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1、NUMERICALANALYSISFORENGINEERINGTERMPAPER:SolvingtheSchrodingerEquationforaParticleinanInfinitePotentialWellEleanorKaufmanDecember9,1999CONTENTS:1.Introduction2.DiscussionofAnalyticalSolution3.DiscussionofNumericalTechniques4.DiscussionofNumericalNormalizationT
2、echniques5.ListofSymbolsUsed6.AppendixA:AlgorithmstofindEigenvalues7.AppendixB:AlgorithmstoSolveInitialValueProblems8.AppendixC:NumericallyDeterminedEigenfunctions9.AppendixD:SelectedNormalizedEigenfunctions10.AppendixE:ErrorinNormalizedEigenfunctionsfor1stEne
3、rgyEigenvalue11.AppendixF:ErrorinNormalizedEigenfunctionsfor2ndEnergyEigenvalueINTRODUCTIONConsideraparticlemovinginathreedimensionalpotentialfield.Thewavefunction,y(r,t),isadescriptionoftheprobabilitythattheparticlewillbewithinagivenspatialvolumeatagiventime.
4、Thepositionprobabilitydensityisgivenby:.Introducingtheconceptofwave-particleduality,aparticleofmassm,welldefinedmomentumpandenergyEisdescribedbythiswavefunction.TheSchrödingerequationisasecondorderdifferentialequationdescribingthewavefunctionYthatcanbederivedb
5、ydifferentiatingfirstwithrespecttotimeandthentwicewithrespecttothespatialcoordinates.Thentheclassicalrelationcanbeusedtoequatethetworesults.(Foramorecompletedescriptionofthisderivation,pleaseseeReference2).Mostgenerally,theSchrödingerequationinvolvesa3-dimensi
6、onaltimedependentwavefunctionY(x,y,z,t)=Y(r,t):The1dimensional,time-independentSchrödingerequationisSinceydependsonlyonx,thederivativesarenolongerpartial.Theobjectiveofthispaperistoexplorethe1-dimensionaltime-independentSchrödingerequationasitappliestoaparticl
7、einsideofaninfinitesquarepotentialwell.Thepotentialinsideofthewelliszero,whilethepotentialoutsideofthewellisinfinite.Forconvenience,thewellhasbeenchosentobecenteredaboutx=0.Forthegivenpotential,weseethatthewavefunctionmustvanishoutsideofthewell,sincetheprobabi
8、litythataparticlewillexistoutsideofthewelliszero:.Insidethewell,thewavefunctionmustbedeterminedbysolvingtheSchrödingerequation:.Thereiszeropotentialinsidethewell,andinfinitepotenti
