www.elsolucionario.net www.elsolucionario.net MODERN QUANTUM MECHANICS Second Edition www.elsolucionario.net www.elsolucionario.net MODERN QUANTUM MECHANICS Second Edition s ·Addison.:wesle- y Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo www.elsolucionario.net Publisher: Jim Smith Director of Development: Michael Gillespie Editorial Manager: Laura Kenney Senior Project Editor: Katie Conley Editorial Assistant: Dyan Menezes Managing Editor: Corinne Benson Production Project Manager: Beth Collins Production Management, Composition, and Art Creation: Techsetters, Inc Copyeditor: Connie Day Cover Designer: Blake Kim; Seventeenth Street Studios Photo Editor: Donna Kalal Manufacturing Buyer: Jeff Sargent Senior Marketing Manager: Kerry Chapman Cover Photo Illustration: Blake Kim Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this textbook appear on the appropriate page within the text Copyright© 1994, 201 Pearson Education, Inc., publishing as Addison-Wesley, 1301 Sansome Street, San Francisco, CA 941 1 All rights reserved Manufactured in the United States of America This publication is protected by Copyright and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, 1900 E Lake Ave., Glenview, IL 60025 For information regarding permissions, call (847) 486-2635 Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps Library of Congress Cataloging-in-Publication Data Sakurai, J J (Jun John), 1933-1982 Modern quantum mechanics - 2nd ed I J.J Sakurai, Jim Napolitano p cm ISBN 978-0-8053-8291-4 (alk paper) Quantum theory-Textbooks I Napolitano, Jim II Title QC174 12.S25 201 530 12 dc22 2010022349 ISBN 10: 0-8053-8291-7; ISBN 13: 978-0-8053-8291-4 10-CRK-14 13 12 1 10 Addison-Wesley is an imprint of I PEARSON www.pearsonhighered.com www.elsolucionario.net Contents Foreword to the First Edition IX Preface to the Revised Edition XI Preface to the Second Edition XIII In Memoriam XVII • Fundamental Concepts 1.1 The Stem-Gerlach Experiment Kets, Bras, and Operators Base Kets and Matrix Representations Measurements, Observables, and the Uncertainty Relations Change of Basis 35 Position, Momentum, and Translation 40 Wave Functions in Position and Momentum Space 50 23 66 • Quantum Dynamics 2.2 2.3 2.4 2.5 2.6 2.7 Time-Evolution and the Schrodinger Equation 66 The Schrodinger Versus the Heisenberg Picture 80 Simple Harmonic Oscillator 89 SchrOdinger's Wave Equation 97 Elementary Solutions to SchrOdinger's Wave Equation Propagators and Feynman Path Integrals 16 Potentials and Gauge Transformations 29 103 • Theory of Angular Momentum Rotations and Angular-Momentum Commutation Relations 3.1 Spin Systems and Finite Rotations 63 3.2 3.3 S0(3), SU(2), and Euler Rotations 172 � 57 157 v www.elsolucionario.net Contents VI 3.5 3.6 3.7 3.8 3.9 3.10 3.1 Density Operators and Pure Versus Mixed Ensembles 178 Eigenvalues and Eigenstates of Angular Momentum Orbital Angular Momentum 99 Schrodinger's Equation for Central Potentials 207 Addition of Angular Momenta 217 Schwinger's Oscillator Model of Angular Momentum 232 Spin Correlation Measurements and Bell's Inequality 238 Tensor Operators 246 • Symmetry in Quantum Mechanics 4.2 4.3 4.4 Symmetries, Conservation Laws, and Degeneracies 262 Discrete Symmetries, Parity, or Space Inversion 269 Lattice Translation as a Discrete Symmetry 280 The Time-Reversal Discrete Symmetry 284 262 303 • Approximation Methods 5.1 Time-Independent Perturbation Theory: Nondegenerate Case 303 5.2 Time-Independent Perturbation Theory: The Degenerate Case 16 Hydrogen-Like Atoms: Fine Structure and the Zeeman Effect 321 5.3 5.4 Variational Methods 332 5.5 Time-Dependent Potentials: The Interaction Picture 336 5.6 Hamiltonians with Extreme Time Dependence 345 5.7 Time-Dependent Perturbation Theory 355 Applications to Interactions with the Classical Radiation Field 365 5.8 5.9 Energy Shift and Decay Width 371 • Scattering Theory • Identical Particles 6.1 6.2 6.3 6.4 6.5 6.6 6.8 6.9 7.1 7.2 Scattering as a Time-Dependent Perturbation 386 The Scattering Amplitude 391 The Born Approximation 399 Phase Shifts and Partial Waves 404 Eikonal Approximation 417 Low-Energy Scattering and Bound States 423 Resonance Scattering 430 Symmetry Considerations in Scattering 433 Inelastic Electron-Atom Scattering 436 Permutation Symmetry 446 Symmetrization Postulate 450 386 446 www.elsolucionario.net vii Contents 7.3 7.4 7.5 Two-Electron System 452 The Helium Atom 455 Multiparticle States 459 Quantization of the Electromagnetic Field • Relativistic Quantum Mechanics Paths to Relativistic Quantum Mechanics 8.1 8.2 The Dirac Equation 494 Symmetries of the Dirac Equation 501 Solving with a Central Potential 506 8.4 Relativistic Quantum Field Theory 8.5 A • Electromagnetic Units A A.2 Coulomb's Law, Charge, and Current Converting Between Systems 520 472 486 19 B • Brief Summary of Elementary Solutions to Schrodinger's Wave Equation B.l B B.3 B B.5 B.6 Free Particles ( V 0) 523 Piecewise Constant Potentials in One Dimension 524 Transmission-Reflection Problems 525 Simple Harmonic Oscillator 526 The Central Force Problem [Spherically Symmetrical Potential V = V(r)] 527 Hydrogen Atom 486 519 523 = C • Proof of the Angular-Momentum Addition Rule Given by Equation (3.8.38) 533 Bibliography 535 Index 537 www.elsolucionario.net www.elsolucionario.net Index A Abelian, definition of, 47 Absorption, in classical radiation fields, 365-367 Absorption-emission cycle, 341-342 Adiabatic approximation, 346-348 Aharonov-Bohm effect, 141-145, 353-355 Airy function, 109-1 10, 1 3-1 15 Alkali atoms, 323-326 Ambler, E., 278 Ampere (unit), Ampere's law, 521 