From Classical to Quantum Mechanics This book provides a pedagogical introduction to the formalism, foundations and appli- cations of quantum mechanics. Part I covers the basic material that is necessary to an understanding of the transition from classical to wave mechanics. Topics include classical dynamics, with emphasis on canonical transformations and the Hamilton–Jacobi equation; the Cauchy problem for the wave equation, the Helmholtz equation and eikonal approxi- mation; and introductions to spin, perturbation theory and scattering theory. The Weyl quantization is presented in Part II, along with the postulates of quantum mechanics.
The Weyl programme provides a geometric framework for a rigorous formulation of canonical quantization, as well as powerful tools for the analysis of problems of current interest in quantum physics. In the chapters devoted to harmonic oscillators and angular momentum operators, the emphasis is on algebraic and group-theoretical methods. Quantum entan- glement, hidden-variable theories and the Bell inequalities are also discussed. Part III is devoted to topics such as statistical mechanics and black-body radiation, Lagrangian and phase-space formulations of quantum mechanics, and the Dirac equation.
This book is intended for use as a textbook for beginning graduate and advanced undergraduate courses. It is self-contained and includes problems to advance the reader’s understanding. Giampiero Esposito received his PhD from the University of Cambridge in 1991 and has been INFN Research Fellow at Naples University since November 1993. His research is devoted to gravitational physics and quantum theory.
His main contributions are to the boundary conditions in quantum field theory and quantum gravity via func- tional integrals. Giuseppe Marmo has been Professor of Theoretical Physics at Naples University since 1986, where he is teaching the first undergraduate course in quantum mechanics. His research interests are in the geometry of classical and quantum dynamical systems, deformation quantization, algebraic structures in physics, and constrained and integrable systems. George Sudarshan has been Professor of Physics at the Department of Physics of the University of Texas at Austin since 1969.
His research has revolutionized the understanding of classical and quantum dynamics. He has been nominated for the Nobel Prize six times and has received many awards, including the Bose Medal in 1977.com ii www.com FROM CLASSICAL TO QUANTUM MECHANICS An Introduction to the Formalism, Foundations and Applications Giampiero Esposito, Giuseppe Marmo INFN, Sezione di Napoli and Dipartimento di Scienze Fisiche, Università Federico II di Napoli George Sudarshan Department of Physics, University of Texas, Austin iii www.com Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , UK Published in the United States of America by Cambridge University Press, New York www.org Information on this title: www. Sudarshan 2004 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.
First published in print format 2004 - ---- eBook (NetLibrary) - --- eBook (NetLibrary) - ---- hardback - --- hardback Cambridge University Press has no responsibility for the persistence or accuracy of s 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 For Michela, Patrizia, Bhamathi, and Margherita, Giuseppina, Nidia v www.com vi www.com Contents Preface page xiii Acknowledgments xvi Part I From classical to wave mechanics 1 1 Experimental foundations of quantum theory 3 1.1 The need for a quantum theory 3 1.2 Our path towards quantum theory 6 1.6 Atomic spectra and the Bohr hypotheses 22 1.7 The experiment of Franck and Hertz 26 1.8 Wave-like behaviour and the Bragg experiment 27 1.9 The experiment of Davisson and Germer 33 1.10 Position and velocity of an electron 37 1.A The phase 1-form 41 2 Classical dynamics 43 2.3 Generating functions of canonical transformations 49 2.4 Hamilton and Hamilton–Jacobi equations 59 2.5 The Hamilton principal function 61 2.6 The characteristic function 64 2.7 Hamilton equations associated with metric tensors 66 2.8 Introduction to geometrical optics 68 2.A Vector fields 74 vii www.com viii Contents Appendix 2.B Lie algebras and basic