A First Course on Symmetry, Special Relativity and Quantum Mechanics

Preface
Acknowledgements
Contents
Conventions and Notations
List of Figures
List of Tables
1 Introduction
	1.1 The Goal of Physics
	1.2 The Connection Between Physics and Mathematics
	1.3 Paradigm Shifts
	1.4 The Correspondence Principle
2 Symmetry and Physics
	2.1 Learning Outcomes
	2.2 What Is Symmetry?
	2.3 Role of Symmetry in Physics
		2.3.1 Symmetry as a Guiding Principle
		2.3.2 Symmetry and Conserved Quantities
		2.3.3 Symmetry as a Tool for Simplifying Problems
	2.4 Symmetries Were Made to Be Broken
		2.4.1 Spacetime Symmetries
		2.4.2 Parity Violation
		2.4.3 Spontaneously Broken Symmetries
		2.4.4 Variational Calculations: Lifeguards and Light Rays
	Reference
3 Formal Aspects of Symmetry
	3.1 Learning Outcomes
	3.2 Symmetries as Operations
		3.2.1 Definition of a Symmetry Operation
		3.2.2 Rules Obeyed by Symmetry Operations
		3.2.3 Multiplication Tables
		3.2.4 Symmetry and Group Theory
	3.3 Examples
		3.3.1 The Identity Operation
		3.3.2 Permutations of Two Identical Objects
		3.3.3 Permutations of Three Identical Objects
		3.3.4 Rotations of Regular Polygons
	3.4 Continuous Versus Discrete Symmetries
	3.5 Noether's Theorem
	3.6 Supplementary: Variational Mechanics and the Proof of Noether's Theorem
		3.6.1 Variational Mechanics: Principle of Least Action
		3.6.2 Euler-Lagrange Equations
		3.6.3 Proof of Noether's Theorem
4 Symmetries and Linear Transformations
	4.1 Learning Outcomes
	4.2 Review of Vectors
		4.2.1 Coordinate Free Definitions
		4.2.2 Cartesian Coordinates
		4.2.3 Vector Operations in Component Form
		4.2.4 Position Vector
		4.2.5 Velocity and Acceleration: Differentiation of Vectors
	4.3 Linear Transformations
		4.3.1 Definition
		4.3.2 Translations
		4.3.3 Rotations
		4.3.4 Reflections
	4.4 Linear Transformations and Matrices
		4.4.1 General Discussion
		4.4.2 Identity Transformation and Inverse
		4.4.3 Rotations
		4.4.4 Reflections
		4.4.5 Matrix Representation of Permutations of Three Objects
	4.5 Pythagoras and Geometry
5 Special Relativity I: The Basics
	5.1 Learning Outcomes
	5.2 Preliminaries
		5.2.1 Frames of Reference
		5.2.2 Spacetime Diagrams
		5.2.3 Newtonian Relativity and Galilean Transformations
	5.3 Derivation of Special Relativity
		5.3.1 The Fundamental Postulate
		5.3.2 The Problem with Galilean Relativity
		5.3.3 Michelson-Morley Experiment
		5.3.4 Maxwell's Equations
	5.4 Summary of Consequences
	5.5 Relativity of Simultaneity
		5.5.1 Surface of Simultaneity: What Time Did It Happen?
	5.6 Time Dilation
		5.6.1 Derivation
		5.6.2 Properties of Time Dilation
		5.6.3 Proper Time
		5.6.4 Experimental Confirmation of Time Dilation
		5.6.5 Examples
	5.7 Lorentz Contraction
		5.7.1 Derivation of Lorentz Contraction
		5.7.2 Properties
		5.7.3 Proper Length and Proper Distance
	5.8 Death Star Betrayal: An Example
6 Special Relativity II: In Depth
	6.1 Learning Outcomes
	6.2 Lorentz Transformations
		6.2.1 Derivation of General Form
		6.2.2 Properties of Lorentz Transformations
		6.2.3 Lorentzian Geometry
	6.3 The Light Cone
	6.4 Proper Time Revisited
	6.5 Relativistic Addition of Velocities
	6.6 Relativistic Doppler Shift
		6.6.1 Non-relativistic Doppler Shift Review
		6.6.2 Relativistic Doppler Shift
	6.7 Relativistic Energy and Momentum
		6.7.1 Relativistic Energy-Momentum Conservation
		6.7.2 Relativistic Inertia
		6.7.3 Relativistic Energy
		6.7.4 Relativistic Three-Momentum
		6.7.5 Relationship Between Relativistic Energy and Momentum
		6.7.6 Kinetic Energy
		6.7.7 Massless Particles
	6.8 Spacetime Vectors
		6.8.1 Position Four-Vector
		6.8.2 Momentum Four-Vector
		6.8.3 Null Four-Vectors
		6.8.4 Relativistic Scattering
	6.9 Relativistic Units
	6.10 Symmetry Redux
		6.10.1 Matrix Form of Lorentz Transformations
		6.10.2 Lorentz Transformations as a Symmetry Group
	6.11 Supplementary: Four-Vectors and Tensors in Covariant Form
7 General Relativity
	7.1 Learning Outcomes
	7.2 Problems with Newtonian Gravity
		7.2.1 Review of Newtonian Gravity
		7.2.2 Perihelion Precession of Mercury
		7.2.3 Action at a Distance
		7.2.4 The Puzzle of Inertial Versus Gravitational Mass
	7.3 Strong Equivalence Principle
	7.4 Geometry of Spacetime
	7.5 Some Consequences of General Relativity
	7.6 Gravitational Waves
		7.6.1 Introduction
		7.6.2 Detection
		7.6.3 Early Observations
	7.7 Black Holes