Amplitude(s) Born, 400, 419, 443, 523 and bound states, 429-430 correlation, 78-80 partial-wave, 410 scattering, see Scattering amplitude transition, 86-89, 120-122, 387 Anderson, Carl, 500 Angular integration, in helium atom, 456 Angular momentum, 57-255 and angular-velocity vector, 5-6 commutation relations for, 57-163 density operator and ensembles for, 178-19 Dirac equation for, 501-502 orbital, see Orbital angular momentum rotations and commutation relations in, 57-172 and Schrodinger's equation for central potentials, 207-2 17 Schwinger's oscillator model of, 232-238 of silver atoms, 23 and S0(3)/SU(2)/Euler rotations, 172-178 spin correlation measurements and Bell's inequality for, 238-245 tensor operator for, 246-255 and uncoupled oscillators, 232-235 Angular-momentum addition, 217-23 Clebsch-Gordan coefficients for, 223-23 examples of, 8-221 formal theory of, 221-224 and rotation matrices, 230-23 rule for, 533-534 Angular-momentum barriers, 208, 209 Angular-momentum commutation relations and eigenvalues/eigenstates, 1-192 and ladder operator, 92 and rotations, 57-163 x matrix realizations, 169 Angular-momentum eigenkets, 93-194 Angular-momentum eigenvalues and eigenstates and commutation relations/ladder operator, 19 1-192 constructing, 93-195 and matrix elements of angular-momentum operator, 95-196 and rotation operator, 196-199 and time reversal, 298 and Wigner-Eckart theorem and, 252-253 Angular-momentum operator, , 95-196, 258 Angular velocity vector, angular momentum and, 5-6 Annihilation operator, 89-9 , 97, 152, 232-233, 465 Anomalous Zeeman effect, 328 Anticommutation relations, 28, 469 Antilinear operator, 287, 29 1-292 Antiparticles, in Klein-Gordon equation, 493, 494, 503 Antisymmetrical states, 275 Antiunitary operator, 287, 291 , 296, 434-436, 504-505 Anyons, 450n Approximation methods, 303-375 for classical radiation field, 365-371 for degenerate energy eigenkets, 6-321 for energy shifts and decay widths, 37 1-375 for hydrogen-like atoms, 321-336 537 538 www.elsolucionario.net Index for nondegenerate energy eigenkets, 303-3 16 for time-dependent Hamiltonians, 345-355 time-dependent perturbation theory, 355-365 for time-dependent potentials, 336-345 time-independent perturbation theory, 303-321 variational, 332-336 Argand diagram, 413 Argon, Ramsauer-Townsend effect and, 425-426 Associative axiom of multiplication, 16-17 Atom(s), See also specific types Bohr, one-electron, 0-5 14 polarizability of, 297 Atomic force microscope, 479-480 Atomic spectroscopy, 163 Axial vectors, 272 B Baker-Hausdorff formula, 95 Balmer formula, 216, Balmer splittings, fine structure splittings and, 326 Base kets, 17-20 change of basis in, 35-36 eigenkets as, 8-19 and eigenkets of observables, 17-18 in Heisenberg and SchrOdinger pictures, 86-89 and spin � systems, 22 in spin � systems, 22-23 and transition amplitudes, 86-89 Basis change of, 35-40 position, 52-53 Baym, G., 250 Bell's inequality, 241-245 and Einstein's locality principle, 241-243 and quantum mechanics, 243-245 Bennett, G W., 76 Berry, M V., 348 Berry's Phase and gauge transformations, 353-355 and time-dependent Hamiltonians, 348-353 Bessel functions properties of, 529-530 spherical, 210-2 1 Bethe, H A., 439 Biedenham, L C., 232 Big box normalization, 104, 388-389 Bitter, T., 352 Bloch, F , 439 Bloch's theorem, 283 Bohr, N., 73, 397, 440 Bohr atom, Bohr model, 216 Bohr radius, 217 Boltzmann constant, 87, 487 Born, M., 1, 48, 89, 99, 191 Born amplitude, first-order, 400, 419, 443, 523 Born approximation, 399-404, 442 Bose-Einstein condensation, 452, 464 Bose-Einstein statistics, 450 Bosons, 450-452, 462-464, 476 Bouncing ball example, 10 Bound states, 423-43 and amplitude, 429-430 and low-energy scattering, 423-430 quasi-, 43 and zero-energy scattering, 426-429 Bowles, T J., 450 Bra, matrix representation of, 21 Bra-ket algebra, 59 Bra-ket notation, Dirac, 292 Bra space, 12-14 Breit-Wigner Formula, 433 Bressi, G., 480 Brillouin, L., 1 Brillouin zone, 284 c Cannonical (fundamental) commutation relations, 48-49 Canonical ensembles, 89-190 Canonical momentum, 36, 138, 140, 262 Cartesian tensors, 247-250 Casimir effect, 476-480 Cauchy principal value, 397 Cayley-Klein parameters, 174 Central force problem, Schrodinger wave equation and, 527-5 Central potentials, 506-5 14 in eigenvalue problem, 506-5 10 and Hamiltonians, 207, 1 for one-electron atom, 10-5 14 Schrodinger equation for, see Schrodinger equation for central potentials solving for, 506-5 14 Cesium atoms, spin manipulation of, 10 CGS system of units, 519 Charge, units for, 9-520 Charge conjugation, 503-504, 506 Chiao, R., 351 Classical physics, symmetry in, 262-263 Classical radiation field, 365-371 absorption and stimulated emission in, 365-367 electric dipole approximation for, 367-369 photoelectric effect in, 369-371 Clebsch-Gordan coefficients, 220 properties of, 223-224 www.elsolucionario.net 539 Index recursion relations for, 224-229 and rotation matrices, 230-23 and tensors, 25 1-253 Clebsch-Gordan series, 230-23 Clebsch-Gordan series formula, 25 Closure, Cobalt atoms parity nonconservation for, 278-279 transition energy of, Coherent state for annihilation operator, 97 in quantum optics, 48 Collective index, 30, 14 Column vector function, 491 Commutation relations, 28 angular-momentum, 57-163, 169, 1-192 cannonical, 48 49 and eigenvalues/eigenstates, 191-192 in second quantization, 462 463 Commutators, 48 49, 64, 85 Compatible observables, 28-3 Completely random ensembles, 179, 186 Completeness relation, Complex conjugate transposed, 20 Complex contour integration, 392-394, 397-398 