Quantum mechanics with applications to nanotechnology and information science

993 Pages·2013·15.24 MB·English

by  Yehuda B. Band

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QUANTUM MECHANICS
WITHAPPLICATIONSTONANOTECHNOLOGYANDINFORMATIONSCIENCE
EDELKAMP 19-ch15-671-700-9780123725127 2011/5/28 14:50 Page 672 #2
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QUANTUM MECHANICS
WITHAPPLICATIONSTONANOTECHNOLOGYANDINFORMATIONSCIENCE
Yehuda B. Band
DepartmentofChemistry,DepartmentofElectro-OpticsandDepartmentofPhysics,and
IlseKatzInstituteforNanoscaleScienceandTechnology
Ben-GurionUniversity
Beer-Sheva,Israel
and
Yshai Avishai
DepartmentofPhysics,and
IlseKatzInstituteforNanoscaleScienceandTechnology
Ben-GurionUniversity
Beer-Sheva,Israel
AMSTERDAM•WALTHAM•HEIDELBERG•LONDON
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CONTENTS
Preface xvii
CHAPTER1 INTRODUCTIONTOQUANTUMMECHANICS 1
1.1 WhatisQuantumMechanics? 2
1.1.1 ABriefEarlyHistoryofQuantumMechanics 2
1.1.2 EnergyQuantization 2
1.1.3 Waves,Light,andBlackbodyRadiation 6
1.1.4 Wave–ParticleDuality 8
1.1.5 AngularMomentumQuantization 12
1.1.6 Tunneling 13
1.1.7 PhotoelectricEffect 15
1.2 NanotechnologyandInformationTechnology 17
1.2.1 STMandAFMMicroscopies 18
1.2.2 MolecularElectronics 18
1.2.3 QuantumDots,WiresandWells,andNanotubes 19
1.2.4 Bio-Nanotechnology 21
1.2.5 InformationTechnology 21
1.3 AFirstTasteofQuantumMechanics 21
1.3.1 QuantumStatesandProbabilityDistributions 22
1.3.2 ObservableOperators 24
1.3.3 QuantumEntanglement 26
1.3.4 ThePostulatesofQuantumMechanics 28
1.3.5 Time-Dependentand-IndependentSchro¨dingerEquations 29
1.3.6 Momentum,Energy,andAngularMomentum 32
1.3.7 DiracDeltaFunctions 36
1.3.8 PositionandMomentumStates,|x(cid:105)and|p(cid:105) 38
1.3.9 Ehrenfest’sTheorem 39
1.3.10 One-DimensionalWaveEquations 41
1.3.11 Particle-in-a-BoxandPiecewise-ConstantPotentials 42
1.3.12 TheDeltaFunctionPotential 52
1.3.13 WavePackets 53
1.3.14 TheLinearPotentialandQuantumTunneling 54
1.3.15 TheHarmonicOscillator 55
v
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vi Contents
CHAPTER2 THEFORMALISMOFQUANTUMMECHANICS 61
2.1 HilbertSpaceandDiracNotation 61
2.1.1 PositionandMomentumRepresentations 63
2.1.2 Basis-StateExpansions 64
2.2 HermitianandAnti-HermitianOperators 66
2.2.1 CompatibleOperatorsandDegeneracy 66
2.3 TheUncertaintyPrinciple 67
2.4 TheMeasurementProblem 69
2.5 MixedStates:DensityMatrixFormulation 70
2.5.1 Many-ParticleSystems:CorrelationFunctions 75
2.5.2 PurityandvonNeumannEntropy 77
2.5.3 DistanceBetweenStates 77
2.5.4 TheMeasurementProblemRevisited 78
2.6 TheWignerRepresentation 79
2.7 Schro¨dingerandHeisenbergRepresentations 85
2.7.1 InteractionRepresentation 87
2.7.2 HarmonicOscillatorRaising–LoweringOperators 88
2.7.3 CoherentStatesandSqueezedStates 92
2.8 TheCorrespondencePrincipleandtheClassicalLimit 97
