Bücher
Service
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.
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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.3 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 . . . . . . . . . . . . . . 393.4 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 . . . . . . . . . . . . . . . . . . . . . . . . 524.2 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 . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.4 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 775.1 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 . . . . . . . . . . . . . . . 1276.7.1 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 . . . . . . . . . . . . . . . . . . . . . . . . 1336.8.1 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 . . . . . . . . . . . . . . . . . . . 1527.2.4 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 . . . . . . . . 22610.5 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 . . . . . . . . . . . . . . . . 23810.8 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 27012.1 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 . . . . 27612.4 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
