量子力学英文课件：Chapt1 Basic Concepts and Principles of Quantum Mechanics( A Brief Review)

QuantumMechanicsChapter1.BasicConceptsandPrinciplesofQuantumMechanics (ABriefReview)§1.1Wave-ParticleDualityExperimentalbackground1.BlackbodyradiationPlanck’squantumassumption:=nh(n=1,2,3….)2.PhotoelectriceffectEinstein’sassumptionofphotons:=h3.ComptoneffectThemomentumofphotons:

Wave-particleduality1.deBroglie’shypothesis:deBrogliewave2.Heisenberg’suncertaintyprinciple

§1.2BasicPostulatesofQuantumMechanicsBasicPostulatesofQuantumMechanics1.Thestateofaparticlewiththreepositionvariablesq1,q2,q3(whichmaybetheCartesian（笛卡尔）coordinatesx,y,z,orthesphericalpolarcoordinates,orsomeothersetsofcoordinates)isspecifiedbyastateorwavefunction(q1,q2,q3,t).Allpossibleinformationabouttheparticlecanbederivedfromthisstatefunction.•2.Toeveryobservabletherecorrespondsa

Hermitianoperatorgivenbythefollowingrules:(1)TheoperatorcorrespondingtotheCartesianpositionxisx—similarlyforthecoordinatesyandz.(2)Theoperatorcorrespondingto,thexcomponentoflinearmomentum,is----similarlyfortheyandzcomponents.(3)Toobtaintheoperatorcorrespondingtoanyotherobservable,firstwritedowntheclassicalexpressionfortheobservableintermsofx,y,z,,andthenreplaceeachofthesequantitiesbyitscorrespondingoperatoraccordingtorules(1)and(2).3.TheonlypossibleresultwhichcanbeobtainedwhenameasurementismadeofanobservablewhoseoperatorisÂisaneigenvalueofÂ.4.LetAbeanobservablewhoseoperatorÂhasasetofeigenfunctionswithcorrespondingeigenvalues.IfalargenumberofmeasurementsofAaremadeonasysteminthestate,thentheexpectationvalue（期望值）ofAforthestate(I.e.thearithmeticmean（算术平均）oftheeigenvaluesobtained)isgivenby,wheredisanelementofvolume,andtheintegralistakenoverallspace.5.Thetimevariationofthestatefunctionofasystemisgivenby,whereĤisoperatorformedfromtheclassicalHamiltonianofthesystem.•Basicdeductionfromthepostulates:Probabilityofresultofmeasurement

•SupposetheeigenvaluesofÂinpostulates4and5arediscrete,andthatthestatefunctionandalltheeigenfunctionsofÂarenormalized.TofindtheprobabilitythattheresultofameasurementoftheobservableAisaparticular,expandintermsofthe,i.e.put•Then

•Thecoefficientisobtainedfrom•Ifthecoefficientsareknown,aconvenientexpressionfortheexpectationvalueis.Chapt.2ThreeDimensionsandAngularMomentum§2.1TheSchroedingerEquationinThreeDimensionsThetime-independentSchreodingerequationThetime-independentSchreodingerequationmayalwaysbewrittenastheoperatorequationwhereE-Vis，asusual，thekineticenergy．Ifisafunctionofcartesiancoordinatesx，y，andz，thentheoperatormaybewritten

(6.2)•Schroedingerequationisthen(6.3)(6.4)

•andwecanwriteastheproductofaspacedependentpartandatime-dependentpart：

(6.5)TheThree-DimensionalHarmonicOscillatorThepotentialenergyofasphericallysymmetricthree-dimensionalharmonicoscillatormaybewritten(6.16)Therefore,writingthetime-independentSchroedingerequationintermsoftheHamiltonianoperatorH,wehave(6.17)wheretheindividualoperatorsHx,Hy,andHzaregivenby,,

•WecannoweasilyseparatethevariablesinEq.(6.16)bywritingthewavefunctionas,therebyobtainingthreeequationsthatareidenticaltothatforonedimensionaloscillator.•Eachsolution,likethesolutionsofonedimensionaloscillator,maybewrittenas (6.18)•whereweapplytheraisingoperatortimes,and=0,1,2,……•SubstitutionintoEq.(6.17)givesus,aftercarryingouttheoperations,(6.19)

whereeachofthethreequantumnumbers,andcanhaveanyofthevalues0,l,2,....Thusthetotalenergyisgivenby (6.20)•wherenisanypositiveintegerorzero.•Whenn=0wehavetheground-stateenergy,andtheunnormalizedeigenfunctionistheproduct

(6.21)Noticethecontrastbetweenthesequantumnumbersandthequantumnumbersforthethree-dimensionalsquarewell.Inthethree-dimensionalsquarewellnoneofthevaluesnx,ny,ornzcanbeequaltozero;intheharmonicoscillatoranyorallofthesecanequalzero,asisclearfromEq.(6.21).

