Quantum Mechanics

758 Pages·2009·10 MB·English

This page intentionally left blank
Quantum Mechanics
Theimportantchangesquantummechanicshasundergoneinrecentyearsarereflectedin
thisnewapproachforstudents.
A strong narrative and over 300 worked problems lead the student from experiment,
throughgeneralprinciplesofthetheory,tomodernapplications.Steppingthroughresults
allows students to gain a thorough understanding. Starting with basic quantum mechan-
ics, the book moves on to more advanced theory, followed by applications, perturbation
methods and special fields, and ending with new developments in the field. Historical,
mathematical, and philosophical boxes guide the student through the theory. Unique to
this textbook are chapters on measurement and quantum optics, both at the forefront of
currentresearch.Advancedundergraduateandgraduatestudentswillbenefitfromthisnew
perspectiveonthefundamentalphysicalparadigmanditsapplications.
Online resources including solutions to selected problems and 200 figures, with color
versionsofsomefigures,areavailableatwww.cambridge.org/Auletta.
GennaroAulettaisScientificDirectorofScienceandPhilosophyatthePontificalGrego-
rianUniversity,Rome.Hismainareasofresearcharequantummechanics,logic,cognitive
sciences,informationtheory,andapplicationstobiologicalsystems.
MauroFortunatoisaStructureratCassadepositieprestitiS.p.A.,Rome.Heisinvolvedin
financial engineering, applying mathematical methods of quantum physics to the pricing
ofcomplexfinancialderivativesandtheconstructionofstructuredproducts.
GiorgioParisiisProfessorofQuantumTheoriesattheUniversityofRome“LaSapienza.”
Hehaswonseveralprizes,notablytheBoltzmannMedal,theDiracMedalandPrize,and
the Daniel Heineman prize. His main research activity deals with elementary particles,
theory of phase transitions and statistical mechanics, disordered systems, computers and
verylargescalesimulations,non-equilibriumstatisticalphysics,optimization,andanimal
behavior.

