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Quantum Mechanics A New Introduction K. Konishi University of Pisa and INFN, Pisa G. Paffuti University of Pisa and INFN, Pisa CLARENDON PRESS 2009 . OXFORD Preface A student’s first encounter with quantum mechanics could be a traumatic one. Instead of the solid differential equation with respect to time (t) which is Newton’s classical equation of motion, with its inevitable consequences, she or he learns that the new mechanics predicts as a rule only certain probabilities (!), and that electrons behave like a sort of wave, a bizarre notion—but an empirical fact.1 When the student makes some progress in her or his study, however, she (he) will realize that, after all, things are not that bad: the fundamental equation of the new mechanics—the Schrödinger equation—is a well-defined, perfectly respectable linear differential equation in t, and when left alone, the microscopic system evolves in a rigorously deterministic fashion. Not only that, but due to the quantization of finite motions, and intimately related to this, to the existence of a new fundamental constant of Nature, the Planck constant, h, quantum mechanics provides a much sharper (and sometimes, far simpler) explanation of the properties of atoms than does classical mechanics. For instance, all atoms of the same kind, in their normal state, have rigorously identical properties. This fact is fundamental, for instance, to the regular structures, and in the working, of the macroscopic world (solids, crystals, biological phenomena, etc). The advantage of the new mechanics over the classical one is, of course, not limited to atoms. There are many phenomena in our daily life, such as electrical conduction, the laser, electronics, quantum optics, and all other related contemporary technologies, which require quantum mechanics for a proper understanding. Later on in her or his study, the student might find out that physicists today are debating the validity of the standard model predictions sometimes to the eleventh digit, for instance concerning the anomalous magnetic moment of the muon (a kind of heavy electron). Of course, here we are comparing a particular model of Nature with experiments; however, the standard model of fundamental interactions—quantum chromodynamics for strong interactions and the Glashow–Weinberg–Salam theory of electroweak interactions—are all based on relativistic quantum mechanics. In atomic physics, the agreement between theory and experiment can be equally good and sometimes even more impressive. All this, finally, will convince her (him) that we are indeed dealing with one of the most precise and perhaps most elegant theories ever known in physics. One day she or he might become a researcher or a teacher, and may start giving a course on quantum mechanics. Perhaps, after many years, 1 The background picture on the frontcover page represents electron wave ripples, formed by 50 kV electron beams going through a collodion thin film with tiny holes. The magnification is such that the full page width corresponds to about 0.6 microns. (Courtesy of Dr. Akira Tonomura, Hitachi Advanced Research Laboratory, Saitama, Japan.) vi Preface she (he) will continue to marvel at the simplicity and beauty of quantum mechanics, and at the same time its subtle and far-reaching consequences. One of the main aims of this book is to try to convey this sense of wonder to young students who are starting to appreciate the beauty of physics. This book is, in fact, meant to be an introductory textbook on quantum mechanics: it should be adequate for those who are learning it for the first time, as well as for slightly more advanced students. Standard courses on classical physics, including classical mechanics, electromagnetism, statistical mechanics and thermodynamics, plus basic mathematics, should provide a sufficient background. At the same time, however, we hope that this book, with its many examples of solved problems, and the diverse subjects discussed, will be a useful reference tool for more advanced students, active researchers and teachers alike. Let us illustrate some of the innovative features of this book. We took great pains to try to present quantum mechanics pedagogically, and at the same time with as much logical