An Introduction to Quantum Physics
An Introduction to Quantum Physics
A First Course for Physicists, Chemists, Materials Scientists, and Engineers
Stefanos Trachanas Authors All books published by Wiley-VCH are carefully produced. Nevertheless, authors, Stefanos Trachanas editors, and publisher do not warrant the Foundation for Research & Technology– information contained in these books, Hellas (FORTH) including this book, to be free of errors. Crete University Press Readers are advised to keep in mind that 100 Nikolaou Plastira statements, data, illustrations, procedural Vassilika Vouton details or other items may inadvertently 70013 Heraklion be inaccurate. Greece Library of Congress Card No.: applied for and
University of Crete British Library Cataloguing-in-Publication Department of Physics Data P.O. Box 2208 A catalogue record for this book is 71003 Heraklion available from the British Library. Greece Bibliographic information published by Manolis Antonoyiannakis the Deutsche Nationalbibliothek The American Physical Society The Deutsche Nationalbibliothek Editorial Office lists this publication in the Deutsche 1ResearchRoad Nationalbibliografie; detailed Ridge, NY 11961 bibliographic data are available on the United States Internet at
Cover Design Schulz Grafik-Design, Fußgönheim, Germany Typesetting SPi Global, Chennai, India Printing and Binding
Printed on acid-free paper to Maria
vii
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 Orderof 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 viii Contents
1.3.11 Numerical Calculations with Matter Waves: Practical Formulas and Physical Applications 31 1.3.12 A Direct Confirmation of the Existence of Matter Waves: The Davisson–Germer Experiment 33 1.3.13 The Double-Slit Experiment: Collapse of the Wavefunction Upon Measurement 34 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)|2 as 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 ℏ2𝜋2 2 2 4.2.2.1 Dimensional Analysis of the Formula En =( ∕2mL )n . Do We Need an Exact Solution to Predict the Energy Dependence on ℏ, m,andL? 135 4.2.2.2 Dependence of the Ground-State Energy on ℏ, m,andL : The Classical Limit 136 4.2.2.3 The Limit of Large Quantum Numbers and Quantum Discontinuities 137 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 Parameter 𝜆 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 x Contents
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 Particle 159 5.3.2.3 A Simple Approximate Expression for the Transmission Coefficient 160 5.3.2.4 Exponential Sensitivity of the Tunneling Effect to the Mass of the Particle 162 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 Forbidden Region? 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: Bidimensional Equations 194 7.3 The Polynomial Method in Action: Exact Solution of the Kratzer and Morse Potentials 197 7.4 Mathematical Afterword 202
8 The Hydrogen Atom. I: Spherically Symmetric Solutions 207 8.1 Introduction 207 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 Limit ℏ → 0orm → ∞ 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 DistanceoftheElectronfromtheNucleus 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:TheFirstExcitedStates 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 xii Contents
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 WhatisSpin? 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 Momenta 295 10.7.1 The Eigenvalues 295 10.7.2 The Eigenfunctions 300
11 Identical Particles and the Pauli Principle 305 11.1 Introduction 305 11.2 The Principle of Indistinguishability of Identical Particles in Quantum Mechanics 305 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 12.3.6 Examples 338 12.4 Approximate Calculations in Atoms: Perturbation Theory and the Variational Method 341 12.4.1 Perturbation Theory 342 12.4.2 Variational Method 346
13 Molecules. I: Elementary Theory of the Chemical Bond 351 13.1 Introduction 351 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 Molecule2 H 360 13.3.2 The Helium “Molecule” He2 363 13.3.3 The Lithium Molecule Li2 364 13.3.4 TheOxygenMoleculeO2 364 13.3.5 The Nitrogen Molecule2 N 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 Molecule3 NH 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 CH4 Molecule According to the Elementary Theory: An Erroneous Prediction 393
14.2.2 Hybridized Orbitals and the CH4 Molecule 395 14.2.3 Total and Partial Hybridization 401
14.2.4 The Need for Partial Hybridization: The Molecules2 C H4,C2H2,and C2H6 404 14.2.5 Application of Hybridization Theory to Conjugated Hydrocarbons 408 xiv Contents
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 aFermiGas 460 15.5.4 A Classic Application: The “Anomaly” of the Electronic Specific Heat of Metals 463
16 Matter and Light: The Interaction of Atoms with Electromagnetic Radiation 469 16.1 Introduction 469 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 Contents xv
16.5.2 Main Properties of Laser Light 491 16.5.2.1 Phase Coherence 491 16.5.2.2 Directionality 491 16.5.2.3 Intensity 491 16.5.2.4 Monochromaticity 492 16.6 Spontaneous Emission 494 16.7 Theory of Time-dependent Perturbations: Fermi’s Rule 499
