Advanced Concepts in Quantum Mechanics Introducing a geometric view of fundamental physics, starting from quantum mechanics and its experimental foundations, this book is ideal for advanced undergraduate and graduate students in quantum mechanics and mathematical physics. Focusing on structural issues and geometric ideas, this book guides readers from the concepts of classical mechanics to those of quantum mechanics. The book features an original presentation of classical mechanics, with the choice of topics motivated by the subsequent development of quantum mechanics, especially wave equations, Poisson brack- ets and harmonic oscillators. It also presents new treatments of waves and particles and the symmetries in quantum mechanics, as well as extensive coverage of the experimental foundations.
Giampiero Esposito is Primo Ricercatore at the Istituto Nazionale di Fisica Nucleare, Naples, Italy. His contributions have been devoted to quantum gravity and quantum field theory on manifolds with boundary. Giuseppe Marmo is Professor of Theoretical Physics at the University of Naples Federico II, Italy. His research interests are in the geometry of classical and quantum dynamical systems, deformation quantization and constrained and integrable systems.
Gennaro Miele is Associate Professor of Theoretical Physics at the University of Naples Federico II, Italy. His main research interest is primordial nucleosynthesis and neutrino cosmology. George Sudarshan is Professor of Physics in the Department of Physics, University of Texas at Austin, USA. His research has revolutionized the understanding of classical and quantum dynamics.com 19:06:19 Advanced Concepts in Quantum Mechanics GIAMPIERO ESPOSITO GIUSEPPE MARMO GENNARO MIELE GEORGE SUDARSHAN www.com 19:06:19 University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge.
It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence.org Information on this title: www. Sudarshan 2015 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2015 Printed in the United Kingdom by TJ International Ltd.
Padstow Cornwall A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Esposito, Giampiero, author. Advanced concepts in quantum mechanics / Giampiero Esposito, Giuseppe Marmo, Gennaro Miele, George Sudarshan. Includes bibliographical references. Marmo, Giuseppe, author.
Miele, Gennaro, author.12–dc23 2014014735 ISBN 978-1-107-07604-4 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.com 19:06:19 for Gennaro and Giuseppina; Patrizia; Arianna, Davide and Matteo; Bhamathi www.com 19:06:19 Contents Preface page xiii 1 Introduction: the need for a quantum theory 1 1.1 Introducing quantum mechanics 1 2 Experimental foundations of quantum theory 5 2.1 Black-body radiation 5 2.2 Electromagnetic field in a hollow cavity 7 2.3 Stefan and displacement laws 9 2.5 Contributions of Einstein 17 2.6 Dynamic equilibrium of the radiation field 19 2.2 Quantum theory of the effect 23 2.4 Particle-like behaviour and the Heisenberg picture 30 2.1 Atomic spectra and the Bohr hypotheses 30 2.5 Corpuscular character: the experiment of Franck and Hertz 34 2.6 Wave-like behaviour and the Bragg experiment 35 2.1 Connection between the wave picture and the discrete-level picture 35 2.7 Experiment of Davisson and Germer 39 2.8 Interference phenomena among material particles 41 Appendix 2.A Classical electrodynamics and the Planck formula 46 3 Waves and particles 51 3.1 Waves: d’Alembert equation 51 3.2 Particles: Hamiltonian equations 58 3.1 Poisson brackets among velocity components for a charged particle 62 3.3 Homogeneous linear differential operators and equations of motion 64 3.4 Symmetries and conservation laws 65 www.com 19:17:31 viii Contents 3.1 Homomorphism between SU(2) and SO(3) 67 3.5 Motivations for studying harmonic oscillators 72 3.6 Complex coordinates for harmonic oscillators 74 3.8 Time-dependent Hamiltonian formalism 76 3.9 Hamilton–Jacobi equation 78 3.10 Motion of surfaces 81 Appendix 3.A Space–time picture 83 3.1 Inertial frames and comparison dynamics 84 3.2 Lagrangian descriptions of second-order differential equations 85 3.3 Symmetries and constants of motion 88 3.4 Symmetries and constants of motion in the Hamiltonian formalism 91 3.5 