Thermodynamics,
Statistical Physics, and
Quantum Mechanics
A GUIDE
TO PHYSICS
PROBLEMS
part 2
This page intentionally left blank
Thermodynamics,
Statistical Physics, and
Quantum Mechanics
Sidney B. Cahn
New York University
New York, New York
Gerald D. Mahan
University of Tennessee
Knoxville, Tennessee, and
Oak Ridge National Laboratory
Oak Ridge, Tennessee
and
Boris E. Nadgorny
Naval Research Laboratory
Washington, D.C.
KLUWER ACADEMIC PUBLISHERS
NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW
part 2
eBook ISBN: 0-306-48401-3
Print ISBN: 0-306-45291-X
©2004 Kluwer Academic Publishers
New York, Boston, Dordrecht, London, Moscow

before it is written.)
In contrast, A Guide to Physics Problems, Part 2 not only serves an
important function, but is a pleasure to read. By selecting problems from
different universities and even different scientific cultures, the authors have
effectively avoided a one-sided approach to physics. All the problems are
good, some are very interesting, some positively intriguing, a few are crazy;
but all of them stimulate the reader to think about physics, not merely to
train you to pass an exam. I personally received considerable pleasure in
working the problems, and I would guess that anyone who wants to be a
professional physicist would experience similar enjoyment. I must confess
v
Foreword
vi
with some embarrassment that some of the problems gave me more trouble
than I had expected. But, of course, this is progress. The coming generation
can do with ease what causes the elder one trouble. This book will be a
great help to students and professors, as well as a source of pleasure and
enjoyment.
Max Dresden
Stanford
Preface
Part 2 of A Guide to Physics Problems contains problems from written
graduate qualifying examinations at many universities in the United States
and, for comparison, problems from the Moscow Institute of Physics and
Technology, a leading Russian Physics Department. While Part 1 presented
problems and solutions in Mechanics, Relativity, and Electrodynamics, Part
2 offers problems and solutions in Thermodynamics, Statistical Physics, and
Quantum Mechanics.
The main purpose of the book is to help graduate students prepare for
this important and often very stressful exam (see Figure P.1). The

gan (Michigan), Michigan State University (Michigan State), Michigan Tech-
nological University (Michigan Tech), Princeton University (Princeton),
Rutgers University (Rutgers), Stanford University (Stanford), State Univer-
sity of New York at Stony Brook (Stony Brook), University ofTennessee at
Knoxville (Tennessee), and University of Wisconsin (Wisconsin-Madison).
The Moscow Institute ofPhysics and Technology (Moscow Phys-Tech) does
not give this type of qualifying exam in graduate school. Some of their prob-
lems came from the final written exam for the physics seniors, some of the
others, mostly introductory problems, are from their oral entrance exams or
Sidney Cahn
New York
Gerald Mahan
Oak Ridge
Boris Nadgorny
Washington, D.C.
magazines such as Kvant. A few of the problems were compiled by the authors
and have never been published before.
We were happy to hear many encouraging comments about Part 1 from
our colleagues, and we are grateful to everybody who took their time to re-
view the book. We wish to thank many people who contributed some of the
problems to Part 2, or discussed solutions with us, in particular Dmitri Averin
(Stony Brook), Michael Bershadsky (Harvard), Alexander Korotkov (Stony
Brook), Henry Silsbee (Stony Brook), and Alexei Stuchebrukhov (UC Davis).
We thank Kirk McDonald (Princeton) and Liang Chen (British Columbia)
for their helpful comments to some problems in Part 1; we hope to include
them in the second edition of Part 1, coming out next year. We are indebted
to Max Dresden for writing the Foreword, to Tilo Wettig (Münich) who read
most, of the manuscript, and to Vladimir Gitt and Yair Minsky who drew the
humorous pictures.
Preface

tivistic Theory, Volume 3 of Course of Theoretical Physics, 3rd ed.,
Elmsford, New York: Pergamon Press, 1977
xi
2)
1)
xii
Textbooks Used in the Preparation of this Volume
Sakurai, J. J., Modern Quantum Mechanics, Menlo Park: Benjamin/
Cummings, 1985
Sakurai, J. J., Advanced Quantum Mechanics, Menlo Park: Benja-
min/Cummings, 1967
Schiff, L. I., Quantum Mechanics, 3rd ed., New York: McGraw-Hill,
1968
Shankar, R., Principles of Quantum Mechanics, New York: Plenum
Press, 1980
3)
4)
5)
6)
Contents
PART I: PROBLEMS
Thermodynamics and Statistical Physics
4.
Introductory Thermodynamics
Why Bother?
(
Moscow Phys-Tech)
Space Station Pressure (MIT)
Baron von Münchausen and Intergalactic Travel (Moscow
Phys-Tech)

Hydrogen Rocket
(
Moscow Phys-Tech
)
Maxwell–Boltzmann Averages (MIT)
Slowly Leaking Box
(
Moscow Phys-Tech, Stony Brook
(a,b))
Surface Contamination
(
Wisconsin-Madison
)
Bell Jar
(
Moscow Phys-Tech
)
Hole in Wall (Princeton)
Ballast Volume Pressure
(
Moscow Phys-Tech
)
Rocket in Drag (Princeton)
Adiabatic Atmosphere (Boston, Maryland)
xiii
4.15.
4.16.
4.17.
4.18.
4.19.