Complex numbers, quantum mechanics and, 27 Complex vector space, spin states and, Compton effect, Compton wavelength, 489 Confluent Hypergeometric Function, 215 Conservation laws, 262-263 Conserved current, 492, 496 497, 16 Conserved Vector Current (CVC) hypothesis, 449 450 Constant perturbation, 359-363 Constant potential and gauge transformations, 29- in one dimension, 524-525 Continuity equation, 496 Continuous spectra, 40 41 Continuous symmetry, 262-263, 265-269 Continuum generalizations, for density operator, 85-186 Correlation amplitude, energy-time uncertainty relation and, 78-80 Correlation function, Coulomb (unit), 520 Coulomb gauge, 473 Coulomb potential first-order energy shift for, 327 and Schr6dinger's equation for central potentials, 3-21 screened, 467 symmetry in, 265-269 Coulomb's law, 19, 520 Covariant derivative, 49 Covariant Dirac equation, 494, 495 Covariant vector operator, 489, 490n Covariant wave equations, 488, 489 CPT operator combination, 506 Creation operator, 89-9 , 52, 232-233, 465 Cross sections, for scattering, 88-389 Current conserved, 492, 496 497, 16 eve hypothesis, 449 450 units of, 19-520 Cutoff frequency, Casimir effect and, 477, 480 CVC (Conserved Vector Current) hypothesis, 449 450 D Dalgarno, A., Davisson-Germer-Thompson experiment, de Broglie, L., 46, 66, 99 de Broglie's matter waves, Decay width, energy shift and, 371-375 Degeneracy, 59 of eigenvalues, 29, 217 exchange, 447 Kramers, 299 and symmetries, 264-265 Degenerate electron gases, 467 472 Degenerate energy eigenkets, 6-321 Degenerate time-independent perturbation theory, 6-321 Density matrix, of completely random ensemble, 86 and continuum generalizations, 85-1 86 Density of states, for free particles, 105 Density operator, 80-1 continuum generalizations for, 85-1 86 definition of, and ensemble averages, 80-185 Hermitian, 82 and pure/mixed ensembles, 178-19 and quantum statistical mechanics, 86-1 time evolution of, 257 and time evolution of ensembles, 185 Detailed balance, 365 , 436 Deuterium atom, energy levels of, 17-5 Diagonalization, 38-39, 64, 90 Diamagnetic susceptibility, 380 Dipole operator, 368 www.elsolucionario.net 540 Index Dirac, P A M., 1, 10-1 1, 23, 49, 50, 83, 14, 24-125, 148, 356, 362, 494 Dirac bra-ket notation, 292 Dirac function, 40 Dirac equation, 494-507 for angular momentum, 501-502 for central potentials, 507 and charge conjugation, 503-504 conserved current in, 496-497 and CPT operator combination, 506 described, 494-496 and electromagnetic interactions, 500-501 free-particle solutions of, 497-499 and negative energies, 499-500 parity of, 502-503 symmetries of, 501-506 time-reversal symmetry of, 504-505 Dirac Hamiltonians, 495, 501 Dirac notation, 8, 223 Dirac picture, 338 Dirac quantization condition, 354-355 Direction eigenkets, 202-203 Discrete symmetries, 269-300, see also specific types and Dirac equation, 504-505 lattice translation as, 280-284 parity as, 269-280 properties of symmetry operations, 287-289 time-reversal discrete, 284-300 Dispersion, 33-34 Double-bar matrix element, 252 Dual correspondence, Dubbers, D , 352 Dyadic tensors, 247-248 Dynamical variables, in second quantization approach, 463-467 Dyson, F J., , 357 Dyson series, 71, 355-357 E Effective potential, 208, 209 Ehrenfest, P., 86 Ehrenfest's theorem, 86, 32, 136 Eichinvarianz, 141 Eigenbras, 12-13 Eigenfunctions, , 523 Eigenkets angular-momentum, 93-194 and base kets, 17-19 direction, 202-203 and eigenbras, 2-13 energy, see Energy eigenkets and Hermitian operator, 59 and observables, 17-18 parity, 273 position, 41-42 and simple harmonic oscillator, 89-93 simultaneous, 30 in spin i systems, 12 zeroth-order, Eigenspinors, 296 Eigenstates angular-momentum, see Angular-momentum eigenvalues and eigenstates energy, 96, 273-274 mass, 77 in spin i systems, 12 zeroth-order, 377 Eigenvalues angular-momentum, see Angular-momentum eigenvalues and eigenstates and central potential, 506-5 10 degeneracy of, 29, 217 energ� 77-78, 89-93, 217 and energy eigenkets, 71 and expectation values, 24-25 and Hermitian operator, 17 of hydrogen atom, 268 and orbital angular momentum squared, 30 and simple harmonic oscillator, 89-93 in spin ! systems, Eikonal approximation, 417-423 described, 417-420 and partial waves, 420-423 Einstein, A., 241 Einstein-Debye theory, Einstein-Podolsky-Rosen paradox, 241 Einstein's locality principle, 241-243 Elastic scattering, 436, 445 Electric dipole approximation, 367-369 Electric fields, time-reversal symmetry and, 298-300 Electromagnetic fields and Casimir effect, 480 and Dirac equation, 500-501 energy of, 474 and momentum, 480-48 polarization vectors of, quantization of, see Quantization of electromagnetic field Electromagnetic units, 9-522 Electromagnetism, gauge transformations in, 134-141 Electron-atom scattering, inelastic, 436-44 Electron gases, degenerate, 467-472 Electron spin, magnetic moment and, 2-4 Emission, in classical radiation fields, 365-367 Energy(-ies) of electromagnetic field, 474 Fermi, 464, 470 of free particles, 487-488 kinetic, 321-323 www.elsolucionario.net 541 Index negative, 492-494, 499-500 quantization of, 475-476 transition, 17 zero-point (vacuum), 476 Energy eigenkets degenerate, 6-321 nondegenerate, 303-3 16 and simple harmonic oscillator, 89-93 and time-evolution operator, 1-73 Energy eigenstates parity properties of, 273-274 superposition of, 96 Energy eigenvalues degeneracy of, 217 of neutrinos, 77-78 and simple harmonic oscillator, 89-93 Energy levels, of hydrogen and deuterium atoms, 3-5 14, 17-5 Energy shifts for Coulomb potentials, 327 and decay width, 37 1-375 