Introduction to Quantum Physics

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.

• Foreword xixPreface xxiiiEditors’ Note xxviiPart I Fundamental Principles 11 The Principle of Wave–Particle Duality: An Overview 31.1 Introduction 31.2 The Principle of Wave–Particle Duality of Light 41.2.1 The Photoelectric Effect 41.2.2 The Compton Effect 71.2.3 A Note on Units 101.3 The Principle of Wave–Particle Duality of Matter 111.3.1 From Frequency Quantization in Classical Waves to Energy Quantization in Matter Waves: The Most Important General Consequence of Wave–Particle Duality of Matter 121.3.2 The Problem of Atomic Stability under Collisions 131.3.3 The Problem of Energy Scales: Why Are Atomic Energies on the Order of eV, While Nuclear Energies Are on the Order of MeV? 151.3.4 The Stability of Atoms and Molecules Against External Electromagnetic Radiation 171.3.5 The Problem of Length Scales: Why Are Atomic Sizes on the Order of Angstroms, While Nuclear Sizes Are on the Order of Fermis? 191.3.6 The Stability of Atoms Against Their Own Radiation: Probabilistic Interpretation of Matter Waves 211.3.7 How Do Atoms Radiate after All? Quantum Jumps from Higher to Lower Energy States and Atomic Spectra 221.3.8 Quantized Energies and Atomic Spectra: The Case of Hydrogen 251.3.9 Correct and Incorrect Pictures for the Motion of Electrons in Atoms: Revisiting the Case of Hydrogen 251.3.10 The Fine Structure Constant and Numerical Calculations in Bohr’s Theory 291.3.11 Numerical Calculations with Matter Waves: Practical Formulas and Physical Applications 311.3.12 A Direct Confirmation of the Existence of Matter Waves: The Davisson–Germer Experiment 331.3.13 The Double-Slit Experiment: Collapse of the Wavefunction Upon Measurement 341.4 Dimensional Analysis and Quantum Physics 411.4.1 The Fundamental Theorem and a Simple Application 411.4.2 Blackbody Radiation Using Dimensional Analysis 441.4.3 The Hydrogen Atom Using Dimensional Analysis 472 The Schrödinger Equation and Its Statistical Interpretation 532.1 Introduction 532.2 The Schrödinger Equation 532.2.1 The Schrödinger Equation for Free Particles 542.2.2 The Schrödinger Equation in an External Potential 572.2.3 Mathematical Intermission I: Linear Operators 582.3 Statistical Interpretation of Quantum Mechanics 602.3.1 The “Particle–Wave” Contradiction in Classical Mechanics 602.3.2 Statistical Interpretation 612.3.3 Why Did We Choose P(x) = |𝜓(x)|2 as the Probability Density? 622.3.4 Mathematical Intermission II: Basic Statistical Concepts 632.3.4.1 Mean Value 632.3.4.2 Standard Deviation (or Uncertainty) 652.3.5 Position Measurements: Mean Value and Uncertainty 672.4 Further Development of the Statistical Interpretation: The Mean-Value Formula 712.4.1 The General Formula for the Mean Value 712.4.2 The General Formula for Uncertainty 732.5 Time Evolution of Wavefunctions and Superposition States 772.5.1 Setting the Stage 772.5.2 Solving the Schrödinger Equation. Separation of Variables 782.5.3 The Time-Independent Schrödinger Equation as an Eigenvalue Equation: Zero-Uncertainty States and Superposition States 812.5.4 Energy Quantization for Confined Motion: A Fundamental General Consequence of Schrödinger’s Equation 852.5.5 The Role of Measurement in Quantum Mechanics: Collapse of the Wavefunction Upon Measurement 862.5.6 Measurable Consequences of Time Evolution: Stationary and Nonstationary States 912.6 Self-Consistency of the Statistical Interpretation and the Mathematical Structure of Quantum Mechanics 952.6.1 Hermitian Operators 952.6.2 Conservation of Probability 982.6.3 Inner Product and Orthogonality 992.6.4 Matrix Representation of Quantum Mechanical Operators 1012.7 Summary: Quantum Mechanics in a Nutshell 1033 The Uncertainty Principle 1073.1 Introduction 1073.2 The Position–Momentum Uncertainty Principle 1083.2.1 Mathematical