Quantum Untangling

Simon Sherwood studied Natural Sciences at Cambridge University and has an MBA from Harvard Business School. Following four years as a strategy consultant with the Boston Consulting Group he embarked on a career in the hospitality industry as CEO of Orient-Express Hotels and more recently, Chairman of Elegant Hotels PLC. Simon Sherwood lives in Oxfordshire with his wife and two daughters.

Introduction xii

Acknowledgements xiii

Module I Special Relativity 1

1 Special Relativity 3

1.1 Special Relativity: Simple, Yet Baffling 3

1.2 The Speed of Light Is Constant: So What? 4

1.3 The Invariant Interval Equation 5

1.4 Time Distortion Quantified 6

1.5 Length Distortion 8

1.6 Leading Clocks Lag 9

1.7 Lorentz Transformations and Invariance 10

1.8 Summary: Are You Joking Mr Einstein? 11

2 Paradoxes of Special Relativity 13

2.1 Journey to a Distant Planet (1) 13

2.2 Journey to a Distant Planet (2) 14

2.3 The Twin Paradox 16

2.4 Experimental Proof 18

3 Einstein’s Famous Equation 20

3.1 Mass, Energy, Momentum – and Particle Time 20

3.2 How Did Albert Figure It Out? 21

3.2.1 The Ingredients 21

3.2.2 The Calculation 21

3.2.3 The Intuition 22

3.3 Three Beautiful Equations 23

3.4 How Wrong Were We? 24

3.5 One Further Equation 25

3.6 Summary 26

Module II Essential Quantum Mechanics 27

4 Wave-particle Duality 29

4.1 Classical Physics Cannot Explain… 29

4.2 Quanta of Light and the Photoelectric Effect 30

4.3 De Broglie’s Crazy Idea 31

4.4 The Double-slit Experiment 32

4.5 Schrödinger’s Mistreated Cat 34

4.6 Summary 35

5 Superpositions and Uncertainty 37

5.1 The Free Particle Wave Function 37

5.1.1 The Phase of the Wave 38

5.1.2 Derivatives of the Free Particle Wave Function 38

5.1.3 Linking Back to Special Relativity 39

5.1.4 Consider a Rocket 40

5.2 From Sinusoid to Uncertainty 41

5.3 Superposition 42

5.3.1 Superposition Saves the Day 42

5.3.2 Combining Eigenstates 43

5.4 Heisenberg’s Uncertainty Principle 44

5.5 In Praise of Fuzziness 45

5.6 God Plays Dice: The Role of Probability 46

5.7 Summary 47

5.8 What Is This Wave Function? 47

5.9 The Role of Rest Mass 48

6 Everything Happens … Kind of 49

6.1 The Feynman Path Integral 49

6.2 Change in Phase of the Wave Function 50

6.3 Simplified Path Integral Model 51

6.4 The Principle of Stationary Action 53

6.5 Action and the Lagrangian 54

6.6 From the Lagrangian to the Equations of Motion 55

6.7 The Uncertainty Relationship: A Different Perspective 56

6.8 Feynman Diagrams 57

6.9 Summary 58

7 Measurement and Interaction 60

7.1 What Can You Know about a Quantum System? 60

7.2 Collapse of the Wave Function 61

7.3 When a Body Meets a Body … 63

7.4 An Electron in a Box 63

7.5 Collapse of the Wave Function – a Twist 65

7.6 Decoherence and the Measurement Problem 66

7.7 When a Body Leaves a Body – Entanglement at a Distance 67

7.8 Summary 68

8 Module Summary and Schrödinger 70

8.1 Module Summary 70

8.2 Adding up the Implications 73

8.3 The Path to Schrödinger’s Equation 73

8.3.1 The Klein-Gordon Equation 74

8.3.2 A Taste of Schrödinger’s Equation 75

8.3.3 Incorporating Potential Energy 76

8.4 Module Memory Jogger 78

Module III Complex Quantum Mechanics 79

9 Introducing Complex Numbers 81

9.1 Welcome to Complex Numbers 81

9.1.1 We Have a Problem 82

9.1.2 Complex Notation for Phase 82

9.1.3 Interference Calculations 83

9.1.4 A Friend with Benefits 84

9.1.5 Not a Free Lunch 84

9.2 Representing the Wave Function with Complex Notation 85

9.3 Summary 85

10 Superpositions and Fourier Transforms 86

10.1 The Maths of Fourier Transforms 87

10.1.1 Example 1: Fourier Transform of a Position Eigenstate 88

10.1.2 Example 2: Fourier Transform of ∂Ψ 88

10.2 Heisenberg’s Uncertainty Principle and the Gaussian Distribution 89

10.3 The Quantum Footprint 90

10.4 Time and Energy 92

10.5 Summary 93

11 Schrödinger’s Equation 95

