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

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 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

4.2.2.1 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

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 PotentialWell 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 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

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 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

9.2.3.1 Solving the Equation for 238

9.2.3.2 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

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 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

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 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

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**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.