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Compendium Master’s Review Examinations 2018-current Physics Department Physics University of Washington
Preface: This is a compendium of problems from the Master’s Review Examinations for physics graduate students at the University of Washington. Problems are grouped by year. The four exams are in Thermodynamics and Statistical Mechanics, Classical Mechanics, Quantum Mechanics, and Electromagnetism. This compendium covers the period after 2017. The MRE’s for 2012-2017 are listed in a separate PDF file. UW physics Ph.D. students are strongly encouraged to study all the problems in these two compendia. Students should not be surprised to see a mix of new and old problems on future exams. Some bits of advice:
- Try to view your time spent studying for the Qual as an opportunity to integrate all the physics you have learned in that specific topic.
- Read problems in their entirety first, and try to predict qualitatively how things will work out before doing any calculations in detail. Use this as a means to improve your physical intuition and understanding.
- Some problems are easy. Some are harder. Try to identify the easiest way to do a problem, and don’t work harder than you have to. Make yourself do the easy problems fast, so that you will have more time to devote to harder problems. Make sure you recognize when a problem is easy.
- Always include enough explanation so that a reader can understand your reasoning.
- At the end of every problem, or part of a problem, look at your result and ask yourself if there is any way to show quickly that it is wrong. Dimensional analysis, and considera- tion of simplifying limits with known behavior, are both enormously useful techniques for identifying errors. Make the use of these techniques an ingrained habit.
- Recognize that good techniques for studying Qual problems, such as those just mentioned, are also good techniques for real research.
Master’s Review Examination Final Exam, Phys524, Autumn 2017
1 Throttling (30 points total)
Consider a (non-ideal) gas that has a volume V , temperature T and pressure P in the initial state. It then undergoes a Joule-Thomson process (is throttled through a porous medium). As a result of this process the pressure changes by P. Assume P to be small.
A. [ 5 points] Is there a thermodynamic potential that is conserved during the process? If so state what it is and briefly explain why it is conserved.
B. [ 10 points] Find the entropy change S.
Master’s Review Examination Final Exam, Phys524, Autumn 2017
2 Thermal fluctuations [20 points]
Consider a container of volume V filled with N 1 atoms of an ideal Boltzmann gas at temperature T. Determine the probability of finding an empty void of volume V 0 ⌧ V in the center of the container. Explain your reasoning.
Master’s Review Examination Final Exam, Phys524, Autumn 2017
3 Ideal Bose gas (50 points total)
A monoatomic ideal Bose gas is placed in a thermally isolated container of volume V. The atoms are spinless and have a mass m. The number of atoms is N. In the initial state the temperature is T , and the gas is Bose condensed, with half its atoms in the condensate.
A. [ 40 points] Determine the following quantities (you may express your answers in terms of dimensionless definite integrals, which you do not need to evaluate).
i. [ 10 points] The condensation temperature T 0. ii. [ 15 points] The energy of the gas. iii. [ 15 points] The pressure.
2018 Master’s Review Examination Classical Mechanics-
This exam consists of three parts, Problem 1, Problem 2 (with sections A-D) and Problem 3 (with sections A-C). Write your solutions for each section in the indicated space.
Some useful formulas:
Newton’s law: F~ = m A~
Euler-Lagrange: d dt
@L
@ q˙ (^) a
@L
@q (^) a
Rotating Frame: ~¨r = ~! ⇥ (~! ⇥ ~r) 2 ~! ⇥ ~r˙ F~ (^) ext /m
Moment of Inertia Tensor:
T (^) ij =
Z
d 3 x ⇢(x 2 (^) ij x (^) i x (^) j ).
Euler Equations (i, j, k cyclic, not summed over):
I (^) i !˙i (I (^) j I (^) k )!j !k = ⌧ (^) i.
Euler-angles: R = R 3 ( )R 1 (✓)R 3 ( ).
