Foreword xix
Preface xxiii
Editors’ Note xxvii
Part I Fundamental Principles 1
1 The Principle of Wave–Particle Duality: An Overview 3
1.1 Introduction 3
1.2 The Principle of Wave–Particle Duality of Light 4
1.2.1 The Photoelectric Effect 4
1.2.2 The Compton Effect 7
1.2.3 A Note on Units 10
1.3 The Principle of Wave–Particle Duality of Matter 11
1.3.1 From Frequency Quantization in Classical Waves to Energy Quantization in Matter Waves: The Most Important General Consequence of Wave–Particle Duality of Matter 12
1.3.2 The Problem of Atomic Stability under Collisions 13
1.3.3 The Problem of Energy Scales: Why Are Atomic Energies on the Order of eV, While Nuclear Energies Are on the Order of MeV? 15
1.3.4 The Stability of Atoms and Molecules Against External Electromagnetic Radiation 17
1.3.5 The Problem of Length Scales: Why Are Atomic Sizes on the Order of Angstroms, While Nuclear Sizes Are on the Order of Fermis? 19
1.3.6 The Stability of Atoms Against Their Own Radiation: Probabilistic Interpretation of Matter Waves 21
1.3.7 How Do Atoms Radiate after All? Quantum Jumps from Higher to Lower Energy States and Atomic Spectra 22
1.3.8 Quantized Energies and Atomic Spectra: The Case of Hydrogen 25
1.3.9 Correct and Incorrect Pictures for the Motion of Electrons in Atoms: Revisiting the Case of Hydrogen 25
1.3.10 The Fine Structure Constant and Numerical Calculations in Bohr’s Theory 29
1.3.11 Numerical Calculations with Matter Waves: Practical Formulas and 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 Schrodinger Equation and Its Statistical Interpretation 53
2.1 Introduction 53
2.2 The Schrodinger Equation 53
2.2.1 The Schrodinger Equation for Free Particles 54
2.2.2 The Schrodinger 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) = |?(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 Schrodinger Equation. Separation of Variables 78
2.5.3 The Time-Independent Schrodinger 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 Schrodinger’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– omentum 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 Schrodinger Equation 132
4.2.2 Discussion of the Results 134
4.2.2.1 Dimensional Analysis of the Formula En = (?2?2?2mL2)n2. Do We Need an Exact Solution to Predict the Energy Dependence on ?, m, and L? 135
4.2.2.2 Dependence of the Ground-State Energy on ?, m, and L : 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 Schrodinger 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 ? 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 Schrodinger Equation and Calculation of the Reflection Coefficient 150
5.2.2 Discussion of the Results 153
5.2.2.1 The Phenomenon of Classically Forbidden Reflection 153
5.2.2.2 Transmission Coefficient in the “Classical Limit” of High Energies 154
5.2.2.3 The Reflection Coefficient Depends neither on Planck’s Constant nor on the Mass of the Particle: Analysis of a Paradox 154
5.2.2.4 An Argument from Dimensional Analysis 155
5.3 Rectangular Potential Barrier: Tunneling Effect 156
5.3.1 Solution of the Schrodinger 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 Schrodinger 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
8.2 Solving the Schrodinger 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 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 Schrodinger 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 Schrodinger Equation: Physical Interpretation of the Centrifugal Term and Connection to the Angular Equation 235
9.2.3 Solution of the Angular Equation: Eigenvalues and Eigenfunctions of Angular Momentum 237
9.2.3.1 Solving the Equation for Φ 238
9.2.3.2 Solving the Equation for Θ 239
9.2.4 Summary of Results for an Arbitrary Central Potential 243
9.3 The Hydrogen Atom 246
9.3.1 Solution of the Radial Equation for the Coulomb Potential 246
9.3.2 Explicit Construction of the First Few Eigenfunctions 249
9.3.2.1 n = 1 : The Ground State 250
9.3.2.2 n = 2 : The First Excited States 250
9.3.3 Discussion of the Results 254
9.3.3.1 The Energy-Level Diagram 254
9.3.3.2 Degeneracy of the Energy Spectrum for a Coulomb Potential: Rotational and Accidental Degeneracy 255
9.3.3.3 Removal of Rotational and Hydrogenic Degeneracy 257
9.3.3.4 The Ground State is Always Nondegenerate and Has the Full Symmetry of the Problem 257
9.3.3.5 Spectroscopic Notation for Atomic States 258
9.3.3.6 The “Concept” of the Orbital: s and p Orbitals 258
9.3.3.7 Quantum Angular Momentum: A Rather Strange Vector 261
9.3.3.8 Allowed and Forbidden Transitions in the Hydrogen Atom: Conservation of Angular Momentum and Selection Rules 263
10 Atoms in a Magnetic Field and the Emergence of Spin 267
10.1 Introduction 267
10.2 Atomic Electrons as Microscopic Magnets: Magnetic Moment and Angular Momentum 270
10.3 The Zeeman Effect and the Evidence for the Existence of Spin 274
10.4 The Stern–Gerlach Experiment: Unequivocal Experimental Confirmation of the Existence of Spin 278
10.4.1 Preliminary Investigation: A Plausible Theoretical Description of Spin 278
10.4.2 The Experiment and Its Results 280
10.5 What is Spin? 284
10.5.1 Spin is No Self-Rotation 284
10.5.2 How is Spin Described Quantum Mechanically? 285
10.5.3 What Spin Really Is 291
10.6 Time Evolution of Spin in a Magnetic Field 292
10.7 Total Angular Momentum of Atoms: Addition of Angular 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
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 Molecule H2 360
13.3.2 The Helium “Molecule” He2 363
13.3.3 The Lithium Molecule Li2 364
13.3.4 The Oxygen Molecule O2 364
13.3.5 The Nitrogen Molecule N2 366
13.3.6 The Water Molecule H2O 367
13.3.7 Hydrogen Bonds: From the Water Molecule to Biomolecules 370
13.3.8 The Ammonia Molecule NH3 373
13.4 Molecular Spectra 377
13.4.1 Rotational Spectrum 378
13.4.2 Vibrational Spectrum 382
13.4.3 The Vibrational–Rotational Spectrum 385
14 Molecules. II: The Chemistry of Carbon 393
14.1 Introduction 393
14.2 Hybridization: The First Basic Deviation from the Elementary Theory of the Chemical Bond 393
14.2.1 The 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 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 a Fermi Gas 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
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 Schrodinger 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
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
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
