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
Print ©1997 Kluwer Academic/Plenum Publishers
It is only rarely realized how important the design of suitable, interesting
problems is in the educational process. This is true for the professor — who
periodically makes up exams and problem sets which test the effectiveness
of his teaching — and also for the student — who must match his skills
and acquired knowledge against these same problems. There is a great need
for challenging problems in all scientific fields, but especially so in physics.
Reading a physics paper requires familiarity and control of techniques which
can only be obtained by serious practice in solving problems. Confidence
in performing research demands a mastery of detailed technology which
requires training, concentration, and reflection — again, gained only by
working exercises.
In spite of the obvious need, there is very little systematic effort made
to provide balanced, doable problems that do more than gratify the ego of
the professor. Problems often are routine applications of procedures mentioned
in lectures or in books. They do little to force students to reflect
seriously about new situations. Furthermore, the problems are often excruciatingly
dull and test persistence and intellectual stamina more than
insight, technical skill, and originality. Another rather serious shortcoming
is that most exams and problems carry the unmistakable imprint of the
teacher. (In some excellent eastern U.S. universities, problems are catalogued
by instructor, so that a good deal is known about an exam even
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 confesswith 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
Textbooks Used in the
Preparation of this
Volume
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 and
Sons, 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 Wiley
and Sons, Inc., 1987
Pathria, R. K., Statistical Mechanics, Oxford: Pergamon Press, 1972
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, Nonrelativistic
Theory, Volume 3 of Course of Theoretical Physics, 3rd ed.,
Elmsford, New York: Pergamon Press, 1977
Sakurai, J. J., Modern Quantum Mechanics, Menlo Park: Benjamin/
Cummings, 1985
Sakurai, J. J., Advanced Quantum Mechanics, Menlo Park: Benjamin/
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
Contents
PART I: PROBLEMS
Thermodynamics and Statistical Physics
Introductory Thermodynamics
Why Bother? (Moscow Phys-Tech)
Space Station Pressure (MIT)
Baron von Münchausen and Intergalactic Travel (Moscow
Phys-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 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
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 (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)
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
Applications to Solid State
Quantum Mechanics
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), 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 (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 Potentials
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, Stony
Brook)
One-Dimensional Coulomb Potential (Princeton)
Two Electrons in a Box (MIT)
Square Well (MIT)
Given the Eigenfunction (Boston, MIT)
Combined Potential (Tennessee) 56
Harmonic Oscillator
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 Spin
Given 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 Calculations
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)
Perturbation Theory
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, Stony
Brook (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)
WKB
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)
Scattering Theory
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)
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)
PART II: SOLUTIONS
Thermodynamics and Statistical Physics
Introductory Thermodynamics
Why Bother? (Moscow Phys-Tech)
Space Station Pressure (MIT)
Baron von Münchausen and Intergalactic Travel (Moscow
Phys-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 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
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 (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)
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)
Isothermal Compressibility and Mean Square Fluctuation
(Stony Brook)
Energy Fluctuation in Canonical Ensemble (Colorado,
Stony Brook)
Number Fluctuations (Colorado (a,b), Moscow
Phys-Tech (c))
Wiggling Wire (Princeton)
LC Voltage Noise (MIT, Chicago)
Applications to Solid State
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)
Quantum Mechanics
One-Dimensional Potentials
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, Stony
Br0ook)
One-Dimensional Coulomb Potential (Princeton)
Two Electrons in a Box (MIT)
Square Well (MIT)
Given the Eigenfunction (Boston, MIT)
Combined Potential (Tennessee)
Harmonic Oscillator
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)
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, Stony
Brook (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)
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)
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)
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