Classical Mechanics Qualifier (Fall 2009)
George Mason University
You will have two hours to complete all of the following problems.
Short Answers (4 x 5 pts = 20 pts):
S1 (5 pts). The kinetic energy and the potential energy of a spherical pendulum can be
written in terms of the generalized coordinates  and  as:


1
m l 2 2  l 2 sin 2  2
2
U   mgl cos 
where m is the mass and l is a length of the pendulum. Are either  and/or cyclic?
What are the conserved quantities for this system?
T
S2 (5 pts). Planet X is orbiting its Star in a circular orbit. If the Star’s mass suddenly
decreases by half, what orbit will Planet X now have? Will Planet X still be bounded to
the Star?
S3 (5 pts). A particle of mass m is moving under the influence of a central force given by
f (r )  kr  where k and  are positive constants. Using the plane polar coordinates (r, )
as your generalized coordinates, find Hamilton’s equations of motion.
S4 (5 pts). A long thin cylindrical rod with length l and mass m rotates around a fixed
axis with frequency  as shown. Find the torque (in the body axes) with respect to O
(CM of the rod) required to maintain the motion around ω .
ω
v

l
O
Problem (35 pts):
A disk of mass m and radius r rolls without slipping inside a circular opening, of radius
3r, within a block of mass M. The block slides without friction on a horizontal surface.
(Take U = 0 when the disk is at the bottom of the well.)
g
3r

contact point
when at bottom
r m
current contact point
(disk rolls without
slipping)

no friction
M
X
a) (5 pts) Write down the Lagrangian for this system using the generalized
coordinates (X, , and ) indicated in the illustration above.
b) (5 pts) Write down the constraint condition for the disk rolling without slipping
inside the circular opening.
c) (10 pts) Obtain the equation of motion for the generalized coordinates (X and ).
d) (10 pts) Assuming small angular deviations and X and  to be small, find the
frequency of small oscillations of the disk inside the block.
e) (5 pts)If M is not allowed to move, what will the frequency of small oscillations
be?
Classical Mechanics Qualifier (Fall 2010)
George Mason University
You will have THREE hours to complete the exam. Choose three out of
the following four problems. You are allowed to use your graduate
textbook during the exam.
Problem 1 (20pts)
A particle of mass m is constrained to move under the
influence of gravity on the inside of a smooth parabolic
surface of revolution given by r 2  az . Use the Lagrange
undetermined multiplier method to derive the constraint
force for this problem. Write your answer as a vector in
cylindrical coordinates. (Hint: You might want to use the
two constants of motion E and l to simplify some of your
expressions. The magnitude of the constraint force is
 4r 2 
proportional to 1  2 
a 

z
z
z
3/2
x
.)

r2
a
y
r
Problem 2 (20pts)
A distant star is surrounded by a dust cloud. A planet at a distance r away moves under
the influence of the star with the familiar inverse-square potential V0 (r )   k r and the
dust cloud contributes a small additional factor V '(r )  ar 2 2 . The planet is observed to
revolve around the star in a nearly circular orbit with an average radius r0 .
a) For the given central force potential
k 1
V (r )  V0 (r )  V '(r )    ar 2
r 2
show that the radius for the circular orbit r0 is given by the following expression:
l2
m
where m is the reduced mass of the system and l is the angular momentum of the
system.
b) Consider the observed orbit as a small deviation from this circular orbit, show that
3a
the apsides will advance approximately by
per revolution, where 0 is the
m0 2
angular frequency for the circular orbit.
r0  k  ar03  
Problem 3 (20pts)
Let I1 , I 2 , I 3 be the three principal moments of inertia relative to the center of mass of a
rigid body and suppose that all these moments are different and they are ranked according
to I1  I 2  I 3 . The rigid body is set to spin around one of its principle axes in free space
(with no external force) with an angular velocity ω . Show that the motion is stable if the
object is spinning about the principal axes corresponding to I1 and I 3 (the largest and the
smallest moments of inertia) and unstable about the principal axis corresponding to I 2 .
Explain this analytically using the Euler’s equations.
Problem 4 (20pts)
A particle of mass m described by one generalized coordinate q moves under the
influence of a potential V(q) and a damping force 2m q proportional to its velocity.
a) Show that the following Lagrangian gives the desired equation of motion.
1

L  e 2 t  mq 2  V (q ) 
2

b) Obtain the Hamiltonian H(q,p,t) for this system.
c) Consider the following generating function:
F ( p, q, Q, P, t )  e t qP  QP
obtain the canonical transformation from (q, p) to (Q, P) and the transformed
Hamiltonian K(Q, P, t).
1
d) Pick V (q )  m 2 q 2 as a harmonic potential with a natural frequency  Show
2
that the transformed Hamiltonian yields a constant of motion
P2 1
 m 2Q 2   QP
K
2m 2
e) Obtain the solution Q(t) for the damped oscillator in the under-damped case
   by solving Hamilton’s equations in the transformed coordinates. Then,
write down the solution q(t) using the canonical transformation obtained in part c.
Classical Mechanics Qualifier (August 2012)
George Mason University
You will have THREE hours to complete the exam.
You are allowed to use your graduate textbook during the exam.
Short Problems – do both (20 points each):
Problem 1
A uniform thin rod of length l and mass m is initially placed with one end on a
frictionless table making an angle ! 0 as shown. i) Write down the Lagrangian for
the rod. ii) Determine the time it takes for the rod to fall to the table after it is
released from its initial slanted position. (The answer can be left as a definite
integral.) iii) How far will the lower end of the rod move horizontally during this
time?
)
Problem 2
p2
1
! 2 . Write down the
2 2q
pq
! Ht is a
Hamilton’s equation of motion for this system. Show that F =
2
constant of motion for this system. Here t is time.
A system is described by the Hamiltonian, H =
Long Problems – do two of the following three (30 points each):
Problem 3
A bobsled is sliding from an initial height h0 on an ice
track with a shape given by a cubic function
x3
h (x ) = x ! 2
3a
parameterized by a parameter a as shown. Use the method
of the Lagrangian multiplier to find the algebraic equation
for the horizontal location x at which the bobsled will
lose contact with the track. If the bobsled loses contact
with the track at x = a , what is the initial height h0 in
terms of the system parameter a?
Problem 4
Two pendula with equal mass m and length l are connected by a spring with a spring
constant k. The unstretched length of the spring is equal to the distance L between the
support points of the two pendula. Considering small oscillations along the line between
the two masses (consider only stretching of the spring horizontally and ignore the small
vertical stretching), find the eigenfrequencies (resonant frequencies) and the normal
modes for the system.
Problem 5
A spaceship lost its power in deep space (far
away from any stars). The spaceship was
originally spinning along one of its principal axes
with a period T. A small perturbation to the
angular velocity was seen to grow exponentially
in time, "# (t ) ~ e!t . Assume the spaceship to
have a uniform density ! , a total mass M , and
an ellipsoidal shape with three semi-principal
axes a < b < c (see figure).
i) Calculate the Principal Moment of Inertia for the ellipsoid with respect to its center of
mass. [Hint: If one rescales the axes, x = au , y = bv, z = cw , one can change the ellipsoid
to a unit sphere for the integration, i.e.,
4!
V = " dxdydz = " (abc )dudvdw =
abc ]
3
ellipsoid
sphere
ii) Around which axis (xˆ , yˆ , zˆ ) was the spaceship originally spinning?
iii) What is the time constant ! in terms of the parameters: a, b, c, T ?
