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Compendium Masters Review Examinations 2011- Physics Department Physics University of Washington
Preface: This is a compendium of problems from the Masters Review Examinations for physics graduate students at the University of Washington. This compendium covers the period between 2011 and 2017. In Autumn 2011 the Department changed the format from a classic stand alone Qualifying Exam (held late Summer and early Spring) into the current course integrated Masters Review Exam (MRE) format. 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. The level of diculty of the problems on the previous old style Qualifying Exams and the current Masters Review Exams is the same. Problems are grouped here by year. The four exams are in Thermodynamics and Statistical Mechanics, Classical Mechanics, Quantum Mechanics, and Electromagnetism. Here are 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 dont 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. Thats the point of the Qual!
2011/2012 Master’s Review Examination Thermo & Statistical Mechanics
1 classical ideal gas in a central potential
[ 50 points]
A cloud of dilute classical mono-atomic gas molecules is confined to a region of 3 dimensional space by a central potential of the form V (~r) = Ar 3 , with r the distance from the force center at ~r = ~0, and with A > 0.
E =
X^ N
i=
⇥ (^) ~p (^2) i 2 m
i
A rigid spherical object of radius R 0 is located at the center, such that the gas cloud extends from R 0 < r < 1.
A. [ 10 points] Evaluate the canonical partition function.
B. [ 10 points] Determine the heat capacity of the gas for when the radius R 0 and particle number N are held constant.
C. [ 10 points] Determine the average volume hV i = 43 ⇡hr 3 i of the gas cloud.
D. [ 10 points] Determine the pressure exerted by the gas onto the surface of the R 0 -sphere.
E. [ 10 points] Determine the chemical potential of the gas cloud.
2011/2012 Master’s Review Examination Classical Mechanics
1 circular motion in general central potential
[ 30 points]
A particle of mass m moves in a central potential of the form U (r) = (^) rk n , where k > 0 and n > 0 are constants.
A. [ 15 points] Find a radius of a circular orbit R as a function of its orbital momentum M , n, and k.
B. [ 15 points] Find a criterium of stability of the circular orbit. (15 points.)
2011/2012 Master’s Review Examination Classical Mechanics
2 adiabatic theorem and pendulum
[ 70 points]
Consider a simple pendulum of length l and mass m 2 which is attached to a mass m 1 , which can move without friction horizontally along a bar. (See Fig.1.)
A. [ 15 points] Find a Lagrangian of the system and the Lagrange equations.
B. [ 15 points] How many vibrational modes does the system have? Find the eigenfrequency ⌦ of small amplitude oscillation.
C. [ 20 points] Suppose the gravitational constant g(t) changes in time slowly compared to the inverse eigenfrequency ⌦ ^1. In this case both the amplitude and the frequency of oscillations slowly change in time. Write the adiabatic invariant of the problem I in terms of the energy and the frequency the oscillations. Express the amplitude of the oscillations 0 (t) in terms of I, l and g(t). Assume that the amplitude of oscillations of the angle 0 is small.
D. [ 20 points] Suppose that g(t) = g 0 + g 1 cos t. Find an interval of frequencies where the parametric resonance takes place. Consider the case where m 1 = 1 and the first mass does not move.
The following formulas may be useful:
cos a cos b =
cos(a b) + cos(a + b) 2 sin a cos b =
sin(a + b) + sin(a b) 2 sin a sin b =
cos(a b) cos(a + b) 2
2011/2012 Master’s Review Examination Quantum Mechanics
2 Time Evolution
Consider a particle of mass m moving in a one-dimensional potential
V (x) = ↵ (x) ,
where ↵ is a positive constant.
A. [ 15 points] Find the energy level(s) and the normalized wave function(s) of the bound state(s).
B. [ 25 points] At time zero, the wavefunction of the particle (which is not necessarily an eigenfunction) is: (t=0, x) = A e |x|^. with being an arbitrary positive parameter not related to ↵.
i. [ 5 points] Explain qualitatively what happens to the wave function in the limit t! 1. ii. [ 10 points] Find the probability W (x) dx of finding the particle in the interval (x, x + dx) in the limit t! 1. iii. [ 5 points] Evaluate the integral
R L
L dx W^ (x). iv. [ 5 points] Consider the L! 1 limit of the integral you evaluated in part (iii). What is the physical interpretation of this quantity? Compare with the analogous quantity at t = 0 and qualitatively explain the result.