Quantum Mechanics
GENNARO AULETTA
Pontifical Gregorian University, Rome
MAURO FORTUNATO
Cassa Depositi e Prestiti S.p.A., Rome
GIORGIO PARISI
‘‘La Sapienza’’ University, Rome
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521869638
© G. Auletta, M. Fortunato and G. Parisi 2009
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 2009
ISBN-13 978-0-511-53363-1 eBook (EBL)
ISBN-13 978-0-521-86963-8 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.
Contents
Listoffigures pagexi
Listoftables xvii
Listofdefinitions,principles,etc. xviii
Listofboxes xx
Listofsymbols xxi
Listofabbreviations xxxii
Introduction 1
Part I Basic features of quantum mechanics
1 Fromclassicalmechanicstoquantummechanics 7
1.1 Reviewofthefoundationsofclassicalmechanics 7
1.2 Aninterferometryexperimentanditsconsequences 12
1.3 Stateasvector 20
1.4 Quantumprobability 28
1.5 Thehistoricalneedofanewmechanics 31
Summary 40
Problems 41
Furtherreading 42
2 Quantumobservablesandstates 43
2.1 Basicfeaturesofquantumobservables 43
2.2 Wavefunctionandbasicobservables 68
2.3 Uncertaintyrelation 82
2.4 Quantumalgebraandquantumlogic 92
Summary 96
Problems 97
Furtherreading 99
3 Quantumdynamics 100
3.1 TheSchrödingerequation 101
3.2 PropertiesoftheSchrödingerequation 107
3.3 SchrödingerequationandGalileitransformations 111
3.4 One-dimensionalfreeparticleinabox 113
3.5 Unitarytransformations 117
vi Contents
(cid:2)
3.6 Differentpictures 125
3.7 TimederivativesandtheEhrenfesttheorem 129
3.8 Energy–timeuncertaintyrelation 130
3.9 Towardsatimeoperator 135
Summary 138
Problems 139
Furtherreading 140
4 Examplesofquantumdynamics 141
4.1 Finitepotentialwells 141
4.2 Potentialbarrier 145
4.3 Tunneling 150
4.4 Harmonicoscillator 154
4.5 Quantumparticlesinsimplefields 165
Summary 169
Problems 170
5 Densitymatrix 174
5.1 Basicformalism 174
5.2 Expectationvaluesandmeasurementoutcomes 177
5.3 Timeevolutionanddensitymatrix 179
5.4 Statisticalpropertiesofquantummechanics 180
5.5 Compoundsystems 181
5.6 Pure-andmixed-staterepresentation 187
Summary 188
Problems 189
Furtherreading 190
Part II More advanced topics
6 Angularmomentumandspin 193
6.1 Orbitalangularmomentum 193
6.2 Specialexamples 207
6.3 Spin 217
6.4 Compositionofangularmomentaandtotalangularmomentum 226
6.5 Angularmomentumandangle 239
Summary 241
Problems 242
Furtherreading 244
7 Identicalparticles 245
7.1 Statisticsandquantummechanics 245
7.2 Wavefunctionandsymmetry 247
7.3 Spinandstatistics 249
vii Contents
(cid:2)
7.4 Exchangeinteraction 254
7.5 Tworecentapplications 255
Summary 257
Problems 257
Furtherreading 258
8 Symmetriesandconservationlaws 259
8.1 Quantumtransformationsandsymmetries 259
8.2 Continuoussymmetries 264
8.3 Discretesymmetries 266
8.4 Abriefintroductiontogrouptheory 267
Summary 275
Problems 275
Furtherreading 276
9 Themeasurementprobleminquantummechanics 277
9.1 Statementoftheproblem 278
9.2 Abriefhistoryoftheproblem 284
9.3 Schrödingercats 291
9.4 Decoherence 297
9.5 Reversibility/irreversibility 308
9.6 Interaction-freemeasurement 315
9.7 Delayed-choiceexperiments 320
9.8 QuantumZenoeffect 322
9.9 Conditionalmeasurementsorpostselection 325
9.10 Positiveoperatorvaluedmeasure 327
9.11 Quantumnon-demolitionmeasurements 335
9.12 Decisionandestimationtheory 341
Summary 349
Problems 351
Furtherreading 353
Part III Matter and light
10 Perturbationsandapproximationmethods 357
10.1 Stationaryperturbationtheory 357
10.2 Time-dependentperturbationtheory 366
10.3 Adiabatictheorem 369
10.4 Thevariationalmethod 371
10.5 Classicallimit 372
10.6 SemiclassicallimitandWKBapproximation 378
10.7 Scatteringtheory 384
10.8 Pathintegrals 389
Summary 398
viii Contents
(cid:2)
Problems 399
Furtherreading 399
11 Hydrogenandheliumatoms 401
11.1 Introduction 401
11.2 Quantumtheoryofthehydrogenatom 403
11.3 Atomandmagneticfield 413
11.4 Relativisticcorrections 423
11.5 Heliumatom 426
11.6 Many-electroneffects 431
Summary 436
Problems 437
Furtherreading 438
12 Hydrogenmolecularion 439
12.1 Themolecularproblem 439
12.2 Born–Oppenheimerapproximation 440
12.3 Vibrationalandrotationaldegreesoffreedom 443
12.4 TheMorsepotential 447
12.5 Chemicalbondsandfurtherapproximations 449
Summary 453
Problems 453
Furtherreading 454
13 Quantumoptics 455
13.1 Quantizationoftheelectromagneticfield 457
13.2 Thermodynamicequilibriumoftheradiationfield 462
13.3 Phase–numberuncertaintyrelation 463
13.4 Specialstatesoftheelectromagneticfield 465
13.5 Quasi-probabilitydistributions 474
13.6 Quantum-opticalcoherence 481
13.7 Atom–fieldinteraction 484
13.8 Geometricphase 497
13.9 TheCasimireffect 501
Summary 506
Problems 507
Furtherreading 509
Part IV Quantum information: state and correlations
14 Quantumtheoryofopensystems 513
14.1 Generalconsiderations 514
14.2 Themasterequation 516