Classical Mechanics Qualifier (January 2012)
George Mason University
You will have THREE hours to complete the exam.
You are allowed to use your graduate textbook during the exam.
Short Problems – do both (20 points each):
Problem 1
A mass m is constrained to slide frictionlessly on the surface obtained by rotating
around the z axis. A is a constant greater than zero. Gravity is in
the curve
the negative z direction.
a) Using the method of Lagrangian multipliers, write down the Lagrangian
for the mass using the cylindrical coordinates  r , , z  and find the equations of
motion.
b) Identify any cyclic coordinates. What is the corresponding conserved
quantity?
c) Eliminate the constraints and the cyclic coordinate(s) to obtain a single
equation of motion in a single variable. You do not need to solve this equation.
Problem 2
Consider the rigid object formed by the four point masses (m, 2m, 3m, and 4m)
placed at the corners of a square of side 2a, as shown.
a) Find the moment of inertia tensor of the system with respect to the origin (not
with respect to the center of mass).
b) Find the principal axes and principal moments of inertia, again with respect to
the origin.
Long Problems – do two of the following three (30 points each):
Problem 3
A physical system with two degrees of freedom is described by the following
Hamiltonian,
2
 p  p2 
2
H  1
  p2   q1  q2 
 2q1 
a) Are there any constants of motion for the system described by this
Hamiltonian?
b) Let f1  q1 , q2  , f 2  q1 , q2  , and g  q1 , q2  be three smooth functions of the
generalized coordinates  q1 , q2  . Use the following generating function,
F2  q1 , q2 , P1 , P2  
 f q , q  P  g q , q 
i 1,2
i
1
2
i
1
2
to calculate the resulting canonical transformation in terms of f1, f2, and g:
Q1  Q1  q1 , q2 , p1 , p2  , P1  P1  q1 , q2 , p1 , p2 
Q2  Q2  q1 , q2 , p1 , p2  , P2  P2  q1 , q2 , p1 , p2 
c) Let f1  q1 , q2   q12 and f 2  q1 , q2   q1  q2 and find an expression for
g  q1 , q2  such that Q1 and Q2 are cyclic in the transformed Hamiltonian K?
d) What are the constants of motion in this transformed Hamiltonian K?
e) Solve Q1  t  and Q2  t  as functions of time explicitly using Hamilton’s
equations.
Problem 4
, where k
Consider the orbits of a mass m in a central inverse-cube force,
is a positive constant. Solve the radial equation of motion for r as a function of
the angle,  . You will need to consider three cases:
a) Large angular momentum:
√ .
b) Small angular momentum:
√ .
c)
√ .
For each case, describe the orbits and state whether they are bound or unbound.
Identify any asymptotes of the orbits.
Problem 5
Three beads with equal mass m slide frictionlessly on a thin hoop of radius R.
They are connected by springs that wrap around the hoop. The springs all have
force constant k. (The springs also slide frictionlessly around the hoop.) When
the beads are equally spaced around the hoop, the springs are unstretched. There
is no gravity.
a) Write the Lagrangian of the system using the angles θ1, θ2, and θ3, which are
measured from equally spaced, but arbitrary, positions around the ring 120
apart.
b) Write down the matrix you would use to find the resonant frequencies.
3k
c) The resonant frequencies are   0 and  
. (You do not need to
mR 2
show this.) What are the corresponding normal modes? (Note: the non-zero
frequency is degenerate (a double root). It is sufficient that you find two
independent eigenvectors for this frequency and they do not need to be
orthogonal to each other.)
d) Give a physical interpretation of each of the normal modes found in (c).
Qualifying Exam for PHYS 685: Dec 12, 2008
1. [25] A thin spherical shell with radius a is centered on the origin. The “Northern Hemisphere”
(i.e., 0 ≤ θ < π/2) carries electric charge area density σ0 and the “Southern Hemisphere” (i.e.,
π/2 < θ ≤ π) carries electric charge area density −σ0 .
a) [15] Find the electrostatic potential Φ(r, θ) for r > a as a series involving powers of r and Legendre
R1
polynomials. Note: Your answer should include a factor 0 du Pl (u). You need not evaluate this
integral.
b) [5] Find the electric dipole moment p~ of the shell.
c) [5] Show that your results in parts (a) and (b) are consistent.
2. [25] A thin disk of radius a lies in the x − y plane with its center at the origin. It carries an
electric charge area density
5r
σ0 1 −
4a
and is spinning about ẑ with angular speed ω.
a) [10] Find its magnetic dipole moment m.
~
~
b) [15] Find the magnetic induction B(z)
along the z-axis for z a.
3. [10] Two square metal plates of side length L are separated by a distance d (d L). A dielectric
slab of size L × L × d just slides between the plates. It is inserted a distance x (with one side of the
dielectric slab parallel to one side of the metal plates) and held there. The metal plates are then
charged to a potential difference V and disconnected from the battery. Find the electric force on
the slab.
4. [10] A rectangular loop of wire with non-zero resitance is turned through 180◦ in a region with
static, uniform magnetic induction. Show that the total charge transported through the loop as it
is flipped is independent of the speed of flipping.
Classical Electrodynamics Qualifying Exam: August 25, 2009
1. [15] Consider the electrostatic potential Φ on the x-y plane in the region y > 0, with
Φ(x, y = 0) = V cos(2πx/L) and Φ → 0 as y → ∞. Find Φ(x, y) when y > 0.
2. [15] A point electric dipole with moment p ẑ is located at the origin. A grounded, conducting,
spherical shell has radius a and is centered on the origin. Find the electrostatic potential Φ for
r < a.
3. [10] A sphere with radius a and made of a linear dielectric material with dielectric constant ǫ/ǫ0
is placed in a region where there is initially a uniform electric field E0 ẑ. As a result, the potential
inside the sphere becomes (in spherical coordinates)
Φ(r, θ) = −
3ǫ0
E0 r cos θ
ǫ + 2ǫ0
Find the bound charge volume density inside the sphere and the bound charge surface density on
its surface.
4. [20] A circular loop with radius a is centered on the origin, lies in the x-y plane, and carries a
current I.
a) [5] What is the magnetic dipole moment m
~ of the loop?
~
b) [15] Find the magnetic vector potential A(~x) in the limit that |~x| ≫ a.
5. [20] Consider a long, straight wire with circular cross section. Each end is held at constant
potential, with potential difference V . Although current may flow down the wire, the charge
density vanishes throughout.
a) [10] Show that the electric field inside the wire is uniform.
b) [10] Use the Poynting vector to show that heat is dissipated in the wire at rate V I when current
I flows.
Classical Electrodynamics Qualifying Exam: August 24, 2010
1. [30] A neutral conducting sphere with radius a is centered on the origin. A line charge with
uniform charge per unit length λ lies on the z-axis between z = b and z = c (a < b < c). Find the
electrostatic potential Φ(r, θ, φ) for r > c as a series involving Pl (cos θ) and powers of r.
2. [20] A dielectric cylinder has length L and radius a. The z-axis is the symmetry axis and the two
end faces are at z = 0 and z = L. The cylinder has a uniform polarization P ẑ. Find the electric
field on the z-axis within the cylinder.
3. [20] A cylinder has length L and radius a and carries a surface-charge density σ on its curved
face. (There is no charge on the end faces.) The z-axis is the symmetry axis and the two end faces
are at z = 0 and z = L. The cylinder spins about its axis with angular speed ω. Find the magnetic
~
induction B(z)
on the z-axis for z ≫ L.
