Quantum Physics in the Nanoworld: Schrödinger's Cat and the Dwarfs

472 Pages·2013·6.356 MB·English

by  Hans Lüth (auth.)

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Hans Lüth
Quantum Physics
in the Nanoworld
Schrödinger’s Cat and the Dwarfs
HansLüth
ForschungszentrumJülichGmbH
PeterGrünbergInstitut(PGI)
PGI-9:SemiconductorNanoelectronicsand
JülichAachenResearchAlliance(JARA)
Jülich,Germany
TranslationfromtheGermanlanguageedition:QuantenphysikinderNanowelt
byHansLüth,©2009Springer-VerlagBerlinHeidelberg.Allrightsreserved
ISSN1868-4513 ISSN1868-4521(electronic)
GraduateTextsinPhysics
ISBN978-3-642-31237-3 ISBN978-3-642-31238-0(eBook)
DOI10.1007/978-3-642-31238-0
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To Roswitha
Preface
The original German edition of this book was published in 2009. Because of the
positiveresponseIhavegotfromstudentsandcolleaguesItranslatedthebookinto
Englishandfurthermoreaddedsomenewproblems,thelastchapter“synopsis”and
anadditionalAppendixaboutthereduceddensitymatrix.
What was the reason to write this book? There are a large number of excellent
textbooksonquantummechanicsonthemarket.Nearlyallofthesebookshavein
common, that quantum mechanics is presented as one of the most important and
successfultheoriestosolvephysicalproblems.Thisistotallyinthesenseofmost
physicists,whoapplied,untilthe1970softhe20thcentury,inafirstquantumrevo-
lutionquantummechanicswithoverwhelmingsuccessnotonlytoatomandparticle
physicsbutalsotonearlyallothersciencebranchesaschemistry,solidstatephysics,
biology or astrophysics. Because of the success in answering essential questions
in these fields fundamental open problems concerning the theory itself were ap-
proachedonlyinrarecases.Thissituationhaschangedsincethelastdecadeofthe
20th century. Since then there are new sophisticated experimental tools in quan-
tumoptics,atomandionphysicsandinnanoelectronics,whichcantouchinherent
quantum physical questions and allow interesting tests of the theory itself. Such
questions,asforexample,originandconsequencesofsuperpositionandentangle-
ment,areofpredominantimportanceforfieldsasquantumteleportation,quantum
computingandquantuminformationingeneral.
Fromthis“secondquantumrevolution”asthiscontinuingfurtherdevelopmentof
quantumphysicalthinkingiscalledbyAlainAspect,oneofthepioneersinthisfield,
oneexpectsadeeperunderstandingofquantumphysicsitselfbutalsoapplications
in engineering. There is already the term “quantum engineering” which describes
scientificactivitiestoapplyparticlewavedualityorentanglementforpracticalpur-
poses,forexample,nano-machines,quantumcomputersetc.
This background in mind I have written the present book. Particular quantum
phenomenaaremoreatthecenterofinterestratherthanthemathematicalformal-
ism.Ipreferamorepictorialandsometimesintuitivedescriptionofthephenomena,
andrecentexperimentalfindingsfromresearchonnanoelectronicsystemsareoften
presentedtosupportthetheory.Alsoconnectionstoothersciencebranchessuchas
vii
viii Preface
elementaryparticle,quantumelectronicsornuclearmagneticresonanceinbiology
andmedicinearemade.
ConcerningtheformalismIgenerallyrestrictmyselftofirstapproximationsteps,
which are relevant for experimental physicists and engineers in applying the the-
ory or to estimate the order of magnitude of experimental results or data. On the
otherhand,theDiracbra-ketnotationisintroducedinanalogytothree-dimensional
vectors and it is used for simplicity reasons in many cases. Similarly commutator
algebra is introduced as essentially adding or subtraction of symbols (operators).
Themathematicalbackgroundnecessarytoreadthebookisquitesimple.Onlythe
knowledge of simple functions, simple differential equations and basics of matrix
algebraisrequired.
RatherthanaxiomaticallyintroducingimportantquantitiesandequationsIhave
preferredtomaketheinventionofbasicequationsorthemathematicaltoolsforfield
quantizationplausiblebyphysicallyreasonableconclusionsandextrapolations.
The book was written on the basis of manuscripts of lectures about quantum
physicsandnanoelectronics,whichIhavegiventophysicsandelectricalengineer-
ingstudentsattheAachenUniversityofTechnology(RWTH).Essentialextensions
are, of course, due to my own research in quantum electronics. In particular, su-
pervisingPhDstudentsinthisfieldandthemanydiscussionswiththemhadgreat
influenceonthewayofpresentation.Iwanttothankallofthemfortheinteresting
discussions which also helped me to a deeper insight into the fascinating field of
quantumphysics.
