Distorted Heisenberg Algebra and Coherent States f

arXiv:hep-th/9501035v111Jan1995DistortedHeisenbergAlgebraandCoherentStatesforIsospectralOscillatorHamiltoniansDavidJ.Fern´andezC.1∗,LuisM.Nieto2†andOscarRosas-Ortiz1‡1DepartamentodeF´ısica,CINVESTAV-IPN,A.P.14-740,07000M´exicoD.F.,Mexico2DepartamentodeF´ısicaTe´orica,UniversidaddeValladolid47011Valladolid,SpainFebruary1,2008AbstractThedynamicalalgebraassociatedtoafamilyofisospectraloscillatorHamil-toniansisstudiedthroughtheanalysisofitsrepresentationinthebasisofenergyeigenstates.Itisshownthatthisrepresentationbecomessimilartothatofthestan-dardHeisenbergalgebra,anditisdependentofaparameterw≥0.WenameitdistortedHeisenbergalgebra,wherewisthedistortionparameter.Thecorrespond-ingcoherentstatesforanarbitrarywarederived,andsomeparticularexamplesarediscussedinfulldetail.Aprescriptiontoproducethesqueezing,byadequatelyselectingtheinitialstateofthesystem,isgiven.Key-Words:Coherentstates,Heisenbergalgebra,squeezedstatesPACS:03.65.-w,11.30.Pb,42.50.DvCIEA-GR-9501∗e-mail:david@fis.cinvestav.mx†e-mail:[e-mail]‡e-mail:orosas@fis.cinvestav.mx1IntroductionThewellknowncoherentstatesoftheharmonicoscillatorturnedoutoneofmostusefultoolsofquantumtheory[1,2,3].IntroducedlongagobySchr¨odinger[4],theywereemployedlateronbyGlauberandotherauthorsinquantumoptics[5,6,7].Furtherdevelopmentsofthesubjectmadepossibletosetupsomespecificdefinitions,applicabletovariousphysicalsystems.Onepossibilityistodefinethecoherentstatesaseigenstatesofanannihilationop-erator.Followingthisidea,thecoherentstatesforafamilyofHamiltoniansisospectraltotheharmonicoscillatorwererecentlyderived[8].Asthereisacertainarbitrarinessintheselectionoftheannihilationandcreationoperatorsforthatsystem,themostobviousrealizationwaschosen:theoperatorsareadjointtoeachotherbuttheircommutatorisnottheidentity.Inthesamepaperadifferentoptionofconstructingtheloweringandrisingoperatorswasalsopursued:thecreatorwasalteredwhiletheannihilatorwasnot;theywerenotadjointtoeachotheranymore,buttheircommutatorwasequaltotheidentity.Thismodifiedpair,inprinciple,couldinducenewcoherentstates,consistentwiththeapplicationofa“displacementoperator”totheextremalstate.However,thestatessoderivedturnedouttobeidenticaltotheonespreviouslydefinedaseigenstatesoftheannihilator.Inthelightofthoseresults,itisinterestingtoposethefollowingquestions:canbothideasbeunifiedtoyieldloweringandrisingoperatorswhichwouldbeadjointtoeachotherandwouldcommutetotheidentity,imitatingthentheHeisenbergalgebra?Ifso,whatkindofcoherentstateswouldtheygenerate?Thegoalofthispaperistofindouttheanswers.InSection2wewillsketchthederivationofthefamilyofHamiltoniansisospectraltotheharmonicoscillator[9,8].Section3containstheconstructionofnewcouplesofannihilationandcreationoperatorsforthoseHamiltonians;wewillbuildthosecouplesfromthegeneratorsofthestandardHeisenbergalgebra.Indeed,wewillseethatthereisafamilyofsuchapairsdependingonaparameterw≥0.InSection4twosetsofcoherentstateswillbefoundforarbitraryvaluesofw:theonesderivableaseigenstatesoftheannihilationoperatorandtheonesresultingfromtheapplicationofa“displacement”operatorontheextremalstate.Byfixingsomespecificvaluesofw,wewillattainthreeparticularlyinterestingcaseswhichwillbediscussedinSection5.WeconcludewithsomegeneralremarksinSection6.12TheisospectraloscillatorHamiltoniansHλWeareinterestedinafamilyofHamiltoniansHλwhichcanbederivedfromtheharmonicoscillatorHamiltonianHusingavariantofthefactorizationmethod[9].ThestandardfactorizationexpressesHastwoproductsH=aa†−12,H=a†a+12,(2.1)whereHandtheannihilationaandcreationa†operatorsaregivenbyH=−12d2dx2+x22,a=1√2ddx+x!,a†=1√2−ddx+x!.(2.2)Itcanbeprovedthatthefirstdecompositionin(2.1)isnotunique.Indeed,thereexistmoregeneraloperatorsbandb†generatingH:H=bb†−12,b=1√2ddx+β(x)!,b†=1√2−ddx+β(x)!.(2.3)Hence,β(x)obeystheRiccatiequationβ′+β2=x2+1,whosegeneralsolutionisβ(x)=x+e−x2λ+Rx0e−y2dy,λ∈R.(2.4)Theinvertedproductb†bisnotrelatedtoH,butinducesadifferentHamiltonianHλ=b†b+12=−12d2dx2+Vλ(x),(2.5)Vλ(x)=x22−ddxe−x2λ+Rx0e−y2dy#,|λ|√π2.(2.6)TheHamiltoniansHandHλareconnectedbythefollowingrelation:Hλb†=b†(H+1).(2.7)Therefore,if|ψniarethestandardeigenstatesofHverifyingH|ψni=(n+1/2)|ψni,thestatesdefinedas|θni=b†|ψn−1i√n,n=1,2,3,···(2.8)arenormalizedorthogonaleigenstatesofHλwitheigenvaluesEn=n+1/2respectively.ThegroundstateofHλisdisconnectedfromtheothereigenstates,ithaseigenvalueE0=1/2andsatisfiesb|θ0i=0.Inthecoordinaterepresentationitisgivenbyθ0(x)∝e−x2/2λ+Rx0e−y2dy.(2.9)2Summarizingthissection,{Hλ,|λ|√π/2}representsafamilyofHamiltonianswiththesamespectrumastheharmonicoscillator.TherelationsnecessarytosetupthecreationandannihilationoperatorsofHλareb|θni=√n|ψn−1i,b†|ψni=√n+1|θn+1i,a|ψni=√n|ψn−1i,a†|ψni=√n+1|ψn+1i.(2.10)3DistortedHeisenbergalgebraofHλItisimportanttoidentifynowasuitablepairofannihilationandcreationoperatorsforHλ.Theobviouschoicefollowsimmediatelyfrom(2.10)[9,8]:A=b†ab,A†=b†a†b.(3.1)Theeffectiveactionof,letussay,theannihilationoperatorAcomesafterthreeinterme-diatetransformations:wetakeaneigenstate|θniofHλandtransformit,bytheactionofb,in|ψn−1i,aneigenstateofH;then,atransform