Quantum Physics

Publisher's description: Quantum physics allows us to understand the nature of the physical phenomena which govern the behavior of solids, semi-conductors, lasers, atoms, nuclei, subnuclear particles and light. In Quantum Physics, Le Bellac provides a thoroughly modern approach to this fundamental theory. Throughout the book, Le Bellac teaches the fundamentals of quantum physics using an original approach which relies primarily on an algebraic treatment and on the systematic use of symmetry principles. In addition to the standard topics such as one-dimensional potentials, angular momentum and scattering theory, the reader is introduced to more recent developments at an early stage. These include a detailed account of entangled states and their applications, the optical Bloch equations, the theory of laser cooling and of magneto-optical traps, vacuum Rabi oscillations, and an introduction to open quantum systems. This is a textbook for a modern course on quantum physics, written for advanced undergraduate and graduate students Cover......Page 1 Half-title......Page 3 Title......Page 5 Copyright......Page 6 Contents......Page 7 Foreword......Page 15 Preface......Page 17 Acknowledgments......Page 19 Addendum for the English edition......Page 20 Units and physical constants......Page 21 1.1.1 Length scales from cosmology to elementary particles......Page 23 1.1.2 States of matter......Page 24 1.1.3 Elementary constituents......Page 27 1.1.4 The fundamental interactions......Page 29 1.2 Classical and quantum physics......Page 31 1.3.1 Black-body radiation......Page 35 1.3.2 The photoelectric effect......Page 38 1.4.1 The de Broglie hypothesis......Page 39 1.4.2 Diffraction and interference of cold neutrons......Page 40 1.4.3 Interpretation of the experiments......Page 43 1.4.4 Heisenberg inequalities I......Page 46 1.5.1 Energy levels in classical mechanics and classical models of the atom......Page 49 1.5.2 The Bohr atom......Page 51 1.5.3 Orders of magnitude in atomic physics......Page 53 1.6.1 Orders of magnitude......Page 55 1.6.2 The black body......Page 56 1.6.4 Neutron diffraction by a crystal......Page 57 1.6.6 The Mach–Zehnder interferometer......Page 59 1.6.7 Neutron interferometry and gravity......Page 60 1.7 Further reading......Page 62 2.1 Hilbert spaces of finite dimension......Page 64 2.2.1 Linear, Hermitian, unitary operators......Page 66 2.2.2 Projection operators and Dirac notation......Page 68 2.3.1 Diagonalization of a Hermitian operator......Page 70 2.3.2 Diagonalization of a 2×2 Hermitian matrix......Page 72 2.3.3 Complete sets of compatible operators......Page 73 2.3.4 Unitary operators and Hermitian operators......Page 74 2.3.5 Operator-valued functions......Page 75 2.4.1 The scalar product and the norm......Page 76 2.4.3 The determinant and the trace......Page 77 2.4.6 Properties of projectors......Page 78 2.4.9 Normal matrices......Page 79 2.4.11 Operator identities......Page 80 2.4.12 A beam splitter......Page 81 2.5 Further reading......Page 82 3.1.1 The polarization of an electromagnetic wave......Page 83 3.1.2 The photon polarization......Page 90 3.1.3 Quantum cryptography......Page 95 3.2.1 Angular momentum and magnetic moment in classical physics......Page 97 3.2.2 The Stern–Gerlach experiment and Stern–Gerlach filters......Page 99 3.2.3 Spin states of arbitrary orientation......Page 102 3.2.4 Rotation of spin 1/2......Page 104 3.2.5 Dynamics and time evolution......Page 109 3.3.1 Decomposition and recombination of polarizations......Page 111 3.3.4 Other solutions of (3.45)......Page 113 3.3.6 Exponentials of Pauli matrices and rotation operators......Page 114 3.3.8 A 2 rotation of spin 1/2......Page 115 3.3.9 Neutron scattering by a crystal: spin-1/2 nuclei......Page 116 3.4 Further reading......Page 117 4.1.1 The superposition principle......Page 118 4.1.2 Physical properties and measurement......Page 120 4.1.3 Heisenberg inequalities II......Page 126 4.2.1 The