Energy-time uncertainty relation, correlation amplitude and, 78-80 Ensemble average definition of, 80-1 and density operator, 80-184 Ensembles, 178-185 canonical, 89-190 completely random, 79, 86 mixed, 80 and polarized vs unpolarized beams, 178-180 pure, 24, 179, 180 time evolution of, 185 Entropy, 187 Equation of motion Euler, 256 Heisenberg, 82-84, 94, 256, 263 Euclidian space, 34 Euler angle notation, 236 Euler-Maclaurin summation formula, 478 Euler rotations, 175-178, 256 Exchange degeneracy, 447 Exchange density, 454 Expectation values, 24-25, 164-165 and Hermitian operator, 34-35 time dependence of, 73 F Faraday's law, 521 Feenberg, Eugene, 397 Fermi-Dirac statistics, 450, 484-485 Fermi energy, 464, 470 Fermions, 450-452, 462-465 Fermi's golden rule, 362, 387, 388 Feshbach, H., 19 Fetter, Alexander L., 467, 469, 15 Feynman, R P., 122, 24, 5 Feynman's formulation, 123-129 Feynman's path integral, 27-129, 143, 5 Filtration, 25 Fine structure, 323-327, 0, 17-5 Finite-range potentials, 394-395 Finite rotations, 166-172 and infinitesimal rotations, 157-160 and neutron interferometry, 166-168 noncommutativity of, 57-158 Pauli two-component formalism for, 168-172 rotation operator for spin -! systems, 163-165 and spin -! systems, 163-172 Finite square wells, 400-40 Finkelstein, R J., 155 Pock, V., 136 Pock space, 46 Form factor, 439 Fortun, M., 476 Fourier decomposition, 375 Fourier inversion formula, 375 Fourier transform, 438 Fractional population, 79 Franck-Hertz experiment, Frauenfelder, H., 298 Free particles and Dirac equation, 497-499 energy of, 487-488 in Heisenberg and Schrodinger pictures, 84-86 and infinite spherical well, 210-2 1 scattering by, 404-409 and Schri:idinger wave equation, 103-105, 523-524 in three dimensions, 103-105 Fundamental commutation relations, 48-49 G Garvey, G T., 450 Gauge invariance, 141 Gauge transformations and Berry's Phase, 353-355 and constant potentials, 129-1 definition of, 130 and electromagnetism, 34-141 Gaussian potential, 444 Gaussian system of units, 9-522 Gaussian wave packets, 55-57, 62, 65, 99-100, 1 8-1 19 Gauss's law, 146, 520-521 Gauss's theorem, 1 Generating functions, 105-108 Geometric phase, 348-353 Gerlach, W., Glauber, Roy, 48 Goldstein, H., 37, 176, 264 Gottfried, K., 25, 52, 33 , 378, 379 Gravity, quantum mechanics and, 1-134 www.elsolucionario.net 542 Index Green's function, 1 8, 394, 404, 442 Griffiths, D J., 346 H Hamilton, W R., 99 Hamiltonian matrix, for two-state systems, 378 Hamiltonian operator, 148-150 for simple harmonic oscillator, 89-90 time-dependent, 70-71 and time-dependent wave equation, 97, 98 and time-evolution operator, 69 time-independent, 70 and two-state systems, 60 Hamiltonians, see also Time-dependent Hamiltonians and central potentials, 207, 211 Dirac, 495, 501 Hamilton-Jacobi theory, 102, 154, Hamilton's characteristic function, 103 Hankel functions, 414, 529, 530 Hard-sphere scattering, 416 423 Harmonic oscillators, 21 1-214, 376, see also Simple harmonic oscillator Harmonic perturbation, 363-365 Heisenberg, W., 1, 46, 48, 99, 191 Heisenberg equation of motion, 82-84, 94, 256, 263 Heisenberg picture, 148-150 and base kets, 86-89 free particles in, 84 and Heisenberg equation of motion, 82-84 and propagators, 20-121 and Schrodinger picture, 80-89 state kets and observables in, 82 and time-dependent potentials, 337-339 and time-evolution of ensembles, 185 unitary operator in, 80-8 Heisenberg uncertainty principle, 3, 56 Helium, 452, 455 459, 483 Helmholtz equation, 394, 404 Henley, E M., 298 Hermite polynomials, 106-108, 527 Hermitian adjoint, 15 Hermitian operator, 63-64, 150 anticommute, 61 definition of, 44 and density operator/ensembles, 82-1 83 and Ehrenfest's theorem, 84 eigenvalues of, 17 and energy eigenkets, 89 expectation values of, 34 and infinitesimal rotations, 161 and simple harmonic oscillators, 95, 97 in spin systems, 26 as time-evolution operator, 69 and time-reversal operator, 292, 298 Hermiticity, 39, 82 Higher-order Born approximation, 403 404 Hilbert, D., 1 , 99 Hilbert space, 1 Holstein, B R., 349 Hooke's law, 89 Hydrogen atom eigenvalues of, 268 energy levels of, 3-514, 7-5 and linear Stark effect, 9-321 polarizability of, and Schrodinger wave equation, 531-532 Hydrogen-like atoms, 32 1-336 and fine structure, 323-326 fine structure of, relativistic correction to kinetic energy for, 321-323 spin-orbit and fine structure of, 323-327 van der Waals interaction in, 33 1-332 and Zeeman effect, 328-33 I Identical particles, 446 483 and helium atoms, 455 459 in multiparticle states, 459 472 permutation symmetry for, 446 450 and quantization of electromagnetic field, 472 483 symmetrization postulate for, 450 452 in two-electron systems, 452 455 Identity matrix, 5 Identity operator, 19, 22, 28 Incoherent mixtures, 179 Incompatible observables, 28-29, 1-33 Inelastic electron-atom scattering, 436 441 Inertia, moment of, computation of, 5-6 Infinitesimal rotation operator, , 199-200 Infinitesimal rotations, 57-163 commutativity of, 159 and finite rotation, 157-160 and quantum mechanics, 160-163 and vector operator, 246 Infinitesimal time-evolution operator, 68 Infinitesimal translation, 42 43 Infinite spherical well, free particles in, 210-2 1 www.elsolucionario.net 543 I ndex Inner products, 13 Integral equation for scattering, 392-396 Interaction picture, 337-339 Irreducible tensors, 247-250 Isomers, optical, 277 Isospin, 235 Isotropic harmonic oscillator, 21 1-214, 376 J Jackson, J D., 324, 369 Jacobi identity, 49 Jaffe, R L., 480 Jenkins, D A., Jordan, P., 48, 99, 191 K KamLAND experiment, 78 Kepler problem, 265 Kets, 