An introduction to quantum physics

Contents Foreword xix Preface xxiii Editors’ Note xxvii Part I Fundamental Principles 1 1 The Principle of Wave–Particle Duality: An Overview 3 1.1 Introduction 3 1.2 The Principle of Wa

An Introduction to Quantum Physics

An Introduction to Quantum Physics

A First Course for Physicists, Chemists,

Materials Scientists, and Engineers

Stefanos Trachanas

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Contents

Foreword xix

Preface xxiii

Editors’ Note xxvii

Part I Fundamental Principles 1

1 The Principle of Wave–Particle Duality: An Overview 3

1.1 Introduction 3

1.2 The Principle of Wave–Particle Duality of Light 4

1.2.1 The Photoelectric Effect 4

1.2.2 The Compton Effect 7

1.2.3 A Note on Units 10

1.3 The Principle of Wave–Particle Duality of Matter 11

1.3.1 From Frequency Quantization in Classical Waves to Energy

Quantization in Matter Waves: The Most Important General

Consequence of Wave–Particle Duality of Matter 12

1.3.2 The Problem of Atomic Stability under Collisions 13

1.3.3 The Problem of Energy Scales: Why Are Atomic Energies on the Order

of eV, While Nuclear Energies Are on the Order of MeV? 15

1.3.4 The Stability of Atoms and Molecules Against External

Electromagnetic Radiation 17

1.3.5 The Problem of Length Scales: Why Are Atomic Sizes on the Order of

Angstroms, While Nuclear Sizes Are on the Order of Fermis? 19

1.3.6 The Stability of Atoms Against Their Own Radiation: Probabilistic

Interpretation of Matter Waves 21

1.3.7 How Do Atoms Radiate after All? Quantum Jumps from Higher to

Lower Energy States and Atomic Spectra 22

1.3.8 Quantized Energies and Atomic Spectra: The Case of Hydrogen 25

1.3.9 Correct and Incorrect Pictures for the Motion of Electrons in Atoms:

Revisiting the Case of Hydrogen 25

1.3.10 The Fine Structure Constant and Numerical Calculations in Bohr’s

Theory 29

1.3.11 Numerical Calculations with Matter Waves: Practical Formulas and

1.4 Dimensional Analysis and Quantum Physics 41

1.4.1 The Fundamental Theorem and a Simple Application 41

1.4.2 Blackbody Radiation Using Dimensional Analysis 44

1.4.3 The Hydrogen Atom Using Dimensional Analysis 47

2 The Schrödinger Equation and Its Statistical Interpretation 53

2.1 Introduction 53

2.2 The Schrödinger Equation 53

2.2.1 The Schrödinger Equation for Free Particles 54

2.2.2 The Schrödinger Equation in an External Potential 57

2.2.3 Mathematical Intermission I: Linear Operators 58

2.3 Statistical Interpretation of Quantum Mechanics 60

2.3.1 The “Particle–Wave” Contradiction in Classical Mechanics 60

2.3.2 Statistical Interpretation 61

2.3.3 Why Did We Choose P(x) = |𝜓(x)|2as the Probability Density? 62

2.3.4 Mathematical Intermission II: Basic Statistical Concepts 63

2.3.4.1 Mean Value 63

2.3.4.2 Standard Deviation (or Uncertainty) 65

2.3.5 Position Measurements: Mean Value and Uncertainty 67

2.4 Further Development of the Statistical Interpretation: The Mean-Value

Formula 71

2.4.1 The General Formula for the Mean Value 71

2.4.2 The General Formula for Uncertainty 73

2.5 Time Evolution of Wavefunctions and Superposition States 77

2.5.1 Setting the Stage 77

2.5.2 Solving the Schrödinger Equation Separation of Variables 78

2.5.3 The Time-Independent Schrödinger Equation as an Eigenvalue

Equation: Zero-Uncertainty States and Superposition States 81

2.5.4 Energy Quantization for Confined Motion: A Fundamental General

Consequence of Schrödinger’s Equation 85

2.5.5 The Role of Measurement in Quantum Mechanics: Collapse of the

Wavefunction Upon Measurement 86

2.5.6 Measurable Consequences of Time Evolution: Stationary and

Nonstationary States 91

2.6 Self-Consistency of the Statistical Interpretation and the Mathematical

Structure of Quantum Mechanics 95

2.6.1 Hermitian Operators 95

2.6.2 Conservation of Probability 98

2.6.3 Inner Product and Orthogonality 99

2.6.4 Matrix Representation of Quantum Mechanical Operators 101

2.7 Summary: Quantum Mechanics in a Nutshell 103

Contents ix

3 The Uncertainty Principle 107

3.1 Introduction 107

3.2 The Position–Momentum Uncertainty Principle 108

3.2.1 Mathematical Explanation of the Principle 108

3.2.2 Physical Explanation of the Principle 109

3.2.3 Quantum Resistance to Confinement A Fundamental Consequence of

the Position–Momentum Uncertainty Principle 112

3.3 The Time–Energy Uncertainty Principle 114

3.4 The Uncertainty Principle in the Classical Limit 118

3.5 General Investigation of the Uncertainty Principle 119

3.5.1 Compatible and Incompatible Physical Quantities and the Generalized

Uncertainty Relation 119

3.5.2 Angular Momentum: A Different Kind of Vector 122

Part II Simple Quantum Systems 127

4 Square Potentials I: Discrete Spectrum—Bound States 129

4.1 Introduction 129

4.2 Particle in a One-Dimensional Box: The Infinite Potential Well 132

4.2.1 Solution of the Schrödinger Equation 132

4.2.2 Discussion of the Results 134

4.2.2.1 Dimensional Analysis of the Formula E n= (ℏ2𝜋2∕2mL2)n2

Do We Need an Exact Solution to Predict the Energy Dependence on

4.2.2.4 The Classical Limit of the Position Probability Density 138

4.2.2.5 Eigenfunction Features: Mirror Symmetry and the Node

Theorem 139

4.2.2.6 Numerical Calculations in Practical Units 139

4.3 The Square Potential Well 140

4.3.1 Solution of the Schrödinger Equation 140

4.3.2 Discussion of the Results 143

4.3.2.1 Penetration into Classically Forbidden Regions 143

4.3.2.2 Penetration in the Classical Limit 144

4.3.2.3 The Physics and “Numerics” of the Parameter𝜆 145

5 Square Potentials II: Continuous Spectrum—Scattering

States 149

5.1 Introduction 149

5.2 The Square Potential Step: Reflection and Transmission 150

5.2.1 Solution of the Schrödinger Equation and Calculation of the Reflection

Coefficient 150

5.2.2 Discussion of the Results 153

5.2.2.1 The Phenomenon of Classically Forbidden Reflection 153

5.2.2.2 Transmission Coefficient in the “Classical Limit” of High

Energies 154

5.2.2.3 The Reflection Coefficient Depends neither on Planck’s Constant nor