Explanation of the Principle 1083.2.2 Physical Explanation of the Principle 1093.2.3 Quantum Resistance to Confinement. A Fundamental Consequence of the Position– omentum Uncertainty Principle 1123.3 The Time–Energy Uncertainty Principle 1143.4 The Uncertainty Principle in the Classical Limit 1183.5 General Investigation of the Uncertainty Principle 1193.5.1 Compatible and Incompatible Physical Quantities and the Generalized Uncertainty Relation 1193.5.2 Angular Momentum: A Different Kind of Vector 122Part II Simple Quantum Systems 1274 Square Potentials. I: Discrete Spectrum—Bound States 1294.1 Introduction 1294.2 Particle in a One-Dimensional Box: The Infinite Potential Well 1324.2.1 Solution of the Schrödinger Equation 1324.2.2 Discussion of the Results 1344.2.2.1 Dimensional Analysis of the Formula En = (ℏ2𝜋2∕2mL2)n2. Do We Need an Exact Solution to Predict the Energy Dependence on ℏ, m, and L? 1354.2.2.2 Dependence of the Ground-State Energy on ℏ, m, and L : The Classical Limit 1364.2.2.3 The Limit of Large Quantum Numbers and Quantum Discontinuities 1374.2.2.4 The Classical Limit of the Position Probability Density 1384.2.2.5 Eigenfunction Features: Mirror Symmetry and the Node Theorem 1394.2.2.6 Numerical Calculations in Practical Units 1394.3 The Square Potential Well 1404.3.1 Solution of the Schrödinger Equation 1404.3.2 Discussion of the Results 1434.3.2.1 Penetration into Classically Forbidden Regions 1434.3.2.2 Penetration in the Classical Limit 1444.3.2.3 The Physics and “Numerics” of the Parameter 𝜆 1455 Square Potentials. II: Continuous Spectrum—Scattering States 1495.1 Introduction 1495.2 The Square Potential Step: Reflection and Transmission 1505.2.1 Solution of the Schrödinger Equation and Calculation of the Reflection Coefficient 1505.2.2 Discussion of the Results 1535.2.2.1 The Phenomenon of Classically Forbidden Reflection 1535.2.2.2 Transmission Coefficient in the “Classical Limit” of High Energies 1545.2.2.3 The Reflection Coefficient Depends neither on Planck’s Constant nor on the Mass of the Particle: Analysis of a Paradox 1545.2.2.4 An Argument from Dimensional Analysis 1555.3 Rectangular Potential Barrier: Tunneling Effect 1565.3.1 Solution of the Schrödinger Equation 1565.3.2 Discussion of the Results 1585.3.2.1 Crossing a Classically Forbidden Region: The Tunneling Effect 1585.3.2.2 Exponential Sensitivity of the Tunneling Effect to the Energy of the Particle 1595.3.2.3 A Simple Approximate Expression for the Transmission Coefficient 1605.3.2.4 Exponential Sensitivity of the Tunneling Effect to the Mass of the Particle 1625.3.2.5 A Practical Formula for T 1636 The Harmonic Oscillator 1676.1 Introduction 1676.2 Solution of the Schrödinger Equation 1696.3 Discussion of the Results 1776.3.1 Shape of Wavefunctions. Mirror Symmetry and the Node Theorem 1786.3.2 Shape of Eigenfunctions for Large n: The Classical Limit 1796.3.3 The Extreme Anticlassical Limit of the Ground State 1806.3.4 Penetration into Classically Forbidden Regions: What Fraction of Its “Lifetime” Does the Particle “Spend” in the Classically Forbidden Region? 1816.3.5 A Quantum Oscillator Never Rests: Zero-Point Energy 1826.3.6 Equidistant Eigenvalues and Emission of Radiation from a Quantum Harmonic Oscillator 1846.4 A Plausible Question: Can We Use the Polynomial Method to Solve Potentials Other than the Harmonic Oscillator? 