Modern Physicsomeza/assets/modphy-serway.pdf · 2019-02-01 · This book is intended as a modern...

Modern PhysicsThird Edition

RAYMOND A. SERWAYEmeritus

James Madison University

CLEMENT J. MOSESEmeritus

Utica College of Syracuse University

CURT A. MOYERUniversity of North Carolina-Wilmington

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Raymond A. Serway received his doctorate at Illinois Institute of Technology andis Professor Emeritus at James Madison University. Dr. Serway began his teachingcareer at Clarkson University, where he conducted research and taught from1967 to 1980. His second academic appointment was at James Madison Univer-sity as Professor of Physics and Head of the Physics Department from 1980 to1986. He remained at James Madison University until his retirement in 1997. Hewas the recipient of the Madison Scholar Award at James Madison University in1990, the Distinguished Teaching Award at Clarkson University in 1977, and theAlumni Achievement Award from Utica College in 1985. As Guest Scientist at theIBM Research Laboratory in Zurich, Switzerland, he worked with K. Alex Müller,1987 Nobel Prize recipient. Dr. Serway also held research appointments at RomeAir Development center from 1961 to 1963, at IIT Research Institute from 1963to 1967, and as a visiting scientist at Argonne National Laboratory, where he col-laborated with his mentor and friend, Sam Marshall. In addition to earlier edi-tions of this textbook, Dr. Serway is the co-author of Physics for Scientists and Engi-neers, 6th edition, Principles of Physics, 3rd edition, College Physics, 6th edition, andthe high-school textbook Physics, published by Holt, Rinehart, and Winston. Inaddition, Dr. Serway has published more than 40 research papers in the field ofcondensed matter physics and has given more than 60 presentations at profes-sional meetings. Dr. Serway and his wife Elizabeth enjoy traveling, golfing, fish-ing, and spending quality time with their four children and seven grandchildren.

Clement J. Moses is Emeritus Professor of Physics at Utica College. He wasborn and brought up in Utica, New York, and holds an A.B. from HamiltonCollege, an M.S. from Cornell University, and a Ph.D. from State University ofNew York at Binghamton. He has over 30 years of science writing and teachingexperience at the college level, and is a co-author of College Physics, 6th edition,with Serway and Faughn. His research work, both in industrial and universitysettings, has dealt with defects in solids, solar cells, and the dynamics of atomsat surfaces. In addition to science writing, Dr. Moses enjoys reading novels,gardening, cooking, singing, and going to operas.

Curt A. Moyer has been Professor and Chair of the Department of Physics andPhysical Oceanography at the University of North Carolina-Wilmington since1999. Before his appointment to UNC-Wilmington, he taught in the PhysicsDepartment at Clarkson University from 1974 to 1999. Dr. Moyer earned a B.S.from Lehigh University and a Ph.D. from the State University of New York atStony Brook. He has published more than 45 research articles in the fields ofcondensed matter physics and surface science. In addition to being an experi-enced teacher, Dr. Moyer is an advocate for the uses of computers in educa-tion and developed the Web-based QMTools software that accompanies thistext. He and his wife, V. Sue, enjoy traveling and the special times they spendwith their four children and three grandchildren.

About the Authors

iii


This book is intended as a modern physics text for science majors and engi-neering students who have already completed an introductory calculus-basedphysics course. The contents of this text may be subdivided into two broad cat-egories: an introduction to the theories of relativity, quantum and statisticalphysics (Chapters 1 through 10) and applications of elementary quantum the-ory to molecular, solid-state, nuclear, and particle physics (Chapters 11through 16).

OBJECTIVES

Our basic objectives in this book are threefold:

1. To provide simple, clear, and mathematically uncomplicated explana-tions of physical concepts and theories of modern physics.

2. To clarify and show support for these theories through a broad range ofcurrent applications and examples. In this regard, we have attempted toanswer questions such as: What holds molecules together? How do elec-trons tunnel through barriers? How do electrons move through solids?How can currents persist indefinitely in superconductors?

3. To enliven and humanize the text with brief sketches of the historical de-velopment of 20th century physics, including anecdotes and quotationsfrom the key figures as well as interesting photographs of noted scientistsand original apparatus.

COVERAGE

Topics. The material covered in this book is concerned with fundamentaltopics in modern physics with extensive applications in science and engineer-ing. Chapters 1 and 2 present an introduction to the special theory of relativ-ity. Chapter 2 also contains an introduction to general relativity. Chapters 3through 5 present an historical and conceptual introduction to early develop-ments in quantum theory, including a discussion of key experiments that showthe quantum aspects of nature. Chapters 6 through 9 are an introduction tothe real “nuts and bolts” of quantum mechanics, covering the Schrödingerequation, tunneling phenomena, the hydrogen atom, and multielectron

Preface

iv


atoms, while Chapter 10 contains an introduction to statistical physics. The re-mainder of the book consists mainly of applications of the theory set forth inearlier chapters to more specialized areas of modern physics. In particular,Chapter 11 discusses the physics of molecules, while Chapter 12 is an introduc-tion to the physics of solids and electronic devices. Chapters 13 and 14 covernuclear physics, methods of obtaining energy from nuclear reactions,and medical and other applications of nuclear processes. Chapter 15 treatselementary particle physics, and Chapter 16 (available online at http://info.brookscole.com/mp3e) covers cosmology.

CHANGES TO THE THIRD EDITION

The third edition contains two major changes from the second edition: First,this edition has been extensively rewritten in order to clarify difficult concepts,aid understanding, and bring the text up to date with rapidly developing tech-nical applications of quantum physics. Artwork and the order of presentationof certain topics have been revised to help in this process. (Many new photosof physicists have been added to the text, and a new collection of color pho-tographs of modern physics phenomena is also available on the Book Com-panion Web Site.) Typically, each chapter contains new worked examples andfive new end-of-chapter questions and problems. Finally, the Suggestions for Fur-ther Reading have been revised as needed.

Second, this edition refers the reader to a new, online (platform indepen-dent) simulation package, QMTools, developed by one of the authors, CurtMoyer. We think these simulations clarify, enliven, and complement the analyt-ical solutions presented in the text. Icons in the text highlight the problemsdesigned for use with this software, which provides modeling tools to help stu-dents visualize abstract concepts. All instructions about the general use of thesoftware as well as specific instructions for each problem are contained on theBook Companion Web Site, thereby minimizing interruptions to the logicalflow of the text. The Book Companion Web Site at http://info.brookscole.mp3e also contains appendices and much supplemental information on cur-rent physics research and applications, allowing interested readers to digdeeper into many topics.

Specific changes by chapter in this third edition are as follows:

• Chapter 1 in the previous editions, “Relativity,” has been extensively revisedand divided into two chapters. The new Chapter 1, entitled “Relativity I,”contains the history of relativity, new derivations of the Lorentz coordinateand velocity transformations, and a new section on spacetime and causality.

• Chapter 2, entitled “Relativity II,” covers relativistic dynamics and energyand includes new material on general relativity, gravitational radiation,and the applications GPS (Global Positioning System) and LIGO (theLaser Interferometer Gravitational-wave Observatory).

• Chapter 3 has been streamlined with a more concise treatment of theRayleigh-Jeans and Planck blackbody laws. Material necessary for a com-plete derivation of these results has been placed on our Book CompanionWeb Site.

• Chapter 5 contains a new section on the invention and principles of op-eration of transmission and scanning electron microscopes.

PREFACE v


• Chapter 6, “Quantum Mechanics in One Dimension,” features a newapplication on the principles of operation and utility of CCDs (Charge-Coupled Devices).

• Chapter 8, “Quantum Mechanics in Three Dimensions,” includes a newdiscussion on the production and spectroscopic study of anti-hydrogen, astudy which has important consequences for several fundamental physicalquestions.

• Chapter 10 presents new material on the connection of wavefunctionsymmetry to the Bose-Einstein condensation and the Pauli exclusion prin-ciple, as well as describing potential applications of Bose-Einstein conden-sates.

• Chapter 11 contains new material explaining Raman scattering, fluores-cence, and phosphorescence, as well as giving applications of theseprocesses to pollution detection and biomedical research. This chapterhas also been streamlined with the discussion of overlap integrals beingmoved to the Book Companion Web Site.

• Chapter 12 has been carefully revised for clarification and features newmaterial on semiconductor devices, in particular MOSFETs and chips. Inaddition, the most important facts about superconductivity have beensummarized, updated, and included in Chapter 12. For those desiringmore material on superconductivity, the entire superconductivity chapterfrom previous editions is available at the Book Companion Web Sitealong with essays on the history of the laser and solar cells.

• Chapter 13 contains new material on MRI (Magnetic Resonance Imag-ing) and an interesting history of the determination of the age of theEarth.

• Chapter 14 presents updated sections on fission reactor safety and wastedisposal, fusion reactor results, and applications of nuclear physics totracing, neutron activation analysis, radiation therapy, and other areas.

• Chapter 15 has been extensively rewritten in an attempt to convey thethrust toward unification in particle physics. By way of achieving this goal,new discussions of positrons, neutrino mass and oscillation, conservationlaws, and grand unified theories, including supersymmetry and string the-ory, have been introduced.