on the Mass of the Particle: Analysis of a Paradox 154

5.2.2.4 An Argument from Dimensional Analysis 155

5.3 Rectangular Potential Barrier: Tunneling Effect 156

5.3.1 Solution of the Schrödinger Equation 156

5.3.2 Discussion of the Results 158

5.3.2.1 Crossing a Classically Forbidden Region: The Tunneling Effect 158

5.3.2.2 Exponential Sensitivity of the Tunneling Effect to the Energy of the

5.3.2.5 A Practical Formula for T 163

6 The Harmonic Oscillator 167

6.1 Introduction 167

6.2 Solution of the Schrödinger Equation 169

6.3 Discussion of the Results 177

6.3.1 Shape of Wavefunctions Mirror Symmetry and the Node

Theorem 178

6.3.2 Shape of Eigenfunctions for Large n: The Classical Limit 179

6.3.3 The Extreme Anticlassical Limit of the Ground State 180

6.3.4 Penetration into Classically Forbidden Regions: What Fraction of Its

“Lifetime” Does the Particle “Spend” in the Classically ForbiddenRegion? 181

6.3.5 A Quantum Oscillator Never Rests: Zero-Point Energy 182

6.3.6 Equidistant Eigenvalues and Emission of Radiation from a Quantum

Harmonic Oscillator 184

6.4 A Plausible Question: Can We Use the Polynomial Method to Solve

Potentials Other than the Harmonic Oscillator? 187

7 The Polynomial Method: Systematic Theory and

Applications 191

7.1 Introduction: The Power-Series Method 191

7.2 Sufficient Conditions for the Existence of Polynomial Solutions:

Contents xi

8.2 Solving the Schrödinger Equation for the Spherically Symmetric

Eigenfunctions 209

8.2.1 A Final Comment: The System of Atomic Units 216

8.3 Discussion of the Results 217

8.3.1 Checking the Classical Limitℏ → 0 or m → ∞ for the Ground State of

the Hydrogen Atom 217

8.3.2 Energy Quantization and Atomic Stability 217

8.3.3 The Size of the Atom and the Uncertainty Principle: The Mystery of

Atomic Stability from Another Perspective 218

8.3.4 Atomic Incompressibility and the Uncertainty Principle 221

8.3.5 More on the Ground State of the Atom Mean and Most Probable

Distance of the Electron from the Nucleus 221

8.3.6 Revisiting the Notion of “Atomic Radius”: How Probable is It to Find

the Electron Within the “Volume” that the Atom Supposedly

Occupies? 222

8.3.7 An Apparent Paradox: After All, Where Is It Most Likely to Find the

Electron? Near the Nucleus or One Bohr Radius Away from It? 223

8.3.8 What Fraction of Its Time Does the Electron Spend in the Classically

Forbidden Region of the Atom? 223

8.3.9 Is the Bohr Theory for the Hydrogen Atom Really Wrong? Comparison

with Quantum Mechanics 225

8.4 What Is the Electron Doing in the Hydrogen Atom after All? A First

Discussion on the Basic Questions of Quantum Mechanics 226

9 The Hydrogen Atom II: Solutions with Angular

Dependence 231

9.1 Introduction 231

9.2 The Schrödinger Equation in an Arbitrary Central Potential:

Separation of Variables 232

9.2.1 Separation of Radial from Angular Variables 232

9.2.2 The Radial Schrödinger Equation: Physical Interpretation of the

Centrifugal Term and Connection to the Angular Equation 235

9.2.3 Solution of the Angular Equation: Eigenvalues and Eigenfunctions of

Angular Momentum 237

9.2.3.1 Solving the Equation for Φ 238

9.2.3.2 Solving the Equation for Θ 239

9.2.4 Summary of Results for an Arbitrary Central Potential 243

9.3 The Hydrogen Atom 246

9.3.1 Solution of the Radial Equation for the Coulomb Potential 246

9.3.2 Explicit Construction of the First Few Eigenfunctions 249

9.3.2.1 n =1 : The Ground State 250

9.3.2.2 n =2 : The First Excited States 250

9.3.3 Discussion of the Results 254

9.3.3.1 The Energy-Level Diagram 254

9.3.3.2 Degeneracy of the Energy Spectrum for a Coulomb Potential:

Rotational and Accidental Degeneracy 255

9.3.3.3 Removal of Rotational and Hydrogenic Degeneracy 257

9.3.3.4 The Ground State is Always Nondegenerate and Has the Full

Symmetry of the Problem 257

9.3.3.5 Spectroscopic Notation for Atomic States 258

9.3.3.6 The “Concept” of the Orbital: s and p Orbitals 258

9.3.3.7 Quantum Angular Momentum: A Rather Strange Vector 261

9.3.3.8 Allowed and Forbidden Transitions in the Hydrogen Atom:

Conservation of Angular Momentum and Selection Rules 263

10 Atoms in a Magnetic Field and the Emergence of Spin 267

10.1 Introduction 267

10.2 Atomic Electrons as Microscopic Magnets: Magnetic Moment and

Angular Momentum 270

10.3 The Zeeman Effect and the Evidence for the Existence of Spin 274

10.4 The Stern–Gerlach Experiment: Unequivocal Experimental

Confirmation of the Existence of Spin 278

10.4.1 Preliminary Investigation: A Plausible Theoretical Description

of Spin 278

10.4.2 The Experiment and Its Results 280

10.5 What is Spin? 284

10.5.1 Spin is No Self-Rotation 284

10.5.2 How is Spin Described Quantum Mechanically? 285

10.5.3 What Spin Really Is 291

10.6 Time Evolution of Spin in a Magnetic Field 292

10.7 Total Angular Momentum of Atoms: Addition of Angular

11.3 Indistinguishability of Identical Particles and the Pauli Principle 306

11.4 The Role of Spin: Complete Formulation of the Pauli Principle 307

11.5 The Pauli Exclusion Principle 310

11.6 Which Particles Are Fermions and Which Are Bosons 314

11.7 Exchange Degeneracy: The Problem and Its Solution 317

Part III Quantum Mechanics in Action: The Structure

of Matter 321

12 Atoms: The Periodic Table of the Elements 323

12.1 Introduction 323

12.2 Arrangement of Energy Levels in Many-Electron Atoms:

The Screening Effect 324

Contents xiii

12.3 Quantum Mechanical Explanation of the Periodic Table:

The “Small Periodic Table” 327

12.3.1 Populating the Energy Levels: The Shell Model 328

12.3.2 An Interesting “Detail”: The Pauli Principle and Atomic

Magnetism 329

12.3.3 Quantum Mechanical Explanation of Valence and Directionality of

Chemical Bonds 331

12.3.4 Quantum Mechanical Explanation of Chemical Periodicity: The Third

Row of the Periodic Table 332

12.3.5 Ionization Energy and Its Role in Chemical Behavior 334

13.2 The Double-Well Model of Chemical Bonding 352

13.2.1 The Symmetric Double Well 352

13.2.2 The Asymmetric Double Well 356