16.7.1 Approximate Calculation of Transition Probabilities Pn→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 Eigenvalues OS4.3 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 xvi Contents
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 Potentials OS7.3 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
xix
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 prenatal form. In the late 1970s, while a graduate student at Harvard, Trachanas was working 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) to friends who appreciated his fascination with Nature’s wonders. At the time, I was an undergraduate student at the technical school down the river, struggling to learn 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, for their 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 language also helped). It is a great pleasure to see that in the latest version of his book on Quantum Physics, this freshness is intact, enriched from decades of teaching experience. This latest version is of course a long way from his original set of notes; it is a thorough account of the theory of quantum mechanics, expertly translated by Manolis Antonoyiannakis and Leonidas Tsetseris, in the form of a comprehensive 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 is important; one does expect a textbook to contain a complete treatment of the subject and to serve as a reference for key results and expressions. But there are also 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 classical pictures 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 of quantum behavior can be elucidated. Finally, the application of key concepts to realistic 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” (my quotes) concepts of the quantum realm, such as wave–particle duality and the uncertainty principle. Then, in Part II, the language is used to describe the xx Foreword
standard simple problems, the square well, the harmonic oscillator, and the Coulomb potential. It is also applied to the hydrogen atom, illustrating how this language can capture the behavior of Nature at the level of fundamental particles—electrons and protons. Finally, in Part III, the student is given a thorough training in the use of the quantum language to address problems relevant to real applications in modern life, which is dominated by quantum devices, for better or for worse. Many everyday activities, from using a cell phone to call friends to employing photovoltaics for powering your house, are directly related to quintessentially quantum phenomena, that is, the physics of semiconductors, conductors, and insulators, and their interaction with light. All these phenomena are explained thoroughly and clearly in Trachanas’ book. The reader of the book will certainly develop a deep appreciation of the principles on which many everyday devices are based. There is also a lovely discussion of the properties of molecules and the nature of the chemical bond. The treatment ranges from the closed sixfold hydrocarbon ring (“benzene”) to the truncated icosahedron formed by 60 carbon atoms (“fullerene”), with several other 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 mathematical and physical sophistication. The two cases that stand out are Chapter 7on the polynomial method and Chapters 10 and 11 on the nature of spin and on identical particles. The treatment of the polynomial method is quite unusual for an introductory text on quantum physics, but it is beautifully explained in simple steps. Although the author suggests that this chapter can be skipped at first reading, in my view, it is not to be missed. Anyone who wondered why all books deal with just three standard problems (square well, harmonic oscillator, Coulomb potential), will find here some very enticing answers, and a wonderful discussion of which types of problem yield closed analytical solutions. For the practitioners of numerical simulations, this approach provides elegant insights to the well-known Kratzer and Morse potentials. It is satisfying to see that these familiar tools for simulating the properties of complex systems have simple analytical solutions. Finally, Trachanas argues that the nature of the electron’s “spin” is related to the essence of quantum measurement, and this is nicely connected to the character of elementary particles, “fermions” or “bosons,” and to their interaction with magnetic fields. The concepts are deep, yet their explanation is elegant and convincing. It is presented through a playful set of questions and answers, with no recourse to technical jargon. The effect of this approach is powerful and empowering: The reader is left with the impression that 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 understand quantum mechanics, but simply gets used to it.” To an undergraduate student being exposed to quantum physics for the first time, this phrase may come very close 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 succeeds brilliantly.