Hamilton’s equations:
H(p (^) i , qi ) = p (^) i q˙ (^) i L, q˙ (^) i =
@H
@p (^) i
, p˙ (^) i =
@H
@q (^) i
Poisson brackets:
{f, g} =
@f @q (^) i
@g @p (^) i
@f @p (^) i
@g @q (^) i
df dt
= {f, H} +
@f @t
Adiabatic Invariant / Action variable:
I =
I
pdq with p =
p 2 m(E V ).
2018 Master’s Review Examination Classical Mechanics-
1 Small oscillations (30 points)
Four massless rods of length L are hinged together at their ends to form a rhombus. A particle of mass M is attached at each joint. The opposite corners of the rhombus are joined by springs, each with a spring con- stant k. In the equilibrium (square) configuration the springs are unstretched. The motion is confined to a horizontal plane, and the particles only move along the diagonals of the rhombus. Introduce suitable general- ized coordinates and find the Lagrangian of the system. Deduce the equations of motion and find the frequency of small oscillations about the equilibrium configuration.
L k M
2018 Master’s Review Examination Classical Mechanics-
1 Hamiltonian Dynamics and Adiabatic Invariants (25 points total)
A. [ 15 points] Find the action variable I for a free particle moving in a one dimensional box of width 2a. That is, it is subject to a potential V (x)=0 for |x| a and V (x) = 1 for |x| > a. Derive the angular frequency! = dE/dI of the corresponding angle variable and give a physical interpretation of your result.
B. [ 5 points] Show that if the size a of the box changes adiabatically, the energy goes as 1 /a^2.
C. [ 5 points] Find the time averaged force the particle applies to a single one of the walls.
2018 Master’s Review Examination Quantum Mechanics
Possibly useful equations, in the right context:
J ±^ |j, mi =
p j(j + 1) m(m ± 1) |j, m ± 1 i
r =
p x 2 + y 2 + z 2 , x = r sin ✓ cos , y = r sin ✓ sin , z = r cos ✓ ⇢ ~ 2 2 mr
d 2 dr 2
r +
~ 2 ( + 1)
2 mr 2
(r) = E (^) n (r)
Y 00 =
p 4 ⇡
, Y 11 =
r 3 8 ⇡
sin ✓e i ^ , Y 10 =
r 3 4 ⇡
cos ✓, Y m = ( 1) m^ Y (^)⇤m
Y 22 =
r 15 32 ⇡
sin 2 ✓e 2 i ^ , Y 21 =
r 15 8 ⇡
sin ✓ cos ✓e i ^ , Y 20 =
r 5 16 ⇡
(3 cos^2 ✓ 1)
i~ c˙ (^) m =
X
n
V (^) mn e i!^ mn^ t^ c (^) n , !mn ⌘
E (^) m E (^) n ~
, V (^) mn ⌘ hm|V |ni
c (1) f =
i ~
Z (^) t
0
hf |V |iie i!^ f i^ t^ c (^) n = c (0) n + c (1) n + · · ·
a =
⇣m! 2 ~
x +
i (2m~!) 1 /^2
p, a†^ =
⇣m! 2 ~
x
i (2m~!) 1 /^2
p, [a, a†^ ] = 1
2018 Master’s Review Examination Quantum Mechanics
C. [ 6 points] Define S~ (^) T ⌘ S~ (^) A + S~ (^) B. Mark each of the the following statements True or False and explain your answers.
(c1)It is possible to find a basis in which all the basis states are simultaneously eigen- states of H and eigenstates of S (^) T x.
(c2)It is possible to find a basis in which all the basis states are simultaneously eigen- states of H and eigenstates of S (^) Ax.
D. [ 9 points] Choose one of (All, Some, None) to make each of the following statements true and explain.
(d1)(All, Some, None) of the eigenstates of S~ (^) A · S~ (^) T are eigenstates of H.
(d2)(All, Some, None) of the eigenstates of H are eigenstates of S~ (^) T z.