4. [20] a) Show that a plane electromagnetic wave
~ =E
~ 0 ei(~k·~x−ωt)
E
;
~ =B
~ 0 ei(~k·~x−ωt)
B
,
~ 0 and B
~ 0 constants, satisfies Maxwell’s equations in vacuum if k̂ × E
~ = cB;
~ c is the speed of
with E
light.
b) Show that c2 ǫ0 µ0 = 1.
NAME:
QM, Qualifying Exam: 2012
Note: This is an open book exam and you are allowed to bring Sakurai or Shankar’s book on
quantum mechanics. If a formula appears in the book, please use that as a starting point,
there is no need to show the derivation of that formula.
(I) (10 points) Consider a particle of mass m, at absolute zero temperature, confined to a
rectangular box of sides (a, a, a/2).
(a)(5 pts) Using the uncertainty principle, derive the energy of the particle.
(b)(2 pts)How does your answer compare with the exact calculation?
(c)(3 pts)For the exact calculation, please give the first two energy levels and the degeneracy
associated with each level.
(II) (20 points) Consider a particle of unit mass in a potential V (x, y) = A {x2 + (y − 1)2 },
A > 0.
(a)(11 pts) Write down for the ground and first excited state: (1) energy (2) degeneracy and
(3) the wavefunction.
(b)(6 pts)What is the expectation value of position and momentum of the particle in the
ground state?
(c)(3 pts)If at an initial time t0 , the particle is equally likely to be in the first and second
excited state, what is the wave function at some later time t.
(III) (10 points) At a given instance in time, a free particle of mass m is described by a
Gaussian wave packet of width unity. The average position of the particle is 0.5 and average
momentum is p0 . In this problem you need not worry about normalization.
(a)(3 pts) What is the wave function of the particle?
(b)(4 pts) Is the wave function an eigenfunction of (1) position (2) momentum (3) energy?
Explain why or why not.
(c)(3 pts) Under what condition does this wave function describes an eigenfunction of a
harmonic oscillator?
(IV) (20 points) Consider a particle in a potential V (r) = 1/(r2 + 1). The particle is in
an eigenfunction of the Hamiltonian such that the square of the net angular momentum is
20h̄2 .
(a)(5 pts) What is the most general angular part of the wave function?
(b) If in addition it is in an eigenstate of Lz with quantum number m = 2 calculate:
1)(4 pts)the expectation value of L2x + L2y .
2)(5 pts)the expectation value of Lx Ly . Is Lx Ly an observable?
3)(6 pts)the expectation value of px and py . Hint: What is the commutation relation between
px and py with Lz ?
(V) (20 points) Suppose a particle has the wave function, ψ(x, y, z) = A(r)[1 + (2x + z)/r],
where A(r) is a radial wave function chosen so that ψ is normalized.
(a)(5 pts)What are the possible angular momentum quantum numbers of the system?
(b)(5 pts)What are the probabilities for the different possible outcomes of the measurement
of Lz ?
(c)(5 pts)What is the expectation value of Lz ?
(d)(5 pts)What is the rms uncertainty in Lz ?
2
George Mason University
Physics PhD Qualifying Exam
8 December 08
Instructions: Answer all the questions. Writing must be legible to get credit.
1. The Hamiltonian operator describing a certain quantum mechanical system has a
matrix representation:
!
1 −1
H = E0
−1 1
H has eigenvalues 2E0 and 0.
Another operator, A, has the representation
2 1
1 0
A=a
A has eigenvalues (1 +
√
2)a and (1 −
√
!
2)a.
(a) Find the eigenvectors of H and A in this representation.
(b) Suppose that a measurement of the energy of the system is made, and the
result is 2E0 . The
√ quantity A is√then measured. Calculate the probabilities of
the results (1 + 2)a and (1 − 2)a.
2.
A particle of mass m is in a 1-dimensional infinite square well, of total width a, as
V
a
0
x
shown in the figure. At time t = 0 its normalized wave function is:
ψ(x, t = 0) =
q
8/5a[1 + cos (πx/a)] sin (πx/a)
(a) Which energy levels are occupied for this wave function?
(b) What is the wave function at some later time, t?
(c) What is the average energy at t = 0, and later at t?
1
3.
δ(x)
V
I
II
x
0
A particle represented by a plane wave is incident, from the left, on a 1-dimensional
potential barrier:
V = V0 δ(x)
Calculate the probability that the particle is reflected from the barrier. Use the
continuity of the wave function at x = 0 and following relation for the discontinuity
in the derivative of the wave function, at a δ-function:
∆(
dψ
2mV0
ψ(0)
)=
dx
h̄2
4. A spinless particle is represented by a wave function
ψ(x, y, z) = Kxe−αr
√
where K and α are real constants and r = x2 + y 2 + z 2 By writing x in terms of
spherical harmonics,
(a) What is the total angular momentum of the particle?
(b) What is the expectation value of Lz ?
(c) What is the probability that the particle’s θ is less than 45◦ ?
Some spherical harmonics:
Y00
Y10
=
s
s
1
4π
Y1±1
3
cos θ
4π
Y2±1
=
2
s
3
sin θe±iφ
8π
s
15
sin θ cos θe±iφ
8π
=∓
=∓
NAME:
QM, Qualifying Exam: 2009
Note: If a formula appears in the Shankar’s book, please use
that as a starting point, there is no need to show the derivation
of that formula.
(1) ( 20 points )
Consider a system whose Hamiltonian H and an operator C are
given by the following matrices:



H = ǫ




C = c

0 4 0
4 0 1
0 1 0
1 −1 0
−1 1 0
0 0 −1










where ǫ has the dimensions of energy.
(a) If we measure the energy, what values will we obtain?
(b) Suppose that when we measure energy, we obtain a value of
−ǫ. Immediately afterwords, we measure C. What values will
we obtain for C and what are the probabilities corresponding to
each value??
(2) ( 20 points)
Consider a system of three non-interacting particles ( each of
mass m ) confined in a two-dimensional potential V (x, y) =
(x2 + y 2 ). Calculate the total ground state energy and the
first excited state energy of the system if, (a) Particles are noninteracting neutrons. (b) Particles are non-interacting He atoms.
(c) Two particles are electrons and the other is a positron.
(3) (10 points)
Wave function of a particle moving in three dimension is ψ(r, θ, φ) =
Ae−r . Calculate the value of r where the particle is most likely
to be found.
(4) (20 points)
At t = 0, a particle of mass m is equally likely to be in the
ground and the first excited state of the system described by
V (x) = 2x2 for x > 0 , V (x) = ∞ at x = 0. What is the wave
function of the system? at t = 0. What is the wave function of
the particle at t = 1 sec.
(5) (15 points )
An electron moving in a harmonic potential V (x) = 2x2 is subjected to a constant electric field E. What is the ground state
energy and the ground state wave function of the system.
(6) (15 points)
A particle in a spherically symmetric potential is described by a
wave function, ψ(x, y, z) = A(x + z) where A is a normalization
constant. Calculate the possible angular momentum quantum
numbers of the system and the probability of being found in
those states.
2
NAME:
QM, Qualifying Exam: 2010
Note: This is an open book exam and you are allowed to bring ”
Principles of Quantum Mechanics” by Shankar. If a formula appears in the Shankar’s book, please use that as a starting point,
there is no need to show the derivation of that formula.
