New solutions of relativistic wave equations in magnetic fie

1、new solutions of relativistic wave equations in magnetic fie 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,

2、which reduces eff 1 2 t c o 5 1 3 v 7 3 1 1 /0h t - p e :hv i x r anewsolutionsofrelativisticlongitudinalwaveequations elds.inmagnetic eldsandv.g.bagrov ,m.c.baldiotti ,d.m.gitman ,andi.v.shirokovinstitutodef sica,universidadedes aopaulo,c.p.66318,05315-970s aopaulo,sp,brasil(february1,2021)abstract

3、wedemonstratehowonecandescribeexplicitlythepresentarbitrarinessinsolutionsofrelativisticwaveequationsinexternalelectromagnetic eldsofspecialform.thisarbitrarinessisconnectedtotheexistenceofatransforma-tion,whichreducese ectivelythenumberofvariablesintheinitialequations.thenweusethecorrespondingrepre

4、sentationstoconstructnewsetsofex-actsolutions,ly,wepresentnewsetsofstationaryandnonstationarysolutionsinmagnetic eldandinsomesuperpositionsofelectricandmagnetic elds.i.introductionrelativisticwaveequations(diracandklein-gordon)provideabasisforrelativisticquantummechanicsandquantumelectrodynamicsofsp

5、inorandscalarparticles1.inrelativisticquantummechanics,solutionsofrelativisticwaveequationsarereferredtoasone-particlewavefunctionsoffermionsandbosonsinexternalelectromagnetic elds.inquantumelectrodynamics,suchsolutionsallowthedevelopmentoftheperturbationexpansionknownasthefurrypicture,whichincorpor

6、atestheinteractionwiththeexternal eldexactly, whiletreatingtheinteractionwiththequantizedelectromagnetic eldperturbatively2.thephysicallymostimportantexactsolutionsoftheklein-gordonandthediracequationsare:anelectroninacoulomb eld,auniformmagnetic eld,the eldofaplanewave,the eldofamagneticmonopole,th

7、e 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 arbi

8、trariness is connected to the existence of a transformation, which reduces eff simpleone-dimensionalelectric elds(foracompletereviewofsolutionsofrelativisticwaveequationssee3). considering,forexample,stationarysolutionsofrelativisticwaveequations,wecanseethatinthegeneralcase,thereexistdi erentsetsof

9、stationarysolutionsforoneandthesamehamiltonian.thepossibilitytogetdi erentsetsofstationarystatesre ectstheex-istenceofanarbitrarinessinthesolutionsoftheeigenvalueproblemforahamiltonian.consideringnonstationarysolutions,wealsoencounterthepossibilityofconstructingdif-ferentcompletesetsofsuchsolutions.

10、thereisnoregularmethodofdescribingsuchanarbitrarinessexplicitly.especiallyinthepresenceofanexternal eldtheproblemappearstobenontrivial. inthepresentarticlewedemonstratehowonecandescribeexplicitlythepresentarbi-trarinessinsolutionsoftherelativisticwaveequationsforsometypesofexternalelectro-magnetic e

11、lds,namely,foruniformmagnetic eldsandcombinationofthese eldswithsomeelectric elds.thisarbitrarinessisconnectedtotheexistenceofatransformation,whichreducese ectivelythenumberofvariablesintheinitialequations.thenweusethecorrespondingrepresentationstoconstructnewsetsofexactsolutions,whichmayhaveaphysic

12、alinterest.insect.iiweconsiderrelativisticwaveequationsinpureuniformmagnetic elds.herewederivearepresentationfortheexactsolutions,inwhichtheabovementionedarbitrarinessisdescribedexplicitlybyanarbitraryfunction.fromasuitablechoiceofthisfunction,wegetboththewell-knownsetofsolutionsandnewones.thissecti

13、oncontainsthemostcomplete(atthepresent)descriptionoftheproblemofauniformmagnetic eldinrelativisticquantummechanics.amongnewsetsofsolutionstherearebothstationary,gen-eralizedcoherentsolutionsandnonstationarysolutions.then,insect.iii,weconsidermorecomplicatedcon gurationsofexternalelectromagnetic elds

14、,namely,longitudinalelectro-magnetic elds.herewedescribeallthearbitrarinessinthesolutions,andonthisbasepresentvarioussetsofnewexactsolutions.insect.ivweinterprettheaboveresultsfromthepointofviewofthegeneraltheoryofdi erentialequations. ii.uniformmagneticfield a.arbitrarinessinsolutionsofrelativistic

