Waves, Particles and Fields: Introducing Quantum Field Theory

Description

This book fills a gap in the middle ground between quantum mechanics of a single electron to the concept of a quantum field. In doing so, the book is divided into two parts; the first provides the necessary background to quantum theory extending from Planck’s formulation of black body radiation to Schrodinger’s equation; and the second part explores Dirac’s relativistic electron to quantum fields, finishing with an description of Feynman diagrams and their meaning. Much more than a popular account, yet not too heavy so as to be inaccessible, this book assumes no prior knowledge of quantum physics or field theory and provides the necessary foundations for readers to then progress to more advanced texts on quantum field theory. It will be of interest to undergraduate students in physics and mathematics, in addition to an interested, general audience. Features: Provides an extensive yet accessible background to the concepts Contains numerous, illustrative diagrams Presents in-depth explanations of difficult subjects Cover Half Title Title Page Copyright Page Contents Acknowledgements 1: Mathematics 1.1 Introduction 1.2 Complex Numbers 1.2.1 Complex Numbers 1.2.2 Complex Quantities 1.2.3 Complex Functions 1.3 Scalars and Vectors 1.3.1 Scalars 1.3.2 Vectors 1.3.3 Dot and Cross Product 1.3.4 Vector Differentiation 1.4 Differential Equations 1.4.1 Differential Equations 1.4.2 Solutions to Differential Equations 1.4.3 Differential Operators 1.5 Partial Derivatives 1.6 Matrices 2: Waves 2.1 Introduction 2.2 Periodic Motion 2.3 Simple Harmonic Motion 2.4 Wave Function 2.5 Wave Equation 2.6 Complex Representation of a Wave 2.7 Energy Carried by a Wave 2.8 Superposition 2.9 Standing Waves 2.10 Beats 2.11 Superposition in Complex Form 3: Electromagnetic Waves 3.1 Electromagnetism 3.2 Energy in Electromagnetic Waves 4: Kinetic Theory of Gases 4.1 Introduction 4.2 Pressure and Temperature 4.2.1 Pressure 4.2.2 Temperature 4.2.3 Degrees of Freedom 4.2.4 Equipartition of Energy 4.2.5 Internal Energy 4.3 Statistical Mechanics 4.3.1 Statistical Weight 4.3.2 Boltzmann Distribution 4.3.3 Velocity Distribution 4.3.4 Partition Function 4.3.5 Properties of the Partition Function 4.3.6 Energy Density of States 4.3.7 Energy in a Harmonic Oscillator System 4.3.8 Average Energy in a Harmonic Oscillator System 5: Quantum Theory 5.1 Introduction 5.2 Black Body Radiation 5.3 Cavity Radiation 5.4 Frequency Density of States 5.4.1 Density of States – 1D 5.4.2 Density of States – 1D Revisited 5.4.3 Density of States – 2D 5.4.4 Density of States – 3D 5.5 Rayleigh–Jeans Radiation Law 5.6 The Birth of Quantum Physics 5.6.1 Introduction 5.6.2 Boltzmann Statistics 5.6.3 Rayleigh–Jeans Radiation Law 5.6.4 Planck’s Radiation Law 5.6.5 Forms of the Radiation Laws 5.6.6 Stefan–Boltzmann Radiation Law 5.6.7 Wien Displacement Law 6: The Bohr Atom 6.1 Introduction 6.2 The Photoelectric Effect 6.3 Line Spectra 6.4 The Bohr Atom 6.5 The Rydberg Constant 6.6 Matter Waves 6.7 The Photon 7: The New Quantum Theory 7.1 Introduction 7.2 The Schrödinger Equation 7.3 Solutions to the Schrödinger equation 7.3.1 Separation of Variables 7.3.2 Solution to the Time-Dependent Schrödinger Equation 7.3.3 The Wave Function 7.3.4 Normalisation 7.3.5 Solutions to the Time-Independent Schrödinger Equation 7.3.5.1 The Zero-Potential 7.3.5.2 The Infinite Square Well Potential 7.4 Significance of the Boundaries 7.4.1 Free Electron 7.4.2 Bound Electron 7.5 Wave Functions and Photons 7.6 Spin 7.6.1 Spin Angular Momentum 7.6.2 Quantum Numbers 7.7 Significance of the Schrödinger Equation 8: Relativity 8.1 Introduction 8.2 Special Relativity 8.2.1 The Michelson–Morley Experiment 8.2.2 The Principle of Relativity 8.2.3 Frames of Reference 8.2.3.1 Distance 8.2.3.2 Velocity and Acceleration 8.2.4 