量子化學(xué)與群論基礎(chǔ)(5)課件

BriefReviewThemostimportantpropertiesofparticle1Thequantizatione.gquantizationofenergyenergylevels2Particle-WaveDualityΕ＝hνP＝h/λPlanck-Eistain-deBroglie

relationsParticleWaveInterferenceandDiffractionΔxΔPx≥h/4impossibletospecifysimultaneouslytheprecisepositionandmomentum.state—wavefunctionDynamic

equation—waveequationamplitudeψ*ψtheprobabilityoffindingtheparticleProbabilitywaveMathematicalBackground

andPostulatesofQuantumMechanics2.1OperatorsOperatorAnoperatorisasymbolthattellsyoutodosomethingwithwhateverfollowsthesymbol.e.g.,,,,ln,sin,d/dx

……Anoperatorisarulethattransformsagivenfunctionorvectorintoanotherfunctionorvector.e.g.2.1.1BasicPropertiesofOperatorsTwooperatorsareequalifThesumanddifferenceoftwooperatorsTheproductoftwooperatorsisdefinedbyTheidentityoperatordoesnothing(ormultipliesby1)Acommonmathematicaltrickistowritethisoperatorasasumoveracompletesetofstates(moreonthislater).TheassociativelawholdsforoperatorsThecommutativelawdoesnotgenerallyholdforoperators.Ingeneral,

Itisconvenienttodefinethequantitywhichiscalledthecommutatorofand.Notethattheordermatters,Ifandhappentocommute,thenThen-thpowerofanoperator

isdefinedasnsuccessiveapplicationsoftheoperator,e.g.Theexponentialofanoperator

isdefinedviathepowerseries2.1.2LinearOperators

Almostalloperatorsencounteredinquantummechanicsarelinearoperators.Alinearoperatorisanoperatorwhichsatisfiesthefollowingtwoconditions:

wherecisaconstantandfandgarefunctions.Asanexample,considertheoperatorsd/dxand()2.Wecanseethatd/dxisalinearoperatorbecauseHowever,()2isnotalinearoperatorbecauseTheonlyothercategoryofoperatorsrelevanttoquantummechanicsisthesetofantilinearoperators,forwhichTime-reversaloperatorsareantilinear.2.1.3EigenfunctionsandEigenvalues

Aneigenfunctionofanoperatorisafunctionusuchthattheapplicationofonugivesuagain,timesaconstantMatrixdescriptionofaneigenvalueequation2.1.4OperatorExpressionoftheTime-IndependentSchr?dingerEquationDefiniteLapacianthenDefiniteHamiltonianthen2.2PostulatesofQuantumMechanicsPostulate1Thestateofaquantummechanicalsystemiscompletelyspecifiedbyafunction(r,

t)thatdependsonthecoordinatesoftheparticle(s)andontime.Thisfunction,calledthewavefunctionorstatefunction,hastheimportantpropertythat

*(r,

t)(r,

t)distheprobabilitythattheparticleliesinthevolumeelementdlocatedatrattimet.Thewavefunctionmustbesingle-valued,continuous,andfinite.Postulate2Inanymeasurementoftheobservableassociatedwithoperator

,theonlyvaluesthatwilleverbeobservedaretheeigenvaluesa,whichsatisfytheeigenvalueequationPostulate3.Ifasystemisinastatedescribedbyawavefunction

,thentheaveragevalueoftheobservablecorrespondingto

isgivenbyPostulate4.Toeveryobservableinclassicalmechanicstherecorrespondsalinear,Hermitianoperatorinquantummechanics.Table1:Physicalobservablesandtheircorrespondingquantumoperators(singleparticle)ObservableObservableOperatorOperatorNameSymbolSymbolOperationPosition

r

Multiplyby

r

Momentum

Pi

KineticenergyT

Potentialenergy

V(r)

MultiplybyV(r)

TotalenergyE

Angularmomentumlx

ly

lz

Postulate4.Anarbitrarystatecanbeexpandedinthecompletesetofeigenvectorsof

aswherenmaygotoinfinity.InthiscaseweonlyknowthatthemeasurementofAwillyieldoneofthevaluesai,butwedon'tknowwhichone.However,wedoknowtheprobabilitythateigenvalueaiwilloccur--itistheabsolutevaluesquaredofthecoefficient,|ci|2

2.3HermitianOperatorsandUnitaryOperators2.3.1

HermitianOperatorsAsmentionedpreviously,theexpectationvalueofanoperator

isgivenbyandallphysicalobservablesarerepresentedbysuchexpectationvalues.Obviously,thevalueofaphysicalobservablesuchasenergyordensitymustbereal,sowerequire<A>tobereal.Thismeansthatwemusthave<A>=<A>*,orOperators

whichsatisfythisconditionarecalledHermitian.2.3.2UnitaryOperators

Alinearoperatorwhoseinverseisitsadjointiscalledunitary.Theseoperatorscanbethoughtofasgeneralizationsofcomplexnumberswhoseabsolutevalueis1.