1877 The Polynomial Method: Systematic Theory and Applications 1917.1 Introduction: The Power-Series Method 1917.2 Sufficient Conditions for the Existence of Polynomial Solutions: Bidimensional Equations 1947.3 The Polynomial Method in Action: Exact Solution of the Kratzer and Morse Potentials 1977.4 Mathematical Afterword 2028 The Hydrogen Atom. I: Spherically Symmetric Solutions 2078.1 Introduction 2078.2 Solving the Schrödinger Equation for the Spherically Symmetric Eigenfunctions 2098.2.1 A Final Comment: The System of Atomic Units 2168.3 Discussion of the Results 2178.3.1 Checking the Classical Limit ℏ → 0 or m → ∞ for the Ground State of the Hydrogen Atom 2178.3.2 Energy Quantization and Atomic Stability 2178.3.3 The Size of the Atom and the Uncertainty Principle: The Mystery of Atomic Stability from Another Perspective 2188.3.4 Atomic Incompressibility and the Uncertainty Principle 2218.3.5 More on the Ground State of the Atom. Mean and Most Probable Distance of the Electron from the Nucleus 2218.3.6 Revisiting the Notion of “Atomic Radius”: How Probable is It to Find the Electron Within the “Volume” that the Atom Supposedly Occupies? 2228.3.7 An Apparent Paradox: After All, Where Is It Most Likely to Find the Electron? Near the Nucleus or One Bohr Radius Away from It? 2238.3.8 What Fraction of Its Time Does the Electron Spend in the Classically Forbidden Region of the Atom? 2238.3.9 Is the Bohr Theory for the Hydrogen Atom Really Wrong? Comparison with Quantum Mechanics 2258.4 What Is the Electron Doing in the Hydrogen Atom after All? A First Discussion on the Basic Questions of Quantum Mechanics 2269 The Hydrogen Atom. II: Solutions with Angular Dependence 2319.1 Introduction 2319.2 The Schrödinger Equation in an Arbitrary Central Potential: Separation of Variables 2329.2.1 Separation of Radial from Angular Variables 2329.2.2 The Radial Schrödinger Equation: Physical Interpretation of the Centrifugal Term and Connection to the Angular Equation 2359.2.3 Solution of the Angular Equation: Eigenvalues and Eigenfunctions of Angular Momentum 2379.2.3.1 Solving the Equation for Φ 2389.2.3.2 Solving the Equation for Θ 2399.2.4 Summary of Results for an Arbitrary Central Potential 2439.3 The Hydrogen Atom 2469.3.1 Solution of the Radial Equation for the Coulomb Potential 2469.3.2 Explicit Construction of the First Few Eigenfunctions 2499.3.2.1 n = 1 : The Ground State 2509.3.2.2 n = 2 : The First Excited States 2509.3.3 Discussion of the Results 2549.3.3.1 The Energy-Level Diagram 2549.3.3.2 Degeneracy of the Energy Spectrum for a Coulomb Potential: Rotational and Accidental Degeneracy 2559.3.3.3 Removal of Rotational and Hydrogenic Degeneracy 2579.3.3.4 The Ground State is Always Nondegenerate and Has the Full Symmetry of the Problem 2579.3.3.5 Spectroscopic Notation for Atomic States 2589.3.3.6 The “Concept” of the Orbital: s and p Orbitals 2589.3.3.7 Quantum Angular Momentum: A Rather Strange Vector 2619.3.3.8 Allowed and Forbidden Transitions in the Hydrogen Atom: Conservation of Angular Momentum and Selection Rules 26310 Atoms in a Magnetic Field and the Emergence of Spin 26710.1 Introduction 26710.2 Atomic Electrons as Microscopic Magnets: Magnetic Moment and Angular Momentum 27010.3 The Zeeman Effect and the Evidence for the Existence of Spin 27410.4 The Stern–Gerlach Experiment: Unequivocal Experimental Confirmation of the Existence of Spin 27810.4.1 Preliminary Investigation: A Plausible Theoretical Description of Spin 27810.4.2 The Experiment and Its Results 28010.5 What is Spin? 28410.5.1 Spin is No Self-Rotation 28410.5.2 How is Spin Described Quantum Mechanically? 28510.5.3 What Spin Really Is 29110.6 Time Evolution of Spin in a Magnetic Field 29210.7 Total Angular Momentum of Atoms: Addition of Angular Momenta 29510.7.1 The Eigenvalues 29510.7.2 The Eigenfunctions 30011 Identical Particles and the Pauli Principle 30511.1 Introduction 30511.2 The Principle of Indistinguishability of Identical Particles in Quantum Mechanics 30511.3 Indistinguishability of Identical Particles and the Pauli Principle 30611.4 The Role of Spin: Complete Formulation of the Pauli Principle 30711.5 The Pauli Exclusion Principle 31011.6 Which Particles Are Fermions and Which Are Bosons 31411.7 Exchange Degeneracy: The Problem and Its Solution 