Ph.D. Comprehensive Exam
Department of Physics
Georgetown University
Part I: Tuesday, July 10, 2018, 12:00pm - 4:00pm
Proctors: Mak Paranjape and Ed Van Keuren
Instructions:
•
•
•
•
•
•
Please put your assigned number 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.
Electricity and Magnetism
Comprehensive Exam Question
Note: The percentage of the grade allocated for each question in this exam has been explicitly
stated.
Q1. To ask a physicist to state the value of the acceleration due to gravity, g, is trivial. So too is
knowing the value of the AC voltage from a common North American household electrical outlet.
When answering the following questions, be absolutely precise with your result, and express
your answer with only 2 significant figures:
a. State the value of “g”.
g=________________________
(4%)
b. i) Clearly and unambiguously state the numerical value of the electrical outlet voltage.
v=_____________(zero-to-peak/peak-to-peak/effective value/average value?)
(8%)
ii) Since electrical outlets are AC, what is the frequency of the voltage?
f=_________________________
(8%)
Q2. For the following resistive network, you are to determine the equivalent resistance between
points a-b and find the current, i (shown), assuming the network is connected to a 17 vAC source.
But, you have a choice in the way you will solve the problem! You can either solve it:
1. using theoretical calculations of series and parallel resistors (Hint - you may want to consider
replacing the orange-colored 30 k𝛺 resistor with two 15 k𝛺 series resistors. This may seem
like a pointless exercise, but critically examine the form of the resulting network),
or,
2. by physically performing an experimental measurement, since the resistive network has been
physically wired up and connected to an AC voltage source (17 vAC) using a simple breadboard
system (the blue box), described later.
i
a
30 kΩ
30 kΩ
30 kΩ
30 kΩ
30 kΩ
30 kΩ
30 kΩ
b
30 kΩ
30 kΩ
The (blue) breadboard circuit layout is as follows:
current-protected LED
power indicator
breadboard power
switch
step-down
transformer
(line voltage to
17 vAC)
i
30 kΩ
30 kΩ
30 kΩ
vAC
30 kΩ
30 kΩ
30 kΩ
30 kΩ
30 kΩ
30 kΩ
Using the provided digital multi-meter (DMM), measure the value for current, i, shown in the
figure. Measure the equivalent resistance Rab seen by the 17 vAC source by using the resistance
setting on the DMM.
Your grade will be determined based on only one solution method that you explicitly select to be
graded. Of course, you may elect to do both methods as a means to confirm your answers,
however, only one method will be graded. Select which method you would like graded:
theoretical
experimental
i= _________________________
(25%)
Rab = _________________________
(25%)
Q3. Derive the wave equations for both electric field and magnetic field for an electromagnetic
wave propagating through conductive matter, where typically, 𝜌=0 but J ≠ 0. The following
vector identities may (or may not) be useful:
Ñ • (Ñ x F) = 0
Ñ x (Ñ x F) = Ñ(Ñ • F) - Ñ2F
Ñ • (F x G) = G • (Ñ x F) - F • (Ñ x G)
(30%)
Physics 2018 Comprehensive Exam: Electricity and Magnetism
Consider a raindrop of radius R that is suspended in air. Assume that a fraction of the
water molecules in the drop are charged, i.e., one electron was removed from each of these
molecules and they became electrically charged. These charged molecules will migrate to
the surface of the drop and form a shell of charge Q0 of radius R. Assume that |Q0 | is much
larger than |qe |, where qe is the charge of a single electron.
1. What is the electrostatic potential, V (r), of the drop for r > R?
2. The work done to remove the electrons and charge the drop is equal and opposite to
the work needed to bring all the electrons back and neutralize the drop. Calculate the
work done to charge the drop using the following steps:
a. Calculate the work dW needed to add a charge dq to the charged drop (assume that
the additional charge dq is brought from r = ∞).
b. Calculate the total work W needed to add a total charge −Q0 and neutralize the
charged drop.