C. [ 10 points] Now put the system in a box of width 2L. That is the potential is as above for |x| < L, but V = 1 for |x| L. Qualitatively describe the spectrum of normalizable eigenstates in this case. How does this change a↵ect the answer to problem B(iv)? Explain.
2011/2012 Master’s Review Examination Electromagnetism
1 Electrostatic line charge with dielectrcum
A. [ 6 points] Write Maxwells equations for D, E, H, and B, in the presence of a free charge density ⇢ and free current density j.
B. [ 5 points] Define the electric displacement D in terms of the electric field E and polar- ization P.
C. [ 14 points] Consider an infinite line in empty space (in the absence of a dielectric) carrying a constant charge density located at x = 0, y = 0 and running in the z- direction. State Gauss’s law and use it to find the electric field E a distance r from the line. Show that everywhere except at the line, the field E can be written in terms of a scalar potential, E = r , where (up to an additive constant)
(x) =
Re [log(x + iy)].
An infinitely long line of linear charge density is placed a distance d above a semi-infinite dielectric medium of permitivity ✏, see figure.
D. [ 10 points] Given that there are no free charges or currents at the boundary of the dielectric medium, state the relations between components of E just above and just below the boundary. Then state these relations in terms of the scalar potential just above the boundary, (^) > , and just below the boundary, (^) <.
E. [ 15 points] Determine the electric field E in the region above the dielectric.
2012/2013 Master’s Review Examination Statistical Mechanics
1 Fluctuating Pendulum (30 points total)
Consider a classical pendulum shown schematically in the figure below. Assume that it is in equilibrium with a thermostat.
You may find useful the following Gaussian integrals:
r ↵ (2⇡)
Z 1
exp(
↵x 2 )dx = 1 and
Z 1
x 2
r ↵ (2⇡)
exp(
↵x 2 )dx =
A. [ 10 points] Write an expression for the probability density to find the pendulum at the angle ✓.
B. [ 10 points] Calculate the variance of small fluctuations of the angle of the pendulum h✓ 2 i in terms of R, m and the gravitational constant g.
C. [ 10 points] State the equipartition theorem and calculate the value of the variance of the velocity hv 2 i. Here v is the velocity of the mass of the pendulum.
2012/2013 Master’s Review Examination Statistical Mechanics
2 Two dimensional Electron Gas (70 points total)
Consider a non-interacting gas of electrons of mass m and energy ✏ (^) p = p^2 / 2 m in 2 dimensions. The total number of electrons is N and the area of the sample is S.
A. [ 10 points] Write the expression for the Fermi distribution of the occupation number of electrons n (^) p in terms of the chemical potential μ, temperature T , and the energy ✏ (^) p.
B. [ 10 points] What is the sign of the chemical potential μ at T = 0? What is the sign of the chemical potential at high temperatures T E (^) F (Boltzmann gas)? Here E (^) F is the Fermi energy at T = 0.
C. [ 10 points] Write an expression for the number of electrons N in terms of the Fermi momentum p (^) F.
D. [ 10 points] Write an expression for the total electron energy E at T = 0 in terms of the number of electrons N and the area of the sample S.
E. [ 10 points] Write an expression for the pressure P at T = 0 in terms of N and S.
F. [ 10 points] Consider the case where there is a magnetic field B (^) k parallel to the 2D sample. Therefore it acts only on electron spins. Estimate the value of the magnetic field B ⇤k which completely polarizes the gas at T = 0 in terms of the Fermi energy E (^) F and the Bohr magneton . Here E (^) F is the Fermi energy in the absence of the magnetic field.
G. [ 10 points] Write an expression for the Helmholtz free energy F at high temperatures T E (^) F , where the translational degrees of freedom of electrons are classical (Boltzmann gas) and, using this result, calculate the heat capacity C (^) V. You do not have to calculate dimensionless integrals which do not contain physical parameters. (In this problem there is no external magnetic field.)
2012/2013 Master’s Review Examination Mechanics
2 (60 points total)
Consider a simple pendulum of length l and mass m whose point of support oscillates vertically according to the law y = A cos ⌦t. (See Fig.1a.)
A. [ 10 points] Derive the Lagrangian of the system. (It is convenient, but not necessary, to omit the total derivative.)