•Theeigenfunctionsu100,u010,andu001forn=1arerespectively, (6.22)where

r2=x2+y2+z2.InSection4.lweshallseehowtheharmonicoscillatorcanbesolvedinthesphericalcoordinatesr,,and.§2.2SphericallySymmetricPotentialSphericalCoordinatesTheprinciplesandtechniquesoftheprevioussectionareapplicabletoallcoordinatesystems.Forthestudyofatoms,sphericalcoordinatesr,θ,andφaremostuseful.(SeeFigure6.2.)Thesecoordinatesarerelatedtothecoordinatesx,y,andzby(6.23)TowritetheSchroedingerequationinthesecoordinates,weneedtoreplacederivativeswithrespecttox,y,andzbyderivativeswithrespecttor,θ,andφ.Wedothisbyusingthechainruletocompute∂u/∂x,∂u/∂y,and∂u/∂z;forexample,

∂u/∂x=(∂u/∂r)(∂r/∂x)+(∂u/∂θ)(∂θ/∂x)+(∂u/∂φ)(∂φ/∂x)(6.24)andcorrespondingequationsfor∂u/∂yand∂u/∂z.•Thenecessarypartialderivativescanbeshown,withtheaidofEqs.(6.23),tobe•Wefindthesecondderivativesbyusingthechainruleagain,replacinguby∂u/∂x,toobtain

(6.25)andcorrespondingequationsfor•ItisunderstoodthatthefirstpartialderivativesinEq.(6.25)arealreadyexpressedintermsofsphericalcoordinatesandderivativeswithrespecttothosecoordinates.Forexample,(6.26)

•

CompletingtheoperationsofEq.(6.24)forallthreecoordinates,andsubstitutingintoEq.(6.3),wefinallyhavethetime-independentSchroedingerequationinsphericalcoordinates:(6.27)

•

whereuisnowafunctionofr,θ,andφ.ThereadershouldverifythateachterminEq.(6.27)hasthesamedimensions.

•Beforeattemptingtogeneratesolutionstothisequation,letusdiscusssomegeneralprinciples.§2.3AngularMomentumOperatorsandEigenvaluesAngularmomentumoperatorsThelawofconservationofangularmomentumisobeyedinallsystemsinwhichthepotentialenergyisindependentoftheangularcoordinates.Thisistrueinquantummechanicsaswellasinclassicalmechanics.Inbothclassicalandquantummechanics,thedefinitionofangularmomentumisthevectorequationL=r

p(6.28)However,inquantummechanicsthevectorpisanoperator.ThusthevectorLmustalsobeanoperator,whosecomponentscanbededucedfromthedefinitionofthecrossproduct.InCartesiancoordinatesthesecomponentsare

(6.29)•Usingthechainrule,wecanwriteEqs.(6.29)insphericalcoordinates:(6.30)

FromEqs.(6.30)wecannowfindthemagnitudeofthevectorL.FirstweconstructtheL2operatorasfollows:(6.31)with(6.32)Theresultis(6.33)

YoucanseethattheoperatorhereisthesameoperatorthatappearsintheSchroedingerequation[Eq.(6.27)].WecanrewriteEq.(6.27)intermsofas(6.34)•

SeparationofVariablesinSphericalCoordinates•AswedidwithCartesiancoordinates,letusnowseparatethevariables.Wewritethewavefunctionuasaproductofthreefunctions,oneforeachcoordinate.•Inthiscasewehaveu(r,θ,φ)=R(r)Y(θ,φ)=R(r)Θ(θ)Φ(φ)(6.35)whereY(θ,φ)is,asweshallsee,asphericalharmonic,afunctionwellknowninsolutionsofLaplace'sequation.•Equation(6.34)nowbecomes•Rememberingthat[definedinEq.(6.33)]operatesonlyonangularcoordinates,wecannowseparatethevariablerfromtheangularvariablesθandφ.Wedividebothsidesof(6.36)byR(r)Y(θ,φ)toobtain(6.37)•Multiplyingbothsidesbyr2andcombiningtermsgivesus(6.38)•Followingthesamereasoningusedbefore,wededucethatthelasttermontheright-handsideofEq.(6.38)mustbeconstant,becauseitvariesonlywiththeangularcoordinates,andtheothertermsareindependentofangle.Thuswemaywrite(6.39)whereαisadimensionlessconstanttobedetermined.•UsingEq.(6.33).wecannowfurtherseparatethevariablesbywritingY(θ,φ)=Θ(θ)Φ(φ),sothatEq.(6.39)becomes(6.40)•or(6.41)Asbefore,wecannowobtaintwoseparateequations.WedividebothsidesbyΘ(θ)Φ()andmultiplybothsidesbysin2,obtaining(6.42)Thesecondterm,theonlytermthatisdependentonthevariable,mustbeconstant,sowemaywrite(6.43)Theformoftheconstantisdictatedbyareasonableconditiononthecoordinate.Werequirethat(6.44)becausetheangle+2describesthesamepointinspaceastheangle,ifrandareunchanged.If2werenotequalto,thenthewavefunctionwouldbedoublevalued.(Althoughtherearecertainconditionsunderwhichadouble-valuedwavefunctionisacceptable,thatisnotpossibleinthiscase;seeProblem9.)Whencondition(6.44)issatisfied,isaperiodicfunctionwithperiod2.ThuswecanwriteEq.(6.43)as(6.45)andEq.(6.42)becomes