Equivalent reference frames 92 4 Schrödinger picture, Heisenberg picture and probabilistic aspects 94 4.1 From classical to wave mechanics 94 4.1 Properties of the Schrödinger equation 96 4.2 Physical interpretation of the wave function 100 4.4 Eigenstates and eigenvalues 106 4.2 Probability distributions associated with vectors in Hilbert spaces 106 4.3 Uncertainty relations for position and momentum 109 4.4 Transformation properties of wave functions 111 4.1 Direct approach to the transformation properties of the Schrödinger equation 113 4.2 Width of the wave packet 114 4.6 States in the Heisenberg picture 119 4.7 ‘Conclusions’: relevant mathematical structures 120 5 Integrating the equations of motion 122 5.1 Green kernel of the Schrödinger equation 122 5.1 Discrete version of the Green kernel by using a fundamental set of solutions 125 5.2 General considerations on how we use solutions of the evolution equation 127 5.2 Integrating the equations of motion in the Heisenberg picture: harmonic oscillator 129 6 Elementary applications: one-dimensional problems 131 6.1 Particle confined by a potential 132 6.2 A closer look at improper eigenfunctions 134 www.com 19:17:31 ix Contents 6.2 Reflection and transmission 135 6.3 Step-like potential 139 6.4 One-dimensional harmonic oscillator 143 6.A Wave-packet behaviour at large time values 148 7 Elementary applications: multi-dimensional problems 151 7.1 The Schrödinger equation in a central potential 151 7.1 Use of symmetries and geometrical interpretation 158 7.2 Angular momentum operators and spherical harmonics 159 7.3 Angular momentum eigenvalues: algebraic treatment 162 7.4 Radial part of the eigenvalue problem in a central potential 163 7.1 Runge–Lenz vector 168 7.3 s-Wave bound states in the square-well potential 170 7.4 Isotropic harmonic oscillator in three dimensions 172 7.5 Multi-dimensional harmonic oscillator: algebraic treatment 174 7.1 An example: two-dimensional isotropic harmonic oscillator 175 7.6 Problems 177 8 Coherent states and related formalism 180 8.1 General considerations on harmonic oscillators and coherent states 180 8.2 Quantum harmonic oscillator: a brief summary 182 8.3 Operators in the number operator basis 185 8.4 Representation of states on phase space, the Bargmann–Fock representation 186 8.1 The Weyl displacement operator 188 8.5 Basic operators in the coherent states’ basis 190 8.8 Problems 194 9 Introduction to spin 195 9.1 Stern–Gerlach experiment and electron spin 195 9.2 Wave functions with spin 199 9.3 Addition of orbital and spin angular momenta 201 9.4 The Pauli equation 203 9.5 Solutions of the Pauli equation 205 9.1 Another simple application of the Pauli equation 207 9.7 Spin–orbit interaction: Thomas precession 210 9.com 19:17:31 x Contents 10 Symmetries in quantum mechanics 214 10.1 Meaning of symmetries 214 10.1 Transformations that preserve the description 216 10.2 Transformations of frames and corresponding quantum symmetries 222 10.7 Problems 232 11 Approximation methods 234 11A Perturbation theory 235 11A.1 Approximation of eigenvalues and eigenvectors 235 11A.2 Hellmann–Feynman theorem 239 11A.5 Secular equation for problems with degeneracy 248 11A.8 Anomalous Zeeman effect 254 11A.9 Relativistic corrections (α 2 ) to the hydrogen atom 256 11A.11 Time-dependent formalism 259 11A.13 Fermi golden rule 263 11A.14 Towards limiting cases of time-dependent theory 263 11A.15 Adiabatic switch on and off of the perturbation 266 11A.16 Perturbation suddenly switched on 266 11A.17 Two-level system 267 11A.18 The quantum K 0 –K 0 system 269 11A.19 The quantum system of three active neutrinos 271 11B Jeffreys–Wentzel–Kramers–Brillouin method 274 11B.1 The JWKB method 274 11B.3 Energy levels in a potential well 278 11B.4 α-decay 279 11C Scattering theory 282 11C.1 Aims and problems of quantum scattering theory 282 11C.2 Time-dependent scattering 282 11C.3 An example: classical scattering 284 www.com 19:17:31 xi Contents 11C.4 Time-independent scattering 287 11C.1 One-dimensional stationary description of scattering 287 11C.5 Integral equation for scattering problems 289 11C.6 The Born series 293 11C.7 Partial wave expansion 295 11C.8 s-Wave scattering states in the square-well potential 298 