Furthermore,Iwanttothankmyformercoworkers,meanwhileallinacademic
teaching and research positions, Arno Förster, Michel Marso, Michael Indlekofer
andThomasSchäpersformanyexcitingdisputes,whichcontributedtofurtherelu-
cidationofdifficultquestions.
During the translation of the original German edition into English Margrit
Klöcker sometimes improved and corrected my English grammar; also thanks to
her.
I owe very special thanks to my late wife Roswitha. She supported me all the
time during which I wrote the original German manuscript and she invented the
subtitle“Schrödinger’sCatandtheDwarfs”.Thissubtitleaccuratelyexpressesthe
mainfocusofthebook,namelyamorethoroughdivingintothephysicalandphilo-
sophical content of quantum mechanics (paradigm: Schrödinger’s cat), and this in
thecontextofthenanoworld(worldofthedwarfs).Roswithafoundtherightwords
forthisaspectofthebookthatIlacked.
AachenandJülich,Germany HansLüth
September2012
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 GeneralandHistoricalRemarks . . . . . . . . . . . . . . . . . . . 1
1.2 ImportanceforScienceandTechnology. . . . . . . . . . . . . . . 3
1.3 PhilosophicalImplications . . . . . . . . . . . . . . . . . . . . . . 5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 SomeFundamentalExperiments . . . . . . . . . . . . . . . . . . . . 9
2.1 PhotoelectricEffect . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 ComptonEffect . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 DiffractionofMassiveParticles . . . . . . . . . . . . . . . . . . . 15
2.4 ParticleInterferenceattheDoubleSlit . . . . . . . . . . . . . . . 19
2.4.1 DoubleSlitExperimentswithElectrons. . . . . . . . . . . 21
2.4.2 ParticleInterferenceand“Which-Way”Information . . . . 23
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Particle-WaveDuality . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 TheWaveFunctionandItsInterpretation . . . . . . . . . . . . . . 27
3.2 WavePacketandParticleVelocity . . . . . . . . . . . . . . . . . . 30
3.3 TheUncertaintyPrinciple . . . . . . . . . . . . . . . . . . . . . . 34
3.4 AnExcursionintoClassicalMechanics . . . . . . . . . . . . . . . 37
3.5 Observables,OperatorsandSchrödingerEquation . . . . . . . . . 40
3.6 SimpleSolutionsoftheSchrödingerEquation . . . . . . . . . . . 46
3.6.1 “Locked-Up”Electrons:ConfinedQuantumStates . . . . . 46
3.6.2 ParticleCurrents . . . . . . . . . . . . . . . . . . . . . . . 54
3.6.3 ElectronsRunAgainstaPotentialStep . . . . . . . . . . . 55
3.6.4 ElectronsTunnelThroughaBarrier . . . . . . . . . . . . . 58
3.6.5 ResonantTunneling . . . . . . . . . . . . . . . . . . . . . 63
3.7 SingleElectronTunneling . . . . . . . . . . . . . . . . . . . . . . 72
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4 QuantumStatesinHilbertSpace . . . . . . . . . . . . . . . . . . . . 79
4.1 EigenvectorsandMeasurementofObservables . . . . . . . . . . . 79
ix
x Contents
4.2 CommutationofOperators:Commutators . . . . . . . . . . . . . 85
4.3 RepresentationofQuantumStatesandObservables . . . . . . . . 87
4.3.1 VectorsofProbabilityAmplitudesandMatrices
asOperators . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.2 RotationsofHilbertSpace . . . . . . . . . . . . . . . . . . 92
4.3.3 QuantumStatesinDiracNotation . . . . . . . . . . . . . . 95
4.3.4 QuantumStateswithaContinuousEigenvalueSpectrum . 99
4.3.5 TimeEvolutioninQuantumMechanics . . . . . . . . . . . 103
4.4 GameswithOperators:TheOscillator . . . . . . . . . . . . . . . 106
4.4.1 TheClassicalHarmonicOscillator . . . . . . . . . . . . . 107
4.4.2 Upstairs-Downstairs:StepOperatorsandEigenvalues . . . 108
4.4.3 TheAnharmonicOscillator . . . . . . . . . . . . . . . . . 115
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5 AngularMomentum,SpinandParticleCategories . . . . . . . . . . 119
5.1 TheClassicalCircularMotion . . . . . . . . . . . . . . . . . . . . 119
5.2 QuantumMechanicalAngularMomentum . . . . . . . . . . . . . 121
5.3 RotationalSymmetryandAngularMomentum;Eigenstates . . . . 128
5.4 CirculatingElectronsinaMagneticField . . . . . . . . . . . . . . 134
5.4.1 TheLorentzForce . . . . . . . . . . . . . . . . . . . . . . 134
5.4.2 TheHamiltonOperatorwithMagneticField . . . . . . . . 135
