First Course in Mathematical Physics

Colm T. Whelan is a Professor of Physics and an Eminent Scholar at Old Dominion University, USA. He received his PhD in Theoretical Atomic Physics from the University of Cambridge in 1985 and his ScD in 2001. He is a fellow of both the American Physical Society and the Institute of Physics (UK). He has over 25 years of experience in undergraduate teaching in both the UK and the US.

• Preface xvPart I Mathematics 11 Functions of One Variable 31.1 Limits 31.2 Elementary Calculus 51.2.1 Differentiation Products and Quotients 61.2.2 Chain Rule 71.2.3 Inverse Functions 81.3 Integration 101.4 The Binomial Expansion 141.5 Taylor’s Series 151.6 Extrema 171.7 Power Series 171.8 Basic Functions 191.8.1 Exponential 191.8.2 Logarithm 221.9 First-Order Ordinary Differential Equations 241.10 Trigonometric Functions 251.10.1 L’Hôpital’s Rule 27Problems 272 Complex Numbers 292.1 Exponential Function of a Complex Variable 302.2 Argand Diagrams and the Complex Plane 322.3 Complex Logarithm 342.4 Hyperbolic Functions 342.5 The Simple Harmonic Oscillator 362.5.1 Mechanics in One Dimension 382.5.2 Damped and Driven Oscillations 40Problems 473 VectorsinR 3 513.1 Basic Operation 513.1.1 Scalar Triple Product 553.1.2 Vector Equations of Lines and Planes 563.2 Kinematics in Three Dimensions 573.2.1 Differentiation 573.2.2 Motion in a Uniform Magnetic Field 573.3 Coordinate Systems 593.3.1 Polar Coordinates 593.4 Central Forces 603.5 Rotating Frames 643.5.1 Larmor Effect 66Problems 674 VectorSpaces 714.1 Formal Definition of a Vector Space 714.2 Fourier Series 754.3 Linear Operators 784.4 Change of Basis 89Problems 915 Functions of Several Variables 955.1 Partial Derivatives 955.1.1 Definition of the Partial Derivative 955.1.2 Total Derivatives 985.1.3 Elementary Numerical Methods 1045.1.4 Change of Variables 1075.1.5 Mechanics Again 1095.2 Extrema under Constraint 1115.3 Multiple Integrals 1135.3.1 Triple Integrals 1165.3.2 Change of Variables 117Problems 1216 Vector Fields and Operators 1256.1 The Gradient Operator 1256.1.1 Coordinate Systems 1276.2 Work and Energy in Vectorial Mechanics 1306.2.1 Line Integrals 1336.3 A Little Fluid Dynamics 1356.3.1 Rotational Motion 1386.3.2 Fields 1416.4 Surface Integrals 1426.5 The Divergence Theorem 1466.6 Stokes’ Theorem 1496.6.1 Conservative Forces 153Problems 1547 Generalized Functions 1597.1 The Dirac Delta Function 1597.2 Green’s Functions 1637.3 Delta Function in Three Dimensions 165Problems 1698 Functions of a Complex Variable 1738.1 Limits 1748.2 Power Series 1788.3 Fluids Again 1798.4 Complex Integration 1808.4.1 Application of the Residue Theorem 186Problems 192Part II Physics 1959 Maxwell’s Equations: A Very Short Introduction 1979.1 Electrostatics: Gauss’s Law 1979.1.1 Conductors 2039.2 The No Magnetic Monopole Rule 2049.3 Current 2059.4 Faraday’s Law 2069.5 Ampère’s Law 2089.6 The Wave Equation 2109.7 Gauge Conditions 211Problems 21310 Special Relativity: Four-Vector Formalism 21710.1 Lorentz Transformation 21710.1.1 Inertial Frames 21710.1.2 Properties and Consequences of the Lorentz Transformation 22010.2 Minkowski Space 22010.2.1 Four Vectors 22010.2.2 Time Dilation 22610.3 Four-Velocity 22710.3.1 Four-Momentum 22910.4 Electrodynamics 23410.4.1 Maxwell’s Equations in Four-Vector Form 23410.4.2 Field of a Moving Point Charge 23710.5 Transformation of the Electromagnetic Fields 239Problems 24011 Quantum Theory 24311.1 Bohr Atom 24311.2 The de Broglie Hypothesis 24611.3 The Schrödinger Wave Equation 24611.4 Interpretation of the Wave function 24911.5 Atom 25111.5.1 The Delta Function Potential 25211.5.2 Molecules 25411.6 Formalism 25711.6.1 Dirac Notation 25711.7 Probabilistic Interpretation 25811.7.1 Commutator Relations 25911.7.2 Functions of Observables 26111.7.3 Block’s Theorem 26111.7.4 Band Structure 26311.8 Time Evolution 26611.9 The Stern–Gerlach Experiment 26911.9.1 Successive Measurements 27011.9.2 Spin Space 27111.9.3 Explicit Matrix Representation 27211.9.4 Larmor Precession 27411.9.5 EPR Paradox 27511.9.6 Bell’s Theorem 27611.9.7 The Harmonic Oscillator 279Problems 28012 An Informal Treatment of Variational Principles and their History 28712.1 Sin and Death 28712.2 The Calculus of Variations 28812.3 Constrained Variations 29312.4 Hamilton’s Equations 29312.5 Phase Space 29612.6 Fixed Points 296Problems 298A Conic Sections 301A.1 Polar Coordinates 303A.2 Intersection of a Cone and a Plane 304B Vector Relations 305B.1 Products 305B.2 Differential Operator Relations 305B.3 Coordinates 306Cylindrical Polar 306Spherical Polar 307Bibliography 309Index 311