A guide to physics problems part 2

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Foreword
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 men-
tioned in lectures or in books. They do little to force students to reflect
seriously about new situations. Furthermore, the problems are often ex-
cruciatingly 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 cata-
logued 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 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
difficulty
and scope of the qualifying exam varies from school to school, but not too
dramatically. Our goal was to present a more or less universal set of problems
that would allow students to feel confident at these exams, regardless of the
graduate school they attended. We also thought that physics majors who are
considering going on to graduate school may be able to test their knowledge
of physics by trying to solve some of the problems, most of which are not
above the undergraduate level. As in Part 1 we have tried to provide as many
details in our solutions as possible, without turning to a trade expression of
an exhausted author who, after struggling with the derivation for a couple of
hours writes, “As it can be easily shown ”
Most of the comments to Part 1 that we have received so far have come not
from the students but from the professors who have to give the exams. The
most typical comment was, “Gee, great, now I can use one of your problems
for our next comprehensive exam.” However, we still hope that this does not
make the book counterproductive and eventually it will help the students to
transform from the state shown in Figure P.1 into a much more comfortable
stationary state as in Figure P.2. This picture can be easily attributed to the
present state of mind of the authors as well, who sincerely hope that Part 3
will not be forthcoming any time soon.
Some of the schools do not have written qualifying exams as part of their
requirements: Brown, Cal-Tech, Cornell, Harvard, UT Austin, University
of Toronto, and Yale. Most of the schools that give such an exam were
vii
Preface
viii
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 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
ix
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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
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: 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)
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
4
5
5
6
7
7
8
8
9
9
9
9
10
10
11
11
12
13
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