5.4.3 AngularMomentumandMagneticMoment . . . . . . . . 137
5.4.4 GaugeInvarianceandAharanov–Bohm-Effect . . . . . . . 140
5.5 TheSpin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.5.1 Stern–GerlachExperiment. . . . . . . . . . . . . . . . . . 147
5.5.2 TheSpinandIts2DHilbertSpace . . . . . . . . . . . . . 151
5.5.3 SpinPrecession . . . . . . . . . . . . . . . . . . . . . . . 155
5.6 ParticleCategories:FermionsandBosons. . . . . . . . . . . . . . 157
5.6.1 TwoandMoreParticles . . . . . . . . . . . . . . . . . . . 157
5.6.2 SpinandParticleCategories:ThePauliExclusionPrinciple 161
5.6.3 TwoDifferentWorlds:FermiandBoseStatistics . . . . . . 166
5.6.4 TheZooofElementaryParticles . . . . . . . . . . . . . . 173
5.7 AngularMomentuminNanostructuresandAtoms . . . . . . . . . 183
5.7.1 ArtificialQuantumDotAtoms . . . . . . . . . . . . . . . 183
5.7.2 AtomsandPeriodicTable . . . . . . . . . . . . . . . . . . 190
5.7.3 QuantumRings . . . . . . . . . . . . . . . . . . . . . . . 196
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6 ApproximateSolutionsforImportantModelSystems . . . . . . . . . 201
6.1 ParticlesinaWeaklyVaryingPotential:TheWKBMethod . . . . 202
6.1.1 Application:TunnelingThroughaSchottkyBarrier . . . . 204
6.2 CleverGuessofaWaveFunction:TheVariationalMethod. . . . . 206
6.2.1 ExampleoftheHarmonicOscillator . . . . . . . . . . . . 210
6.2.2 TheGroundStateoftheHydrogenAtom . . . . . . . . . . 212
6.2.3 MoleculesandCoupledQuantumDots . . . . . . . . . . . 215
Contents xi
6.3 SmallStationaryPotentialPerturbations:TheTime-Independent
PerturbationMethod . . . . . . . . . . . . . . . . . . . . . . . . . 223
6.3.1 PerturbationofDegenerateStates . . . . . . . . . . . . . . 227
6.3.2 Example:TheStarkEffectinaSemiconductorQuantum
Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
6.4 Transitions Between Quantum States: The Time-Dependent
PerturbationMethod . . . . . . . . . . . . . . . . . . . . . . . . . 232
6.4.1 PeriodicPerturbation:Fermi’sGoldenRule. . . . . . . . . 234
6.4.2 Electron–LightInteraction:OpticalTransitions . . . . . . . 237
6.4.3 OpticalAbsorptionandEmissioninaQuantumWell. . . . 239
6.4.4 DipoleSelectionRulesforAngularMomentumStates . . . 243
6.5 ElectronicTransitionsin2-LevelSystems:TheRotatingWave
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6.5.1 2-Level Systems in Resonance with Electromagnetic
Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
6.5.2 SpinFlip . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
6.5.3 Nuclear Spin Resonance in Chemistry, Biology and
Medicine . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
6.6 ScatteringofParticles . . . . . . . . . . . . . . . . . . . . . . . . 265
6.6.1 ScatteredWavesandDifferentialScatteringCrossSection . 267
6.6.2 ScatteringAmplitudeandBornApproximation . . . . . . . 269
6.6.3 CoulombScattering . . . . . . . . . . . . . . . . . . . . . 274
6.6.4 ScatteringonCrystals,onSurfacesandonNanostructures . 278
6.6.5 InelasticScatteringonaMolecule. . . . . . . . . . . . . . 285
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
7 Superposition,EntanglementandOtherOddities . . . . . . . . . . . 291
7.1 SuperpositionofQuantumStates . . . . . . . . . . . . . . . . . . 291
7.1.1 Scattering of Two Identical Particles: A Special
SuperpositionState . . . . . . . . . . . . . . . . . . . . . 294
7.2 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
7.2.1 Bell’sInequalityandItsExperimentalCheck . . . . . . . . 302
7.2.2 “WhichWay”InformationandEntanglement:AGedanken
Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . 309
7.3 PureandMixedStates:TheDensityMatrix . . . . . . . . . . . . . 313
7.3.1 QuantumMechanicalandClassicalProbabilities . . . . . . 313
7.3.2 TheDensityMatrix . . . . . . . . . . . . . . . . . . . . . 317
7.4 QuantumEnvironment,MeasurementProcessandEntanglement . 321
7.4.1 SubsystemandEnvironment . . . . . . . . . . . . . . . . 321
7.4.2 OpenQuantumSystems,DecoherenceandMeasurement
Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
7.4.3 Schrödinger’sCat . . . . . . . . . . . . . . . . . . . . . . 328
7.5 SuperpositionStatesforQuantum-BitsandQuantumComputing . 329
7.5.1 CoupledQuantumDotsasQuantum-Bits . . . . . . . . . . 331