Exact solution for a fermion in the background of a scalar i(2)

OneshouldrealizethattheDiracenergylevelsaresymmetricalaboutE=0(see,e.g.,Refs.[22]and[23]).Thisconclusioncanbeobtaineddirectlyfrom(5)aswellasfromthechargeconjugation.Indeed,ifΨisasolutionwithenergyEthenσ3Ψ isalsoasolutionwithenergy Efortheverysamepotential.Itmeansthatthepotentialcouplestothepositive-energycomponentofthespinorinthesamewayitcouplestothenegative-energycomponent.Inotherwords,thissortofpotentialcouplestothemassofthefermioninsteadofitschargesothatthereisnoatmosphereforthespontaneousproductionofparticle-antiparticlepairs.Nomattertheintensityandsignofthecouplingparameter,thepositive-andthenegative-energysolutionsnevermeet.Thusthereisnoroomfortransitionsfrompositive-tonegative-energysolutions.ThisallmeansthatKlein´sparadoxnevercomestothescenario.

ItisworthtonotethattheDiracequationiscovariantunderx→ xifV(x)doesnotchangesignandΨ±( x)=Ψ (x)orΨ±( x)= Ψ (x).ThisisbecausetheparityoperatorP=exp(iη)P0σ1,whereηisaconstantphaseandP0changesxinto x,changessignofαbutnotofβ.Foran± even-paritypotentialonecannoticethatVeff( x)=Veff(x)whereasEeffremainsunchanged.

NowletusconsiderascalarpotentialintheformintheformV=µ|x|δ,thenthee ectivepotentialbecomes

(7)2mc

whereε(x)standsforthesignfunction.Inthesubsequentsurveyforthepossibleexistenceofboundedsolutionswetakeadvantageofthesymmetry±forVeffandonlyconsiderthepositivesideofthex-axis.Whenδ>0thee ectivepotentialgoestoin nityasx→∞anditis niteattheoriginforδ≥1whereasintherange0<δ<1ithasasingularitygivenby + h¯µδ/2mc|x|1 δ,implyinginapotential-wellstructureforVeff(Veff)when+µ>0(µ<0)andanattractivepotentiallesssingularthanx 1forVeff (Veff).Therefore,forδ>0thepowerpotentialleadstoe ectivepotentialsful llingthekeyconditionstofurnishdiscretespectra.Ontheotherhand,whenδ<0thee ectivepotentialvanishesasx→∞andwhenµ>0the+e ectivepotentialVeffisalwaysrepulsiveattheorigininsuchawaythatitisrepulsiveeverywhere.Therefore,forδ<0andµ>0thepower-lawpotentialdoesnotleadtobound-statesolutions.Forδ<0andµ<0, though,thereisapotential-wellstructureforVeff.Thesameistruefor

4±Veff=µ2|x|δ 1

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.

+Veffontheconditionthatδ< 1.For 1<δ<0andµ<0thereis

+anattractivepotentialVeffwithsingularitygivenby h¯|µ||δ|/2mc|x|1+|δ|.+Forδ= 1andµ<0thereisalsoapotential-wellstructureforVeffwhen+µ< h¯c,andanattractivepotentialVeffwithsingularityattheorigingiven

by |µ|/|x|whenµ= h¯c,andanattractiveinverse-squaresingularitygivenby h¯|µ|(1 |µ|/h¯c)/2mcx2when h¯c<µ<0.Thereisnocollapsetothecenterbecausethepotentialisnotmoreattractivethan h¯2/8mx2[24].Asamatteroffact,whenµ= h¯c/2thesingularpotentialassumesitscriticalvalue.Therefore,forδ<0andµ<0thepower-lawpotentialalsoleadstoe ectivepotentialsful llingthekeyconditionstofurnishdiscretespectra.

UptothispointwehaveconsideredsolutionsforE=0.Thissupposi-tionhasbeenexplicitlyassumedtoobtain(4)-(6).Nevertheless,onecouldalsoaskforpossiblezero-energysolutions.Thesezero-modeenergiescanbeobtaineddirectlyfromtheDiracequation(3).Inthiscasethe rst-orderdi erentialequationsarefullyuncoupled.Thentheupperandlowercompo-nentsoftheDiracspinorforthepower-lawpotentialareexpressedby

Ψ±=N±F±(x)exp

h¯c mc,δ= 1

δ+1

andN±isanormalizationconstant.Normalizableeigenspinorsontheposi-tivesideofthex-axisaregivenby

exp µ (9),forδ= 1Ψ+Ψ= 1

1 ,forµ>0andδ> 1orµ<0andδ≤ 1,forµ<0andδ> 1Ψ

(10)

Notethattheprobabilitypositiondensityofthezero-modespinorhasalonelyhump.Notealsothattheconditionsfortheexistenceofboundedsolutionsforzero-energieshasnothingtodowiththoseonesforE=0.Inthecaseofδ= 1anotherrestrictionmustbeaddedtothetoplineof(10):µ≤ h¯c.Thisrestrictionisnecessaryforobtainingadi erentiablespinor

5

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.

attheoriginanditmeansthattheinverselylinearpotentialmustbeenoughstrongtoholdazero-modesolution.

Itissurprisingto ndDiraceigenspinorswithavanishinglowercom-ponentinatheorywithoutanonrelativisticlimit.Moresurprisingisto ndavanishinguppercomponent.Bothdramaticcircumstancesmaketheirappearanceduetotheparticularrepresentationsofthematricesαandβadoptedinthispaper.Itisinstructiveatthispointtoconsiderforamomentarepresentationwheretheeigenspinorpresentsamorefamiliarbehaviour.LetuswritetheDiracequation(1)as

cσ

1p+σ3mc+V

bytheunitarytransformationψ=Uψ ,Theoriginalspinorisrelatedtoψ

where

U=

√ sothatφ=φ iχ /

approximation(11)becomes 2 =Eψ ψ(11)12 1 i1+i (12)2.Inthenonrelativistic

χ =

d2

2mp

|x|

6(15)

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.

whereqisarealparameter.Thenthee ectivepotentialbecomestheKratzerpotentialh¯cq±Veff= (16)x2

whereh¯2qA±=

h¯

mc2Ψ±=0(19)h¯2ξ2

Nowtheprimedenotesdi erentiationwithrespecttoξ.Thenormalizableasymptoticformofthesolutionasξ→∞ise ξ/2.Asξ→0,whentheter …… 此处隐藏：5696字，全部文档内容请下载后查看。喜欢就下载吧 ……