15、waveequations. considerauniformmagnetic eldh=(0,0,h)directedalongthex3axis(h0).theelectromagneticpotentialsarechoseninthesymmetricgauge a0=a3=0,a1=1 2hx1.(2.1) wewritetheklein-gordonandthediracequationsintheform k=0,2h2k=p2 m2 0c,p=ihe we demonstrate how one can describe explicitly the present arbit

16、rariness 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=ihx 1 x 2,lz,p0=lz,p3=k,lz=0, 21 canbeincluded(togethe

17、rwithp0andp3)inthecompletesetofintegralsofmotion,whereasforthediracequationcase,theoperatorjz, jz=lz+h (2.3) dx1dx2=2cos , 2x2=y=ch0,dd ,x+iy= ei (x+iy+ x+i y)=(+i +2 ),2 11 12+=p2+ip1+hx ixa2=2h2 e i (x iy+ x i y)=( i +2 ),2 11(ip p)=a+=1212h2 d=h 1 p0+p3 03 2 i 21 h,21a1+i a+1 m.(2.9) we demonstra

18、te 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,plusitisanintegralofmo

19、tioninthecaseoftheklein-gordonequation.itsgeneralizationforthediracequationhastheformnd=n+1 2=x+k, 2x=+,2a1=+ ,2a2=+ , i m. onecanseethatthelatteroperatorsdonotcontainthevariable.noticethatbothoperatorslzandjzcontainvariables,.forexample, 222lz=2 2+ . 21 (2.15)(2.16) theintegrationoverkin (2.10)canb

20、ereplacedbyanintegrationover, eixy (x,y)= e i2x . (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, whic

21、h reduces eff besides,onecanwrite (,)= dxdy (x,y)(x,y)= ,d (,) = d (,).(2.18) theindependenceoftheoperators(2.15)onthevariablewillallowustoseparateexplicitlythefunctionalarbitrarinessinthesolutions(2.17),aswillbeseenbelow. b.stationarystates knownsetsofstationarysolutionsinauniformmagnetic eld(thatw

22、erefoundinthe rstworks48)areeigenfunctionsoftheoperatorsp0,p3,ninthescalarcaseandoftheoperatorsp0,p3,ndinthespinorcase.thusforscalarwavefunctionswehavetheconditions p0=hk0,p3=hk3,n=n,n=0,1,2,.,(2.19) andfordiracwavefunctionstheconditions p0=hk0,p3=hk3,nd= n 1 2x .(2.23) hereeqs.(2.19),(2.14)wereused

23、.un()arehermitfunctions; correspondingpolynomialshn()asun(x)=(2nn!theyarerelatedtothe 2exp( x2/2)hn(x)14.the function()isarbitrary.thefunctionsn(x,y)from(2.22)obeytherelations a1n=n+1a+n 1 n+1,n(x,y)= (n +1)0(x,y),(2.24) 3 0(x,y)= 2+ we demonstrate how one can describe explicitly the present arbitra

24、riness 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 tn,k3(x,y)=(c1n 1(x,y),ic2n(x,y),c3n 1(x,y),ic4n(x,y).(2.26)thefunctionsn(x,y)arede nedbytherelations(2.17),(

25、2.23),whereastheconstantbispinorc(withtheelementsck)obeysanalgebraicsystemofequations ac=0,a=0k0+3k3 2n1 k33)v ,c+c=2k0(k0+m)v+v,(2.29) wherevisanarbitraryconstantbispinorandarepaulimatrices.wecanspecifyvselectingaspinintegralofmotion(see3).thestaten=0isaspecialcase.herewemustsetc1=c3=0,thatcorrespo

26、ndstothechoicevt=(0,c2),c2 meansthat3d= d.thus,forn=0,theelectronspincanonlypoint=to0.thethedirectionlatteroppositetothemagnetic eld. expressionsforn(x,y)inthesemi-momentumrepresentationcontainexplicitlyafunc-tionalarbitrariness,whichmeansthateveryenergylevelisin nitelydegenerated.letusdemandthatthe

27、scalarandspinorwavefunctionsbeeigenvectorsoftheoperatorslzandjzrespectively.accordingto(2.4)and(2.8)thatmeansthatthefunctionsn(x,y)havetoobeyanadditionalcondition a+2a2n(x,y)=sn(x,y),s=0,1,2,., lz=h(n s)=hl,l=n s,nl ,jz=h l 1 n s x iy 2x sn,s 1,a+2n,s= 2(x2+y2) =e we demonstrate how one can describe