Postulates of Special Relativity 8.2.5 Time Dilation 8.2.6 Length Contraction 8.2.7 Lorentz Transformations 8.2.7.1 Lorentz Distance Transformation 8.2.7.2 Lorentz Time Transformation 8.2.7.3 Lorentz Velocity Transformation 8.2.7.4 Momentum and Mass Transformations 8.2.7.5 Mass and Energy Transformations 8.2.8 Consequences of Special Relativity 8.2.8.1 Energy and Momentum 8.2.8.2 Kinetic Energy 8.2.8.3 Photons 8.2.9 Summary of Special Relativity 8.2.9.1 Length 8.2.9.2 Time 8.2.9.3 Velocity 8.2.9.4 Mass 8.2.9.5 Momentum 8.2.9.6 Energy 8.3 General Relativity 8.3.1 Introduction 8.3.2 Space-Time 8.4 Conclusion 9: Advanced Mathematics 9.1 Vector Calculus 9.1.1 Vector Differential Operator 9.1.2 Line Integral 9.1.3 Multiple Integrals 9.1.4 Surface and Volume Integrals 9.1.5 Stokes’ Theorem 9.2 Gauss’ Law 9.3 Continuity Equation 9.4 Four-Vectors 9.4.1 Four-Position 9.4.2 Four-Velocity 9.4.3 Four-Momentum 9.4.4 Dot Product of Four-Vectors 9.4.5 Four-Differential Operator, and the d’Alembertian 9.5 The Hamiltonian 9.6 The Lagrangian 9.6.1 Action 9.6.2 Variational Calculus 9.6.3 Equations of Motion 10: Relativistic Quantum Mechanics 10.1 The Dirac Equation 10.2 Solutions to the Dirac Equation 10.2.1 At Rest 10.2.2 Constant Velocity 10.3 Antimatter 10.4 Natural Units 10.5 Single Particle Dirac Equation 11: Probability Flow 11.1 Introduction 11.2 Probability Current 11.3 The Adjoint Dirac Equation 12: Wave Functions and Spinors 12.1 Particles 12.2 Dirac Spinors 12.3 Antiparticles 13: Classical Field Theory 13.1 Classical Field Theory 13.2 Action 13.3 The Lagrangian 13.4 The Euler–Lagrange Equation 13.5 Lagrangian for a Free Particle 13.6 Lagrangian for a Free Particle in a Scalar Field 13.7 Lagrangian for the Dirac Field 14: Lorentz Invariance 14.1 Introduction 14.2 Transformations 14.3 Contravariant and Covariant Notation 14.4 Transformation Matrix 15: The Electromagnetic Field 15.1 Introduction 15.2 The Scalar Potential 15.3 The Vector Potential 15.4 Maxwell’s Equations in Potential Form 15.4.1 Maxwell’s Equations and the Vector Potential 15.4.2 The Four-Potential 15.5 Transformations of the Four-Potential 15.6 Lagrangian for the Electromagnetic Field 15.6.1 The Lagrangian for a Field 15.6.2 The Lagrangian for the Electromagnetic Field 15.7 The Electromagnetic Field Tensor 15.7.1 The Electromagnetic Field Tensor 15.7.2 The Lagrangian for the Electromagnetic Field Tensor 15.8 Charged Particle in an Electromagnetic Field 15.9 Transformations of the Electromagnetic Field 15.10 The Electromagnetic Wave 16: The Quantum Field 16.1 Introduction 16.2 Classical Fields 16.2.1 Scalar Field (Spin 0) 16.2.2 Dirac Field (Spin ½) 16.2.3 Vector Field (Spin 1) 16.3 The Harmonic Oscillator 16.3.1 Commutator 16.3.2 Energy Levels 16.3.2.1 Energy Levels 16.3.2.2 Power Series Method 16.3.2.3 Operator Method 16.3.3 The Harmonic Oscillator Field 16.3.4 Particles 16.3.5 The Quantum Field 16.4 Propagators 16.5 Interactions 17: Feynman Diagrams 17.1 Introduction 17.2 Quantum Electrodynamics 17.2.1 Path Taken by a Photon 17.2.2 Alternate Paths 17.2.3 Successive Steps 17.3 The Behaviour of Light 17.3.1 Straight-Line Path 17.3.2 Single Slit Diffraction 17.3.3 Double Slit Interference 17.4 Action 17.5 Feynman Diagrams 17.5.1 Feynman Diagrams 17.5.2 External Particles 17.5.2.1 Particles 17.5.2.2 Antiparticles 17.5.2.3 Photons 17.5.3 Interactions 17.5.3.1 Vertex Factor 17.5.3.2 Photon Propagator 17.5.4 Electron–Photon Interactions 17.6 Components of a Feynman Diagram 17.7 A Feynman Diagram 17.8 Renormalisation 17.9 Connection to Experimental Results 18: Conclusion Appendix Appendix for Chapter 2 Euler’s Formula Appendix for Chapter 8 Integration by Parts Binomial Theorem Appendix for Chapter 9 Commutators Appendix for Chapter 10 The Gamma Matrices Appendix for Chapter 11 Probability Flow Appendix for Chapter 15 Maxwell’s Equations Index