11.1 Understanding Schrödinger’s Equation 95

11.1.1 Incorporating Potential Energy 96

11.1.2 Superpositions 96

11.1.3 Schrödinger’s Equation in Words 96

11.2 Operators, Eigenstates and Eigenvalues 97

11.3 Commutation Relations 100

11.4 Expectation Values and Dirac Notation 101

11.5 Energy Eigenstates are Stationary 102

11.6 Time-independent Schrödinger Equation 102

12 Schrödinger’s Equation in Action 104

12.1 Free Particle Wave Function (E > V) 104

12.2 Creeping into Forbidden Places (E < V) 105

12.3 The Finite Potential Well 106

12.4 Quantum Tunnelling and the Sun 106

12.5 Dodging Potential Obstacles (E > V) 108

12.6 Quantum Biology 110

12.7 Wave Packets: A Model for Localised Particles 110

12.8 Summary 113

13 Quantum Harmonic Oscillator 114

13.1 Introduction 114

13.1.1 The Simple Harmonic Oscillator 114

13.1.2 The SHO and QHO: Why Do We Care? 115

13.2 Penetration Model for the QHO 116

13.3 Schrödinger’s Equation for the QHO 117

13.3.1 Ground State of the QHO 118

13.3.2 A Trick to Find the Other Energy Eigenstates of the QHO 119

13.3.3 The QHO Energy Eigenstate Ladder 120

13.3.4 QHO Superpositions 121

13.4 The QHO in Three Dimensions 122

13.5 Formal Definition of the Creation and Annihilation Operators 123

13.6 The Path to Quantum Field Theory (QFT) 125

14 Angular Momentum 126

14.1 A Primer on Classical Angular Momentum 126

14.2 Quanta of Angular Momentum 128

14.3 Angular Momentum’s Intricate Dance 128

14.4 Angular Kinetic Energy and Angular Momentum 129

14.5 The Pattern of Angular Momentum Eigenstates 130

14.5.1 Ground State: l = 0 131

14.5.2 First Energy Level: l = 1 131

14.5.3 Three Distinct First Level States: l = 1, m = −1, 0, + 1 131

14.5.4 Resulting in the Pattern 132

14.6 The Angular Momentum Creation Operator 133

14.7 Summary 134

15 Coulomb Potential 136

15.1 The Hydrogen Emission Spectrum 136

15.2 The Challenge of the Coulomb Potential 137

15.3 A Primitive Model 138

15.4 Schrödinger’s Equation for Hydrogen 139

15.4.1 Spherical Harmonics – merci Monsieur Laplace 139

15.4.2 The Angular Equation 141

15.4.3 The Shape of the Atomic Orbitals 142

15.4.4 Radial Kinetic Energy 143

15.4.5 The Radial Equation 144

15.5 Discussion 146

16 The Periodic Table 149

16.1 Introduction 149

16.2 Adding More Protons 150

16.3 The Periodic Table 150

16.4 Molecular Bonds 152

16.4.1 Ionic Bonds 152

16.4.2 Covalent Bonds 153

16.5 Bonds in the Nucleus 154

16.6 Virtual Particles 154

16.7 Fusion and Fission 155

16.8 Module Summary 156

16.9 Module Memory Jogger 157

Module IV Relativistic Quantum Mechanics 159

17 Spin 161

17.1 Intrinsic Angular Momentum: Spin 161

17.2 Spin-half Particles and the Pauli Exclusion Principle 162

17.2.1 The Stern-Gerlach Experiment 162

17.2.2 Spin-half and Spinors 163

17.2.3 The Pauli Exclusion Principle 164

17.2.4 The Pauli Matrices 165

17.3 Integer-spin: The Photon 168

17.3.1 Photon Polarisation 169

17.4 Bell’s Inequality and the Aspect Experiment 170

17.5 Summary 172

18 The Dirac Equation 173

18.1 Yet Another Equation? 173

18.2 Bi-spinors and Four-component Wave Functions 174

18.3 The Dirac Equation 175

18.3.1 The Ingredients 175

18.3.2 Dirac’s Crazy Insight 176

18.3.3 Dirac’s Matrices 177

18.3.4 We Are Finally There: Dirac’s Equation 179

18.4 Spin-half Is Built in 180

18.5 Interpreting the Dirac Equation 182

18.5.1 Zero Momentum: Distinct Spin and Antiparticles 182

18.5.2 The Dirac Equation and Minkowski Spacetime 182

18.5.3 Particle and Antiparticle States 183

18.5.4 Moving Frame 184

18.6 The Dirac Equation and Hydrogen 185

18.7 Dirac Equation: Modern Formulation 186

18.8 The Aftermath: Physics Falls Apart Again 186

19 Quantum Field Theory 189

19.1 Changing the Question 190

19.2 Quantum Fields Win the Day 190

19.2.1 The Quantum Field Structure 191

19.2.2 Quantum Fields and Spin 192

19.2.3 Creation and Annihilation 192

19.2.4 Bosons Like to Party 193

