Consistent Quantum Theory

407 Pages·2002·1.532 MB·English

by  Robert B. Griffiths

Consistent
Quantum Theor y
Robert B.Grif(cid:222)ths
CONSISTENT QUANTUM THEORY
Quantummechanicsisoneofthemostfundamentalyetdifficultsubjectsinmodern
physics. Inthisbook,nonrelativisticquantumtheoryispresentedinaclearandsys-
tematicfashionthatintegratesBorn’sprobabilisticinterpretationwithSchro¨dinger
dynamics.
Basicquantumprinciplesareillustratedwithsimpleexamplesrequiringnomath-
ematics beyond linear algebra and elementary probability theory, clarifying the
mainsourcesofconfusionexperiencedbystudentswhentheybeginaseriousstudy
ofthesubject. Thequantummeasurementprocessisanalyzedinaconsistentway
using fundamental quantum principles that do not refer to measurement. These
same principles are used to resolve several of the paradoxes that have long per-
plexed quantum physicists, including the double slit and Schro¨dinger’s cat. The
consistenthistoriesformalismusedinthisbookwasfirstintroducedbytheauthor,
andextendedbyM.Gell-Mann,J.B.Hartle,andR.Omne`s.
Essential for researchers, yet accessible to advanced undergraduate students in
physics, chemistry, mathematics, andcomputerscience, thisbookmaybeusedas
a supplement to standard textbooks. It will also be of interest to physicists and
philosophersworkingonthefoundationsofquantummechanics.
ROBERT B. GRIFFITHS is the Otto Stern University Professor of Physics at
Carnegie-Mellon University. In 1962 he received his PhD in physics from Stan-
fordUniversity. CurrentlyaFellowoftheAmericanPhysicalSocietyandmember
of the National Academy of Sciences of the USA, he received the Dannie Heine-
manPrizeforMathematicalPhysicsfromtheAmericanPhysicalSocietyin1984.
Heistheauthororcoauthorof130papersonvarioustopicsintheoreticalphysics,
mainlystatisticalandquantummechanics.
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Consistent Quantum Theory
RobertB.Griffiths
Carnegie-MellonUniversity
PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING)
FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF
CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge CB2 IRP
40 West 20th Street, New York, NY 10011-4211, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia

http://www.cambridge.org

© R. B. Griffiths 2002
This edition © R. B. Griffiths 2003

First published in printed format 2002

A catalogue record for the original printed book is available
from the British Library and from the Library of Congress
Original ISBN 0 521 80349 7 hardback

