Pseudospin symmetry as a relativistic dynamical symmetry

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1、PseudospinsymmetryasarelativisticdynamicalsymmetryinthenucleusManuelMalheiroUniversidadeFederalFluminenseNiterói-RiodeJaneiroIntroduction■Theideaofpseudospin:◆Toexplainthequasi-degeneracy:13n,l,j=l+andn−1,l+2,j=l+22◆Theselevelshavethesame“pse

2、udo”orbitalangularmomentum~l=l+1anda“pseudo”spinquantumnumber.~1s=2■Itisexactwhendoubletswith~~j=l±saredegenerate.K.T.HechtandA.Adler,Nucl.Phys.A137,129(1969)AArima,M.HarveyandK.Shimizu,Phys.Lett.B30,517(1969)Shellmodelenergylevelswithaspinorbitterm3p3/2

3、-2f5/22f7/2-1h9/22d5/2-1g7/22p3/2-1f5/2Pseudospinsymmetry■TheHamiltonianofaDiracparticleofmassminaexternalscalar,S,andvector,V,potentialsisgivenbyrrH=α⋅p+β()m+S+VwhereaandbaretheusualDiracmatrices.■WedenotetheupperandlowercomponentsoftheDiracspinorby1Ψ=(

4、)1±βΨ±2rrα⋅pΨ−()ε+m−∆Ψ=0+−rrα⋅pΨ−()ε−m−UΨ=0−+■TheDiracHamiltonianisinvariantunderspecialSU(2)transformationswhen:V−S=0=∆orS+V=0=UB.SmithandL.J.Tassie,Ann.Phys.65,352(1971)J.S.BellandH.Ruegg,Nucl.Phys.B98,151(1975)†ParticularCaseD=0(S=V)1Ψ=α⋅pΨE=ε−m−+ε+m2

5、pΨ+UΨ=EΨ+++2m+ESchroendinger-likeequationfortheuppercomponentwithnospin-orbitcoupling-()n,lbasis.†ParticularCaseU=0(S=-V)1Ψ=α⋅pΨ+−ε−m2pEΨ+∆Ψ=EΨ−−−2m+E2m+ESchroendinger-likeequationforthelowercomponentwithnopseudospin-orbitcoupling-~~()n,lbasis.■Pseudospi

6、nsymmetryisanexactSU(2)symmetryfortheDiracHamiltonianwhenU=0.θ⋅σθ⋅σΨ′−=1+Ψ−δΨ−=Ψ−2i2irrrrrrα⋅pα⋅pΨ−()ε−mα⋅pΨ=0−+ε−mrrΨ=α⋅pΨ-p2+ε−mθ⋅σrrδΨ=α⋅pΨ−p22i+rrα⋅pδΨ−()ε−mδΨ=0−+1rrδΨ=α⋅pδΨ+−ε−mrrrvα⋅pθ⋅σα⋅pδΨ=Ψ++p2ipJ.N.Ginocchio,Phys.Rev.Lett.78,436(1997);i

7、bid,Phys.Rept.315,231(1999)J.N.GinocchioandA.Leviatan,Phys.Lett.B245,1(1998ÉSU(2)generatorsofthePseudospinSymmetryrrrrˆα⋅pα⋅p(1+β)(1−β)S=s+siipp22)Us)U0ˆpipsi0S=)=)0s0siirrσ⋅pwhereU=ppisthemomentumhelicityoperator.†SU(2)algebra[Si,Sj]=iεi

8、jkSkrrs⋅p†Pseudospinsˆ=UsˆU=2p−spip2iip†Hamiltoniancommutator[]U,s0[]H,S=D00J.N.Ginocchio,Phys.Rev.Lett.78,436(1997);ibid,Phys.Rept.315,231(1999)J.N.GinocchioandA.Leviatan,Phys.Lett.B245,1(1998)HarmonicOscillatorwi