New solutions of relativistic wave equations in magnetic fields and longitudinal fields

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

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aNewsolutionsofrelativisticlongitudinalwaveequations elds.inmagnetic eldsandV.G.Bagrov ,M.C.Baldiotti ,D.M.Gitman ,andI.V.Shirokov§InstitutodeF´ sica,UniversidadedeS aoPaulo,C.P.66318,05315-970S aoPaulo,SP,Brasil(February1,2008)AbstractWedemonstratehowonecandescribeexplicitlythepresentarbitrarinessinsolutionsofrelativisticwaveequationsinexternalelectromagnetic eldsofspecialform.Thisarbitrarinessisconnectedtotheexistenceofatransforma-tion,whichreducese ectivelythenumberofvariablesintheinitialequations.Thenweusethecorrespondingrepresentationstoconstructnewsetsofex-actsolutions,ly,wepresentnewsetsofstationaryandnonstationarysolutionsinmagnetic eldandinsomesuperpositionsofelectricandmagnetic elds.I.INTRODUCTIONRelativisticwaveequations(DiracandKlein-Gordon)provideabasisforrelativisticquantummechanicsandquantumelectrodynamicsofspinorandscalarparticles[1].Inrelativisticquantummechanics,solutionsofrelativisticwaveequationsarereferredtoasone-particlewavefunctionsoffermionsandbosonsinexternalelectromagnetic elds.Inquantumelectrodynamics,suchsolutionsallowthedevelopmentoftheperturbationexpansionknownastheFurrypicture,whichincorporatestheinteractionwiththeexternal eldexactly,

whiletreatingtheinteractionwiththequantizedelectromagnetic eldperturbatively[2].ThephysicallymostimportantexactsolutionsoftheKlein-GordonandtheDiracequationsare:anelectroninaCoulomb eld,auniformmagnetic eld,the eldofaplanewave,the eldofamagneticmonopole,the eldofaplanewavecombinedwithauniformmagneticandelectric eldsparalleltothedirectionofwavepropagation,crossed elds,andsome

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

simpleone-dimensionalelectric elds(foracompletereviewofsolutionsofrelativisticwaveequationssee[3]).

Considering,forexample,stationarysolutionsofrelativisticwaveequations,wecanseethatinthegeneralcase,thereexistdi erentsetsofstationarysolutionsforoneandthesameHamiltonian.Thepossibilitytogetdi erentsetsofstationarystatesre ectstheex-istenceofanarbitrarinessinthesolutionsoftheeigenvalueproblemforaHamiltonian.Consideringnonstationarysolutions,wealsoencounterthepossibilityofconstructingdif-ferentcompletesetsofsuchsolutions.Thereisnoregularmethodofdescribingsuchanarbitrarinessexplicitly.Especiallyinthepresenceofanexternal eldtheproblemappearstobenontrivial.

Inthepresentarticlewedemonstratehowonecandescribeexplicitlythepresentarbi-trarinessinsolutionsoftherelativisticwaveequationsforsometypesofexternalelectro-magnetic elds,namely,foruniformmagnetic eldsandcombinationofthese eldswithsomeelectric elds.Thisarbitrarinessisconnectedtotheexistenceofatransformation,whichreducese ectivelythenumberofvariablesintheinitialequations.Thenweusethecorrespondingrepresentationstoconstructnewsetsofexactsolutions,whichmayhaveaphysicalinterest.InSect.IIweconsiderrelativisticwaveequationsinpureuniformmagnetic elds.Herewederivearepresentationfortheexactsolutions,inwhichtheabovementionedarbitrarinessisdescribedexplicitlybyanarbitraryfunction.Fromasuitablechoiceofthisfunction,wegetboththewell-knownsetofsolutionsandnewones.ThisSectioncontainsthemostcomplete(atthepresent)descriptionoftheproblemofauniformmagnetic eldinrelativisticquantummechanics.Amongnewsetsofsolutionstherearebothstationary,gen-eralizedcoherentsolutionsandnonstationarysolutions.Then,inSect.III,weconsidermorecomplicatedcon gurationsofexternalelectromagnetic elds,namely,longitudinalelectro-magnetic elds.Herewedescribeallthearbitrarinessinthesolutions,andonthisbasepresentvarioussetsofnewexactsolutions.InSect.IVweinterprettheaboveresultsfromthepointofviewofthegeneraltheoryofdi erentialequations.