3. Assume that the radius of the drop is 1 mm and the fraction of charged water molecules
is 1%. Calculate Q0 and the work done to charge the drop.
You will need the following parameters:
1
= 9 · 109 Nm2 /C−2
4π0
qe = 1.6 · 10−19 C
Avogadro’s number: NA = 6.02 x 1023
Density of water: 1000 kg / m3
Atomic mass of oxygen is 16 (One mole of oxygen weighs 0.016 kg)
1
1. Consider an electron of a linear triatomic molecule formed by three equidistant atoms
(A, B, C) as shown below:
A
B
C
Use |φA i, |φB i, and |φC i to denote three orthonormal states of the electron, corresponding to three wavefunctions localized, respectively, about the nuclei of atoms A,
B, and C.
When the possibility of the electron jumping from one nucleus to another is neglected,
the energy (of the electron) is described by the Hamiltonian Ĥ0 , whose eigenstates are
|φA i, |φB i, and |φC i with the same eigenvalue E0 .
The couplings between the nuclei are described by Ŵ , an additional contribution to
the Hamiltonian defined by:
Ŵ |φA i = −a|φB i
Ŵ |φB i = −a|φA i − a|φC i
Ŵ |φC i = −a|φB i,
where a is a real, positive constant.
(a) Write the Hamiltonian Ĥ = Ĥ0 + Ŵ in the {|φA i, |φB i, |φC i} representation.
(b) Calculate the energies and stationary states of Ĥ.
(c) At time t = 0, a measurement is made and the electron is found to be localized
on nucleus A.
i. Determine the state of the system at an arbitrary time t ≥ 0.
ii. Let D̂ be the observable whose eigenstates are |φA i, |φB i, and |φC i with
corresponding eigenvalues −d, 0, and d. D̂ is measured at time t ≥ 0. What
values can be found and with what probabilities?
iii. Are there times when the electron is localized perfectly on atom B?
iv. Are there times when the electron is localized perfectly on atom C?
(d) When the initial state of the electron is arbitrary, list all non-zero frequencies that
could appear in a Fourier decomposition of the time-evolution of the expectation
value of a generic, time-independent operator Ô.
(e) When the initial state of the electron is arbitrary, what non-zero frequencies
can appear in a Fourier decomposition of the evolution of hD̂i(t), where D̂ is
the operator defined above? (Be sure the justify which ones are included and
excluded.)
Comprehensive exam question 2018
Quantum Mechanics
Consider a two-dimensional isotropic simple harmonic oscillator given by the following Hamiltonian:
H = ~ω(â†1 â1 + â†2 â2 + 1).
(1)
The raising and lowering operators satisfy the usual commutation relations, with the only nonzero
commutators being
[â1 , â†1 ] = 1 and [â2 , â†2 ] = 1.
(2)
All other commutators vanish (including those between any 1 and 2 operators).
The (normalized) energy eigenstates are given by
n2
1 † n1
1
â1
|0i1 ⊗ a†2
|0i2 ,
|n1 , n2 i = √ √
n1 ! n2 !
(3)
where âi |0i = 0 for i = 1, 2.
In the 1950s, Julian Schwinger devised a way to use these operators to represent angular momentum states. You will explore this mapping in this problem.
(a) (9 points) Define the operators Jˆ+ = ~â†1 â2 , Jˆ− = ~â†2 â1 , and Jˆz = ~(â†1 â1 − â†2 â2 )/2. Verify
the angular momentum commutator algebra given by
[Jˆ+ , Jˆ− ] = 2~Jˆz ,
[Jˆz , Jˆ+ ] = ~Jˆ+ , and [Jˆz , Jˆ− ] = −~Jˆ− .
(4)
(b) (6 points) Define Jˆx = (Jˆ+ + Jˆ− )/2 and Jˆy = (Jˆ+ − Jˆ− )/(2i) and compute the commutator
[Jˆx , Jˆy ] =?. Be sure to express your answer in terms of the Cartesian Jˆ operators and show
all of your work. You may use commutators calculated in (a) in your derivation, if desired.