clarity and organization as possible. Concepts and methods are introduced gradually, and each of them is elaborated better and more precisely as the pages go on. We start, in fact, from the very basic concepts illustrated by elementary applications, and move on to more structural issues such as symmetry, statistics, and formal aspects of quantum mechanics, and then explore several standard approximation methods. Various applications of physical interest are then discussed, taking full advantage of the artillery we have armed ourselves with. As far as the content goes, for the most part, it is fairly standard, even though some of the discussions in the main text, e.g. in Chapter 12 (systems with general time-dependent Hamiltonians), in Chapter 13 (metastable states), and in Chapter 15 (atoms), and several topics treated in Supplements (Chapter 20), may often not be found in a standard textbook. At the end of each chapter, there are a number of problems to be solved analytically, as well as some others to be solved by numerical methods. The solutions to both types of problems are provided in an accompanying CD, in the form of PDF files (analytical problems) or in the form of Mathematica notebooks (there are 88 of these). The latter contain self-explanatory expositions of the solutions proposed, as well as an elementary guide to the Mathematica commands used, so that they should be easily usable even by those who are seeing a Mathematica program for the first time. The reader is encouraged to run the program, enjoy observing how the wave functions evolve, for example, modify and extend the problems as she (he) pleases, try to improve the precision of the calculation, etc. (Here are practical tips for the beginner: first, carefully read the ReadmeFirst file before starting; second, make a copy of each nb file before proceeding, and keep the original intact. Use a copy, when actually running the program, and making modifications vii and extensions.) In some cases the analysis is pushed a little deeper into the heart of the problem than is ordinarily done in a quantum mechanics textbook (such as the problem of the divergences of perturbation series and resummation; the study of metastable systems; concrete determination of atomic spectra for general elements, etc.), but always in a concrete, physical fashion, never going too much into mathematics. All in all, this is meant to be a contemporary, but at the same time relatively self-contained and comprehensive, textbook on quantum mechanics. The book is organized as follows. Part I is an elementary introduction to the basics of quantum mechanics. Together with some initial sections on perturbation theory and variational methods in Part II, Part I could correspond to standard material for an introductory semester course on quantum mechanics in most universities. Part II is dedicated to the three standard methods of approximation, perturbation theory, the variational method, and the semiclassical approximation, through which the concepts in the theory are further developed and the range of applicability vastly increased. In Part III the formalism and methods of analyses developed are applied to various physical situations, from general timedependent Hamiltonians, general discussions of metastable systems, the motion of electrically charged particles in electromagnetic fields, atoms, the scattering problem, atomic nuclei, and elementary particles. Part IV is dedicated to two fundamental issues of a conceptual nature: quantum entanglement and the measurement problems. Part V—the Supplements—is a collection of discussions of various natures, ranging from a review of useful formulas and tables, to some advanced topics, technical issues, and mathematical appendices. They are independent of each other, there is no ordering among them, and many are even independent of the main text, so that each of them can be read at leisure in a convenient moment for each reader. The accompanying CD, as already anticipated, contains the Mathematica notebooks and PDF files in which the problems proposed at the end of each chapter are solved and discussed. The subfiles for each chapter contain all the notebooks of that chapter, accompanied by a file called Guide-to-NB.nb. In this file a list of all the Mathematica