Thermodynamics,Statistical Physics, and

Quantum Mechanics

A GUIDETO PHYSICSPROBLEMS

part 2

Thermodynamics,Statistical Physics, and

Quantum Mechanics

Sidney B. CahnNew York University

New York, New York

Gerald D. MahanUniversity of Tennessee

Knoxville, Tennessee, andOak Ridge National Laboratory

Oak Ridge, Tennessee

andBoris E. Nadgorny

Naval Research LaboratoryWashington, D.C.

KLUWER ACADEMIC PUBLISHERSNEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

part 2

eBook ISBN: 0-306-48401-3Print ISBN: 0-306-45291-X

©2004 Kluwer Academic PublishersNew York, Boston, Dordrecht, London, Moscow

Print ©1997 Kluwer Academic/Plenum Publishers

All rights reserved

No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,mechanical, recording, or otherwise, without written consent from the Publisher

Created in the United States of America

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New York

Foreword

It is only rarely realized how important the design of suitable, interestingproblems is in the educational process. This is true for the professor — whoperiodically makes up exams and problem sets which test the effectivenessof his teaching — and also for the student — who must match his skillsand acquired knowledge against these same problems. There is a great needfor challenging problems in all scientific fields, but especially so in physics.Reading a physics paper requires familiarity and control of techniques whichcan only be obtained by serious practice in solving problems. Confidencein performing research demands a mastery of detailed technology whichrequires training, concentration, and reflection — again, gained only byworking exercises.

In spite of the obvious need, there is very little systematic effort madeto provide balanced, doable problems that do more than gratify the ego ofthe professor. Problems often are routine applications of procedures men-tioned in lectures or in books. They do little to force students to reflectseriously about new situations. Furthermore, the problems are often ex-cruciatingly dull and test persistence and intellectual stamina more thaninsight, technical skill, and originality. Another rather serious shortcomingis that most exams and problems carry the unmistakable imprint of theteacher. (In some excellent eastern U.S. universities, problems are cata-logued by instructor, so that a good deal is known about an exam evenbefore it is written.)

In contrast, A Guide to Physics Problems, Part 2 not only serves animportant function, but is a pleasure to read. By selecting problems fromdifferent universities and even different scientific cultures, the authors haveeffectively avoided a one-sided approach to physics. All the problems aregood, some are very interesting, some positively intriguing, a few are crazy;but all of them stimulate the reader to think about physics, not merely totrain you to pass an exam. I personally received considerable pleasure inworking the problems, and I would guess that anyone who wants to be aprofessional physicist would experience similar enjoyment. I must confess

v

Forewordvi

with some embarrassment that some of the problems gave me more troublethan I had expected. But, of course, this is progress. The coming generationcan do with ease what causes the elder one trouble. This book will be agreat help to students and professors, as well as a source of pleasure andenjoyment.

Max DresdenStanford

Preface

Part 2 of A Guide to Physics Problems contains problems from writtengraduate qualifying examinations at many universities in the United Statesand, for comparison, problems from the Moscow Institute of Physics andTechnology, a leading Russian Physics Department. While Part 1 presentedproblems and solutions in Mechanics, Relativity, and Electrodynamics, Part2 offers problems and solutions in Thermodynamics, Statistical Physics, andQuantum Mechanics.

The main purpose of the book is to help graduate students prepare forthis important and often very stressful exam (see Figure P.1). The difficultyand scope of the qualifying exam varies from school to school, but not toodramatically. Our goal was to present a more or less universal set of problemsthat would allow students to feel confident at these exams, regardless of thegraduate school they attended. We also thought that physics majors who areconsidering going on to graduate school may be able to test their knowledgeof physics by trying to solve some of the problems, most of which are notabove the undergraduate level. As in Part 1 we have tried to provide as manydetails in our solutions as possible, without turning to a trade expression ofan exhausted author who, after struggling with the derivation for a couple ofhours writes, “As it can be easily shown....”