• Chapter 16 is a new chapter devoted exclusively to the exciting topic ofthe origin and evolution of the universe. Topics covered include the dis-covery of the expanding universe, primordial radiation, inflation, the fu-ture evolution of the universe, dark matter, dark energy, and the acceler-ating expansion of the universe. This cosmology chapter is available onour Book Companion Web Site.

FEATURES OF THIS TEXT

QMTools Five chapters contain several new problems requiring the use ofour simulation software, QMTools. QMTools is a sophisticated interactive learn-ing tool with considerable flexibility and scope. Using QMTools, students cancompose matter-wave packets and study their time evolution, find stationarystate energies and wavefunctions, and determine the probability for particletransmission and reflection from nearly any potential well or barrier. Access toQMTools is available online at http://info.brookscole.com/mp3e.

vi PREFACE


PREFACE vii

Style. We have attempted to write this book in a style that is clear and suc-cinct yet somewhat informal, in the hope that readers will find the text appeal-ing and enjoyable to read. All new terms have been carefully defined, and wehave tried to avoid jargon.

Worked Examples. A large number of worked examples of varying difficultyare presented as an aid in understanding both concepts and the chain of rea-soning needed to solve realistic problems. In many cases, these examples willserve as models for solving some end-of-chapter problems. The examples areset off with colored bars for ease of location, and most examples are given ti-tles to describe their content.

Exercises Following Examples. As an added feature, many of the workedexamples are followed immediately by exercises with answers. These exercisesare intended to make the textbook more interactive with the student, andto test immediately the student’s understanding of key concepts and problem-solving techniques. The exercises represent extensions of the worked examplesand are numbered in case the instructor wishes to assign them for homework.

Problems and Questions. An extensive set of questions and problems is in-cluded at the end of each chapter. Most of the problems are listed by sectiontopic. Answers to all odd-numbered problems are given at the end of thebook. Problems span a range of difficulty and more challenging problemshave colored numbers. Most of the questions serve to test the student’s under-standing of the concepts presented in a given chapter, and many can be usedto motivate classroom discussions.

Units. The international system of units (SI) is used throughout the text.Occasionally, where common usage dictates, other units are used (such as theangstrom, Å, and cm�1, commonly used by spectroscopists), but all such unitsare carefully defined in terms of SI units.

Chapter Format. Each chapter begins with a preview, which includes a briefdiscussion of chapter objectives and content. Marginal notes set in color are usedto locate important concepts and equations in the text. Important statements areitalicized or highlighted, and important equations are set in a colored box foradded emphasis and ease of review. Each chapter concludes with a summary,which reviews the important concepts and equations discussed in that chapter.

In addition, many chapters contain special topic sections which are clearlymarked optional. These sections expose the student to slightly more advancedmaterial either in the form of current interesting discoveries or as fuller devel-opments of concepts or calculations discussed in that chapter. Many of thesespecial topic sections will be of particular interest to certain student groupssuch as chemistry majors, electrical engineers, and physics majors.

Guest Essays. Another feature of this text is the inclusion of interesting ma-terial in the form of essays by guest authors. These essays cover a wide range oftopics and are intended to convey an insider’s view of exciting current devel-opments in modern physics. Furthermore, the essay topics present extensionsand/or applications of the material discussed in specific chapters. Some of the


essay topics covered are recent developments in general relativity, the scan-ning tunneling microscope, superconducting devices, the history of the laser,laser cooling of atoms, solar cells, and how the top quark was detected. Theguest essays are either included in the text or referenced as being on our Website at appropriate points in the text.

Mathematical Level. Students using this text should have completed a com-prehensive one-year calculus course, as calculus is used throughout the text.However, we have made an attempt to keep physical ideas foremost so as not toobscure our presentations with overly elegant mathematics. Most steps are shownwhen basic equations are developed, but exceptionally long and detailed proofswhich interrupt the flow of physical arguments have been placed in appendices.

Appendices and Endpapers. The appendices in this text serve several pur-poses. Lengthy derivations of important results needed in physical discussionshave been placed on our Web site to avoid interrupting the main flow of argu-ments. Other appendices needed for quick reference are located at the end ofthe book. These contain physical constants, a table of atomic masses, and a listof Nobel prize winners. The endpapers inside the front cover of the book con-tain important physical constants and standard abbreviations of units used inthe book, and conversion factors for quick reference, while a periodic table isincluded in the rear cover endpapers.

Ancillaries. The ancillaries available with this text include a Student Solu-tions Manual, which has solutions to all odd-numbered problems in the book,an Instructor’s Solutions Manual, consisting of solutions to all problems in thetext, and a Multimedia Manager, a CD-ROM lecture tool that contains digitalversions of all art and selected photographs in the text.

TEACHING OPTIONS

As noted earlier, the text may be subdivided into two basic parts: Chapters 1through 10, which contain an introduction to relativity, quantum physics, andstatistical physics, and Chapters 11 through 16, which treat applications tomolecules, the solid state, nuclear physics, elementary particles, and cosmol-ogy. It is suggested that the first part of the book be covered sequentially. How-ever, the relativity chapters may actually be covered at any time because E2 �p2c2 � m2c4 is the only formula from these chapters which is essential for sub-sequent chapters. Chapters 11 through 16 are independent of one anotherand can be covered in any order with one exception: Chapter 14, “NuclearPhysics Applications,” should follow Chapter 13, “Nuclear Structure.”

A traditional sophomore or junior level modern physics course for science,mathematics, and engineering students should cover most of Chapters 1through 10 and several of the remaining chapters, depending on the studentmajor. For example, an audience consisting mainly of electrical engineering stu-dents might cover most of Chapters 1 through 10 with particular emphasis ontunneling and tunneling devices in Chapter 7, the Fermi-Dirac distribution inChapter 10, semiconductors in Chapter 12, and radiation detectors in Chapter14. Chemistry and chemical engineering majors could cover most of Chapters 1through 10 with special emphasis on atoms in Chapter 9, classical and quantum

viii PREFACE


PREFACE ix

statistics in Chapter 10, and molecular bonding and spectroscopy in Chapter 11.Mathematics and physics majors should pay special attention to the unique de-velopment of operator methods and the concept of sharp and fuzzy observablesintroduced in Chapter 6. The deep connection of sharp observables with classi-cally conserved quantities and the powerful role of sharp observables in shapingthe form of system wavefunctions is developed more fully in Chapter 8.

Our experience has shown that there is more material contained in thisbook than can be covered in a standard one semester three-credit-hourcourse. For this reason, one has to “pick-and-choose” from topics in the sec-ond part of the book as noted earlier. However, the text can also be used in atwo-semester sequence with some supplemental material, such as one of manymonographs on relativity, and/or selected readings in the areas of solid state,nuclear, and elementary particle physics. Some selected readings are sug-gested at the end of each chapter.

ACKNOWLEDGMENTS

We wish to thank the users and reviewers of the first and second editions whogenerously shared with us their comments and criticisms. In preparing thisthird edition we owe a special debt of gratitude to the following reviewers:

Melissa Franklin, Harvard University

Edward F. Gibson, California State University, Sacramento

Grant Hart, Brigham Young University

James Hetrick, University of the Pacific

Andres H. La Rosa, Portland State University

Pui-tak (Peter) Leung, Portland State University

Peter Moeck, Portland State University

Timothy S. Sullivan, Kenyon College

William R. Wharton, Wheaton College

We thank the professional staff at Brooks-Cole Publishing for their fine workduring the development and production of this text, especially Jay Campbell,Chris Hall, Teri Hyde, Seth Dobrin, Sam Subity, Kelley McAllister, StaceyPurviance, Susan Dust Pashos, and Dena Digilio-Betz. We thank Suzon O.Kister for her helpful reference work, and all the authors of our guest essays:Steven Chu, Melissa Franklin, Roger A. Freedman, Clark A. Hamilton, Paul K.Hansma, David Kestenbaum, Sam Marshall, John Meakin, and Clifford M. Will.

Finally, we thank all of our families for their patience and continual support.

