Exact solution for a fermion in the background of a scalar i

The problem of a fermion subject to a general scalar potential in a two-dimensional world is mapped into a Sturm-Liouville problem for nonzero eigenenergies. The searching for possible bounded solutions is done in the circumstance of power-law potentials.

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aExactsolutionforafermioninthebackgroundofascalarinverselylinearpotentialAntonioS.deCastroUNESP-CampusdeGuaratinguet´aDepartamentodeF´ sicaeQu´ micaCaixaPostal20512516-410Guaratinguet´aSP-BrasilE-mailaddress:castro@feg.unesp.br(A.S.deCastro)

The problem of a fermion subject to a general scalar potential in a two-dimensional world is mapped into a Sturm-Liouville problem for nonzero eigenenergies. The searching for possible bounded solutions is done in the circumstance of power-law potentials.

Abstract

Theproblemofafermionsubjecttoageneralscalarpotentialinatwo-dimensionalworldismappedintoaSturm-Liouvilleproblemfornonzeroeigenenergies.Thesearchingforpossibleboundedsolutionsisdoneinthecircumstanceofpower-lawpotentials.Thenormalizablezero-eigenmodeso-lutionsarealsosearched.Forthespeci ccaseofaninverselylinearpotential,whichgivesrisetoane ectiveKratzerpotential,exactboundedsolutionsarefoundinclosedform.ThebehaviouroftheupperandlowercomponentsoftheDiracspinorisdiscussedindetailandsomeunusualresultsarerevealed.

The problem of a fermion subject to a general scalar potential in a two-dimensional world is mapped into a Sturm-Liouville problem for nonzero eigenenergies. The searching for possible bounded solutions is done in the circumstance of power-law potentials.

TheCoulombpotentialofapointelectricchargeina1+1dimension,con-sideredasthetimecomponentofaLorentzvector,islinear(～|x|)andsoitprovidesaconstantelectric eldalwayspointingto,orfrom,thepointcharge.Thisproblemisrelatedtothecon nementoffermionsintheSchwingerandinthemassiveSchwingermodels[1]-[2]andintheThirring-Schwingermodel[3].Itisfrustratingthat,duetothetunnelinge ect(Klein´sparadox),therearenoboundstatesforthiskindofpotentialregardlessofthestrengthofthepotential[4]-[5].Thelinearpotential,consideredasaLorentzscalar,isalsorelatedtothequarkoniummodelinone-plus-onedimensions[6]-[7].Recentlyitwasincorrectlyconcludedthateveninthiscasethereissolelyoneboundstate[8].Later,thepropersolutionsforthislastproblemwerefound[9]-[11].Themixedvector-scalarpotentialhasalsobeenanalyzedforalinearpotential[12]aswellasforageneralpotentialwhichgoestoin nityas|x|→∞[13].Inbothofthoselastreferencesithasbeenconcludedthatthereiscon nementifthescalarcouplingisofsu cientintensitycomparedtothevectorcoupling.

Theproblemofaparticlesubjecttoaninverselylinearpotentialinonespatialdimension(～|x| 1),knownastheone-dimensionalhydrogenatom,hasreceivedconsiderableattentionintheliterature.Thisproblempresentssomeconundrumsandthemostperplexingisthatoneregardingthegroundstate.ThenonrelativisticSchr¨odingerequationprovidesaground-stateso-lutionwithin niteeigenenergyandarelatedeigenfunctiongivenbyadeltafunctioncenteredabouttheorigin.ThisproblemwasalsoanalyzedwiththeKlein-Gordonequationandthereitwasrevealeda niteeigenenergyandaexponentiallydecreasingeigenfunction[14].Byusingthetechniqueofcon-tinuousdimensionalitytheproblemwasapproachedwiththeSchr¨odinger,Klein-GordonandDiracequations.TheconclusioninthismorerecentworkisthattheKlein-GordonequationprovidesunacceptablesolutionswhiletheDiracequation,withtheinteractingpotentialconsideredasatimecompo-nentofavector,hasnoboundedsolutionsatall[15].ThisproblemwasalsosketchedforaLorentzscalarinteractingpotentialintheDiracequation[16],buttheanalysisisincomplete.

Inthepresentpaperitisshownthattheproblemofafermionunderthein uenceofageneralscalarpotentialfornonzeroeigenenergiescanbemappedintoaSturm-Liouvilleproblem.Nextthekeyconditionsfortheexistenceofbound-statesolutionsaresettledforpower-lawpotentials.Thepossiblezero-modesolutionsareshowntoconformwiththeultrarelativisticlimitofthetheory.Inaddition,thesolutionforaninverselylinearpotential

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The problem of a fermion subject to a general scalar potential in a two-dimensional world is mapped into a Sturm-Liouville problem for nonzero eigenenergies. The searching for possible bounded solutions is done in the circumstance of power-law potentials.

isobtainedinclosedform.Thee ectivepotentialresultingfromthemappinghastheformoftheKratzerpotential[17].Itisnoticeablethatthisproblemhasanin nitenumberofacceptableboundedsolutions,neverthelessithasnononrelativisticlimitforsmallquantumnumbers.Itisalsoshownthatintheregimeofstrongcouplingadditionalzero-energysolutionscanbeobtainedasalimitcaseofnonzero-energysolutions.Theideasofsupersymmetryhavealreadybeenusedtoexplorethetwo-dimensionalDiracequationwithascalarpotential[18]-[19],neverthelessthepower-lawpotentialhasbeenexcludedofsuchdiscussions.

Thetwo-dimensionalDiracequationcanbeobtainedfromthefour-dimen-sionalonewiththemixtureofsphericallysymmetricscalar,vectorandanomalousmagnetic-like(tensor)interactions.Ifwelimitthefermiontomoveinthex-direction(py=pz=0)thefour-dimensionalDiracequationdecomposesintotwoequivalenttwo-dimensionalequationswith2-componentspinorsand2×2matrices[20].Then,thereresultsthatthescalarandvectorinteractionspreservetheirLorentzstructureswhereastheanomalousmag-neticinteractionturnsouttobeapseudoscalarinteraction.Furthermore,inthe1+1worldthereisnoangularmomentumsothatthespinisabsent.Therefore,the1+1dimensionalDiracequationallowustoexplorethephys-icalconsequencesofthenegative-energystatesinamathematicallysimplerandmorephysicallytransparentway.

Inthepresenceofatime-independentscalarpotentialthe1+1dimen-sionaltime-independentDiracequationforafermionofrestmassmreads

cαp+βmc+V

whereEistheenergyofthefermion,cisthevelocityoflightandpisthemomentu …… 此处隐藏：5839字，全部文档内容请下载后查看。喜欢就下载吧 ……