Quantum Mechanics and Quantum Information

866 Pages·2013·8.45 MB·English

by  Moses Fayngold

www.wiley-vch.de
A F Moses Fayngold, Vadim Fayngold
longside a thorough definition of basic concepts and their a
interrelations, backed by numerous examples, this textbook y
n
features a rare discussion of quantum mechanics and informa-
g
tion theory combined in one text. It deals with important topics
o
hardly found in regular textbooks, including the Robertson- l Quantum Mechanics
d
Schrödinger relation, incompatibility between angle and angular

•
momentum, “dispersed indeterminacy”, interaction-free mea-

F
surements, “submissive quantum mechanics”, and many
a
Moses Fayngold graduated from the others. With its in-depth discussion of key concepts complete y and Quantum
State University of Samarkand and with problems and exercises, this book is poised to become n
got his PhD at the Nuclear Research the standard textbook for advanced undergraduate and begin- g
Institute of Academy of Science in o
ning graduate quantum mechanics courses and an essential
Uzbekistan (former USSR). He has ld
reference for physics students and physics professionals.
combined teaching and research in Information
colleges of USSR and USA, most
recently as a Senior University Lectur-
er at the Physics Department of the From the contents:
New Jersey Institute of Technology.
He has lectured on Quantum Mech- • Embryonic Quantum Mechanics: Basic Features
anics and Special Relativity to both • Playing with the Amplitudes
undergraduate and graduate students.
• Representations and the Hilbert Space
His research interests and areas of
activity include Special and General • Angular Momentum
Relativity, Quantum Mechanics, • Evolution of Quantum States
Optics and optical imaging, particle • Indeterminacy Revisited
scattering and propagation in peri- • “Submissive” Quantum Mechanics
odic structures. • Quantum Statistics aQ
n
• Second Quantization u
d
• Quantum Mechanics and Measurements a
• Quantum Non-Locality Qn
t
• Quantum Measurements and POVMs uu
• Quantum Information am
n
• Quantum Gates t M
• Quantum Key Distribution u
me
c

Vadim Fayngold holds two degrees – Ih
n
M.S. in Physics and B.S. in Computer a
Science. While working as a research fon
assistant at the Department of Com- ric
m
puter Engineering (Polytechnic Uni- s
versity, New York), he focused on a
computer simulation of complex t
i
processes in fluid dynamics. The o
combined expertise he developed n
there has spurred his interest in the
Quantum Information theory.
Vadim came to the idea of writing
this book while working on computer
animations of various relativistic and ISBN 978-3-527-40647-0
quantum-mechanical phenomena.
PHYSICS TEXTBOOK

