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首页 03 正交函数与多极子展开.pdf

03 正交函数与多极子展开.pdf

03 正交函数与多极子展开.pdf

上传者: zenger2017 2018-03-03 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《03 正交函数与多极子展开pdf》,可适用于自然科学领域,主题内容包含OrthogonalFunctionsandMultipoleExpansionExpansionofarbitraryfunctionsinter符等。

OrthogonalFunctionsandMultipoleExpansionExpansionofarbitraryfunctionsintermsofacompletesetoffunctionsInmathematicalphysicstheexpansionofanarbitraryfunctionintermsofanorthogonalsystemoffunctionsplaysanimportantroleTheFourierseriesencounteredintheproblemofthevibratingstringisanexampleofsuchanexpansionLetusconsidernowafiniteorinfinitesystemofrealorcomplexfunctions(),(),,(),nUxUxUxintheinterval,abThissystemoffunctionsiscalledtobeorthogonalifthefunctionssatisfytheorthogonalityrelation:()()bnmnnmaUxUxdxs()wheremUisthefunctioncomplexconjugatetomUIfnsfor,,n,thenthesystemissaidtobenormalized(orthonormalsystem)If()Uxisnotidenticalwiththefunction,thatis,()()baUxUxdx()thenthisintegralisdenotedthenormofthefunction()UxSo,afunctionissaidtobenormalizedif()()baUxUxdx()Anyfunctiondifferentfromthefunctioncanbenormalizedbymultiplyingwithanappropriateconstantsothat()()()()baUxUxUxUxdx()isthenormalizedfunctioncorrespondingto()UxTheanalogytotheorthogonalvectorsewitheeinnisobviousTherelation()correspondstothenormalizationofavectoraeaaathelefthandsideofequation()correspondstothescalarproductofvectorsNow,wewanttoexpandanarbitraryfunction()fxin,abintermsofanorthogonalsystemoffunctions()nUxThisexpansionisgivenby()()NnnnfxaUx()Thiscorrespondstotheexpansionofavectorainthebasise()aaeaeeAnalogoustothecorrespondingnotioninthetheoryofFourierseries,thenumbersnaareoftendenotedasgeneralizedFouriercoefficientsNowithastobeinvestigatedwhetherthisseries()convergesto()fxasN,thatis,whethertheequation()()nnnfxaUxisvalid,andwhatthecoefficientsnalooklikeLetthefunction()fxbedescribedby()WetakethesquareerrorNM:(,,)()()bNNNnnnaMaafxaUxdx()anddeterminethecoefficientsnainsuchawaythatweminimizeNMSquaringtheintegrandweobtain,NNNNjkjkjjjjjkjjMaalaladwith()()bjkjkaUxUxdx,()()bjjalUxfxdx,()()bjjalUxfxdxand()()badfxfxdxasthenormof()fxNMbecomesminimalforthosevaluesofjaforwhichNjjjMala,NjjjMalaie,()()bjjjaalUxfxdx(a)isvalidThisminimalvaluemin()NMis()()|()|||bbNNNnnjjjnjjaafxaUxdxdaafxdxa(b)andequation()followsimmediatelyInthesenseoftheleastsquaresmethod,theseriesformedwiththeka()()NNkkkFxaUx()isthebestapproximationto()fxwhichcanbeachievedbyNfunctionsofanorthonormalsystemBut,thesystemoffunctionsisasuitableapproximationto()fxonlyifminlim()lim()()bNNNNaMFxfxdx()ThisistheconditionforthesocalledconvergenceinsquaremeanIngeneral,thisnotationistransferredtoanyfunctionsequenceinthefunctionspace,ie,lim()()NNFxfx()in,ab,iflim()()bNaNFxfxdx()Thisyieldsimmediatelythestatement()()nnnfxaUx()Formingtheproductofthisrelationand()mUx,duetotheorthonormalityrelation,oneobtainstheexpression()()bmmaafxUxdx()forthegeneralizedFouriercoefficientsThisisagainequation(a)Now,westillwanttoconsiderBessel#sinequality|||()|bnnaafxdx()knownfromanalysisTruncatingtheexpansionofthefunction()fxafterNterms,inanycasewehave()()bNnnnafxaUxdx()Takingintoaccount()andthepossibilitytointerchangeintegrationandsummation,wehave()()|()|||bbNNnnnnnaafxaUxdxfxdxaandhence|||()|bNnnaafxdx()Sincetherighthandsideof()isanexpressionindependentofN,onthelefthandsideNcanbechosenarbitrarily(alsoN)so()isdemonstratedBessel#sinequalitycanbeconcludedalsodirectlyfrom(b)takingintoaccountthefactNMIfinthisrelationtheequalitysignisvalid,thenthesystembelongingtothenadescribesthefunction()fxcompletely:()()|()|||bbnnjnjaafxaUxdxfxdxaie,()()nnnfxaUxForthatreasonthis(Parseval#s)equation|||()|bnnaafxdxisalsocalledthecompletenessrelationAnalogousto()weobtainanexpansionofafunctionoftwoindependentvariables(,)fxyintheintervals:,xaband,ycd:(,)()()mnmnmnfxyaUxVy()with(,)()()bdmnmnacadxfxyUxVydy()Here,()mUxand()nVyareorthonormalAsimilargeneralizationholdsinhigherdimensionalspacesFourierseriesSinceforpositiveintegers,nmtherelationsinsinnmnxmxdxisvalid,thesequenceoffunctions()sinnUxnx(,,n)representsanorthonormalsystemintheintervalxHowever,asthissequenceconsistsofoddfunctionsonly(()()fxfx)itcanbeusedtoapproximateonlyoddfunctionsIfonetakes,inaddition,theorthonormalsystemofevenfunctions(()()fxfx):()cosnVxnx(,,n),()Vx(n)thenarbitraryfunctionsin,canberepresentedby()cossinnnnafxanxbnx(a)Forevenfunctions()fx,nbvanishesforoddfunctionsnavanishesTheFouriercoefficientsare,from(),()cosnafxnxdx(,,n),()sinnbfxnxdx(,,n)Fromthefunctions,nnUV,thefunctions,nnuvbeingorthonormalintheintervaldxd:()sinnnxuxdd(,,n)()cosnnxvxdd(,,n),()vxd(n)maybeconstructedbythecoordinatetransformationxxdForthesefunctionswehave()()dnnnnduxuxdx,()()dnnnndvxvxdx,()()dnnduxvxdxandfortheexpansion()()()()nnnnnfxavxavxbux(b)oneobtainsthecoefficients()()()ddddafxvxdxfxdxd(n)()()()cosddnnddnxafxvxdxfxdxdd(,,n)()()()sinddnnddnxbfxuxdxfxdxdd(,,n)FourierintegralsAnyfunction()fxintheintervaldxdcanberepresentedalsointheform()Thefollowingelementaryrelationshold:()()()cosinxdinxdnnngxgxnxeevxddd()and()()()sininxdinxdnnngxgxnxeeuxdddii()Byrewritingwefindthatthefunctions()inxdngxed(,,,n)()alsoformanorthogonalsystemintheintervaldxdforwhich()()dnnnndgxgxdx(a)Namely,()()()nnnvxiuxgx,()()()nnnvxiuxgx(n)(b)andtherefore,eg,()()dnndgxgxdxand()()dnnnndgxgxdxThefactthatthefactorappearsin()and(),andnot,guaranteesthenormalizationofthefunctions()ngx(a)Foranyfunction()fxintheintervaldxdthatcanbeconsideredperiodic,wehave()()inxdnnnnnefxagxad()with()()()ddinxdnnddeafxgxdxfxdxd(,,,n)Inparticular,forthe()xxfunctionwithdxdwehave()()()()()indxxnnnnexxgxgxd(a)LetusconsidernowthelimitdTheperiodicityintervaloftheFourierseriesofdtodisthenextendedtothewholeregionxObviously,()inxdinxdnnnneefxaandd(b)wheren,andforthe()xxfunction()()()()()()nikxxindxxindxxnnnneenexxnkdd()ikxxikxikxikxikxeeeeedkdkdk(c)Here,thevariablenkndhasbeenintroduced,whichinthelimitdconvertstothecontinuousvariablekObviously,thefunctions(,)ikxegkx()formanorthonormalcompletebasisdependingonthecontinuousparameterkDuetothesymmetryofkandxinikxe,analogoustoequation(c)weobtain()ikxikxeekkdx()Foranyfunction()fx,()()(,)()ikxefxAkgkxdkAkdkwith()()(,)()ikxeAkfxgkxdxfxdx()since()()()ikxikxikxikxikxeeeeefxdxdxAkdkAkdkdx()()()AkkkdkAk()()()ikxikxikxikxikxeeeeeAkdkdkfxdxfxdxdk()()()fxxxdxfxThesymmetrybetween()fxand()Axisaremarkablefact()AkiscalledtheFouriertransformorthespectralfunctionof()fxTheexpansionofaperiodicfunction(withtheperiodd)inaFourierseriesisarepresentationintermsofasumofharmonicoscillatorswiththefrequenciesnknd,()nikxnnfxaed,()ndikxndafxedxdwhiletheFourierintegralexpressesthefunction()fxasasumofaninfinitenumberofoscillatorswithdifferentfrequencies,()()ikxfxAkedk,()()ikxAkfxedxOnesaysthattheFourierintegralyieldsanexpansionofthefunctionintermsofacontinuousspectrum,wherethedistributionofthecontinuousfrequencieskcorrespondstothespectraldensity()AkTorepresentafunction()fxasaFourierintegralonehastoassumetheconvergenceofthefollowingintegral:|()|fxdx,thus,lim()lim()xxfxfxFurthermore,thefunctionhasalsotoobeyDirichletboundaryconditionsItshouldbementionedthattheFourierintegralsimplifieswhen()fxisanevenoroddfunction(Thenoneobtainscosineorsinetermsonly)Furthermore,wenoteParseval#sidentity:let()()ikxfxFkedk,()()ikxgxGkedkthen()()()()ikxikxeefxgxdxFkdkGkdkdx()()()FkdkGkkkdk()()FkGkdkandsimilarlyalso()()()()ikxefugxuduFkGkdkThelastrelationisknownastheconvolutiontheoremExercise:OrthonormalizationofthepolynomialsShowthat(a)Inthespace(,)Lofthefunctionsquadraticintegrableintheinterval(,)thesetofallpolynomials(,,,)nxnislinearlyindependentwiththescalarproduct(,)()()fgfxgxdxApplySchmidt#sorthogonalizationproceduretothisset(b)Showthatthefunctions()imemformanorthogonalsysteminthespace(,)LAretheynormalizedSolution(a)Linearindependence:AsetoffunctionsislinearlyindependentifanyfinitesubsetislinearlyindependentThus,weconsidertheset{,,}NnnMxxandthelinearcombinationNnnNaxax(,,Nnnpairwisedistinct)However,suchapolynomialvanishesonlyifNaa(identitytheoremforpowerseries)implyingthelinearindependenceofMSinceMisanarbitraryfinitesubsetof{:,,}nxnThissetisalsolinearlyindependentOrthonormalization:TheSchmidtorthonormalizationprocedurechangesthelinearlyindependentvectors,,aaofavectorspacewiththescalarproduct(,)toanorthonormalsystem||||aaa,(,)||(,)||aaaaaaaaa,(,)(,)||(,)(,)||aaaaaaaaaaaaaaaetcInourcase,()||||Px(,())()()||(,())()||xxPxPxPxxxxPxPx(,())()(,())()()||(,())()(,())()||xxPxPxxPxPxPxxxxPxPxxPxPxetcAlthoughthepolynomials()(,,,)nPxnformanorthonormalsystem,theyarenotofuseinphysicsInstead,thesocalledLegendrepolynomialsareused,whicharedefinedby()()()nnnPxPxPwhichyields()Px,()Pxx,()()Pxx,()()Pxxx,()Remark:Thepolynomials()nPx,aswellastheLegendrepolynomials,formevenacompleteorthonormalsystemororthogonalsystemin(,)L,thatmeans,anyfunctionofthisspacecanberepresentedas()()()nnnnnnnnnfxaPxaPxbxHere,theconvergenceisdefinedbythescalarproductingeneral,itisnotpointwiseTheproofofthatisnottrivialThenx,()nPx,or()nPxdonotformabasisof(,)L,sinceabasis{,}cbcIofavectorspacehastohavethepropertythatanyvectorfofthisvectorspacemayberepresentedasafinitelinearcombinationnNncnfcbThisisnotguaranteedforthevectorsystemsmentionedbecausenotallfunctionsof(,)LarepolynomialsofdegreenBrequentlythenotionvectororthonormalsystemisusedsynonymouslywithbasis注可以证明,数学物理中通常出现的正交函数集都是完备的(b)Wehave()(,)iminiminimnmneeeededTherefore,thefunctionsimeareorthogonaltheirnormisSphericalharmonicsWewanttoconsiderthreefurtherorthogonalsystemsoffunctionsofparticularimportanceinmathematicalphysicsInthesolutionofpotentialproblems,oneencountersalwaysthefunction(Figure)||rr()whichforapointchargertakestheformcosrrrr()TheroothastobeexpandedinapowerseriesintheratioofthedistancesrandrWedenotebyrandrthesmallerandthegreatervalue,respectively,ofrandrThen,rr,andweobtainthefollowingexpansion:cos()cos()rrrrrrrrrcos(cos)(coscos)rrrrrrrThecoefficientsoccurringaretheLegendrefunctionscosrrrr(cos)(cos)(cos)rrrPPPPrrrr(cos)llllrPr()Settingcosx,thefirstfewLegendrepolynomialsare:()Px,()Pxx,()()Pxx,()()Pxxx()TheLegendrepolynomialswithevenindicesareevenfunctions,thosewithoddindicesareoddfunctionsTheLegendrepolynomialsmaybecalculatedalsobytherelation(Rodriguesformula)()()!llllldPxxldx()Theyformacompleteorthogonalsystemoffunctionsintheinterval,x,(,),and()()llllPxPxdxl()Ifwewanttoexpandthepotentialofarbitraryextendedchargedistributions,wehavetoperformtwofurtherstepsgivenherewithoutadetailedproofAcompleteorthonormalsystemoffunctionsdefinedontheunitsphereisrepresentedbythesphericalharmonicsHere,wepointoutbrieflytheirrelationshipwiththeLegendrepolynomials,,()!