distin
guishes the up and down states. The new Hamiltonian is
H
=
H
0
+
H
0
+
H
00
, where
H
00
=
1
0
0
2
with
1
6
=
2
. What are the exact energy eigenvalues of the new Hamiltonian
H
?
(c) (20 points) When

1

,

2


A

, what answer does timeindependent perturbation
theory give for the energies of
H
to lowest nonvanishing order in
i
? Compare your result
to the exact answer.
(d) (30 points) When

A


1

,

2

,

2

1

, what answer does timeindependent per
turbation theory give for the energies of
H
to lowest nonvanishing order in
A
? Compare
your result to the exact answer.
3
QUALIFYING EXAMINATION, Part 4
2:00 pm – 4:30 pm, Friday August 30, 2019
Attempt all parts of both problems.
Please begin your answer to each problem on a separate sheet, write your 3 digit code
and the problem number on each sheet, and then number and staple together the sheets
for each problem. Each problem is worth 100 points; partial credit will be given.
Calculators and cell phones may NOT be used.
1
Problem 1: Statistical Mechanics I
Consider a noninteracting classical twodimensional (2D) gas of
N
nonrelativistic iden
tical particles confined in a harmonic trap with hard walls at
r
=
R
V
trap
(
r
) =
1
2
mω
2
r
2
for
r < R
∞
for
r
≥
R
,
where
m
is the particles mass,
ω
is the trap angular frequency and
r
is the radial coordi
nate.
(a) (20 points) Show that the density distribution
n
(
r
) of this gas in thermal equilibrium
at temperature
T
is
n
(
r
) =
Ce

βV
trap
(
r
)
, with
β

1
=
k
B
T
. Compute
C
such that
n
(
r
) is
normalized as
R
n
(
r
)d
2
r
=
N
. Sketch
n
(
r
)
/n
(0) as a function of
r/R
.
Next, this gas is set to rotate at a fixed angular velocity Ω around the
z
axis, defined as
perpendicular to the gas plane and intersecting it at the center of the trap
r
= 0. The ther
modynamics of a noninteracting gas in equilibrium in the rotating frame can be calculated
by replacing the singleparticle Hamiltonian in the lab frame
h
lab
=
p
2
/
(2
m
)+
V
trap
(
r
) by
h
rot
=
h
lab

Ω
·
L
, where
Ω
= Ω
ˆ
z
(
ˆ
z
is the unit vector along the
z
axis) and
L
=
r
×
p
is the angular momentum of the particle (note that
L
is collinear with
ˆ
z
here).
(b) (25 points) Show that
h
rot
can be written in the form
h
rot
=
1
2
m
(
p

m
Ω
×
r
)
2
+
V
eff
(
r
),
where
V
eff
(
r
) =
1
2
m
(
ω
2

Ω
2
)
r
2
for
r < R
(and
∞
otherwise), and interpret this result.
(c) (20 points) Suppose the rotating gas is in equilibrium at a temperature
T
. Show that its
density distribution is
n
Ω
(
r
) =
De

βV
eff
(
r
)
, and calculate
D
to normalize this distribution
as in (a). For simplicity, express your result in terms of the parameter
α
=
m
(
ω
2

Ω
2
)
2
k
B
T
R
2
.
Sketch
n
Ω
(
r
)
/n
Ω
(0) as a function of
r/R
for slow (Ω
ω
), fast (Ω =
ω
), and ultrafast
rotation (Ω
ω
).
(d) (15 points) Show that the canonical partition function of the classical gas has the
form
Z
=
Z
0
(1

e

α
)
N
where
Z
0
is a proportionality factor independent of
R
(you do
not need to calculate
Z
0
).
(e) (20 points) Using the partition function in (d), calculate the force
F
=
k
B
T
∂
log
Z
∂R
exerted by the gas on the disk walls, and thus the pressure
P
=
F
2
πR
. Express your results
for
PπR
2
Nk
B
T
as a function of
α
.
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