Cambridge, Massachusetts Efthimios Kaxiras March, 2017 Harvard University
xxiii
Preface
Learning quantum mechanics is like learning a foreign language. To speak it well one needs to relocate to the country where it is spoken, and settle there for a while—to make it one’s day-to-day language. This book has been designed so that the teaching of quantum mechanics as a “foreign language” satisfies this residence requirement. 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 talk the language of quantum mechanics in its natural habitat. So, in a way, this is a double 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 prior exposure to quantum theory, except perhaps from a general physics course. From a mathematical perspective, all that is required from the readers is to have taken a Calculus I course and to be simply familiar with matrix diagonalization in linear algebra. Those readers with some previous exposure to quantum mechanical concepts—say, the wave–particle duality principle—can readily proceed to Chapter 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 our everyday 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 a more formal course on quantum mechanics, and also some further applications. Installed in an open-source platform—Open edX—designed for massive open online courses (MOOCs), the online supplement offers an interactive online learning 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 Antonoyian- nakis and Leonidas Tsetseris to undertake the translation and editing of the original Greek edition, this book would not have seen the light of day. My deepest thanks therefore go to them. They tirelessly plowed through the original text and my own continuous—and extensive—revisions, as well as two entirely xxiv Preface
new chapters (Chapters 1 and 7). Throughout this process, their comments and feedback were critical in helping me finalize the text. In the last stages of the editing process, I have benefited from a particularly fruitful collaboration with Manolis Antonoyiannakis, and also with our younger colleague Tasos Epitropakis, who copyedited the last version of the text, and undertook the translation and editing of the online supplement. I am also grateful to Manolis for managing the whole project, from the book proposal, to peer review, to negotiating and liaising with Wiley-VCH, to market research 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 manuscript from cover to cover and made numerous comments that helped improve the text 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 New York), Nikos Kopidakis (Macquarie University), Daniel Esteve (CEA-Saclay), Che Ting Chan (Hong Kong University of Science and Technology), and Pak Wo Leung (Hong Kong University of Science and Technology) also made various useful comments and recommendations, as did Jessica Thomas (American Physical Society) on two early chapters. I also thank Dimitrios Psaltis (University of Arizona) and Demetrios Christodoulides (University of Central Florida) for their 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) for his constant encouragement, support, and endorsement. Prior to this English edition, the book has benefited enormously from its wide use as the main textbook for quantum mechanics courses in most universities and polytechnics in Greece and Cyprus. I owe a lot to the readers and instructors who supported the book and provided feedback throughout these years. But my greatest debt is to John Iliopoulos (Ecole Normale Superieure) for his invaluable advice and comments during the first writing of this book, and his generous support of its original Greek version. At a technical level, the skillful typesetting by Sofia Vlachou (Crete University Press), the design of the figures by Iakovos Ouranos, and the installation of the online 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 Kunchi Parambathu, for a smooth and productive collaboration.
Heraklion, Crete Stefanos Trachanas 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 us that 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 approach that perhaps sets this book apart from existing textbooks can be summed up in the 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 understand its plausibility, how it behaves in various limits, and what are the broader conclusions that can be derived from it. These extensive discussions aim at gradually familiarizing students with the quantum mechanical concepts, and developing 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 mechanical problems but also extract important results, even when the underlying theory is not known in detail or is quite cumbersome. For example, to be able to show—on purely dimensional grounds, and in one line of algebra—that the ultraviolet catastrophe is a consequence of the universality of thermal radiation, is, in our opinion, more interesting and profound than the detailed calculation 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 transparent numerical calculations—in appropriate units—in order to understand a physical result and decide whether it makes sense is of primary importance in this 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 quan- tities involved in a problem, the use of suitable practical formulas in helpful units, the construction of dimensionless combinations of quantities that allow numerical calculations to have universal validity, or even the use of a natural system of units (such as the atomic system), are some of the tools widely employed in the book. In contrast to classical theories, quantum mechanics cannot be properly understood via its equations alone. Numbers matter here. 4. Exact Solutions: Is There a Method? The exactly solvable quantum mechanical problems are important because—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 on their own, and thus reach the level of self-confidence that is necessary for a demanding course such as quantum mechanics. But the traditional way of presenting methods of exact solution, especially the power-series method, does not serve this purpose in our opinion. The method itself has never been popular among students (and many teachers also), while the application—in the hydrogen atom, for example—of finely xxvi Preface