8, see also Base kets; Eigenkets definition of, 1 and electromagnetic field polarization vectors, normalized, 14 null, 1 and operator, 14-15 perturbed, normalization of, 0-3 1 spin, 165 state, 67-68, 82 vacuum, 232-233 Ket space, 1-15, 63 Kinematic momentum, 136, 138, 140 Kinetic energy, relativistic correction for, 321-323 Klein-Gordon equation, 488-494 Kramers, H A., 10 Kramers degeneracy, 299 Kronecker symbol, 40, 469 Krypton, Ramsauer-Townsend effect and, 425-426 Kummer's equation, 215, 259 Kunselman, R., 17 L Ladder operator, angular momentum commutation relations and, 1-192 Lagrange equation, 262 Lagrangian, classical, 123, 143 Laguerre polynomials, 259, 53 Lamb shift, 321, 379, Lamoreaux, S., 476 Landau, Rubin, 46 1, 467, Lande's interval rule, 325-326 Laplace-Fourier transform, 120 Laporte's rule, 278 Lattice translation, as discrete symmetry, 280-284 Lattice translation operator, 281-282 Legendre function, 443 Lenz vector, 265 Lewis, J T., Light, polarization of, 6-10 Linear potential, 08-1 Linear Stark effect, 9-321 Liouville's theorem, 85 Lipkin, H J., 148 Lippmann-Schwinger equation, 390-39 , 442, 444 Local potentials, 394 London, F , 136 Lorentz contraction factor, 497 Lorentz force, 136, 143, 285 Lorentz force law, 490n, 521 Lorentz invariance, 506 Lorentz transformations, 489 Loudon, R., 472 Low-energy scattering, 423-429 M Magnetic fields and Aharonov-Bohm effect, 353-354 and Stem-Gerlach experiment, 2-4 and time-reversal discrete symmetry, 298-300 Magnetic flux, fundamental unit of, 144 Magnetic moment, 2-4, 501 Magnetic monopoles, 145-148, 353-355 Marcus, George E., 476 Masers, 344-345 Mass eigenstates, 77 Matrices, see specific types Matrix elements of angular-momentum operator, 195-196 double bar, 252 reduced, 255 tensors, 252-255 Matrix mechanics, 48 Matrix representations, 20-23 Matter waves, de Broglie's, Maxwell-Boltzmann statistics, 45 Maxwell's equations, 145, 285, 472-475, 521 McKeown, R D., 449, 450 Mean square deviation, 34 Measurements position, 41-42 quantum theory of, 23-25 selective, 25 spin-correlation, 238-245 Melissinos, A., 351 Merzbacher, E., 5, 377, 379, 380, 46 , 467, 472, 5 Minimum uncertainty wave packets, 56 Mixed ensembles, 80 MKS system of units, Momentum, see also Angular momentum canonical, 136, 138, 140, 262 definition of, 52 and electromagnetic field, 480-48 kinematic, 136, 138, 140 position-momentum uncertainty relation, 46 and translation generation, 45-48 Momentum operator, 52-53, 58, 64 Momentum-space wave function, 53-55, 65, Morse, P M., 1 544 Index Motion Euler equation of, 256 Heisenberg equation of, 82-84, 94, 256, 263 Multiparticle states, 459-472 degenerate electron gases as, 467-472 described, 459-460 second quantization approach, 460-467 Multiplication, of operators, 5-17, 250-25 Muons, spin precession of, 76-77, 166 N National Institute of Standards and Technology (NIST), 17-5 18 Natural units, 487 Negative energies and Dirac equation, 499-500 relativistic quantum mechanics, 492-494 Neutrino oscillations, 77-78 Neutron interferometry, 56, 66-168 Neutrons, ultra-cold, 352-353 Newton, R G., 397 Newton's second law, 86, 129, 144-145 NIST (National Institute of Standards and Technology), 17-5 No-level crossing theorem, Non-Abelian, definition of, 62 Nonconservation of parity, 278-279 Nondegenerate time-independent perturbation theory, 303-3 16 Nonlocal wave equations, 488 Nonstationary states, 73, 275 Norm, 14 Normalization big box, 104, 388-389 of perturbed kets, 0-3 1 www.elsolucionario.net Normalization constant, 108, 204 Normalized kets, 14, 0-3 1 Normal ordering, 465 Nuclear form factor, inelastic scattering and, 440-441 Nuclear magnetic resonance, 163 Nuclear shell model, 213, 214 Null kets, 1 Number operator, 462, 469 Observables, 1 , 28-33 compatible, 28-3 eigenkets of, 17-18 in Heisenberg and Schrodinger pictures, 82 incompatible, 28-29, 1-33, 35-36 matrix representation of, 22 and transformation operator, 35-36 unitary equivalent, 39-40 Occupation number notation, for state vectors, 46 One-electron atoms, central potential for, 0-5 14 Operator equation, 246 Operator identity, 44 Operators, 1 , 14-17, see also specific types associative axiom of, 16-17 definition of, 33, 63 multiplication of, 5-17, 250-25 for spin systems, 25-28, 163-165 and time reversal, 29 1-293 trace of, 37-38 and uncertainty relation, 33-35 Optical isomers, 277 Optical theorem, 397-399 Orbital angular momentum, 199-206 eigenvalues of, 30 parity eigenket of, 273 quenching of, 302 and rotation generation, 99-202 and rotation matrices, 205-206 and spherical harmonics, 202-206 Orthogonal groups, 172-173, 175 Orthogonality and Clebsch-Gordan coefficients, 224, 23 definition of, 14 of eigenkets, 17 and inelastic scattering, 439 and simple harmonic oscillator, 108 in spin systems, 26 and wave functions, 50, 52 Orthogonal matrices, 157-1 59, 173 Orthohelium, 458, 459 Orthonormality and Clebsch-Gordan coefficients, 224 definition of, and degeneracy, 30 of Dirac function, 126 of eigenkets, 8-19 in spin systems, 22 and unitary operator, 36, 59, 63 Oscillations, neutrino, 77-78 Oscillation strength, 368 Oscillators, see also Simple harmonic oscillator isotropic harmonic, 21 1-214, 376 Schwinger's model of, 232-238 uncoupled, 232-235 Outer products, matrix representation of, 21-22 p Pair distribution operator, 465 Parahelium, 458, 459 Parametric down conversions, 482 www.elsolucionario.net 545 Index Parity (space inversion), 269-280 and central potentials, 507 described, 269-272 of Dirac equation, 502-503 nonconservation of, 278-279 parity-selection rule, 277-278 for symmetrical