(6.46)wheremisaninteger.ThesolutiontoEq.(6.44)maybewrittenintheform

Aeim(6.47)whereAisanormalizationconstant.NoticethatthissolutionisaneigenfunctionoftheLzoperator,for(6.48)•Theeigenvaluesaremħ,andthereforetheonlypossibleresultsofameasurementofanycomponentofangularmomentumareintegralmultiplesofħ.•ThisisreminiscentoftheBohrcondition,butwithonehighlysignificantdifference:mcanbezero.Ifm=0,thenEq.(6.46)becomes(6.49)•Equation(6.49)isknownasLegendre'sequation,andthesolutionsarecalledLegendrepolynornials.TheLegendrepolynomialscanbewrittenintheform(6.50)•ThesolutionstoEq.(6.49)arethesepolynomialswiththevariablecosinsteadofx.YoushouldverifyfromEq.(6.50)thatthepolynomialsforl=0,l,2,and3(withx=cos)are,respectively,P0=1,P1=cos,P2=(3cos2-1)/2,P3=(5cos2-3cos)/2(6.51)•Severalfeaturesofthesepolynomialsaresignificant.•

Eachpolynomialcontainsonlyeven(odd)powersofcosifliseven(odd).•Eachpolynomialisnormalizedtobeequalto1when=0.•Ineachpolynomialthehighestpowerofcosisequaltol.•SubstitutionofaLegendrepolynomialintoEq.(6.49)nowallowsustofindthevalueof,andthustofindthevalueofL2.Thefirstthreeresultsare:•Whenl=0,()=l,and=0;thetotalangularmomentumiszero.•Whenl=l,()=cos;substitutionintoEq.(6.49)showsthat=2.•Whenl=2,()=(3cos2-l)/2;Eq.(6.49)showsthat=6.Theseresultswerebaseduponthevaluem=0.ItisshowninAppendixDthat,ingeneral,=l(l+1);thisistrueforanyvalueofm,aslongasml.WecannowsummarizetheresultsofEqs.(6.39)asfollows:ThepossibleresultsofameasurementofLzaremħ; m=-l,-l+l,...,+l.ThepossibleresultsofameasurementofL2arel(l+l)ћ2;l=0,l,2,....EigenfunctionsofL2•EacheigenfunctionofL2correspondstoanallowedcombinationofthequantumnumberslandm

and

maybewrittenasasphericalharmonic:(lm)(6.52)wherethefunctionsPlm(cos),foundbysolvingEq.(6.39a),arecalledassociatedLegendrefunctions.•NoticethatPlm(cos)dependsonlyontheabsolutevalueofm,becauseonlym2appearsinEq.(6.39a).AfewnormalizedeigenfunctionsareshowninTable6.l.3•ProbabilityandSphericalCoordinates•ThenormalizationfactorsshowninTable6.1arebasedonthefactthattheintegraloftheprobabilitydensityu(r,,)overallspacemustbeequaltounity,•or(6.53)wherer2sindrddisthevolumeelementinsphericalcoordinates.•Itisconvenienttonormalizetheradialandangularpartsofthewavefunctionseparately.Thuswerequirethat(6.54)•Thisrequirementenablesustointerpretl,m2asaprobabilitydensityfortheangularcoordinatesofaparticle.IftheangulardependenceofthewavefunctionisYl,m(,),then

equalstheprobabilitythattheparticlewillbeobservedwhere1<<2and

1<<2.•Noticethatl,m2isafunctionofonly,becauseeim2=1.Figure6.3showspolarplotsof2,0