Raymond A. SerwayLeesburg, VA 20176

Clement J. MosesDurham, NC 27713

Curt A. MoyerWilmington, NC 28403

December 2003


1 Relativity I 1

2 Relativity II 41

3 The Quantum Theory of Light 65

4 The Particle Nature of Matter 106

5 Matter Waves 151

6 Quantum Mechanics in One Dimension 191

7 Tunneling Phenomena 231

8 Quantum Mechanics in Three Dimensions 260

9 Atomic Structure 295

10 Statistical Physics 334

11 Molecular Structure 372

12 The Solid State 404

13 Nuclear Structure 463

14 Nuclear Physics Applications 503

15 Elementary Particles 547

16 Cosmology (Web Only)

Appendix A Best Known Values for Physical Constants A.1

Appendix B Table of Selected Atomic Masses A.2

Appendix C Nobel Prizes A.7

Answers to Odd-Numbered Problems A.12

Index I.1

Contents Overview

x


Contents

1 RELATIVITY I 1

1.1 Special Relativity 21.2 The Principle of Relativity 3

The Speed of Light 6

1.3 The Michelson–Morley Experiment 7Details of the Michelson–Morley

Experiment 8

1.4 Postulates of Special Relativity 101.5 Consequences of Special Relativity 13

Simultaneity and the Relativity of Time 14Time Dilation 15Length Contraction 18The Twins Paradox (Optional) 21The Relativistic Doppler Shift 22

1.6 The Lorentz Transformation 25Lorentz Velocity Transformation 29

1.7 Spacetime and Causality 31Summary 35

2 RELATIVITY II 41

2.1 Relativistic Momentum and the Relativistic Form of Newton’s Laws 41

2.2 Relativistic Energy 442.3 Mass as a Measure of Energy 482.4 Conservation of Relativistic

Momentum and Energy 522.5 General Relativity 53

Gravitational Radiation, or a Good Wave Is Hard to Find 56

Summary 59Web Essay The Renaissance of General Relativity

Clifford M. Will

3 THE QUANTUM THEORYOF LIGHT 65

3.1 Hertz’s Experiments—Light as an Electromagnetic Wave 66

3.2 Blackbody Radiation 68Enter Planck 72The Quantum of Energy 74

3.3 The Rayleigh–Jeans Law and Planck’s Law (Optional) 77Rayleigh–Jeans Law 77Planck’s Law 79

3.4 Light Quantization and the Photoelectric Effect 80

3.5 The Compton Effect and X-Rays 86X-Rays 86The Compton Effect 89

3.6 Particle–Wave Complementarity 943.7 Does Gravity Affect Light? (Optional) 95

Summary 98Web Appendix Calculation of the Number of Modes

of Waves in a CavityPlanck’s Calculation of the AverageEnergy of an Oscillator

4 THE PARTICLE NATUREOF MATTER 106

4.1 The Atomic Nature of Matter 1064.2 The Composition of Atoms 108

Millikan’s Value of the Elementary Charge 113Rutherford’s Model of the Atom 119

4.3 The Bohr Atom 125Spectral Series 126Bohr’s Quantum Model of the Atom 130

xi


xii CONTENTS

4.4 Bohr’s Correspondence Principle, or Why Is Angular Momentum Quantized? 139

4.5 Direct Confirmation of Atomic Energy Levels: The Franck–Hertz Experiment 141

Summary 143

5 MATTER WAVES 151

5.1 The Pilot Waves of De Broglie 152De Broglie’s Explanation of

Quantization in the Bohr Model 153

5.2 The Davisson–Germer Experiment 154The Electron Microscope 159

5.3 Wave Groups and Dispersion 164Matter Wave Packets 169

5.4 Fourier Integrals (Optional) 170Constructing Moving Wave Packets 173

5.5 The Heisenberg Uncertainty Principle 173A Different View of the Uncertainty Principle 175

5.6 If Electrons Are Waves, What’s Waving? 178

5.7 The Wave–Particle Duality 179The Description of Electron

Diffraction in Terms of � 179A Thought Experiment: Measuring

Through Which Slit the Electron Passes 184

5.8 A Final Note 186Summary 186

6 QUANTUM MECHANICS INONE DIMENSION 191

6.1 The Born Interpretation 1916.2 Wavefunction for a Free Particle 1946.3 Wavefunctions in the Presence

of Forces 1976.4 The Particle in a Box 200

Charge-Coupled Devices (CCDs) 205

6.5 The Finite Square Well (Optional) 2096.6 The Quantum Oscillator 2126.7 Expectation Values 2176.8 Observables and Operators 221

Quantum Uncertainty and the Eigenvalue Property (Optional) 222

Summary 224

7 TUNNELING PHENOMENA 231

7.1 The Square Barrier 2317.2 Barrier Penetration: Some

Applications 238Field Emission 239� Decay 242Ammonia Inversion 245Decay of Black Holes 247

Summary 248Essay The Scanning Tunneling Microscope

Roger A. Freedman and Paul K. Hansma 253

8 QUANTUM MECHANICS INTHREE DIMENSIONS 260

8.1 Particle in a Three-Dimensional Box 2608.2 Central Forces and Angular

Momentum 2668.3 Space Quantization 2718.4 Quantization of Angular Momentum and

Energy (Optional) 273Lz Is Sharp: The Magnetic Quantum Number 275�L� Is Sharp: The Orbital Quantum Number 276E Is Sharp: The Radial Wave Equation 276

8.5 Atomic Hydrogen and Hydrogen-like Ions 277The Ground State of Hydrogen-like Atoms 282Excited States of Hydrogen-like Atoms 284

8.6 Antihydrogen 287Summary 289

9 ATOMIC STRUCTURE 295

9.1 Orbital Magnetism and the Normal Zeeman Effect 296

9.2 The Spinning Electron 3029.3 The Spin–Orbit Interaction and

Other Magnetic Effects 3099.4 Exchange Symmetry and the

Exclusion Principle 3129.5 Electron Interactions and Screening

Effects (Optional) 3169.6 The Periodic Table 3199.7 X-Ray Spectra and Moseley’s Law 325

Summary 328


CONTENTS xiii

10 STATISTICAL PHYSICS 334

10.1 The Maxwell–Boltzmann Distribution 335The Maxwell Speed Distribution for

Gas Molecules in Thermal Equilibrium at Temperature T 341

The Equipartition of Energy 343

10.2 Under What Physical Conditions Are Maxwell–Boltzmann Statistics Applicable? 344

10.3 Quantum Statistics 346Wavefunctions and the Bose–Einstein

Condensation and Pauli Exclusion Principle 346

Bose–Einstein and Fermi–Dirac Distributions 347

10.4 Applications of Bose–Einstein Statistics 351Blackbody Radiation 351Einstein’s Theory of Specific Heat 352

10.5 An Application of Fermi–Dirac Statistics: The Free-Electron Gas Theory of Metals 356

Summary 360Essay Laser Manipulation of Atoms

Steven Chu 366

11 MOLECULAR STRUCTURE 372

11.1 Bonding Mechanisms: A Survey 373Ionic Bonds 374Covalent Bonds 374van der Waals Bonds 375The Hydrogen Bond 377

11.2 Molecular Rotation and Vibration 377Molecular Rotation 378Molecular Vibration 381

11.3 Molecular Spectra 38511.4 Electron Sharing and the

Covalent Bond 390The Hydrogen Molecular Ion 390The Hydrogen Molecule 396

11.5 Bonding in Complex Molecules (Optional) 397

Summary 399Web Appendix Overlap Integrals of Atomic

Wavefunctions

12 THE SOLID STATE 404

12.1 Bonding in Solids 405Ionic Solids 405Covalent Solids 408Metallic Solids 409Molecular Crystals 409Amorphous Solids 410

12.2 Classical Free Electron Model of Metals 413Ohm’s Law 414Classical Free Electron Theory

of Heat Conduction 418

12.3 Quantum Theory of Metals 420Replacement of vrms with vF 421Wiedemann–Franz Law Revisited 422Quantum Mean Free Path of Electrons 423

12.4 Band Theory of Solids 425Isolated-Atom Approach to Band Theory 425Conduction in Metals, Insulators, and

Semiconductors 426Energy Bands from Electron Wave Reflections 429

12.5 Semiconductor Devices 433The p -n Junction 433Light-Emitting and -Absorbing

Diodes—LEDs and Solar Cells 436The Junction Transistor 437The Field-Effect Transistor (FET) 439The Integrated Circuit 441

12.6 Superconductivity 44312.7 Lasers 447

Absorption, Spontaneous Emission, and Stimulated Emission 447

Population Inversion and Laser Action 449Semiconductor Lasers 451

Summary 454Web Essay The Invention of the Laser

S. A. Marshall

Web Essay Photovoltaic Conversion John D. Meakin

Web Chapter Superconductivity

13 NUCLEAR STRUCTURE 463

13.1 Some Properties of Nuclei 464Charge and Mass 465Size and Structure of Nuclei 466Nuclear Stability 468Nuclear Spin and Magnetic Moment 469Nuclear Magnetic Resonance and Magnetic

Resonance Imaging 470


xiv CONTENTS

13.2 Binding Energy and Nuclear Forces 47213.3 Nuclear Models 476

Liquid-Drop Model 476Independent-Particle Model 478Collective Model 479

13.4 Radioactivity 47913.5 Decay Processes 484

Alpha Decay 484Beta Decay 487Carbon Dating 489Gamma Decay 491

13.6 Natural Radioactivity 492Four Radioactive Series 492Determining the Age of the Earth 493

Summary 495

14 NUCLEAR PHYSICS APPLICATIONS 503

14.1 Nuclear Reactions 50314.2 Reaction Cross Section 50614.3 Interactions Involving Neutrons 50814.4 Nuclear Fission 51014.5 Nuclear Reactors 513

Neutron Leakage 515Regulating Neutron Energies 515Neutron Capture 515Control of Power Level 515Safety and Waste Disposal 516