double-well potential, 274-277 for wave functions, 272-274 Parity eigenkets, 273 Parity operator, 269, 502, 506 Parity-selection rule, 277-278 Partially polarized beams, 80 Partial-wave amplitude, 410 Partial-wave expansion, 409 1 Partial waves and determination of phase shifts, 14 15 and eikonal approximation, 420 423 and hard-sphere scattering, 416 417 partial-wave expansion, 409 41 and phase shifts, 414 415 and scattering, 409 16 and unitarity, 1 414 Particles, in Klein-Gordon equation, 493, 494, 503 Paschen-Back limit, 330 Path integrals, 122-129, 5 Pauli, W., 168 Pauli exclusion principle, 284, 45 , 462, 470, 499 Pauli matrices, 168-169, 49 492, 496 Pauli two-component formalism, 68-172 Peierls, R., 397 Permutation operator, 447 Permutation symmetry, 446 450 Perturbation, 303 constant, 359-363 harmonic, 363-365 Perturbation expansion, formal development of, 306-3 Perturbation theory, see Time-dependent perturbation theory; Time-independent perturbation theory Perturbed kets, 0-3 1 Peshkin, M., 148 Phase shifts determination of, 14 41 for free-particle states, 404 409 and hard-sphere scattering, 210n, 416 417 and unitarity, 41 414 Photoelectric effect, 369-371 Photons, 475 476, 48 483 Pinder, D N., 345 Placzek, G., 397 Planck, M., 14 Planck-Einstein relation, angular frequence and, 69 Planck's radiation law, Podolsky, B., 241 Poisson bracket, 48 49, 64, 83 Polarizability, of atom, 297 Polarization, of light, 6-10 Polarized beams, 178-180 Polaroid filters, 6-9 Polar vectors, 272 Position basis, 52-53 Position eigenkets, 41 42 Position-momentum uncertainty relation, 46 Position-space wave functions, 50-52 Positive definite metric, 13 Positrons, 499, 500 Potassium atom, fine structure and, 323-326 Potential differences, 130 Potentials, 129-134, 141-148, see also specific types and Aharonov-Bohm effect, 141-145 and gauge transformations, 129-148 and gravity, 13 1-134 and magnetic monopoles, 145-148 and Schrodinger wave equation, 524 Preston, M., 428 Principal quantum number, 213, 216 Principle of unitary symmetry, 463n Probability charge density, 492 Probability conservation, 412 Probability current density, 493 Probability density, 100, 490 492, 496 Probability flux, 100, 208, 389, 490 Projection operator, Projection theorem, 254-255 Propagators, 16-122 and transition amplitude, 120-122 and wave mechanics, 16-120 Pseudoscalar, examples of, 272 Pseudovectors, 272 Pure ensembles, 24, 179, 180 Q Quadratic Stark effect, 3-3 14 Quadrature squeezed states, 482 Quantization condition, 1 Quantization of electromagnetic field, 472 483 and Casimir effect, 476 480 and Maxwell's equations, 472 475 and photons, 475 476 and quantum optics, 48 483 Quantization of energy, 475 476 Quantum dynamics, 66-148 potentials and gauge transformations, 29-148 propagators and path integrals, 1 6-129 Schrodinger and Heisenberg pictures, 80-89 546 Index Schrodinger wave equation, 97-1 16 simple harmonic oscillator, 89-97 time-evolution and Schr6dinger equation, 66-80 Quantum electrodynamics, covariant, 357 Quantum field theory, 14-5 Quantum interference, gravity-induced, 33-134 Quantum mechanics and Bell's inequality, 243-245 and complex numbers, 27 gravity in, 1-134 and infinitesimal rotations, 160-163 symmetry in, 263 tunneling in, 276 Quantum optics, 48 1-483 Quantum statistical mechanics, 86-19 Quarkonium, 10 Quenching, 302 R Rabi, I I., 340, 343 Rabi's formula, 340 Radial equation, 207-21 Radial integration, helium atom and, 456 Radiation field, classical, see Classical radiation field Radiation law, Planck's, Ramsauer-Townsend effect, 425-426 Rayleigh-Schr6dinger perturbation theory, 303, 331 Rectangular wells, low-energy scattering for, 424-426 Recursion relations, Clebsch-Gordan coefficients and, 224-229 Reduced matrix element, 255 www.elsolucionario.net Relativistic quantum mechanics, 486-5 central potential in, 506-5 14 development of, 486-494 and Dirac equation, 494-506 and energy of free particles, 487-488 kinetic energy in, 321-323 and Klein-Gordon equation, 488-492 natural units for, 487 and negative energies, 492-494 quantum field theory of, 14-5 15 Renormalization, wave-function, 10-3 1 Resonance, 163, 341-344, 430 Resonance scattering, 430-433 Richardson, D J., 352-353 Rigid-wall potential, Schrodinger wave equation and, 524 Rosen, N., 241 Rotational invariance, 412 Rotation generation, orbital angular momentum and, 99-202 Rotation matrices and Clebsch-Gordan coefficients, 230-23 and orbital angular momentum, 205-206 Schwinger's oscillator model for, 236-238 Rotation operator, 160-162 effect on general kets, 165 irreducible representation of, 178 representations of, 96-199 S0(4) group of, 265-267 for spin ! systems, 163-165 x matrix representation of, 170-171 Rotations, see also specific types and angular momentum commutation relations, 57-163 finite vs infinitesimal, 57-163 and Pauli two-component formalism, 170-172 structure constants for, 269 2n , 166-168 Runge-Lenz vector, 265 Rutherford scattering, 402 s Saxon, D S., 19 Scattering amplitude, 391-404 and Born approximation, 399-404 described, 39 1-396 and optical theorem, 397-399 wave-packet description of, 396-397 Scattering length, 426 Scattering processes, 386-441 amplitude of, see Scattering amplitude and Born approximation, 399-404 and eikonal approximation, 417-423 elastic, 436 from future to past, 391 and hard-sphere, 416-423 inelastic electron-atom, 436-441 and Lippmann-Schwinger equation, 390-391 low-energy, rectangular well/barrier, 424-426 and low-energy, bound states, 423-430 and optical theorem, 397-399 and phase shifts/partial waves, 404-4 17 resonance, 430-433 