2.9 SymmetryandConservationLawsinQuantumMechanics 98
2.9.1 ExchangeSymmetry 98
2.9.2 InversionSymmetry 100
2.9.3 Time-ReversalSymmetry 100
2.9.4 AdditionalGeneratorsofGalileanTransformations 102
CHAPTER3 ANGULARMOMENTUMANDSPHERICALSYMMETRY 105
3.1 AngularMomentuminQuantumMechanics 105
3.1.1 AngularMomentumRaisingandLoweringOperators 107
3.1.2 ElectronSpin:j=1/2 110
3.1.3 AngularMomentuminSphericalCoordinates 111
3.1.4 SphericalHarmonics 112
3.2 SphericallySymmetricSystems 116
3.2.1 AngularMomentumDecompositionofPlaneWaves 119
3.2.2 SphericalQuantumDot 120
3.2.3 The3DHarmonicOscillator 121
3.2.4 TheMorseOscillator 122
3.2.5 vanderWaalsandLennard-JonesPotentials 123
3.2.6 TheHydrogenAtom 124
3.3 RotationsandAngularMomentum 132
3.3.1 Euleranglesα,β,γ andtheRotationMatrix 133
3.3.2 RotationandDFunctions 134
3.3.3 Rigid-RotorEigenfunctions 137
3.4 Addition(Coupling)ofAngularMomenta 139
3.4.1 Clebsch–GordanCoefficientsand3jSymbols 140
3.4.2 Clebsch–GordanSeries 143
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Contents vii
3.5 TensorOperators 144
3.5.1 IrreducibleRepresentationsoftheDensityMatrix 146
3.5.2 VectorFields 148
3.5.3 SpinorFields 149
3.5.4 MultipoleExpansions 150
3.6 SymmetryConsiderations 151
3.6.1 SelectionRules 152
3.6.2 InversionSymmetry 152
3.6.3 Time-ReversalSymmetry 153
3.6.4 Wigner–EckartTheorem 153
3.6.5 6jandHigherCoefficients 156
CHAPTER4 SPIN 159
4.1 SpinAngularMomentum 159
4.2 Spinors 160
4.2.1 PauliMatrices 161
4.2.2 RotationofSpinors 164
4.2.3 Spin-Orbitals 165
4.3 ElectroninaMagneticField 166
4.3.1 ChargedParticleinaMagneticField:OrbitalEffects 169
4.4 Time-ReversalPropertiesofSpinors 172
4.5 Spin–OrbitInteractioninAtoms 175
4.6 HyperfineInteraction 178
4.6.1 ElectricQuadrupoleHyperfineInteraction 181
4.6.2 ZeemanSplittingofHyperfineStates 182
4.7 Spin-DipolarInteractions 183
4.8 IntroductiontoMagneticResonance 185
4.8.1 TheRotating-WaveApproximation 187
4.8.2 SpinRelaxationandtheBlochEquation 188
4.8.3 NuclearSpinHamiltonian 189
4.8.4 ChemicalShifts 189
4.8.5 FourierTransformNMR 190
CHAPTER5 QUANTUMINFORMATION 193
5.1 ClassicalComputationandClassicalInformation 194
5.1.1 InformationandEntropy 194
5.1.2 ShannonEntropy 196
5.1.3 DataCompression 198
5.1.4 ClassicalComputersandGates 200
5.1.5 ClassicalCryptography 202
5.1.6 ComputationalComplexity 204
5.2 QuantumInformation 205
5.2.1 Qubits 205
5.2.2 QuantumEntanglementandBellStates 207
5.2.3 QuantumGates 213
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viii Contents
5.2.4 No-CloningTheorem 218
5.2.5 DenseCoding 219
5.2.6 DataCompressionofQuantumInformation 220
5.2.7 QuantumTeleportation 221
5.2.8 QuantumCryptography 222
5.2.9 QuantumCircuits 223
5.2.10 QuantumComputingDespiteMeasurement 224
5.3 QuantumComputingAlgorithms 224
5.3.1 DeutschandDeutsch–JozsaAlgorithms 225
5.3.2 TheGroverSearchAlgorithm 228
5.3.3 QuantumFourierTransform 232
