com An Introduction to Quantum Physics www.com An Introduction to Quantum Physics A First Course for Physicists, Chemists, Materials Scientists, and Engineers Stefanos Trachanas www.com 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.
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 1 Research Road Nationalbibliografie; detailed Ridge, NY 11961 bibliographic data are available on the United States Internet at <http://dnb. and © 2018 Wiley-VCH Verlag GmbH & Co. 12, 69469 Weinheim, Columbia University Germany Department of Applied Physics & Applied Mathematics All rights reserved (including those of 500 W.
120th Street translation into other languages). No part New York, NY 10027 of this book may be reproduced in any United States form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language Leonidas Tsetseris without written permission from the National Technical University of Athens publishers. Registered names, trademarks, Department of Physics etc. used in this book, even when not Zografou Campus specifically marked as such, are not to be 15780 Athens considered unprotected by law.
Greece Print ISBN: 978-3-527-41247-1 ePDF ISBN: 978-3-527-67665-1 ePub ISBN: 978-3-527-67668-2 Mobi ISBN: 978-3-527-67667-5 Cover Design Schulz Grafik-Design, Fußgönheim, Germany Typesetting SPi Global, Chennai, India Printing and Binding Printed on acid-free paper www.com to Maria www.com vii Contents Foreword xix Preface xxiii Editors’ Note xxvii Part I Fundamental Principles 1 1 The Principle of Wave–Particle Duality: An Overview 3 1.2 The Principle of Wave–Particle Duality of Light 4 1.1 The Photoelectric Effect 4 1.2 The Compton Effect 7 1.3 A Note on Units 10 1.3 The Principle of Wave–Particle Duality of Matter 11 1.1 From Frequency Quantization in Classical Waves to Energy Quantization in Matter Waves: The Most Important General Consequence of Wave–Particle Duality of Matter 12 1.2 The Problem of Atomic Stability under Collisions 13 1.3 The Problem of Energy Scales: Why Are Atomic Energies on the Order of eV, While Nuclear Energies Are on the Order of MeV? 15 1.4 The Stability of Atoms and Molecules Against External Electromagnetic Radiation 17 1.5 The Problem of Length Scales: Why Are Atomic Sizes on the Order of Angstroms, While Nuclear Sizes Are on the Order of Fermis? 19 1.6 The Stability of Atoms Against Their Own Radiation: Probabilistic Interpretation of Matter Waves 21 1.7 How Do Atoms Radiate after All? Quantum Jumps from Higher to Lower Energy States and Atomic Spectra 22 1.8 Quantized Energies and Atomic Spectra: The Case of Hydrogen 25 1.9 Correct and Incorrect Pictures for the Motion of Electrons in Atoms: Revisiting the Case of Hydrogen 25 1.10 The Fine Structure Constant and Numerical Calculations in Bohr’s Theory 29 www.com viii Contents 1.11 Numerical Calculations with Matter Waves: Practical Formulas and Physical Applications 31 1.12 A Direct Confirmation of the Existence of Matter Waves: The Davisson–Germer Experiment 33 1.13 The Double-Slit Experiment: Collapse of the Wavefunction Upon Measurement 34 1.4 Dimensional Analysis and Quantum Physics 41 1.1 The Fundamental Theorem and a Simple Application 41 1.2 Blackbody Radiation Using Dimensional Analysis 44 1.3 The Hydrogen Atom Using Dimensional Analysis 47 2 The Schrödinger Equation and Its Statistical Interpretation 53 2.2 The Schrödinger Equation 53 2.1 The Schrödinger Equation for Free Particles 54 2.2 The Schrödinger Equation in an External Potential 57 2.3 Mathematical Intermission I: Linear Operators 58 2.3 Statistical Interpretation of Quantum Mechanics 60 2.1 The “Particle–Wave” Contradiction in Classical Mechanics 60 2.3 Why Did We Choose P(x) = |𝜓(x)|2 as the Probability Density? 62 2.4 Mathematical Intermission II: Basic Statistical Concepts 63 2.5 Position Measurements: Mean Value and Uncertainty 67 2.4 Further Development of the Statistical Interpretation: The Mean-Value Formula 71 2.1 The General Formula for the Mean Value 71 2.2 The General Formula for Uncertainty 73 2.5 Time Evolution of Wavefunctions and Superposition States 77 2.1 Setting the Stage 77 2.2 Solving the Schrödinger Equation. Separation of Variables 78 2.3 The Time-Independent Schrödinger Equation as an Eigenvalue Equation: Zero-Uncertainty States and Superposition States 81 2.4 Energy Quantization for Confined Motion: A Fundamental General Consequence of Schrödinger’s Equation 85 2.5 The Role of Measurement in Quantum Mechanics: Collapse of the Wavefunction Upon Measurement 86 2.6 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 2.2 Conservation of Probability 98 2.3 Inner Product and Orthogonality 99 2.4 Matrix Representation of Quantum Mechanical Operators 101 2.7 Summary: Quantum Mechanics in a Nutshell 103 www.com Contents ix 3 The Uncertainty Principle 107 3.2 The Position–Momentum Uncertainty Principle 108 3.1 Mathematical Explanation of the Principle 108 3.2 Physical Explanation of the Principle 109 3.3 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 3.1 Compatible and Incompatible Physical Quantities and the Generalized Uncertainty Relation 119 3.2 Angular Momentum: A Different Kind of Vector 122 Part II Simple Quantum Systems 127 4 Square Potentials. I: Discrete Spectrum—Bound States 129 4.2 Particle in a One-Dimensional Box: The Infinite Potential Well 132 4.1 Solution of the Schrödinger Equation 132 4.2 Discussion of the Results 134 4.1 Dimensional Analysis of the Formula En = (ℏ2 𝜋 2 ∕2mL2 )n2.
