A First Course on Symmetry, Special Relativity and Quantum Mechanics: The Foundations of Physics

ISBN-13: 9783030923457 / Angielski / Miękka / 2022 / 424 str.

Saurya Das

Spis treści

1 Introduction 9

1.1 The goal of physics . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 The connection between physics and mathematics . . . . . . . 10

1.3 Paradigm shifts . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 The Correspondence Principle . . . . . . . . . . . . . . . . . . 16

2 Symmetry and Physics 17

2.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 What is Symmetry? . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Role of Symmetry in Physics . . . . . . . . . . . . . . . . . . . 18

2.3.1 Symmetry as a guiding principle . . . . . . . . . . . . . 18

2.3.2 Symmetry and Conserved Quantities: Noether's Theorem

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.3 Symmetry as a tool for simplifying problems . . . . . . 19

2.4 Symmetries were made to be broken . . . . . . . . . . . . . . 20

2.4.1 Spacetime symmetries . . . . . . . . . . . . . . . . . . 20

2.4.2 Parity violation . . . . . . . . . . . . . . . . . . . . . . 21

2.4.3 Spontaneously broken symmetries . . . . . . . . . . . . 24

2.4.4 Variational calculations: Lifeguards and light rays . . . 27

3 Formal Aspects of Symmetry 30

3.1 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Symmetries and Operations . . . . . . . . . . . . . . . . . . . 30

3.2.1 Denition of a symmetry operation . . . . . . . . . . . 30

3.2.2 Rules obeyed by symmetry operations . . . . . . . . . 32

3.2.3 Multiplication tables . . . . . . . . . . . . . . . . . . . 35

3.2.4 Symmetry and group theory . . . . . . . . . . . . . . . 36

3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.1 The identity operation . . . . . . . . . . . . . . . . . . 37

3.3.2 Permutations of two identical objects . . . . . . . . . . 37

3.3.3 Permutations of three identical objects . . . . . . . . . 38

3.3.4 Rotations of regular polygons . . . . . . . . . . . . . . 39

3.4 Continuous vs discrete symmetries . . . . . . . . . . . . . . . 40

3.5 Symmetries and Conserved Quantities:

Noether's Theorem . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6 Supplementary: Variational Mechanics and the Proof of Noether's

Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.6.1 Variational Mechanics: Principle of Least Action . . . . 42

3.6.2 Euler-Lagrange Equations . . . . . . . . . . . . . . . . 47

3.6.3 Proof of Noether's Theorem . . . . . . . . . . . . . . . 48

4 Symmetries and Linear Transformations 52

4.1 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Review of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1 Coordinate free denitions . . . . . . . . . . . . . . . . 53

4.2.2 Cartesian Coordinates . . . . . . . . . . . . . . . . . . 58

4.2.3 Vector operations in component form . . . . . . . . . . 59

4.2.4 Position vector . . . . . . . . . . . . . . . . . . . . . . 60

4.2.5 Dierentiation of vectors: velocity and acceleration . . 62

4.3 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . 63

4.3.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3.2 Translations . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.4 Re

ections . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4 Linear Transformations and matrices . . . . . . . . . . . . . . 68

4.4.1 Linear transformations as matrices . . . . . . . . . . . 68

4.4.2 Identity Transformation and Inverses . . . . . . . . . . 70

4.4.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4.4 Re

ections . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4.5 Matrix Representation of Permutations of Three Objects 73

4.5 Pythagoras and Geometry . . . . . . . . . . . . . . . . . . . . 74

5 Special Relativity I: The Basics 77

5.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2.1 Frames

5.2.2 Spacetime Diagrams . . . . . . . . . . . . . . . . . . . 78

5.2.3 Newtonian Relativity and Galilean Transformations . . 83

5.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3.1 The Fundamental Postulate . . . . . . . . . . . . . . . 85

5.3.2 The problem with Galilean Relativity . . . . . . . . . . 85

5.3.3 Michelson-Morley Experiment . . . . . . . . . . . . . . 87

5.3.4 Maxwell's Equations . . . . . . . . . . . . . . . . . . . 90

5.4 Summary of Consequences . . . . . . . . . . . . . . . . . . . . 91

5.5 Relativity of Simultaneity . . . . . . . . . . . . . . . . . . . . 92

5.6 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6.1 Derivation: . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6.2 Proper Time . . . . . . . . . . . . . . . . . . . . . . . . 99

5.6.3 Experimental Conrmation . . . . . . . . . . . . . . . 101

5.6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.7 Lorentz Contraction . . . . . . . . . . . . . . . . . . . . . . . 104

