Ph.D. Comprehensive Exam Department of Physics Georgetown University

Part I: Tuesday, July 11, 2017, 12:00pm - 4:00pm

Proctors: Peter Olmsted and Amy Liu

Instructions:

• Please put your name on the first page of every problem!

• This is a closed-book, closed-notes exam. The only electronic devices allowed are

calculators provided by the department.

• Each problem is worth 50 points.

• You should submit work for all of the problems. In many cases, even if you get stuck on

one part of a problem, you may be able to make progress on subsequent parts.

• Please write your solution for each problem on separate sheets of paper.

• Show all your work.

E&M Comprehensive Exam Question

A conductor is aligned along the z-axis from −1.5 ≤ z ≤ 1.5m as shown in the diagrambelow. The wire is free to move along two infinitely long straight wires that extend in the+ex direction. The wires (circuit) carry a fixed current of 10.0 A which travels in the -ezdirection along the vertical conductor. For a magnetic field given by the following

~B = 3 × 10−4 e(−x/5.0m) ey T. (1)

Calculate:

A. The work required to move the conductor at a constant speed to a position [2.0, 0.0, 0.0]m.

B. The power required to move the conductor the same distance.

Assume the the motion is parallel to the z-axis as shown below and that the time requiredfor the motion is 5 × 10−3s.

B

I1.5

1.52.0

x

y

z

1

1

I

I

1

Q. An infinitely long conducting cylinder of radius “a” has a positive line charge density,+λ, and is oriented with its axis parallel to the z axis. The axis of the cylinder lies adistance “d” above an infinite, grounded conducting sheet parallel to the x-z plane.

a) Find the capacitance per unit length of the cylinder and grounded sheet. Makethe approximation that the charge on the cylinder is uniformly distributed aroundthe circumference, as shown schematically in the figure inset.

b) Provide a rough sketch of the actual charge distribution around thecircumference of the cylinder. Do you expect the actual capacitance (takingproper account of the non-uniform charge distribution on the cylinder) to begreater than, less than, or equal to the result obtained in part a? Explain.

a

x

y

z

d

+λ

∞

∞

d

+

+

+ +

+

++

+inset

Consider a system with three spins on lattice sites 1, 2, 3 (you can picture them on the cornersof an equilateral triangle).The theory of angular momentum tells us we have one s = 3/2 multiplet of states and twoindependent s = 1/2 multiplets. All of the problems presented here can be done best by employingoperators in the appropriate fashion, but they can be done in other ways as well. If we define

Stot = S1 + S2 + S3 (1)

and recall the spins operators on different sites commute, then working with raising, lowering,S2 and Sz will help you in the following problems.

(a) (4 points) There are 2 × 3

2+ 1 = 4 states with s = 3

2. Determine all four of them (with

different Sz eigenvalues) expressed in terms of linear combinations of the following eightz-basis vectors

| ↑↑↑〉 | ↑↑↓〉 | ↑↓↑〉 | ↑↓↓〉 | ↓↑↑〉 | ↓↑↓〉 | ↓↓↑〉 | ↓↓↓〉 (2)

Be sure to normalize your vectors!

(b) (4 points) Determine the two independent s = 1

2multiplets grouped according to their Sz

eigenvalues. Use the same basis as above. Be sure to normalize your vectors!

(c) (8 points) A Hamiltonian given by

H = −J

h2(S1 · S2 + S2 · S3 + S3 · S1) − gµbB (S1z + S2z + S3z) (3)

is applied to the spins. Determine the energy eigenvalues, their degeneracies, and theground state for J > 0 and B > 0. (Hint: the sum of operator products in the Hamiltoniancan be expressed in terms of S

2

totand S

2

i, where i = 1, 2, 3).

(d) (4 points) A perturbation V = γS1x is added to the Hamiltonian. Determine the first-order(in γ) change in energy for the ground state.

1

The Hamiltonian of a three-level system is represented by the matrix:

=

200020001

εH

Another observable is represented by the matrix:

=

010100002

aA

1) Find the eigenvalues and normalized eigenvectors of H and A.

2) Can you find a common basis of eigenvectors?

3) A particle is in a state |ψ>, which is unknown. You measure H and get the result2ε. Do you know the state of the particle soon after the measurement? Explain.

4) Soon after measuring H, with outcome 2ε, you measure A. Can the state of theparticle change?

5) Suppose that the system starts in the state

=>ψ

212

12

1

|

If H is measured and you obtain the result 2ε, what is the state |ψ0> of the particle after the measurement? If now you measure A, what are the possible outcomes and their probabilities?

6) Suppose again that the system starts in the state |ψ0> defined in 5). Find the state|ψ(t)>. If you measured the energy in this state at the time t, what are the possibleoutcomes and their probabilities?

Ph.D. Comprehensive Exam Department of Physics Georgetown University

Part II: Thursday, July 13, 2017, 12:00pm - 4:00pm

Proctors: Ed Van Keuren and Peter Olmsted

Instructions:

• Please put your name on the first page of every problem!

• This is a closed-book, closed-notes exam. The only electronic devices allowed are

calculators provided by the department.

• Each problem is worth 50 points.

• You should submit work for all of the problems. In many cases, even if you get stuck on

one part of a problem, you may be able to make progress on subsequent parts.

• Please write your solution for each problem on separate sheets of paper.