and symmetry, 433-436 and time-dependent perturbation, 386-393 and T matrix, 389-391 transition rates and cross sections for, 88-389 zero-energy, 426-429 Schiff, L., 1 3, 265 www.elsolucionario.net 547 I ndex Schlitt, D W., 345 Schrodinger equation, 346 Schrodinger, E., , 66, 99, 101 Schrodinger equation, see also Schr6dinger equation for central potentials; Schrodinger wave equation and Aharonov-Bohm effect, 142, 143 described, 69-7 and Ehrenfest theorem, 132 and Klein-Gordon equation, 490, 49 and Kramers degeneracy, 299 for linear potential, 109 and momentum-space wave function, 54 in three dimensions, 415 and time-evolution operator, 66-80, 85, 345, 486-487 and time-independent perturbation, for two particles, 455 Schrodinger equation for central potentials, 207-217 and Coulomb potential, 213-217 for free particles and infinite spherical well, 210-2 1 for isotropic harmonic oscillator, 1-214 and radial equation, 207-2 10 Schrodinger picture, 149-150 base kets in, 86-89 and energy shifts, 374 free particles in, 84-86 and Heisenberg picture, 80-89 state kets and observables in, 82 and time-dependent potentials, 337-339 and time-evolution of ensembles, 185 and transition probability, 357 unitary operator in, 80-8 Schrodinger wave equation, 94-1 16, 1 , 36, 140, 285 for central force problem, 527-53 and classical limit of wave mechanics, 102-103 for constant potentials in one dimension, 524-525 for free particles, 523-524 for free particles in three dimensions, 03-105 for hydrogen atoms, 53 1-532 interpretations of, 100-102 for linear potential, 108-1 for simple harmonic oscillator, 105-108, 526-527 solutions to, 03- 1 6, 523-532 time-dependent, 97-98 and time development, 94-97 time-independent, 98-100 for transmission-reflection problems, 525-526 WKB approximation of, 10-1 16 Schrodinger wave function, 100-102, 294 Schwartz inequality, 34, 62 Schwinger, J., 25, 45, 232, 236, 343 Schwinger action principle, 155 Schwinger's oscillator model, 232-238 described, 232-235 for rotation matrices, 236-238 Screened Coulomb potential, 467 Second quantization approach, 460-472, 15 for degenerate electron gas, 467-472 described, 460-463 dynamical variables in, 463-467 Selective measurement, 25 Semiclassical (WKB) approximation of wave equations, 10-1 16 Separation of variables technique, 104 Shankar, R., 322 Silver atoms polarized vs unpolarized beams, 178-180 spin states of, 8-9 and Stem-Gerlach experiment, 2-4 Similarity transformation, 37 Simple harmonic oscillator, 89-97, 50-1 , 92 energy eigenkets and eigenvalues of, 89-93 ground state of, one-dimensional, ground-state energy of, 380 parity properties of, 274 and perturbation, 1-3 13, 376 and Schrodinger wave equation, 05-108, 526-527 time development of, 94-97 Simultaneous eigenkets, 30 Singlets, 383 SI system of units, 9-522 Slowly varying potentials, 12 Sodium atoms, fine structure and, 323-326 Sodium D lines, 326 S0(3) groups, 172-173, 175 Sommerfled, A., 14 S0(4) symmetry, 265-269 Space inversion, see Parity Space quantization, Spatial displacement, see Translation Specific heats, Einstein-Debye theory of, Spherical Bessel functions, 210-2 1 Spherical harmonics and helium atom, 456 Laguerre times, 445 www.elsolucionario.net 548 Index and orbital angular momentum, 202-206 orthogonality of, 23 Spherical tensors, 248-250 Spherical-wave states, 405 Spin ! particles, spin operator for, 21 Spin ! systems, 22-23, 25-28, 59 and anticommutation relations, 28 base kets in, 22-23 Berry's Phase for, 35 1-353 and canonical ensembles, 190 dispersion in, 34 eigenvalues-eigenket relations in, matrix representations in, 22-23 operators for, 25-28, 163-165 rotations of, 163-172 and spin precession, 74-76 and time-evolution operator, 67 time reversal for, 295-298 and x matrix, 174 Spin-angular functions, definition of, 229, 503 Spin correlations, spin-singlet states and, 238-240 Spin kets, 165 Spin magnetic resonance, 342-344 Spin operator, 165, 19 Spin-orbit interaction, fine structure and, 323-327 Spinors, two-component, 168 Spin precession, 74-77, 165-166, 324, 343 Spin-singlet states, spin correlations in, 238-240 Spin states, 8-9 Square-well potential, Schrodinger wave equation and, 525 Squeezed states, 482, 483 Squeeze parameter, 482, 483 Statcoulomb (unit), State kets, 67-68, 82 State vectors, 1, 46 Stationary states, 73 Stem, 0., 1-2 Stem-Gerlach experiment, 1-10 description of, and light polarization, 6-10 sequential, 4-6 Stimulated emission, 365-367 Stoke's theorem, 142, 349n Stopping power, inelastic-scattering and, 439 String theory, 5 Structure constants, 269 Sturm-Liouville theory, 205 Stutz, C., 345 Sudden approximation for time-dependent Hamiltonians, 345-346 SU(2) groups, 174-175 Superposition of energy eigenstates, 96 Symmetrical double-well potential, 274-277 Symmetrical states, 274-275 Symmetrization postulate, 450 45 Symmetry(-ies), 262-300 in classical physics, 262-263 and conservation laws/degeneracies, 262-269 continuous, 262-263, 265-269 and Coulomb potential, 265-269 of Dirac equation, 501-506 discrete, 269-300, 504-505, see also specific types for identical particles, 446 45 lattice translation as, 280-284 parity as, 269-280 permutation, 446 45 properties of symmetry operations, 287-289 in quantum mechanics, 263 and scattering, 433 436 S0(4), 265-269 time-reversal discrete, 284-300 Symmetry operator, 263 T Taylor expansion, 198 Tensors, 246-255, see also specific types Cartesian vs irreducible, 247-250 product of, 250-25 rank of, 247-248 and vector operator, 246-247 Thomas, L H., 324 Thomas precession, 324 Thomas-Reiche-Kuhn sum rule, 368 Threshold