28、 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 erentf

29、rom(2.30).thisresultsinadi erentformforthefunction(). takingintoaccountthattheoperatorsa+2,a2areintegralsofmotion,wemayconstructstationarystates,whichareeigenvectorsofalinearcombinationa,oftheseoperators,2 a,=a2+a+22.(2.33) here,arearbitrarycomplexnumbers.onehastodistinguishherethreenonequivalentcas

30、es: if|2|2,thendonotexistanynormalizableeigenvectorsoftheoperator(2.33).wearenotgoingtoconsidersuchcase. if|2=|2,thena,is,infact,reducedtoahermitianoperator2 +a2=a2+a2,a+2=a2,=0,(2.34) whereisanarbitrarycomplexnumber. haveif|2|2,thenwithoutlossofgeneralitywecanassumethatoperatorsa,2theform a,2=a2+a+

31、2,| |=1,22 ,thena+,aarecreationandannihilationoperators,whicharerelatedtoa+222,a2byacanonicaltransformation +,a2= a,2 a2+,a+ a,2.2=a2 a,2,a+2 =1.(2.35)(2.36) considereigenvectorsoftheoperator(2.34),i.e.,az=z .this2n,z(x,y)=zn,z(x,y), equationresultsintheequationa2z()=zz()forthefunction().takinginto

32、account(2.13),onecan ndthatsolutionsofthelatterequationare z()= 2|( ) 1 2z z2(+ )| 2. thesesolutionsobeytheorthonormalityandcompletenessrelations z()z()(2.37)d=(z z), z()z()dz=( ).(2.38) theiroverlappinghastheform r,(z,z)= z()z()d=n1exp q2 q2=z 22|( ) z , z 2 2+ z .(2.39) we demonstrate how one can

33、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 z()= ,(z,z)dz.z()r(2.40) thecoordinater

34、epresentation(2.17)forthesolutionsunderconsiderationhastheform n,z(x,y)= 2 2 un(p1)expiq3, 4|2q3=i( )x+(+ )y(+ )x+i( )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 ar

35、bitrariness is connected to the existence of a transformation, which reduces eff ,s,z()= 2 2eq4us(p2),4| |2q4 2z( ) =2( )+22 theoverlapping, s,z()s,z()=( ),d2z=drezdimz.(2.47) ,;,rs(z,z),s= , s,z()s,z()d,(2.48) allowsusto ndmutualdecompositions ,s,z()= ,;,rs(z,z),ss,z(), s=0,s,z()= ,;,d2zrs(z,z),ss,

36、z().(2.49) unfortunately,theoverlapping(2.48)hasacomplicatedformviaa nitesumofhermitfunctions.insomeparticularcasesthissumcanbesimpli ed.forexample,if=,=,thentheoverlappingdoesnotdependon,andhastheform ,;,rs(z,z),s=rs,s(z,z)= z z2 exp 1 2q5=2expq5,z2( )+(z )2( )+2zz x iy z n s we demonstrate how one

37、 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 1n ,n,0,z(x,y)=,n,z(x,y)=( 1)n n,z(x,y)=(x+iy z)n (n+1)exp 2|z|2

38、z(x iy) 1 , s,n,s 1,z,a+2,n,s,z=z ,n,s,z+ 2 q2is,n(q) =( 1)nn z expx iy n s e i =(x iy)(x+iy z). forn=1theabovesetobeys(besides(2.24)therelations a2n,s,z=zn,s,z+ zn,s,z= (s+1) (s+1)(2.56) we demonstrate how one can describe explicitly the present arbitrariness in solutions of relativistic wave equat

39、ions 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)= (s+k+1) d2z (k+s+1) k!n,s+k,z(x,y).(2.59) thatmeans,inparticular,that(2.56)isacompletesetsincetheset(2.31)iscomplete.selectingdi erentformsfo

40、rthefunction(),wecangetothersetsofstationarystatesforachargeinauniformmagnetic eld. c.nonstationarystates themostinterestingnonstationarysolutionsofrelativisticwaveequationsforachargeinauniformmagnetic eldarecoherentstates;forthe rsttimesuchsolutionswerepresentedin1013,seealso3.belowwepresentanewfam

41、ilyofnonstationarysolutions,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=4h 2p30 2n m 2(2.