II.UNIFORMMAGNETICFIELD

A.Arbitrarinessinsolutionsofrelativisticwaveequations.

Considerauniformmagnetic eldH=(0,0,H)directedalongthex3axis(H>0).Theelectromagneticpotentialsarechoseninthesymmetricgauge

A0=A3=0,A1=1

2Hx1.(2.1)

WewritetheKlein-GordonandtheDiracequationsintheform

KΨ=0,2h¯2K=P2 m2¯ µ 0c,Pµ=ihe

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

InthecaseoftheKlein-Gordonequation,theoperatorLz,

Lz=ih¯x 1 x 2,[Lz,P0]=[Lz,P3]=[K,Lz]=0, 21

canbeincluded(togetherwithP0andP3)inthecompletesetofintegralsofmotion,whereasfortheDiracequationcase,theoperatorJz,

Jz=Lz+h¯ (2.3)

γ

dx1dx2=2ρcos ,

γ 2x2=y=√√ch¯>0,dρd ,x+iy=

ei √(x+iy+ x+i y)=(ρ+i +2ρ ρ),2ρ

11 12+=P2+iP1+h¯γx ixa2=√2γh¯2

e i √(x iy+ x i y)=(ρ i +2ρ ρ),2ρ

11(iP P)=a+=√1212γh¯2

D=h¯ 1 γP0+γP3 03 2 γ iγ

21 h¯,21a1+γ+iγ a+1 m.(2.9)

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

TheoperatorNcommuteswithP0,P3,Lz,plusitisanintegralofmotioninthecaseoftheKlein-Gordonequation.ItsgeneralizationfortheDiracequationhastheformND=N+1

√

2ξ=x+k,

√√2x=ξ+η,√√2a1=ξ+ ξ,2a2=η+ η,

γγξ iγ ξ m.

Onecanseethatthelatteroperatorsdonotcontainthevariableη.NoticethatbothoperatorsLzandJzcontainvariablesξ,η.Forexample,

222Lz=ξ2 ξ η2+ η. 21 (2.15)(2.16)

Theintegrationoverkin

(2.10)canbereplacedbyanintegrationoverη,

eixy

Ψ(x,y)=π∞ e√ i2x η.

(2.17) ∞

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

Besides,onecanwrite

(Ψ,Φ)= ∞dxdyΨ (x,y)Φ(x,y)= Ψ,ΦdηΨ (ξ,η)Φ

∞ ∞ ∞ = ∞dξ ∞ ∞ ∞ (ξ,η).(2.18)

Theindependenceoftheoperators(2.15)onthevariableηwillallowustoseparateexplicitlythefunctionalarbitrarinessinthesolutions(2.17),aswillbeseenbelow.

B.Stationarystates

Knownsetsofstationarysolutionsinauniformmagnetic eld(thatwerefoundinthe rstworks[4–8])areeigenfunctionsoftheoperatorsP0,P3,NinthescalarcaseandoftheoperatorsP0,P3,NDinthespinorcase.ThusforscalarwavefunctionsΨwehavetheconditions

P0Ψ=h¯k0Ψ,P3Ψ=h¯k3Ψ,NΨ=nΨ,n=0,1,2,...,(2.19)

andforDiracwavefunctionsΨtheconditions

P0Ψ=h¯k0Ψ,P3Ψ=h¯k3Ψ,NDΨ= n 1

2x η.(2.23)