notebooks of that chapter is given, as well as a brief description of each notebook. All analyses have been done by using Mathematica 6, Wolfram Research, and tested with Mathematica 7, which has just come out. For updates and corrections, consult our webpages: http://www.df.unipi.it/e konishi http://www.df.unipi.it/e paffuti We are grateful to Mark Seymour of OUP for his brave attempt at polishing our English and for his invaluable help in improving the look of the whole book. Of course, the responsibility for any errors in the text or formulas, or for any misleading expressions, which may undoubtedly still remain or might have been introduced during the course of corrections, viii Preface is ours and ours only. Thanks are also due to Charlotte Green of OUP for her crisp approach to editorial help, and to Sonke Adlung, the senior physics editor, for his admirable patience during this book’s long period of gestation. Our hearty gratitude goes to our friends and colleagues who helped us at various moments and in various—small and big—ways. A short list includes: A. Bonaccorso, D. M. Brink, P. Calabrese, P. Cecchi, R. Collina, E. D’Emilio, A. Di Giacomo, G. Dunne, T. Elze, M. Fukugita, C. Giannessi, V. Gracco, R. Guida, R. Jackiw, F. Maccarrone, G. Marchesini, M. Matone, P. Menotti, M. Mintchev, F. M. Miranda, G. Morchio, E. Onofri, L. Picasso, M. Rocca, S. Shore, A. Toncelli, M. Tonelli, A. Tonomura, P. Truini, A. Vainshtein, and G. Veneziano. A final message to all of you (especially to the young): Read and Enjoy! Pisa, February 2009 K. Konishi and G. Paffuti Contents I Basic quantum mechanics 1 Introduction 1.1 The quantum behavior of the electron 1.1.1 Diffraction and interference—visualizing the quantum world 1.1.2 The stability and identity of atoms 1.1.3 Tunnel effects 1.2 The birth of quantum mechanics 1.2.1 From the theory of specific heat to Planck’s formula 1.2.2 The photoelectric effect 1.2.3 Bohr’s atomic model 1.2.4 The Bohr–Sommerfeld quantization condition; de Broglie’s wave Further reading Guide to the Supplements Problems Numerical analyses 2 Quantum mechanical laws 2.1 Quantum states 2.1.1 Composite systems 2.1.2 Photon polarization and the statistical nature of quantum mechanics 2.2 The uncertainty principle 2.3 The fundamental postulate 2.3.1 The projection operator and state vector reduction 2.3.2 Hermitian operators 2.3.3 Products of operators, commutators, and compatible observables 2.3.4 The position operator, the momentum operator, fundamental commutators, and Heisenberg’s relation 2.3.5 Heisenberg’s relations 2.4 The Schrödinger equation 2.4.1 More about the Schrödinger equations 2.4.2 The Heisenberg picture 2.5 The continuous spectrum 2.5.1 The delta function 2.5.2 Orthogonality 1 5 5 5 6 8 9 9 14 15 17 18 18 19 19 21 21 24 24 26 29 31 32 33 35 36 37 38 40 40 41 43 x Contents 2.5.3 The position and momentum eigenstates; momentum as a translation operator 2.6 Completeness Problems Numerical analyses 3 The Schrödinger equation 3.1 General properties 3.1.1 Boundary conditions 3.1.2 Ehrenfest’s theorem 3.1.3 Current density and conservation of probability 3.1.4 The virial and Feynman–Hellman theorems 3.2 One-dimensional systems 3.2.1 The free particle 3.2.2 Topologically nontrivial space 3.2.3 Special properties of one-dimensional Schrödinger equations 3.3 Potential wells 3.3.1 Infinitely deep wells (walls) 3.3.2 The finite square well 3.3.3 An application 3.4 The harmonic oscillator 3.4.1 The wave function and Hermite polynomials 3.4.2 Creation and annihilation operators 3.5 Scattering problems and the tunnel effect 3.5.1 The potential barrier and the tunnel effect 3.5.2 The delta function potential 3.5.3 General aspects of the scattering problem 3.6 Periodic potentials 3.6.1 The band structure of the energy spectrum 3.6.2 Analysis Guide to the Supplements Problems Numerical analyses 4 Angular momentum 4.1 Commutation relations 4.2 Space rotations 4.3 Quantization 4.4 The Stern–Gerlach experiment 4.5 Spherical harmonics 4.6 Matrix elements of J 4.6.1 Spin- 12 and Pauli matrices 4.7 The composition rule 4.7.1 The Clebsch–Gordan coefficients 4.8 Spin 4.8.1 Rotation matrices for spin 21 Guide to the Supplements 43 45 47 48 49 49 49 50 51 52 53 54 55 56 58 58 59 61 63 63 67 71 71 74 78 80 80 82 84 85 87 89 89 91 92 