42、61)whereasthediracequationreads(isadiracbispinor)4h 2 p= (p1,p2,0), p p30( )=2nd+m =(+)+( ), ( ), 2p3m( ),3(+)=(p)+h()=p,2p=13.(2.62)hereand3arediracmatrices3,andpprojectionoperators. ,p areinthecaseoftheuniformmagnetic eldunderconsideration,theoperatorsp30 integralsofmotion.thus,wewillconsidersolut

43、ionsthatareeigenvectorsofp3, =hp3 3 2u im 2 .(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 red

44、uces eff supposeeq.(2.63)holds,then( )canbepresentedintheform: ( )(x)=nexp i 2u0w(1 3)c(u0,x,y). (2.66) herecisanarbitraryconstantbispinor,andwisaunitarymatrix( 0isaconstantphase), w=cos i3sin,2=u0+ 0,w+w=i,(2.67) and(u0,x,y)isascalarfunction.thelatterfunctionobeystheequation(2.65).then,the(+)projec

45、tioncanbefoundfrom(2.62),(+)=(h) 1(p)+hm3( ). thus,bothinthescalarandspinorcaseswehavetosolvethesameequation(2.65). (u0,)obeystheinthesemi-momentumrepresentation,thecorrespondingfunction sameequation(2.65),where,however,onehastousetheexpression(2.14)fortheoperator 0 0n=a+1a1.therelationbetweenthefun

46、ctions(u,)and(u,)stillhastheform (2.17). letusintroducetheoperators +af,g1=fa1+ga1,f,g a+=f a+11+ga1,(2.68) wherethecomplexquantitiesfandgcandependonu0.theseoperatorsareintegralsofmotionwheneverf,gobeytheequations(bydotsabovearedenotedderivativeswithrespecttou0) if+f=0, itiseasyto nd f=f0expiu 0ig g

47、=0. (2.69)wheref0,g0aresomecomplexconstants.bearinginmindconsiderationsrelatedtothe operators(2.33),wearegoingtoconsidertwononequivalentcasesonly.the rstonecorrespondsto|f|2=|g|2orequivalentlyto|f0|2=|g0|2.inthiscasewecan,infact,onlyconsiderthehermitianoperator +a1=a1+a1, ,g=g0exp iu0,(2.70)=0eiu,00

48、=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)intothecomp

49、letesetofoperators.then ,a1z1,z2=z1z1,z2,a2z1,z2=z2z1,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 arbitrarine

50、ss is connected to the existence of a transformation, which reduces eff wherefunctionsz1arede nedin(2.37).thecorrespondingcoordinaterepresentationreads ,0z(u,x,y)=,z12 u,=z1()z2(),0 (2.74) 2exp q6 nf,gn ;1,s;z1,z2,1 z 1f,gn,s;,z1,z2= sf,gn,s; ,1;z1,z2, a+f,ga+2, z 2 f,gn,s;,z1,z2= we demonstrate how

51、 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,wegetthecoordinaterepresentationforthesqueezedcohere

52、ntstatesintheform ;g;,0fz1,z2(u,x,y) ( 1)x+iy z1 z22m1=eis,n(p4), 2m1=(z1 z2)(x+iy) (z1 z2)(x iy)+z1z2 z1z2 2inu0, p4=|x+iy z1 z2|2,z1=z1exp( iu0),0;1,001n,s;z1,z2(u,x,y)n (2.80)= f2expq7, q7= solutionsfrom1013areparticularcasesof(2.81)forf0=1,g=0. calculatingmeanvaluesinthestates(2.78),weget1 (f g

53、)z1+(f g)z1.(2.82)p2= h2 herewehavetakenintoaccounttherelations(2.6),(2.36),(2.79),andtheorthogonalityofthestateswithrespecttotheindicesn,s.remembernowthatinclassicaltheorythe clclcorrespondingmomentap1,p2havethefollowingparametricrepresentation(withu0being theevolutionparameter,rradiusoftheclassicalorbit,andisgivenby(2.67)itiseasytoseethat(2.82)coincideswith(2.83)forz1=(/2)1/2r(f0e i 0+g0ei 0).cal- x2,we ndthattheyevolveasthecorrespondingculatingmeanvaluesofthecoordinates 2classicalquantitiesx1cl,x2cl(x1(0),x(0)arecoordinatesoftheorbitcen