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A compendium of problems from the Masters Review Examinations for physics graduate students at the University of Washington. It covers the period between 2011 and 2017 and includes problems in Thermodynamics and Statistical Mechanics, Classical Mechanics, Quantum Mechanics, and Electromagnetism. advice on how to study for the exams and encourages students to study all the problems in the compendia. The problems are of varying difficulty, and students are advised to identify the easiest way to solve a problem and not work harder than necessary. The document also emphasizes the importance of good techniques for studying Qual problems, such as dimensional analysis and consideration of simplifying limits with known behavior, which are also useful for real research.
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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:
2011/2012 Master’s Review Examination Thermo & Statistical Mechanics
[ 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 =
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
[ 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
[ 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
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