HereEqs.(2.19),(2.14)wereused.Un(ξ)areHermitfunctions;

correspondingpolynomialsHn(ξ)asUn(x)=(2nn!√theyarerelatedtothe

2exp( x2/2)Hn(x)[14].The

functionΦ(η)isarbitrary.ThefunctionsΨn(x,y)from(2.22)obeytherelations

a1Ψn=√n+1Ψa+n

1

n+1,Ψn(x,y)=Γ (n +1)Ψ0(x,y),(2.24)

Ψ3

0(x,y)=π 2+√

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

ΨTn,k3(x,y)=(c1Ψn 1(x,y),ic2Ψn(x,y),c3Ψn 1(x,y),ic4Ψn(x,y)).(2.26)ThefunctionsΨn(x,y)arede nedbytherelations(2.17),(2.23),whereastheconstantbispinorC(withtheelementsck)obeysanalgebraicsystemofequations

AC=0,A=γ0k0+γ3k3

2γnσ1 k3σ3)v ,C+C=2k0(k0+m)v+v,(2.29)

wherevisanarbitraryconstantbispinorandσarePaulimatrices.Wecanspecifyvselectingaspinintegralofmotion(see[3]).Thestaten=0isaspecialcase.Herewemustsetc1=c3=0,thatcorrespondstothechoicevT=(0,c2),c2

meansthatΣ3ΨD= ΨD.Thus,forn=0,theelectronspincanonlypoint=to0.theThedirectionlatteroppositetothemagnetic eld.

ExpressionsforΨn(x,y)inthesemi-momentumrepresentationcontainexplicitlyafunc-tionalarbitrariness,whichmeansthateveryenergylevelisin nitelydegenerated.LetusdemandthatthescalarandspinorwavefunctionsbeeigenvectorsoftheoperatorsLzandJzrespectively.Accordingto(2.4)and(2.8)thatmeansthatthefunctionsΨn(x,y)havetoobeyanadditionalcondition

a+2a2Ψn(x,y)=sΨn(x,y),s=0,1,2,...,

Lz=h¯(n s)=h¯l,l=n s,n≥l> ∞,Jz=h¯ l 1

n s

√√x iy

α2x

sΨn,s 1,a+2Ψn,s=√

√2(x2+y2) =e√ ρ

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

Belowwearegoingto ndnewsetsofsolutionsimposingcomplementaryconditionsdi erentfrom(2.30).Thisresultsinadi erentformforthefunctionΦ(η).

Takingintoaccountthattheoperatorsa+2,a2areintegralsofmotion,wemayconstructstationarystates,whichareeigenvectorsofalinearcombinationAα,βoftheseoperators,2

Aα,β=αa2+βa+22.(2.33)

Hereα,βarearbitrarycomplexnumbers.Onehastodistinguishherethreenonequivalentcases:

If|α|2<|β|2,thendonotexistanynormalizableeigenvectorsoftheoperator(2.33).Wearenotgoingtoconsidersuchcase.

If|α|2=|β|2,thenAα,βis,infact,reducedtoaHermitianoperator2

+Aµ2=µa2+µa2,µµA+2=A2,µ=0,(2.34)

whereµisanarbitrarycomplexnumber.

haveIf|α|2>|β|2,thenwithoutlossofgeneralitywecanassumethatoperatorsAα,β2theform

Aα,β2=αa2+βa+2,|α| |β|=1,22

α,βα,βThenA+,Aarecreationandannihilationoperators,whicharerelatedtoa+222,a2byacanonicaltransformation

+α,βa2=α Aα,β,2 βA2+α,βa+ β Aα,β2.2=αA2 Aα,β2,α,βA+2 =1.(2.35)(2.36)

µµConsidereigenvectorsoftheoperator(2.34),i.e.,Aµz=z .This2Ψn,z(x,y)=zΨn,z(x,y),

µµequationresultsintheequationAµ2Φz(η)=zΦz(η)forthefunctionΦ(η).Takinginto

account(2.13),onecan ndthatsolutionsofthelatterequationare

Φµz(η)= µ

2π|µ|(µ µ ) 1

2zη z2(µ+µ )|µ| 2.

