Non-technical and accessible primer providing key foundational knowledge on quantum mechanics and quantum field theory
Quantum Untangling introduces the readers to the fascinating and strange realm of quantum mechanics and quantum field theory, written in an accessible manner while not shying away from using mathematics where necessary. The book goes into sufficient depth and conveys basic and more intricate concepts such as wave-particle duality, wave functions, the superposition principle, quantum tunneling, the quantum harmonic oscillator, the Dirac equation, and Feynman diagrams. It also covers the physics of the Higgs boson and provides a glimpse into string theory and loop quantum gravity.
Overall, the author introduces complex concepts of quantum mechanics in an accessible and fun-to-read manner while laying the groundwork for mastering an advanced level of treatment in standard quantum mechanics textbooks and university courses.
Quantum Untangling includes information on: - Special relativity, time and length distortion, Einstein’s famous equation, how Einstein figured it out, and the implications for energy, mass and momentum - Wave particle duality, discussing what classical physics cannot explain, quanta of light and the photoelectric effect, De Broglie’s crazy idea, and the double-slit experiment - Making sense of Schrödinger’s equation, angular momentum and the wave function, angular rotational energy, atomic structure and molecular bonds - Spin, Quantum Electrodynamics, gauge invariance, the strong and weak forces, plus a step-by-step description of the Higgs mechanism
With Quantum Untangling, any reader with a good grasp of and an above-average interest in mathematics at advanced high-school level can follow the presentation and acquaint themselves with the fundamental and advanced topics of quantum mechanics and quantum field theory, making it a helpful resource for many different students.
Table of Contents
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