Most of the comments to Part 1 that we have received so far have come notfrom the students but from the professors who have to give the exams. Themost typical comment was, “Gee, great, now I can use one of your problemsfor our next comprehensive exam.” However, we still hope that this does notmake the book counterproductive and eventually it will help the students totransform from the state shown in Figure P.1 into a much more comfortablestationary state as in Figure P.2. This picture can be easily attributed to thepresent state of mind of the authors as well, who sincerely hope that Part 3will not be forthcoming any time soon.

Some of the schools do not have written qualifying exams as part of theirrequirements: Brown, Cal-Tech, Cornell, Harvard, UT Austin, Universityof Toronto, and Yale. Most of the schools that give such an exam were

vii

Prefaceviii

happy to trust us with their problems. We wish to thank the Physics Depart-ments of Boston University (Boston), University of Colorado at Boulder (Col-orado), Columbia University (Columbia), University of Maryland (Mary-land), Massachusetts Institute of Technology (MIT), University of Michi-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 of Tennessee atKnoxville (Tennessee), and University of Wisconsin (Wisconsin-Madison).The Moscow Institute ofPhysics and Technology (Moscow Phys-Tech) doesnot 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 theothers, mostly introductory problems, are from their oral entrance exams or

Sidney CahnNew York

Gerald MahanOak Ridge

Boris NadgornyWashington, D.C.

magazines such as Kvant. A few of the problems were compiled by the authorsand have never been published before.

We were happy to hear many encouraging comments about Part 1 fromour 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 theproblems to Part 2, or discussed solutions with us, in particular Dmitri Averin(Stony Brook), Michael Bershadsky (Harvard), Alexander Korotkov (StonyBrook), 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 includethem in the second edition of Part 1, coming out next year. We are indebtedto Max Dresden for writing the Foreword, to Tilo Wettig (Münich) who readmost, of the manuscript, and to Vladimir Gitt and Yair Minsky who drew thehumorous pictures.

Preface ix

Textbooks Used in thePreparation of thisVolume

Chapter 4 — Thermodynamics and Statistical Physics

Landau, L. D., and Lifshitz, E. M., Statistical Physics, Volume 5,part 1 of Course of Theoretical Physics, 3rd ed., Elmsford, New York:Pergamon Press, 1980

Kittel, C., Elementary Statistical Physics, New York: John Wiley andSons, Inc., 1958

Kittel, C., and Kroemer, H., Thermal Physics, 2nd ed., New York:Freeman and Co., 1980

Reif, R., Fundamentals of Statistical and Thermal Physics, New York:McGraw-Hill, 1965

Huang, K., Statistical Mechanics, 2nd ed., New York: John Wileyand Sons, Inc., 1987

Pathria, R. K., Statistical Mechanics, Oxford: Pergamon Press, 1972

1)

2)

3)

4)

5)

6)

Chapter 5 — Quantum Mechanics

Liboff, R. L., Introductory Quantum Mechanics, 2nd ed., Reading,MA: Pergamon Press, 1977

Landau, L. D., and Lifshitz, E. M., Quantum Mechanics, Nonrela-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: PlenumPress, 1980

3)

4)

5)

6)

Contents

PART I: PROBLEMS

Thermodynamics and Statistical Physics4.Introductory Thermodynamics

Why Bother? (Moscow Phys-Tech)Space Station Pressure (MIT)Baron von Münchausen and Intergalactic Travel (MoscowPhys-Tech)Railway Tanker (Moscow Phys-Tech)Magic Carpet (Moscow Phys-Tech)Teacup Engine (Princeton, Moscow Phys-Tech)Grand Lunar Canals (Moscow Phys-Tech)Frozen Solid (Moscow Phys-Tech)Tea in Thermos (Moscow Phys-Tech)Heat Loss (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)

xiii

4.15.4.16.4.17.4.18.4.19.4.20.

4.10.4.11.4.12.4.13.4.14.

4.4.4.5.4.6.4.7.4.8.4.9.

4.1.4.2.4.3.