(c) (10 points) Consider the state |j, m = ji in the angular momentum representation that
satisfies Jˆ+ |j, m = ji = 0. Use this condition on the set of states {|n1 , n2 i}, for all
nonnegative integers n1 and n2 , to determine what states are |j, m = ji states. Next,
by using your knowledge about the |j, mi multiplet, determine formulas for j and m in
terms of n1 and n2 . Hint: you should be able to verify that the states |n1 , n2 i are already
eigenstates of Jˆz ; use the properties of the raising and lowering operators acting on these
states to infer the relationship between the quantum numbers n1 and n2 and j and m.
(d) (6 points) Using Jˆ2 = (Jˆ+ Jˆ− + Jˆ− Jˆ+ )/2 + Jˆz2 , compute Jˆ2 |j, mi = Jˆ2 |n1 , n2 i, for the values
of n1 and n2 which give the general |j, mi state that you found in part (c). You must
show all of the algebra, perform the algebra with the simple harmonic oscillator raising
and lowering operators, and express your final answer for the eigenvalue as a function of j
and m. Do not just quote the known answer.
(e) (15 points) We now work with the multiplet given by all states of the form (Jˆ+ )α |n1 =
0, n2 = 2i and α any of the allowed nonnegative integers consistent with your results in
part (c). We define the angular momentum matrices via
0
0
ˆ
ˆ
ˆ
ˆ
Jz
= hj, m|Jz |j, m i, J+
= hj, m|J+ |j, m i, and Jˆ−
= hj, m|Jˆ− |j, m0 i.
m,m0
m,m0
m,m0
(5)
Determine the three angular momentum matrices for the representation created from the
|n1 = 0, n2 = 2i state. Use the simple harmonic oscillator algebra and recall (or rederive)
what â and ↠do on simple harmonic oscillator number states.
(f) (4 points) Construct the matrices for Jˆx and Jˆy , using the results from part (e) and the
relationship between the raising and lowering operators and the Cartesian operators.
Ph.D. Comprehensive Exam
Department of Physics
Georgetown University
Part II: Thursday, July 12, 2018, 12:00pm - 4:00pm
Proctors: Ed Van Keuren and Amy Liu
Instructions:
•
•
•
•
•
•
Please put your assigned number 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 wor
Comprehensive exam
Condensed Matter Physics
Graphene is a two-dimensional crystal made of a single layer of carbon atoms arranged in a hexagonal
honeycomb lattice. The valence and conduction band cross at two special points in the first Brillouin zone,
K and K’. The dispersion relation near the Fermi energy (which is set at 𝜖 = 0) and for wavevectors near
̃|, where 𝒌
̃ =𝒌−𝑲
the K and K’ points is sketched in Fig. 1 and it can be approximated by 𝜖𝐾 = ±ℏ𝑣𝐹 |𝒌
for the valence and conduction band near K (the same dispersion relation holds for values of k near K’) and
𝑣𝐹 = 8 × 105 𝑚/𝑠.
1) Find the density of states near the Fermi energy with the following steps:
̃ |) within a circle of radius |𝒌
̃| centered at K. (Hint on
a) Find the number of states N(|𝒌
degeneracy: note that for a given value of 𝜖 there are two bands (at K and K’).)
̃|) to find D(𝜖 ) for 𝜖 > 0 and show that it is proportional to 𝜖 .
b) Use N(|𝒌
2) The effective mass of electrons and holes in graphene is not infinite, as it may naively appear by
̃|−3 (note that (m
the linear dispersion relation. As an example, derive that (mxx)-1 = ħ-1vF ̃𝒌2𝑦 |𝒌
𝜕2 𝜖𝐾
ħ-2
𝜕𝑘𝛼 𝜕𝑘𝛽
)
3) When charges are added to graphene, the Fermi energy shifts. If n0 electrons per unit area are added
to the graphene the Fermi energy shifts in the conduction band. Show how n0 is related to the Fermi
energy via the density of states.