95 96 98 100 101 104 105 107 108 Contents xi Problems 109 5 Symmetry and statistics 111 5.1 Symmetries in Nature 111 5.2 Symmetries in quantum mechanics 113 5.2.1 The ground state and symmetry 116 5.2.2 Parity (P) 117 5.2.3 Time reversal 121 5.2.4 The Galilean transformation 123 5.2.5 The Wigner–Eckart theorem 125 5.3 Identical particles: Bose–Einstein and Fermi–Dirac statistics 127 5.3.1 Identical bosons 130 5.3.2 Identical fermions and Pauli’s exclusion principle 132 Guide to the Supplements 133 Problems 134 6 Three-dimensional problems 6.1 Simple three-dimensional systems 6.1.1 Reduced mass 6.1.2 Motion in a spherically symmetric potential 6.1.3 Spherical waves 6.2 Bound states in potential wells 6.3 The three-dimensional oscillator 6.4 The hydrogen atom Guide to the Supplements Problems Numerical analyses 135 135 135 136 137 140 141 143 148 149 150 7 Some finer points of quantum mechanics 7.1 Representations 7.1.1 Coordinate and momentum representations 7.2 States and operators 7.2.1 Bra and ket; abstract Hilbert space 7.3 Unbounded operators 7.3.1 Self-adjoint operators 7.4 Unitary transformations 7.5 The Heisenberg picture 7.5.1 The harmonic oscillator in the Heisenberg picture 7.6 The uncertainty principle 7.7 Mixed states and the density matrix 7.7.1 Photon polarization 7.8 Quantization in general coordinates Further reading Guide to the Supplements Problems 151 151 152 155 155 158 160 167 169 171 172 173 176 178 182 182 182 8 Path integrals 8.1 Green functions 183 183 xii Contents 8.2 Path integrals 8.2.1 Derivation 8.2.2 Mode expansion 8.2.3 Feynman graphs 8.2.4 Back to ordinary (Minkowski) time 8.2.5 Tunnel effects and instantons Further reading Numerical analyses II Approximation methods 186 186 190 192 197 198 201 202 203 9 Perturbation theory 207 9.1 Time-independent perturbations 207 9.1.1 Degenerate levels 212 9.1.2 The Stark effect on the n = 2 level of the hydrogen atom 214 9.1.3 Dipole interactions and polarizability 217 9.2 Quantum transitions 219 9.2.1 Perturbation lasting for a finite interval 221 9.2.2 Periodic perturbation 223 9.2.3 Transitions in a discrete spectrum 223 9.2.4 Resonant oscillation between two levels 225 9.3 Transitions in the continuum 226 9.3.1 State density 228 9.4 Decays 228 9.5 Electromagnetic transitions 233 9.5.1 The dipole approximation 234 9.5.2 Absorption of radiation 237 9.5.3 Induced (or stimulated) emission 238 9.5.4 Spontaneous emission 239 9.6 The Einstein coefficients 240 Guide to the Supplements 242 Problems 242 Numerical analyses 244 10 Variational methods 10.1 The variational principle 10.1.1 Lower limits 10.1.2 Truncated Hilbert space 10.2 Simple applications 10.2.1 The harmonic oscillator 10.2.2 Helium: an elementary variational calculation 10.2.3 The virial theorem 10.3 The ground state of the helium Guide to the Supplements Problems Numerical analyses 245 245 247 249 250 250 252 254 255 261 261 262 Contents xiii 11 The semi-classical approximation 11.1 The WKB approximation 11.1.1 Connection formulas 11.2 The Bohr–Sommerfeld quantization condition 11.2.1 Counting the quantum states 11.2.2 Potentials defined for x > 0 only 11.2.3 On the meaning of the limit ~ → 0 11.2.4 Angular variables 11.2.5 Radial equations 11.2.6 Examples 11.3 The tunnel effect 11.3.1 The double well 11.3.2 The semi-classical treatment of decay processes 11.3.3 The Gamow–Siegert theory 11.4 Phase shift Further reading Guide to the Supplements Problems Numerical analyses 265 265 268 271 273 275 276 276 279 282 283 285 289 292 295 300 300 301 302 III 303 Applications 12 Time evolution 307 12.1 General features of time evolution 307 12.2 Time-dependent unitary transformations 309 12.3 Adiabatic processes 311 12.3.1 The Landau–Zener transition 313 12.3.2 The impulse approximation 315 12.3.3 The Berry phase 316 12.3.4 Examples 318 12.4 Some nontrivial systems 320 12.4.1 A particle within moving walls 320 12.4.2 Resonant oscillations 324 12.4.3 A particle encircling a solenoid 327 12.4.4 A ring with a defect 328 12.5 The cyclic harmonic oscillator: a theorem 331 12.5.1 Inverse linear variation of the frequency 335 12.5.2 The Planck distribution inside an oscillating cavity 336 12.5.3 General power-dependent frequencies 338 12.5.4 Exponential dependence 339 12.5.5 Creation and annihilation operators; coupled