3

4

3

45567788999

9101011111213

3

4.21.4.22.

xiv

Atmospheric Energy (Rutgers)Puncture (Moscow Phys-Tech)

Heat and WorkCylinder with Massive Piston (Rutgers, MoscowPhys-Tech)Spring Cylinder (Moscow Phys-Tech)Isothermal Compression and Adiabatic Expansion ofIdeal Gas (Michigan)Isochoric Cooling and Isobaric Expansion (MoscowPhys-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 StatisticsPoisson 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 GasHeat Capacities (Princeton)Return of Heat Capacities (Michigan)Nonideal Gas Expansion (Michigan State)van der Waals (MIT)

4.49.4.50.4.51.4.52.

4.45.4.46.4.47.4.48.

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

1314

14

1415

15

161616161717181819

19192020202121212122

2324242425

2626262727

Contents xv

4.53. Critical Parameters (Stony Brook)

Mixtures and Phase SeparationEntropy 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 StatisticsFermi 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)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)

4.69.4.70.4.71.4.72.4.73.4.74.4.75.4.76.4.77.4.78.4.79.4.80.4.81.4.82.4.83.4.84.4.85.4.86.4.87.4.88.

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

2828282829303030313131

323232

3233

333334343435353636373738393940404041414142

xvi Contents

42424343

43

44

444444

4545454546464747474849

51515152525354

5454

5555555656

Fluctuations4.89.4.90.4.91.4.92.

4.93.

4.94.

4.95.4.96.

Applications to Solid State4.97.4.98.4.99.

4.100.4.101.4.102.4.103.4.104.4.105.4.106.

5. Quantum Mechanics

5.1.5.2.5.3.5.4.5.5.5.6.

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,Stony Brook)Number Fluctuations (Colorado (a,b), MoscowPhys-Tech (c))Wiggling Wire (Princeton)LC Voltage Noise (MIT, Chicago)

Thermal Expansion and Heat Capacity (Princeton)Schottky Defects (Michigan State, MIT)Frenkel Defects (Colorado, MIT)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)

One-Dimensional PotentialsShallow 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)Transmission Through a Delta Function Potential(Michigan State, MIT, Princeton)Delta Function in a Box (MIT)Particle in Expanding Box (Michigan State, MIT, StonyBrook)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 Oscillator5.14.5.15.5.16.5.17.5.18.

5656575758

5.19.5.20.5.21.

58595960

60606061616162626363

636363646464

6465

6666666667

6767

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)Time-Dependent Harmonic Oscillator I(Wisconsin-Madison)Time-Dependent Harmonic Oscillator II (Michigan State)Switched-on Field (MIT)Cut the Spring! (MIT)

Angular Momentum and SpinGiven Another Eigenfunction (Stony Brook)Algebra of Angular Momentum (Stony Brook)Triplet Square Well (Stony Brook)Dipolar Interactions (Stony Brook)Spin-Dependent Potential (MIT)Three Spins (Stony Brook)Constant Matrix Perturbation (Stony Brook)Rotating Spin (Maryland, MIT)Nuclear Magnetic Resonance (Princeton, Stony Brook)

Variational CalculationsAnharmonic Oscillator (Tennessee)Linear Potential I (Tennessee)Linear Potential II (MIT, Tennessee)Return of Combined Potential (Tennessee)Quartic in Three Dimensions (Tennessee)Halved Harmonic Oscillator (Stony Brook, Chicago (b),Princeton (b))Helium Atom (Tennessee)

Perturbation TheoryMomentum Perturbation (Princeton)Ramp in Square Well (Colorado)Circle with Field (Colorado, Michigan State)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, StonyBrook (a))

xviii Contents

5.45.5.46.5.47.5.48.5.49.5.50.5.51.5.52.

6868696969707070

5.53.5.54.5.55.5.56.5.57.5.58.5.59.5.60.

717171717272737373

5.61.5.62.5.63.5.64.5.65.5.66.5.67.5.68.5.69.