MosesFayngoldandVadim
Fayngold
QuantumMechanicsand
QuantumInformation
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Moses Fayngold and Vadim Fayngold
Quantum Mechanics and Quantum
Information
AGuidethroughtheQuantumWorld
The Authors All books published by Wiley-VCH are carefully
produced . Neverthel ess, authors, editors, and
publisher do not warrant the information contained
Moses Fayngold in these books, inclu ding this book, to be free of
NJIT errors. Rea ders are advised to keep in mind that
Dept. of Physi cs statemen ts, data, illust rations, procedural details or
Newark, NJ 07102-1982 other items may inadver tently be inaccurate .
fayngold@mai laps.org
Library of Congress Car d No.: applied for
Vadim Fayngold British Library Catalogu ing-in-Publica tion Dat a
vadim.resear [email protected] A catalogu e record for this book is av ailable from the
British Library.
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The D eutsche Na tionalbibliothe k lists this
publicatio n in the Deutsche Nationalbibli ografi e;
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V
Contents
Preface XIII
AbbreviationsandNotations XIX
1 TheFailureofClassicalPhysics 1
1.1 BlackbodyRadiation 1
1.2 HeatCapacity 4
1.3 ThePhotoelectricEffect 9
1.4 AtomsandTheirSpectra 12
1.5 TheDouble-SlitExperiment 14
Problem 19
References 19
2 TheFirstStepsintotheUnknown 21
2.1 TheBBRandPlanck’sFormula 21
2.2 Einstein’sLightQuantaandBBR 24
2.2.1 Discussion 27
2.3 PEERevisited 30
2.4 TheThirdBreakthrough:deBroglieWaves 31
2.4.1 Exercise 33
Problems 35
References 35
3 EmbryonicQuantumMechanics:BasicFeatures 37
3.1 AGlimpseoftheNewRealm 37
3.2 Quantum-MechanicalSuperpositionofStates 39
3.3 WhatIsWavingThere(theMeaningoftheC-Function)? 42
3.4 ObservablesandTheirOperators 47
3.5 Quantum-MechanicalIndeterminacy 49
3.6 IndeterminacyandtheWorld 53
3.7 QuantumEntanglementandNonlocality 58
3.8 Quantum-MechanicalPhaseSpace 62
3.9 DeterminismandCausalityinQuantumWorld 63
j
VI Contents
3.9.1 Discussion 63
Problems 66
References 66
4 PlayingwiththeAmplitudes 69
4.1 CompositionofAmplitudes 69
4.2 DoubleSlitRevisedI 74
4.3 DoubleSlitRevisedII 77
4.4 NeutronScatteringinCrystals 78
4.5 BosonicandFermionicStates 81
4.6 PathIntegrals 89
Problems 93
References 93
5 BasicFeaturesandMathematicalStructureofQM 95
5.1 Observables:theDomainofClassicalandQuantum
Mechanics 95
5.2 Quantum-MechanicalOperators 97
5.3 AlgebraofOperators 100
5.4 EigenvaluesandEigenstates 102
5.5 OrthogonalityofEigenstates 107
5.6 TheRobertson–Schr€odingerRelation 110
5.7 TheWaveFunctionandMeasurements(Discussion) 112
Problems 116
References 117
6 RepresentationsandtheHilbertSpace 119
6.1 VariousFacesofaStateFunction 119
6.2 UnitaryTransformations 121
6.3 OperatorsintheMatrixForm 125
6.4 TheHilbertSpace 129
6.5 OperationsintheHilbertSpace 135
6.6 NonorthogonalStates 142
Problems 147
References 148
7 AngularMomentum 149
7.1 OrbitalandSpinAngularMomenta 149
7.2 TheEigenstatesandEigenvaluesofL^ 151
7.3 OperatorL^andItsCommutationProperties 154
7.4 SpinasanIntrinsicAngularMomentum 164
7.5 AngularMomentumofaCompoundSystem 183
7.6 SphericalHarmonics 188
Problems 196
References 197
j
Contents VII
8 TheSchro€dingerEquation 199
8.1 TheSchr€odingerEquation 199
8.2 StateFunctionandtheContinuityEquation 200
8.3 SeparationofTemporalandSpatialVariables:Stationary
States 203
8.4 TheHelmholtzEquationandDispersionEquationforaFree
Particle 205
8.5 SeparationofSpatialVariablesandtheRadialSchr€odinger
Equation 207
8.6 SuperpositionofDegenerateStates 209
8.7 PhaseVelocityandGroupVelocity 212
8.8 deBroglie’sWavesRevised 218
8.9 TheSchr€odingerEquationinanArbitraryBasis 222
Problems 226
References 226
9 ApplicationstoSimpleSystems:OneDimension 227
9.1 AQuasi-FreeParticle 227
9.2 PotentialThreshold 232
9.3 TunnelingthroughaPotentialBarrier 236
9.4 ColdEmission 241
9.5 PotentialWell 244
9.6 QuantumOscillator 249
9.7 OscillatorintheE-Representation 254
9.8 TheOriginofEnergyBands 257
9.9 PeriodicStructures 260
Problems 269
References 271
10 Three-DimensionalSystems 273
10.1 AParticleina3DBox 273
10.2 AFreeParticlein3D(SphericalCoordinates) 274
10.2.1 Discussion 277
10.3 SomePropertiesofSolutionsinSphericallySymmetric
Potential 277
10.4 SphericalPotentialWell 278
10.5 StatesintheCoulombFieldandaHydrogenAtom 281
10.6 AtomicCurrents 287
10.7 PeriodicTable 290
Problems 293
References 294
11 EvolutionofQuantumStates 295
11.1 TheTimeEvolutionOperator 295
11.2 EvolutionofOperators 299
j
VIII Contents
11.3 SpreadingofaGaussianPacket 301
11.4 TheB-FactorandEvolutionofanArbitraryState 303
11.5 TheFraudulentLifeofan“Illegal”Spike 306
11.6 JinneeOutoftheBox 311
11.7 InadequacyofNonrelativisticApproximationinDescription
ofEvolvingDiscontinuousStates 315
11.7.1 Discussion 316
11.8 Quasi-StationaryStates 317
11.8.1 Discussion 323
11.9 3DBarrierandQuasi-StationaryStates 324
11.10 TheTheoryofParticleDecay 327
11.11 Particle–AntiparticleOscillations 331
11.11.1 Discussion 337
11.12 AWatchedPotNeverBoils(QuantumZenoEffect) 339
11.13 AWatchedPotBoilsFaster 344
Problems 350
References 352
12 QuantumEnsembles 355
12.1 PureEnsembles 355
12.2 Mixtures 356
12.3 TheDensityOperator 358
12.4 TimeEvolutionoftheDensityOperator 366
12.5 CompositeSystems 368
Problems 376
References 376
13 IndeterminacyRevisited 377
13.1 IndeterminacyUnderScrutiny 377
13.2 TheHeisenbergInequalityRevised 380
13.3 TheIndeterminacyofAngularMomentum 382
13.4 TheRobertson–Schr€odingerRelationRevised 384
13.5 TheN–fIndeterminacy 388
13.6 DispersedIndeterminacy 390
Problems 394
References 395
14 QuantumMechanicsandClassicalMechanics 397
14.1 RelationshipbetweenQuantumandClassicalMechanics 397
14.2 QMandOptics 400
14.3 TheQuasi-ClassicalStateFunction 401
14.4 TheWKBApproximation 404
14.5 TheBohr–SommerfeldQuantizationRules 406
Problems 409
References 410