This book is a comprehensive account of five extended modules covering the key branches of twentieth-century theoretical physics, taught by the author over a period of three decades to students on bachelor and master university degree courses in both physics and theoretical physics.

The modules cover nonrelativistic quantum mechanics, thermal and statistical physics, many-body theory, classical field theory (including special relativity and electromagnetism), and, finally, relativistic quantum mechanics and gauge theories of quark and lepton interactions, all presented in a single, self-contained volume.

In a number of universities, much of the material covered (for example, on Einstein’s general theory of relativity, on the BCS theory of superconductivity, and on the Standard Model, including the theory underlying the prediction of the Higgs boson) is taught in postgraduate courses to beginning PhD students.

A distinctive feature of the book is that full, step-by-step mathematical proofs of all essential results are given, enabling a student who has completed a high-school mathematics course and the first year of a university physics degree course to understand and appreciate the derivations of very many of the most important results of twentieth-century theoretical physics.

Notation xiii

Preface xv

I NONRELATIVISTIC QUANTUM MECHANICS 1

1 Basic Concepts of Quantum Mechanics 3

1.1 Probability interpretation of the wave function 3

1.2 States of definite energy and states of definite momentum 4

1.3 Observables and operators 5

1.4 Examples of operators 5

1.5 The time-dependent Schr¨odinger equation 6

1.6 Stationary states and the time-independent Schr¨odinger equation 7

1.7 Eigenvalue spectra and the results of measurements 8

1.8 Hermitian operators 8

1.9 Expectation values of observables 10

1.10 Commuting observables and simultaneous observability 10

1.11 Noncommuting observables and the uncertainty principle 11

1.12 Time dependence of expectation values 12

1.13 The probability-current density 12

1.14 The general form of wave functions 12

1.15 Angular momentum 15

1.16 Particle in a three-dimensional spherically symmetric potential 17

1.17 The hydrogen-like atom 18

2 Representation Theory 23

2.1 Dirac representation of quantum mechanical states 23

2.2 Completeness and closure 27

2.3 Changes of representation 28

2.4 Representation of operators 29

2.5 Hermitian operators 31

2.6 Products of operators 31

2.7 Formal theory of angular momentum 32

3 Approximation Methods 39

3.1 Time-independent perturbation theory for nondegenerate states 39

3.2 Time-independent perturbation theory for degenerate states 44

3.3 The variational method 50

3.4 Time-dependent perturbation theory 54

4 Scattering Theory 63

4.1 Evolution operators and Møller operators 63

4.2 The scattering operator and scattering matrix 66

4.3 The Green operator and T operator 70

4.4 The stationary scattering states 76

4.5 The optical theorem 83

4.6 The Born series and Born approximation 85

4.7 Spherically symmetric potentials and the method of partial waves 87

4.8 The partial-wave scattering states 92

II THERMAL AND STATISTICAL PHYSICS 97

5 Fundamentals of Thermodynamics 99

5.1 The nature of thermodynamics 99

5.2 Walls and constraints 99

5.3 Energy 100

5.4 Microstates 100

5.5 Thermodynamic observables and thermal fluctuations 100

5.6 Thermodynamic degrees of freedom 102

5.7 Thermal contact and thermal equilibrium 103

5.8 The zeroth law of thermodynamics 104

5.9 Temperature 104

5.10 The International Practical Temperature Scale 107

5.11 Equations of state 107

5.12 Isotherms 108

5.13 Processes 109

5.13.1 Nondissipative work 109

5.13.2 Dissipative work 111

5.13.3 Heat flow 112