13.3 Examples of Simple Molecules 360

13.3.1 The Hydrogen Molecule H2 360

13.3.2 The Helium “Molecule” He2 363

13.3.3 The Lithium Molecule Li2 364

13.3.4 The Oxygen Molecule O2 364

13.3.5 The Nitrogen Molecule N2 366

13.3.6 The Water Molecule H2O 367

13.3.7 Hydrogen Bonds: From the Water Molecule to Biomolecules 370

13.3.8 The Ammonia Molecule NH3 373

13.4 Molecular Spectra 377

13.4.1 Rotational Spectrum 378

13.4.2 Vibrational Spectrum 382

13.4.3 The Vibrational–Rotational Spectrum 385

14 Molecules II: The Chemistry of Carbon 393

14.1 Introduction 393

14.2 Hybridization: The First Basic Deviation from the Elementary Theory

of the Chemical Bond 393

14.2.1 The CH4Molecule According to the Elementary Theory: An

Erroneous Prediction 393

14.2.2 Hybridized Orbitals and the CH4Molecule 395

14.2.3 Total and Partial Hybridization 401

14.2.4 The Need for Partial Hybridization: The Molecules C2H4, C2H2, and

C2H6 404

14.2.5 Application of Hybridization Theory to Conjugated

Hydrocarbons 408

14.2.6 Energy Balance of Hybridization and Application to Inorganic

Molecules 409

14.3 Delocalization: The Second Basic Deviation from the Elementary

Theory of the Chemical Bond 414

14.3.1 A Closer Look at the Benzene Molecule 414

14.3.2 An Elementary Theory of Delocalization: The Free-Electron

Model 417

14.3.3 LCAO Theory for Conjugated Hydrocarbons I: Cyclic Chains 418

14.3.4 LCAO Theory for Conjugated Hydrocarbons II: Linear Chains 424

14.3.5 Delocalization on Carbon Chains: General Remarks 427

14.3.6 Delocalization in Two-dimensional Arrays of p Orbitals: Graphene and

Fullerenes 429

15 Solids: Conductors, Semiconductors, Insulators 439

15.1 Introduction 439

15.2 Periodicity and Band Structure 439

15.3 Band Structure and the “Mystery of Conductivity.” Conductors,

Semiconductors, Insulators 441

15.3.1 Failure of the Classical Theory 441

15.3.2 The Quantum Explanation 443

15.4 Crystal Momentum, Effective Mass, and Electron Mobility 447

15.5 Fermi Energy and Density of States 453

15.5.1 Fermi Energy in the Free-Electron Model 453

15.5.2 Density of States in the Free-Electron Model 457

15.5.3 Discussion of the Results: Sharing of Available Space by the Particles of

16.2 The Four Fundamental Processes: Resonance, Scattering, Ionization,

and Spontaneous Emission 471

16.3 Quantitative Description of the Fundamental Processes: Transition

Rate, Effective Cross Section, Mean Free Path 473

16.3.1 Transition Rate: The Fundamental Concept 473

16.3.2 Effective Cross Section and Mean Free Path 475

16.3.3 Scattering Cross Section: An Instructive Example 476

16.4 Matter and Light in Resonance I: Theory 478

16.4.1 Calculation of the Effective Cross Section: Fermi’s Rule 478

16.4.2 Discussion of the Result: Order-of-Magnitude Estimates and Selection

Rules 481

16.4.3 Selection Rules: Allowed and Forbidden Transitions 483

16.5 Matter and Light in Resonance II: The Laser 487

16.5.1 The Operation Principle: Population Inversion and the Threshold

Condition 487

16.7 Theory of Time-dependent Perturbations: Fermi’s Rule 499

16.7.1 Approximate Calculation of Transition Probabilities P n→m(t)for an

Arbitrary “Transient” Perturbation V (t) 499

16.7.2 The Atom Under the Influence of a Sinusoidal Perturbation:

Fermi’s Rule for Resonance Transitions 503

16.8 The Light Itself: Polarized Photons and Their Quantum Mechanical

Description 511

16.8.1 States of Linear and Circular Polarization for Photons 511

16.8.2 Linear and Circular Polarizers 512

16.8.3 Quantum Mechanical Description of Polarized Photons 513

Online Supplement

1 The Principle of Wave–Particle Duality: An Overview

OS1.1 Review Quiz

OS1.1 Determining Planck’s Constant from Everyday Observations

2 The Schrödinger Equation and Its Statistical Interpretation

OS2.1 Review Quiz

OS2.2 Further Study of Hermitian Operators: The Concept of the

Adjoint Operator

OS2.3 Local Conservation of Probability: The Probability Current

3 The Uncertainty Principle

OS3.1 Review Quiz

OS3.2 Commutator Algebra: Calculational Techniques

OS3.3 The Generalized Uncertainty Principle

OS3.4 Ehrenfest’s Theorem: Time Evolution of Mean Values and the

Classical Limit

4 Square Potentials I: Discrete Spectrum—Bound States

OS4.1 Review Quiz

OS4.2 Square Well: A More Elegant Graphical Solution for Its EigenvaluesOS4.3 Deep and Shallow Wells: Approximate Analytic Expressions for Their

Eigenvalues

5 Square Potentials II: Continuous Spectrum—Scattering

States

OS5.1 Review Quiz

OS5.2 Quantum Mechanical Theory of Alpha Decay

6 The Harmonic Oscillator

OS6.1 Review Quiz

OS6.2 Algebraic Solution of the Harmonic Oscillator: Creation and

Annihilation Operators

7 The Polynomial Method: Systematic Theory and Applications

OS7.1 Review Quiz

OS7.2 An Elementary Method for Discovering Exactly Solvable PotentialsOS7.3 Classic Examples of Exactly Solvable Potentials: A Comprehensive

List

8 The Hydrogen Atom I: Spherically Symmetric Solutions

OS8.1 Review Quiz

9 The Hydrogen Atom II: Solutions with Angular Dependence

OS9.1 Review Quiz

OS9.2 Conservation of Angular Momentum in Central Potentials, and Its

Consequences

OS9.3 Solving the Associated Legendre Equation on Our Own

10 Atoms in a Magnetic Field and the Emergence of Spin

OS10.1 Review Quiz

OS10.2 Algebraic Theory of Angular Momentum and Spin

11 Identical Particles and the Pauli Principle

OS11.1 Review Quiz

OS11.2 Dirac’s Formalism: A Brief Introduction

12 Atoms: The Periodic Table of the Elements

OS12.1 Review Quiz

OS12.2 Systematic Perturbation Theory: Application to the Stark Effect and

Atomic Polarizability

13 Molecules I: Elementary Theory of the Chemical Bond

OS13.1 Review Quiz

14 Molecules II: The Chemistry of Carbon

OS14.1 Review Quiz

OS14.2 The LCAO Method and Matrix Mechanics

OS14.3 Extension of the LCAO Method for Nonzero Overlap

15 Solids: Conductors, Semiconductors, Insulators

OS15.1 Review Quiz

OS15.2 Floquet’s Theorem: Mathematical Study of the Band Structure for an

Arbitrary Periodic Potential V(x)