2,2,12and

|2,22.•Eachoftheserepresentsaparticlewiththesametotalangularmomentum,butwithvaluesof0,ћ,and2ћ,respectively,forLz.•AsLz

increases,wefindthatthemaximumprobabilitymovesfromthezaxis(θ=0)tothe

xyplane(θ=π/2).•Thisbehavioragreeswithwhatwewouldexpectclassically;anorbitinthexyplanemust

correspondtoalargevalueofLz,andaparticleonthezaxismusthaveLz

equaltozero.•Summary

•ThefunctionsYl,m(θ,φ)areeigenfunctionsofboththeLzoperatorandtheL2operator,givenbyEqs.(6.28)and(6.31),respectively.•TheL2operatoristhesumofthethreeoperatorsLx2,Ly2,andLz2,givenbythreeequations(6.30).•WehavenotyetconsideredtheeigenfunctionsofLxandLy,butlogicdictatesthat,foragivenvalueofl,thesetof EigenvaluesofLxandLyshouldbethesameasthoseofLz.Letusnowseewherethisleads.§2.4StatisticalAnalysisofAngularMomentumEigenfunctions

Section2.3posestwoquestionsfortheinquiringreader:(l).WhathappenswhenyoumeasureLxorLy?(2).IsthereaphysicalreasonwhyLzisequaltol(l+l)ћ2,whenLz2hasamaximumvalueofonlyl2ћ2?•Thequickanswertoquestion(1)isthatthesamethinghappenswhenyoumeasureLx,Ly,orLz,becausex,y,andzarearbitrarylabels.Asweshallseeinsection2.5,wecanonlymeasureonecomponentofLatatime.andthecomponentthatwehavemeasurediscustomarilycalledthezcomponent.ThelongeranswerhastodowithmeasuringasecondcomponentofL,sayLx,aftermeasuringLz.WedealwiththatinSection2.5.Inanswertoquestion2,letusassumethatwehaveacollectionofatomsallhavingthesamevalueofl,andconsiderwhathappenswhenwemeasureLz.IfweknownothingaboutthedirectionofthevectorLforeachatom,wemustassumethateachofthepossiblevaluesofmisequallyprobable.Thereare21+1possiblevaluesofm;thuswecancomputetheexpectationvalueofLz2byaddingupallofthepossibleresultsanddividingby21+1:(6.55)•Itiseasilyshownbymathematicalinductionthattheseriessumisequaltol(l+1)(2l+l)/6.Substitutioninto(6.55)thenshowsthat

Lz2=l(l+1)ћ2/3(6.56)•IntheabsenceofinformationaboutthedirectionofL,wemustassumethatEq.(6.52)holdsfortheothercomponentsofLaswell;thus

andtherefore(6.57)•Then,sinceeachparticlehasthesamevalueofL2,(6.58)•AfterwehavemeasuredL2andLz,weknowthatthevectorLliessomewhereonaconewhoseaxisisthezaxisandwhoseapexangleis(Figure6.4),where(6.59)

•Allazimuthalanglesareequallylikely,becausetheprobabilitydensityfactorYl,misindependentof.ThusitcannottellusthevalueofeitherLxorLy..•However,thevaluesofL2andLzdotellussomethingaboutLxandLy..FIGURE6.4VisualrepresentationoftheangularmomentumvectorL,makinganangleηwiththez

axis...•

ExampleProblem3.2

Wehaveacollectionofparticles,allofwhichareknowntohaveLz=3ћandL2=12ћ2.FindLxandLx2fortheseparticles.

•

Solution.SinceallallowedvaluesofLx,positiveandnegative,areequallylikely,expectationvalueLxmustbezero.•TofindLx2,werewriteEq.(6.57)asLx2Ly2L2-Lz2=12ћ2-9ћ2=3ћ2.

•Sincealldirectionsinthexyplaneareequallyunknown,thereisnodifferencebetweenLx2andLy2,andLx2=Ly2.ThusLx2+Ly2=2Lx2=3ћ2,andLx2=1.5ћ2.•ExampleProblem3.3

•Determinetheangularmomentumofeacheigenfunctionu100,u010,andu001ofthethree-dimensionalharmonicoscillator.

Solution.RefertoEqs.(6.22).Thefunctionsare,,•Theangulardependenceofeachfunctioniscontainedinthefactorx,y,orz.Table6.lshowsthatY1,0=(3/4π)1/2

cosθ=(3/4π)1/2z/r.•ThereforezisproportionaltorY1,0,makingtheangulardependenceofu001equaltoY1,0.Thusforu001wehavel=1andm=0,orandLz=0.