		7.7.1 Properties of Black Holes
		7.7.2 Observational Evidence for Black Holes
	7.8 Cosmology
	Reference
8 Introduction to the Quantum
	8.1 Learning Outcomes
	8.2 Light as Particles
		8.2.1 Review: Light as Waves
		8.2.2 Photoelectric Effect
		8.2.3 Compton Scattering
	8.3 Blackbody Radiation and the Ultraviolet Catastrophe
		8.3.1 Blackbody Radiation
		8.3.2 Derivation of the Rayleigh-Jeans Law
		8.3.3 The Ultraviolet Catastrophe
		8.3.4 Quantum Resolution
		8.3.5 The Early Universe: The Ultimate Blackbody
	8.4 Particles as Waves
		8.4.1 Electron Waves
		8.4.2 de Broglie Wavelength
		8.4.3 Consequences
	8.5 The Heisenberg Uncertainty Principle
	References
9 The Wave Function
	9.1 Learning Outcomes
	9.2 Quantum Versus Newtonian Mechanics
		9.2.1 Newtonian Description of the State of a Particle
		9.2.2 Quantum Description of the State of a Particle
	9.3 Measurements of Position
	9.4 Example: Gaussian Wave Function
	9.5 Momentum in Quantum Mechanics
		9.5.1 Pure Waves
		9.5.2 The Momentum Operator
	9.6 Energy in Quantum Mechanics
		9.6.1 Energy-Time Uncertainty Relation
10 The Schrödinger Equation
	10.1 Learning Outcomes
	10.2 The Time Independent Schrödinger Equation
		10.2.1 Stationary States
	10.3 Examples of Stationary States
		10.3.1 Free Particle in One Dimension
		10.3.2 Particle in a Box with Impenetrable Walls
		10.3.3 Simple Harmonic Oscillator
	10.4 Absorption and Emission
	10.5 Tunnelling
		10.5.1 Particle in a Box with Penetrable Walls
		10.5.2 Tunnelling Through a Potential Barrier of Finite Width
		10.5.3 Applications of Tunnelling
	10.6 The Quantum Correspondence Principle
		10.6.1 Recovering the Everyday World
		10.6.2 The Bohr Correspondence Principle
	10.7 The Time Dependent Schrödinger Equation
		10.7.1 Heuristic Derivation
		10.7.2 Coherent States
	10.8 Observables as Linear Operators
	10.9 Symmetry in Quantum Mechanics
	10.10 Supplementary: Quantum Mechanics and Relativity
	Reference
11 The Hydrogen Atom
	11.1 Learning Outcomes
	11.2 Newtonian Dynamics
	11.3 Supplementary: Symmetries of the Hydrogen Atom
		11.3.1 Spherical Symmetry
		11.3.2 Accidental Symmetry of the Hydrogen Atom
	11.4 The Bohr Atom
	11.5 Emission and Absorption Spectra
	11.6 Three Dimensional Hydrogen Atom
		11.6.1 The Schrödinger Equation
		11.6.2 Solutions: Symmetry to the Rescue
		11.6.3 Probability Densities
		11.6.4 Shells, Orbitals and Degeneracy
		11.6.5 Fermions and the Spin Quantum Number
		11.6.6 Summary of the 3D Hydrogen Atom
	11.7 The Periodic Table
		11.7.1 Hydrogen-Like Atoms
		11.7.2 Chemical Properties and the Periodic Table
12 Nuclear Physics
	12.1 Learning Outcomes
	12.2 Properties of the Nucleus
		12.2.1 The Constituents
		12.2.2 Structure of Nucleus
		12.2.3 The Nuclear Force
	12.3 Radioactivity
		12.3.1 Isotopes
		12.3.2 Neutrinos
		12.3.3 Types of Radioactive Decay
		12.3.4 Decay Rates
		12.3.5 Carbon Dating
	12.4 Fission and Fusion
		12.4.1 Binding Energy
		12.4.2 Binding Energy per Nucleon
		12.4.3 Formation of Elements
	Reference
13 Mysteries of the Quantum World
	13.1 Learning Outcomes
	13.2 What Is Real?—A Quantum Conundrum
	13.3 Bell's Theorem and the Nature of Quantum Reality
	13.4 More on Spin
		13.4.1 Overview
		13.4.2 Mathematical Details
		13.4.3 Summary
	13.5 Experimental Confirmation of Quantum Weirdness
	13.6 Entanglement: The Key to Unlocking Quantum Weirdness
	13.7 Quantum Computation: Entanglement as a Resource
		13.7.1 Quantum Recap
		13.7.2 Classical Computers, in Brief
		13.7.3 What Is a Quantum Computer?
		13.7.4 Examples of Quantum Algorithms
		13.7.5 Two Important Technical Details
	13.8 Interpretations of Quantum Mechanics: What Does It All Mean?
		13.8.1 The Copenhagen Interpretation
		13.8.2 The Many-Worlds Interpretation
		13.8.3 Hidden Variables and Non-locality
	References
14 Conclusions
15 Appendix: Mathematical Background
	15.1 Complex Numbers
	15.2 Probabilities and Expectation Values
		15.2.1 Discrete Distributions
		15.2.2 Continuous Probability Distributions
		15.2.3 Dirac Delta Function
	15.3 Fourier Series and Transforms
		15.3.1 Fourier Series
		15.3.2 Fourier Transforms
		15.3.3 The Mathematical Uncertainty Principle
		15.3.4 Dirac Delta Function Revisited
		15.3.5 Parseval's Theorem
	15.4 Waves
		15.4.1 Moving Pure Waves
		15.4.2 Complex Waves
		15.4.3 Group Velocity and Phase Velocity
		15.4.4 Wave Packets
		15.4.5 Wave Number and Momentum
Index