19.2.5 Conservation of Energy and Momentum 194

19.3 Non-relativistic Path Integrals and Action 195

19.4 QFT Path Integrals: A Relativistic Twist 197

19.5 Energy and Time 197

19.6 QFT Field Development Pathways 198

19.7 The Klein-Gordon Lagrangian as a Model 199

19.8 Global Gauge Invariance to Phase 200

19.9 Summary 201

20 Local Gauge Invariance 202

20.1 Introduction to Local Gauge Invariance 202

20.2 The Infinity Swimming Pool – an Analogy 204

20.3 Refresher in Electromagnetics (EM) 205

20.3.1 EM Refresher (1): The Basics 205

20.3.2 EM Refresher (2): The Vector Potential 206

20.4 The EM Quantum Field and Lagrangian 208

20.5 EM Gauge Invariance 210

20.6 U(1) Local Gauge Invariance: Putting Together the Pieces 210

20.6.1 The Swimming Pool: The Electron Field 210

20.6.2 The Balancing Tank: The EM Field 211

20.6.3 The Connection 211

20.6.4 The Interaction 211

20.6.5 The Infinity Pool: Combined Electron and EM Fields 211

20.7 The Dirac Lagrangian 212

20.8 Interaction and the Pathway of Stationary Action 213

20.9 The Photon Must Be Massless 214

20.10 Summary 214

21 QED and Feynman Diagrams 216

21.1 Feynman Diagrams 216

21.2 Example: Electron-positron Annihilation 218

21.3 Off-shell Drift and the QED Interaction 219

21.4 Feynman Rules 221

21.4.1 The Vertex and the Coupling Constant 221

21.4.2 The Propagator 222

21.4.3 Illustrative QED Calculation (Simplified) 223

21.4.4 From Amplitude to Cross Section 224

21.5 Resonance and the Search for New Particles 225

21.6 Do Virtual Particles Exist? 225

22 Renormalisation and EFT 227

22.1 Troublesome Loops 227

22.2 The Dressed Electron 228

22.3 Using Feynman Diagrams 229

22.4 Renormalisation 230

22.5 Ken Wilson’s Effective Field Theory (EFT) 232

22.6 Summary 232

23 The Strong Force 234

23.1 The Elementary Particles 234

23.2 The Strong Force: An Overview 235

23.2.1 Colour Charge 236

23.2.2 QCD, Gluons and Confinement 236

23.2.3 Strong Force Coupling Constant 237

23.3 QCD Local Gauge Invariance 238

23.3.1 SU(3) Symmetry and Colour 238

23.3.2 A Short Detour into Group Theory 240

23.3.3 The QCD Lagrangian 241

23.3.4 Gluons and the Generators 242

23.3.5 Summary: QCD As an Infinity Swimming Pool 243

23.4 The Residual Strong Force 244

23.5 Oh No! Here Comes Jill Again! 245

24 The Weak Force and Higgs Field (1) 246

24.1 Idealised Weak Force and SU(2) Symmetry 246

24.2 The Real Weak Force 248

24.2.1 Weak Isospin 248

24.2.2 Weak Interactions 249

24.2.3 Massive Weak Bosons 250

24.2.4 Wu and the Weak Left-handed Bias 250

24.3 What About SU(2) Gauge Symmetry? 251

24.4 Mass, Chirality and the Higgs Field 252

24.4.1 Mass as an Interaction 252

24.4.2 Chirality Versus Helicity 253

24.4.3 Chiral Dirac Equation 254

24.5 The Story So Far 255

25 The Weak Force and Higgs Field (2) 257

25.1 The Higgs Interaction 257

25.2 The Higgs Field and Mechanism 258

25.3 The Maths of the Higgs Field 259

25.4 Visualising the Higgs Field 259

25.5 Spontaneous Symmetry Breaking 260

25.6 The Maths of the Higgs Mechanism 260

25.6.1 The Starting Point 261

25.6.2 The Potential of the Higgs Field 261

25.6.3 Rotational Fluctuations of the Higgs Field 262

25.6.4 Putting It All Together 262

25.7 The Discovery of the Higgs Boson 264

25.8 Electroweak Unification 264

25.8.1 The Z Boson 265

25.8.2 The Photon 266

25.9 Summary 266

26 The Standard Model and Beyond 269

26.1 The Standard Model Lagrangian 269

26.2 From Einstein and de Broglie to Higgs 271

26.3 Questions and Problems 271

26.4 General Relativity and Quantum Mechanics 272

26.5 Supersymmetry (SUSY) 273

26.6 String Theory 274

26.6.1 Gravity in String Theory 274

26.6.2 Difficulties with String Theory 275

26.7 Loop Quantum Gravity (LQG) 276

26.7.1 LQG Space as a Quantum Entity 277

26.7.2 LQG Background Independence: Spin Networks 278

26.7.3 Difficulties with LQG 279

26.8 That’s All Folks! 280

26.9 Module Memory Jogger 280

Index 282