A First Course in Mathematical Physics

The book assumes next to no prior knowledge of the topic. The first part introduces the core mathematics, always in conjunction with the physical context. In the second part of the book, a series of examples showcases some of the more conceptually advanced areas of physics, the presentation of which draws on the developments in the first part. A large number of problems helps students to hone their skills in using the presented mathematical methods. Solutions to the problems are available to instructors on an associated password-protected website for lecturers.

Table of Contents:
Preface xv Part I Mathematics 1 1 Functions of One Variable 3 1.1 Limits 3 1.2 Elementary Calculus 5 1.2.1 Differentiation Products and Quotients 6 1.2.2 Chain Rule 7 1.2.3 Inverse Functions 8 1.3 Integration 10 1.4 The Binomial Expansion 14 1.5 Taylor’s Series 15 1.6 Extrema 17 1.7 Power Series 17 1.8 Basic Functions 19 1.8.1 Exponential 19 1.8.2 Logarithm 22 1.9 First-Order Ordinary Differential Equations 24 1.10 Trigonometric Functions 25 1.10.1 L’Hôpital’s Rule 27 Problems 27 2 Complex Numbers 29 2.1 Exponential Function of a Complex Variable 30 2.2 Argand Diagrams and the Complex Plane 32 2.3 Complex Logarithm 34 2.4 Hyperbolic Functions 34 2.5 The Simple Harmonic Oscillator 36 2.5.1 Mechanics in One Dimension 38 2.5.2 Damped and Driven Oscillations 40 Problems 47 3 VectorsinR 3 51 3.1 Basic Operation 51 3.1.1 Scalar Triple Product 55 3.1.2 Vector Equations of Lines and Planes 56 3.2 Kinematics in Three Dimensions 57 3.2.1 Differentiation 57 3.2.2 Motion in a Uniform Magnetic Field 57 3.3 Coordinate Systems 59 3.3.1 Polar Coordinates 59 3.4 Central Forces 60 3.5 Rotating Frames 64 3.5.1 Larmor Effect 66 Problems 67 4 VectorSpaces 71 4.1 Formal Definition of a Vector Space 71 4.2 Fourier Series 75 4.3 Linear Operators 78 4.4 Change of Basis 89 Problems 91 5 Functions of Several Variables 95 5.1 Partial Derivatives 95 5.1.1 Definition of the Partial Derivative 95 5.1.2 Total Derivatives 98 5.1.3 Elementary Numerical Methods 104 5.1.4 Change of Variables 107 5.1.5 Mechanics Again 109 5.2 Extrema under Constraint 111 5.3 Multiple Integrals 113 5.3.1 Triple Integrals 116 5.3.2 Change of Variables 117 Problems 121 6 Vector Fields and Operators 125 6.1 The Gradient Operator 125 6.1.1 Coordinate Systems 127 6.2 Work and Energy in Vectorial Mechanics 130 6.2.1 Line Integrals 133 6.3 A Little Fluid Dynamics 135 6.3.1 Rotational Motion 138 6.3.2 Fields 141 6.4 Surface Integrals 142 6.5 The Divergence Theorem 146 6.6 Stokes’ Theorem 149 6.6.1 Conservative Forces 153 Problems 154 7 Generalized Functions 159 7.1 The Dirac Delta Function 159 7.2 Green’s Functions 163 7.3 Delta Function in Three Dimensions 165 Problems 169 8 Functions of a Complex Variable 173 8.1 Limits 174 8.2 Power Series 178 8.3 Fluids Again 179 8.4 Complex Integration 180 8.4.1 Application of the Residue Theorem 186 Problems 192 Part II Physics 195 9 Maxwell’s Equations: A Very Short Introduction 197 9.1 Electrostatics: Gauss’s Law 197 9.1.1 Conductors 203 9.2 The No Magnetic Monopole Rule 204 9.3 Current 205 9.4 Faraday’s Law 206 9.5 Ampère’s Law 208 9.6 The Wave Equation 210 9.7 Gauge Conditions 211 Problems 213 10 Special Relativity: Four-Vector Formalism 217 10.1 Lorentz Transformation 217 10.1.1 Inertial Frames 217 10.1.2 Properties and Consequences of the Lorentz Transformation 220 10.2 Minkowski Space 220 10.2.1 Four Vectors 220 10.2.2 Time Dilation 226 10.3 Four-Velocity 227 10.3.1 Four-Momentum 229 10.4 Electrodynamics 234 10.4.1 Maxwell’s Equations in Four-Vector Form 234 10.4.2 Field of a Moving Point Charge 237 10.5 Transformation of the Electromagnetic Fields 239 Problems 240 11 Quantum Theory 243 11.1 Bohr Atom 243 11.2 The de Broglie Hypothesis 246 11.3 The Schrödinger Wave Equation 246 11.4 Interpretation of the Wave function 249 11.5 Atom 251 11.5.1 The Delta Function Potential 252 11.5.2 Molecules 254 11.6 Formalism 257 11.6.1 Dirac Notation 257 11.7 Probabilistic Interpretation 258 11.7.1 Commutator Relations 259 11.7.2 Functions of Observables 261 11.7.3 Block’s Theorem 261 11.7.4 Band Structure 263 11.8 Time Evolution 266 11.9 The Stern–Gerlach Experiment 269 11.9.1 Successive Measurements 270 11.9.2 Spin Space 271 11.9.3 Explicit Matrix Representation 272 11.9.4 Larmor Precession 274 11.9.5 EPR Paradox 275 11.9.6 Bell’s Theorem 276 11.9.7 The Harmonic Oscillator 279 Problems 280 12 An Informal Treatment of Variational Principles and their History 287 12.1 Sin and Death 287 12.2 The Calculus of Variations 288 12.3 Constrained Variations 293 12.4 Hamilton’s Equations 293 12.5 Phase Space 296 12.6 Fixed Points 296 Problems 298 A Conic Sections 301 A.1 Polar Coordinates 303 A.2 Intersection of a Cone and a Plane 304 B Vector Relations 305 B.1 Products 305 B.2 Differential Operator Relations 305 B.3 Coordinates 306 Cylindrical Polar 306 Spherical Polar 307 Bibliography 309 Index 311