317Part III Quantum Mechanics in Action: The Structure of Matter 32112 Atoms: The Periodic Table of the Elements 32312.1 Introduction 32312.2 Arrangement of Energy Levels in Many-Electron Atoms: The Screening Effect 32412.3 Quantum Mechanical Explanation of the Periodic Table: The “Small Periodic Table” 32712.3.1 Populating the Energy Levels: The Shell Model 32812.3.2 An Interesting “Detail”: The Pauli Principle and Atomic Magnetism 32912.3.3 Quantum Mechanical Explanation of Valence and Directionality of Chemical Bonds 33112.3.4 Quantum Mechanical Explanation of Chemical Periodicity: The Third Row of the Periodic Table 33212.3.5 Ionization Energy and Its Role in Chemical Behavior 33412.3.6 Examples 33812.4 Approximate Calculations in Atoms: Perturbation Theory and the Variational Method 34112.4.1 Perturbation Theory 34212.4.2 Variational Method 34613 Molecules. I: Elementary Theory of the Chemical Bond 35113.1 Introduction 35113.2 The Double-Well Model of Chemical Bonding 35213.2.1 The Symmetric Double Well 35213.2.2 The Asymmetric Double Well 35613.3 Examples of Simple Molecules 36013.3.1 The Hydrogen Molecule H2 36013.3.2 The Helium “Molecule” He2 36313.3.3 The Lithium Molecule Li2 36413.3.4 The Oxygen Molecule O2 36413.3.5 The Nitrogen Molecule N2 36613.3.6 The Water Molecule H2O 36713.3.7 Hydrogen Bonds: From the Water Molecule to Biomolecules 37013.3.8 The Ammonia Molecule NH3 37313.4 Molecular Spectra 37713.4.1 Rotational Spectrum 37813.4.2 Vibrational Spectrum 38213.4.3 The Vibrational–Rotational Spectrum 38514 Molecules. II: The Chemistry of Carbon 39314.1 Introduction 39314.2 Hybridization: The First Basic Deviation from the Elementary Theory of the Chemical Bond 39314.2.1 The CH4 Molecule According to the Elementary Theory: An Erroneous Prediction 39314.2.2 Hybridized Orbitals and the CH4 Molecule 39514.2.3 Total and Partial Hybridization 40114.2.4 The Need for Partial Hybridization: The Molecules C2H4, C2H2, and C2H6 40414.2.5 Application of Hybridization Theory to Conjugated Hydrocarbons 40814.2.6 Energy Balance of Hybridization and Application to Inorganic Molecules 40914.3 Delocalization: The Second Basic Deviation from the Elementary Theory of the Chemical Bond 41414.3.1 A Closer Look at the Benzene Molecule 41414.3.2 An Elementary Theory of Delocalization: The Free-Electron Model 41714.3.3 LCAO Theory for Conjugated Hydrocarbons. I: Cyclic Chains 41814.3.4 LCAO Theory for Conjugated Hydrocarbons. II: Linear Chains 42414.3.5 Delocalization on Carbon Chains: General Remarks 42714.3.6 Delocalization in Two-dimensional Arrays of p Orbitals: Graphene and Fullerenes 42915 Solids: Conductors, Semiconductors, Insulators 43915.1 Introduction 43915.2 Periodicity and Band Structure 43915.3 Band Structure and the “Mystery of Conductivity.” Conductors, Semiconductors, Insulators 44115.3.1 Failure of the Classical Theory 44115.3.2 The Quantum Explanation 44315.4 Crystal Momentum, Effective Mass, and Electron Mobility 44715.5 Fermi Energy and Density of States 45315.5.1 Fermi Energy in the Free-Electron Model 45315.5.2 Density of States in the Free-Electron Model 45715.5.3 Discussion of the Results: Sharing of Available Space by the Particles of a Fermi Gas 46015.5.4 A Classic Application: The “Anomaly” of the Electronic Specific Heat of Metals 46316 Matter and Light: The Interaction of Atoms with Electromagnetic Radiation 46916.1 Introduction 46916.2 The Four Fundamental Processes: Resonance, Scattering, Ionization, and Spontaneous Emission 47116.3 Quantitative Description of the Fundamental Processes: Transition Rate, Effective Cross Section, Mean Free Path 47316.3.1 Transition Rate: The Fundamental Concept 47316.3.2 Effective Cross Section and Mean Free Path 47516.3.3 Scattering Cross Section: An Instructive Example 47616.4 Matter and Light in Resonance. I: Theory 47816.4.1 