U-1=U?

UU?=U?U=IAunitaryoperatorpreservesthe``lengths''and``angles''betweenvectors,anditcanbeconsideredasatypeofrotationoperatorinabstractvectorspace.LikeHermitianoperators,theeigenvectorsofaunitarymatrixareorthogonal.However,itseigenvaluesarenotnecessarilyreal.Wavefunctionψ：1Thestatedescription2ψ*ψ

Probabilitydensity3Thevalueofobservable4TheaveragevalueoftheobservableTheproblemisHowtogetWavefunction?Theonlywayis3SomeAnalyticallySolubleProblemsThemotionsofparticleTranslationalmotionRotationalmotionVibrationalmotionElectronicmotionNuclearmotionTheEnergyoftheparticle：3.1TheFreeParticleAfreeparticleisonewhichmovesthroughspacewithoutexperiencinganyforces.Henceittravelsinastraightline.Itspotentialenergyiseverywhereconstant,andsocanbeassignedtobe0.TheenergystatesareNOTquantized,butanyvalueisallowed.3.2TheParticleinaBox3.2.1The1-DimensionalParticle-in-a-Box(1)Schr?dingerEquationTheparticleofmassmisconfinedbetweentwowalls:V(x)=0(0<x<l)V(x)＝∞(x≤0andx≥0)letBoundaryconditionsx=0,(0)＝Asin0＋Bcos0=0;B=0(x)＝Asinkxx=l,(l)＝Asinkl=0;sinkl=0,kl=nπsquare,

n=1,2,3……quantumnumberThegeneralsolutionsare

(x)＝Asinkx＋Bcoskx

n=1,2,3…...(2)PropertiesofthesolutionsTherefore,thecompletesolutiontotheproblemis(i)Thequantizationofenergy

n=1,2,3…...quantumnumberThislowest,irremovableenergyiscalledthezero-pointenergy.E=T+VThe1-DimensionalParticle-in-a-Box,V=0,E=T（a）Zero-pointenergy(b)Elorm，EClassicalorfreeparticle，E0.(ii)WavefunctionandquantumnumbernGroundstateandexcitatedstate(iii)Probabilitydistributions(iv)Applications1,3-butadieneb-carotenel=210.140nm=3.08nm.Andthelowest11energylevelswillbefilled.Carrotsareorangebecausetheabsorptionoftheshortwavelength(blue)lightleavesonlythered-orangetoreflect.(v)OrthogonalityandthebracketnotationTwowavefuctionsareorthogonaliftheirproductvanishes.e.g.Theintegralisoftenwritten<n|n’>=0(n’n)Diracbracketnotation

<n|bra|n>ketNormalizedwavefuctions<n|n>=1Thesetwoexpressionscanbecombinedintoasingleexpression:Kroneckerdelta3.3TheTwoandThree-DimensionalParticle-in-a-Box3.3.1Motionintwodimensions(1)Schr?dingerEquationInbox，V＝0(2)Separationofvariablesψ＝X(x)·Y(y)E＝Ex＋Ey(3)Thesolution(4)DegeneracyConsiderthecasenx=1,ny=2andnx=2,ny=1Whena=bWesaythatthestates|1,2>and|2,1>aredegenerate.3.3.2Motioninthreedimensions(1)Schr?dingerEquationInbox，V＝0Separationofvariablesψ＝X(x)·Y(y)·Z(z)E＝Ex＋Ey＋Ez(2)Solution

(3)DegeneracyCubic，a

＝b＝c112121211E112=E121=E2113.4Vibrationmotion3.4.1TheHarmonicOscillator

(1)Schr?dingerEquationConsideraparticlesubjecttoarestoring

forceF=-kx,thepotentialisthenZero-point：(2)Thesolutions(i)Theenergylevelsv=0,1,2,3…(ii)Thewavefunctions3.5RotationalMotionR=ra+rbxyzrarbBAOTherigidrotorisasimplemodelofarotatingdiatomicmolecule.Weconsiderthediatomictoconsistoftwopointmassesatafixedinternucleardistance.(1)Schr?dingerEquationForarigidrotorso(2)ThesolutionsAfteralittleeffort,theeigenf