5.3.4 ShorFactorizationAlgorithm 234
5.3.5 QuantumSimulation 239
5.4 Decoherence 240
5.5 QuantumErrorCorrection 240
5.6 ExperimentalImplementations 243
5.6.1 IonTraps 243
5.6.2 NeutralAtomsinOpticalLattices 245
5.6.3 CavityBasedQuantumComputing 246
5.6.4 NuclearMagneticResonanceSystems 247
5.6.5 All-OpticalQuantumComputers 248
5.6.6 Solid-StateQubits 248
5.7 TheEPRParadox 250
5.8 Bell’sInequalities 251
5.8.1 Bell’sInequalitiesandtheEPRParadox 252
5.8.2 Bell’sAnalysisusingHiddenVariables 253
5.8.3 GeneralAspectsofBell’sInequalities 256
CHAPTER6 QUANTUMDYNAMICSANDCORRELATIONS 259
6.1 Two-LevelSystems 259
6.1.1 Two-LevelDynamics(SpinDynamics) 261
6.1.2 TheBlochSpherePicture 262
6.1.3 CouplingtoaBath:Decoherence 266
6.1.4 PeriodicallyDrivenTwo-LevelSystem 268
6.1.5 AtomsinanElectromagneticField:DispersionandAbsorption 273
6.1.6 DopplerCoolingofAtoms 275
6.1.7 OpticalTrappingofAtoms 278
6.1.8 TwoorMoreCorrelated“Spins” 279
6.1.9 TheN-Two-LevelSystemBlochSphere 282
6.1.10 RamseyFringeSpectroscopy 285
6.2 Three-LevelSystems 287
6.2.1 TwoorMoreThree-LevelCorrelatedSystems 288
6.3 ClassificationofCorrelationandEntanglement 290
6.3.1 EntanglementWitnessOperators 292
6.4 Three-LevelSystemDynamics 293
6.5 Continuous-VariableSystems 295
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Contents ix
6.6 WavePacketDynamics 297
6.7 Time-DependentHamiltonians 299
6.8 QuantumOptimalControlTheory 300
CHAPTER7 APPROXIMATIONMETHODS 303
7.1 Basis-StateExpansions 303
7.1.1 Time-DependentBasisSetExpansions 303
7.2 SemiclassicalApproximations 304
7.2.1 TheWKBApproximation 304
7.2.2 SemiclassicalTreatmentofDynamics 309
7.2.3 SemiclassicalHamilton–JacobiExpansion 309
7.3 PerturbationTheory 309
7.3.1 NondegeneratePerturbationTheory 310
7.3.2 DegeneratePerturbationTheory 313
7.3.3 Time-DependentPerturbationTheory 315
7.4 DynamicsinanElectromagneticField 321
7.4.1 SpontaneousandStimulatedEmissionofRadiation 323
7.4.2 ElectricDipoleandMultipoleRadiation 324
7.4.3 Thomson,Rayleigh,Raman,andBrillouinTransitions 326
7.4.4 DecayWidth 329
7.4.5 DopplerShift 330
7.5 ExponentialandNonexponentialDecay 331
7.6 TheVariationalMethod 331
7.7 TheSuddenApproximation 333
7.8 TheAdiabaticApproximation 335
7.8.1 ChirpedPulseAdiabaticPassage 336
7.8.2 StimulatedRamanAdiabaticPassage 338
7.8.3 TheLandau–ZenerProblem 339
7.8.4 GeneralizedDisplacementsandForces 343
7.8.5 BerryPhase 344
7.9 LinearResponseTheory 349
7.9.1 Susceptibilities 350
7.9.2 KuboFormulas 352
7.9.3 OnsagerReciprocalRelations 360
7.9.4 Fluctuation–DissipationTheorem 362
CHAPTER8 IDENTICALPARTICLES 367
8.1 PermutationSymmetry 367
8.1.1 TheSymmetricGroupS 369
N
8.1.2 YoungTableaux 370
8.2 ExchangeSymmetry 371
8.2.1 SymmetrizationPostulate 372
8.3 PermanentsandSlaterDeterminants 374
8.4 SimpleTwo-andThree-ElectronStates 375
8.5 ExchangeSymmetryforTwoTwo-LevelSystems 377
8.6 Many-ParticleExchangeSymmetry 378