A First Course in Mathematical Physics

Preface page ix

Part I: Mathematics 1

1 Functions of one variable 2
1.1 Limits 2
1.2 Elementary Calculus 6
1.2.1 Chain Rule 7
1.2.2 Differentiation products and quotients 8
1.2.3 Inverse Functions 9
1.3 Integration 10
1.4 The Binomial Expansion 13
1.5 Taylor's series 15
1.6 Extrema 17
1.7 Power Series 18
1.8 Basic Functions 19
1.8.1 Exponential 19
1.8.2 Logarithm 23
1.9 1st order ordinary differential equations 24
1.10 Trigonometric Functions 26
1.10.1 L'Hopital's rule 28
1.11 Problems 30

2 Complex numbers 33
2.1 Exponential function of a complex variable 34
2.2 Argrand Diagrams and the Complex Plane 36
2.3 Hyperbolic functions 38
2.4 The simple harmonic oscillator 40
2.4.1 Mechanics in one dimension 42
2.5 Problems 45

3 Functions of Several Variables 48
3.1 Partial derivatives 48
3.1.1 Definition of the partial derivative 48
3.1.2 Total derivatives 51
3.1.3 Some relations 53
3.1.4 Change of variables 55
3.1.5 Mechanics again 56
3.1.6 Exact differentials and thermodynamics 58
3.2 Extrema under constraint 60
3.3 Multiple Integrals 62
3.3.1 Triple Integrals 66
3.3.2 Change of variables 67
3.4 Problems 69

4 Vectors in R3 72
4.1 Basic operation 72
4.1.1 scalar triple product 79
4.1.2 Vector equations of lines and planes 80
4.2 Kinematics in three dimensions 81
4.2.1 Differentiation 81
4.2.2 Motion in a uniform magnetic field 81
4.3 Coordinate systems 83
4.3.1 Polar coordinates 83
4.4 Central Forces 84
4.5 Rotating Frames 88
4.5.1 Larmor Effect 91
4.6 Problems 93

5 Vector fields and operators 96
5.1 The gradient operator 96
5.1.1 Coordinate Systems 97
5.2 Work and energy in vectorial mechanics 101
5.2.1 Line integrals 104
5.3 A little fluid dynamics 107
5.3.1 Rotational motion 111
5.3.2 Fields 114
5.3.3 Surface integrals 115
5.4 The divergence theorem 118
5.5 Stokes' Theorem 121
5.5.1 Conservative Forces 123
5.6 Problems 126

6 Generalized Functions. 130
6.1 The Dirac delta function 130
6.2 Problems 139

7 Vector Spaces 140
7.1 Formal Definition of a vector space 140
7.2 Fourier Series 145
7.3 Linear Operators 148
7.4 Change of basis 160
7.5 Problems 163

Part II: Physics 168

8 Maxwell's Equations: A very short Introduction 169
8.1 Electrostatic: Gauss's Law 169
8.2 Gauss's Law for a magnetic field 173
8.3 Ampere's Law 173
8.3.1 Gauge conditions 174
8.4 Problems 177

9 Special Relativity:4-vector formalism 179
9.1 Lorentz transformation 179
9.1.1 Inertial frames 179
9.1.2 Properties and consequences of the Lorentz transformation
182
9.2 Minkowski space 183
9.2.1 Four vectors 183
9.2.2 Time Dilation 189
9.3 Four velocity 190
9.3.1 Four momentum 191
9.4 Electrodynamics 197
9.4.1 Maxwell's equations in 4-vector form 197
9.4.2 Field of a moving point charge 200
9.5 Problems 202

10 Quantum Theory 205
10.1 Formalism 205
10.1.1 Dirac notation 206
10.2 Probabilistic interpretation 207
10.2.1 Commutator relations 208
10.2.2 Functions of observables 210
10.3 The Stern-Gerlach experiment 210
10.3.1 Spin space 210
viii Contents
10.3.2 Explicit matrix representation 210
10.3.3 EPR paradox 210
10.3.4 Bell's Theorem 210
10.4 Quantization 210
10.4.1 Time evolution 210
10.4.2 The harmonic oscillator, coherent states 210
10.5 Problems 212

11 Atoms, molecules, solids, wave mechanics in one dimension
216
11.1 Atom 217
11.1.1 The square well 218
11.1.2 The delta function potential 219
11.2 Molecules 221
11.3 Solids 223
11.3.1 Block's Theorem 224
11.3.2 Band structure 226
12 An informal treatment of variational principles and their
history 229

Appendix 1 Conic Sections 230
Appendix 2 Vector Relations 232
Index 237

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.