A First Course on Symmetry, Special Relativity and Quantum Mechanics

1 Introduction 91.1 The goal of physics . . . . . . . . . . . . . . . . . . . . . . . . 91.2 The connection between physics and mathematics . . . . . . . 101.3 Paradigm shifts . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 The Correspondence Principle . . . . . . . . . . . . . . . . . . 162 Symmetry and Physics 172.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 172.2 What is Symmetry? . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Role of Symmetry in Physics . . . . . . . . . . . . . . . . . . . 182.3.1 Symmetry as a guiding principle . . . . . . . . . . . . . 182.3.2 Symmetry and Conserved Quantities: Noether's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.3 Symmetry as a tool for simplifying problems . . . . . . 192.4 Symmetries were made to be broken . . . . . . . . . . . . . . 202.4.1 Spacetime symmetries . . . . . . . . . . . . . . . . . . 202.4.2 Parity violation . . . . . . . . . . . . . . . . . . . . . . 212.4.3 Spontaneously broken symmetries . . . . . . . . . . . . 242.4.4 Variational calculations: Lifeguards and light rays . . . 273 Formal Aspects of Symmetry 303.1 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Symmetries and Operations . . . . . . . . . . . . . . . . . . . 303.2.1 Denition of a symmetry operation . . . . . . . . . . . 303.2.2 Rules obeyed by symmetry operations . . . . . . . . . 323.2.3 Multiplication tables . . . . . . . . . . . . . . . . . . . 353.2.4 Symmetry and group theory . . . . . . . . . . . . . . . 363.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.1 The identity operation . . . . . . . . . . . . . . . . . . 373.3.2 Permutations of two identical objects . . . . . . . . . . 373.3.3 Permutations of three identical objects . . . . . . . . . 383.3.4 Rotations of regular polygons . . . . . . . . . . . . . . 393.4 Continuous vs discrete symmetries . . . . . . . . . . . . . . . 403.5 Symmetries and Conserved Quantities:Noether's Theorem . . . . . . . . . . . . . . . . . . . . . . . . 413.6 Supplementary: Variational Mechanics and the Proof of Noether'sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6.1 Variational Mechanics: Principle of Least Action . . . . 423.6.2 Euler-Lagrange Equations . . . . . . . . . . . . . . . . 473.6.3 Proof of Noether's Theorem . . . . . . . . . . . . . . . 484 Symmetries and Linear Transformations 524.1 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . 524.2 Review of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 534.2.1 Coordinate free denitions . . . . . . . . . . . . . . . . 534.2.2 Cartesian Coordinates . . . . . . . . . . . . . . . . . . 584.2.3 Vector operations in component form . . . . . . . . . . 594.2.4 Position vector . . . . . . . . . . . . . . . . . . . . . . 604.2.5 Dierentiation of vectors: velocity and acceleration . . 624.3 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . 634.3.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.2 Translations . . . . . . . . . . . . . . . . . . . . . . . . 644.3.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.4 Reections . . . . . . . . . . . . . . . . . . . . . . . . . 674.4 Linear Transformations and matrices . . . . . . . . . . . . . . 684.4.1 Linear transformations as matrices . . . . . . . . . . . 684.4.2 Identity Transformation and Inverses . . . . . . . . . . 704.4.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 704.4.4 Reections . . . . . . . . . . . . . . . . . . . . . . . . . 724.4.5 Matrix Representation of Permutations of Three Objects 734.5 Pythagoras and Geometry . . . . . . . . . . . . . . . . . . . . 745 Special Relativity I: The Basics 775.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2.1 Frames5.2.2 Spacetime Diagrams . . . . . . . . . . . . . . . . . . . 