14.6 Nuclear Fusion 517Fusion Reactions 518Magnetic Field Confinement 521Inertial Confinement 523Fusion Reactor Design 524Advantages and Problems of Fusion 526

14.7 Interaction of Particles with Matter 526Heavy Charged Particles 526Electrons 528Photons 528

14.8 Radiation Damage in Matter 53014.9 Radiation Detectors 53214.10 Uses of Radiation 536

Tracing 536Neutron Activation Analysis 537Radiation Therapy 538Food Preservation 539

Summary 539

15 ELEMENTARY PARTICLES 547

15.1 The Fundamental Forces in Nature 54815.2 Positrons and Other Antiparticles 550

15.3 Mesons and the Beginning of Particle Physics 553

15.4 Classification of Particles 556Hadrons 556Leptons 557The Solar Neutrino Mystery and

Neutrino Oscillations 558

15.5 Conservation Laws 559Baryon Number 560Lepton Number 560

15.6 Strange Particles and Strangeness 56115.7 How Are Elementary Particles Produced

and Particle Properties Measured? 563Resonance Particles 564Energy Considerations in Particle Production 568

15.8 The Eightfold Way 57115.9 Quarks 574

The Original Quark Model 574Charm and Other Developments 575

15.10 Colored Quarks, or Quantum Chromodynamics 577Experimental Evidence for Quarks 578Explanation of Nuclear Force in Terms

of Quarks 579

15.11 Electroweak Theory and the Standard Model 580

15.12 Beyond the Standard Model 582Grand Unification Theory and Supersymmetry 582String Theory—A New Perspective 582

Summary 583Essay How to Find a Top Quark 590

Melissa Franklin and David Kestenbaum

16 COSMOLOGY (Web Only)

APPENDIX A BEST KNOWN VALUESFOR PHYSICALCONSTANTS A.1

APPENDIX B TABLE OF SELECTEDATOMIC MASSES A.2

APPENDIX C NOBEL PRIZES A.7

ANSWERS TO ODD-NUMBEREDPROBLEMS A.12

INDEX I.1


Chapter 6Section 6.2, after Example 6.4Exercise 3, following Example 6.8

Problems 22, 27, 36

Chapter 7Exercise 1, following Example 7.1Section 7.2, after Example 7.6Subsection on Ammonia Inversion in Section 7.2

Problems 8, 9, 10, 19, 20

Chapter 8Problems 27, 28, 32, 33

Chapter 9Problems 19, 20

Chapter 11Subsection on The Hydrogen Molecular Ion in Section 11.4

Problems 16, 17, 22, 23

QM ToolsText References to the Software

xv


1. A. Piccard 2. E. Henriot 3. P. Ehrenfest 4. E. Herzen 5. Th. de Donder 6. E. Schroedinger 7. E. Verschaffelt 8. W. Pauli 9. W. Heisenberg10. R.H. Fowler

11. L. Brillouin12. P. Debye13. M. Knudsen14. W.L. Bragg15. H.A. Kramers16. P.A.M. Dirac17. A.H. Compton18. L.V. de Broglie 19. M. Born20. N. Bohr

21. I. Langmuir22. M. Planck23. M. Curie24. H.A. Lorentz25. A. Einstein26. P. Langevin27. C.E. Guye28. C.T.R. Wilson29. O.W. Richardson

The “architects” of modern physics. This unique photograph shows many eminent scientists who participated in the Fifth International Congress of Physics held in 1927by the Solvay Institute in Brussels. At this and similar conferences, held regularly from1911 on, scientists were able to discuss and share the many dramatic developmentsin atomic and nuclear physics. This elite company of scientists includes fifteen Nobelprize winners in physics and three in chemistry. (Photograph courtesy of AIP Niels BohrLibrary)


1

1Relativity I

1.1 Special Relativity1.2 The Principle of Relativity

The Speed of Light

1.3 The Michelson–MorleyExperimentDetails of the Michelson–Morley

Experiment

1.4 Postulates of Special Relativity1.5 Consequences of Special Relativity

Simultaneity and the Relativity of TimeTime DilationLength ContractionThe Twins Paradox (Optional)The Relativistic Doppler Shift

1.6 The Lorentz TransformationLorentz Velocity Transformation

1.7 Spacetime and CausalitySummary

Chapter Outline

At the end of the 19th century, scientists believed that they had learnedmost of what there was to know about physics. Newton’s laws of motion andhis universal theory of gravitation, Maxwell’s theoretical work in unifyingelectricity and magnetism, and the laws of thermodynamics and kinetic the-ory employed mathematical methods to successfully explain a wide variety ofphenomena.

However, at the turn of the 20th century, a major revolution shook theworld of physics. In 1900 Planck provided the basic ideas that led to the quan-tum theory, and in 1905 Einstein formulated his special theory of relativity.The excitement of the times is captured in Einstein’s own words: “It was a mar-velous time to be alive.” Both ideas were to have a profound effect on ourunderstanding of nature. Within a few decades, these theories inspired newdevelopments and theories in the fields of atomic, nuclear, and condensed-matter physics.

Although modern physics has led to a multitude of important technologicalachievements, the story is still incomplete. Discoveries will continue to bemade during our lifetime, many of which will deepen or refine our under-standing of nature and the world around us. It is still a “marvelous time tobe alive.”


1.1 SPECIAL RELATIVITY

Light waves and other forms of electromagnetic radiation travel through freespace at the speed c � 3.00 � 108 m/s. As we shall see in this chapter, thespeed of light sets an upper limit for the speeds of particles, waves, and thetransmission of information.

Most of our everyday experiences deal with objects that move at speedsmuch less than that of light. Newtonian mechanics and early ideas on spaceand time were formulated to describe the motion of such objects, and thisformalism is very successful in describing a wide range of phenomena. Al-though Newtonian mechanics works very well at low speeds, it fails when ap-plied to particles whose speeds approach that of light. Experimentally, onecan test the predictions of Newtonian theory at high speeds by acceleratingan electron through a large electric potential difference. For example, it ispossible to accelerate an electron to a speed of 0.99c by using a potentialdifference of several million volts. According to Newtonian mechanics, ifthe potential difference (as well as the corresponding energy) is increasedby a factor of 4, then the speed of the electron should be doubled to 1.98c.However, experiments show that the speed of the electron — as well as thespeeds of all other particles in the universe — always remains less than thespeed of light, regardless of the size of the accelerating voltage. In part be-cause it places no upper limit on the speed that a particle can attain, New-tonian mechanics is contrary to modern experimental results and is there-fore clearly a limited theory.

In 1905, at the age of 26, Albert Einstein published his special theory of rela-tivity. Regarding the theory, Einstein wrote,

The relativity theory arose from necessity, from serious and deep contradictions inthe old theory from which there seemed no escape. The strength of the new theorylies in the consistency and simplicity with which it solves all these difficulties, usingonly a few very convincing assumptions. . . .1

Although Einstein made many important contributions to science, the theoryof relativity alone represents one of the greatest intellectual achievements ofthe 20th century. With this theory, one can correctly predict experimental ob-servations over the range of speeds from rest to speeds approaching the speedof light. Newtonian mechanics, which was accepted for over 200 years, is infact a limiting case of Einstein’s special theory of relativity. This chapter andthe next give an introduction to the special theory of relativity, which dealswith the analysis of physical events from coordinate systems moving with con-stant speed in straight lines with respect to one another. Chapter 2 also in-cludes a short introduction to general relativity, which describes physicalevents from coordinate systems undergoing general or accelerated motionwith respect to each other.

In this chapter we show that the special theory of relativity follows from twobasic postulates:

1. The laws of physics are the same in all reference systems that moveuniformly with respect to one another. That is, basic laws such as

2 CHAPTER 1 RELATIVITY I

1A. Einstein and L. Infeld, The Evolution of Physics, New York, Simon and Schuster, 1961.


�F � dp/dt have the same mathematical form for all observers movingat constant velocity with respect to one another.

2. The speed of light in vacuum is always measured to be 3 � 108 m/s, andthe measured value is independent of the motion of the observer or ofthe motion of the source of light. That is, the speed of light is the samefor all observers moving at constant velocities.

Although it is well known that relativity plays an essential role in theoreticalphysics, it also has practical applications, for example, in the design of particleaccelerators, global positioning system (GPS) units, and high-voltage TV dis-plays. Note that these devices simply will not work if designed according toNewtonian mechanics! We shall have occasion to use the outcomes of relativityin many subsequent topics in this text.

1.2 THE PRINCIPLE OF RELATIVITY

To describe a physical event, it is necessary to establish a frame of reference,such as one that is fixed in the laboratory. Recall from your studies in mechan-ics that Newton’s laws are valid in inertial frames of reference. An inertial frameis one in which an object subjected to no forces moves in a straight line at constantspeed—thus the name “inertial frame” because an object observed from such aframe obeys Newton’s first law, the law of inertia.2 Furthermore, any frame orsystem moving with constant velocity with respect to an inertial system mustalso be an inertial system. Thus there is no single, preferred inertial frame forapplying Newton’s laws.