oscillators 340 Guide to the Supplements 341 Problems 341 Numerical analyses 342 13 Metastable states 343 xiv Contents 13.1 Green functions 343 13.1.1 Analytic properties of the resolvent 345 13.1.2 Free particles 349 13.1.3 The free Green function in general dimensions 351 13.1.4 Expansion in powers of HI 352 13.2 Metastable states 356 13.2.1 Formulation of the problem 356 13.2.2 The width of a metastable state; the mean halflifetime 358 13.2.3 Formal treatment 361 13.3 Examples 368 13.3.1 Discrete–continuum coupling 368 13.4 Complex scale transformations 370 13.4.1 Analytic continuation 372 13.5 Applications and examples 374 13.5.1 Resonances in helium 375 13.5.2 The potential V0 r2 e−r 375 13.5.3 The unbounded potential; the Lo Surdo–Stark effect376 Further reading 379 Problems 379 Numerical analyses 379 14 Electromagnetic interactions 14.1 The charged particle in an electromagnetic field 14.1.1 Classical particles 14.1.2 Quantum particles in electromagnetic fields 14.1.3 Dipole and quadrupole interactions 14.1.4 Magnetic interactions 14.1.5 Relativistic corrections: LS coupling 14.1.6 Hyperfine interactions 14.2 The Aharonov–Bohm effect 14.2.1 Superconductors 14.3 The Landau levels 14.3.1 The quantum Hall effect 14.4 Magnetic monopoles Guide to the Supplements Problems Numerical analyses 381 381 381 383 385 388 388 390 392 395 397 399 401 404 404 404 15 Atoms 405 15.1 Electronic configurations 405 15.1.1 The ionization potential 408 15.1.2 The spectrum of alkali metals 410 15.1.3 X rays 410 15.2 The Hartree approximation 412 15.2.1 Self-consistent fields and the variational principle 415 15.2.2 Some results 417 15.3 Multiplets 418 Contents xv 15.3.1 Structure of the multiplets 15.4 Slater determinants 15.5 The Hartree–Fock approximation 15.5.1 Examples 15.6 Spin–orbit interactions 15.6.1 The hydrogen atom 15.7 Atoms in external electric fields 15.7.1 Dipole interaction and polarizability 15.7.2 Quadrupole interactions 15.8 The Zeeman effect 15.8.1 The Zeeman effect in quantum mechanics Further reading Guide to the Supplements Problems Numerical analyses 16 Elastic scattering theory 16.1 The cross section 16.2 Partial wave expansion 16.2.1 The semi-classical limit 16.3 The Lippman–Schwinger equation 16.4 The Born approximation 16.5 The eikonal approximation 16.6 Low-energy scattering 16.7 Coulomb scattering: Rutherford’s formula 16.7.1 Scattering of identical particles Further reading Guide to the Supplements Problems Numerical analyses 419 424 427 430 433 436 438 438 442 443 444 450 451 451 452 453 453 457 459 460 461 463 465 468 473 474 475 476 476 17 Atomic nuclei and elementary particles 477 17.1 Atomic nuclei 477 17.1.1 General features 477 17.1.2 Isospin 478 17.1.3 Nuclear forces, pion exchange, and the Yukawa potential 480 17.1.4 Radioactivity 482 17.1.5 The deuteron and two-nucleon forces 483 17.2 Elementary particles: the need for relativistic quantum field theories 485 17.2.1 The Klein–Gordon and Dirac equations 487 17.2.2 Quantization of the free Klein–Gordon fields 490 17.2.3 Quantization of the free Dirac fields and the spin– statistics connection 491 17.2.4 Causality and locality 492 17.2.5 Self-interacting scalar fields 494 17.2.6 Non-Abelian gauge theories: the Standard Model 495 xvi Contents Further reading IV Entanglement and Measurement 496 499 18 Quantum entanglement 503 18.1 The EPRB Gedankenexperiment and quantum entanglement 503 18.2 Aspect’s experiment 508 18.3 Entanglement with more than two particles 511 18.4 Factorization versus entanglement 512 18.5 A measure of entanglement: entropy 514 Further reading 516 19 Probability and measurement 517 19.1 The probabilistic nature of quantum mechanics 517 19.2 Measurement and state preparation: from PVM to POVM 519 19.3 Measurement “problems” 521 19.3.1 The EPR “paradox” 522 19.3.2 Measurement as a physical process: decoherence and the classical limit 525 19.3.3 Schrödinger’s cat 527 19.3.4 The fundamental postulate versus Schrödinger’s equation 529 19.3.5 Is quantum mechanics exact? 