10
10
11
11
12
13
3
4.21.
4.22.
xiv
Atmospheric Energy (Rutgers)
Puncture (Moscow Phys-Tech)
Heat and Work
Cylinder with Massive Piston (Rutgers, Moscow
Phys-Tech)
Spring Cylinder (Moscow Phys-Tech)
Isothermal Compression and Adiabatic Expansion of
Ideal Gas (Michigan)
Isochoric Cooling and Isobaric Expansion (Moscow
Phys-Tech
)
Venting (Moscow Phys-Tech)
Cylinder and Heat Bath (Stony Brook)
Heat Extraction (MIT, Wisconsin-Madison)
Heat Capacity Ratio (Moscow Phys-Tech)
Otto Cycle (Stony Brook)
Joule Cycle (Stony Brook)
Diesel Cycle (Stony Brook)
Modified Joule–Thomson (Boston)
Ideal Gas and Classical Statistics

4.48.
4.35.
4.36.
4.37.
4.38.
4.39.
4.40.
4.41.
4.42.
4.43.
4.44.
4.27.
4.28.
4.29.
4.30.
4.31.
4.32.
4.33.
4.34.
4.26.
4.24.
4.25.
4.23.
Contents
13
14
14
14
15
15

xv
4.53.
Critical Parameters (Stony Brook)
Mixtures and Phase Separation
Entropy of Mixing (Michigan, MIT)
Leaky Balloon (Moscow Phys-Tech)
Osmotic Pressure (MIT)
Clausius–Clapeyron (Stony Brook)
Phase Transition (MIT)
Hydrogen Sublimation in Intergalactic Space (Princeton)
Gas Mixture Condensation (Moscow Phys-Tech)
Air Bubble Coalescence (Moscow Phys-Tech)
Soap Bubble Coalescence (Moscow Phys-Tech)
Soap Bubbles in Equilibrium (Moscow Phys-Tech)
Quantum Statistics
Fermi Energy of a 1D Electron Gas (Wisconsin-Madison)
Two-Dimensional Fermi Gas (MIT, Wisconson-Madison)
Nonrelativistic Electron Gas (Stony Brook,
Wisconsin-Madison, Michigan State)
Ultrarelativistic Electron Gas (Stony Brook)
Quantum Corrections to Equation of State (MIT,
Princeton, Stony Brook)
Speed of Sound in Quantum Gases (MIT)
Bose Condensation Critical Parameters (MIT)
Bose Condensation (Princeton, Stony Brook)
How Hot the Sun? (Stony Brook)
Radiation Force (Princeton, Moscow Phys-Tech, MIT)
Hot Box and Particle Creation (Boston, MIT)
D-Dimensional Blackbody Cavity (MIT)
Fermi and Bose Gas Pressure (Boston)

4.86.
4.87.
4.88.
4.67.
4.68.
4.64.
4.65.
4.66.
4.54.
4.55.
4.56.
4.57.
4.58.
4.59.
4.60.
4.61.
4.62.
4.63.
28
28
28
28
28
29
30
30
30
31
31
31

43
43
43
44
44
44
44
45
45
45
45
46
46
47
47
47
48
49
51
51
51
52
52
53
54
54
54
55
55
55

5.7.
5.8.
5.9.
5.10.
5.11.
5.12.
Magnetization Fluctuation (Stony Brook)
Gas Fluctuations
(Moscow Phys-Tech)
Quivering Mirror
(MIT, Rutgers, Stony Brook)
Isothermal Compressibility and Mean Square Fluctuation
(Stony Brook)
Energy Fluctuation in Canonical Ensemble
(Colorado,
Ston
y
Brook)
Number Fluctuations
(Colorado (a,b), Moscow
Phys-Tech (c))
Wiggling Wire
(Princeton)
LC
Voltage Noise
(MIT, Chicago)
Thermal Expansion and Heat Capacity (Princeton)
Schottky Defects
(Michigan State, MIT)
Frenkel Defects