785.2.3 Newtonian Relativity and Galilean Transformations . . 835.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.3.1 The Fundamental Postulate . . . . . . . . . . . . . . . 855.3.2 The problem with Galilean Relativity . . . . . . . . . . 855.3.3 Michelson-Morley Experiment . . . . . . . . . . . . . . 875.3.4 Maxwell's Equations . . . . . . . . . . . . . . . . . . . 905.4 Summary of Consequences . . . . . . . . . . . . . . . . . . . . 915.5 Relativity of Simultaneity . . . . . . . . . . . . . . . . . . . . 925.6 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.6.1 Derivation: . . . . . . . . . . . . . . . . . . . . . . . . 975.6.2 Proper Time . . . . . . . . . . . . . . . . . . . . . . . . 995.6.3 Experimental Conrmation . . . . . . . . . . . . . . . 1015.6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 1025.7 Lorentz Contraction . . . . . . . . . . . . . . . . . . . . . . . 1045.7.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 1045.7.2 Properties: . . . . . . . . . . . . . . . . . . . . . . . . . 1045.7.3 Proper Length and Proper Distance. . . . . . . . . . . 1045.7.4 Examples: . . . . . . . . . . . . . . . . . . . . . . . . . 1056 Special Relativity II: In Depth 1106.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 1106.2 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . 1106.2.1 Derivation of general form . . . . . . . . . . . . . . . . 1106.2.2 Properties of Lorentz Transformations . . . . . . . . . 1136.2.3 Lorentzian Geometry . . . . . . . . . . . . . . . . . . . 1166.3 The Light Cone . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.4 Proper time revisited . . . . . . . . . . . . . . . . . . . . . . . 1206.5 Relativistic Addition of Velocities . . . . . . . . . . . . . . . . 1226.6 Relativistic Doppler Shift . . . . . . . . . . . . . . . . . . . . . 1246.6.1 Non-relativistic Doppler Shift Review . . . . . . . . . . 1246.6.2 Relativistic Doppler Shift . . . . . . . . . . . . . . . . 1246.7 Relativistic Energy and Momentum . . . . . . . . . . . . . . . 1276.7.1 Relativistic Energy Momentum Conservation . . . . . . 1276.7.2 Relativistic Inertia . . . . . . . . . . . . . . . . . . . . 1286.7.3 Relativistic Energy . . . . . . . . . . . . . . . . . . . . 1296.7.4 Relativistic Three-Momentum . . . . . . . . . . . . . . 1296.7.5 Relationship Between Relativistic Energy and Momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.7.6 Kinetic energy: . . . . . . . . . . . . . . . . . . . . . . 1306.7.7 Massless particles . . . . . . . . . . . . . . . . . . . . 1316.8 Space-time Vectors . . . . . . . . . . . . . . . . . . . . . . . . 1336.8.1 Position Four-Vector: . . . . . . . . . . . . . . . . . . . 1346.8.2 Four-momentum: . . . . . . . . . . . . . . . . . . . . . 1356.8.3 Null four-vectors . . . . . . . . . . . . . . . . . . . . . 1376.8.4 Relativistic Scattering . . . . . . . . . . . . . . . . . . 1376.8.5 More Examples . . . . . . . . . . . . . . . . . . . . . . 1386.9 Relativistic Units . . . . . . . . . . . . . . . . . . . . . . . . . 1396.10 Symmetry Redux . . . . . . . . . . . . . . . . . . . . . . . . . 1406.10.1 Matrix form of Lorentz Transformations . . . . . . . . 1406.10.2 Lorentz Transformations as a Symmetry Group . . . . 1426.11 Supplementary: Four vectors and tensors in covariant form . . 1437 General Relativity 1497.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 1497.2 Problems with Newtonian Gravity . . . . . . . . . . . . . . . . 1497.2.1 Review of Newtonian Gravity . . . . . . . . . . . . . . 1497.2.2 Perihelion Shift of Mercury . . . . . . . . . . . . . . . 1517.2.3 Action at a Distance . . . . . . . . . . . . . . . . . . . 1527.2.4 The Puzzle of Inertial vs Gravitational Mass . . . . . . 1537.3 Einstein's Thinking: the Strong Principle of Equivalence . . . 1537.4 Geometry of Spacetime . . . . . . . . . . . . . . . . . . . . . . 