OS15.3 Compressibility of Condensed Matter: The Bulk Modulus

OS15.4 The Pauli Principle and Gravitational Collapse: The Chandrasekhar

Limit

Contents xvii

16 Matter and Light: The Interaction of Atoms with

Electromagnetic Radiation

OS16.1 Review Quiz

OS16.2 Resonance Transitions Beyond Fermi’s Rule: Rabi Oscillations

OS16.3 Resonance Transitions at Radio Frequencies: Nuclear Magnetic

Resonance (NMR)

Appendix 519

Bibliography 523

Index 527

Foreword

As fate would have it, or perhaps due to some form of quantum interference,

I encountered Stefanos Trachanas’ book on Quantum Physics in its prenatalform In the late 1970s, while a graduate student at Harvard, Trachanas wasworking on a set of notes on quantum physics, written in his native language

He occasionally lent his handwritten notes (the file-sharing mode of that era) tofriends who appreciated his fascination with Nature’s wonders At the time, I was

an undergraduate student at the technical school down the river, struggling tolearn quantum physics, and was very grateful to have access to Trachanas’ notes

I still remember the delight and amazement I felt when reading his notes, fortheir clarity and freshness, and for the wonderful insights, not to be found in any

of the classic physics texts available at the time (our common native languagealso helped) It is a great pleasure to see that in the latest version of his book onQuantum Physics, this freshness is intact, enriched from decades of teachingexperience This latest version is of course a long way from his original set ofnotes; it is a thorough account of the theory of quantum mechanics, expertlytranslated by Manolis Antonoyiannakis and Leonidas Tsetseris, in the form of acomprehensive and mature textbook

It is an unusual book All the formulas and numbers and tables that you find

in any other textbook on the subject are there This level of systematic detail isimportant; one does expect a textbook to contain a complete treatment of thesubject and to serve as a reference for key results and expressions But there arealso many wonderful insights that I have not found elsewhere, and numerous

elaborate discussions and explanations of the meaning of the formulas, a crucial ingredient for developing an understanding of quantum physics.

The detailed examples, constantly contrasting the quantum and the classicalpictures for model systems, are the hallmark of the book Another key charac-teristic is the use of dimensional analysis, through which many of the secrets ofquantum behavior can be elucidated Finally, the application of key concepts torealistic problems, including atoms, molecules, and solids, makes the treatment

of the subject not only pedagogically insightful but also of great practical value.The book is nicely laid out in three parts: In Part I, the student is intro-duced to “the language of quantum mechanics” (the author’s astute definition

of the subject, as mentioned in the Preface), including all the “cool” (myquotes) concepts of the quantum realm, such as wave–particle duality and theuncertainty principle Then, in Part II, the language is used to describe the

standard simple problems, the square well, the harmonic oscillator, and theCoulomb potential It is also applied to the hydrogen atom, illustrating howthis language can capture the behavior of Nature at the level of fundamentalparticles—electrons and protons Finally, in Part III, the student is given athorough training in the use of the quantum language to address problemsrelevant to real applications in modern life, which is dominated by quantumdevices, for better or for worse Many everyday activities, from using a cellphone to call friends to employing photovoltaics for powering your house, aredirectly related to quintessentially quantum phenomena, that is, the physics

of semiconductors, conductors, and insulators, and their interaction withlight All these phenomena are explained thoroughly and clearly in Trachanas’book The reader of the book will certainly develop a deep appreciation of theprinciples on which many everyday devices are based There is also a lovelydiscussion of the properties of molecules and the nature of the chemical bond.The treatment ranges from the closed sixfold hydrocarbon ring (“benzene”) tothe truncated icosahedron formed by 60 carbon atoms (“fullerene”), with severalother important structures in between This discussion touches upon the origin

of chemical complexity, including many aspects related to carbon, the “element

of life” (again my quotes), and occupies, deservedly, a whole chapter

For the demanding reader, there are several chapters of higher mathematicaland physical sophistication The two cases that stand out are Chapter 7 onthe polynomial method and Chapters 10 and 11 on the nature of spin and onidentical particles The treatment of the polynomial method is quite unusualfor an introductory text on quantum physics, but it is beautifully explained insimple steps Although the author suggests that this chapter can be skipped atfirst reading, in my view, it is not to be missed Anyone who wondered why allbooks deal with just three standard problems (square well, harmonic oscillator,Coulomb potential), will find here some very enticing answers, and a wonderfuldiscussion of which types of problem yield closed analytical solutions For thepractitioners of numerical simulations, this approach provides elegant insights

to the well-known Kratzer and Morse potentials It is satisfying to see that thesefamiliar tools for simulating the properties of complex systems have simpleanalytical solutions Finally, Trachanas argues that the nature of the electron’s

“spin” is related to the essence of quantum measurement, and this is nicelyconnected to the character of elementary particles, “fermions” or “bosons,”and to their interaction with magnetic fields The concepts are deep, yet theirexplanation is elegant and convincing It is presented through a playful set ofquestions and answers, with no recourse to technical jargon The effect of thisapproach is powerful and empowering: The reader is left with the impressionthat even the most puzzling concepts of quantum physics can actually be grasped

in simple, intuitive terms

Foreword xxi

A famous joke among physicists is that “One does not really understandquantum mechanics, but simply gets used to it.” To an undergraduate studentbeing exposed to quantum physics for the first time, this phrase may come veryclose to how it feels to speak Nature’s language of the atomic scale Trachanas’Quantum Physics aims to remove this feeling and in my opinion it succeedsbrilliantly

Cambridge, Massachusetts

March, 2017

Efthimios Kaxiras Harvard University

Preface

Learning quantum mechanics is like learning a foreign language To speak it wellone needs to relocate to the country where it is spoken, and settle there for awhile—to make it one’s day-to-day language This book has been designed so thatthe teaching of quantum mechanics as a “foreign language” satisfies this residencerequirement Once the readers become familiar with the fundamental principles(Part I) and study some simple quantum systems (Part II), they are invited to “set-tle” in the atomic world (Part III) and learn quantum mechanics in action To talkthe language of quantum mechanics in its natural habitat So, in a way, this is adouble book: Quantum Mechanics and Structure of Matter It includes a com-plete introduction to the basic structures of nonnuclear matter—not simply as

“applications” but as a necessary final step toward understanding the theory itself.This is an introductory book, aimed at undergraduate students with no priorexposure to quantum theory, except perhaps from a general physics course.From a mathematical perspective, all that is required from the readers is to havetaken a Calculus I course and to be simply familiar with matrix diagonalization

in linear algebra

Those readers with some previous exposure to quantum mechanicalconcepts—say, the wave–particle duality principle—can readily proceed toChapter 2 But a quick browse through Chapter 1 may prove useful for them also,since this is a quite conceptual chapter that prepares the ground for acceptance

of the rather bizarre quantum mechanical concepts, which are so alien to oureveryday experience

An integral part of the book is the online supplement It contains review

quizzes, theory supplements that cover the few additional topics taught in amore formal course on quantum mechanics, and also some further applications.Installed in an open-source platform—Open edX—designed for massive openonline courses (MOOCs), the online supplement offers an interactive onlinelearning environment that may become an integral part of academic textbooks

in the future It can be freely accessed at http://www.mathesis.org

Had it not been for the generous decision of colleagues Manolis nakis and Leonidas Tsetseris to undertake the translation and editing of theoriginal Greek edition, this book would not have seen the light of day Mydeepest thanks therefore go to them They tirelessly plowed through the originaltext and my own continuous—and extensive—revisions, as well as two entirely

Antonoyian-new chapters (Chapters 1 and 7) Throughout this process, their comments andfeedback were critical in helping me finalize the text.