B. [ 10 points] Write the Lagrange equation.
C. [ 10 points] Suppose ⌦ ⌧
p g/l and the amplitude of the oscillation of the pendulum is small. So the equation of motion is linear. In this case both the amplitude and the frequency !(t) of oscillation of the pendulum slowly change in time. Write the adiabatic invariant of the problem, I in terms of the energy E and !. How does E(t) depend on time?
D. [ 10 points] Find an interval of frequency ⌦ ± ⌦, where parametric resonance takes place. Do your calculations to lowest order in the amplitude A. Assume that the oscillations are linear.
E. [ 10 points] Suppose now that ⌦
p g/l. Write the equation of motion and the e↵ective potential averaged over the period of oscillation 2⇡/⌦.
F. [ 10 points] What is the condition on the frequency for stability of the vertical inverted position of the pendulum, as shown in Fig.1b?
m
y=A cos t y=A cos t
l
a) b)
2012/2013 Master’s Review Examination Quantum Mechanics
1 Symmetry in Quantum Mechanics (50 points total)
The parity operator in one dimension, P 1 , reverses the sign of the position coordinate, x:
P 1 (x) = ( x).
A. [ 10 points] Find the eigenvalues and eigenfunctions of the parity operator. Is the parity operator Hermitian? Explain.
B. [ 10 points] Calculate the commutator of P 1 with the position operators, [P 1 , xˆ], and with the position operator squared, [P 1 , xˆ 2 ].
C. [ 10 points] For a 1d particle moving in a potential V (x), what condition does the po- tential V (x) have to satisfy for parity to be a symmetry (that is, [H, P 1 ] = 0)? Explain. If parity is a symmetry, what are the implications for degeneracy of the eigenstates of H?
D. [ 5 points] For the potential V (x) = k|x|, is the lowest energy eigenstate an eigenfunction of P 1? If so, what is its parity? Explain briefly.
The angular momentum operator in 3d is given by L~ = ~x ⇥ ~p.
E. [ 10 points] Calculate the commutator of L~ with the position operators ~x, [L (^) i , x (^) j ], where i, j run over the 3 Cartesian directions x, y and z. Also calculate the commutator of ~L with ~x 2.
F. [ 5 points] For a spinless particle moving in a 3d potential V (~x), what condition does the potential V (~x) have to satisfy for angular momentum to correspond to a symmetry (that is [H, ~L] = 0)? Explain. If angular momentum commutes with the Hamiltonian, what are the implications for the eigenstates of H?
2012/2013 Master’s Review Examination Electricity and Magnetism
1 Electrostatics (50 points total)
You may find the following equations to be relevant:
|r r 0 |
X^1
l=
X^ l
m= l
2 l + 1
Y (^) lm⇤ (br 0 )Y (^) lm (br)
r (^) +
, Y (^) l 0 (✓, ) = (^) m 0
r 2 l + 1 4 ⇡
P (^) l (cos ✓)
A. [ 15 points] A ring of charge Q, of radius r 0 , is centered at a distance z = d above the origin and lies parallel to the xy plane. Determine the scalar potential (r) as a series in Legendre polynomials.
B. [ 11 points] A Green’s function is given by G(r, r 0 ) =
1 |r r 0 | ^
a r 0
1 |r r 00 |
, with r 00 = a 2 r 0 /|r 0 | 2. Define the properties that a Green’s function for electrostatics must have for a Dirichlet boundary value problem and explain why the given G(r, r 0 ) satisfies those properties for a sphere of radius a.
C. [ 12 points] A grounded conducting sphere of radius a < d is placed at the origin and embedded in a dielectric of infinite extent with dielectric constant ". Derive the bound- ary conditions satisfied by the electrostatic field at the surface of the dielectric from Maxwell’s equations.
D. [ 12 points] Now consider the ring of part A a distance d > a above the center of a conducting sphere as in the figure. Express (r) in the dielectric medium resulting from the loop-sphere system as series involving Legendre polynomials.