(1) ( 20 points )
Let ψ1 , ψ−1 and ψ0 are the eigenstates of the Lz operator with
eigenvalues 1, −1, 0 respectively. Consider a particle in a normalized state described by,
1
1
1
ψ = eiθ1 ψ1 + eiθ−1 ψ−1 + √ eiθ0 ψ0
2
2
2
where θi , (i = 1, −1, 0) are real constants.
(a) Calculate the probability of finding the particle in a state
with Lx = 0.
(b) Calculate the probability of finding the particle in a state
with Lz = 0.
(c) If at t = 0, measurement of Lz gives 1, what will the measurement of Lz give at a later time ??
(2) ( 20 points)
Consider a system of three non-interacting particles ( each of
unit mass ) confined in a two-dimensional square box of unit
dimensions. Calculate the total ground state energy and the
first excited state energy of the system if, (a) Particles are
non-interacting neutrons. (b) Particles are non-interacting He4
atoms. (c) Two particles are electrons and the other is a positron.
(d)Consider a composite object such as the H-atom. Will it
behave as a boson or fermion??
(3) (10 points)
Use uncertainty principle to obtain ground state energy of a particle of unit mass in a one-dimensional box of length unity. Compare the resulting answer with the actual value of the ground
state energy of the system.
(4) (20 points )
A particle is moving in a harmonic potential,
V (x) = 3(x2 + y 2 + z 2 ). Solve the problem in two different coordinate systems and write down the ground and the first excited
state wave functions in these two coordinates. Are the solutions
degenerate?? If yes, what is the degeneracy of the ground and
the first excited states ??
(5) (10 points)
2
If a particle is described by a wave function, ψ(x, y, z) = Ae−2r (x+
z) where A is a normalization constant. Calculate the possible
angular momentum quantum numbers of the system and the
probability of being found in those states.
(6) (20 points )
Consider a free particle , with initial wave function given by the
2
wave packet,
2
ψ(x, t = 0) = Aeix e−(x−2)
where A is a normalization constant and both x and t are dimensionless. (a)What is the mean position and mean momentum of
the particle at t = 0.??
(b)What is the mean position and mean momentum of the particle at t = 5.
(c)What is the wave function of the particle at t = 5??
(d) How does the answers to (a)-(c) change if ψ(x, t = 0) describes the ground state wave function of a harmonic oscillator.
3
Classical Mechanics Qualifier (August 2011)
George Mason University
You will have THREE hours to complete the exam.
You are allowed to use your graduate textbook during the exam.
Problem 1 (30pts)
Two beads with equal mass m connected by a
spring are restricted to slide on a circular hoop
with radius R as shown. The spring has an
equilibrium length of 2r0 and a force constant k .
r0 is assumed to be less than R. The hoop rotates
about the z-axis with a constant angular velocity ω.
For simplicity, we assume that the spring will
remain parallel to the xy-plane as the hoop spins.
(This is a reasonable assumption for two equal
masses starting with the same z coordinates.)
There is no gravity.
z
!
m
m
x
R
a) (8 pts) Using cylindrical coordinates (r , ! , z ) , show that the Lagrangian can be
written in the following one-dimensional form in terms of a single generalized
coordinate z:
1
L = µ (z ) z& 2 ! Veff (z )
2
where
µ(z) = 2m(1 " z 2 /R 2 ) "1 and
Veff (z) = 2k
(
R 2 " z 2 " r0
)
2
" m(R 2 " z 2 )# 2
!
b) (8 pts) Use the Euler-Lagrange equation to derive an equation of motion for the
system.
!
c) (14 pts) Use the effective potential Veff ( z ) (calculated in a) to determine the
equilibrium coordinates ze for the beads. Show that there exists a critical angular
velocity
!c =
r0
2k " r0 #
% 1 $ & = !0 1 $
m ' R(
R
where !0 2 " 2k m such that there are three possible equilibria for ! < !c
and only one possible equilibrium for ! > !c .
y
Problem 2 (30pts)
a) (15 pts) Use the Poisson brackets to find the values of ! and ! such that the
following transformation is canonical:
Q = q! cos (" p ),
P = q! sin (" p )
b) (15 pts) A physical system is characterized by a time-independent Hamiltonian
H 0 ( p, q ) with ( p, q ) as the conjugate pair of canonical variables for the system.
Now, the system is being perturbed by an external oscillating field so that the
Hamiltonian becomes:
H = H 0 ( p, q ) # ! q sin ("t )
where ! and ! are respectively the magnitude and angular frequency of the
periodic perturbation.
i.
Write down the Hamilton equation of motion for this system.
ii.
Given the following generating function:
!q
F2 (q, P, t ) = qP # cos ("t )
"
calculate the resulting canonical transformation (q,p)  (Q,P).
iii.
What is the new Hamiltonian K (Q, P ) in the new canonical variables?
Show that the Hamilton equation of motion in these new canonical variables
is in the standard form:
!K (P, Q )
Q& =
!P
!K (P, Q )
P& = "
!Q
Problem 3 (30pts)
“Jacks” is a childhood game involving metal pieces that can be thought of as six small
equal masses m connected by a set of orthogonal massless rods of length 2l as shown.
z'
m
z
s
!
2l
o
a) (8 pts) Choosing the origin O of the body axis as the contact point between the
jack and the ground, calculate the principal moments of inertia for the jack.
b) (8 pts) Using the three Euler’s angles, write out the components of the torque on
the jack due to gravity in the body frame if it is tilted slightly off the vertical as
shown above.
" g above) so that
c) (8 pts) If you spin the jack around the principal axis z’ (see figure
there is a steady precession around the vertical, what is the relation between the
spin angular velocity s, the precession rate Ω, and the angle θ between the z-axis
in the space frame and the z’–axis in the body frame?
d) (6 pts) Assuming the jack is spinning near the vertical ( ! << 1), show that the
precession is stable if the spin rate s satisfies the following condition:
3! 3 g
s>
+
2 2l !
Classical Mechanics Qualifier (January 2011)
George Mason University
You will have THREE hours to complete the exam.
You MUST COMPLETE Problems 1 & 2.
CHOOSE TO COMPLETE one of the two among Problems 3 &4.
You are allowed to use your graduate textbook during the exam.
Problem 1 (30pts)
Two masses m1, m2 = m are connected by two massless
rigid rods of length l as shown. The top end of the rod
is fixed at the origin O and m2 is constrained to move
along the z-axis. The whole system is rotating with a
constant angular frequency Ω about the z-axis.
a) (10 pts) Choosing a convenient generalized
coordinate, calculate the Lagrangian for the
system. Use the Euler-Lagrange equation to
derive an equation of motion for the system.
b) (10 pts) Find the three equilibrium states for this
system. Explicitly specify the condition or conditions under which these
equilibria can exist.
c) (10 pts) Examine the stability for these equilibria.
Problem 2 (30pts)
An orbit for an object in a central force problem is given by
1
r (θ ) =
, a, b > 0
a + b cos ( 3θ 2 )
a) (5 pts) Under what condition on a and b will the orbit be a bounded orbit?
b) (5 pts) What are the pericenter and apocenter for the bounded orbit?
c) (5 pts) What is the angular advance for the apsis from one apocenter to the next
apocenter for this bounded orbit?
d) (5 pts) Sketched this bounded orbit. Is the orbit closed?
e) (5 pts) Determined the central potential V(r) that produces this orbit.
f) (5 pts) What is the total energy E for the circular orbit in this system and what is
the radius for this circular orbit?