In the last stages of the editing process, I have benefited from a particularlyfruitful collaboration with Manolis Antonoyiannakis, and also with our youngercolleague Tasos Epitropakis, who copyedited the last version of the text, andundertook the translation and editing of the online supplement

I am also grateful to Manolis for managing the whole project, from the bookproposal, to peer review, to negotiating and liaising with Wiley-VCH, to marketresearch and outreach

The book was fortunate to have Jerry Icban Dadap (Columbia University)

as the very first critical reader of its English edition He read the manuscriptfrom cover to cover and made numerous comments that helped improve thetext considerably Vassilis Charmandaris (University of Crete), Petros Ditsas(University of Crete), Eleftherios Economou (University of Crete, and Founda-tion for Research & Technology–Hellas), Themis Lazaridis (City College of NewYork), Nikos Kopidakis (Macquarie University), Daniel Esteve (CEA-Saclay),Che Ting Chan (Hong Kong University of Science and Technology), and Pak WoLeung (Hong Kong University of Science and Technology) also made varioususeful comments and recommendations, as did Jessica Thomas (AmericanPhysical Society) on two early chapters I also thank Dimitrios Psaltis (University

of Arizona) and Demetrios Christodoulides (University of Central Florida) fortheir encouragement

I am indebted to—and humbled by—Efthimios Kaxiras (Harvard University)

for his artfully crafted Foreword I could not have hoped for a better introduction

to the book! I am also grateful to Nader Engheta (University of Pennsylvania) forhis constant encouragement, support, and endorsement

Prior to this English edition, the book has benefited enormously from its wideuse as the main textbook for quantum mechanics courses in most universitiesand polytechnics in Greece and Cyprus I owe a lot to the readers and instructorswho supported the book and provided feedback throughout these years But mygreatest debt is to John Iliopoulos (Ecole Normale Superieure) for his invaluableadvice and comments during the first writing of this book, and his generoussupport of its original Greek version

At a technical level, the skillful typesetting by Sofia Vlachou (Crete UniversityPress), the design of the figures by Iakovos Ouranos, and the installation of theonline supplement on Open edX by Nick Gikopoulos (Crete University Press)have contributed critically to the quality of the final product At the proofs stage,the assistance of Katerina Ligovanli (Crete University Press) was invaluable.Finally, I am grateful to Wiley’s production team, and especially Sujisha KunchiParambathu, for a smooth and productive collaboration

March 2017 Foundation for Research & Technology–Hellas (FORTH)

Preface xxv

The Teaching Philosophy of the Book

The long—and ever-increasing—list of quantum mechanics textbooks tells usthat there are wide-ranging views about how to teach the subject and what topics

to include in a course Aside from our topic selection, the pedagogic approachthat perhaps sets this book apart from existing textbooks can be summed up inthe following six themes:

1 Extensive Discussions of Results: The Physics Behind the Formulas

Throughout the book—and especially in Part II—we discuss in detail the result

of every solved problem, to highlight its physical meaning and understandits plausibility, how it behaves in various limits, and what are the broaderconclusions that can be derived from it These extensive discussions aim atgradually familiarizing students with the quantum mechanical concepts, anddeveloping their intuition

2 Dimensional Analysis: A Valuable Tool

Unlike most textbooks in the field, dimensional analysis is a basic tool for

us It not only helps us simplify the solution of many quantum mechanicalproblems but also extract important results, even when the underlying theory

is not known in detail or is quite cumbersome For example, to be able toshow—on purely dimensional grounds, and in one line of algebra—thatthe ultraviolet catastrophe is a consequence of the universality of thermalradiation, is, in our opinion, more interesting and profound than the detailedcalculation of the corresponding classical formula

3 Numerical Calculations and Order-of-Magnitude Estimates: Numbers Matter

The ability to make order-of-magnitude estimates and execute transparentnumerical calculations—in appropriate units—in order to understand aphysical result and decide whether it makes sense is of primary importance inthis book This is a kind of an art—with much physics involved—that needs to

be taught, too In this spirit, the order-of-magnitude estimates of basic tities involved in a problem, the use of suitable practical formulas in helpfulunits, the construction of dimensionless combinations of quantities that allownumerical calculations to have universal validity, or even the use of a naturalsystem of units (such as the atomic system), are some of the tools widelyemployed in the book In contrast to classical theories, quantum mechanicscannot be properly understood via its equations alone Numbers matter here

quan-4 Exact Solutions: Is There a Method?

The exactly solvable quantum mechanical problems are importantbecause—thanks to the explicit form of their solutions—they allow students

to develop familiarity with the quantum mechanical concepts and methods

It is therefore important for students to be able to solve these problems ontheir own, and thus reach the level of self-confidence that is necessary for ademanding course such as quantum mechanics

But the traditional way of presenting methods of exact solution, especiallythe power-series method, does not serve this purpose in our opinion Themethod itself has never been popular among students (and many teachersalso), while the application—in the hydrogen atom, for example—of finely

tuned transformations in order to arrive at a specific eponymous equationthat has known solutions, dispels any hope that one might ever be able tosolve the problem on one’s own.

On this topic, at least, this book can promise something different: Namely,that the exact solution and the calculation of eigenvalues for problems such

as (but not only) the harmonic oscillator or the hydrogen atom will not beharder than for the infinite-well potential And even further, that readers will

be able to execute such calculations on their own, in a few minutes, with noprior knowledge of any eponymous differential equation or the correspondingspecial function The pertinent ideas and techniques are presented, at a verybasic level, in Chapters 6, 8, and 9, while those interested in a systematicpresentation can consult Chapter 7

5 The Weirdness of Quantum Mechanics: Discussion of Conceptual Problems

Quantum mechanics is not just a foreign language It is a very strangelanguage, often at complete odds with the language of our classical world.Therefore, it cannot be taught in the same manner as any classical theory.Aside from its equations and calculational rules, quantum mechanics alsorequires a radical change in how we perceive physical reality and the kind

of knowledge we can draw from it Thus, innocuous questions like “what isthe electron really doing in the ground state of the hydrogen atom” or “whatexactly is the spin of an electron” cannot be properly answered without theappropriate conceptual gear The development of the pertinent conceptsbegins in Chapter 1 and continues in Chapters 8 and 10, where we discuss

questions such as those mentioned and discover the central role of the measurement processin quantum mechanics—both for the very definition ofphysical quantities (e.g., spin) and for the fundamental distinction to be madebetween questions that are valid in the quantum mechanical context (i.e.,experimentally testable) and those that are not

