3D Schrodinger Equation - NIU - Northern Illinois University 三维薛定谔方程-牛-北伊利诺斯大学

3DSchrodingerEquationSimplysubstitutemomentumoperatordoparticleinboxandHatomaddeddimensionsgivemorequantumnumbers.Canhavedegeneracies(morethan1statewithsameenergy).Addedcomplexity.SolvebyseparatingvariablesP460-3DS.E.IfVwell-behavedcanseparatefurther:V(r)orVx(x)+Vy(y)+Vz(z).Lookingatsecondone:LHSdependsonx,yRHSdependsonzS=separationconstant.RepeatforxandyP460-3DS.E.Example:2D(~sameas3D)particleinaSquareBoxsolve2differentialequationsandgetsymmetryassquare.“broken”ifrectangleP460-3DS.E.2Dgives2quantumnumbers.LevelnxnyEnergy1-1112E01-2125E02-1215E02-2228E0fordegeneratelevels,wavefunctionscanmix(unless“something”breaksdegeneracy:externalorinternalB/Efield,deformation….)thisstillsatisfiesS.E.withE=5E0P460-3DS.E.SphericalCoordinatesCansolveS.E.ifV(r)functiononlyofradialcoordinatevolumeelementissolvebyseparationofvariablesmultiplyeachsidebyP460-3DS.E.SphericalCoordinates-PhiLookatphiequationfirstconstant(knowinganswerallowsform)mustbesinglevaluedthethetaequationwilladdaconstraintonthemquantumnumberP460-3DS.E.SphericalCoordinates-ThetaTakephiequation,pluginto(theta,r)andrearrangeknowinganswergivesformofconstant.Givesthetaequationwhichdependson2quantumnumbers.AssociatedLegendreequation.Canuseeitheranalytical(calculus)oralgebraic(grouptheory)tosolve.Doanalytical.StartwithLegendreequationP460-3DS.E.SphericalCoordinates-ThetaGetassociatedLegendrefunctionsbytakingthederivativeoftheLegendrefunction.ProvebysubstitutionintoLegendreequationNotethatpowerofPdetermineshowmanyderivativesonecando.SolveLegendreequationbyseriessolutionP460-3DS.E.SolvingLegendreEquationPlugseriestermsintoLegendreequationletk-1=j+2infirstpartandk=jinsecond(thinkofitashavingtwoindependentsums).Combinealltermswithsamepowergivesrecursionrelationshipseriesendsifavalueequals0L=j=integerendupwithodd/even(Parity)seriesP460-3DS.E.SolvingLegendreEquationCanstartmakingLegendrepolynomials.BeinascendingpowerordercannowformassociatedLegendrepolynomials.CanonlyhavelderivativesofeachLegendrepolynomial.Givesconstraintonm(thetasolutionconstrainsphisolution)P460-3DS.E.SphericalHarmonicsTheproductofthethetaandphitermsarecalledSphericalHarmonics.AlsooccurinE&M.TheyholdwheneverVisfunctionofonlyr.SeenrelatedtoangularmomentumP460-3DS.E.3DSchr.Eqn.-RadialEqn.ForVfunctionofradiusonly.Lookatradialequationcanberewrittenas(usuallymuchbetter...)noteL(L+1)term.Angularmomentum.Actslikerepulsivepotentialandgoestoinfinityatr=0(alaclassicalmechanics)energyeigenvaluestypicallydependon2quantumnumbers(nandL).Only1/rpotentialsdependonlyonn(andtrueforhydrogenatomonlyinfirstorder.Afteraddingperturbationsduetospinandrelativity,dependsonnandj=L+s).P460-3DS.E.ParticleinsphericalboxGoodfirstmodelfornucleiplugintoradialequation.Canguesssolutionslookfirstatl=0boundaryconditions.R=u/randmustbefiniteatr=0.GivesB=0.Forcontinuity,musthaveR=u=0atr=a.givessin(ka)=0andnoteplanewavesolution.Supplement8-Bdiscussesscattering,phaseshifts.GeneraltermsareP460-3DS.E.ParticleinsphericalboxForLl>0solutionsareBesselfunctions.Oftenarisesinscatteringoffsphericallysymmetricpotentials(likenuclei…..).Canguessshape(alsocanguessfinitewell)energywilldependonbothquantumnumbersandso1s1p1d2s2p2d3s3d…………….andordering(excepthigherEforhighern,l)dependingondetailsgiveswhatnuclei(whatZorN)havefilled(sub)shellsbeingdifferentthanwhatatomshavefilledelectronicshells.Inatoms:innuclei(withjsubshells)P460-3DS.E.HAtomRadialFunctionForV=a/rget(usereducedmass)Laguerreequation.SolutionsareLaguerrepolynomials.Solveusingseriessolution(afterpullingoutanexponentialfactor),getrecursionrelation,geteigenvaluesbyhavingtheseriesend……nisanyinteger>0andL1ignorespinfornowEnergynlmD-13.6eV10(S)01-3.4eV20011(P)-1,0,13-1.5eV30011-1,0,132(D)-2,-1,0,1,251GroundState4Firstexcitedstates9secondexcitedstatesP460-3DS.E.ProbabilityDensityPisradialprobabilitydensitysmallrnaturallysuppressedbyphasespace(novolume)cangetaverage,mostprobableradius,andwidth(inr)fromP(r).(Supplement8-A)P460-3DS.E.MostprobableradiusFor1SstateBohrradius(scaledfordifferentlevels)isagoodapproximationoftheaverageormostprobablevalue---dependsonnandLbutelectronprobability“spreadout”withwidthaboutthesamesizeP460-3DS.E.RadialProbabilityDensityP460-3DS.E.RadialProbabilityDensitynote#nodesP460-3DS.E.AngularProbabilitiesnophidependence.If(arbitrarily)havephibeanglearoundz-axis,thismeansnox,ydependencetowavefunction.We’llseeinangularmomentumquantizationL=0statesaresphericallysymmetric.ForL>0,individualstatesare“squished”butinarbitrarydirection(unlessbrokenbyanexternalfield)AddupprobabilitiesforallmsubshellsforagivenLgetasphericallysymmetricprobabilitydistributionP460-3DS.E.Orthogonalityeachindividualeigenfunctionisalsoorthogonal.Manyrelationshipsbetweensphericalharmonics.Importantin,e.g.,matrixelementcalculations.OruseraisingandloweringoperatorsexampleP460-3DS.E.Wavefunctionsbuildupwavefunctionsfromeigenfunctions.examplewhataretheexpectationvaluesfortheenergyandthetotalandz-componentsoftheangularmomentum?havewavefunctionineigenfunctioncomponentsP460-3DS.E.