group theory 76 Appendix 2.C Some basic geometrical operations 80 Appendix 2.D Space–time 83 Appendix 2.E From Newton to Euler–Lagrange 83 3 Wave equations 86 3.1 The wave equation 86 3.2 Cauchy problem for the wave equation 88 3.4 Symmetries of wave equations 91 3.6 Fourier analysis and dispersion relations 92 3.7 Geometrical optics from the wave equation 99 3.8 Phase and group velocity 100 3.9 The Helmholtz equation 104 3.10 Eikonal approximation for the scalar wave equation 105 3.11 Problems 114 4 Wave mechanics 115 4.1 From classical to wave mechanics 115 4.2 Uncertainty relations for position and momentum 128 4.3 Transformation properties of wave functions 131 4.4 Green kernel of the Schrödinger equation 136 4.5 Example of isometric non-unitary operator 142 4.8 JWKB solutions of the Schrödinger equation 155 4.9 From wave mechanics to Bohr–Sommerfeld 162 4.A Glossary of functional analysis 167 Appendix 4.B JWKB approximation 172 Appendix 4.C Asymptotic expansions 174 5 Applications of wave mechanics 176 5.1 Reflection and transmission 176 5.2 Step-like potential; tunnelling effect 180 5.4 The Schrödinger equation in a central potential 191 5.6 Introduction to angular momentum 201 5.7 Homomorphism between SU(2) and SO(3) 211 5.8 Energy bands with periodic potentials 217 5.com Contents ix Appendix 5.A Stationary phase method 221 Appendix 5.B Bessel functions 223 6 Introduction to spin 226 6.1 Stern–Gerlach experiment and electron spin 226 6.2 Wave functions with spin 230 6.3 The Pauli equation 233 6.4 Solutions of the Pauli equation 235 6.A Lagrangian of a charged particle 242 Appendix 6.B Charged particle in a monopole field 242 7 Perturbation theory 244 7.1 Approximate methods for stationary states 244 7.2 Very close levels 250 7.4 Occurrence of degeneracy 255 7.8 Time-dependent formalism 269 7.9 Limiting cases of time-dependent theory 274 7.10 The nature of perturbative series 280 7.11 More about singular perturbations 284 7.A Convergence in the strong resolvent sense 295 8 Scattering theory 297 8.1 Aims and problems of scattering theory 297 8.2 Integral equation for scattering problems 302 8.3 The Born series and potentials of the Rollnik class 305 8.4 Partial wave expansion 307 8.5 The Levinson theorem 310 8.6 Scattering from singular potentials 314 8.8 Separable potential model 320 8.9 Bound states in the completeness relationship 323 8.10 Excitable potential model 324 8.11 Unitarity of the Möller operator 327 8.12 Quantum decay and survival amplitude 328 8.com x Contents Part II Weyl quantization and algebraic methods 337 9 Weyl quantization 339 9.1 The commutator in wave mechanics 339 9.2 Abstract version of the commutator 340 9.3 Canonical operators and the Wintner theorem 341 9.4 Canonical quantization of commutation relations 343 9.5 Weyl quantization and Weyl systems 345 9.6 The Schrödinger picture 347 9.7 From Weyl systems to commutation relations 348 9.8 Heisenberg representation for temporal evolution 350 9.9 Generalized uncertainty relations 351 9.10 Unitary operators and symplectic linear maps 357 9.11 On the meaning of Weyl quantization 363 9.12 The basic postulates of quantum theory 365 9.13 Problems 372 10 Harmonic oscillators and quantum optics 375 10.1 Algebraic formalism for harmonic oscillators 375 10.2 A thorough understanding of Landau levels 383 10.4 Weyl systems for coherent states 390 10.5 Two-photon coherent states 393 10.6 Problems 395 11 Angular momentum operators 398 11.1 Angular momentum: general formalism 398 11.2 Two-dimensional harmonic oscillator 406 11.3 Rotations of angular momentum operators 409 11.4 Clebsch–Gordan coefficients and the Regge map 412 11.5 Postulates of quantum mechanics with spin 416 11.6 Spin and Weyl systems 419 11.8 Problems 426 12 Algebraic methods for eigenvalue problems 429 12.1 Quasi-exactly solvable operators 429 12.2 Transformation operators for the hydrogen atom 432 12.3 Darboux maps: general framework 435 12.4 SU (1, 1) structures in a central potential 438 12.5 The Runge–Lenz vector 441 12.com Contents xi 13 From density matrix to geometrical phases 445 13.1 The density matrix 446 13.2 Applications of the density matrix 450 13.4 Hidden variables and the Bell inequalities 455 13.5 