According to the principle of Newtonian relativity, the laws of mechanicsmust be the same in all inertial frames of reference. For example, if you per-form an experiment while at rest in a laboratory, and an observer in a passingtruck moving with constant velocity performs the same experiment, Newton’slaws may be applied to both sets of observations. Specifically, in the laboratoryor in the truck a ball thrown up rises and returns to the thrower’s hand. More-over, both events are measured to take the same time in the truck or in thelaboratory, and Newton’s second law may be used in both frames to computethis time. Although these experiments look different to different observers(see Fig. 1.1, in which the Earth observer sees a different path for the ball)and the observers measure different values of position and velocity for the ballat the same times, both observers agree on the validity of Newton’s laws andprinciples such as conservation of energy and conservation of momentum.This implies that no experiment involving mechanics can detect any essentialdifference between the two inertial frames. The only thing that can bedetected is the relative motion of one frame with respect to the other. That is,the notion of absolute motion through space is meaningless, as is the notion ofa single, preferred reference frame. Indeed, one of the firm philosophicalprinciples of modern science is that all observers are equivalent andthat the laws of nature must take the same mathematical form for allobservers. Laws of physics that exhibit the same mathematical form forobservers with different motions at different locations are said to be covariant.Later in this section we will give specific examples of covariant physical laws.

1.2 THE PRINCIPLE OF RELATIVITY 3

Inertial frame of reference

2An example of a noninertial frame is a frame that accelerates in a straight line or rotates with re-spect to an inertial frame.


In order to show the underlying equivalence of measurements made in dif-ferent reference frames and hence the equivalence of different frames for do-ing physics, we need a mathematical formula that systematically relates mea-surements made in one reference frame to those in another. Such a relationis called a transformation, and the one satisfying Newtonian relativity is the so-called Galilean transformation, which owes its origin to Galileo. It can bederived as follows.

Consider two inertial systems or frames S and S�, as in Figure 1.2. Theframe S� moves with a constant velocity v along the xx� axes, where v is mea-sured relative to the frame S. Clocks in S and S� are synchronized, and theorigins of S and S� coincide at t � t� � 0. We assume that a point event, a phys-ical phenomenon such as a lightbulb flash, occurs at the point P. An observerin the system S would describe the event with space–time coordinates (x, y, z,t), whereas an observer in S� would use (x�, y�, z�, t�) to describe the sameevent. As we can see from Figure 1.2, these coordinates are related bythe equations

(1.1)

These equations constitute what is known as a Galilean transformation ofcoordinates. Note that the fourth coordinate, time, is assumed to be thesame in both inertial frames. That is, in classical mechanics, all clocks run at thesame rate regardless of their velocity, so that the time at which an event occursfor an observer in S is the same as the time for the same event in S�. Conse-quently, the time interval between two successive events should be the same

x� � x � vt

y� � y

z� � z

t� � t


(a) (b)

Figure 1.1 The observer in the truck sees the ball move in a vertical path whenthrown upward. (b) The Earth observer views the path of the ball as a parabola.

Galilean transformation of

coordinates

y

0 x

y′

0′ x ′

x

vt x ′

P (event)

v

S ′S

Figure 1.2 An event occurs ata point P. The event is observedby two observers in inertialframes S and S�, in which S�moves with a velocity v relativeto S.


for both observers. Although this assumption may seem obvious, it turns outto be incorrect when treating situations in which v is comparable to thespeed of light. In fact, this point represents one of the most profounddifferences between Newtonian concepts and the ideas contained inEinstein’s theory of relativity.

Exercise 1 Show that although observers in S and S� measure different coordinatesfor the ends of a stick at rest in S, they agree on the length of the stick. Assume the stickhas end coordinates x � a and x � a � l in S and use the Galilean transformation.

An immediate and important consequence of the invariance of the distancebetween two points under the Galilean transformation is the invariance of

force. For example if gives the electric force between two

charges q,Q located at x1 and x2 on the x-axis in frame S, F �, the force mea-

sured in S�, is given by since x�2 � x�1 � x2 � x1. In fact

any force would be invariant under the Galilean transformation as long as it involved only the relative positions of interacting particles.

Now suppose two events are separated by a distance dx and a time intervaldt as measured by an observer in S. It follows from Equation 1.1 that thecorresponding displacement dx� measured by an observer in S� is given bydx� � dx � v dt, where dx is the displacement measured by an observer in S.Because dt � dt�, we find that

or

(1.2)

where ux and u�x are the instantaneous velocities of the object relative to Sand S�, respectively. This result, which is called the Galilean addition law forvelocities (or Galilean velocity transformation), is used in everyday observa-tions and is consistent with our intuitive notions of time and space.

To obtain the relation between the accelerations measured by observers inS and S�, we take a derivative of Equation 1.2 with respect to time and use theresults that dt � dt� and v is constant:

(1.3)

Thus observers in different inertial frames measure the same acceleration foran accelerating object. The mathematical terminology is to say that lengths(�x), time intervals, and accelerations are invariant under a Galilean transfor-mation. Example 1.1 points up the distinction between invariant and covariantand shows that transformation equations, in addition to converting mea-surements made in one inertial frame to those in another, may be usedto show the covariance of physical laws.

du�xdt�

� a�x � ax

u�x � ux � v

dx�

dt��

dx

dt� v

F � �kqQ

(x �2 � x �1)2� F

F �kqQ

(x2 � x1)2

1.2 THE PRINCIPLE OF RELATIVITY 5

Galilean addition law for

velocities


Exercise 2 Conservation of Linear Momentum Is Covariant Under the Galilean Transforma-tion. Assume that two masses m�1 and m�2 are moving in the positive x direction with ve-locities v�1 and v�2 as measured by an observer in S� before a collision. After the colli-sion, the two masses stick together and move with a velocity v� in S�. Show that if anobserver in S� finds momentum to be conserved, so does an observer in S.

The Speed of Light

It is natural to ask whether the concept of Newtonian relativity and theGalilean addition law for velocities in mechanics also apply to electricity, mag-netism, and optics. Recall that Maxwell in the 1860s showed that the speed oflight in free space was given by c � (�00)�1/2 � 3.00 � 108 m/s. Physicists ofthe late 1800s were certain that light waves (like familiar sound and waterwaves) required a definite medium in which to move, called the ether,3 andthat the speed of light was c only with respect to the ether or a frame fixed inthe ether called the ether frame. In any other frame moving at speed v relativeto the ether frame, the Galilean addition law was expected to hold. Thus, thespeed of light in this other frame was expected to be c � v for light travelingin the same direction as the frame, c � v for light traveling opposite to theframe, and in between these two values for light moving in an arbitrary direc-tion with respect to the moving frame.

Because the existence of the ether and a preferred ether frame would showthat light was similar to other classical waves (in requiring a medium), consid-erable importance was attached to establishing the existence of the specialether frame. Because the speed of light is enormous, experiments involvinglight traveling in media moving at then attainable laboratory speeds had notbeen capable of detecting small changes of the size of c v prior to the late1800s. Scientists of the period, realizing that the Earth moved rapidly around


m� � m to obtain Fx � m�a�x . If we now assume that Fx de-pends only on the relative positions of m and the particlesinteracting with m, that is, Fx � f(x2 � x1, x3 � x1, . . .),then Fx � F �x , because the �x’s are invariant quantities.Thus we find F �x � m�a�x and establish the covariance ofNewton’s second law in this simple case.

EXAMPLE 1.1 Fx � max Is Covariant Under aGalilean Transformation

Assume that Newton’s law Fx � max has been shown tohold by an observer in an inertial frame S. Show thatNewton’s law also holds for an observer in S� or is covari-ant under the Galilean transformation, that is, has theform F �x � m�a�x . Note that inertial mass is an invariantquantity in Newtonian dynamics.

Solution Starting with the established law Fx � max, weuse the Galilean transformation a�x � ax and the fact that

3It was proposed by Maxwell that light and other electromagnetic waves were waves in a luminifer-ous ether, which was present everywhere, even in empty space. In addition to an overblownname, the ether had contradictory properties since it had to have great rigidity to support thehigh speed of light waves yet had to be tenuous enough to allow planets and other massive ob-jects to pass freely through it, without resistance, as observed.


the Sun at 30 km/s, shrewdly decided to use the Earth itself as the movingframe in an attempt to improve their chances of detecting these small changesin light velocity.

From our point of view of observers fixed on Earth, we may say that we arestationary and that the special ether frame moves past us with speed v. Deter-mining the speed of light under these circumstances is just like determiningthe speed of an aircraft in a moving air current or wind, and consequently wespeak of an “ether wind” blowing through our apparatus fixed to the Earth.If v is the velocity of the ether relative to the Earth, then the speed of lightshould have its maximum value, c � v, when propagating downwind, asshown in Figure 1.3a. Likewise, the speed of light should have its minimumvalue, c � v, when propagating upwind, as in Figure 1.3b, and an intermediatevalue, (c2 � v2)1/2, in the direction perpendicular to the ether wind, as inFigure 1.3c. If the Sun is assumed to be at rest in the ether, then the velocity of theether wind would be equal to the orbital velocity of the Earth around the Sun,which has a magnitude of about 3 � 104 m/s compared to c � 3 � 108 m/s.Thus, the change in the speed of light would be about 1 part in 104 for mea-surements in the upwind or downwind directions, and changes of this sizeshould be detectable. However, as we show in the next section, all attempts todetect such changes and establish the existence of the ether proved futile!

