Quantum mechanics - modern development

309 Pages·2002·2.455 MB·English

by  Alastair I. M. Rae

Quantum Mechanics
Quantum Mechanics
Fourthedition
Alastair I. M. Rae
Department of Physics
University of Birmingham
UK
Institute of Physics Publishing
Bristol and Philadelphia
c IOPPublishingLtd2002
(cid:13)
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AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN0750308397
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Firstedition1980
Secondedition1986
Reprinted1987
Reprintedwithcorrections1990
Reprinted1991
Thirdedition1992
Reprinted1993
Reprintedwithcorrections1996
Reprinted1998,2001
Fourthedition2002
CommissioningEditor:JamesRevill
ProductionEditor:SimonLaurenson
ProductionControl:SarahPlenty
CoverDesign:Fre´de´riqueSwist
Published by Institute of Physics Publishing, wholly owned by The Institute of
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To Angus and Gavin

Contents
PrefacetoFourthEdition xi
PrefacetoThirdEdition xiii
PrefacetoSecondEdition xv
PrefacetoFirstEdition xvii
1 Introduction 1
1.1 Thephotoelectriceffect 2
1.2 TheComptoneffect 3
1.3 Linespectraandatomicstructure 5
1.4 deBrogliewaves 6
1.5 Wave–particleduality 7
1.6 Therestofthisbook 12
Problems 13
2 Theone-dimensionalSchro¨dingerequations 14
2.1 Thetime-dependentSchro¨dingerequation 14
2.2 Thetime-independentSchro¨dingerequation 18
2.3 Boundaryconditions 19
2.4 Examples 20
2.5 Quantummechanicaltunnelling 27
2.6 Theharmonicoscillator 33
Problems 38
3 Thethree-dimensionalSchro¨dingerequations 39
3.1 Thewaveequations 39
3.2 SeparationinCartesiancoordinates 41
3.3 Separationinsphericalpolarcoordinates 45
3.4 Thehydrogenicatom 53
Problems 59
viii Contents
4 Thebasicpostulatesofquantummechanics 60
4.1 Thewavefunction 61
4.2 Thedynamicalvariables 62
4.3 Probabilitydistributions 68
4.4 Commutationrelations 74
4.5 Theuncertaintyprinciple 76
4.6 Thetimedependenceofthewavefunction 81
4.7 Degeneracy 83
4.8 Theharmonicoscillatoragain 86
4.9 ThemeasurementofmomentumbyComptonscattering 88
Problems 92
5 AngularmomentumI 94
5.1 Theangular-momentumoperators 95
5.2 Theeigenvaluesandeigenfunctions 96
5.3 Theexperimentalmeasurementofangularmomentum 100
5.4 Generalsolutiontotheeigenvalueproblem 103
Problems 108
6 AngularmomentumII 109
6.1 Matrixrepresentations 109
6.2 Paulispinmatrices 112
6.3 Spinandthequantumtheoryofmeasurement 114
6.4 Diracnotation 118
6.5 Spin–orbitcouplingandtheZeemaneffect 119
6.5.1 Thestrong-fieldZeemaneffect 121
6.5.2 Spin–orbitcoupling 122
6.5.3 Theweak-fieldZeemaneffect 124
6.6 Amoregeneraltreatmentofthecouplingofangularmomenta 126
Problems 132
7 Time-independentperturbationtheoryandthevariationalprinciple 134
7.1 Perturbationtheoryfornon-degenerateenergylevels 135
7.2 Perturbationtheoryfordegeneratelevels 141
7.2.1 Nearlydegeneratesystems 143
7.3 Thevariationalprinciple 151
Problems 155
8 Timedependence 157
8.1 Time-independentHamiltonians 158
8.2 Thesuddenapproximation 163
8.3 Time-dependentperturbationtheory 165
8.4 Selectionrules 170
8.5 TheEhrenfesttheorem 174
8.6 Theammoniamaser 176
Problems 179
Contents ix
9 Scattering 181
9.1 Scatteringinonedimension 181
9.2 Scatteringinthreedimensions 186
9.3 TheBornapproximation 189
9.4 Partialwaveanalysis 193
Problems 203
10 Many-particlesystems 205
10.1 Generalconsiderations 205
10.2 Isolatedsystems 206
10.3 Non-interactingparticles 208
10.4 Indistinguishableparticles 208
10.5 Many-particlesystems 212
10.6 Theheliumatom 216
10.7 Scatteringofidenticalparticles 223
Problems 224
11 Relativityandquantummechanics 226
11.1 Basicresultsinspecialrelativity 226
