A First Course on Symmetry, Special Relativity and Quantum Mechanics: The Foundations of Physics (Undergraduate Lecture Notes in Physics)

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. 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 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