ISBN 0 511 01894 0 virtual (netLibrary Edition)
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Contents
Preface pagexiii
1 Introduction 1
1.1 Scopeofthisbook 1
1.2 Quantumstatesandvariables 2
1.3 Quantumdynamics 3
1.4 MathematicsI.Linearalgebra 4
1.5 MathematicsII.Calculus,probabilitytheory 5
1.6 Quantumreasoning 6
1.7 Quantummeasurements 8
1.8 Quantumparadoxes 9
2 Wavefunctions 11
2.1 Classicalandquantumparticles 11
2.2 Physicalinterpretationofthewavefunction 13
2.3 Wavefunctionsandposition 17
2.4 Wavefunctionsandmomentum 20
2.5 Toymodel 23
3 LinearalgebrainDiracnotation 27
3.1 Hilbertspaceandinnerproduct 27
3.2 Linearfunctionalsandthedualspace 29
3.3 Operators,dyads 30
3.4 Projectorsandsubspaces 34
3.5 Orthogonalprojectorsandorthonormalbases 36
3.6 Columnvectors,rowvectors,andmatrices 38
3.7 DiagonalizationofHermitianoperators 40
3.8 Trace 42
3.9 Positiveoperatorsanddensitymatrices 43
vii
viii Contents
3.10 Functionsofoperators 45
4 Physicalproperties 47
4.1 Classicalandquantumproperties 47
4.2 Toymodelandspinhalf 48
4.3 Continuousquantumsystems 51
4.4 Negationofproperties(NOT) 54
4.5 Conjunctionanddisjunction(AND,OR) 57
4.6 Incompatibleproperties 60
5 Probabilitiesandphysicalvariables 65
5.1 Classicalsamplespaceandeventalgebra 65
5.2 Quantumsamplespaceandeventalgebra 68
5.3 Refinement,coarsening,andcompatibility 71
5.4 Probabilitiesandensembles 73
5.5 Randomvariablesandphysicalvariables 76
5.6 Averages 79
6 Compositesystemsandtensorproducts 81
6.1 Introduction 81
6.2 Definitionoftensorproducts 82
6.3 Examplesofcompositequantumsystems 85
6.4 Productoperators 87
6.5 Generaloperators,matrixelements,partialtraces 89
6.6 Productpropertiesandproductofsamplespaces 92
7 Unitarydynamics 94
7.1 TheSchro¨dingerequation 94
7.2 Unitaryoperators 99
7.3 Timedevelopmentoperators 100
7.4 Toymodels 102
8 Stochastichistories 108
8.1 Introduction 108
8.2 Classicalhistories 109
8.3 Quantumhistories 111
8.4 Extensionsandlogicaloperationsonhistories 112
8.5 Samplespacesandfamiliesofhistories 116
8.6 Refinementsofhistories 118
8.7 Unitaryhistories 119
9 TheBornrule 121
9.1 Classicalrandomwalk 121
Contents ix
9.2 Single-timeprobabilities 124
9.3 TheBornrule 126
9.4 Wavefunctionasapre-probability 129
9.5 Application: Alphadecay 131
9.6 Schro¨dinger’scat 134
10 Consistenthistories 137
10.1 Chainoperatorsandweights 137
10.2 Consistencyconditionsandconsistentfamilies 140
10.3 Examplesofconsistentandinconsistentfamilies 143
10.4 Refinementandcompatibility 146
11 Checkingconsistency 148
11.1 Introduction 148
11.2 Supportofaconsistentfamily 148
11.3 Initialandfinalprojectors 149
11.4 Heisenbergrepresentation 151
11.5 Fixedinitialstate 152
11.6 Initialpurestate. Chainkets 154
11.7 Unitaryextensions 155
11.8 Intrinsicallyinconsistenthistories 157
12 Examplesofconsistentfamilies 159
12.1 Toybeamsplitter 159
12.2 Beamsplitterwithdetector 165
12.3 Time-elapsedetector 169
12.4 Toyalphadecay 171
13 Quantuminterference 174
13.1 Two-slitandMach–Zehnderinterferometers 174
13.2 ToyMach–Zehnderinterferometer 178
13.3 Detectorinoutputofinterferometer 183
13.4 Detectorininternalarmofinterferometer 186
13.5 Weakdetectorsininternalarms 188
14 Dependent(contextual)events 192
14.1 Anexample 192
14.2 Classicalanalogy 193
14.3 Contextualpropertiesandconditionalprobabilities 195
14.4 Dependenteventsinhistories 196
15 Densitymatrices 202
15.1 Introduction 202
x Contents
15.2 Densitymatrixasapre-probability 203
15.3 Reduceddensitymatrixforsubsystem 204
15.4 Timedependenceofreduceddensitymatrix 207
15.5 Reduceddensitymatrixasinitialcondition 209
15.6 Densitymatrixforisolatedsystem 211
15.7 Conditionaldensitymatrices 213
16 Quantumreasoning 216
16.1 Somegeneralprinciples 216
16.2 Example: Toybeamsplitter 219
16.3 Internalconsistencyofquantumreasoning 222
16.4 Interpretationofmultipleframeworks 224
17 MeasurementsI 228
17.1 Introduction 228
17.2 Microscopicmeasurement 230
17.3 Macroscopicmeasurement,firstversion 233
17.4 Macroscopicmeasurement,secondversion 236
17.5 Generaldestructivemeasurements 240
18 MeasurementsII 243
18.1 Beamsplitterandsuccessivemeasurements 243
18.2 Wavefunctioncollapse 246
18.3 NondestructiveStern–Gerlachmeasurements 249
18.4 Measurementsandincompatiblefamilies 252
18.5 Generalnondestructivemeasurements 257
19 Coinsandcounterfactuals 261
19.1 Quantumparadoxes 261
19.2 Quantumcoins 262
19.3 Stochasticcounterfactuals 265
19.4 Quantumcounterfactuals 268
20 Delayedchoiceparadox 273
20.1 Statementoftheparadox 273
20.2 Unitarydynamics 275
20.3 Someconsistentfamilies 276
20.4 Quantumcointossandcounterfactualparadox 279
20.5 Conclusion 282
21 Indirectmeasurementparadox 284
21.1 Statementoftheparadox 284
21.2 Unitarydynamics 286