Do We Need an Exact Solution to Predict the Energy Dependence on ℏ, m, and L? 135 4.2 Dependence of the Ground-State Energy on ℏ, m, and L : The Classical Limit 136 4.3 The Limit of Large Quantum Numbers and Quantum Discontinuities 137 4.4 The Classical Limit of the Position Probability Density 138 4.5 Eigenfunction Features: Mirror Symmetry and the Node Theorem 139 4.6 Numerical Calculations in Practical Units 139 4.3 The Square Potential Well 140 4.1 Solution of the Schrödinger Equation 140 4.2 Discussion of the Results 143 4.1 Penetration into Classically Forbidden Regions 143 4.2 Penetration in the Classical Limit 144 4.3 The Physics and “Numerics” of the Parameter 𝜆 145 5 Square Potentials. II: Continuous Spectrum—Scattering States 149 5.2 The Square Potential Step: Reflection and Transmission 150 5.1 Solution of the Schrödinger Equation and Calculation of the Reflection Coefficient 150 5.2 Discussion of the Results 153 www.1 The Phenomenon of Classically Forbidden Reflection 153 5.2 Transmission Coefficient in the “Classical Limit” of High Energies 154 5.3 The Reflection Coefficient Depends neither on Planck’s Constant nor on the Mass of the Particle: Analysis of a Paradox 154 5.4 An Argument from Dimensional Analysis 155 5.3 Rectangular Potential Barrier: Tunneling Effect 156 5.1 Solution of the Schrödinger Equation 156 5.2 Discussion of the Results 158 5.1 Crossing a Classically Forbidden Region: The Tunneling Effect 158 5.2 Exponential Sensitivity of the Tunneling Effect to the Energy of the Particle 159 5.3 A Simple Approximate Expression for the Transmission Coefficient 160 5.4 Exponential Sensitivity of the Tunneling Effect to the Mass of the Particle 162 5.5 A Practical Formula for T 163 6 The Harmonic Oscillator 167 6.2 Solution of the Schrödinger Equation 169 6.3 Discussion of the Results 177 6.1 Shape of Wavefunctions. Mirror Symmetry and the Node Theorem 178 6.2 Shape of Eigenfunctions for Large n: The Classical Limit 179 6.3 The Extreme Anticlassical Limit of the Ground State 180 6.4 Penetration into Classically Forbidden Regions: What Fraction of Its “Lifetime” Does the Particle “Spend” in the Classically Forbidden Region? 181 6.5 A Quantum Oscillator Never Rests: Zero-Point Energy 182 6.6 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.com Contents xi 8.2 Solving the Schrödinger Equation for the Spherically Symmetric Eigenfunctions 209 8.1 A Final Comment: The System of Atomic Units 216 8.3 Discussion of the Results 217 8.1 Checking the Classical Limit ℏ → 0 or m → ∞ for the Ground State of the Hydrogen Atom 217 8.2 Energy Quantization and Atomic Stability 217 8.3 The Size of the Atom and the Uncertainty Principle: The Mystery of Atomic Stability from Another Perspective 218 8.4 Atomic Incompressibility and the Uncertainty Principle 221 8.5 More on the Ground State of the Atom.
Mean and Most Probable Distance of the Electron from the Nucleus 221 8.6 Revisiting the Notion of “Atomic Radius”: How Probable is It to Find the Electron Within the “Volume” that the Atom Supposedly Occupies? 222 8.7 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 8.8 What Fraction of Its Time Does the Electron Spend in the Classically Forbidden Region of the Atom? 223 8.9 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 9.2 The Schrödinger Equation in an Arbitrary Central Potential: Separation of Variables 232 9.1 Separation of Radial from Angular Variables 232 9.2 The Radial Schrödinger Equation: Physical Interpretation of the Centrifugal Term and Connection to the Angular Equation 235 9.3 Solution of the Angular Equation: Eigenvalues and Eigenfunctions of Angular Momentum 237 9.1 Solving the Equation for Φ 238 9.2 Solving the Equation for Θ 239 9.4 Summary of Results for an Arbitrary Central Potential 243 9.3 The Hydrogen Atom 246 9.1 Solution of the Radial Equation for the Coulomb Potential 246 9.2 Explicit Construction of the First Few Eigenfunctions 249 9.1 n = 1 : The Ground State 250 9.2 n = 2 : The First Excited States 250 9.3 Discussion of the Results 254 9.1 The Energy-Level Diagram 254 9.2 Degeneracy of the Energy Spectrum for a Coulomb Potential: Rotational and Accidental Degeneracy 255 9.3 Removal of Rotational and Hydrogenic Degeneracy 257 www.com xii Contents 9.4 The Ground State is Always Nondegenerate and Has the Full Symmetry of the Problem 257 9.5 Spectroscopic Notation for Atomic States 258 9.6 The “Concept” of the Orbital: s and p Orbitals 258 9.7 Quantum Angular Momentum: A Rather Strange Vector 261 9.8 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 10.2 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 10.1 Preliminary Investigation: A Plausible Theoretical Description of Spin 278 10.2 The Experiment and Its Results 280 10.5 What is Spin? 284 10.1 Spin is No Self-Rotation 284 10.2 How is Spin Described Quantum Mechanically? 285 10.3 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 10.2 The Eigenfunctions 300 11 Identical Particles and the Pauli Principle 305 11.