73737475757676767777

7777787878

797979

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 (Maryland, Michigan State)

of Tritium (Michigan State)

WKBBouncing Ball (Moscow Phys-Tech, Chicago)Truncated Harmonic Oscillator (Tennessee)Stretched Harmonic Oscillator (Tennessee)Ramp Potential (Tennessee)Charge and Plane (Stony Brook)Ramp Phase Shift (Tennessee)Parabolic Phase Shift (Tennessee)Phase Shift for Inverse Quadratic (Tennessee)

Scattering TheoryStep-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)

GeneralSpherical Box with Hole (Stony Brook)Attractive Delta Function in 3D (Princeton)Ionizing Deuterium (Wisconsin-Madison)Collapsed Star (Stanford)Electron in Magnetic Field (Stony Brook, MoscowPhys-Tech)Electric and Magnetic Fields (Princeton)Josephson Junction (Boston)

Contents

PART II: SOLUTIONS

xix

4. Thermodynamics and Statistical PhysicsIntroductory Thermodynamics

4.1.4.2.4.3.

4.4.4.5.4.6.4.7.4.8.4.9.

4.10.4.11.4.12.4.13.4.14.

4.15.4.16.4.17.4.18.4.19.4.20.4.21.4.22.

4.23.

4.24.4.25.

4.26.

4.27.4.28.4.29.4.30.

83838384

8485878990929294959697

99101102103104106107108110

112

112113

115

117118119120122

Why Bother? (Moscow Phys-Tech)Space Station Pressure (MIT)Baron von Münchausen and Intergalactic Travel (MoscowPhys-Tech)Railway Tanker (Moscow Phys-Tech )Magic Carpet (Moscow Phys-Tech )Teacup Engine (Princeton, Moscow Phys-Tech)Grand Lunar Canals (Moscow Phys-Tech)Frozen Solid (Moscow Phys-Tech)Tea in Thermos (Moscow Phys-Tech)Heat Loss (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 WorkCylinder with Massive Piston (Rutgers, MoscowPhys-Tech)Spring Cylinder (Moscow Phys-Tech)Isothermal Compression and Adiabatic Expansion ofIdeal Gas (Michigan)Isochoric Cooling and Isobaric Expansion (MoscowPhys-Tech)Venting (Moscow Phys-Tech)Cylinder and Heat Bath (Stony Brook)Heat Extraction (MIT, Wisconsin-Madison)Heat Capacity Ratio (Moscow Phys-Tech)

xx Contents

123125126127

4.31.4.32.4.33.4.34.

128128130131133135136137138141

4.35.4.36.4.37.4.38.4.39.4.40.4.41.4.42.4.43.4.44.

142146147148149

4.45.4.46.4.47.4.48.

151151152154155156

4.49.4.50.4.51.4.52.4.53.

4.54.4.55.4.56.4.57.4.58.4.59.4.60.4.61.4.62.4.63.

158158159160162163164165166167168

170170171

4.64.4.65.

Otto Cycle (Stony Brook)Joule Cycle (Stony Brook)Diesel Cycle (Stony Brook)Modified Joule–Thomson (Boston)

Ideal Gas and Classical StatisticsPoisson 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 GasHeat Capacities (Princeton)Return of Heat Capacities (Michigan)Nonideal Gas Expansion (Michigan State)van der Waals (MIT)Critical Parameters (Stony Brook)

Mixtures and Phase SeparationEntropy 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 StatisticsFermi Energy of a 1D Electron Gas (Wisconsin-Madison)Two-Dimensional Fermi Gas (MIT, Wisconson-Madison)

Contents xxi

4.66.

4.67.4.68.

4.69.4.70.4.71.4.72.4.73.4.74.4.75.4.76.4.77.4.78.4.79.4.80.4.81.4.82.4.83.4.84.4.85.4.86.4.87.4.88.

4.89.4.90.4.91.4.92.

4.93.

207207209210

210

212

216219221

223223226226

174177180181182183185189189191192194196197200203204204205205206

172173

4.95.4.96.

4.94.

4.97.4.98.4.99.

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)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)

FluctuationsMagnetization 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,Stony Brook)Number Fluctuations (Colorado (a,b), MoscowPhys-Tech (c))Wiggling Wire (Princeton)LC Voltage Noise (MIT, Chicago)

Applications to Solid StateThermal Expansion and Heat Capacity (Princeton)Schottky Defects (Michigan State, MIT)Frenkel Defects (Colorado, MIT)

xxii Contents

4.100.4.101.4.102.4.103.4.104.4.105.4.106.