Problem 3 (20pts)
A torsion pendulum consists of a vertical wire attached to a mass m which may rotate
about the vertical axis. k is the torque constant for the wire. Consider three torsion
pendula consisting of identical wires from which identical homogeneous solid cubes are
hung. All cubes have side a. One cube is hung from the middle of a face, one from a
corner, and one from midway along an edge:
a) (10 pts) Calculate the moment of inertia tensor for a cube with side a with respect
to its center of mass.
b) (10 pts) What are the natural frequencies of oscillations for these three torsion
pendulua. What is the relation between the natural frequencies for these three
cases?
Problem 4 (20pts)
a) (5 pts)You are given the following transformation between two sets of phase
space variables (x,p) and (Q,P)
αp
Q=
P = β x2 ,
where α ,β are two real parameters
x
What is the condition on α,β so that this transformation is canonical?
For the remainder of this problem, suppose β = 1/ 2 .
b) (5 pts) You are given the Hamiltonian K(Q,P),
PQ 2
K ( Q, P ) =
+ kP
m
Find the Hamiltonian H(x,p) under this canonical transformation. What physical
system does this Hamiltonian represent?
c) (5 pts) Consider the following function:
u ( x, p, t ) = ln ( p + imω x ) − iωt
Use the Poisson bracket to show that u is a constant of motion for a onedimensional harmonic oscillator with natural frequency ω = k m .
d) (5 pts) What does this constant of motion u correspond to physically?
Classical Electrodynamics Qualifying Exam: August, 2011
1. [15] Charge q is uniformly distributed around a circular ring of radius a. The ring’s axis is the
z-axis and its center is located at z = b. Find the potential Φ(r, θ) in spherical coordinates as a
series involving Legendre polynomials in cos θ and powers of r.
2. [15] The region above the x-y plane (where z > 0) contains a linear isotropic dielectric with
dielectric constant ǫ1 /ǫ0 . The region below the x-y plane (where z < 0) contains a linear isotropic
dielectric with dielectric constant ǫ2 /ǫ0 . A point charge q is located on the z-axis at z = d.
a) [10] Find the electrostatic potential Φ(r, φ, z) in cylindrical coordinates everywhere in space.
b) [5] Find the bound charge surface density σb (r, φ) on the x-y plane.
3. [15] Consider a thick, hemispherical shell of ferromagnetic material. With the z-axis as the polar
axis for a spherical coordinate system, the shell occupies a ≤ r ≤ b, 0 ≤ θ ≤ π/2, and 0 ≤ φ ≤ 2π.
~ at the origin.
The shell has a magnetization Az ẑ, with A a constant. Find the magnetic field H
4. [15] A point charge q moves with constant velocity βcẑ and is at the origin at time t = 0. Find
~
the electric field at the origin E(t)
for time t > 0.
Classical Electrodynamics Qualifying Exam: August, 2012
1. [10] A thin charged disk of radius R has uniform area charge density σ.
Find the electrostatic potential Φ(z) along ẑ, its symmetry axis. Check that
your result makes sense when z ≫ R.
2. [10] A neutral, conducting sphere of radius a is placed in a uniform
external electric field E0 ẑ. Charge is induced on the sphere, modifying the
external field. Find the electrostatic potential Φ(r, θ) outside the sphere.
Adopt spherical coordinates with ẑ the polar axis.
3. [10] Show that, for a spherically symmetric charge distribution, all multipole moments beyond the monopole vanish. (Hint: Recall that the spherical
harmonics are orthogonal.)
2
4. [20] A paraboloidal surface z = r⊥
/r0 , extending from r⊥ = 0 to r⊥ = r0
(r⊥ and z are cylindrical coordinates), spins with angular velocity ω ẑ and
carries a surface-charge density
Q
σ(r⊥ ) = 2
r0
4r2
1 + 2⊥
r0
!−1/2
where ω, Q and r0 are constants.
a) [10] Find the magnetic dipole moment m.
~
~ θ, φ) in spherical coordinates, in the
b) [10] Find the magnetic induction B(r,
limit r ≫ r0 .
5. [10] Starting with Maxwell’s equations in vacuum (i.e., ǫ = ǫ0 and µ = µ0 ,
but free charges and currents are allowed), derive the wave equations satisfied
by the scalar and vector potentials in the Lorenz gauge.
1
Classical Electrodynamics Qualifying Exam: January 19, 2011
1. [30] A line charge with uniform charge density lies along the z-axis between z = 0 and z = b
and has total charge Q.
a) [10] Find an exact expression for the electrostatic potential Φ(r, z) in cylindrical coordinates.
√
b) [5] Show that your result in part (a) has the correct asymptotic form as r2 + z 2 /b → ∞.
c) [10] Find the potential Φ(r, θ) in spherical coordinates (r, θ, φ) as a series involving Legendre
polynomials and powers of r, for r > b.
d) [5] Show that your results in parts (a) and (c) are equivalent for observation points on the z-axis
with z > b. Recall the Taylor series
∞
X
xn+1
ln(1 − x) = −
.
n+1
(1)
n=0
2. [25] In this problem, you will find the electrostatic potential Φ(r, z) inside a circular cylinder
with radius a and height L, adopting cylindrical coordinates (r, φ, z). The bottom of the cylinder
is at z = 0 and the cylinder’s axis is the z-axis. The potential is zero on the surface at z = 0 and
on the curved surface and is a constant V0 on the surface at z = L.
a) [15] Show that the potential inside the cylinder can be expressed in the form
Φ(r, z) =
∞
X
n=1
An sinh
x z x r
0n
0n
J0
a
a
(2)
where x0n is the nth zero of the Bessel function J0 (x).
b) [10] Find the coefficients An . Recall that
d ν
[u Jν (u)] = uν Jν−1 (u)
du
(3)
3. [15] A sphere has radius a, is centered on the origin, and is made up of a uniform, linear dielectric
material with dielectric constant ǫ/ǫ0 . A point charge q is located at the origin. Find the surface
and volume bound charge densities.
4. [30] An infinite, conducting plane at z = 0 carries a uniform current per unit transverse length,
K ŷ.
~ everywhere outside of the plane.
a) [10] Find the magnetic induction B
b) [10] A second infinite, conducting plane at z = −d carries a uniform current per unit transverse
length, −K ŷ. Use the Lorentz force law to find the pressure that the first plane exerts on the
second. Is it attractive or repulsive?
c) [10] Repeat part (b), this time using the Maxwell stress tensor.
Classical Electrodynamics Qualifying Exam: January, 2012
1. [10] Show how to obtain the differential form of Gauss’s Law from the integral form using the
Divergence Theorem.
2. [20] An infinite, hollow, rectangular, conducting pipe runs along the z-axis and extends from
x = 0 to x = a and y = 0 to y = b. All the faces are grounded except the face at x = a, which
is held at constant potential V and insulated from the other faces. Find the potential Φ(x, y, z)
inside the pipe.
3. [20] Two spherical shells with radii a and b (a < b) are centered on the origin. The inner
shell is held at zero potential, while the outer shell is observed to be at potential Φ(r = b, θ) =
V0 (3 cos2 θ − 1) where V0 is a constant. Find the potential between the 2 shells (i.e., for a ≤ r ≤ b).
4. [30] a) [10] A circular loop of radius R lies in the x-y plane, is centered on the origin, and carries
a current I. The current flows counterclockwise when viewed from above the x-y plane (z > 0).
