New solutions of relativistic wave equations in magnetic fields and longitudinal fields(13)

We demonstrate how one can describe explicitly the present arbitrariness in solutions of relativistic wave equations in external electromagnetic fields of special form. This arbitrariness is connected to the existence of a transformation, which reduces eff

¯n,s+k,z′(x,y)=Ψ Γ(s+k+1) d2z

Γ(k+s+1)

k!¯n,s+k,z(x,y).Ψ(2.59)

Thatmeans,inparticular,that(2.56)isacompletesetsincetheset(2.31)iscomplete.Selectingdi erentformsforthefunctionΦ(η),wecangetothersetsofstationarystatesforachargeinauniformmagnetic eld.

C.Nonstationarystates

Themostinterestingnonstationarysolutionsofrelativisticwaveequationsforachargeinauniformmagnetic eldarecoherentstates;forthe rsttimesuchsolutionswerepresentedin[10–13],seealso[3].Belowwepresentanewfamilyofnonstationarysolutions,whichincludestheabovecoherentstatesasaparticularcase.

Herewearegoingtouselight-conevariablesu0=x0 x3,u3=x0+x3,andthecorrespondingmomentumoperators

1 =ih P¯ =002(P0+P3),(2.60)

0= / u0,where

form 3= / u3.ThentheKlein-Gordonoperatorcanbepresentedinthe

2 P ,K=4¯h 2P30 2γN m

2(2.61)whereastheDiracequationreads(ΨisaDiracbispinor)4¯h 2

P⊥= (P1,P2,0), P P30Ψ( )=2γND+m Ψ=Ψ(+)+Ψ( ), Ψ( ), Ψ2P¯ρ3m]Ψ( ),3(+)=[(αP⊥)+hΨ(±)=p±Ψ,2p±=1±α3.(2.62)Hereαandρ3areDiracmatrices[3],andp±projectionoperators. ,P areInthecaseoftheuniformmagnetic eldunderconsideration,theoperatorsP30 integralsofmotion.Thus,wewillconsidersolutionsthatareeigenvectorsofP3,

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