Thesesolutionsobeytheorthonormalityandcompletenessrelations

∞

∞µµΦ ′z(η)Φz(η)(2.37)dη=δ(z z),′ ∞

∞µ′µ′Φ z(η)Φz(η)dz=δ(η η).(2.38)

Theiroverlappinghastheform

Rµ′,µ(z′,z)=∞ µ′ µΦ z′(η)Φz(η)dη=N1exp

∞ Q2

Q2=z 2π2|µ′||µ|(µµ′ µ′µ )µ z′ ,

z′ 2

µ′ 2+ z µ µ′ .(2.39)

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

Itde nesthemutualdecomposition

Φµz(η)=∞ ′µ,µΦµ(z′,z)dz′.z′(η)R′(2.40)

∞

Thecoordinaterepresentation(2.17)forthesolutionsunderconsiderationhastheform

Ψµn,z(x,y)= √2 µ 2

Un(p1)expiQ3,

4√|µ|2Q3=[i(µ µ)x+(µ+µ )y][(µ+µ )x+i(µ µ )y 2z],

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

Φα,βs,z(η)= α

2 α β √2eQ4Us(p2),4|α β|2Q4

2z(α β ) √=2(αβ αβ)η+22

π

Theoverlappingα,β′α,β′Φ s,z(η)Φs,z(η)=δ(η η),d2z=dRezdImz.(2.47)

α′,β′;α,β′Rs(z,z)′,s= ∞

∞α,βα,βΦ s′,z′(η)Φs,z(η)dη,′′(2.48)

allowsusto ndmutualdecompositions

Φα,βs,z(η)=∞ α′,β′;α,β′α′,β′Rs(z,z)Φ′,ss′,z′(η),

s′=0Φα,βs,z(η)= α,β;α,β′,βd2z′Rs(z,z)Φα′,ss′,z′(η).′′′′(2.49)

Unfortunately,theoverlapping(2.48)hasacomplicatedformviaa nitesumofHermitfunctions.Insomeparticularcasesthissumcanbesimpli ed.Forexample,ifα′=α,β′=β,thentheoverlappingdoesnotdependonα,βandhastheform

α,β;α,β′Rs(z,z)′,s=Rs′,s(z′,z)= z z′2

exp 1

αα′

2Q5=2expQ5,z2(α′ β α β′ )+(z′ )2(αβ′ α′β)+2zz′

√x iy z n s

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

π1n

Ψα,βn,0,z(x,y)=Ψα,βn,z(x,y)=( 1)n α √

n,z(x,y)=(x+iy z)n

πΓ(n+1)exp 2|z|2

z(x iy) 1 ,

sΨα,βn,s 1,z,A+2α,βΨα,βn,s,z=z Ψα,βn,s,z+√

√2 q2Is,n(q)

=( 1)nN

z

πexpx iy n s

ρe i =(x iy)(x+iy z).

ForN=1theabovesetobeys(besides(2.24))therelations

a2Ψ¯n,s,z=zΨ¯n,s,z+√ zΨ¯n,s,z=√

Γ(s+1)

Γ(s′+1)(2.56)

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,

Ψ=hP¯3

3λ

2u im 2

λ

.(2.65)

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

SupposeEq.(2.63)holds,thenΨ( )canbepresentedintheform:

Ψ( )(xµ)=Nexp i λ

2λu0W(1 α3)Cψ(u0,x,y). (2.66)

HereCisanarbitraryconstantbispinor,andWisaunitarymatrix( 0isaconstantphase),

W=cosκ iΣ3sinκ,2κ=ωu0+ 0,W+W=I,(2.67)

andψ(u0,x,y)isascalarfunction.Thelatterfunctionobeystheequation(2.65).Then,theΨ(+)projectioncanbefoundfrom(2.62),Ψ(+)=(¯hλ) 1[(αP⊥)+h¯mρ3]Ψ( ).

