What is included with this book?
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