Calculation of the Effective Cross Section: Fermi’s Rule 47816.4.2 Discussion of the Result: Order-of-Magnitude Estimates and Selection Rules 48116.4.3 Selection Rules: Allowed and Forbidden Transitions 48316.5 Matter and Light in Resonance. II: The Laser 48716.5.1 The Operation Principle: Population Inversion and the Threshold Condition 48716.5.2 Main Properties of Laser Light 49116.5.2.1 Phase Coherence 49116.5.2.2 Directionality 49116.5.2.3 Intensity 49116.5.2.4 Monochromaticity 49216.6 Spontaneous Emission 49416.7 Theory of Time-dependent Perturbations: Fermi’s Rule 49916.7.1 Approximate Calculation of Transition Probabilities Pn→m(t) for an Arbitrary “Transient” Perturbation V(t) 49916.7.2 The Atom Under the Influence of a Sinusoidal Perturbation: Fermi’s Rule for Resonance Transitions 50316.8 The Light Itself: Polarized Photons and Their Quantum Mechanical Description 51116.8.1 States of Linear and Circular Polarization for Photons 51116.8.2 Linear and Circular Polarizers 51216.8.3 Quantum Mechanical Description of Polarized Photons 513Online Supplement1 The Principle of Wave–Particle Duality: An OverviewOS1.1 Review QuizOS1.1 Determining Planck’s Constant from Everyday Observations2 The Schrödinger Equation and Its Statistical InterpretationOS2.1 Review QuizOS2.2 Further Study of Hermitian Operators: The Concept of the Adjoint OperatorOS2.3 Local Conservation of Probability: The Probability Current3 The Uncertainty PrincipleOS3.1 Review QuizOS3.2 Commutator Algebra: Calculational TechniquesOS3.3 The Generalized Uncertainty PrincipleOS3.4 Ehrenfest’s Theorem: Time Evolution of Mean Values and the Classical Limit4 Square Potentials. I: Discrete Spectrum—Bound StatesOS4.1 Review QuizOS4.2 Square Well: A More Elegant Graphical Solution for Its EigenvaluesOS4.3 Deep and Shallow Wells: Approximate Analytic Expressions for Their Eigenvalues5 Square Potentials. II: Continuous Spectrum—Scattering StatesOS5.1 Review QuizOS5.2 Quantum Mechanical Theory of Alpha Decay6 The Harmonic OscillatorOS6.1 Review QuizOS6.2 Algebraic Solution of the Harmonic Oscillator: Creation and Annihilation Operators7 The Polynomial Method: Systematic Theory and ApplicationsOS7.1 Review QuizOS7.2 An Elementary Method for Discovering Exactly Solvable PotentialsOS7.3 Classic Examples of Exactly Solvable Potentials: A Comprehensive List8 The Hydrogen Atom. I: Spherically Symmetric SolutionsOS8.1 Review Quiz9 The Hydrogen Atom. II: Solutions with Angular DependenceOS9.1 Review QuizOS9.2 Conservation of Angular Momentum in Central Potentials, and Its ConsequencesOS9.3 Solving the Associated Legendre Equation on Our Own10 Atoms in a Magnetic Field and the Emergence of SpinOS10.1 Review QuizOS10.2 Algebraic Theory of Angular Momentum and Spin11 Identical Particles and the Pauli PrincipleOS11.1 Review QuizOS11.2 Dirac’s Formalism: A Brief Introduction12 Atoms: The Periodic Table of the ElementsOS12.1 Review QuizOS12.2 Systematic Perturbation Theory: Application to the Stark Effect and Atomic Polarizability13 Molecules. I: Elementary Theory of the Chemical BondOS13.1 Review Quiz14 Molecules. II: The Chemistry of CarbonOS14.1 Review QuizOS14.2 The LCAO Method and Matrix MechanicsOS14.3 Extension of the LCAO Method for Nonzero Overlap15 Solids: Conductors, Semiconductors, InsulatorsOS15.1 Review QuizOS15.2 Floquet’s Theorem: Mathematical Study of the Band Structure for an Arbitrary Periodic Potential V(x)OS15.3 Compressibility of Condensed Matter: The Bulk ModulusOS15.4 The Pauli Principle and Gravitational Collapse: The Chandrasekhar Limit16 Matter and Light: The Interaction of Atoms with Electromagnetic RadiationOS16.1 Review QuizOS16.2 Resonance Transitions Beyond Fermi’s Rule: Rabi OscillationsOS16.3 Resonance Transitions at Radio Frequencies: Nuclear Magnetic Resonance (NMR)Appendix 519Bibliography 523Index 527