228230232234236238241

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 MechanicsOne-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)Transmission Through a Delta Function Potential(Michigan State, MIT, Princeton)Delta Function in a Box (MIT)Particle in Expanding Box (Michigan State, MIT, StonyBr0ook)One-Dimensional Coulomb Potential (Princeton)Two Electrons in a Box (MIT)Square Well (MIT)Given the Eigenfunction (Boston, MIT)Combined Potential (Tennessee)

Harmonic Oscillator5.14.5.15.5.16.5.17.5.18.

5.19.5.20.5.21.

Given a Gaussian (MIT)Harmonic Oscillator ABCs (Stony Brook)Number States (Stony Brook)Coupled Oscillators (MIT)Time-Dependent Harmonic Oscillator I(Wisconsin-Madison)Time-Dependent Harmonic Oscillator II (Michigan State)Switched-on Field (MIT)Cut the Spring! (MIT)

243243243244245247248

250250

251253253255255256

257257258260262

263263264265

266266267269271272

Angular Momentum and Spin5.22.5.23.5.24.5.25.5.26.

Given Another Eigenfunction (Stony Brook)Algebra of Angular Momentum (Stony Brook)Triplet Square Well (Stony Brook)Dipolar Interactions (Stony Brook)Spin-Dependent Potential (MIT)

5.

5.7.5.8.

Contents xxiii

5.27.5.28.5.29.5.30.

Variational Calculations5.31.5.32.5.33.5.34.5.35.5.36.

5.37.

Perturbation Theory5.38.5.39.5.40.5.41.5.42.

5.43.5.44.

5.45.5.46.5.47.5.48.5.49.5.50.5.51.5.52.

WKB5.53.5.54.5.55.5.56.5.57.5.58.5.59.5.60.

Three Spins (Stony Brook)Constant Matrix Perturbation (Stony Brook)Rotating Spin (Maryland, MIT)Nuclear Magnetic Resonance (Princeton, Stony Brook)

Anharmonic Oscillator (Tennessee)Linear Potential I (Tennessee)Linear Potential II (MIT, Tennessee)Return of Combined Potential (Tennessee)Quartic in Three Dimensions (Tennessee)Halved Harmonic Oscillator (Stony Brook, Chicago (b),Princeton (b))Helium Atom (Tennessee)

Momentum Perturbation (Princeton)Ramp in Square Well (Colorado)Circle with Field (Colorado, Michigan State)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, StonyBrook (a))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 (Maryland, Michigan State)

of Tritium (Michigan State)

Bouncing Ball (Moscow Phys-Tech, Chicago)Truncated Harmonic Oscillator (Tennessee)Stretched Harmonic Oscillator (Tennessee)Ramp Potential (Tennessee)Charge and Plane (Stony Brook)Ramp Phase Shift (Tennessee)Parabolic Phase Shift (Tennessee)Phase Shift for Inverse Quadratic (Tennessee)

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278278279280281282

283286

287287288289290

290292

293297297298299300302303305

305305306307308309310311311

xxiv Contents

Scattering Theory 312312312313315316317318320321

5.61.5.62.5.63.5.64.5.65.5.66.5.67.5.68.5.69.

322322323324324

5.70.5.71.5.72.5.73.5.74.

5.75.5.76.

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335336336337337338341342342342343344344345345

347

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)

GeneralSpherical Box with Hole (Stony Brook)Attractive Delta Function in 3D (Princeton)Ionizing Deuterium (Wisconsin-Madison)Collapsed Star (Stanford)Electron in Magnetic Field (Stony Brook, MoscowPhys-Tech)Electric and Magnetic Fields (Princeton)Josephson Junction (Boston)

PART III: APPENDIXESApproximate Values of Physical ConstantsSome Astronomical DataOther Commonly Used UnitsConversion Table from Rationalized MKSA to Gaussian UnitsVector IdentitiesVector Formulas in Spherical and Cylindrical CoordinatesLegendre PolynomialsRodrigues’ FormulaSpherical HarmonicsHarmonic OscillatorAngular Momentum and SpinVariational CalculationsNormalized Eigenstates of Hydrogen AtomConversion Table for Pressure UnitsUseful Constants

Bibliography

PROBLEMS