1557.5 Some Consequences of General Relativity: . . . . . . . . . . . 1587.6 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . 1597.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1597.6.2 Detection . . . . . . . . . . . . . . . . . . . . . . . . . 1607.6.3 Recent Observations . . . . . . . . . . . . . . . . . . . 1617.7 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.7.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . 1637.7.2 Properties: . . . . . . . . . . . . . . . . . . . . . . . . . 1637.7.3 Observational Evidence . . . . . . . . . . . . . . . . . . 1647.7.4 Further Information . . . . . . . . . . . . . . . . . . . 1667.8 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1668 Introduction to the Quantum 1708.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 1708.2 Light as particles . . . . . . . . . . . . . . . . . . . . . . . . . 1718.2.1 Review: Light as Waves . . . . . . . . . . . . . . . . . 1718.2.2 Photoelectric Eect . . . . . . . . . . . . . . . . . . . . 1718.2.3 Compton Scattering . . . . . . . . . . . . . . . . . . . 1758.3 Blackbody Radiation and the Ultraviolet Catastrophe . . . . . 1798.3.1 Blackbody Radiation . . . . . . . . . . . . . . . . . . . 1798.3.2 Derivation of Rayleigh-Jeans Law . . . . . . . . . . . . 1818.3.3 The ultraviolet catastrophe . . . . . . . . . . . . . . . 1888.3.4 Quantum resolution: . . . . . . . . . . . . . . . . . . . 1898.3.5 The Early Universe: the ultimate blackbody . . . . . . 1918.4 Particles as waves . . . . . . . . . . . . . . . . . . . . . . . . . 1968.4.1 Electron waves . . . . . . . . . . . . . . . . . . . . . . 1968.4.2 de Broglie Wavelength . . . . . . . . . . . . . . . . . . 1978.4.3 Observational Evidence . . . . . . . . . . . . . . . . . . 1998.5 The Heisenberg Uncertainty Principle . . . . . . . . . . . . . . 2029 The Wave Function 2049.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 2049.2 Quantum vs Newtonian description of physical states . . . . . 2049.2.1 Newtonian description of the state of a particle . . . . 2059.2.2 Quantum description of the state of a particle . . . . . 2059.3 Physical Consequences and Interpretation . . . . . . . . . . . 2079.4 Measurements of position . . . . . . . . . . . . . . . . . . . . 2089.5 Example: Gaussian wavefunction . . . . . . . . . . . . . . . . 2099.6 \Spooky Action at a Distance: Non-Locality in QuantumMechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2119.6.1 The EPR \Paradox . . . . . . . . . . . . . . . . . . . 2119.6.2 Bell's Theorem and the Experimental Repudiation ofEPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21410 The Schr odinger Equation 21710.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 21710.2 Momentum in Quantum Mechanics . . . . . . . . . . . . . . . 21810.2.1 Pure Waves . . . . . . . . . . . . . . . . . . . . . . . . 21810.2.2 The Momentum Operator . . . . . . . . . . . . . . . . 22010.3 Energy in Quantum Mechanics . . . . . . . . . . . . . . . . . 22310.4 The Time Independent Schr odinger Equation . . . . . . . . . 22410.4.1 Stationary States . . . . . . . . . . . . . . . . . . . . . 22410.4.2 The \Quantum in Quantum Mechanics . . . . . . . . 22610.5 Examples of Stationary States . . . . . . . . . . . . . . . . . . 22610.5.1 Free particle in one dimension . . . . . . . . . . . . . . 22610.5.2 Example 2: Particle in a Box with Impenetrable Walls 22710.5.3 Example 3 : Simple Harmonic Oscillator . . . . . . . . 22910.6 Absorption and emission . . . . . . . . . . . . . . . . . . . . . 23110.7 Tunnelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23310.7.1 Tunnelling through a potential barrier of nite width . 23310.7.2 Particle in a Box with Penetrable Walls . . . . . . . . . 23510.7.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 23710.7.4 Applications of tunnelling . . . . . . . . . . . . . . . . 