- CONDUCTORS IN DIELECTRIC EM
.5 Conductors in Dielectric
loop of radius r 0 and charge Q is located above a ounded, conducting sphere of radius a, as shown in e figure. The plane of the loop is displaced vertically om the center of the sphere by a distance d. The tire system is embedded in a dielectric of infinite tent with dielectric constant ".^1
A. [ 12 points] From Maxwell’s equations, derive the boundary conditions satisfied by an electro- static field at the interface between the dielec- tric and the conducting sphere.
he Green’s function G(r, r^0 ) for (minus) the Laplacian in the region outside the conducting here is given by G(r, r^0 ) = 1 4 ⇡
1 |r r^0 | a |r^0 |
1 |r r^00 |
,
here r is the field position, r^0 is the source position, and r^00 a^2 r /|r^0 |^2 (with the origin osen to be the center of the conducting sphere).
B. [ 12 points] What properties does G(r, r^0 ) possess to make it the appropriate Green’s function. Draw a schematic to show the location of the sphere, a point charge, and any image charges that might arise. C. [ 14 points] Express the potential in the dielectric medium resulting from the charged loop as a sum of Legendre polynomials. D. [ 12 points] Find the charge induced on the conducting sphere.
2012/2013 Master’s Review Examination Electricity and Magnetism
2 Maxwell’s Equations and Radiation (50 points total)
A. [ 10 points] Write Maxwell’s equations for D, E, H and B in the presence of a free charge density ⇢ and free current density J. Define H in terms of the magnetic field B, the magnetization M, and the permeability of free space.
B. [ 10 points] Derive equations for the scalar and vector potentials (r, t), A(r, t) in terms of ⇢ and J. Assume that the magnetization M, and the polarization P vanish.
C. [ 20 points] A current distribution with a time-dependence J(r, t) = J(r) cos !t is con- fined to a region of space of size l. Derive an expression for the electric field E(r, t), valid for distances r l, in terms of the three-dimensional Fourier transform of J(r). Determine the time-averaged radiated power per unit solid angle in terms of the same Fourier transform.
D. [ 10 points] Consider the situation as in the previous question, but now !r/c ⌧ 1. Derive an expression for the vector potential A(r) in the Lorentz gauge for regions outside a localized distributions of charge and currents.
2013/2014 Master’s Review Examination Statistical Mechanics
2 High temperature relativistic electrons (50 points total)
Consider a gas of N non-interacting relativistic electrons, ✏ (^) p = cp, in a three dimensional volume V.
A. [ 5 points] Write down the expression for the Fermi-Dirac distribution of the occupation number of electrons n (^) p in terms of the chemical potential μ, temperature T , and the energy ✏ (^) p.
B. [ 5 points] Write down a formal expression for the total number of electrons in the system involving the density of states using your answer in (A).
C. [ 10 points] Consider an experimental setup with a fixed number of electrons and consider very high temperatures where your expression in (B) simplifies. Does the chemical potential μ increase or decrease when the temperature increases?
D. [ 10 points] At these very high temperatures the electron gas behaves classically. Write down a formal statistical definition of the Helmholtz free energy F and evaluate it. Show it takes the form F = N k (^) B T log AT 3 (2) Here A is a constant independent of T. Do not evaluate the value of A.
E. [ 10 points] Calculate the heat capacity C (^) V of the electrons at these high temperatures and compare it with what it would be for non-relativistic electrons.
F. [ 10 points] Calculate the magnetic susceptibility density of the electrons at these high temperatures in terms of the Bohr magneton , the density of the electrons N/V and T. Take into account only the spin (Pauli) part of the susceptibility and neglect the spin-orbit interaction.
2013/2014 Master’s Review Examination Mechanics
1 A particle in a central potential (60 points total)
A particle of mass m = 1 moves in a potential
V =
r 2
where r is the distance to the center.
A. [ 10 points] Write down the Lagrangian of the system, using the polar coordinates r and .
B. [ 10 points] The particle moves from r = 1 along an orbit with angular momentum M. Show that if M <
p 2 then the particle falls into the center of force (capture), and if M >
p 2 it will escape to infinity.
C. [ 10 points] Suppose the velocity of the particle at t = 1 is v 0. Write the condition for capture in terms of the impact parameter b.
D. [ 10 points] Express ˙r and ˙ in terms of E, M , and r, and find a di↵erential equation for the trajectory of the particle, (r).
E. [ 10 points] Find the trajectory r( ) of the particle parameterized by initial angle 0 in the case M 2 = 2.
F. [ 10 points] How many rotations does the trajectory make before approaching r = 0?