(d3) (All, Some, None) of the eigenstates of S~ (^) T z are eigenstates of H.
2018 Master’s Review Examination Quantum Mechanics
2 Wigner Eckhart Theorem and Selection rules
A. [ 15 points] A certain atom is in a state | i = |↵jmi with total angular momentum quantum number j = 1/2 and z component m which is either 12 or 12. ↵ is used to represent all other quantum numbers. The position operator is ~r, the momentum operator is p~, and the total angular momentum operator is J~. The radial component of position in spherical coordinates is r, and x, y, z are the Cartesian components of ~r. Which of the following matrix elements can be shown to vanish using rotational sym- metry arguments? Explain why or why not.
(a1) h↵ 12 12 |r 2 |↵ 12 12 i
(a2) h↵ 12 12 |z 2 |↵ 12 12 i
(a3) h↵ 12 12 |p 2 3 p (^2) z |↵ 12 12 i
(a4) h↵ 12 12 |J (^) x |↵ 12 12 i
(a5) h↵ 12 12 |J (^) x^2 J (^) z^2 |↵ 12 12 i
2018 Master’s Review Examination Quantum Mechanics
3 Stationary Perturbation Theory [ 30 points]
The Zeroth order Hamiltonian for two non identical particles of spin 1 in a magnetic field in the z direction is H 0 = μ 1 BS (^1) z + μ 2 BS (^2) z
Treat the spin-spin interaction between the particles as a perturbation
H 1 = S~ 1 · S~ (^2)
and find the energy levels of the system with Hamiltonian H = H 0 + H 1 to first order in the perturbation, assuming |μ 1 | 6 = |μ 2 |. Would your answer change in the limit μ 1 = μ 2? Explain why or why not.
2018 Master’s Review Examination Quantum Mechanics
4 Time Dependent Perturbation Theory [ 20 points]
A one dimensional harmonic oscillator is in its ground state at time t = 0. A weak field is turned on at time t = 0 and turned o↵ at time t = T. The Hamiltonian is
H = p^
2 2 m +^
1 2!^
(^2) mx 2 + V (t)
with V (t) = 0, t < 0 or t > T V (t) = ✏x 2 , 0 < t < T.
In the weak field limit, use first order time dependent perturbation theory to find the prob- ability that the oscillator is in the n th^ excited state at time t > T , for all n for which this probability is non zero.
2018 Master’s Review Examination Electromagnetism
II. ( 30 points total) Radiation. A free point charge e at the origin having mass m is subject to a linearly-polarized plane wave of angular frequency " with electric field amplitude E 0 as shown.
a (10 points). Find the radiated (asymptotic) E and B fields. Assume the charge oscillates at a low enough speed where you can ignore the effects of the incident-wave’s magnetic field. Recall the radiation fields emitted by an electric dipole P are
B r, $ = − '( )*
,- .×^
01 021 P (^) ,42 and E r, $ = − 6 .×B r, $
b (10 points). At a field point a distance r from the origin in the x-z plane and at a polar angle # near $/4, sketch in the figure above the directions of radiation E and B fields at the field point. Describe the polarization of the radiation fields.
c (10 points) Find the intensity of the radiation fields (the time-average of the Poynting vector) in terms of the polar angle # and the azimuthal angle %.
fleLo @ Port'tt
r
\a
2018 Master’s Review Examination Electromagnetism
III. ( 35 Points total) Reflection. A plane wave of angular frequency " is normally incident on a conductor having permittivity and permeability that of free space and real conductivity &. The frequency and conductivity are such that within the conductor the magnitude of conduction and displacement currents are equal.
a (10 points). Within the conductor , find the r elation between conduction (tr ue cur r ents) and displacement cur r ents.
b (10 points). Find the (complex) index of r efr action n within the conductor.
c (15 points). Find the r eflection coefficient r (the r atio of time-aver age incident and r eflected power s).