Entangled pairs of photons 459 13.6 Production of statistical mixtures 461 13.7 Pancharatnam and Berry phases 464 13.8 The Wigner theorem and symmetries 468 13.9 A modern perspective on the Wigner theorem 472 13.10 Problems 476 Part III Selected topics 477 14 From classical to quantum statistical mechanics 479 14.1 Aims and main assumptions 480 14.5 Equipartition of energy 485 14.6 Specific heats of gases and solids 486 14.7 Black-body radiation 487 14.8 Quantum models of specific heats 502 14.9 Identical particles in quantum mechanics 504 14.10 Bose–Einstein and Fermi–Dirac gases 516 14.11 Statistical derivation of the Planck formula 519 14.A Towards the Planck formula 522 15 Lagrangian and phase-space formulations 526 15.1 The Schwinger formulation of quantum dynamics 526 15.2 Propagator and probability amplitude 529 15.3 Lagrangian formulation of quantum mechanics 533 15.4 Green kernel for quadratic Lagrangians 536 15.5 Quantum mechanics in phase space 541 15.A The Trotter product formula 548 16 Dirac equation and no-interaction theorem 550 16.1 The Dirac equation 550 16.2 Particles in mutual interaction 554 16.3 Relativistic interacting particles.4 The no-interaction theorem in classical mechanics 556 16.5 Relativistic quantum particles 563 www.com xii Contents 16.6 From particles to fields 564 16.7 The Kirchhoff principle, antiparticles and QFT 565 References 571 Index 588 www.com Preface The present manuscript represents an attempt to write a modern mono- graph on quantum mechanics that can be useful both to expert readers, i. graduate students, lecturers, research workers, and to educated read- ers who need to be introduced to quantum theory and its foundations. For this purpose, part I covers the basic material which is necessary to under- stand the transition from classical to wave mechanics: the key experiments in the development of wave mechanics; classical dynamics with empha- sis on canonical transformations and the Hamilton–Jacobi equation; the Cauchy problem for the wave equation, the Helmholtz equation and the eikonal approximation; physical arguments leading to the Schrödinger equation and the basic properties of the wave function; quantum dynam- ics in one-dimensional problems and the Schrödinger equation in a central potential; introduction to spin and perturbation theory; and scattering theory. We have tried to describe in detail how one arrives at some ideas or some mathematical results, and what has been gained by introducing a certain concept.
Indeed, the choice of a first chapter devoted to the experimental foun- dations of quantum theory, despite being physics-oriented, selects a set of readers who already know the basic properties of classical mechan- ics and classical electrodynamics. Thus, undergraduate students should study chapter 1 more than once. Moreover, the choice of topics in chap- ter 1 serves as a motivation, in our opinion, for studying the material described in chapters 2 and 3, so that the transition to wave mechanics is as smooth and ‘natural’ as possible. A broad range of topics are presented in chapter 7, devoted to perturbation theory.
Within this framework, after some elementary examples, we have described the nature of perturbative series, with a brief outline of the various cases of physical interest: regu- lar perturbation theory, asymptotic perturbation theory and summabil- ity methods, spectral concentration and singular perturbations. Chapter xiii www.com xiv Preface 8 starts along the advanced lines of the end of chapter 7, and describes a lot of important material concerning scattering from potentials. Advanced readers can begin from chapter 9, but we still recommend that they first study part I, which contains material useful in later inves- tigations. The Weyl quantization is presented in chapter 9, jointly with the postulates of the currently accepted form of quantum mechanics.
The Weyl programme provides not only a geometric framework for a rigor- ous formulation of canonical quantization, but also powerful tools for the analysis of problems of current interest in quantum mechanics. We have therefore tried to present such a topic, which is still omitted in many textbooks, in a self-contained form.