Description
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. 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., Universit 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. 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