tuned transformations in order to arrive at a specific eponymous equation that has known solutions, dispels any hope that one might ever be able to solve 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 be harder 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 no prior knowledge of any eponymous differential equation or the corresponding special function. The pertinent ideas and techniques are presented, at a very basic level, in Chapters 6, 8, and 9, while those interested in a systematic presentation 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 strange language, 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 also requires a radical change in how we perceive physical reality and the kind of knowledge we can draw from it. Thus, innocuous questions like “what is the electron really doing in the ground state of the hydrogen atom” or “what exactly is the spin of an electron” cannot be properly answered without the appropriate conceptual gear. The development of the pertinent concepts begins 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 process in quantum mechanics—both for the very definition of physical quantities (e.g., spin) and for the fundamental distinction to be made between 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 Open Online Courses (MOOCs) and his experimentation with various forms of blended 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 as suitable multi-step problems that make it easier for students (thanks also to immediate access to their answers) to identify their own weaknesses and proceed to further study as they deem necessary. The online quizzes will bea living—and evolving—element of the book. At any rate, the fundamental teaching philosophy of this book is what we earlier 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 lingering objections, they may actually turn out to be legitimate, leading some of today’s students to a fundamental revision of the quantum mechanical theory tomorrow. If such a revision is indeed to come. xxvii
Editors’ Note
This book is a labor of love. Officially, the translation project began in 2006. By then, Stefanos had completely rewritten and recast his original, three-volume textbook into one comprehensive volume. But the idea of bringing Stefanos’s work to a global audience was born in the early 1990s, when, as undergradu- ate students, we experienced firsthand—through his books and lectures at a University of Crete summer school—his original style of explaining quantum mechanics, combining a high command of the material with an eagerness to demystify and connect with the students. Since then, we have often mused with fellow physicists and chemists that Stefanos’s work ought to be translated to English 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 than Stefanos’ssignaturebookonquantumphysics? The translation and editing of this book has been a challenge. Both ofushave had 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 slow and intermittent. Over time, we developed a methodology for how to collaborate effectively, utilize online tools, resolve translation issues, and calibrate our prose. And we revised the text relentlessly: Each chapter has been edited at least a dozen times. But it was not all work and no play. While editing the book, we were able to expand our understanding, particularly with the new material accompanying this English 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 digress from 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 well beyond having authored the original book: He oversaw our translation, gave us valuable feedback, and took the opportunity to add two new chapters, revise extensively the rest of the text, redesign dozens of problems, and expand the online supplement. We are fortunate to have Wiley-VCH as our publisher. We xxviii Editors’ Note
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 of the book. Special thanks to: Nina Stadhaus, our project editor at Wiley-VCH, for a smooth collaboration and for her patience during the preparation of the manuscript; Claudia Nussbeck, for critical assistance in the design of the cover; and Sujisha Kunchi Parambathu, our production editor, for the skillful and efficient processing of our manuscript. Manolis: I am indebted to Richard M. Osgood Jr., for hosting me at the Department of Applied Physics & Applied Mathematics at Columbia University for the duration of this project, and for his constant encouragement and support. The Columbia community has provided me with invaluable access to people and resources (library access, online tools, etc.) that critically affected the book. I am deeply grateful to Jerry Icban Dadap (Columbia University) for substantive feedback, encouragement, and his inspiring friendship. My editorial position at the journals of the American Physical Society (APS) has aided my professional 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 for the privilege of working in this historic organization. My writing style owes a lot 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 am thankful 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 de Madrid), Miles Blencowe (Dartmouth College), and Che Ting Chan (Hong Kong University of Science and Technology) for their encouragement. But my deepest gratitude goes to my family: my parents, Yannis and Chrysoula, for their love, support, and wise counsel; and, of course, my wife Katerina and our daughter Nefeli, without whom none of this would be possible—they gave me the sweetest motive to complete the work, while graciously accepting that I had to spend too many evenings, weekends, and vacations away from them. Leonidas: I wish to thank Sokrates Pantelides of Vanderbilt University for host- ing me as a research associate and research assistant professor in the period that overlapped with the first years of this project. But, my deepest thanks go to my wife Nektaria and our son Ioannis for their understanding and patience during all 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 ultimate judges of our work are the readers, of course. We hope they will enjoy reading Stefanos’s book as much as we have.