~ on the axis of the loop.
Find the magnetic induction B
b) [20] A sphere of radius a is centered on the origin and rotates with angular velocity ωẑ. The
sphere carries electric charge with surface-charge density σ(θ) = σ0 sin2 θ, where θ is the angle with
~ along the z-axis in the limit z ≫ a. Hint: Use the
respect to ẑ. Find the magnetic induction B
result from part (a).
Quantum Mechanics Qualifying Exam
August 2011
1. For a complex number z, define the so-called coherent state
2
|zi = e−|z|
/2
∞
X
zn
√ |ni
n!
n=0
where |ni is the n-th eigenstate of the one dimensional
harmonic oscillator. Show that |zi is an eigenstate of the
√
annihilation operator a which satisfies a|ni = n|n − 1i. Find the corresponding eigenvalue of a. [10 pts]
2. Let ψnlm be the energy eigenstate wave functions of the hydrogen atom, with n the principle quantum number
and l, m labeling the angular momentum eigenstates. Neglect the spin degree of freedom and assume the atom is
isolated. At t = 0, the wave function of the atom is
√
√
1
ψ(t = 0) = √ (2ψ100 + ψ210 + 2ψ211 + 3ψ21−1 ).
10
a). What is the expectation value of the energy of the atom in electron volts? [10 pts]
b). Find ψ(t), the wave function of the atom for t > 0. [10 pts]
c). What is the probability of finding the system with angular momentum Lz = −~ at time t? [10 pts]
d). An angular momentum measurement on the atom yields Lz = −~, (l = 1, m = −1). Immediately after the
measurement, what are the expectation values of angular momentum Lx and L2x in terms of ~ and ~2 respectively?
[15 pts]
3. Consider an electron moving on a (smooth) ring of radius R in the xy plane. In cylindrical coordinates, its
position is uniquely specified by the azimuthal angle φ. There is a homogeneous magnetic field B in the z direction.
The Hamiltonian is given by (in the symmetric gauge and CGS units)
H=
1
2me
−i
eBR
~ d
−
R dφ
2c
2
where e is the electron charge, me is the electron mass, and c is the speed of light. Show that angular momentum Lz
is conserved. Find the energy eigenvalues and eigenstates. [15 pts]
4. A particle of mass m is constrained to move along the x axis and is subject to a potential modeled by Dirac’s
delta function (V0 > 0)
V (x) = −V0 δ(x)
a). Show that there exists only one bound state. Find the bound state energy. [15 pts]
b). Find the value x0 such that the probability of finding the particle in the bound state within −x0 < x < x0 is
exactly 1/2. [15 pts]
Quantum Mechanics Qualifying Exam
Jan. 2011
You are allowed to quote results directly from the main text of Shankar, but not from its exercises or solutions.
1. A particle of mass m is confined within a cubic box of side length L. The wave-function vanishes beyond the
box.
a). What is the parity (eigenvalue) of the ground state? Assume the origin is at the center of the box. [5 pts]
b). What is the energy of the first excited state? What is its degeneracy? [10 pts]
c). Assume the particle is in its ground state. Suddenly, the box expands to a cube with side 2L. What is the
probability of finding the particle in the new ground state? [10 pts]
d). Consider 3 non-interacting identical bosons in the box, what is the energy of the first excited state? [5 pts]
2. A molecule is rotating around its center of mass. The Hamiltonian is H = (L2x + L2y )/2Ia + L2z /2Ib , where Ia,b
are the moments of inertia, and Lx,y,z are the orbital angular momentum operators.
a). Find the energy eigenvalues and eigenstates. [10 pts]
p
Now consider a state described by angular wave function ψ(θ, φ) = 3/4π sin θ cos φ, where θ and φ are the polar
and azimuthal angles respectively. Using the Dirac ket notation |lmi may prove convenient for subsequent problems
2b) and 2c).
b). Compute the expectation value of Lz in state ψ. [10 pts]
c). Suppose Lz is measured in state ψ and result ~ is obtained. Immediately afterwards, Lx is measured, find the
uncertainty (standard deviation) ∆Lx . [10 pts]
3. In April (O’Connell et al, Nature 464, 697, 2010), a group of physicists at UCSB succeeded in quantum control
of a macroscopic mechanical system. A mechanical resonator made of aluminum nitride and aluminum was cooled to
25mK. A superconducting qubit was coupled to the resonator to prepare and measure its quantum states.
Treat the mechanical resonator as a one-dimensional harmonic oscillator of frequency ω. Let |ψ0 i and |ψ1 i be the
normalized energy eigenstate of the ground state and the first excited state respectively. At time t = 0, the system
is prepared at state A|ψ0 i + B|ψ1 i. A and B are in general complex numbers. Then, the dynamics of the system is
monitored in the experiment to determine the relaxation and coherence time.
a). Compute the average value of energy E and position x at t = 0. [10 pts]
b). Find the state vector at later time t > 0. [10 pts]
c). What is the oscillation frequency of hxi as a function of time? [10 pts]
4. A boy drops a marble of mass m from height H (above the ground) and tries to hit a marked point on the
ground. Show that no matter how hard he tries, the marble is going to miss the point by a distance ∆x on the order
of (~2 H/m2 g)1/4 , where g is the gravitational acceleration. [10 pts]
Hint: the initial (horizontal) position and velocity are constrained by Heisenberg’s uncertainty relation. Treat the
motion of the marble after release as classical. This problem is a mock version of 87 Rb atoms released from an optical
trap.
Quantum Mechanics, Qualifying Exam, Jan. 2012
Name:
Note: This is an open book exam and you are allowed to bring Sakurai or
Shankar’s book on Quantum Mechanics. If a formula appears in the book,
please use that as a starting point; there is no need to show the derivation
of that formula.
(1)
Consider a system described by the Hamiltonian H,
H=b
0 2
2 0
!
where b is a constant with dimension of energy.
(a) At t = 0, we measure the energy of the system. What possible values
will we obtain? [5 pts]
(b) At later time t, we measure the energy again. How is it related to its
value we obtained at t = 0? [5 pts]
(c) Suppose at t = 0, the system is equally likely to be in its two possible
energy eigenstates. Write down the most general state of the system at t = 0.
Taking this state as the initial state, find the state at t = 10 h̄/b. What is
the probability that the system at t = 10 h̄/b is in a state different from its
initial state? [10pts]
(2)
At t = 0, a particle of mass m confined in a one-dimensional potential well
2
is in an energy eigenstate with wave function ψ(x) = Ae−(x/b+3) , where b is
a constant with dimension of length. You can use the following integrals:
Z +∞
−∞
−αx2
e
r
dx =
r
π Z +∞ 2 −αx2
1 π
,
xe
dx =
α
2α α
−∞
(a) Determine the normalization constant A. [5 pts]
(b) Where is the particle most likely to be found? [5 pts]
(c) Calculate hxi, hpi, hx2 i, hp2 i, and the uncertainty ∆x∆p in this state.
[10 pts]
(d) If ψ(x) is the ground state wave function, find the Hamiltonian of the
particle, and the ground state energy. [10 pts]
(3) Short questions [20 pts, 5 pts each].
Let x̂ be the position operator, and p̂ be the momentum operator, in onedimension.
(a) Write down the form of x̂ and p̂ in the x-basis.
(b) Write down the form of x̂ and p̂ in the p-basis.