Thus,bothinthescalarandspinorcaseswehavetosolvethesameequation(2.65). (u0,ξ,η)obeystheInthesemi-momentumrepresentation,thecorrespondingfunctionψ

sameequation(2.65),where,however,onehastousetheexpression(2.14)fortheoperator

0 0N=a+1a1.Therelationbetweenthefunctionsψ(u,ξ,η)andψ(u,ξ,η)stillhastheform

(2.17).

Letusintroducetheoperators

+Af,g1=fa1+ga1,f,g A+=f a+11+ga1,(2.68)

wherethecomplexquantitiesfandgcandependonu0.Theseoperatorsareintegralsofmotionwheneverf,gobeytheequations(bydotsabovearedenotedderivativeswithrespecttou0)

if˙+ωf=0,

Itiseasyto nd

f=f0expiωu 0ig˙ ωg=0. (2.69)wheref0,g0aresomecomplexconstants.Bearinginmindconsiderationsrelatedtothe

operators(2.33),wearegoingtoconsidertwononequivalentcasesonly.The rstonecorrespondsto|f|2=|g|2orequivalentlyto|f0|2=|g0|2.Inthiscasewecan,infact,onlyconsidertheHermitianoperator

+Aν1=νa1+νa1, ,g=g0exp iωu0,(2.70)ν=ν0eiωu,0ν0=const.(2.71)

Thesecondcasecorrespondsto|f|2>|g|2,andherewecansupposethat

|f|2 |g|2=|f0|2 |g0|2=1,(2.72)

withoutthelossofgenerality.Inbothcasestheoperators(2.68)are,withinconstantcomplexfactors,creationandannihilationoperators.

Letusincludeoperators(2.71)and(2.34)(theyareintegralsofmotion)intothecompletesetofoperators.Then

ν,µν,µAν1ψz1,z2=z1ψz1,z2,ν,µν,µAµ2ψz1,z2=z2ψz1,z2, zk=zk,k=1,2.(2.73)

Inthesemi-momentumrepresentationwe nd

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

wherefunctionsΦνz1arede nedin(2.37).Thecorrespondingcoordinaterepresentationreads

ν,µ0ψz(u,x,y)=,z12 ψ µu,ξ,η=Φνz1(ξ)Φz2(η),0 (2.74) µν2exp Q6

µ νµν

nψf,gn ;1α,β,s;z1,z2,1 z 1ψf,gn,s;;α,βz1,z2=√

sψf,gn,s; α,β1;z1,z2, A+f,gA+2α,β z 2 ψf,gn,s;;α,βz1,z2=√

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

Forn=s=0,wegetthecoordinaterepresentationforthesqueezedcoherentstatesintheform

;g;α,β0Ψfz1,z2(u,x,y) ( 1)x+iy z¯1 z22M1=eIs,n(p4),π 2M1=(¯z1 z2)(x+iy) (¯z1 z2)(x iy)+z¯1z2 z¯1z2 2inωu0, p4=|x+iy z¯1 z2|2,z¯1=z1exp( iωu0),,0;1,00Ψ1n,s;z1,z2(u,x,y)n (2.80)= αf2expQ7,

Q7=

Solutionsfrom[10–13]areparticularcasesof(2.81)forf0=α=1,g=β=0.

Calculatingmeanvaluesinthestates(2.78),weget1

γ [(f g )z1+(f g)z1].(2.82)P2= h¯2

Herewehavetakenintoaccounttherelations(2.6),(2.36),(2.79),andtheorthogonalityofthestateswithrespecttotheindicesn,s.Remembernowthatinclassicaltheorythe

clclcorrespondingmomentaP1,P2havethefollowingparametricrepresentation(withu0being

theevolutionparameter,Rradiusoftheclassicalorbit,andκisgivenby(2.67))Itiseasytoseethat(2.82)coincideswith(2.83)forz1=(γ/2)1/2R(f0e i 0+g0ei 0).Cal-

x2,we ndthattheyevolveasthecorrespondingculatingmeanvaluesofthecoordinates

2classicalquantitiesx1cl,x2cl(x1(0),x(0)arecoordinatesoftheorbitcenter)