behavior, 424 Tight-binding approximation, 282, 283 Time-dependent Hamiltonians, 345-355, 386 adiabatic approximation for, 346-348 and Aharonov-Bohm effect/magnetic monopoles, 353-355 and Berry's Phase, 348-353 sudden approximation for, 345-346 Time-dependent perturbation theory, 355-365 for constant perturbation, 359-363 Dyson series in, 355-357 for harmonic perturbation, 363-365 and scattering processes, 386-393 transition probability in, 357-359 Time-dependent potentials, 336-345 interaction picture for, 337-339 for masers, 344-345 www.elsolucionario.net 549 I ndex for spin-magnetic resonance, 342-344 statement of problem for, 336-337 for two-state problems, 340-345 Time-dependent wave equations, 97-98 Time-evolution operator, 66-80, 263, 356 and correlation amplitude/energy-time uncertainty relation, 78-80 described, 66-69 and energy eigenkets, 1-73 and ensembles, 85 and expectation values, 73 and Heisenberg equation of motion, 83 infinitesimal, 68 and neutrino oscillations, 77-78 and Schrodinger equation, 69-7 and spin precession, 74-77 Time-independent perturbation theory, 303-321 degenerate, 6-321 development of expansion for, 306-3 10 examples of, 1-3 and linear Stark effect, 9-321 nondegenerate, 303-3 16 statement of problem for, 303-304 for two-state problem, 304-306 and wave-function renormalization, 0-3 1 Time-independent wave equations, 98-100 Time reversal, 284-300 described, 284-286 of Dirac equation, 504-505 and electric/magnetic fields, 298-300 formal theory of, 289-293 and Kramers degeneracy, 299 and properties of symmetry operations, 287-289 and spin systems, 295-298 for wave function, 294-295 Time reversal operator, 289-295, 505-506 T matrix, 387, 389-391 Tomita, A., 351 Townsend, J S., 322, 327 Trace, definition of, 37-38 Transformation functions, 53-54 Transformation matrix, 36-38, 64 Transformation operator, 35-36 Transition amplitudes, 387 and base kets, 86-89 composition property of, 122 propagators as, 120-122 Transition energies, Transition probability, 357-359 Transition rate, 362, 88-389 Translation, 42-49 and cannonical commutation relations, 48-49 infinitesimal, 42-43 momentum as generator of, 45-48 Translation operator, physical interpretation of, 92 Transmission-reflection, Schrodinger wave equation and, 525-526 Transverse gauge, 473 Trapezoidal rule, 478 2rr rotations, 166-168 x matrix, 169-171, 174, 496 Two-electron systems, 452-455 Two-particle interactions, 464-467 Two-state problems and perturbation theory, 304-306 time-dependent, 340-342 Two-state systems Hamiltonian matrix for, 378 Hamiltonian operator for, 60 Stem-Gerlach, u Ultra-cold neutrons (UCN), 352-353 Uncertainty principle, Heisenberg, 3, 56 Uncertainty relation, 33-35, 78-80 Uncoupled oscillators, 232-235 Unitarity, 1-414 Unitarity relation, 412 Unitary circle, 3-414 Unitary equivalent observables, 39-40 Unitary operator, 36, 80-8 , 263 Unitary symmetry, principle of, 463n Unitary transform, 39 Unitary unimodular matrix, 174-175 Unpolarized beams, 178-180 Unsold, A., 458 v Vacuum energy, 476 Vacuum kets, 232-233 Van Dam, H., 232 Van der Waals' interactions, 33 1-332 Van Vleck, J H., 343 Variance, 34 Variational approximation methods, 332-336 Vector operator, 246-247, 489, 490n Vector potentials, 472 Vectors, see also specific types column vector function, 49 complex vector space, eve hypothesis, 449-450 definition of, 246 transformation properties of, 171 Virtual transitions, 363 von Neumann, J., 80 www.elsolucionario.net 550 Index w Walecka, John Dirk, 467, 469, 515 Wave equations covariant, 488, 489 and Feynman's path integral, 127-129 nonlocal, 488 Schrodinger, see Schrodinger wave equation semiclassical (WKB) approximation of, 10-1 16 and special relativity, 486 time-dependent, 97-98 time-independent, 99-100 Wave functions, 50-58 for Gaussian wave packets, 55-57 momentum-space, 53-55, 65, 151 under parity, 272-274 position-space, 50-52 renormalization of, 0-3 1 Schrodinger, 00-102, 294 in three dimensions, 57-58 and time reversal, 294-295 Wave mechanics, 98 classical limit of, 102-103 probability density in, 100 propagators in, 16-120 Wave packets and eigenfunctions, 523 Gaussian, 55-57, 62, 65, 99-100, 1 8-1 19 minimum uncertainty, 56 and scattering, 396-397 Weisberger, W I., 148 Weisskopf, V., 375 Wentzel, G., 10 Weyl, H., 99 Whiskers, Aharonov-Bohm effect and, 145 White dwarf star, 464 Wiener, N., 89 Wigner, E P., 196, 236, 241 , 278, 299, 375, 428 Wigner-Eckart theorem, 252-255, 261 , 298, 14, 409 Wigner functions, 196 Wigner's - j symbol, 224 Wigner's formula, 238 Wilson, W., 14 WKB (semiclassical) approximation of wave equations, 10-1 16 Wu, C S., 278 Wu, T T., 148 X Xenon, Ramsauer-Townsend effect and, 425-426 y Yang, C N., 148 Yukawa potential, 401-403, 438, 443 z Zee, Anthony, 5 Zeeman effect, 328-331 Zeeman splitting, 377 Zero-energy scattering, bound states and, 426-429 Zero-point (vacuum) energy, 476 Zeroth-order eigenkets, Zeroth-order energy eigenstates, 377 www.elsolucionario.net ... Cataloging-in-Publication Data Sakurai, J J (Jun John), 1933-1982 Modern quantum mechanics - 2nd ed I J.J Sakurai, Jim Napolitano p cm ISBN 978-0-8053-8291-4 (alk paper) Quantum theory-Textbooks I Napolitano, ... Intheshort, quantum mechanics is the ultimate basis, today, by which understand physical world Thus, I was very pleased to be asked to write the next revised edition of Modern Quantum Mechanics, by Sakurai. .. multiparticle quantum velopmentintoof quantum field theory.and Thethisotheris path involvesof incorporatingThespecial relativity quantum mechanics, the subject sub ject is introduced, andDiractheequation