5.14 Internal energy and heat 112

5.14.1 Joule’s experiments and internal energy 112

5.14.2 Heat 113

5.15 Partial derivatives 115

5.16 Heat capacity and specific heat 116

5.16.1 Constant-volume heat capacity 117

5.16.2 Constant-pressure heat capacity 117

5.17 Applications of the first law to ideal gases 118

5.18 Difference of constant-pressure and constant-volume heat capacities 119

5.19 Nondissipative-compression/expansion adiabat of an ideal gas 120

6 Quantum States and Temperature 125

6.1 Quantum states 125

6.2 Effects of interactions 128

6.3 Statistical meaning of temperature 130

6.4 The Boltzmann distribution 134

7 Microstate Probabilities and Entropy 141

7.1 Definition of general entropy 141

7.2 Law of increase of entropy 142

7.3 Equilibrium entropy S 144

7.4 Additivity of the entropy 146

7.5 Statistical–mechanical description of the three types of energy transfer 147

8 The Ideal Monatomic Gas 151

8.1 Quantum states of a particle in a three-dimensional box 151

8.2 The velocity-component distribution and internal energy 153

8.3 The speed distribution 156

8.4 The equation of state 158

8.5 Mean free path and thermal conductivity 160

9 Applications of Classical Thermodynamics 163

9.1 Entropy statement of the second law of thermodynamics 163

9.2 Temperature statement of the second law of thermodynamics 164

9.3 Summary of the basic relations 166

9.4 Heat engines and the heat-engine statement of the second law of thermodynamics 167

9.5 Refrigerators and heat pumps 169

9.6 Example of a Carnot cycle 170

9.7 The third law of thermodynamics 172

9.8 Entropy-change calculations 174

10 Thermodynamic Potentials and Derivatives 177

10.1 Thermodynamic potentials 177

10.2 The Maxwell relations 179

10.3 Calculation of thermodynamic derivatives 180

11 Matter Transfer and Phase Diagrams 183

11.1 The chemical potential 183

11.2 Direction of matter flow 184

11.3 Isotherms and phase diagrams 184

11.4 The Euler relation 187

11.5 The Gibbs–Duhem relation 188

11.6 Slopes of coexistence lines in phase diagrams 188

12 Fermi–Dirac and Bose–Einstein Statistics 191

12.1 The Gibbs grand canonical probability distribution 191

12.2 Systems of noninteracting particles 193

12.3 Indistinguishability of identical particles 194

12.4 The Fermi–Dirac and Bose–Einstein distributions 195

12.5 The entropies of noninteracting fermions and bosons 197

13 Quantum Mechanics and Low-Temperature Thermodynamics of Many-Particle Systems 201

13.1 Introduction 201

13.2 Systems of noninteracting particles 201

13.2.1 Bose systems 202

13.2.2 Fermi systems 204

13.3 Systems of interacting particles 209

13.4 Systems of interacting fermions (the Fermi liquid) 211

13.5 The Landau theory of the normal Fermi liquid 214

13.6 Collective excitations of a Fermi liquid 221

13.6.1 Zero sound in a neutral Fermi gas with repulsive interactions 221

13.6.2 Plasma oscillations in a charged Fermi liquid 221

13.7 Phonons and other excitations 223

13.7.1 Phonons in crystals 223

13.7.2 Phonons in liquid helium-4 232

13.7.3 Magnons in solids 233

13.7.4 Polarons and excitons 233

14 Second Quantization 235

14.1 The occupation-number representation 235

14.2 Particle-field operators 246

15 Gas of Interacting Electrons 251

15.1 Hamiltonian of an electron gas 251

16 Superconductivity 261

16.1 Superconductors 261

16.2 The theory of Bardeen, Cooper and Schrieffer 262

16.2.1 Cooper pairs 267

16.2.2 Calculation of the ground-state energy 269

16.2.3 First excited states 277

16.2.4 Thermodynamics of superconductors 280

IV CLASSICAL FIELD THEORY AND RELATIVITY 287

17 The Classical Theory of Fields 289

17.1 Mathematical preliminaries 289

17.1.1 Behavior of fields under coordinate transformations 289