5.7.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.7.2 Properties: . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.7.3 Proper Length and Proper Distance. . . . . . . . . . . 104

5.7.4 Examples: . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 Special Relativity II: In Depth 110

6.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . 110

6.2.1 Derivation of general form . . . . . . . . . . . . . . . . 110

6.2.2 Properties of Lorentz Transformations . . . . . . . . . 113

6.2.3 Lorentzian Geometry . . . . . . . . . . . . . . . . . . . 116

6.3 The Light Cone . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.4 Proper time revisited . . . . . . . . . . . . . . . . . . . . . . . 120

6.5 Relativistic Addition of Velocities . . . . . . . . . . . . . . . . 122

6.6 Relativistic Doppler Shift . . . . . . . . . . . . . . . . . . . . . 124

6.6.1 Non-relativistic Doppler Shift Review . . . . . . . . . . 124

6.6.2 Relativistic Doppler Shift . . . . . . . . . . . . . . . . 124

6.7 Relativistic Energy and Momentum . . . . . . . . . . . . . . . 127

6.7.1 Relativistic Energy Momentum Conservation . . . . . . 127

6.7.2 Relativistic Inertia . . . . . . . . . . . . . . . . . . . . 128

6.7.3 Relativistic Energy . . . . . . . . . . . . . . . . . . . . 129

6.7.4 Relativistic Three-Momentum . . . . . . . . . . . . . . 129

6.7.5 Relationship Between Relativistic Energy and Momentum

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.7.6 Kinetic energy: . . . . . . . . . . . . . . . . . . . . . . 130

6.7.7 Massless particles . . . . . . . . . . . . . . . . . . . . 131

6.8 Space-time Vectors . . . . . . . . . . . . . . . . . . . . . . . . 133

6.8.1 Position Four-Vector: . . . . . . . . . . . . . . . . . . . 134

6.8.2 Four-momentum: . . . . . . . . . . . . . . . . . . . . . 135

6.8.3 Null four-vectors . . . . . . . . . . . . . . . . . . . . . 137

6.8.4 Relativistic Scattering . . . . . . . . . . . . . . . . . . 137

6.8.5 More Examples . . . . . . . . . . . . . . . . . . . . . . 138

6.9 Relativistic Units . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.10 Symmetry Redux . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.10.1 Matrix form of Lorentz Transformations . . . . . . . . 140

6.10.2 Lorentz Transformations as a Symmetry Group . . . . 142

6.11 Supplementary: Four vectors and tensors in covariant form . . 143

7 General Relativity 149

7.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 149

7.2 Problems with Newtonian Gravity . . . . . . . . . . . . . . . . 149

7.2.1 Review of Newtonian Gravity . . . . . . . . . . . . . . 149

7.2.2 Perihelion Shift of Mercury . . . . . . . . . . . . . . . 151

7.2.3 Action at a Distance . . . . . . . . . . . . . . . . . . . 152

7.2.4 The Puzzle of Inertial vs Gravitational Mass . . . . . . 153

7.3 Einstein's Thinking: the Strong Principle of Equivalence . . . 153

7.4 Geometry of Spacetime . . . . . . . . . . . . . . . . . . . . . . 155

7.5 Some Consequences of General Relativity: . . . . . . . . . . . 158

7.6 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . 159

7.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 159

7.6.2 Detection . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.6.3 Recent Observations . . . . . . . . . . . . . . . . . . . 161

7.7 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.7.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.7.2 Properties: . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.7.3 Observational Evidence . . . . . . . . . . . . . . . . . . 164

7.7.4 Further Information . . . . . . . . . . . . . . . . . . . 166

7.8 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8 Introduction to the Quantum 170