New York Manolis Antonoyiannakis (1) American Physical Society (2) Columbia University
Athens, Greece Leonidas Tsetseris National Technical University of Athens 1
Part I
Fundamental Principles
3
1
The Principle of Wave–Particle Duality: An Overview
1.1 Introduction
In the year 1900, physics entered a period of deep crisis as a number of peculiar phenomena, for which no classical explanation was possible, began to appear one after the other, starting with the famous problem of blackbody radiation. By 1923, when the “dust had settled,” it became apparent that these peculiarities had a common explanation. They revealed a novel fundamental principle of nature that was completely at odds with the framework of classical physics: the celebrated 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 , h, E = hf p = 𝜆 (1.1) or, equivalently, E h f = ,𝜆= . (1.2) h p 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 × 10−27 ergs = 6.62 × 10−34 Js. Actually, the aim here is not to retrace the historical process that led to this fun- damental discovery, but precisely the opposite: Taking wave–particle duality as
An Introduction to Quantum Physics: A First Course for Physicists, Chemists, Materials Scientists, and Engineers, First Edition. Stefanos Trachanas, Manolis Antonoyiannakis and Leonidas Tsetseris. © 2018 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2018 by Wiley-VCH Verlag GmbH & Co. KGaA. 4 1 The Principle of Wave–Particle Duality: An Overview
granted, we aim to show how effortlessly the peculiar phenomena we mentioned earlier can be explained. Incidentally, these phenomena merit discussion not only for their historical role in the discovery of a new physical principle but also because of their continuing significance as fundamental quantum effects. Furthermore, we show that the principle of wave–particle duality should be recognized 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 prob- lems that led to their postulation.
1.2 The Principle of Wave–Particle Duality of Light
According to the preceding discussion, the wave–particle duality says that light—which in classical physics is purely an EM wave—has also a corpuscular character. 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 two key physical phenomena—the photoelectric effect and the Compton effect—that are 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 the end of the nineteenth and the beginning of twentieth century. What makes the effect peculiar from a classical perspective is the failure of classical physics to explain 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 from which the removal occurs (atom, molecule, metal, etc.). We thus have ≥ . f f0 (1.3) In classical physics, a “threshold condition” of the type (1.3) has no physical justification. Whatever the frequency of the incident EM wave, its electric field will always produce work on the electrons, and when this exceeds the work function of the metal—the minimum energy required for extraction—electrons will be ejected from it. In other words, in classical physics, the frequency plays no crucial role in the energy exchanges between light and matter, while the intensity 1.2 The Principle of Wave–Particle Duality of Light 5 oftheelectricfieldoflightisthedecisivefactor.Clearly,theveryexistenceof a threshold frequency in the photoelectric effect leaves no room for a classical explanation. 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 = hf; 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 by definition indivisible, when it “encounters” an electron it will either be absorbed by it or “pass by” without interacting with it.1 Inthefirstcase(absorption),the outcome depends on the relative size of 𝜖 = hf and the work function, W,of the metal. If the energy of the light quantum (i.e., the photon) is greater than the work function, the photoelectric effect occurs; if it is lower, there is no such effect. Therefore, the quantum nature of light points naturally to the existence of a threshold frequency in the photoelectric effect, based on the condition W hf ≥ W ⇒ f ≥ = f , (1.4) h 0 which also determines the value of the threshold frequency f0 = W∕h.For hf >W, the energy of the absorbed photon is only partially spent to extract the electron, while the remainder is turned into kinetic energy K (= m𝑣2∕2) of the electron. We thus have 1 hf = W + K = W + m𝑣2, (1.5) 2 which is known as Einstein’s photoelectric equation.Writtenintheform ≥ , K = hf − W ( f f0) (1.6) 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 quantum nature of light, as manifested via the photoelectric effect. In addition, we can deduce the value of Planck’s constant from the slope of the experimental line. The discussion becomes clearer if in the basic relation 𝜖 = hf = hc∕𝜆 we express energy in electron volts and length in angstroms—the “practical units” of the atomic world (1Å= 10−10 m, 1 eV = 1.6 × 10−19 J = 1.6 × 10−12 erg). 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 12 400 12 000 𝜖(eV)= ≈ . (1.7) 𝜆(Å) 𝜆(Å)
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. 6 1 The Principle of Wave–Particle Duality: An Overview
K (eV) 4 W = 3 eV f = 0.72 × 1015 Hz 3 0
2
1
f (1015 Hz) f 0.5 0 11.52
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.