WealsocanshowfromthetablethatY1,1+Y1,-1=(3/2π)1/2x/rsothatx=(2π/3)1/2r(Y1,1+Y1,-1)andu100=Nr(Y1,1+Y1,-1)e-ar2/2whereNisanormalizationconstant.

•Thusu100isalsoaneigenfunctionofL2withl=1,butitisamixtureofm=+landm=-lwithequalamplitudes.AmeasurementofLzwouldyieldwithequalprobability.Thesameistrueforu010;ittooisamixtureofm=1andm=-1.

6.5ExperimentalTestofTheory:DoubleStern-GerlachExperiment

UncertaintyintheMeasurementofAngularMomentumComponentsFigure6.4illustratesthefactthat

Itisnotpossibletoknowtwocomponentsofangularmomentumsimultaneously.Anotherwayofsayingthisis:

Itisnotpossibletoknowtwocomponentsofangularmomentumsimultaneously.•

IfwecouldknowtwocomponentsofLaswellasthemagnitudeofL,thenwewouldknowtheprecisedirectionofL,sothetwostatementsareequivalenttotestthesestatements,wemustrelyonexperiment.Howdowemeasureangularmomentumontheatomicscale?TheStern-GerlachExperimentTheangularmomentumofatomicstateswasfirstmeasuredbySternandGerlach.4Themeasurementwasbasedonthefactthatanorbitingelectroninanatomformsacurrentloopandthushasamagneticdipolemomentμ.ThismagneticmomentisproportionaltotheorbitalangularmomentumL;fromstandardelectromagnetism5weknowthatμ=-eL/2m.WealsoknowthatifabeamofatomspassesthroughanFIGURE6.5(a)SchematicviewofStern-Gerlachapparatus.Atomstravelalongsharppolepiece.whichproducesafieldthatisstrongernearthenorthpole(N).Theresultingfieldlinesconvergeandarenotexactlyvertical.(b)EdgewiseviewofacurrentloopintheBfieldof(a).BecausetheBfieldlineareinthedirectionsshown.theforcesFallhavedownwardcomponents.resultinginanetdownwardforceontheloop.

inhomogeneousmagneticfieldB(Figure6.5)thedipolemomentofeachatomresultsinaforcewhosezcomponentisFigure6.5bshowstheoriginofthenetforce,whichactedoneachatomwhichhadnonzeroLz.BecauseLzcantakeononlyadiscretesetofvalues,onlycertainanglesofdeflectionareseeninsuchanexperiment.Thefigureillustratesthecasel=1,whenthreedeflectionsareseen,correspondingtoorLz=zero.ThusweknowthevalueofL2andthevalueofLzforeachatomsimplyfromitspresenceinonebeamoranother.

LetusnowfollowthebeaminwhichtheatomshaveandtrytodeterminethevaluesofLxfortheseatomsbysendingthemthroughanothermagnetinwhichthefieldisinthexdirection.Presumablythisbeamwillalsosplitintothreebeams,correspondingtoorLx=zero.ConsidertheatomsforwhichCanwenotsaythatwenowknowthevaluesofbothLxandLz

fortheseatoms?Wheredoestheexperimentaldifficultylie?Classically.therewouldbenodifficulty.LzwouldchangewhenwemeasureLx,butitwouldchangebyaknownamount,andwewouldthenknowthevaluesofbothcomponentsLxandLz.AnalysisofthemechanismforthischangeinLzwillshowwhythechangemustbeuncertainaccordingtoquantummechanics.Whenwesendthebeamthroughthemagnet,themagneticfieldBcausesatorqueofμ×Boneachatominthebeam,andclassicalmechanicstellsusthat

Tosimplifytheanalysis,letussetthethirdangularmomentumcomponent,Ly,equaltozero.ThefirstmeasurementtoldusthatLzwasequalto-ћwhentheatomenteredthesecondmagnet.WhenwemeasureLx,Bisinthexdirection.ThusEq.(6.61)becomeswherejisaunitvectorintheydirection.Therefore,whentheLvectorliesinthexzplaneaswehaveassumed,itmustacquireaycomponent.Inashorttimeδt,thechangeinLisAstimepasses,thedirectionsofLandδLchange,andL

precesses（进动）aboutthexaxis(Figure6.6).

AsthetipoftheLvectordescribesacirclewhoseradiusis–ћ,theangleφxthroughwhichitmovesintimetisgivenbyFIGURE6.6PrecessionofthetipoftheLvectorinamagneticfieldalongthexaxis.LchangesbyδLintimeδt.ThusclassicaltheorytellsusthatLzchangeswhenwemeasureLx,asyoucanseefromFigure6.6.Butclassicaltheorysaysthatwe