1/2clP1=h¯γRsin2κ,clP2= h¯γRcos2κ.,2(αf βg)q= (α+β)(f+g)x2 (α β)(f g)y2+2i(βf αg)xy+2x[(α+β)z1+(f+g)z2]+2iy[(α β)z1 (f g)z2]22+(αg βf )z1+(β f α g)z2 2z1z2.1(2.81)(2.83).(α++i(α forz2=(γ/2)

Thus,mean-valuetrajectoriesintheplanex1,x2donotdependonquantumnumbersn,s.Thesetrajectorieshaveclassicalformsunderaproperchoiceofz1,z2.

Calculatingquadratic uctuationsinthestates(2.78),weget

2

2γ

( x2)2=|f+g|2(2n+1)+|α+β|2(2s+1),

σ1= σ2=i(fg gf )(2n+1),

σk=( P2)2=h¯2γ|f g|2(2n+1), x1cl=Rcosκ+x1(0),β)x1(0)β)x2(0) x2cl=Rsinκ+x2(0),(2.84)

1Onecanobtainthesameresultsusingspinorwavefunctionsforthecalculations.

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

Theydonotdependonz1,z2,butdodependonquantumnumbersn,sandonparametersf0,g0,α,β.Therelations(2.85)implythegeneralizedHeisenberginequalities

4J1=h¯(2n+1)(2n+1)+(2s+1)|(α β)(f+g)|

4J2=h¯(2n+1)(2n+1)+(2s+1)|(α+β)(f g)|22 2

2 ≥h¯2.≥h¯2,

Jk=( P 1k)2

( xk)2or

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

SuchsolutionscanbeconstructedintermsoftheLaguerrefunctionsIn,m(x)withnonintegerindices,

ψpq,l0) βI|l|+s,s(x),

Γ= u,ρ, =Nexp(il Γ)(1+p¯) α(1 p¯ilωu0

2(1 p¯2)ρ,α=p q

2p,

s=q

2,x=2¯pρ

Γ(1+n)2 2Φ( m,n m+1;x).(2.92)

HereΦ(α,β;x)isthedegeneratehypergeometricfunction(see[14],9.210).Forp2=1,theoperator(2.90)isanti-Hermitianandqisimaginary,Req=0.Forp=0,solutionshaveaverysimpleform

ψ0q,l u0,ρ, =N0exp(il +q¯ Γ0)J|l| 2√0

2lωu+ρ

4r =Jα β(x).(2.94)

Thefunctions(2.92)and(2.93)areorthogonalonlywithrespecttoquantumnumbersl,

ψppq′,l′,ψq,l =δl,l′QF( s, s′ ;1+|l|;y),y= 2p

1

p2Γ(1+s)Γ(1+s′ )

ψ0 l|4pΓ(1+|l|)y1+|,q′,l′,ψ0q,l =δ′

l,l′2πN0N0 I|l| 2

L2zh¯ 2+4|a1a2|2 a1a2 a+1a+2.(2.96)

Forp=0,itfollowsfrom(2.91)thata1a2= q¯,Lz=h¯l.Thus,wecanrewrite(2.96)intheform

ρ u0 =ρcl0+q¯+q¯ ,ρcl0=

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

Calculatingthemeanvalueρ¯bymeansofthefunctions(2.93),we nd

ρ¯=ρ0+q¯+q¯ ,ρ0=|l| 1 2|q|I|l| 1(2|q|)

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

B.Klein-Gordonequation

ConsidertheKlein-Gordonequationinthe eldsunderconsideration.Representingthewavefunctionas

Ψ= x,xψ(3.5)

we nd 03 x1,x2 ,

P12+P22 k21

Usingthevariablesx,y ψ,η x1,x2,ξde ned =0,in P02 P32 m2 k21 x0,x3=0.(3.6)(2.5)and(2.12),we can rewrite theequationforψ(x1,x2)inthefollowingform

