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. KLUWER ACADEMIC PUBLISHERS NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW eBookISBN: 0-306-48401-3 Print ISBN: 0-306-45291-X ©2004 Kluwer Academic Publishers NewYork, Boston, Dordrecht, London, Moscow Print ©1997 Kluwer Academic/Plenum Publishers New York All rights reserved No part of this eBook maybe reproducedor transmitted inanyform or byanymeans,electronic, mechanical, recording, or otherwise,withoutwritten consent from the Publisher Createdin the UnitedStates of America Visit Kluwer Online at: http://kluweronline.com and Kluwer's eBookstoreat: http://ebooks.kluweronline.com 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.