8.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 170

8.2 Light as particles . . . . . . . . . . . . . . . . . . . . . . . . . 171

8.2.1 Review: Light as Waves . . . . . . . . . . . . . . . . . 171

8.2.2 Photoelectric Eect . . . . . . . . . . . . . . . . . . . . 171

8.2.3 Compton Scattering . . . . . . . . . . . . . . . . . . . 175

8.3 Blackbody Radiation and the Ultraviolet Catastrophe . . . . . 179

8.3.1 Blackbody Radiation . . . . . . . . . . . . . . . . . . . 179

8.3.2 Derivation of Rayleigh-Jeans Law . . . . . . . . . . . . 181

8.3.3 The ultraviolet catastrophe . . . . . . . . . . . . . . . 188

8.3.4 Quantum resolution: . . . . . . . . . . . . . . . . . . . 189

8.3.5 The Early Universe: the ultimate blackbody . . . . . . 191

8.4 Particles as waves . . . . . . . . . . . . . . . . . . . . . . . . . 196

8.4.1 Electron waves . . . . . . . . . . . . . . . . . . . . . . 196

8.4.2 de Broglie Wavelength . . . . . . . . . . . . . . . . . . 197

8.4.3 Observational Evidence . . . . . . . . . . . . . . . . . . 199

8.5 The Heisenberg Uncertainty Principle . . . . . . . . . . . . . . 202

9 The Wave Function 204

9.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 204

9.2 Quantum vs Newtonian description of physical states . . . . . 204

9.2.1 Newtonian description of the state of a particle . . . . 205

9.2.2 Quantum description of the state of a particle . . . . . 205

9.3 Physical Consequences and Interpretation . . . . . . . . . . . 207

9.4 Measurements of position . . . . . . . . . . . . . . . . . . . . 208

9.5 Example: Gaussian wavefunction . . . . . . . . . . . . . . . . 209

9.6 \Spooky" Action at a Distance: Non-Locality in Quantum

Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

9.6.1 The EPR \Paradox" . . . . . . . . . . . . . . . . . . . 211

9.6.2 Bell's Theorem and the Experimental Repudiation of

EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

10 The Schrodinger Equation 217

10.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 217

10.2 Momentum in Quantum Mechanics . . . . . . . . . . . . . . . 218

10.2.1 Pure Waves . . . . . . . . . . . . . . . . . . . . . . . . 218

10.2.2 The Momentum Operator . . . . . . . . . . . . . . . . 220

10.3 Energy in Quantum Mechanics . . . . . . . . . . . . . . . . . 223

10.4 The Time Independent Schrodinger Equation . . . . . . . . . 224

10.4.1 Stationary States . . . . . . . . . . . . . . . . . . . . . 224

10.4.2 The \Quantum" in Quantum Mechanics . . . . . . . . 226

10.5 Examples of Stationary States . . . . . . . . . . . . . . . . . . 226

10.5.1 Free particle in one dimension . . . . . . . . . . . . . . 226

10.5.2 Example 2: Particle in a Box with Impenetrable Walls 227

10.5.3 Example 3 : Simple Harmonic Oscillator . . . . . . . . 229

10.6 Absorption and emission . . . . . . . . . . . . . . . . . . . . . 231

10.7 Tunnelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

10.7.1 Tunnelling through a potential barrier of nite width . 233

10.7.2 Particle in a Box with Penetrable Walls . . . . . . . . . 235

10.7.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 237

10.7.4 Applications of tunnelling . . . . . . . . . . . . . . . . 238

10.8 The Quantum Correspondence Principle . . . . . . . . . . . . 242

10.8.1 Recovering the everyday world . . . . . . . . . . . . . . 242

10.8.2 The Bohr Correspondence Principle . . . . . . . . . . . 243

10.9 The Time Dependent Schrodinger equation . . . . . . . . . . . 244

10.9.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 246

11 The Hydrogen Atom 249

11.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 249

11.2 Newtonian (Classical) Dynamics . . . . . . . . . . . . . . . . . 249

11.3 The Bohr Atom . . . . . . . . . . . . . . . . . . . . . . . . . . 251

11.4 Semi-classical spectrum from the Bohr correspondence principle254

11.5 Emission and Absorption Spectra . . . . . . . . . . . . . . . . 254

11.6 Three Dimensional Hydrogen Atom . . . . . . . . . . . . . . . 256

11.6.1 Schrodinger Equation . . . . . . . . . . . . . . . . . . . 256

11.6.2 Solutions and Quantum Numbers . . . . . . . . . . . . 258

11.6.3 Fermions and the spin quantum number . . . . . . . . 262

11.7 Periodic Table . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

11.7.1 Hydrogen-like atoms . . . . . . . . . . . . . . . . . . . 265

11.7.2 Chemical Properties and the Periodic Table . . . . . . 266

12 Nuclear Physics 270

12.1 Properties of the Nucleus . . . . . . . . . . . . . . . . . . . . . 270

12.1.1 Mass of Nucleons . . . . . . . . . . . . . . . . . . . . . 270

12.1.2 Structure of Nucleus . . . . . . . . . . . . . . . . . . . 271

12.1.3 The Nuclear Force . . . . . . . . . . . . . . . . . . . . 271

12.2 Binding Energy and Stability . . . . . . . . . . . . . . . . . . 274

12.2.1 Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . 274

12.2.2 Binding Energy . . . . . . . . . . . . . . . . . . . . . . 275

12.2.3 Binding Energy per Nucleon . . . . . . . . . . . . . . . 275

12.3 Formation of Elements: A Brief History of the Universe . . . . 276