f Light
Vacuum tube
I A
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 numerical results for typical wavelength values. For example, for a photon with 𝜆 = 6000 Å—at about the middle of the visible spectrum—we have 𝜖 = 2eV.We remind the readers that the electron volt (eV) is defined as the kinetic energy attained 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 photoelectric effect. Indeed, Einstein’s photoelectric equation is validated by experiment, thus confirming directly that light is quantized, as predicted by the principle of wave–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 𝑣 energy K and the speed of the photoelectrons, (ii) the cutoff potential V0.
Solution: If we set 𝜆 = 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 be calculated by the relation ( ) √ 1 1 𝑣 2 𝑣 2K K = m𝑣2 = mc2 ⇒ = 2 √2 2 c c mc 2 × 4eV 𝑣 = = 4 × 10−3 ⇒ = 4 × 10−3 ⇒ 𝑣 = 1.2 × 108 cm∕s. 1 × 106 eV c 2 Here we wrote 1 m𝑣2 as 1 mc2(𝑣∕c)2 in order to express mc2 in eV (mc2 = 0.5 MeV 2 2 for an electron) and the electronic speed as a fraction of the speed of light (which is useful in several ways: for example, it helps us assess the validity of our nonrelativistic treatment of the problem). As for the cutoff potential V0,itis ⋅ equal to V0 = 4V,sinceK = 4eVandK = e V0. We should pause here to remark how much simpler and more transparent our calculations 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 handy when we express the rest mass of particles in terms of their equivalent energy rather than in g or kg. In this spirit, it is worthwhile to memorize the numbers 2 2 2 mec ≈ 0.5 MeV and mpc ≈ 1836 mec = 960 MeV ≈ 1GeV for electrons and protons, 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 energy 𝜖 = hf and momentum p = h∕𝜆.Andbecauseitcarriesmomentum,thephotoncanberegardedasa particle in the full sense of the term. But how can we verify that a photon has not 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. Whydoweneedthetargetparticlestobeaslightaspossible—thatis,electrons? It is well known that when small moving spheres collide with considerably larger stationary ones, they simply recoil with no significant change in their energy, while the large spheres stay practically still during the collision. Conversely, if the target spheres are also small (or even smaller than the projectile particles), then upon collision they will move, taking some of the kinetic energy of the impinging spheres, 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 8 1 The Principle of Wave–Particle Duality: An Overview
transfer part of their momentum and energy to the target electrons and end up with lower energy than they had before the collision. In other words, we will have 휖′ = hf′ <휖= hf ⇒ f ′ < f ⇒ 휆′ >휆, (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 anx-raybeamwasscatteredoffbytheelectronsofatargetmaterial.Why were x-rays used to study the effect? (Today we actually prefer 훾 rays for this purpose.) Because x- (and 훾) rays have very short wavelength, the momentum p = h∕휆 of the impinging photons is large enough to ensure large momentum and energy transfer to the practically stationary target electrons (whereby the scattered photons suffer a great loss of momentum and energy). In a Compton experiment we measure the wavelength 휆′ of the scattered photon as a function of the scattering angle 휃 between the directions of the impinging and scattered photon. By applying the principles of energy and momentum conservation we can calculate the dependence 휆′ = 휆′(휃) in a typical collision event such as the onedepictedinFigure1.3. Indeed, if we use the conservation equations—see Example 1.2—to eliminate the parameters E, p,and휙 (which we do not observe in the experiment, as they pertain to the electron), we eventually obtain h Δ휆 = 휆′ − 휆 = (1 − cos 휃)=휆 (1 − cos 휃), (1.9) mc C where h 휆 = = 0.02427 Å ≈ 24 × 10−3 Å (1.10) C mc is the so-called Compton wavelength of the electron. It follows from (1.9) that 휆 휆 휆 휆 the fractional shift in the wavelength, Δ ∕ ,isontheorderof C∕ ,soitis considerableinsizeonlywhen휆 is comparable to or smaller than the Compton wavelength. This condition is met in part for hard x-rays and in full for 훾 rays. Compton’s experiment fully confirmed the prediction (1.9) and, concomi- tantly, the relation p = h∕휆 on which it was based. The wave–particle duality of light is thus an indisputable experimental fact. Light—and, more generally, EM radiation—has a wavelike and a corpuscular nature at the same time.