23810.8 The Quantum Correspondence Principle . . . . . . . . . . . . 24210.8.1 Recovering the everyday world . . . . . . . . . . . . . . 24210.8.2 The Bohr Correspondence Principle . . . . . . . . . . . 24310.9 The Time Dependent Schr odinger equation . . . . . . . . . . . 24410.9.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 24611 The Hydrogen Atom 24911.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 24911.2 Newtonian (Classical) Dynamics . . . . . . . . . . . . . . . . . 24911.3 The Bohr Atom . . . . . . . . . . . . . . . . . . . . . . . . . . 25111.4 Semi-classical spectrum from the Bohr correspondence principle25411.5 Emission and Absorption Spectra . . . . . . . . . . . . . . . . 25411.6 Three Dimensional Hydrogen Atom . . . . . . . . . . . . . . . 25611.6.1 Schr odinger Equation . . . . . . . . . . . . . . . . . . . 25611.6.2 Solutions and Quantum Numbers . . . . . . . . . . . . 25811.6.3 Fermions and the spin quantum number . . . . . . . . 26211.7 Periodic Table . . . . . . . . . . . . . . . . . . . . . . . . . . . 26511.7.1 Hydrogen-like atoms . . . . . . . . . . . . . . . . . . . 26511.7.2 Chemical Properties and the Periodic Table . . . . . . 26612 Nuclear Physics 27012.1 Properties of the Nucleus . . . . . . . . . . . . . . . . . . . . . 27012.1.1 Mass of Nucleons . . . . . . . . . . . . . . . . . . . . . 27012.1.2 Structure of Nucleus . . . . . . . . . . . . . . . . . . . 27112.1.3 The Nuclear Force . . . . . . . . . . . . . . . . . . . . 27112.2 Binding Energy and Stability . . . . . . . . . . . . . . . . . . 27412.2.1 Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . 27412.2.2 Binding Energy . . . . . . . . . . . . . . . . . . . . . . 27512.2.3 Binding Energy per Nucleon . . . . . . . . . . . . . . . 27512.3 Formation of Elements: A Brief History of the Universe . . . . 27612.4 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 27912.4.1 Unstable Isotopes . . . . . . . . . . . . . . . . . . . . . 27912.4.2 Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . 28112.4.3 Beta decay . . . . . . . . . . . . . . . . . . . . . . . . . 28212.4.4 Alpha Decay . . . . . . . . . . . . . . . . . . . . . . . 28312.4.5 Decay Rates . . . . . . . . . . . . . . . . . . . . . . . . 28312.4.6 Carbon Dating . . . . . . . . . . . . . . . . . . . . . . 28513 Supplementary: Advanced Topics 28713.1 Quantum Information and Quantum Computation . . . . . . . 28713.2 Relativity and quantum mechanics . . . . . . . . . . . . . . . 28714 Conclusions 28815 Appendix: Mathematical Background 28915.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . 28915.2 Probabilities and expectation values . . . . . . . . . . . . . . . 29115.2.1 Discrete Distributions . . . . . . . . . . . . . . . . . . 29115.2.2 Continuous probability distributions . . . . . . . . . . 29215.2.3 Dirac Delta Function . . . . . . . . . . . . . . . . . . . 29615.3 Supplementary: Fourier Series and Transforms . . . . . . . . . 29815.3.1 Fourier series . . . . . . . . . . . . . . . . . . . . . . . 29815.3.2 Fourier Transforms . . . . . . . . . . . . . . . . . . . . 30015.3.3 The mathematical uncertainty principle . . . . . . . . . 30215.3.4 Dirac Delta Function Revisited . . . . . . . . . . . . . 30315.3.5 Parseval's Theorem . . . . . . . . . . . . . . . . . . . . 30315.4 Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30415.4.1 Moving pure waves . . . . . . . . . . . . . . . . . . . . 30415.4.2 Complex Waves . . . . . . . . . . . . . . . . . . . . . . 30515.4.3 Group velocity and phase velocity . . . . . . . . . . . 30515.4.4 Wave packets . . . . . . . . . . . . . . . . . . . . . . . 30715.4.5 Wave number and momentum . . . . . . . . . . . . . . 30915.5 Derivation of Hydrogen Wave Functions . . . . . . . . . . . . 312