evolution equation......Page 127 4.2.2 The evolution operator......Page 130 4.2.3 Stationary states......Page 131 4.2.4 The temporal Heisenberg inequality......Page 133 4.2.5 The Schrödinger and Heisenberg pictures......Page 136 4.3 Approximations and modeling......Page 137 4.4.1 Dispersion and eigenvectors......Page 138 4.4.4 Time evolution of a two-level system......Page 139 4.4.5 Unstable states......Page 141 4.4.6 The solar neutrino puzzle......Page 142 4.4.8 The system of neutral K mesons......Page 144 4.5 Further reading......Page 146 5.1.1 The ethylene molecule......Page 147 5.1.2 The benzene molecule......Page 150 5.2.1 A spin 1/2 in a periodic magnetic field......Page 154 5.2.2 Rabi oscillations......Page 155 5.2.3 Principles of NMR and MRI......Page 159 5.3.1 The ammonia molecule as a two-level system......Page 161 5.3.2 The molecule in an electric field: the ammonia maser......Page 163 5.3.3 Off-resonance transitions......Page 168 5.4 The two-level atom......Page 171 5.5.3 Butadiene......Page 174 5.5.5 The hydrogen molecular ion H+2......Page 176 5.5.6 The rotating-wave approximation in NMR......Page 177 5.6 Further reading......Page 179 6.1.1 Definition and properties of the tensor product......Page 180 Postulate V......Page 181 6.1.2 A system of two spins 1/2......Page 182 6.2.1 Definition and properties......Page 184 6.2.2 The state operator for a two-level system......Page 186 6.2.3 The reduced state operator......Page 189 6.2.4 Time dependence of the state operator......Page 191 6.3.1 The EPR argument......Page 193 6.3.2 Bell inequalities......Page 196 6.3.3 Interference and entangled states......Page 201 6.3.4 Three-particle entangled states (GHZ states)......Page 204 6.4.1 Measurement and decoherence......Page 207 6.4.2 Quantum information......Page 213 6.5.3 Properties of state operators......Page 220 6.5.4 Fine structure and the Zeeman effect in positronium......Page 221 6.5.5 Spin waves and magnons......Page 222 6.5.6 Spin echo and level splitting in NMR......Page 223 6.5.7 Calculation of E(a, b)......Page 224 6.5.8 Bell inequalities involving photons......Page 225 6.5.9 Two-photon interference......Page 226 6.5.10 Interference of emission times......Page 228 6.6 Further reading......Page 229 7.1.1 Definitions......Page 231 7.1.2 Realizations of separable spaces of infinite dimension......Page 233 7.2.1 The domain and norm of an operator......Page 235 7.2.2 Hermitian conjugation......Page 237 7.3.1 Hermitian operators......Page 238 7.3.2 Unitary operators......Page 241 7.4.3 Canonical commutation relations......Page 242 7.5 Further reading......Page 243 8 Symmetries in quantum physics......Page 244 8.1.1 Invariance of probabilities in a symmetry operation......Page 245 8.1.2 The Wigner theorem......Page 247 8.2.1 Definitions......Page 249 8.2.2 Conservation laws......Page 250 8.2.3 Commutation relations of infinitesimal generators......Page 252 8.3.1 Dimension d = 1......Page 256 8.3.2 Explicit realization and von Neumann’s theorem......Page 258 8.3.3 The parity operator......Page 259 8.4.1 The Hamiltonian in dimension d = 1......Page 262 8.4.2 The Hamiltonian in dimension d = 3......Page 265 8.5.2 Rotations and SU(2)......Page 267 8.5.4 The Lie algebra of a continuous group......Page 268 8.5.5 The Thomas–Reiche–Kuhn sum rule......Page 269 8.5.7 The Galilean transformation......Page 270 8.6 Further reading......Page 271 9.1.1 Diagonalization of X......Page 272 9.1.2 Realization in.........Page 274 9.1.3 Realization in.........Page 276 9.1.4 Evolution of a free wave packet......Page 278 9.2.1 The Hamiltonian of the Schrödinger equation......Page 282 9.2.2 The probability density and the probability current density......Page 283 9.3.1 Generalities......Page 286 9.3.2 Reflection and transmission by a potential step......Page 287 The potential step: total reflection......Page 290 The potential step: reflection and transmission......Page 291 9.3.3 The bound states of the square well......Page 292 