Thermodynamics, Statistical Physics, and Quantum Mechanics

374 Pages·2004·5.71 MB·English

A GUIDE
TO PHYSICS
PROBLEMS
part 2
Thermodynamics,
Statistical Physics, and
Quantum Mechanics
This page intentionally left blank
part 2
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.
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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 scientificfields,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 scientificcultures, the authors have
effectively avoided a one-sided approach to physics. All the problems are
good,some are very interesting,some positivelyintriguing, 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
vi Foreword
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 physicsmajors 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 bookcounterproductive 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
viii Preface
happy totrust uswiththeir problems. WewishtothankthePhysicsDepart-
ments of Boston University (Boston), University of Colorado at Boulder (Col-
orado), Columbia University (Columbia), University of Maryland (Mary-
land), MassachusettsInstitute ofTechnology (MIT), University ofMichi-
gan (Michigan),MichiganState University(MichiganState), MichiganTech-
nologicalUniversity(MichiganTech),Princeton University (Princeton),
Rutgers University(Rutgers), StanfordUniversity(Stanford),State Univer-
sity ofNewYorkatStony Brook(StonyBrook),University ofTennessee at
Knoxville (Tennessee), and University ofWisconsin (Wisconsin-Madison).
TheMoscowInstitute ofPhysicsandTechnology (MoscowPhys-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
Preface ix
magazinessuch 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 solutionswith us, in particular Dmitri Averin
(Stony Brook), Michael Bershadsky (Harvard), Alexander Korotkov (Stony
Brook), Henry Silsbee (StonyBrook), 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.
Sidney Cahn
New York
Gerald Mahan
Oak Ridge
Boris Nadgorny
Washington, D.C.