An Introduction to Quantum PhysicsA First Course for Physicists, Chemists, Materials Scientists, and Engineers
This modern textbook offers an introduction to Quantum Mechanics as a theory that underlies the world around us, from atoms and molecules to materials, lasers, and other applications. The main features of the book are: Emphasis on the key principles with minimal mathematical formalism Demystifying discussions of the basic features of quantum systems, using dimensional analysis and order-of-magnitude estimates to develop intuition Comprehensive overview of the key concepts of quantum chemistry and the electronic structure of solids Extensive discussion of the basic processes and applications of light-matter interactions Online supplement with advanced theory, multiple-choice quizzes, etc.
Foreword xix Preface xxiii Editors’ Note xxvii Part I Fundamental Principles 1 1 The Principle ofWave–Particle Duality: An Overview 3 1.1 Introduction 3 1.2 The Principle ofWave–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 ofWave–Particle Duality of Matter 11 1.3.1 From Frequency Quantization in ClassicalWaves to Energy Quantization in MatterWaves: 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 MatterWaves 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 MatterWaves: Practical Formulas and Physical Applications 31 1.3.12 A Direct Confirmation of the Existence of MatterWaves:The Davisson–Germer Experiment 33 1.3.13 The Double-Slit Experiment: Collapse of theWavefunction 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 DidWe Choose P(x) = |??(x)|2 as the Probability Density? 62 2.3.4 Mathematical Intermission II: Basic Statistical Concepts 63 126.96.36.199 Mean Value 63 188.8.131.52 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 ofWavefunctions 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 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 PotentialWell 132 4.2.1 Solution of the Schrödinger Equation 132 4.2.2 Discussion of the Results 134 184.108.40.206 Dimensional Analysis of the Formula En = (?2??2?M2mL2)n2. DoWe Need an Exact Solution to Predict the Energy Dependence on ?, m, and L? 135 220.127.116.11 Dependence of the Ground-State Energy on ?, m, and L :The Classical Limit 136 18.104.22.168 The Limit of Large Quantum Numbers and Quantum Discontinuities 137 22.214.171.124 The Classical Limit of the Position Probability Density 138 126.96.36.199 Eigenfunction Features: Mirror Symmetry and the Node Theorem 139 188.8.131.52 Numerical Calculations in Practical Units 139 4.3 The Square PotentialWell 140 4.3.1 Solution of the Schrödinger Equation 140 4.3.2 Discussion of the Results 143 184.108.40.206 Penetration into Classically Forbidden Regions 143 220.127.116.11 Penetration in the Classical Limit 144 18.104.22.168 The Physics and “Numerics” of the Parameter ?? 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 22.214.171.124 The Phenomenon of Classically Forbidden Reflection 153 126.96.36.199 Transmission Coefficient in the “Classical Limit” of High Energies 154 188.8.131.52 The Reflection Coefficient Depends neither on Planck’s Constant nor on the Mass of the Particle: Analysis of a Paradox 154 184.108.40.206 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 158 220.127.116.11 Crossing a Classically Forbidden Region: The Tunneling Effect 158 18.104.22.168 Exponential Sensitivity of the Tunneling Effect to the Energy of the Particle 159 22.214.171.124 A Simple Approximate Expression for the Transmission Coefficient 160 126.96.36.199 Exponential Sensitivity of the Tunneling Effect to the Mass of the Particle 162 188.8.131.52 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 ofWavefunctions. 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: CanWe Use the PolynomialMethod 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 PolynomialMethod 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 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 ? ? 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 ElectronWithin 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 ReallyWrong? Comparison with QuantumMechanics 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 237 184.108.40.206 Solving the Equation for ? 238 220.127.116.11 Solving the Equation for ? 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 18.104.22.168 n = 1 : The Ground State 250 22.214.171.124 n = 2 : The First Excited States 250 9.3.3 Discussion of the Results 254 126.96.36.199 The Energy-Level Diagram 254 188.8.131.52 Degeneracy of the Energy Spectrum for a Coulomb Potential: Rotational and Accidental Degeneracy 255 184.108.40.206 Removal of Rotational and Hydrogenic Degeneracy 257 220.127.116.11 The Ground State is Always Nondegenerate and Has the Full Symmetry of the Problem 257 18.104.22.168 Spectroscopic Notation for Atomic States 258 22.214.171.124 The “Concept” of the Orbital: s and p Orbitals 258 126.96.36.199 Quantum Angular Momentum: A Rather Strange Vector 261 188.8.131.52 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 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 andWhich 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 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: PerturbationTheory and the Variational Method 341 12.4.1 PerturbationTheory 342 12.4.2 VariationalMethod 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 DoubleWell 352 13.2.2 The Asymmetric DoubleWell 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 OxygenMolecule O2 364 13.3.5 The Nitrogen Molecule N2 366 13.3.6 TheWater Molecule H2O 367 13.3.7 Hydrogen Bonds: From theWater 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 ElementaryTheory 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 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 ElementaryTheory of Delocalization:The Free-Electron Model 417 14.3.3 LCAOTheory for Conjugated Hydrocarbons. I: Cyclic Chains 418 14.3.4 LCAOTheory 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 ClassicalTheory 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 a FermiGas 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 theThreshold Condition 487 16.5.2 Main Properties of Laser Light 491 184.108.40.206 Phase Coherence 491 220.127.116.11 Directionality 491 18.104.22.168 Intensity 491 22.214.171.124 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 ofWave–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’sTheorem: Time Evolution of Mean Values and the Classical Limit 4 Square Potentials. I: Discrete Spectrum—Bound States OS4.1 Review Quiz OS4.2 SquareWell: A More Elegant Graphical Solution for Its Eigenvalues OS4.3 Deep and ShallowWells: Approximate Analytic Expressions forTheir 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 PolynomialMethod: Systematic Theory and Applications OS7.1 Review Quiz OS7.2 An ElementaryMethod 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 PerturbationTheory: 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 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
Stefanos Trachanas is an educator, author, and publisher. For over 35 years he has taught most of the core undergraduate courses at the Physics Department of the University of Crete. His books on quantum mechanics and differential equations are used as primary textbooks in most Greek University Departments of Physics, Chemistry, Materials Science, and Engineering. He is a cofounder of Crete University Press, which he led as Director from 1984 until his retirement in 2013. His awards include an honorary doctorate from the University of Crete, the Xanthopoulos-Pnevmatikos national award for excellence in academic teaching, and the Knight Commander of the Order of Phoenix, bestowed by the President of Greece. Manolis Antonoyiannakis is an Associate Editor and Bibliostatistics Analyst at the American Physical Society. He is also an Adjunct Associate Research Scientist at the Department of Applied Physics & Applied Mathematics at Columbia University, USA. He received his Master's degree from the University of Illinois at Urbana-Champaign, USA, and his PhD from Imperial College London, UK.His editorial experience in the Physical Review journals stimulated his interest in statistical, sociological, and historical aspects of peer review, but also in scientometrics and information science. He is currently developing data science tools to analyze scientific publishing and enhance research assessment. Leonidas Tsetseris is an Associate Professor at the School of Applied Mathematical and Physical Sciences of the National Technical University of Athens, Greece. He obtained his Master's and PhD degrees in physics from the University of Illinois at Urbana-Champaign, USA.His research expertise is on computational condensed matter physics and materials science, particularly quantum-mechanical studies on emerging materials. He has taught a variety of university physics courses, including classical mechanics, electromagnetism, quantum mechanics, and solid state physics.