11.2 TheDiracequation 227
11.3 Antiparticles 233
11.4 Otherwaveequations 235
11.5 Quantumfieldtheoryandthespin-statisticstheorem 235
Problems 239
12 Quantuminformation 241
12.1 Quantumcryptography 242
12.2 Entanglement 245
12.3 Teleportation 246
12.4 Quantumcomputing 249
Problems 252
13 Theconceptualproblemsofquantummechanics 253
13.1 Theconceptualproblems 253
13.2 Hidden-variabletheories 255
13.3 Non-locality 262
13.4 Thequantum-mechanicalmeasurementproblem 273
13.5 Theontologicalproblem 287
Problems 288
Hintstosolutionofproblems 290
Index 296
Preface to Fourth Edition
When I told a friend that I was working on a new edition, he asked me what
had changed in quantum physics during the last ten years. In one sense very
little: quantum mechanics is a very well established theory and the basic ideas
and concepts are little changed from what they were ten, twenty or more years
ago. However, new applications have been developed and some of these have
revealedaspectsofthesubjectthatwerepreviouslyunknownorlargelyignored.
Muchofthisdevelopmenthasbeeninthefieldofinformationprocessing,where
quantumeffectshavecometothefore. Inparticular,quantumtechniquesappear
tohavegreatpotentialinthefieldofcryptography,bothinthecodingandpossible
de-codingof messages, and I have includeda chapter aimed at introducingthis
topic.
I have also added a short chapter on relativistic quantum mechanics and
introductoryquantum field theory. This is a little more advancedthan many of
the other topics treated, but I hope it will be accessible to the interested reader.
It aims to open the door to the understanding of a number of points that were
previouslystatedwithoutjustification.
Once again, I have largely re-written the last chapter on the conceptual
foundations of the subject. The twenty years since the publication of the first
edition do not seem to have brought scientists and philosophers significantly
closer to a consensus on these problems. However, many issues have
been considerably clarified and the strengths and weaknesses of some of the
explanationsare more apparent. My own understandingcontinues to grow, not
leastbecauseofwhatIhavelearnedfromformalandinformaldiscussionsatthe
annualUKConferencesontheFoundationsofPhysics.
Otherchangesincludeamoredetailedtreatmentoftunnellinginchapter2,
amoregentletransitionfromtheBornpostulatetoquantummeasurementtheory
in chapter4, the introductionof Dirac notationin chapter6 anda discussion of
theBose–Einsteincondensateinchapter10.
I am gratefulto a numberofpeoplewho havehelpedme withthis edition.
Glenn Cox shared his expertise on relativistic quantum mechanics when he
read a draft of chapter 11; Harvey Brown corrected my understanding of the
deBroglie–Bohmhiddenvariabletheorydiscussedinthefirstpartofchapter13;
DemetrisCharalambousreadalatedraftofthewholebookandsuggestedseveral
xi
xii PrefacetoFourthEdition
improvementsand corrections. Of course, I bear fullresponsibilityforthe final
versionandanyremainingerrors.
Moderntechnologymeansthatthepublishersareabletosupportthebookat
the web site http://bookmarkphysics.iop.org/bookpge.htm/book=1107p. This is
whereyouwillfindreferencestothewiderliterature,colourillustrations,linksto
otherrelevantwebsites, etc. Ifanymistakesareidentified,correctionswillalso
belistedthere. Readersarealsoinvitedtocontributesuggestionsonwhatwould
be usefulcontent. The mostconvenientform of communicationis by e-mailto
[email protected].
Finally I should like to pay tribute to Ann for encouragingme to returnto
writingaftersometime. Hersupporthasbeeninvaluable.
AlastairI.M.Rae
2002