Delta Function in a Box
(MIT)
Particle in Expanding Box
(Michigan State, MIT, Stony
Brook
)
One-Dimensional Coulomb Potential
(Princeton)
Two Electrons in a Box (MIT)
Square Well
(MIT)
Given the Eigenfunction
(Boston, MIT)
Contents
xvii
5.13.
Combined Potential
(Tennessee)
56
Harmonic Oscillator
5.14.
5.15.
5.16.
5.17.
5.18.
56
56
57
57
58

67
67
68
5.43.
5.44.
5.38.
5.39.
5.40.
5.41.
5.42.
5.37.
5.31.
5.32.
5.33.
5.34.
5.35.
5.36.
5.22.
5.23.
5.24.
5.25.
5.26.
5.27.
5.28.
5.29.
5.30.
Given a Gaussian (MIT)
Harmonic Oscillator ABCs (Stony Brook)
Number States (Stony Brook)
Coupled Oscillators (MIT)

Rotator in Field (Stony Brook)
Finite Size of Nucleus (Maryland, Michigan State,
Princeton, Stony Brook)
U and Perturbation (Princeton)
Relativistic Oscillator (MIT, Moscow Phys-Tech, Stony
Broo
k
(a))
xviii
Contents
5.45.
5.46.
5.47.
5.48.
5.49.
5.50.
5.51.
5.52.
68
68
69
69
69
70
70
70
5.53.
5.54.
5.55.
5.56.

77
77
77
77
78
78
78
79
79
79
5.70.
5.71.
5.72.
5.73.
5.74.
5.75.
5.76.
Spin Interaction
(Princeton)
Spin–Orbit Interaction
(Princeton)
Interacting Electrons
(MIT)
Stark Effect in Hydrogen
(Tennessee)
Hydrogen with Electric and Magnetic Fields
(MIT)
Hydrogen in Capacitor
(Maryland, Michigan State)
Harmonic Oscillator in Field

(Princeton)
Scattering of Two Electrons
(Princeton)
Spin-Dependent Potentials
(Princeton)
Rayleigh Scattering
(Tennessee)
Scattering from Neutral Charge Distribution
(Princeton)
General
Spherical Box with Hole
(Stony Brook)
Attractive Delta Function in 3D
(Princeton)
Ionizing Deuterium
(Wisconsin-Madison)
Collapsed Star (Stanford)
Electron in Magnetic Field
(Stony Brook, Moscow
Phys-Tech
)
Electric and Magnetic Fields
(Princeton)
Josephson Junction
(Boston)
Contents
PART II: SOLUTIONS
xix
4.
Thermodynamics and Statistical Physics

4.30.
83
83
83
84
84
85
87
89
90
92
92
94
95
96
97
99
101
102
103
104
106
107
108
110
112
112
113
115
117

(Moscow Phys-Tech)
Liquid–Solid–Liquid
(Moscow Phys-Tech)
Hydrogen Rocket
(Moscow Phys-Tech)
Maxwell–Boltzmann Averages (MIT)
Slowly Leaking Box (Moscow Phys-Tech, Stony Brook
(a,b))
Surface Contamination (Wisconsin-Madison)
Bell Jar
(Moscow Phys-Tech)
Hole in Wall
(Princeton)
Ballast Volume Pressure
(Moscow Phys-Tech)
Rocket in Drag
(Princeton)
Adiabatic Atmosphere (Boston, Maryland)
Atmospheric Energy
(Rutgers)
Puncture
(Moscow Phys-Tech)
Heat and Work
Cylinder with Massive Piston
(Rutgers, Moscow
Phys-Tech)
Spring Cylinder
(Moscow Phys-Tech)
Isothermal Compression and Adiabatic Expansion of
Ideal Gas (Michigan)

4.35.
4.36.
4.37.
4.38.
4.39.
4.40.
4.41.
4.42.
4.43.
4.44.
142
146
147
148
149
4.45.
4.46.
4.47.
4.48.
151
151
152
154
155
156
4.49.
4.50.
4.51.
4.52.
4.53.

Brook)
Modified Joule–Thomson
(Boston)
Ideal Gas and Classical Statistics
Poisson Distribution in Ideal Gas (Colorado)
Polarization of Ideal Gas (Moscow Phys-Tech)
Two-Dipole Interaction (Princeton)
Entropy of Ideal Gas (Princeton)
Chemical Potential of Ideal Gas (Stony Brook)
Gas in Harmonic Well (Boston)
Ideal Gas in One-Dimensional Potential (Rutgers)
Equipartition Theorem (Columbia, Boston)
Diatomic Molecules in Two Dimensions (Columbia)
Diatomic Molecules in Three Dimensions (Stony Brook,
Michigan State)
Two-Level System (Princeton)
Zipper (Boston)
Hanging Chain (Boston)
Molecular Chain (MIT, Princeton, Colorado)
Nonideal Gas
Heat Capacities (Princeton)
Return of Heat Capacities (Michigan)
Nonideal Gas Expansion (Michigan State)
van der Waals (MIT)
Critical Parameters (Stony Brook)
Mixtures and Phase Separation
Entropy of Mixing (Michigan, MIT)
Leaky Balloon (Moscow Phys-Tech)
Osmotic Pressure (MIT)
Clausius–Clapeyron