ξ2 2k2

ξ k′12 ψ x1,x2 =0,k′12=1

√

√2yηψ (ξ,η)dη,ξ=√

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

Ax 3 =αx,Ax 3 =αexpβx 3 ,Ax 3 =α

√x iy n s

nψ1,a+1ψ1=i√

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

PECULIARITIESOFINTEGRATIONOFLINEARDIFFERENTIAL

EQUATIONSWITHNONCOMMUTATIVESYMMETRIES

Herewearegoingtoreturntotheaboveresultsfromapointofviewofgeneraltheoryofdi erentialequation.Recallthatwesucceededto ndexplicitlythetransformation(2.10)-(2.13)whichhasreducede ectivelythenumberofthevariablesintheinitialequations.Infact,thatwasthemainstartingpointforallfurtherconstructions.However,onecanseethatthis”reduction”ofvariablesisaparticularexampleofageneralsituation,whichisdescribedbrie ybelow.

Consider rstthecaseofanintegrableclassical2N-dimensionalHamiltoniansystemwiththeHamiltonianH.SupposethissystemhasNindependentintegralsofmotionthatareininvolution.Itiswellknownthatinsuchacasethevariablesofthetypeaction-angle(J, )areavailable,andtheHamiltoniandependsontheactionvariablesonly,H=H(J).LetussupposethatforsuchasystemexistsonemoreindependentintegralofmotionY.SinceYisindependent,itcannotcommutewiththeformerintegrals,and,therefore,Ymustdependontheanglevariables.OnecandemonstratethatinsuchacasetheHamiltoniansystemisdegenerate,det|| H(J)/ Ji Jj

dependonsomecombinationsoftheaction||=variables.0,and,therefore,Forexample,theHamiltoniansupposethedoesintegralnotYdoesnotcommutewiththeintegralJNonly.ThentheHamiltoniancandependonthevariablesJ1,...,JN 1only,otherwiseHcannotcommutewithY.Thus,weseethatthenoncommutativealgebraofintegralsofmotionallowsoneto ly,trajectoriesoftheintegrableHamiltoniansystemwithN-dimensionalcommutativesetofintegralsofmotionform(inthecompactcase)awindingofN-torusin2N-dimensionalphasespace.Ifthesetoftheintegralsisnoncommutativethenthedimensionofthecorrespondingtorusisr<N(see[16]).

Thephenomenonofvariable”reduction”takesplaceinthequantumintegrableHamilto-niansystemaswell.Aswillbedemonstratedbelow,byconstructingaspecialisomorphismoflinearfunctionalspaces,wecantransformtheinitialdi erentialoperatorofanequationintoanotheronewithareducednumberofvariables.Themethodwhichwearegoingtouseforsuchademonstration,is,infact,theharmonicanalysisonthenoncommutativefunctionalalgebras.

Supposeadi erentialequation

H(x, x)ψ(x)=0,ψ∈L C∞(RN)(3.16)

withNindependentvariablesx∈RNadmitsanoncommutativealgebraoffunctionallyin-dependentsymmetryoperatorsF={Xa(x, x)}.Thecorrespondingcommutationrelationsareinthegeneralcasenonlinear,i

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

quadraticalgebra,andsoon.ThealgebraFcorrespondstothealgebraF′={Yα(x, x)}oftheinvariantoperatorsonL:

[Xa,Yα]=0,i

h¯ a,X b]= ab(X ),[X (q, q,j))=κµ(j),Kµ(X κ(j) µ det

h¯ α,Y β]=ωαβ(Y ),[Y′ ′ (q, q,j))=κµ(j).Kµ(Y(q, q′,j))=Kµ(X(3.22)Supposethatinthespacesoffunctionsofx,ofq,andofq′arede nedscalarproducts( ,ψ)= (q)dµ(q),( , )′= (q)ψ ψ

RNQ′

2Via[q]wedenotethenumberofthevariablesq.Similarnotationsareusedbelow.

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