6 Online Quizzes: Student Engagement and Self-Learning

The interactive self-examination of students is another pedagogical feature

of the book, drawing on the author’s growing experience with Massive OpenOnline Courses (MOOCs) and his experimentation with various forms ofblended learning In contrast to the conventional textbook (or take-home)problems, the online quizzes allow and encourage a much greater variety

of targeted questions—many of them of a conceptual character—as well assuitable multi-step problems that make it easier for students (thanks also

to immediate access to their answers) to identify their own weaknesses andproceed to further study as they deem necessary The online quizzes will be aliving—and evolving—element of the book

At any rate, the fundamental teaching philosophy of this book is what weearlier called the residence requirement: the residence, for some time, in the

“country” where quantum mechanics is the spoken language Only in that

“place” can we come to terms with the weirdness of quantum mechanics.And, if, having fulfilled this residence requirement, there remain lingeringobjections, they may actually turn out to be legitimate, leading some oftoday’s students to a fundamental revision of the quantum mechanical theorytomorrow If such a revision is indeed to come

Editors’ Note

This book is a labor of love Officially, the translation project began in 2006 Bythen, Stefanos had completely rewritten and recast his original, three-volumetextbook into one comprehensive volume But the idea of bringing Stefanos’swork to a global audience was born in the early 1990s, when, as undergradu-ate students, we experienced firsthand—through his books and lectures at aUniversity of Crete summer school—his original style of explaining quantummechanics, combining a high command of the material with an eagerness todemystify and connect with the students Since then, we have often mused withfellow physicists and chemists that Stefanos’s work ought to be translated toEnglish one day After all, his books are taught in most departments of physics,chemistry, materials science, and engineering in Greece By 2006, we felt that

it was time to break the language barrier And what better place to start thanStefanos’s signature book on quantum physics?

The translation and editing of this book has been a challenge Both of us havehad demanding full-time jobs, so this had to be a part-time project on our “free”time Translating into a target language other than one’s mother tongue is tricky.And Stefanos’s native prose is highly elaborate, with rich syntax, long sentences,and a playfulness that is challenging to translate Progress has thus been slowand intermittent Over time, we developed a methodology for how to collaborateeffectively, utilize online tools, resolve translation issues, and calibrate our prose.And we revised the text relentlessly: Each chapter has been edited at least adozen times

But it was not all work and no play While editing the book, we were able toexpand our understanding, particularly with the new material accompanying thisEnglish edition, or on historical aspects we had previously overlooked And dur-ing a series of marathon phone calls with Stefanos, we were often able to digressfrom editorial issues and discuss physics, as if we were, once again, young stu-dents at a summer school in Crete, our whole life ahead of us and time on our side.Needless to say, there are many people who helped us complete the project.First and foremost, we are indebted to Stefanos for being an inspiring teacher,

a dear friend, and an extraordinary colleague His contribution goes wellbeyond having authored the original book: He oversaw our translation, gave usvaluable feedback, and took the opportunity to add two new chapters, reviseextensively the rest of the text, redesign dozens of problems, and expand theonline supplement We are fortunate to have Wiley-VCH as our publisher We

are grateful to Valerie Moliere, formerly consultant senior commissioning editor

at Wiley-VCH and currently at the Institution of Engineering and Technology,for her professionalism, support, and enthusiasm during the early stages ofthe book Special thanks to: Nina Stadhaus, our project editor at Wiley-VCH,for a smooth collaboration and for her patience during the preparation of themanuscript; Claudia Nussbeck, for critical assistance in the design of the cover;and Sujisha Kunchi Parambathu, our production editor, for the skillful andefficient processing of our manuscript

Manolis: I am indebted to Richard M Osgood Jr., for hosting me at theDepartment of Applied Physics & Applied Mathematics at Columbia Universityfor the duration of this project, and for his constant encouragement and support.The Columbia community has provided me with invaluable access to peopleand resources (library access, online tools, etc.) that critically affected thebook I am deeply grateful to Jerry Icban Dadap (Columbia University) forsubstantive feedback, encouragement, and his inspiring friendship My editorialposition at the journals of the American Physical Society (APS) has aided myprofessional development in numerous ways since 2003, and I am grateful to my

colleagues—especially from Physical Review B and Physical Review Letters—for

a productive collaboration throughout this time, and to the APS in general forthe privilege of working in this historic organization My writing style owes alot to the influence of Fotis Kafatos, whom I had the honor to advise from 2008

to 2010 in his capacity as President of the European Research Council I amthankful to my math mentor Manolis Maragakis for guidance and for instilling

in me a sense of urgency about this project I must also thank Nader Engheta(University of Pennsylvania), Daniel Esteve (CEA-Saclay), Dimitrios Psaltis(University of Arizona), Francisco-Jose Garcia-Vidal (Universidad Autonoma deMadrid), Miles Blencowe (Dartmouth College), and Che Ting Chan (Hong KongUniversity of Science and Technology) for their encouragement But my deepestgratitude goes to my family: my parents, Yannis and Chrysoula, for their love,support, and wise counsel; and, of course, my wife Katerina and our daughterNefeli, without whom none of this would be possible—they gave me the sweetestmotive to complete the work, while graciously accepting that I had to spend toomany evenings, weekends, and vacations away from them

Leonidas: I wish to thank Sokrates Pantelides of Vanderbilt University for ing me as a research associate and research assistant professor in the period thatoverlapped with the first years of this project But, my deepest thanks go to mywife Nektaria and our son Ioannis for their understanding and patience duringall these countless, multihour sessions I had to “borrow” from our “free” time.While this translation project has been a labor of love for us, the ultimatejudges of our work are the readers, of course We hope they will enjoy readingStefanos’s book as much as we have

(1) American Physical Society (2) Columbia University

National Technical University of Athens

Part I

Fundamental Principles

principle of wave–particle duality, which can be phrased as follows.

The principle of wave–particle duality: All physical entities have a dual character; they are waves and particles at the same time Everything we used to regard as being exclusively a wave has, at the same time, a corpuscular character, while everything we thought of as strictly a particle behaves also as a wave The relations between these two classically irreconcilable points of view—particle versus wave—are

In expressions (1.1) we start off with what we traditionally considered to be solely

a wave—an electromagnetic (EM) wave, for example—and we associate its wave characteristics f and 𝜆 (frequency and wavelength) with the corpuscular charac- teristics E and p (energy and momentum) of the corresponding particle Conversely,

in expressions (1.2), we begin with what we once regarded as purely a particle—say,

an electron—and we associate its corpuscular characteristics E and p with the wave characteristics f and 𝜆 of the corresponding wave Planck’s constant h, which provides the link between these two aspects of all physical entities, is equal to

h =6.62 × 10−27erg s = 6.62 × 10−34J s.