• Show all your work

1

Density of states and energy bands

1) Consider a free-electron gas in a volume V in three dimensions. Derive the expression of the Fermi energy

εF = h2

2m

(

3π2NV

)2/3

and explain its meaning. Derive the expression of the density of states D(E) and discuss

its main features.

2) Consider the case in which the electron gas is confined within a potential well of width δ along z (quasi two-dimensional) and has discrete energy bands. Hint: Consider that the bands at kx = ky = 0 have energy

En = h2

2mπ2

δ2 n2 (with n integer).

Derive (or explain why) the density of states for each band is independent on energy. Derive the total densityof states g(E) as a function of the band energy.

3) Discuss (if they exist) conditions in which g(E)/δ in 2) would have the same expression obtained for D(E) in1).

From the comparison, identify and discuss the range of temperatures for which g(E)/δ has two-dimensionalfeatures clearly distinct from the three-dimensional case.

Acoustical properties of two linear polymer chain materials:

Polyethylene (PE) and Polyacetylene (PA)

Shown below is an ideal ball-and-stick model for linear PE (left) and PA (right). The PE chain has

repeating CH2-CH2 single carbon bonds while the PA has alternating single and double carbon bonds:

CH-CH=CH-CH. Take the separation of the carbon atoms along the linear axis to be a constant distance

“a” for all carbon bonds. Note that this means that the spatial periodicity length for PE is a, while for PA

it is 2a. Consequently, the edge of the 1D Brillouin Zone for PE is at 𝜋

𝑎, while for PA it is at

𝜋

2𝑎 .

Model the longitudinal (along the axis) stretching vibrations of these polymers by a linear chain

(length L) of identical masses M connected by Hooke’s Law springs with force constant K1 for single

carbon bonds and K2 for double carbon bonds.

(a) Carefully write down for each polymer the equations of motion for the stretching modes.

(b) In this model, stretching mode angular frequencies ω for PA as a function of wave vector k are:

𝜔2 =𝐾1+𝐾2

𝑀(1 ± √1 −

4𝐾1𝐾2

(𝐾1+𝐾2)2 sin2 𝑘𝑎 ) [rad/s]2

Give the corresponding expression of the frequencies of PE. For both PE and PA sketch the

dispersions ω(k) and noting carefully the values of ω for the wave vectors k = 0 and k = π/a (for

PE) or k = π/2a (for PA).

(c) In the Debye model of phonons (bosons), the dispersion of the lowest (acoustical) branch ω is

approximated by a linear dispersion ω(k) = cs k, where cs is the speed of sound in the chain:

c-1) Calculate cs in terms of the constants of the linear mass-spring model for PE. From your

knowledge of solids, estimate in [m/s] cs for these materials.

c-2) For a Debye model describing a polymer chain with length L, periodicity “a”, and speed of

sound cs, calculate the total chain internal energy U and the chain specific heat CL =(∂U/∂T)|L at

low temperature (kT << h𝑣D) using the following notation:

𝑣 = 𝜔/2π; h=Planck’s constant; k=Boltzmann’s constant

the one-dimensional density of states for a frequency 𝑣 is g(𝑣) = 2L/cs

the Bose-Einstein distribution function is (exp(h𝑣/kT) -1)-1

the Debye cut-off frequency is 𝑣D =cs/2a

the zero-point internal energy is U0.

Hint: You can use the following approximation

for 1

Some formulae that may simplify your mathematics:

n∑k=0

rk =1− rn+1

1− r

sinhx ≡ ex − e−x

2coshx ≡ ex + e−x

2

tanhx ≡ sinhx

coshxcothx ≡ coshx

sinhx

d

dxsinhx = coshx

d

dxcoshx = sinhx

d

dxtanhx = sech2 x

d

dxcothx = − csch2 x

1. Consider N independent, distinguishable, one-dimensional quantum harmonic oscil-lators in equilibrium at temperature T . The energy levels of single oscillator areε = α

v

(n+ 1

2

), where α is a constant, v ≡ V/N is the (one-dimensional) average

spacing, and n = 0, 1, 2, 3, . . .. For each answer, be sure to evaluate any sumsand simplify where possible.

(a) Determine the partition function for the system.

(b) Determine the internal energy as a function of T and V (or, equivalently, β andv).

(c) Determine the average number of particles in the m-th energy level as a functionof T and V (or, equivalently, β and v).

(d) Determine the average quantum number n as a function of T and V (or, equiva-lently, β and v).

(e) Determine the entropy as a function of T and V (or, equivalently, β and v).

(f) Does the behavior of this system agree with the equipartition theorem in theappropriate limit? (Be sure to explain why or why not.)

Physics 2017 Comprehensive Exam: Statistical Mechanics

What is the entropy of a surface film?

Let a water surface of area A be covered by a thin film containing N organic molecules. Thesurface tension γ and heat capacity CA (for constant area) are as follows:

γ(T,A) = γo −NkBT

A− bN+aN2

A2

CA(T ) = NkB +NkB

( TTo

)2

where a, b, γo and To are constants. Note that in a 2D system, the surface tension is analogousto the pressure in a 3D system, so the work is γdA.

1. Show that: (∂S∂T

)A

=NkBT

+NkBT

T 2o

2. Now, show that: (∂S∂A

)T

=NkBA− bN

3. Finally, show that the entropy, S(T,A) of the surface film, up to an additive constantis:

S(T,A) = NkB ln T +1

2NkB

( TTo

)2

+NkB ln (A− bN) + c

1