9.4.1 The transmission matrix......Page 295 9.4.2 The tunnel effect......Page 299 9.4.3 The S matrix......Page 302 9.5.1 The Bloch theorem......Page 305 9.5.2 Energy bands......Page 307 9.6.1 Generalities......Page 311 9.6.2 The phase space and level density......Page 313 9.6.3 The Fermi Golden Rule......Page 315 9.7.1 The Heisenberg inequalities......Page 319 9.7.2 Wave-packet spreading......Page 320 9.7.4 Heuristic estimates using the Heisenberg inequality......Page 321 9.7.7 A delta-function potential......Page 322 9.7.9 Energy levels of an infinite cubic well in dimension d = 3......Page 324 9.7.13 Study of the Stern–Gerlach experiment......Page 325 9.7.14 The von Neumann model of measurement......Page 326 9.7.15 The Galilean transformation......Page 327 9.8 Further reading......Page 328 10.1 Diagonalization of J2 and Jz......Page 329 10.2 Rotation matrices......Page 333 10.3.1 The orbital angular momentum operator......Page 338 1. Basis on the unit sphere......Page 341 2. Relation to the Legendre polynomials......Page 342 3. Transformation under rotation......Page 343 4. Parity of the spherical harmonics......Page 344 10.4.1 The radial wave equation......Page 345 10.4.2 The hydrogen atom......Page 349 10.5.1 Rotations by pi, parity, and reflection with respect to a plane......Page 353 10.5.2 Dipole transitions......Page 354 10.5.3 Two-body decays: the general case......Page 359 10.6.1 Addition of two spins 1/2......Page 361 10.6.2 The general case: addition of two angular momenta J1 and J2......Page 363 10.6.3 Composition of rotation matrices......Page 366 10.6.4 The Wigner–Eckart theorem (scalar and vector operators)......Page 367 10.7.2 Rotation of angular momentum......Page 369 10.7.5 Orbital angular momentum......Page 370 10.7.7 Independence of the energy from m......Page 371 10.7.10 Matrix elements of a potential......Page 372 10.7.12 Symmetry property of the matrices d(j)......Page 373 10.7.14 Measurement of the Lambda0 magnetic moment......Page 374 10.7.15 Production and decay of the rho+ meson......Page 376 10.7.16 Interaction of two dipoles......Page 377 10.7.17 Sigma0 decay......Page 378 10.8 Further reading......Page 379 11 The harmonic oscillator......Page 380 11.1.1 Creation and annihilation operators......Page 381 11.1.2 Diagonalization of the Hamiltonian......Page 382 11.1.3 Wave functions of the harmonic oscillator......Page 384 11.2 Coherent states......Page 386 11.3.1 Sound waves and phonons......Page 389 11.3.2 Quantization of a scalar field in one dimension......Page 393 11.3.3 Quantization of the electromagnetic field......Page 397 11.3.4 Quantum fluctuations of the electromagnetic field......Page 402 11.4.1 Local gauge invariance......Page 406 11.4.2 A uniform magnetic field: Landau levels......Page 409 11.5.3 Coherent states......Page 412 11.5.4 Coupling to a classical force......Page 413 11.5.5 Squeezed states......Page 416 11.5.7 The scalar and vector potentials in Coulomb gauge......Page 417 11.5.9 Quantization in a cavity......Page 418 11.5.11 Non-Abelian gauge transformations......Page 419 11.5.12 The Casimir effect......Page 421 11.5.13 Quantum computing with trapped ions......Page 422 11.6 Further reading......Page 424 12.1.1 The differential and total cross sections......Page 426 12.1.2 The scattering amplitude......Page 428 12.2.1 The partial-wave expansion......Page 431 12.2.2 Low-energy scattering......Page 435 12.2.3 The effective potential......Page 439 12.2.4 Low-energy neutron–proton scattering......Page 441 12.3.1 The optical theorem......Page 442 12.3.2 The optical potential......Page 445 12.4.1 The integral equation of scattering......Page 447 12.4.2 Scattering of a wave packet......Page 449 12.5.1 The Gamow peak......Page 451 12.5.2 Low-energy neutron scattering by a hydrogen molecule......Page 452 12.5.3 Analytic properties of the neutron–proton scattering amplitude......Page 453 12.5.5 Neutron optics......Page 455 12.5.6 The cross section for neutrino absorption......Page 