Actually, the aim here is not to retrace the historical process that led to this damental discovery, but precisely the opposite: Taking wave–particle duality as

fun-granted, we aim to show how effortlessly the peculiar phenomena we mentionedearlier can be explained Incidentally, these phenomena merit discussion notonly for their historical role in the discovery of a new physical principle butalso because of their continuing significance as fundamental quantum effects.Furthermore, we show that the principle of wave–particle duality should berecognized as the only sensible explanation to fundamental “mysteries” of the

atomic world—such as the extraordinary stability of its structures (e.g., atoms and molecules) and the uniqueness of their form—and not as some whim of

nature, which we are supposed to accept merely as an empirical fact

From its very name, it is clear that the principle of wave–particle duality can be

naturally split in two partial principles: (i) the principle of wave–particle duality

of light and (ii) the principle of wave–particle duality of matter We proceed to

examine both these principles, in relation to the peculiar phenomena and lems that led to their postulation

prob-1.2 The Principle of Wave–Particle Duality of Light

According to the preceding discussion, the wave–particle duality says thatlight—which in classical physics is purely an EM wave—has also a corpuscularcharacter The associated particle is the celebrated quantum of light, the

photon The wavelike features f and 𝜆 of the corresponding EM wave, and the particle-like features E and p of the associated particle, the photon, are related

through expressions (1.1) We will now see how this principle can explain twokey physical phenomena—the photoelectric effect and the Compton effect—thatare completely inexplicable in the context of classical physics

1.2.1 The Photoelectric Effect

With this term we refer today to the general effect of light-induced removal

of electrons from physical systems where they are bound Such systems can

be atoms and molecules—in which case we call the effect ionization—or a

metal, in which case we have the standard photoelectric effect studied at theend of the nineteenth and the beginning of twentieth century What makes theeffect peculiar from a classical perspective is the failure of classical physics toexplain the following empirical fact: The photoelectric effect (i.e., the removal

of electrons) is possible only if the frequency f of the incident EM radiation is greater than (or at least equal to) a value f0 that depends on the system fromwhich the removal occurs (atom, molecule, metal, etc.) We thus have

1.2 The Principle of Wave–Particle Duality of Light 5

of the electric field of light is the decisive factor Clearly, the very existence of

a threshold frequency in the photoelectric effect leaves no room for a classicalexplanation In contrast, the phenomenon is easily understood in quantum

mechanics A light beam of frequency f is also a stream of photons with energy

𝜖 = h f ; therefore, when quantized light—a “rain of light quanta”—impinges

on a metal, only one of two things can happen: Since the light quantum is bydefinition indivisible, when it “encounters” an electron it will either be absorbed

by it or “pass by” without interacting with it.1In the first case (absorption), theoutcome depends on the relative size of𝜖 = h f and the work function, W, of

the metal If the energy of the light quantum (i.e., the photon) is greater thanthe work function, the photoelectric effect occurs; if it is lower, there is no sucheffect Therefore, the quantum nature of light points naturally to the existence of

a threshold frequency in the photoelectric effect, based on the condition

h f ≥ W ⇒ f ≥ W

which also determines the value of the threshold frequency f0=W ∕h For

h f >W, the energy of the absorbed photon is only partially spent to extract the electron, while the remainder is turned into kinetic energy K (= m 𝑣2∕2) of theelectron We thus have

h f = W + K = W +1

which is known as Einstein’s photoelectric equation Written in the form

Equation (1.5) predicts a linear dependence of the photoelectrons’ kinetic energy

on the light frequency f , as represented by the straight line in Figure 1.1.

Therefore, by measuring K for various values of f we can fully confirm—or

disprove—Einstein’s photoelectric equation and, concomitantly, the quantumnature of light, as manifested via the photoelectric effect In addition, we candeduce the value of Planck’s constant from the slope of the experimental line.The discussion becomes clearer if in the basic relation 𝜖 = h f = hc∕𝜆 we

express energy in electron volts and length in angstroms—the “practical units”

of the atomic world (1 Å = 10−10m, 1 eV = 1.6 × 10−19J = 1.6 × 10−12erg) The product hc, which has dimensions of energy times length (since h has dimensions

of energy times time), then takes the value hc = 12 400 eV Å, and the formula for

the energy of the photon is written as

𝜖(eV) = 12 400

𝜆(Å) ≈

12 000

1 For completeness, let us also mention the possibility of scattering Here, the photon “collides”

with an electron, transfers to it part of its energy and momentum, and scatters in another direction

as a photon of different frequency (i.e., a different photon) This is the Compton effect, which we

examine in the coming section But let us note right away that Compton scattering has negligible probability to occur for low-energy photons like those used in the photoelectric effect.

Figure 1.1 The kinetic energy K of electrons as a function of photon frequency f

The experimental curve is a straight line whose slope is equal to Planck’s constant.

Vacuum tube

V

Figure 1.2 The standard experimental setup for studying the photoelectric effect The

photoelectric current occurs only when f > f0 and vanishes when f gets smaller than the threshold frequency f0 The kinetic energy of the extracted electrons is measured by reversing

the polarity of the source up to a value V0—known as the cutoff potential—for which the photoelectric current vanishes and we get K = eV0.

The last expression is often used in this book, since it gives simple numericalresults for typical wavelength values For example, for a photon with

𝜆 = 6000 Å—at about the middle of the visible spectrum—we have 𝜖 = 2 eV We

remind the readers that the electron volt (eV) is defined as the kinetic energyattained by an electron when it is accelerated by a potential difference of 1 V.Figure 1.2 shows a typical setup for the experimental study of the photoelectriceffect Indeed, Einstein’s photoelectric equation is validated by experiment,thus confirming directly that light is quantized, as predicted by the principle ofwave–particle duality of light

1.2 The Principle of Wave–Particle Duality of Light 7

Example 1.1 A beam of radiation of wavelength 𝜆 = 2000 Å impinges on a metal If the work function of the metal is W = 2 eV, calculate: (i) the kinetic energy K and the speed 𝑣 of the photoelectrons, (ii) the cutoff potential V0

Solution:If we set𝜆 = 2000 Å in the relation 𝜖(eV) = 12 000∕𝜆(Å), we obtain

𝜖 = 6 eV So if we subtract the work function 2 eV, we obtain 4 eV for the kinetic

energy of the outgoing electrons The speed of the photoelectrons can then becalculated by the relation

K = 1

2m𝑣2= 1

2mc

2(𝑣 c

our nonrelativistic treatment of the problem) As for the cutoff potential V0, it is

equal to V0=4 V, since K = 4 eV and K = e ⋅V0

We should pause here to remark how much simpler and more transparent ourcalculations become when, instead of using the macroscopic units of one system

or another (cgs or SI), we use the “natural” units defined by the very phenomena

we study For example, we use eV for energy, which also comes in handywhen we express the rest mass of particles in terms of their equivalent energyrather than in g or kg In this spirit, it is worthwhile to memorize the numbers

m e c2≈0.5 MeV and m p c2≈1836 m e c2=960 MeV ≈ 1 GeV for electrons andprotons, respectively We will revisit the topic of units later (Section 1.2.3)

1.2.2 The Compton Effect

According to expressions (1.1), a photon carries energy𝜖 = h f and momentum

p = h∕𝜆 And because it carries momentum, the photon can be regarded as a

particle in the full sense of the term But how can we verify that a photon hasnot only energy but also momentum? Clearly, we need an experiment whereby

photons collide with very light particles—we will shortly see why We can then

apply the conservation laws of energy and momentum during the collision to

check whether photons satisfy a relation of the type p = h∕ 𝜆.