(,)(cos)()!mimlmlllmYPelm()TheindexmisanintegerintheboundslmlWegivetheexplicitformofthefirstfewsphericalharmonics:,Y,,siniYe,,cosY,,siniYe,,sincosiYe,,cosY,,siniYe,,sincosiYe,,sin(cos)iYe,,coscosY()The,lmYwithnegativeindexmresultfromthesefunctionsaccordingtoequation()belowThefunctions()mlPxaretheassociatedLegendrepolynomialsTheyarisefromtheLegendrepolynomials:()()()()mmmmllmdPxxPxdx()Here,theindexmispositiveTheassociatedLegendrepolynomialswithnegativeindexmdifferfromthosewithpositivemonlybyaprefactor:()!()()()()!mmmlllmPxPxlm()Substitutingrelation()into(),()()()()!mlmmmllllmdPxxxldx()ForafixedmtheassociatedLegendrepolynomialsformasetoforthogonalfunctions:()!()()()!mmlllllmPxPxdxllm()ThefunctionsimeformasetoforthogonalfunctionsaswehaveseenaboveTheproductoftheassociatedLegendrepolynomials(cos)mlPandthefunctionimeform,togetherwiththenormalizationfactor,thesphericalharmonics,(,)lmYThesphericalharmonicssatisfytherelation,,(,)()(,)mlmlmYY()Thesefunctionsareorthonormal,,(,)(,)sinlmlmllmmdYYd()ThetermsinoccurssincetheareaelementontheunitsphereissinddThecompletenessofthesetofsphericalharmonicsisexpressedbytherelation,,(,)(,)()(coscos)llmlmlmlYY()Thesphericalharmonics,lmYarerelatedtoLegendrepolynomialslPbyarelationknownastheadditiontheorem(seeFigure):,,(cos)(,)(,)lllmlmmlPYYlTheangleistheanglebetweenthevectorsrandr,cosrrrrExpressingbothofthepositionvectorsinsphericalcoordinatesweobtainfor:(sinsincos()coscos)cosrrrrrrie,cossinsincos()coscos()Theadditiontheoremoffersonethepossibilitytoextendtheexpansionvalidforapointcharge(axiallysymmetricdistribution)toanarbitrarychargedistributionInthiscase,cosisreplacedinequation()accordingto(),andwiththeadditiontheoremweobtain,,(cos)(,)(,)||lllllmlmllllmlrrPYYrlrrr()ThesphericalharmonicsarealsoobtainedinsolvingtheLaplaceequationinsphericalcoordinatesIntheseparationofthevariablesoneobtainsthenumberslandmasseparationconstantsForthefunctionsdependingontheangle,thegeneralizedLegendredifferentialequationisvalid(cosx):()()ddmxxlldxdxxwhosesolutionsaretheassociatedLegendrepolynomials()mlPxFormtheequationbecomestheLegendredifferentialequationpossessingtheLegendrepolynomials()lPxassolutionsInthissectionwehavegatheredthemostimportantpropertiesofthesphericalharmonicsmostlywithoutproofInanothervolumeofthisseries,thelecturesonquantummechanics,thetheoryofsphericalharmonicswillbeoutlinedindetailMultipoleexpansionsIfwehaveaboundedchargedistributionvanishingoutsideasphereofradiusa(seeFigure)abouttheorigin,thenthepotentialintheexternalregion(withoutboundarysurfaceswithboundaryconditionsinfiniteness,theDirichletsurfacetermvanishes),forxa,isgivenby()()||dVrrrr()Sincethechargedistributionisboundedandthepotentialintheexteriorregionisconsidered,withrrwecansubstitutetheexpansionaccordingtoequation():,,()()(,)()(,)||lllmlmllmldVYrYdVlrrrrrr()TheintegralsoccurringinthissumdependonlyontheparticularchargedistributionTheydescribetheoutwardactionofthechargedistributioncompletelyTheexpressions,,(,)()llmlmqYrdVror,,(,)()llmlmqYrdVr()arecalledthemultipolemomentsofthechargedistribution()rInparticular,themostimportantmultipolemomentslmqarelmonopolemomentldipolemomentlquadrupolemomentloctupolemomentlhexadecupolemomentForeachlthemultipolemomentslmqformatensorofrankl(hereinsphericalrepresentationwithlcomponents(,,,,,(),mllll)ToachieveauniquerepresentationweagreetoputtheoriginofthecoordinatesystematthecenterofgravityofthechargedistributionMultipolemomentNow,weinvestigatesystematicallythefirstthreemultipolemomentsDueto,,(,)()(,)mlmlmYYwegetthefollowingrelation:,,,()(,)()()mlmlmlmlmqYrdVqr()Hence,foreachmultipoleoforderlwehavetocalculateonlylmultipolemomentslmqMonopolemomentHere,theonlycomponentis,,(,)()()QqYrdVdVrrObservedfromalargedistancer,anychargedistributionactsapproximatelyasifthetotalchargeQwouldbeconcentratedatonepointsincethedominatingtermin()is,,()QqYrrr()DipolemomentInCartesiancoordinatesthethreecomponentsofthedipolemomentaregivenby()xpxdVr,()ypydVr,()zpzdVrThedefinition()dVprrisidenticalwiththeonegivenpreviously,afterequation()andinExercise()Thiscanbecheckedeasilyforthechargedistributionconsistingofthepositivechargeqataandthenegativechargeqata(SeeFigure)Itreads()qqaarrrandthedipolemomentpis()dVqqqaaprraInsphericalrepresentationoneobtainswiththehelpof(),,(,)()sin()iqYrdVerdVrr(sincossinsin)()rirdVr()WemayrepresentthesphericalcomponentsbyCartesianones:,()()()xyqxiydVpipr()Also,,,(,)()cos()qYrdVrdVrr()zzdVpr,,()xyqqpipHence,thethreedipolemomentsarelinearcombinationsoftheCartesianonesThetworepresentations(sphericalandCartesian)areequivalenttoeachotherWhichformseemstobemoreappropriatewilldependontheproblemQuadrupolemomentTheCartesiancomponentsofthequadrupoletensoraredefinedby()()ijijijQxxrdVr,,,,ij()InthenextsectionthisrelationwillturnouttobemeaningfulintheTaylorexpansionofthepotentialNow,wecalculateagainthesphericalcomponentsandexpressthemintermsoftheCartesianones:,,(,)()sin()iqYrdVerdVrr(sincossinsin)()rirdVr()()xiydVr()Rewriting()()()xiyxrixyyrweobtain,()qQiQQ()Analogously,,,(,)()()()qYrdVzxiydVrr()QiQ()and,,(,)()()qYrdVzrdVQrrSo,withthehelpoftherelation()weobtainatoncethecomponentswithnegativem:,,()qqQiQ*,,()qqQiQQ()MultipoleexpansionsinCartesiancoordinatesTheexpansionofafunction()frinaTaylorseriesaboutthepointris,()()()()!iijiijiijffffxxxxxxr()()!!nnininixffnxnr()Here,thegradientactsonlyuponthefunction()frForthespecialfunction()||()()()fxxxxxxrrr()onecomputes()fr,()iixfxr,()ijijijxxrfxxr()Hence,forthepotentialintheregionrrweobtain,()()()||!ijijiiijiijxxrxdVxxxdVrrrrrrrr,()!ijijijijxxrQxxdVrrrrpr()Thethirdterm(quadrupoleterm)canberewrittenfurther:,,()()!ijijijijijijijijijxxrxxrxxrdVQrrr()because,()()ijijiiijijixxrxxrrdVrdVrrrr()Here,thedefinition()()ijijijQxxrdVr()hasbeensuggestedfortheCartesiancomponentsofthequadrupoletensorbysymmetryreasonsThus,outsidethechargedistribution(rr)thepotentialis,()()()||ijijijijQxxrQdVrrrrrprrr()Onerealizeseasilythat,duetoiiiQ(see()),thequadrupoletermcanbewrittenalsointheform,ijijijQxxrComparingthisserieswiththecorrespondingexpansioninsphericalcoordinates,adifferenceseemstoexistbetweenthequadrupoletermsWhilethereareonlydistinctmq,independentijQseemtoexistThatthisisnotthecaseonecanprovebyformingthesumofthe(,,)iiQi(thetraceofthetensor)whichvanishesidentically:()()

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