12.4 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

12.4.1 Unstable Isotopes . . . . . . . . . . . . . . . . . . . . . 279

12.4.2 Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . 281

12.4.3 Beta decay . . . . . . . . . . . . . . . . . . . . . . . . . 282

12.4.4 Alpha Decay . . . . . . . . . . . . . . . . . . . . . . . 283

12.4.5 Decay Rates . . . . . . . . . . . . . . . . . . . . . . . . 283

12.4.6 Carbon Dating . . . . . . . . . . . . . . . . . . . . . . 285

13 Supplementary: Advanced Topics 287

13.1 Quantum Information and Quantum Computation . . . . . . . 287

13.2 Relativity and quantum mechanics . . . . . . . . . . . . . . . 287

14 Conclusions 288

15 Appendix: Mathematical Background 289

15.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . 289

15.2 Probabilities and expectation values . . . . . . . . . . . . . . . 291

15.2.1 Discrete Distributions . . . . . . . . . . . . . . . . . . 291

15.2.2 Continuous probability distributions . . . . . . . . . . 292

15.2.3 Dirac Delta Function . . . . . . . . . . . . . . . . . . . 296

15.3 Supplementary: Fourier Series and Transforms . . . . . . . . . 298

15.3.1 Fourier series . . . . . . . . . . . . . . . . . . . . . . . 298

15.3.2 Fourier Transforms . . . . . . . . . . . . . . . . . . . . 300

15.3.3 The mathematical uncertainty principle . . . . . . . . . 302

15.3.4 Dirac Delta Function Revisited . . . . . . . . . . . . . 303

15.3.5 Parseval's Theorem . . . . . . . . . . . . . . . . . . . . 303

15.4 Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

15.4.1 Moving pure waves . . . . . . . . . . . . . . . . . . . . 304

15.4.2 Complex Waves . . . . . . . . . . . . . . . . . . . . . . 305

15.4.3 Group velocity and phase velocity . . . . . . . . . . . 305

15.4.4 Wave packets . . . . . . . . . . . . . . . . . . . . . . . 307

15.4.5 Wave number and momentum . . . . . . . . . . . . . . 309

15.5 Derivation of Hydrogen Wave Functions . . . . . . . . . . . . 312

O autorze

Gabor Kunstatter is a theoretical physicist who has worked on general relativity, gauge theory quantization, finite temperature quantum field theory, quantum computing and quantum gravity. His current research focuses on the quantum mechanics of black holes, quantum information and effective theories for non-singular black hole evaporation and evaporation. Dr. Kunstatter is Professor Emeritus at the University of Winnipeg and Adjunct Professor at the University of Victoria, Simon Fraser University and the University of Manitoba. He has been a visiting scientist at a variety of institutions, including M.I.T., Université de Paris (Orsay), UNAM (Mexico), University of Nottingham and CECS (Chile). Dr. Kunstatter has also served as the President of the Canadian Association of Physicists and as Dean of Science at the University of Winnipeg.

Saurya Das is a theoretical physicist whose research areas include quantum gravity theory and phenomenology and cosmology. He has worked on problems in black hole physics, testing signatures of quantum gravity in the laboratory and on dark matter and dark energy, on which he has published more than 80 papers. After doing postdoctoral research at the Pennsylvania State University and the Universities of Winnipeg and New Brunswick, Dr. Das joined the faculty the University of Lethbridge, Canada in 2003, where he is now a full professor.

 

Blurb

This book provides an in-depth and accessible description of special relativity and quantum mechanics which together form the foundation of 21st century physics. A novel aspect  is that symmetry is given its rightful prominence as an integral part of this foundation. The book offers not only a conceptual understanding of symmetry, but also the mathematical tools necessary for quantitative analysis. As such, it provides a valuable precursor to more focused, advanced books on special relativity or quantum mechanics.

Students are introduced to several topics not typically covered until much later in their education.These include space-time diagrams, the action principle, a proof of Noether's theorem, Lorentz vectors and tensors, symmetry breaking and general relativity. The book also provides extensive descriptions on topics of current general interest such as gravitational waves, cosmology, Bell's theorem, entanglement and quantum computing.

Throughout the text, every opportunity is taken to emphasize the intimate connection between physics, symmetry and mathematics.The style remains light despite the rigorous and intensive content. 

The book is intended as a stand-alone or supplementary physics text for a one or two semester course for students who have completed an introductory calculus course and a first-year physics course that includes Newtonian mechanics and some electrostatics. Basic knowledge of linear algebra is useful but not essential, as all requisite mathematical background is provided either in the body of the text or in the Appendices. Interspersed through the text are well over a hundred worked examples and unsolved exercises for the student.