457 12.6 Further reading......Page 459 13.1.1 Symmetry or antisymmetry of the state vector......Page 460 13.1.2 Spin and statistics......Page 463 13.2 The scattering of identical particles......Page 468 13.3 Collective states......Page 470 13.4.1 The Tonos- particle and color......Page 472 13.4.4 Positronium decay......Page 473 13.4.5 Quantum statistics and beam splitters......Page 474 13.5 Further reading......Page 476 14.1.1 Generalities......Page 477 14.1.2 Nondegenerate perturbation theory......Page 479 14.1.3 Degenerate perturbation theory......Page 480 14.1.4 The variational method......Page 481 14.2.1 Energy levels in the absence of spin......Page 482 14.2.2 The fine structure......Page 483 14.2.3 The Zeeman effect......Page 485 14.2.4 The hyperfine structure......Page 487 14.3.1 The semiclassical theory......Page 489 14.3.2 The dipole approximation......Page 491 14.3.3 The photoelectric effect......Page 493 14.3.4 The quantized electromagnetic field: spontaneous emission......Page 495 14.4.1 The optical Bloch equations......Page 500 14.4.2 Dissipative forces and reactive forces......Page 504 14.4.3 Doppler cooling......Page 506 14.4.4 A magneto-optical trap......Page 511 14.5.1 The ground state of the helium atom......Page 513 14.5.2 The excited states of the helium atom......Page 515 14.6.1 Second-order perturbation theory and van der Waals forces......Page 517 14.6.2 Order-alpha2 corrections to the energy levels......Page 518 14.6.3 Muonic atoms......Page 520 14.6.5 The diamagnetic term......Page 521 14.6.6 Vacuum Rabi oscillations......Page 522 14.6.7 Reactive forces......Page 524 14.6.8 Radiative capture of neutrons by hydrogen......Page 526 14.7 Further reading......Page 528 15 Open quantum systems......Page 529 15.1.1 Schmidt’s decomposition......Page 531 15.1.2 Positive operator-valued measures......Page 533 15.1.3 Example: a POVM with spins 1/2......Page 535 15.2.1 Kraus decomposition......Page 539 15.2.2 The depolarizing channel......Page 544 15.2.3 The phase-damping channel......Page 545 15.2.4 The amplitude-damping channel......Page 546 15.3.1 The Markovian approximation......Page 548 15.3.2 The Lindblad equation......Page 549 15.3.3 Example: the damped harmonic oscillator......Page 551 15.4.1 Exact evolution equations......Page 552 15.4.2 The Markovian approximation......Page 555 15.4.3 Relaxation of a two-level system......Page 557 15.4.4 Quantum Brownian motion......Page 560 15.4.5 Decoherence and Schrödinger’s cats......Page 564 15.5.2 Using a POVM to distinguish between states......Page 566 15.5.4 Transposition is not completely positive......Page 567 15.5.7 Superposition of coherent states......Page 568 15.5.8 Dissipation in a two-level system......Page 570 15.5.10 Another choice for the spectral function J(omega)......Page 571 15.6 Further reading......Page 572 Appendix A The Wigner theorem and time reversal......Page 574 A.1 Proof of the theorem......Page 575 A.2 Time reversal......Page 577 B.1 An elementary model of measurement......Page 583 B.2 Ramsey fringes......Page 586 B.3 Interaction with a field inside the cavity......Page 589 B.4 Decoherence......Page 591 Appendix C The Wigner–Weisskopf method......Page 595 References......Page 600 Index......Page 601 "Michel Le Bellac teaches the fundamentals of quantum physics using an original approach which relies primarily on an algebraic treatment and on the systematic use of symmetry principles. In addition to the standard topics such as one-dimensional potentials, angular momentum and scattering theory, the reader is introduced to more recent developments at an early stage. These include a detailed account of entangled states and their applications, the optical Bloch equations, the theory of laser cooling and of magneto-optical traps, vacuum Rabi oscillations, and an introduction to open quantum systems. This is a textbook for a modern course on quantum physics, written for advanced undergraduate and graduate students."--Jacket