17.1.2 Properties of the rotation matrix 293

17.1.3 Proof that a “dot product” is a scalar 295

17.1.4 A lemma on determinants 297

17.1.5 Proof that the “cross product” of two vectors is a “pseudovector” 298

17.1.6 Useful index relations 299

17.1.7 Use of index relations to prove vector identities 300

17.1.8 General definition of tensors of arbitrary rank 301

17.2 Introduction to Einsteinian relativity 302

17.2.1 Intervals 302

17.2.2 Timelike and spacelike intervals 304

17.2.3 The light cone 304

17.2.4 Variational principle for free motion 305

17.2.5 The Lorentz transformation 305

17.2.6 Length contraction and time dilation 307

17.2.7 Transformation of velocities 308

17.2.8 Four-tensors 308

17.2.9 Integration in four-space 314

17.2.10 Integral theorems 316

17.2.11 Four-velocity and four-acceleration 317

17.3 Principle of least action 318

17.3.1 Free particle 318

17.3.2 Three-space formulation 318

17.3.3 Momentum and energy of a free particle 319

17.3.4 Four-space formulation 321

17.4 Motion of a particle in a given electromagnetic field 325

17.4.1 Equations of motion of a charge in an electromagnetic field 326

17.4.2 Gauge invariance 328

17.4.3 Four-space derivation of the equations of motion 329

17.4.4 Lorentz transformation of the electromagnetic field 332

17.4.5 Lorentz invariants constructed from the electromagnetic field 334

17.4.6 The first pair of Maxwell equations 335

17.5 Dynamics of the electromagnetic field 337

17.5.1 The four-current and the second pair of Maxwell equations 338

17.5.2 Energy density and energy flux density of the electromagnetic field 342

17.6 The energy–momentum tensor 345

17.6.1 Energy–momentum tensor of the electromagnetic field 350

17.6.2 Energy–momentum tensor of particles 353

17.6.3 Energy–momentum tensor of continuous media 355

18 General Relativity 361

18.1 Introduction 361

18.2 Space–time metrics 362

18.3 Curvilinear coordinates 364

18.4 Products of tensors 365

18.5 Contraction of tensors 366

18.6 The unit tensor 366

18.7 Line element 366

18.8 Tensor inverses 366

18.9 Raising and lowering of indices 367

18.10 Integration in curved space–time 367

18.11 Covariant differentiation 369

18.12 Parallel transport of vectors 370

18.13 Curvature 374

18.14 The Einstein field equations 376

18.15 Equation of motion of a particle in a gravitational field 381

18.16 Newton’s law of gravity 383

V RELATIVISTIC QUANTUM MECHANICS AND GAUGE THEORIES 385

19 Relativistic Quantum Mechanics 387

19.1 The Dirac equation 387

19.2 Lorentz and rotational covariance of the Dirac equation 391

19.3 The current four-vector 398

19.4 Compact form of the Dirac equation 400

19.5 Dirac wave function of a free particle 401

19.6 Motion of an electron in an electromagnetic field 405

19.7 Behavior of spinors under spatial inversion 408

19.8 Unitarity properties of the spinor-transformation matrices 409

19.9 Proof that the four-current is a four-vector 411

19.10 Interpretation of the negative-energy states 412

19.11 Charge conjugation 413

19.12 Time reversal 414

19.13 PCT symmetry 417

19.14 Models of the weak interaction 422

20 Gauge Theories of Quark and Lepton Interactions 427

20.1 Global phase invariance 427

20.2 Local phase invariance? 427

20.3 Other global phase invariances 429

20.4 SU(2) local phase invariance (a non-abelian gauge theory) 433

20.5 The “gauging” of color SU(3) (quantum chromodynamics) 436

20.6 The weak interaction 436

20.7 The Higgs mechanism 439

20.8 The fermion masses 448

Appendices 451

A.1 Proof that the scattering states |φ+ ≡ Ω+|φ exist for all states |φ in the Hilbert

space H 451

A.2 The scattering matrix in momentum space 452

A.3 Calculation of the free Green function r|G0(z)|r 454

Supplementary Reading 457

Index 459