From Classical to Quantum Mechanics
This book provides a pedagogical introduction to the formalism, foundations and applications
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 approximation;
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 entanglement,
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 functional
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.
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,
Universit`a Federico II di Napoli
George Sudarshan
Department of Physics,
University of Texas, Austin
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, CambridgeUniversity press, UK
First published in print format 2004
ISBN-13 978-0-511-18490-1 eBook (NetLibrary)
© G. Esposito, G. Marmo and E. C. G. Sudarshan 2004
2004
Information on this title: www.cambridge.org/9780521833240
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.
Cambridge University Press has no responsibility for the persistence or accuracy
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.
Contents
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
1.3 Photoelectric effect 7
1.4 Compton effect 11
1.5 Interference experiments 17
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
1.11 Problems 41
Appendix 1.A The phase 1-form 41
2 Classical dynamics 43
2.1 Poisson brackets 44
2.2 Symplectic geometry 45
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
2.9 Problems 73
Appendix 2.A Vector fields 74
vii
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
3.3 Fundamental solutions 90
3.4 Symmetries of wave equations 91
3.5 Wave packets 92
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 Schr¨odinger equation 136
4.5 Example of isometric non-unitary operator 142
4.6 Boundary conditions 144
4.7 Harmonic oscillator 151
4.8 JWKB solutions of the Schr¨odinger equation 155
4.9 From wave mechanics to Bohr–Sommerfeld 162
4.10 Problems 167
Appendix 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
5.3 Linear potential 186
5.4 The Schr¨odinger equation in a central potential 191
5.5 Hydrogen atom 196
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
5.9 Problems 220
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
6.5 Landau levels 239
6.6 Problems 241
Appendix 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
7.3 Anharmonic oscillator 252
7.4 Occurrence of degeneracy 255
7.5 Stark effect 259
7.6 Zeeman effect 263
7.7 Variational method 266
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
7.12 Problems 293
Appendix 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
8.7 Resonances 317
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 M¨oller operator 327
8.12 Quantum decay and survival amplitude 328
8.13 Problems 335
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 Schr¨odinger 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
10.3 Coherent states 386
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
11.7 Monopole harmonics 420
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
12.6 Problems 443
Contents xi
13 From density matrix to geometrical phases 445
13.1 The density matrix 446
13.2 Applications of the density matrix 450
13.3 Quantum entanglement 453
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
14.2 Canonical ensemble 481
14.3 Microcanonical ensemble 482
14.4 Partition function 483
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
14.12 Problems 522
Appendix 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
15.6 Problems 548
Appendix 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. Manifest covariance 555
16.4 The no-interaction theorem in classical mechanics 556
16.5 Relativistic quantum particles 563
xii Contents
16.6 From particles to fields 564
16.7 The Kirchhoff principle, antiparticles and QFT 565
Preface
The present manuscript represents an attempt to write a modern monograph
on quantum mechanics that can be useful both to expert readers,
i.e. graduate students, lecturers, research workers, and to educated readers
who need to be introduced to quantum theory and its foundations. For
this purpose, part I covers the basic material which is necessary to understand
the transition from classical to wave mechanics: the key experiments
in the development of wave mechanics; classical dynamics with emphasis
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 Schr¨odinger
equation and the basic properties of the wave function; quantum dynamics
in one-dimensional problems and the Schr¨odinger 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 foundations
of quantum theory, despite being physics-oriented, selects a set
of readers who already know the basic properties of classical mechanics
and classical electrodynamics. Thus, undergraduate students should
study chapter 1 more than once. Moreover, the choice of topics in chapter
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: regular
perturbation theory, asymptotic perturbation theory and summability
8 starts along the advanced lines of the end of chapter 7, and describes amethods, spectral concentration and singular perturbations. Chapter
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 investigations.
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 rigorous
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. In the chapters devoted to harmonic
oscillators and angular momentum operators the emphasis is on algebraic
and group-theoretical methods. The same can be said about chapter 12,
devoted to algebraic methods for the analysis of Schr¨odinger operators.
The formalism of the density matrix is developed in detail in chapter 13,
which also studies some very important topics such as quantum entanglement,
hidden-variable theories and Bell inequalities; how to transfer the
polarization state of a photon to another photon thanks to the projection
postulate, the production of statistical mixtures and phase in quantum
mechanics.
Part III is devoted to a number of selected topics that reflect the authors’
taste and are aimed at advanced research workers: statistical mechanics
and black-body radiation; Lagrangian and phase-space formulations
of quantum mechanics; the no-interaction theorem and the need for
a quantum theory of fields.
The chapters are completed by a number of useful problems, although
the main purpose of the book remains the presentation of a conceptual
framework for a better understanding of quantum mechanics. Other important
topics have not been included and might, by themselves, be the
object of a separate monograph, e.g. supersymmetric quantum mechanics,
quaternionic quantum mechanics and deformation quantization. But
we are aware that the present version already covers much more material
than the one that can be presented in a two-semester course. The material
in chapters 9–16 can be used by students reading for a master or
Ph.D. degree.
Our monograph contains much material which, although not new by itself,
is presented in a way that makes the presentation rather original with
respect to currently available textbooks, e.g. part I is devoted to and built
around wave mechanics only; Hamiltonian methods and the Hamilton–
Jacobi equation in chapter 2; introduction of the symbol of differential operators
and eikonal approximation for the scalar wave equation in chapter
3; a systematic use of the symbol in the presentation of the Schr¨odinger
equation in chapter 4; the Pauli equation with time-dependent magnetic
fields in chapter 6; the richness of examples in chapters 7 and 8; Weyl
quantization in chapter 9; algebraic methods for eigenvalue problems in
chapter 12; the Wigner theorem and geometrical phases in chapter 13;
and a geometrical proof of the no-interaction theorem in chapter 16.
So far we have defended, concisely, our reasons for writing yet another
book on quantum mechanics. The last word is now with the readers.
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