Exact solution of the Dirac equation for a Coulomb

arXiv:hep-th/9503051v18Mar1995ExactsolutionoftheDiracequationforaCoulombandascalarPotentialinthepresenceofanAharonov-BohmandamagneticmonopolefieldsV´ıctorM.Villalba∗CentredePhysiqueTh´eoriqueC.N.R.S.Luminy-Case907-F-13288MarseilleCedex9,FranceCentrodeF´ısica,InstitutoVenezolanodeInvestigacionesCient´ıficasIVIC,Apdo.21827,Caracas1020-AVenezuelaAbstractInthepresentarticleweanalyzetheproblemofarelativisticDiracelectroninthepresenceofacombinationofaCoulombfield,a1/rscalarpotentialaswellasaDiracmagneticmonopoleandanAharonov-Bohmpotential.Usingthealgebraicmethodofseparationofvariables,theDiracequationexpressedinthelocalrotatingdiagonalgaugeiscompletelyseparatedinsphericalcoor-dinates,andexactsolutionsareobtained.WecomputetheenergyspectrumandanalyzehowitdependsontheintensityoftheAharonov-Bohmandthemagneticmonopolestrengths.TypesetusingREVTEX∗e-mailaddress:[e-mail]tionisasystemoffourcoupledpartialdifferentialwhichdescribestherelativisticelectronandotherspin1/2particles.Despitetheremarkableeffortmadedur-ingthelastdecadesinordertofindexactsolutionsfortherelativisticDiracelectrontheamountofsolvableconfigurationsisrelativelyscarce,beingtheCoulombproblemperhapsthemostrepresentativeexampleandalsooneofthemostdiscussedandanalyzedproblemsinrelativisticquantummechanics.Amongthedifferentapproachesavailableinthelitera-turefordiscussingtheDirac-CoulombprobleminthepresenceofotherinteractionsliketheAharonov-Bohmfieldoranyotherelectromagneticpotentialwehavethequaternionicap-proachproposedbyHautot1,theSt¨ackelseparationmethoddevelopedbyBagrovetal2,thealgebraicmethodofseparationofvariables3,4,theshiftoperatormethod5,andthealgebraicmethodproposedbyKomarovandRomanova6.Recently,LeeVanHoangetal7havesolvedtheDirac-CoulombproblemwhenanAharanov-BohmandaDiracmagneticmonopolefieldsarepresent.Theauthorsuse,fortacklingtheproblem,atwodimensionalcomplexspacewhichresultsafterapplyingtheKustaanheimo-Stiefeltransformation8onthethreespacevariables,reducinginthiswaytheKeplerproblemtoanoscillatorproblem.ThisidealiesontheutilizationofaSU(2)dynam-icalalgebraforcomputingtheresultingenergyspectrum9,which,likethespinorsolution,isexpressedintermsofintrinsiccoordinatesappearingafterusingthecomplexspace.TheutilizationofdifferenttechniquesforstudyingtheDirac-CoulombfieldinthepresenceofanAharonov-Bohmfieldoramagneticmonopolecouldgiverisetotheideathatthisproblemisnotsolublewithoutintroducingnewvariablesoradditionalconservedquantities.HereitisshownthatusingthealgebraicmethodofseparationofvariablesitispossibletosolvetheDiracequationinthepresenceofaCoulombfieldandascalar1/rpotentialwithanAharonov-Bohmandamagneticmonopolefields.Theadvantageofthisapproachisthatdoesnotrequiretheintroductionofnonbijectivequadratictransformations,alsoitbecomescleartheroleplayedbytheDiracmagneticmonopoleaswellastheAharonov-Bohmfield2intheangulardependenceofthespinorΨ(~r)solutionoftheDiracequation.Thearticleisstructuredasfollows:InSec.II,applyingapairwiseschemeofseparation,weseparatevariablesintheDiracequationexpressedinthelocalrotatingframe,weseparatetheradialdependencefromtheangularone.InSec.III.theangulardependenceissolvedintermsofJacobiPolynomials.InSecIV,theseparatedradialequationissolvedandtheenergyspectrumiscomputed.InSec.V,wediscusstheinfluenceoftheAharonov-Bohmfieldandthemagneticmonopoleontheenergyspectrum.II.SEPARATIONOFVARIABLESInthissectionweproceedtoseparatevariablesintheDiracequationwhenaCoulombfield,ascalar1/rpotentialaswellasaDiracmagneticmonopoleandaAharonov-Bohmfieldarepresent.Forthispurpose,wewriteinsphericalcoordinatesthecovariantgeneralizationoftheDiracequationn˜γμ(∂μ−Γμ−iAμ)+M+˜V(r)oΨ=0(1)where˜γμarethecurvedgammamatricessatisfyingtherelation,{˜γμ,˜γν}+=2gμν,andΓμarethespinconnections10.with˜V(r)asthescalar1/rfield˜V(r)=−α′r(2)whereα′isaconstant,andthevectorpotentialAμreadsAμ=A(mon)μ+A(Coul)μ+A(AB)μ(3)wherethecomponentsoftheCoulombpotentialA(Coul)μtaketheformA(Coul)0=V(r)=−αr,A(Coul)i=0,i=1,2,3(4)theAharonovBohmpotentialA(AB)μreadsA(AB)=Frsinϑˆeϕ(5)3andtheDiracmonopolefieldA(mon)μisA(mon)=g(1−cosϑ)rsinϑˆeϕ(6)where,followingtheDirac11prescriptionforquantizingthemagneticcharge,gtakesintegerorhalfintegervaluesg=j2,j=0,±1,±2....(7)IfwechoosetoworkinthefixedCartesiangauge,thespinorconnectionsarezeroandthe˜γmatricestaketheform˜γ0=γ0=¯γ0,˜γ1=h(γ1cosϕ+γ2sinϕ)sinϑ+γ3cosϑi=¯γ1,˜γ2=1rh(γ1cosϕ+γ2sinϕ)cosϑ−γ3sinϑi=¯γ2r,(8)˜γ3=1rsinϑ(−γ1sinϕ+γ2cosϕ)=¯γ3rsinϑwhereγαarethestandardMinkowskigammamatrices,andtheDiracequationinthefixedCartesiantetradframe(8)takestheform(¯γ0(∂t+iV(r))+¯γ1∂r+¯γ2r∂ϑ+¯γ3rsinϑ(∂ϕ−iF−i(1−cosϑ)g)+M+˜V(r))ΨCart=0(9)wherewehaveintroducedthespinorΨCart,solutionoftheDiracequation(9)inthefixedtetradgauge.InordertoseparatevariablesintheDiracequation,wearegoingtoworkinthediagonaltetradgaugewherethegammamatrices˜γdtaketheform˜γ0d=γ0,˜γ1d=γ1,˜γ2d=1rγ2,˜γ3d=1rsinϑγ3(10)Sincethecurvilinearmatrices˜γμand˜γdsatisfythesameanticommutationrelations,theyarerelatedbyasimilaritytransformation,uniqueuptoafactor.Inthepresentcasewechoosethisfactorinordertoeliminat