4) Use the relation above to calculate the Fermi energy shift as a function of n0 at T = 0 (solve the
integral).
5) For any conductor, the Fermi energy is defined at T = 0 and it coincides with the chemical potential.
At T ≠ 0, the chemical potential varies with temperature. Show how at any finite temperature the
chemical potential of a conductor is related to its density of states D(𝜖) and the total number of
electrons N0.
6) In general, would you expect the temperature dependence of the chemical potential of
a conductor to depend on whether it is three-dimensional, two-dimensional or onedimensional? Justify your answer? Explain.
Comprehensive Exam
Condensed Matter Physics
1. Consider a metal whose thermal and electrical conductivity can be accurately modeled as a Free Electron Gas.
(a) Use kinetic theory to derive the thermal conductivity of the metal
due to the electrons in terms of the electronic heat capacity per unit
volume, C, the mean electron speed, v, and the electron mean free
path l. Recall that the thermal conductivity, K, is defined by the
relation
jU = −K
dT
dx
(Hint: Consider how far the electrons will travel in the time between
collisions, τ , and how much energy they carry along the temperature
gradient.)
(b) The electronic specific heat per unit volume is given approximately
by
Cel =
1 2
π nkB T /TF
2
where n is the electron density. The precise calculation is somewhat
involved, but you can get a rough estimate by recognizing that, for
metals at room temperature and below, the Fermi-Dirac distribution
goes from 1 to 0 over a narrow range of width KB T centered around
EF , as sketched in Fig. 1. Argue that Cel ∼ nKB (T /TF ).
Figure 1: Fermi-Dirac Distribution Function.
(c) Using the relation for the conductivity in the free electron gas model,
σ = ne2 τ /m, derive the Wiedemann-Franz law,
K/σ =
π2
3
kB
e
2
T
(1)
(d) Identify at least two physically plausible situations where the WiedemannFranz law would fail, that is where the ratio of the measured conductivities of a material would not be given by Eq. 1.
Quantum spin-0 ultra-relativistic gas
(Bowley Ch4 through 7)
A homogeneous gas of N, non-interacting, indistinguishable atoms of mass m is confined in a
cubic volume V. The gas is in thermal equilibrium at a temperature T which is so high that the kinetic
energy ε of each atom can be approximated by its limiting ultra-relativistic limit: ε = cp where p is the
modulus of the atom’s momentum. You may find useful that for integer values of n:
a) For an atom in the box, what are the wavenumber eigen states, and their corresponding eigen
energies?
b) Calculate the partition function for one atom of the gas, Z1(T, V, N=1).
c) Introduce a relativistic thermal de Broglie wavelength λrel(T) by: Z1≡V λrel-3. Use your result in
part b) above to calculate λrel. What is the physical meaning of the thermal de Broglie
wavelength?
d) State the expression for the nonrelativistic thermal de Broglie wavelength λD(T). Calculate λrel/
λD for atomic hydrogen gas at 1x105K.
Constants: mH =1.67x10-27 [kg], kB=1.38x10-23 [J/K], and c= 3x108 [m/s]
e) Find the partition function for the entire gas, Z(T, V, N).
f) Find the Helmholtz energy of the gas, F(T, V, N). Explain why F is not directly proportional to
the number of atoms N for a given volume V.
g) Find the internal energy of the gas, U(T, V, N), and the isochoric heat capacity CV.
h) Find the pressure, P(T, V, N), and hence the equation of state for the gas.
i)
Find the entropy, S(T, V, N). Under what condition is the entropy extensive?
Page 1 of 1
Stat-Mech Comprehensive Exam Question
In this problem you will calculate the total thermal energy E of a system of N independent,
one-dimensional harmonic oscillators using the canonical ensemble. You can solve this problem quantum mechanically or classically.
In order to solve this problem for full credit, you must first determine the Hamiltonian for
a single oscillator with an angular frequency given by ω and then use it to calculate the
partition function.
1