1.3 THE MICHELSON–MORLEY EXPERIMENT

The famous experiment designed to detect small changes in the speed of lightwith motion of an observer through the ether was performed in 1887 byAmerican physicist Albert A. Michelson (1852–1931) and the Americanchemist Edward W. Morley (1838–1923).4 We should state at the outset thatthe outcome of the experiment was negative, thus contradicting the ether hy-pothesis. The highly accurate experimental tool perfected by these pioneersto measure small changes in light speed was the Michelson interferometer,shown in Figure 1.4. One of the arms of the interferometer was aligned alongthe direction of the motion of the Earth through the ether. The Earth movingthrough the ether would be equivalent to the ether flowing past the Earth inthe opposite direction with speed v, as shown in Figure 1.4. This ether windblowing in the opposite direction should cause the speed of light measured inthe Earth’s frame of reference to be c � v as it approaches the mirror M2 inFigure 1.4 and c � v after reflection. The speed v is the speed of the Earththrough space, and hence the speed of the ether wind, and c is the speed oflight in the ether frame. The two beams of light reflected from M1 and M2would recombine, and an interference pattern consisting of alternating darkand bright bands, or fringes, would be formed.

During the experiment, the interference pattern was observed while the in-terferometer was rotated through an angle of 90°. This rotation would changethe speed of the ether wind along the direction of the arms of the interferom-eter. The effect of this rotation should have been to cause the fringe pattern toshift slightly but measurably. Measurements failed to show any change in the

1.3 THE MICHELSON–MORLEY EXPERIMENT 7

4A. A. Michelson and E. W. Morley, Am. J. Sci. 134:333, 1887.

v

c + v

c

v

c – v

c

v

c

c 2 – v 2

(a) Downwind

(b) Upwind

(c) Across

√

Figure 1.3 If the velocity ofthe ether wind relative to theEarth is v, and c is the velocityof light relative to the ether,the speed of light relative tothe Earth is (a) c � v in thedownwind direction, (b) c � vin the upwind direction, and(c) (c2 � v2)1/2 in the directionperpendicular to the wind.


interference pattern! The Michelson–Morley experiment was repeated byother researchers under various conditions and at different times of the yearwhen the ether wind was expected to have changed direction and magnitude,but the results were always the same: No fringe shift of the magnitude required wasever observed.5

The negative results of the Michelson–Morley experiment not only meantthat the speed of light does not depend on the direction of light propagationbut also contradicted the ether hypothesis. The negative results also meantthat it was impossible to measure the absolute velocity of the Earth withrespect to the ether frame. As we shall see in the next section, Einstein’spostulates compactly explain these and a host of other perplexing questions,relegating the idea of the ether to the ash heap of history. Light is nowunderstood to be a phenomenon that requires no medium for its propagation.As a result, the idea of an ether in which these waves could travel becameunnecessary.

Details of the Michelson–Morley Experiment

To understand the outcome of the Michelson–Morley experiment, let us as-sume that the interferometer shown in Figure 1.4 has two arms of equallength L. First consider the beam traveling parallel to the direction of theether wind, which is taken to be horizontal in Figure 1.4. According to New-tonian mechanics, as the beam moves to the right, its speed is reduced by thewind and its speed with respect to the Earth is c � v. On its return journey, asthe light beam moves to the left downwind, its speed with respect to the Earthis c � v. Thus, the time of travel to the right is L/(c � v), and the time oftravel to the left is L/(c � v). The total time of travel for the round-trip alongthe horizontal path is

Now consider the light beam traveling perpendicular to the wind,as shown in Figure 1.4. Because the speed of the beam relative to theEarth is (c 2 � v 2)1/2 in this case (see Fig. 1.3c), the time of travel foreach half of this trip is L/(c 2 � v 2)1/2, and the total time of travel for theround-trip is

Thus, the time difference between the light beam traveling horizontally andthe beam traveling vertically is

�t � t1 � t2 �2Lc ��1 �

v2

c2 ��1

� �1 � v2

c2 ��1/2

�

t2 �2L

(c2 � v2)1/2�

2Lc �1 �

v2

c2 ��1/2

t1 �L

c � v�

L

c � v�

2Lcc2 � v2

�2Lc �1 �

v2

c2 ��1


Telescope

Ether wind

M1

M2

M0

v

Arm 1

Arm 2

Source

L

L

Figure 1.4 Diagram of theMichelson interferometer. Ac-cording to the ether wind con-cept, the speed of light shouldbe c � v as the beam ap-proaches mirror M2 and c � vafter reflection.

5From an Earth observer’s point of view, changes in the Earth’s speed and direction in the courseof a year are viewed as ether wind shifts. In fact, even if the speed of the Earth with respect to theether were zero at some point in the Earth’s orbit, six months later the speed of the Earth wouldbe 60 km/s with respect to the ether, and one should find a clear fringe shift. None has ever beenobserved, however.


Because v2/c2 �� 1, this expression can be simplified by using the followingbinomial expansion after dropping all terms higher than second order:

(1 � x)n � 1 � nx (for x �� 1)

In our case, x � v2/c2, and we find

(1.4)

The two light beams start out in phase and return to form an interference pat-tern. Let us assume that the interferometer is adjusted for parallel fringes andthat a telescope is focused on one of these fringes. The time difference be-tween the two light beams gives rise to a phase difference between the beams,producing the interference fringe pattern when they combine at the positionof the telescope. A difference in the pattern (Fig. 1.6) should be detectedby rotating the interferometer through 90� in a horizontal plane, such thatthe two beams exchange roles. This results in a net time difference of twicethat given by Equation 1.4. The path difference corresponding to this timedifference is

The corresponding fringe shift is equal to this path difference divided by thewavelength of light, , because a change in path of 1 wavelength correspondsto a shift of 1 fringe.

(1.5)

In the experiments by Michelson and Morley, each light beam was reflectedby mirrors many times to give an increased effective path length L of about11 m. Using this value, and taking v to be equal to 3 � 104 m/s, the speed ofthe Earth about the Sun, gives a path difference of

�d �2(11 m)(3 � 104 m/s)2

(3 � 108 m/s)2� 2.2 � 10�7 m

Shift �2Lv2

c2

�d � c(2�t) �2Lv2

c2

�t � t1 � t2 �Lv2

c3

1.3 THE MICHELSON–MORLEY EXPERIMENT 9

Fixed spacing(one fringe)

(a) (b)

Fixedmarker

Figure 1.6 Interference fringe schematic showing (a) fringes before rotation and(b) expected fringe shift after a rotation of the interferometer by 90�.


Image not available due to copyright restrictions

This extra distance of travel should produce a noticeable shift in the fringepattern. Specifically, using light of wavelength 500 nm, we find a fringe shiftfor rotation through 90� of

The precision instrument designed by Michelson and Morley had the capabil-ity of detecting a shift in the fringe pattern as small as 0.01 fringe. However,they detected no shift in the fringe pattern. Since then, the experiment has beenrepeated many times by various scientists under various conditions, and nofringe shift has ever been detected. Thus, it was concluded that one cannotdetect the motion of the Earth with respect to the ether.

Many efforts were made to explain the null results of the Michelson–Morley experiment and to save the ether concept and the Galilean addition lawfor the velocity of light. Because all these proposals have been shown to bewrong, we consider them no further here and turn instead to an auspiciousproposal made by George F. Fitzgerald and Hendrik A. Lorentz. In the 1890s,Fitzgerald and Lorentz tried to explain the null results by making the followingad hoc assumption. They proposed that the length of an object moving at

speed v would contract along the direction of travel by a factor of .The net result of this contraction would be a change in length of one of thearms of the interferometer such that no path difference would occur as the in-terferometer was rotated.

Never in the history of physics were such valiant efforts devoted to tryingto explain the absence of an expected result as those directed at theMichelson–Morley experiment. The difficulties raised by this null resultwere tremendous, not only implying that light waves were a new kind of wavepropagating without a medium but that the Galilean transformationswere flawed for inertial frames moving at high relative speeds. The stagewas set for Albert Einstein, who solved these problems in 1905 with his specialtheory of relativity.

1.4 POSTULATES OF SPECIAL RELATIVITY

In the previous section we noted the impossibility of measuring the speed ofthe ether with respect to the Earth and the failure of the Galilean velocitytransformation in the case of light. In 1905, Albert Einstein (Fig. 1.7) pro-posed a theory that boldly removed these difficulties and at the same timecompletely altered our notion of space and time.6 Einstein based his specialtheory of relativity on two postulates.

1. The Principle of Relativity: All the laws of physics have the same formin all inertial reference frames.

2. The Constancy of the Speed of Light: The speed of light in vacuum hasthe same value, c � 3.00 � 108 m/s, in all inertial frames, regardless of thevelocity of the observer or the velocity of the source emitting the light.