Why do we need the target particles to be as light as possible—that is, electrons?

It is well known that when small moving spheres collide with considerably largerstationary ones, they simply recoil with no significant change in their energy,while the large spheres stay practically still during the collision Conversely, if thetarget spheres are also small (or even smaller than the projectile particles), thenupon collision they will move, taking some of the kinetic energy of the impingingspheres, which then scatter in various directions with lower kinetic energy.Therefore, if photons are particles in the full sense of the term, they will behave

as such when scattered by light particles, like the electrons of a material: They will

transfer part of their momentum and energy to the target electrons and end upwith lower energy than they had before the collision In other words, we will have

𝜖′=h f′< 𝜖 = h f ⇒ f′< f ⇒ 𝜆′>𝜆, (1.8)where the primes refer to the scattered photons This shift of the wavelength to

greater values when photons collide with electrons is known as the Compton effect It was confirmed experimentally by Arthur H Compton in 1923, when

an x-ray beam was scattered off by the electrons of a target material Whywere x-rays used to study the effect? (Today we actually prefer𝛾 rays for this

purpose.) Because x- (and𝛾) rays have very short wavelength, the momentum

p = h∕ 𝜆 of the impinging photons is large enough to ensure large momentum

and energy transfer to the practically stationary target electrons (whereby thescattered photons suffer a great loss of momentum and energy) In a Comptonexperiment we measure the wavelength𝜆′of the scattered photon as a function

of the scattering angle𝜃 between the directions of the impinging and scattered

photon By applying the principles of energy and momentum conservation wecan calculate the dependence𝜆′=𝜆′(𝜃) in a typical collision event such as the

one depicted in Figure 1.3

Indeed, if we use the conservation equations—see Example 1.2—to eliminate

the parameters E, p, and 𝜙 (which we do not observe in the experiment, as they

pertain to the electron), we eventually obtain

Δ𝜆 = 𝜆′

−𝜆 = h

mc(1 − cos𝜃) = 𝜆C(1 − cos𝜃), (1.9)where

𝜆C= h

mc =0.02427 Å ≈ 24 × 10

is the so-called Compton wavelength of the electron It follows from (1.9) that

the fractional shift in the wavelength, Δ𝜆∕𝜆, is on the order of 𝜆C∕𝜆, so it is

considerable in size only when𝜆 is comparable to or smaller than the Compton

wavelength This condition is met in part for hard x-rays and in full for𝛾 rays.

Compton’s experiment fully confirmed the prediction (1.9) and,

concomi-tantly, the relation p = h∕ 𝜆 on which it was based The wave–particle duality of

light is thus an indisputable experimental fact Light—and, more generally, EMradiation—has a wavelike and a corpuscular nature at the same time

f, λ

y

x f′, λ′

E, p

θ ϕ

Figure 1.3 A photon colliding with

a stationary electron The photon

is scattered at an angle𝜃 with a

wavelength𝜆′ that is greater than its initial wavelength𝜆 The

electron recoils at an angle𝜙 with energy E and momentum p.

1.2 The Principle of Wave–Particle Duality of Light 9

Example 1.2 In a Compton experiment the impinging photons have length𝜆 = 12 × 10−3Å =𝜆C∕2 and some of them are detected at an angle of 60∘with respect to the direction of the incident beam Calculate (i) the wavelength,momentum, and energy of the scattered photons and (ii) the momentum, energy,and scattering angle of the recoiling electrons Express your results as a function

wave-of the electron mass and fundamental physical constants

Solution:For𝜆 = 𝜆C∕2 and𝜃 = 60∘ (⇒ cos 𝜃 = 1∕2), the formula Δ𝜆 = 𝜆′−𝜆 =

𝜆C(1 − cos𝜃) yields 𝜆′=𝜆C, which is twice the initial wavelength The tum and energy of the photon before and after scattering are

where the index “𝛾” in the momentum symbol p denotes the photon (in

custom-ary reference to “𝛾 rays”) to disambiguate it from the symbol p of the electronic

momentum We can now write the conservation laws of energy and momentum

2 −psin𝜙

⇒ p sin 𝜙 = mc

√3

Now that p and E for the electron ( p =√

3 mc, E = 2mc2)have been calculated,one may wonder whether they satisfy the relativistic energy–momentum relation

E2=c2p2+m2c4.2Indeed they do, as the readers can readily verify

2 The use of relativistic formulas here is necessary because the speeds of the recoiling electrons are indeed relativistic.

1.2.3 A Note on Units

At this point we should pause to make some remarks on the system of units Wehave already suggested (see Example 1.1) that both the cgs and SI system of unitsare equally unsuited for the atomic world, since it would be quite unreasonable

to measure, for example, the energy in joules (SI) or erg (cgs) The natural scale

of energies in atoms is the electron volt, a unit that is 19 orders of magnitudesmaller than the joule and 12 orders smaller than the erg! Likewise, the naturallength unit in the atomic world is the angstrom (= 10−10m), since it is the typicalsize of atoms In this spirit, it is inconvenient to express, say, Planck’s constant

in erg s or J s, and hc—another useful constant—in erg cm or J m; instead, it is easier to use the corresponding practical units eV s for h and eV Å for hc There

is, however, one instance in atomic physics where we cannot avoid choosingone system over another: The basic force law governing atomic and molecularstructure—Coulomb’s law—has a much more convenient form in cgs than SIunits, namely,

of the hydrogen atom has the simple form WI=me4∕ ℏ2, while in SI units it

becomes WI=me4∕32𝜋2𝜖2

0ℏ2! Therefore, our choice is to go with the cgs systemfor the mathematical expression of Coulomb’s law, but to make all calculations in

the practical units eV and Å, or even in the so-called atomic system of units, which

we will introduce later As you will soon find out, the practical unit of energy (i.e.,the eV) is much better suited than the joule, even for calculations concerningphysical quantities, like voltage or electric field intensities in atoms, where the

SI units (V or V∕m) are certainly preferable The reason is that the energy unit

eV is directly related both to the fundamental unit of charge e and the SI unit

of volt A pertinent example was the calculation—without much effort!—of thecutoff potential in Example 1.1 The same holds true for electric field intensities

in atoms, where the SI unit V∕m (or V∕cm) arises naturally from the energyunit eV

So, even readers who are adherents of the SI system will find that the practicalenergy unit, eV, is much closer to the SI system than the joule itself

As for the cgs system, we remind the readers that its basic units—length, mass,and time—are the centimeter (cm), the gram (g), and the second (s), while forderivative quantities such as force, energy, and charge, the cgs units are the dyn,the erg, and the esu-q (electrostatic unit of charge), respectively These units arerelated to their SI counterparts as follows:

Charge esu-q C (coulomb) 1 C = 3 × 10 9 esu-q