Evaluate the following,
(c) e−ip̂L/h̄ |xi =
(d) [p̂, e−ikx ] =
where L and k are constants.
(4)
A spinless particle in a spherically symmetric potential is described by a wave
function, ψ(x, y, z) = A[1 + (x + z)/r] where A is a normalization constant.
(a) Find the possible angular momentum quantum numbers, l and m, of the
system. [5 pts]
(b) Calculate the probability of the system being found in each angular momentum eigenstates labeled by l and m. [10 pts]
(5)
A particle moving in three dimensions is subject to a potential V (r) =
V0 log(r/a), where r is the radial distance from the origin, V0 and a are
constants.
(a) What is the angular part of the wave function ψ(θ, φ) for angular momentum l = 3, m = 3? [5 pts]
(b) Does the eigen energy depend on l and m? Explain your answer using
the symmetry of the Hamiltonian. [5pts]
(c) Does the answer to (b) change if we replace V (r) with V (r) = V0 a/r?
Explain why. [5pts]
2
Qualifying exam - August 2011
Statistical Mechanics
You can use one textbook. Please write legibly and show all steps of your derivations.
Problem 1 [15 points]
Consider a substance for which
G = AN T n pm ,
(1)
where G is the Gibbs free energy, T temperature, p pressure, N the number of particles and A a
positive constant. Find the values of n and m, if any, for which the substance is thermodynamically stable.
Problem 2
The water molecule H2 O has a nonlinear structure with a 104.5◦ angle between the H-O
bonds. The molecule has a known mass m and principal moments of inertia I1 , I2 and I3 .
1. [5 points] How many vibrational degrees of freedom does the molecule have?
2. [15 points] Considering water vapor as an ideal gas, derive an expression for the chemical
potential as a function of temperature T , pressure p and the molecular parameters m, Ii (i =
1, 2, 3) and νj (vibration frequencies). Assume that kT � �2 /Ii , where k is the Boltzmann
constant.
3. [5 points] What is the high-temperature limit of the constant-volume specific heat of water
vapor?
Problem 3
Vibrational properties of a solid containing Na atoms can be represented by a set of 3Na
identical (but distinguishable) harmonic quantum oscillators with the same frequency ω. This
model of a solid was proposed by Einstein in 1907. As a generalization of the Einstein model,
assume that the vibration frequency depends on volume per atom v: ω = ω(v).
1. [5 points] Using this generalized model, show that
pV = γE,
(2)
where p is pressure, V is total volume of the solid, E is its total energy, and
γ=−
d ln ω
d ln v
is called the Gruneisen constant.
1
(3)
2. [10 points] Calculate the chemical potential of this solid as a function of v and temperature
T.
Now consider a particular case of a solid with ω = Av −α , where A and α are constants.
3. [10 points] Calculate the high-temperature limit (kT � �ω) of the isothermal compressibility βT = − (∂ ln v/∂p)T .
4. [10 points] A solid containing Na atoms is reversibly expanded from a volume V1 to a
volume V2 at a fixed temperature T . Assuming that the temperature is high, i.e. kT � �ω, what
is the amount of heat absorbed by the solid in this process?
Problem 4
Consider a free electron gas at T = 0 K. Suppose its volume is V and the number of electrons
is N .
1. [5 points] Show that the total kinetic energy of the gas is
3
U0 = N εF ,
5
(4)
where εF is the Fermi energy.
2. [10 points] Derive a relation between the gas pressure p and εF .
3. [10 points] Show that the isothermal compressibility of the gas, βT = − (∂ ln V /∂p)T,N ,
equals
3V
βT =
.
(5)
2N εF
2
Qualifying exam - January 2011
Statistical Mechanics
NAME:____________________________
You can use a graduate level textbook and a calculator. Please write legibly and show all
steps of your derivations and/or calculations.
Problem 1 [20 points]
Consider a hypothetical substance for which
U = AV T n
(1)
S = BV T 3 ,
(2)
and
where U is internal energy, V volume, S entropy, T temperature, and A and B are constant
coefficients.
1. Find the exponent n and the ratio A/B.
2. Find the pressure in the substance.
3. Find the chemical potential of the substance.
Problem 2 [20 points]
Calculate the internal energy (in J/mole) and specific heat at a constant volume (in J/mole/K)
of carbon dioxide CO2 at the temperature of 1000 K. Consider CO2 as an ideal gas and treat the
molecular rotations and vibrations in the classical limit. The CO2 molecule has a linear structure
O=C=O. (The gas constant is R = 8.314 J/mole/K.)
Problem 3 [30 points]
A paramagnetic salt can be modeled as a set of fixed non-interacting magnetic moments µ
oriented either parallel or antiparallel to an applied magnetic field. Suppose the kinetic energy of
the ions carrying the magnetic moments can be neglected.
1. The salt is equilibrated with a thermostat at a temperature T and the magnetic field is
slowly increased from zero to B at the fixed temperature. What is the amount of heat
released by one mole of the salt during this process?
1
2. The salt is now thermally isolated and the magnetic field is adiabatically decreased from
B to B/10. What is the final temperature of the salt?
(This process of adiabatic demagnetization is used for cooling materials to very low temperatures.
The real process is more complex than assumed in this problem.)
Problem 4 [30 points]
1. In the Einstein model, a solid containing N atoms is represented by a set of 3N identical
but distinguishable harmonic oscillators of frequency ν. The chemical bonding between
the atoms lowers the free energy by εN . Assume that ν and ε are independent of the
volume. Present a complete derivation of the chemical potential of atoms in this solid.
2. If the solid evaporates, it forms an ideal atomic gas. Present a complete derivation of the
chemical potential in the vapor.
3. Assuming that the vapor is in thermodynamic equilibrium with the Einstein solid (saturated
vapor), show that the pressure in the vapor equals
p = kT
�
2πmkT
h2
�3/2 �
where m is the atomic mass.
2
hν
2 sinh
kT
�3
e−ε/kT ,
(3)
Qualifying exam - January 2012
Statistical Mechanics
You can use one textbook. Please write legibly and show all steps of your derivations.
Problem 1 [20 points]
Consider a substance for which
E = AV T n ,
(1)
where E is energy, V is volume, T is temperature and A > 0 and n > 1 are constants.
1. What is the entropy of this substance? [5 points]
2. Calculate the pressure p of this substance as a function of temperature. [5 points]
3. Show that pV /E is a constant and determine this constant. [5 points]
4. Is this substance thermodynamically stable if n < 1? [5 points]
Problem 2 [35 points]
Consider a system of N localized non-interacting identical molecules, each having an electric
dipole moment p. The system is placed in an electric field E at a temperature T . Assuming
that the system is classical and disregarding the kinetic energy of the molecules, calculate the
following properties:
1. Partition function of the system. [7 points]
2. Average potential energy "¯ per molecule. [7 points]
3. Average dipole moment p̄ per molecule. [7 points]
4. The dielectric susceptibility (@ p̄/@E)T . [7 points]
5. The specific heat (@ "¯/@T )E . [7 points]
Problem 3 [20 points]
Calculate the internal energy (in J/mole) and specific heat at a constant volume (in J/mole/K)
of hydrogen cyanide HCN at the temperature of 800 K. Consider HCN as an ideal gas and treat the
molecular rotations and vibrations in the classical limit. The HCN molecule has a linear structure
H C⌘N (see figure below). The gas constant is R = 8.314 J/mole/K.