√1 � v2/c2

Shift ��d

�

2.2 � 10�7 m5.0 � 10�7 m

� 0.40


6A. Einstein, “On the Electrodynamics of Moving Bodies,” Ann. Physik 17:891, 1905. For an Englishtranslation of this article and other publications by Einstein, see the book by H. Lorentz,A. Einstein, H. Minkowski, and H. Weyl, The Principle of Relativity, Dover, 1958.

Postulates of special relativity


1.4 POSTULATES OF SPECIAL RELATIVITY 11

A lbert Einstein, one of thegreatest physicists of all time,was born in Ulm, Germany.As a child, Einstein was very un-happy with the discipline of Germanschools and completed his early edu-cation in Switzerland at age 16. Be-cause he was unable to obtain anacademic position following gradua-tion from the Swiss Federal Poly-technic School in 1901, he accepteda job at the Swiss Patent Office inBerne. During his spare time, hecontinued his studies in theoreticalphysics. In 1905, at the age of 26, hepublished four scientific papers that

special theory of relativity. In 1915,Einstein published his work on thegeneral theory of relativity, which re-lates gravity to the structure of spaceand time. One of the remarkablepredictions of the theory is thatstrong gravitational forces in thevicinity of very massive objects causelight beams to deviate from straight-line paths. This and other predic-tions of the general theory of rel-ativity have been experimentallyverified (see the essay on our com-panion Web site by Clifford Will).

Einstein made many other im-portant contributions to the devel-opment of modern physics, includ-ing the concept of the lightquantum and the idea of stimulatedemission of radiation, which led tothe invention of the laser 40 yearslater. However, throughout his life,he rejected the probabilistic inter-pretation of quantum mechanicswhen describing events on theatomic scale in favor of a determin-istic view. He is quoted as saying,“God does not play dice with theuniverse.” This comment is reputedto have been answered by NielsBohr, one of the founders of quan-tum mechanics, with “Don’t tell Godwhat to do!”

In 1933, Einstein left Germany(by then under Nazis control) andspent his remaining years at the In-stitute for Advanced Study in Prince-ton, New Jersey. He devoted most ofhis later years to an unsuccessfulsearch for a unified theory of gravityand electromagnetism.

B I O G R A P H Y

ALBERT EINSTEIN

(1879–1955)

revolutionized physics. One of thesepapers, which won him the Nobelprize in 1921, dealt with the pho-toelectric effect. Another was con-cerned with Brownian motion, theirregular motion of small particlessuspended in a liquid. The remain-ing two papers were concerned withwhat is now considered his mostimportant contribution of all, the


Image not available due to copyright restrictions

The first postulate asserts that all the laws of physics, those dealing withelectricity and magnetism, optics, thermodynamics, mechanics, and so on, willhave the same mathematical form or be covariant in all coordinate framesmoving with constant velocity relative to one another. This postulate is asweeping generalization of Newton’s principle of relativity, which refers only tothe laws of mechanics. From an experimental point of view, Einstein’s princi-ple of relativity means that no experiment of any type can establish anabsolute rest frame, and that all inertial reference frames are experimentallyindistinguishable.

Note that postulate 2, the principle of the constancy of the speed oflight, is consistent with postulate 1: If the speed of light was not the same inall inertial frames but was c in only one, it would be possible to distinguishbetween inertial frames, and one could identify a preferred, absolute framein contradiction to postulate 1. Postulate 2 also does away with the problemof measuring the speed of the ether by essentially denying the existence ofthe ether and boldly asserting that light always moves with speed c with re-spect to any inertial observer. Postulate 2 was a brilliant theoretical insighton Einstein’s part in 1905 and has since been directly confirmed experi-mentally in many ways. Perhaps the most direct demonstration involvedmeasuring the speed of very high frequency electromagnetic waves (gammarays) emitted by unstable particles (neutral pions) traveling at 99.975% ofthe speed of light with respect to the laboratory. The measured gamma rayspeed relative to the laboratory agreed in this case to five significant figureswith the speed of light in empty space.

The Michelson–Morley experiment was performed before Einstein pub-lished his work on relativity, and it is not clear that Einstein was aware of thedetails of the experiment. Nonetheless, the null result of the experiment canbe readily understood within the framework of Einstein’s theory. According tohis principle of relativity, the premises of the Michelson–Morley experimentwere incorrect. In the process of trying to explain the expected results, westated that when light traveled against the ether wind its speed was c � v, in ac-cordance with the Galilean addition law for velocities. However, if the state ofmotion of the observer or of the source has no influence on the value foundfor the speed of light, one will always measure the value to be c. Likewise, thelight makes the return trip after reflection from the mirror at a speed of c, andnot with the speed c � v. Thus, the motion of the Earth should not influencethe fringe pattern observed in the Michelson–Morley experiment, and a nullresult should be expected.

Perhaps at this point you have rightly concluded that the Galilean velocityand coordinate transformations are incorrect; that is, the Galilean transforma-tions do not keep all the laws of physics in the same form for different inertialframes. The correct coordinate and time transformations that preserve the co-variant form of all physical laws in two coordinate systems moving uniformlywith respect to each other are called Lorentz transformations. These are derivedin Section 1.6. Although the Galilean transformation preserves the form ofNewton’s laws in two frames moving uniformly with respect to each other,Newton’s laws of mechanics are limited laws that are valid only for low speeds.In general, Newton’s laws must be replaced by Einstein’s relativistic laws of me-chanics, which hold for all speeds and are invariant, as are all physical laws,under the Lorentz transformations.



1.5 CONSEQUENCES OF SPECIAL RELATIVITY

Almost everyone who has dabbled even superficially with science is aware ofsome of the startling predictions that arise because of Einstein’s approach torelative motion. As we examine some of the consequences of relativity in thissection, we shall find that they conflict with our basic notions of space andtime. We restrict our discussion to the concepts of length, time, and simultane-ity, which are quite different in relativistic mechanics and Newtonian mechan-ics. For example, we will find that the distance between two points and the time in-terval between two events depend on the frame of reference in which they are measured.That is, there is no such thing as absolute length or absolute time in relativity. Further-more, events at different locations that occur simultaneously in one frame are not si-multaneous in another frame moving uniformly past the first.

Before we discuss the consequences of special relativity, we must first under-stand how an observer in an inertial reference frame describes an event. Wedefine an event as an occurrence described by three space coordinates andone time coordinate. In general, different observers in different inertialframes would describe the same event with different spacetime coordinates.

The reference frame used to describe an event consists of a coordinate gridand a set of clocks situated at the grid intersections, as shown in Figure 1.8 intwo dimensions. It is necessary that the clocks be synchronized. This can be ac-complished in many ways with the help of light signals. For example, supposean observer at the origin with a master clock sends out a pulse of light at t � 0.The light pulse takes a time r/c to reach a second clock, situated a distance rfrom the origin. Hence, the second clock will be synchronized with the clockat the origin if the second clock reads a time r/c at the instant the pulsereaches it. This procedure of synchronization assumes that the speed of lighthas the same value in all directions and in all inertial frames. Furthermore, theprocedure concerns an event recorded by an observer in a specific inertial ref-erence frame. Clocks in other inertial frames can be synchronized in a similarmanner. An observer in some other inertial frame would assign differentspacetime coordinates to events, using another coordinate grid with anotherarray of clocks.

1.5 CONSEQUENCES OF SPECIAL RELATIVITY 13

Figure 1.8 In relativity, we use a reference frame consisting of a coordinate grid anda set of synchronized clocks.


Simultaneity and the Relativity of Time

A basic premise of Newtonian mechanics is that a universal time scale existsthat is the same for all observers. In fact, Newton wrote that “Absolute, true,and mathematical time, of itself, and from its own nature, flows equably with-out relation to anything external.” Thus, Newton and his followers simply tooksimultaneity for granted. In his special theory of relativity, Einstein abandonedthis assumption. According to Einstein, a time interval measurement depends onthe reference frame in which the measurement is made.

Einstein devised the following thought experiment to illustrate this point. Aboxcar moves with uniform velocity, and two lightning bolts strike the ends ofthe boxcar, as in Figure 1.9a, leaving marks on the boxcar and ground. Themarks left on the boxcar are labeled A� and B�; those on the ground are la-beled A and B. An observer at O� moving with the boxcar is midway betweenA� and B�, and a ground observer at O is midway between A and B. The eventsrecorded by the observers are the light signals from the lightning bolts.

The two light signals reach the observer at O at the same time, as indicatedin Figure 1.9b. This observer realizes that the light signals have traveled at thesame speed over equal distances. Thus, observer O concludes that the eventsat A and B occurred simultaneously. Now consider the same events as viewedby the observer on the boxcar at O�. By the time the light has reached ob-server O, observer O� has moved as indicated in Figure 1.9b. Thus, the lightsignal from B� has already swept past O�, but the light from A� has not yetreached O�. According to Einstein, observer O� must find that light travels at thesame speed as that measured by observer O. Therefore, observer O� concludes thatthe lightning struck the front of the boxcar before it struck the back. Thisthought experiment clearly demonstrates that the two events, which appear toO to be simultaneous, do not appear to O� to be simultaneous. In other words,


Two events that are simultaneous in one frame are in general notsimultaneous in a second frame moving with respect to the first. Thatis, simultaneity is not an absolute concept, but one that depends on thestate of motion of the observer.

v

A' B'

OA B

v

A' B'O'

OA B

(a) (b)

O'

Figure 1.9 Two lightning bolts strike the ends of a moving boxcar. (a) The eventsappear to be simultaneous to the stationary observer at O, who is midway between Aand B. (b) The events do not appear to be simultaneous to the observer at O�, whoclaims that the front of the train is struck before the rear.