4.81.
4.82.
4.83.
4.84.
4.85.
4.86.
4.87.
4.88.
4.89.
4.90.
4.91.
4.92.
4.93.
207
207
209
210
210
212
216
219
221
223
223
226
226
174
177
180
181

Quantum Corrections to Equation of State (MIT,
Princeton
,
Stony Brook)
Speed of Sound in Quantum Gases (MIT)
Bose Condensation Critical Parameters (MIT)
Bose Condensation (Princeton, Stony Brook)
How Hot the Sun? (Stony Brook)
Radiation Force (Princeton, Moscow Phys-Tech, MIT)
Hot Box and Particle Creation (Boston, MIT)
D
-Dimensional Blackbody Cavity (MIT)
Fermi and Bose Gas Pressure (Boston)
Blackbody Radiation and Early Universe (Stony Brook)
Photon Gas (Stony Brook)
Dark Matter (Rutgers)
Einstein Coefficients (Stony Brook)
Atomic Paramagnetism (Rutgers, Boston)
Paramagnetism at High Temperature (Boston)
One-Dimensional Ising Model (Tennessee)
Three Ising Spins (Tennessee)
N Independent Spins (Tennessee)
N
Independent Spins, Revisited (Tennessee)
Ferromagnetism (Maryland, MIT)
Spin Waves in Ferromagnets (Princeton, Colorado)
Fluctuations
Magnetization Fluctuation (Stony Brook)
Gas Fluctuations (Moscow Phys-Tech)
Quivering Mirror (MIT, Rutgers, Stony Brook)

Two-Dimensional Debye Solid (Columbia, Boston)
Einstein Specific Heat (Maryland, Boston)
Gas Adsorption (Princeton, MIT, Stanford)
Thermionic Emission (Boston)
Electrons and Holes
(Boston, Moscow
Phys-Tech)
Adiabatic Demagnetization (Maryland)
Critical Field in Superconductor (Stony Brook, Chicago)
Quantum Mechanics
One-Dimensional Potentials
5.1.
5.2.
5.3.
5.4.
5.5.
5.6.
5.9.
5.10.
5.11.
5.12.
5.13.
Shallow Square Well I (Columbia)
Shallow Square Well II
(Stony
Brook)
Attractive Delta Function Potential I
(Stony Brook)
Attractive Delta Function Potential II (Stony Brook)
Two Delta Function Potentials (Rutgers)

(Wisconsin-Madison)
Time-Dependent Harmonic Oscillator II
(Michigan
State)
Switched-on Field
(MIT)
Cut the Spring!
(MIT)
243
243
243
244
245
247
248
250
250
251
253
253
255
255
256
257
257
258
260
262
263
263

5.29.
5.30.
Variational Calculations
5.31.
5.32.
5.33.
5.34.
5.35.
5.36.
5.37.
Perturbation Theory
5.38.
5.39.
5.40.
5.41.
5.42.
5.43.
5.44.
5.45.
5.46.
5.47.
5.48.
5.49.
5.50.
5.51.
5.52.
WKB
5.53.
5.54.
5.55.

Finite Size of Nucleus
(Maryland, Michigan State,
Princeton, Stony Brook)
U and Perturbation (Princeton)
Relativistic Oscillator
(MIT, Moscow Phys-Tech, Stony
Brook (a))
Spin Interaction (Princeton)
Spin–Orbi
t
Interaction (Princeton)
Interactin
g
Electrons (MIT)
Star
k
Effect in Hydrogen (Tennessee)
Hydroge
n
with Electric and Magnetic Fields (MIT)
Hydroge
n
in Capacitor
(Maryland,
Michigan State)
Harmoni
c
Oscillator in Field (Maryland, Michigan State)
o
f

287
287
288
289
290
290
292
293
297
297
298
299
300
302
303
305
305
305
306
307
308
309
310
311
311
xxiv
Contents
Scattering
Theory
312

328
329
330
335
336
336
337
337
338
341
342
342
342
343
344
344
345
345
34
7
Step-Down Potential (Michigan State, MIT)
Step-Up Potential (Wisconsin-Madison)
Repulsive Square Well (Colorado)
3D Delta Function
(Princeton)
Two-Delta-Function Scattering (Princeton)
Scattering of Two Electrons (Princeton)
Spin-Dependent Potentials (Princeton)
Rayleigh Scattering (Tennessee)
Scattering from Neutral Charge Distribution (Princeton)