11C.9 Problems 299 12 Modern pictures of quantum mechanics 301 12.1 Quantum mechanics on phase space 301 12.2 Representations of the group algebra 304 12.4 Tomographic picture: preliminaries 309 12.6 Pictures of quantum mechanics for a two-level system 315 12.1 von Neumann picture 317 12.4 A closer look at states in the Heisenberg picture 321 12.6 Probability distributions and states 324 12.1 Inner product in tensor spaces 330 12.2 Complex linear operators in tensor spaces 330 12.3 Composite systems and Kronecker products 331 12.4 Two-electron atoms 335 12.9 Generalized paraFermi and paraBose oscillators 337 12.10 Problems 337 13 Formulations of quantum mechanics and their physical implications 339 13.1 Towards an overall view 339 13.2 From Schrödinger to Feynman 339 13.1 Remarks on the Feynman approach 341 13.3 Path integral for systems interacting with an electromagnetic field 344 13.4 Unification of quantum theory and special relativity 346 13.5 Dualities: quantum mechanics leads to new fundamental symmetries 351 www.com 19:17:31 xii Contents 14 Exam problems 353 14.1 End-of-year written exams 353 15 Definitions of geometric concepts 360 15.5 Various definitions of vector fields 363 15.6 Covariant vectors and 1-form fields 366 15.9 Symplectic vector spaces 370 15.10 Homotopy maps and simply connected spaces 371 15.1 Examples of spaces which are or are not simply connected 372 15.11 Diffeomorphisms of manifolds 372 15.12 Foliations of manifolds 373 References 374 Index 381 www.com 19:17:31 Preface In the course of teaching quantum mechanics at undergraduate and post-graduate level, we have come to the conclusion that there is another original book to be written on the subject. The abstract setting foreseen by Dirac and the geometric view pioneered by von Neumann are finding new realizations, leading to further progress both in physics and mathematics, while the applications to quantum computation are opening a new era in modern science. Our emphasis is mainly on structural issues and geometric ideas, moving the reader gradually from the concepts of classical mechanics to those of quantum mechanics, but we have also inserted many problems for students throughout the text, since the book is written, in the first place, for advanced undergraduate and graduate students, as well as for research workers. The overall picture presented here is original, and also the parts in common with a previous monograph by some of us have been rewritten in most cases.
The analysis of waves and particles (Chapter 3), the treatment of symmetries in quantum mechanics (in particular, the first half of Chapter 10), the assessment of modern pictures of quantum mechanics (Chapter 12) have never appeared before in any monograph, to the best of our knowledge. The material on experimental foundations is rather rich and it cannot easily be found to the same extent elsewhere. Our presentation of classical mechanics is original and the choice of topics is motivated by the subsequent development of quantum mechanics, expecially wave equations, Poisson brackets and harmonic oscillators. The examples in Chapters 6 and 7 are frequently discussed with a care not always used in many introductory presentations in the literature.
We find it also useful to offer an unified view of approximation methods, as we do in Chapter 11, which is divided into three parts: perturbation theory, the JWKB method and scattering theory. We hope that, having acquired familiarity with symbols of differential operators, geometric formulation and tomographic picture, the reader will find it easier to follow the latest developments in quantum theory, which embodies, in the broadest sense, all we know about guiding principles and fundamental interactions in physics. Our friend Eugene Saletan, with whom some of us worked and corresponded on the subject of dynamical systems over many years, is deeply missed. Special thanks are due to our colleagues Fedele Lizzi, Francesco Nicodemi and Luigi Rosa for discussing various aspects of the manuscript, and to our students who, never being satisfied with our writing, helped us a lot in conceiving and completing the present monograph.
Last, but not least, the Cambridge University Press staff, i.