At this point, you might wonder which observer is right concerning the twoevents. The answer is that both are correct, because the principle of relativitystates that there is no preferred inertial frame of reference. Although the two ob-servers reach different conclusions, both are correct in their own referenceframe because the concept of simultaneity is not absolute. This, in fact, is thecentral point of relativity—any uniformly moving frame of reference can beused to describe events and do physics. However, observers in different inertialframes will always measure different time intervals with their clocks and differ-ent distances with their meter sticks. Nevertheless, they will both agree on theforms of the laws of physics in their respective frames, because these laws mustbe the same for all observers in uniform motion. It is the alteration of timeand space that allows the laws of physics (including Maxwell’s equations) to bethe same for all observers in uniform motion.

Time Dilation

The fact that observers in different inertial frames always measure different timeintervals between a pair of events can be illustrated in another way by consider-ing a vehicle moving to the right with a speed v, as in Figure 1.10a. A mirror isfixed to the ceiling of the vehicle, and observer O�, at rest in this system, holds alaser a distance d below the mirror. At some instant the laser emits a pulse of lightdirected toward the mirror (event 1), and at some later time, after reflectingfrom the mirror, the pulse arrives back at the laser (event 2). Observer O� carriesa clock, C�, which she uses to measure the time interval �t� between these twoevents. Because the light pulse has the speed c, the time it takes to travel from O�to the mirror and back can be found from the definition of speed:

(1.6)

This time interval �t�—measured by O�, who, remember, is at rest in the mov-ing vehicle—requires only a single clock, C�, in this reference frame.

�t� �distance traveled

speed of light�

2dc


d

v∆t2

c∆t2

(c)

y ′

v

O ′

d

Mirror

x ′

(a)

O ′ O ′ O ′

v

xO

v∆t

(b)

y ′

Figure 1.10 (a) A mirror is fixed to a moving vehicle, and a light pulse leaves O� atrest in the vehicle. (b) Relative to a stationary observer on Earth, the mirror and O�move with a speed v. Note that the distance the pulse travels measured by the station-ary observer on Earth is greater than 2d. (c) The right triangle for calculating the rela-tionship between �t and �t�.


Now consider the same set of events as viewed by observer O in a second frame(Fig. 1.10b). According to this observer, the mirror and laser are moving to theright with a speed v, and as a result, the sequence of events appears different tothis observer. By the time the light from the laser reaches the mirror, the mirrorhas moved to the right a distance v�t/2, where �t is the time interval requiredfor the light pulse to travel from O� to the mirror and back as measured by O. Inother words, O concludes that, because of the motion of the vehicle, if the light isto hit the mirror, it must leave the laser at an angle with respect to the verticaldirection. Comparing Figures 1.10a and 1.10b, we see that the light must travelfarther in (b) than in (a). (Note that neither observer “knows” that he or she ismoving. Each is at rest in his or her own inertial frame.)

According to the second postulate of special relativity, both observers mustmeasure c for the speed of light. Because the light travels farther according toO, it follows that the time interval �t measured by O is longer than the time in-terval �t� measured by O�. To obtain a relationship between �t and �t�, it isconvenient to use the right triangle shown in Figure 1.10c. The Pythagoreantheorem gives

Solving for �t gives

(1.7)

Because �t� � 2d/c, we can express Equation 1.7 as

(1.8)

where � � (1 � v2/c2)�1/2. Because � is always greater than unity, this resultsays that the time interval �t measured by the observer moving with respect tothe clock is longer than the time interval �t� measured by the observer at restwith respect to the clock. This effect is known as time dilation.

The time interval �t� in Equation 1.8 is called the proper time. In general,proper time, denoted �tp, is defined as the time interval between twoevents as measured by an observer who sees the events occur at thesame point in space. In our case, observer O� measures the proper time.That is, proper time is always the time measured by an observer movingalong with the clock. As an aid in solving problems it is convenient toexpress Equation 1.8 in terms of the proper time interval, �tp, as

�t � ��tp (1.9)

Because the time between ticks of a moving clock, �(2d/c), is observed tobe longer than the time between ticks of an identical clock at rest, 2d/c, onecommonly says, “A moving clock runs slower than a clock at rest by a factor of �.”This is true for ordinary mechanical clocks as well as for the light clock justdescribed. In fact, we can generalize these results by stating that all physicalprocesses, including chemical reactions and biological processes, slowdown when observed from a reference frame in which they are moving. For

�t ��t�

√1 � (v 2/c 2)� ��t�

�t �2d

√c2 � v2�

2d

c√1 � v2/c2

� c�t2 �2

� � v�t2 �2

� d2


Time dilation

A moving clock runs slower


example, the heartbeat of an astronaut moving through space would keeptime with a clock inside the spaceship, but both the astronaut’s clock andher heartbeat appear slow to an observer, with another clock, in any otherreference frame. The astronaut would not have any sensation of life slowingdown in her frame.

Time dilation is a very real phenomenon that has been verified by variousexperiments. For example, muons are unstable elementary particles thathave a charge equal to that of an electron and a mass 207 times that of theelectron. Muons are naturally produced by the collision of cosmic radiationwith atoms at a height of several thousand meters above the surface of theEarth. Muons have a lifetime of only 2.2 �s when measured in a referenceframe at rest with respect to them. If we take 2.2 �s (proper time) as theaverage lifetime of a muon and assume that its speed is close to the speedof light, we would find that these particles could travel a distance of about650 m before they decayed. Hence, they could not reach the Earth fromthe upper atmosphere where they are produced. However, experimentsshow that a large number of muons do reach the Earth. The phenomenonof time dilation explains this effect (see Fig. 1.11a). Relative to an observeron Earth, the muons have a lifetime equal to ��, where � � 2.2 �s is thelifetime in a frame of reference traveling with the muons. For example,for v � 0.99c, � � 7.1 and �� 16 �s. Hence, the average distance traveledas measured by an observer on Earth is �v� � 4700 m, as indicated inFigure 1.11b.

In 1976, experiments with muons were conducted at the laboratory of theEuropean Council for Nuclear Research (CERN) in Geneva. Muons were in-jected into a large storage ring, reaching speeds of about 0.9994c. Electronsproduced by the decaying muons were detected by counters around the ring,enabling scientists to measure the decay rate, and hence the lifetime, of themuons. The lifetime of the moving muons was measured to be about 30 timesas long as that of the stationary muon (see Fig. 1.12), in agreement with theprediction of relativity to within two parts in a thousand.

It is quite interesting that time dilation can be observed directly by com-paring high-precision atomic clocks, one carried aboard a jet, the other


Muon’sframe

τ = 2.2 µs

(a)

650 m

4700 m

(b)

Earth’sframe

τ′ = γ τ ≈ 16 µs

Figure 1.11 (a) Muons travel-ing with a speed of 0.99c travelonly about 650 m as measuredin the muons’ reference frame,where their lifetime is about2.2 �s. (b) The muons travelabout 4700 m as measured byan observer on Earth. Becauseof time dilation, the muons’lifetime is longer as measuredby the Earth observer.

Muonat rest

Muon movingat 0.9994c

50 100 150

0.5

1.0

Frac

tion

of m

uon

s re

mai

nin

g

t(µs)

Figure 1.12 Decay curves for muons traveling at a speed of 0.9994c and for muonsat rest.


remaining in a laboratory on Earth. The actual experiment involved the use ofvery stable cesium beam atomic clocks.7 Time intervals measured with four suchclocks in jet flight were compared with time intervals measured by referenceatomic clocks located at the U.S. Naval Observatory. To compare these resultswith the theory, many factors had to be considered, including periods of accel-eration and deceleration relative to the Earth, variations in direction of travel,and the weaker gravitational field experienced by the flying clocks comparedwith the Earth-based clocks. The results were in good agreement with the pre-dictions of the special theory of relativity and can be completely explained interms of the relative motion between the Earth and the jet aircraft.


clock runs slower than a stationary clock by �, Equation1.8 gives

That is, a moving pendulum slows down or takes longerto complete one period.

T � (3.2)(3.0 s) � 9.6 s

T � �T � �1

√1 � (0.95c)2/c23.0 s

EXAMPLE 1.2 What Is the Period of thePendulum?

The period of a pendulum is measured to be 3.0 s in therest frame of the pendulum. What is the period of thependulum when measured by an observer moving at aspeed of 0.95c with respect to the pendulum?

Solution In this case, the proper time is equal to3.0 s. From the point of view of the observer, the pen-

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