530 19.3.6 Cosmology and quantum mechanics 531 19.4 Hidden-variable theories 532 19.4.1 Bell’s inequalities 532 19.4.2 The Kochen–Specker theorem 535 19.4.3 “Quantum non-locality” versus “locally causal theories” or “local realism” 538 Further reading 539 Guide to the Supplements 539 V Supplements 541 20 Supplements for Part I 545 20.1 Classical mechanics 545 20.1.1 The Lagrangian formalism 545 20.1.2 The Hamiltonian (canonical) formalism 547 20.1.3 Poisson brackets 549 20.1.4 Canonical transformations 550 20.1.5 The Hamilton–Jacobi equation 552 20.1.6 Adiabatic invariants 552 20.1.7 The virial theorem 554 20.2 The Hamiltonian of electromagnetic radiation field in the vacuum 554 Contents xvii 20.3 Orthogonality and completeness in a system with a onedimensional delta function potential 20.3.1 Orthogonality 20.3.2 Completeness 20.4 The S matrix; the wave packet description of scattering 20.4.1 The wave packet description 20.5 Legendre polynomials 20.6 Groups and representations 20.6.1 Group axioms; some examples 20.6.2 Group representations 20.6.3 Lie groups and Lie algebras 20.6.4 The U (N ) group and the quarks 20.7 Formulas for angular momentum 20.8 Young tableaux 20.9 N -particle matrix elements 20.10 The Fock representation 20.10.1 Bosons 20.10.2 Fermions 20.11 Second quantization 20.12 Supersymmetry in quantum mechanics 20.13 Two- and three-dimensional delta function potentials 20.13.1 Bound states 20.13.2 Self-adjoint extensions 20.13.3 The two-dimensional delta-function potential: a quantum anomaly 20.14 Superselection rules 20.15 Quantum representations 20.15.1 Weyl’s commutation relations 20.15.2 Von Neumann’s theorem 20.15.3 Angular variables 20.15.4 Canonical transformations 20.15.5 Self-adjoint extensions 20.16 Gaussian integrals and Feynman graphs 21 Supplements for Part II 21.1 Supplements on perturbation theory 21.1.1 Change of boundary conditions 21.1.2 Two-level systems 21.1.3 Van der Waals interactions 21.1.4 The Dalgarno–Lewis method 21.2 The fine structure of the hydrogen atom 21.2.1 A semi-classical model for the Lamb shift 21.3 Hydrogen hyperfine interactions 21.4 Divergences of perturbative series 21.4.1 Perturbative series at large orders: the anharmonic oscillator 21.4.2 The origin of the divergence 21.4.3 The analyticity domain 556 557 558 560 560 564 566 566 568 570 573 575 581 584 586 586 588 589 590 595 597 598 599 601 604 605 605 606 608 610 611 615 615 615 616 618 619 621 626 630 633 633 635 636 xviii Contents 21.4.4 Asymptotic series 21.4.5 The dispersion relation 21.4.6 The perturbative–variational approach 21.5 The semi-classical approximation in general systems 21.5.1 Introduction 21.5.2 Keller quantization 21.5.3 Integrable systems 21.5.4 Examples 21.5.5 Caustics 21.5.6 The KAM theorem and quantization 638 642 645 648 648 650 653 654 655 655 22 Supplements for Part III 22.1 The K 0 –K 0 system and CP violation 22.2 Level density 22.2.1 The free particle 22.2.2 g(E) and the partition function 22.2.3 g(E) and short-distance behavior 22.2.4 Level density and scattering 22.2.5 The stabilization method 22.3 Thomas precession 22.4 Relativistic corrections in an external field 22.5 The Hamiltonian for interacting charged particles 22.5.1 The interaction potentials 22.5.2 Spin-dependent interactions 22.5.3 The quantum Hamiltonian 22.5.4 Electron–electron interactions 22.5.5 Electron–nucleus interactions 22.5.6 The 1/M corrections 22.6 Quantization of electromagnetic fields 22.6.1 Matrix elements 22.7 Atoms 22.7.1 The Thomas–Fermi approximation 22.7.2 The Hartree approximation 22.7.3 Slater determinants and matrix elements 22.7.4 Hamiltonians for closed shells 22.7.5 Mean energy 22.7.6 Hamiltonians for incomplete shells 22.7.7 Eigenvalues of H 22.7.8 The elementary theory of multiplets 22.7.9 The Hartree–Fock equations 22.7.10 The role of Lagrange multipliers 22.7.11 Koopman’s theorem 22.8 H2+ 22.9 The Gross–Pitaevski equation 22.10 The semi-classical scattering amplitude 22.10.1 Caustics and rainbows 657 657 661 665 666 669 671 673 674 676 678 679 681 682 682 683 686 687 689 692 693 700 703 707 712 712 716 717 719 721 723 726 729 731 732 23 Supplements for Part IV 735 Contents xix 23.1 Speakable and unspeakable in quantum mechanics 23.1.1 Bell’s toy model for hidden variables 23.1.2 Bohm’s pilot waves 23.1.3 The many-worlds interpretation 23.1.4 Spontaneous wave function collapse 735 735 736 739 740 24 Mathematical appendices and tables 24.1 Mathematical appendices 24.1.1 Laplace’s method 24.1.2 The saddle-point method 24.1.3 Airy functions 743 743 743 744 748 References 765 Index 775