Figure 1.3 A photon colliding with y a stationary electron. The photon f′, λ′ is scattered at an angle 휃 with a wavelength 휆′ that is greater than its initial wavelength 휆.The electron recoils at an angle 휙 with θ f, λ energy E and momentum p. x ϕ
E, p 1.2 The Principle of Wave–Particle Duality of Light 9
Example 1.2 In a Compton experiment the impinging photons have wave- 휆 −3 휆 ∘ length = 12 × 10 Å = C∕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 of the electron mass and fundamental physical constants.
휆 휆 휃 ∘ ⇒ 휃 휆 휆′ 휆 Solution: For = C∕2and = 60 ( cos = 1∕2),theformulaΔ = − = 휆 휃 휆′ 휆 C(1 − cos ) yields = C, which is twice the initial wavelength. The momen- tum and energy of the photon before and after scattering are h h h h h p = = = = 2mc, p′ = = = mc 훾 휆 휆 훾 휆′ 휆 C∕2 (h∕mc)∕2 C and hc hc hc hc 휖 = hf = = = 2mc2,휖′ = = = mc2, 휆 휆 휆′ 휆 C∕2 C where the index “훾” in the momentum symbol p denotes the photon (in custom- ary reference to “훾 rays”) to disambiguate it from the symbol p of the electronic momentum. We can now write the conservation laws of energy and momentum as follows: • Conservation of energy 휖 + mc2 = 휖′ + E ⇒ 2mc2 + mc2 = mc2 + E ⇒ E = 2mc2. • Conservation of momentum along the x-axis (Figure 1.3 with 휃 = 60∘)
′ 1 p훾 + 0 = p훾 cos 휃 + p cos 휙 ⇒ 2mc + 0 = mc + p cos 휙 2 3 ⇒ p cos 휙 = mc. (1) 2 • Conservation of momentum along the y-axis √
′ 3 0 + 0 = p훾 sin 휃 − p sin 휙 ⇒ 0 = mc − p sin 휙 2 √ 3 ⇒ p sin 휙 = mc . (2) 2 If we now take the square of (1) and (2) and add them, we get √ p2 = 3m2c2 ⇒ p = 3 mc √ √ and, based on (1), we find that 3 mc cos 휙 =(3∕2)mc ⇒ cos 휙 = 3∕2 ⇒ ∘ 휙 = 30 . √ 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 + m2c4.2 Indeed 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. 10 1 The Principle of Wave–Particle Duality: An Overview
1.2.3 A Note on Units At this point we should pause to make some remarks on the system of units. We have already suggested (see Example 1.1) that both the cgs and SI system of units are 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 magnitude smaller than the joule and 12 orders smaller than the erg! Likewise, the natural length unit in the atomic world is the angstrom (= 10−10 m), since it is the typical size 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 choosing one system over another: The basic force law governing atomic and molecular structure—Coulomb’s law—has a much more convenient form in cgs than SI units, namely, 1 q q q q F = 1 2 (SI) F = 1 2 (cgs), (1.11) 휋휖 2 2 4 0 r r whence we immediately see why the cgs system is preferable over SI in atomic physics. In SI units all mathematical expressions of the basic quantum results for atoms appear much less elegant because they carry the cumbersome factor 휋휖 1∕4 0. For example, in cgs units the quantum formula for the ionization energy 4 ℏ2 of the hydrogen atom has the simple form WI = me ∕2 , while in SI units it 4 휋2휖2ℏ2 becomes WI = me ∕32 0 ! Therefore, our choice is to go with the cgs system for 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 concerning physical 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 the cutoff potential in Example 1.1. The same holds true for electric field intensities in atoms, where the SI unit V∕m(orV∕cm) arises naturally from the energy unit eV. So, even readers who are adherents of the SI system will find that the practical energy 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 for derivative 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 are related to their SI counterparts as follows:
Quantity cgs SI Conversion Force dyn N (newton) 1N = 105 dyn Energy erg J (joule) 1J = 107 erg Charge esu-q C (coulomb) 1C = 3 × 109 esu-q