1
Problem 4 [25 points]
Consider a cavity containing black-body radiation at a temperature T1 . Suppose the volume
of the cavity increases in an equilibrium adiabatic process from an initial value V1 to a final value
V2 = 5V1 .
1. What is the final temperature T2 in the cavity? [5 points]
2. If the initial radiation pressure was p1 , what is the final pressure p2 ? [5 points]
3. If the cavity initially contained a total of N1 photons, what is the final number N2 of
photons in the cavity? Explain the physical meaning of this result. [15 points]
2
1. One mole of an ideal monatomic gas undergoes an isothermal expansion from a
volume of 2 Liters to a volume of 20 Liters. If the initial pressure is 30 atm calculate the:
a, work performed.
b, heat exchanged with the environment.
c, and change in entropy of the gas.
2. A cylindrical column of gas of given temperature rotates about a fixed axis with
constant angular velocity. Find the equilibrium distribution function.
3. A rod-like pollen grain floats in the air at a constant temperature. On average, is the
angular momentum vector nearly parallel to or perpendicular to the long axis of the
grain?
4. A system of two energy levels Eo and E1 is populated by N particles at temperature T.
The particles populate the energy levels according to the classical distribution law.
a, Derive an expression for the average energy per particle.
b, Compute the average energy per particle vs the temperature as T  0 and T  ∞.
c, Derive an expression fro the specific heat of the system of N particles.
5. a, Derive a formula for the maximum kinetic energy of an electron in a noninteracting Fermi gas consisting of N electrons in a volume V at zero absolute
temperature?
b, Calculate the energy gap between the ground state and the first excited state for such a
Fermi gas consisting of the valence electrons in a 100 A cube of copper.
c, Compare the energy gap with kT at 1K.
The density for copper is 8.93 g/cm3 and its atomic weight is 63.6.
6. Consider a classical system of N noninteracting diatomic molecules enclosed in a
box of volume V at temperature T . The Hamiltonian for a single molecule is
1
K
   
2
H (r1 , r2 , p1 , p2 ) 
( p12  p22 )  r1  r2
2m
2
a. Find the Helmholtz free energy of the system.
b. Find U/N, and compare your result to what the equipartition theorem
suggests.
3kT
2
c. Show that the mean-square molecular diameter r1  r2 
K
.
Qualifying exam - 2010
Statistical Mechanics
NAME:____________________________
You can use a graduate level textbook, a calculator and a unit conversion table. Please write
legibly and show all steps of your derivations and/or calculations.
Problem 1 [15 points]
Consider a simple model of a polymer chain composed of N � 1 identical molecules of
length a as in the figure below. Each molecule can be aligned either along the chain or normal
to it. In the latter case its projection on the chain direction is assumed to be zero. Each molecule
can be only in one these two states. Each state is degenerate and has the same energy in the
absence of forces. Kinetic energy of the molecules can be neglected. A tension force f is
applied parallel to the chain and the system is equilibrated with a thermostat.
1. Find the entropy S and length L of the polymer chain as functions of temperature T and
force f .
2. In the limit of small force, show that L becomes a linear function of f at a fixed T . This
relation is similar to Hooke’s law of elasticity.
a
f
f
Problem 2 [25 points]
Consider N fixed non-interacting magnetic moments of magnitude µ. The system is in a
uniform external magnetic field B and in equilibrium with a thermostat at a temperature T .
Assuming that each magnetic moment can be oriented only parallel or anti-parallel to the field,
derive analytical expressions for:
1. Specific heat of the system
2. Average magnetic moment M
3. Fluctuation of the magnetic moment ∆M =
1
�
�
M −M
�2
4. Magnetic susceptibility χ = dM /dB
5. Find the asymptotic behaviors of χ in the limits of high temperatures and low temperatures
Problem 3 [25 points]
Considering ammonia as an ideal gas whose molecules NH3 have the structure of a triangular pyramid,
1. Derive analytical expressions for:
(a) the chemical potential of ammonia at a given temperature T and pressure p,
(b) constant-volume specific heat (CV ) per molecule of ammonia as a function of T .
2. Compute CV per mole of ammonia (in J/mol/K) at the temperatures of 300 K and 1500
K. Explain the difference between the results.
3. Compute the high-temperature limit of CV for ammonia (in J/mol/K).
Include only translational, rotational and vibrational degrees of freedom. The rotations can be
treated in the classical limit. The NH3 molecule has six normal frequencies of atomic vibrations.
For numerical calculations, assume that all normal frequencies are equal to ν = 60 THz.
Problem 4 [35 points]
1. Consider a gas of free electrons at a temperature much smaller than the Fermi temperature. Present a detailed, step by step derivation of the following relations:
3
u = εF
(1)
5
2
p = nu
(2)
3
π2k2T
C=
(3)
2εF
where εF is the Fermi energy, u is the average energy per electron, n is the number density of
electrons, p is pressure of the gas, C is specific heat per electron, and k is Boltzmann’s constant.
2. Suppose the gas contains 15 electrons per cubic nanometer (typical electron density in
metals). Compute the Fermi energy (in eV) and the specific heat per electron (in eV/K) at a
temperature of 300 K. Compare with specific heat of a classical gas per particle. Explain the
difference.
2
Qualifying exam - August 2012
Statistical Mechanics
You can use one textbook. Please write legibly and show all steps of your derivations.
Problem 1 [20 points]
1. [10 points] Two vessels are isolated and contain equal amounts of molecular oxygen O2 at
the same temperature T but different pressures p1 and p2 . The vessels are then connected. Find
the change in entropy and determine its sign.
2. [10 points] Two vessels are isolated and contain equal amounts of molecular oxygen O2 at
the same pressure p but different temperatures T1 and T2 . The vessels are then connected. Find
the change in entropy and determine its sign.
In this problem you can consider oxygen as an ideal gas and treat molecular rotations and
vibrations in the classical approximation.
Problem 2 [32 points]
Consider an extreme relativistic gas of identical classical particles with the energy-momentum
relation " = pc, where c is the speed of light.
1. [8 points] Show that the partition function of the gas is
"
✓ ◆3 # N
1
kT
Z(V, T ) =
8⇡V
.
(1)
N!
hc
2. [8 points] Calculate the specific heats of this gas at constant volume (cv ) and constant
pressure (cp ). Compare your results with the respective specific heats on a non-relativistic ideal
gas of single atoms.
3. [8 points] Show that for the extreme relativistic gas P V = E/3, where P is pressure, V is
volume and E is internal energy.
4. [8 points] If this gas expands reversibly and adiabatically from a volume V to a volume
3V , what is the change in temperature?
Problem 3 [33 points]
According to the model proposed by Einstein (1907), a solid can be represented by a set
of 3Na identical but distinguishable quantum oscillators with the same frequency ! (Na is the
number of atoms in the solid). Suppose the vibration frequency depends on volume per atom, v,
according to the relation
! = !0 e av ,
(2)
where !0 and a are constants.
1
1. [11 points] Calculate the isothermal compressibility, T , of the solid.
2. [11 points] Calculate its heat capacity, Cv , at a constant volume.
3. [11 points] Prove the following relation:
E = Cv T +
Na
,
va2 T
(3)
where E is the total energy of the solid and T is temperature.
Problem 4 [15 points]
Consider a three-dimensional free electron gas at zero temperature (degenerate electron gas).
Calculate the relative root-mean-square deviation of its energy,
⇣
("
")2
"
where " is energy per electron.
2
⌘1/2
,
(4)