physicastatussolidi, 11October2012 Radiation fields for nanoscale systems MingLiangZhang,D.A.Drabold*, DepartmentofPhysicsandAstronomy,OhioUniversity,Athens,Ohio45701 2 1 ReceivedXXXX,revisedXXXX,acceptedXXXX 0 PublishedonlineXXXX 2 t c Keywords: radiation,temporalcoarsegraining,semi-classicalradiationtheory,quantumelectrodynamics. O ∗Correspondingauthor:[email protected], [email protected] 0 1 ] h For a group of charged particles obeying quantum me- Five radiation fields are perpendicular to the canonical c chanics interacting with an electromagnetic field, the momentumof a single chargedparticle. For a group of e charge and current density in a pure state of the sys- charged particles, a new type of radiation field is pre- m tem are expressed with the many-body wave function dictedtobeperpendiculartoA(x ,t)×[∇×(∇ Ψ′)], j j t- of the state. Using these as sources, the microscopic whereΨ′ isthemany-bodywavefunction.Thiskindof a Maxwell equations can be written down for any given radiationdoesnotexistforasinglechargedparticle.The t s purestate ofa many-bodysystem.By employingsemi- macroscopic Maxwell equations are derived from the t. classicalradiationtheorywiththesesources,themicro- correspondingmicroscopicequationsforapurestateby a scopic Maxwell equations can be used to compute the the Russakoff-Robinson procedurewhich requires only m strong radiation fields produced by interacting charged a spatial coarse graining. Because the sources of fields - quantalparticles.Forachargedquantalparticle,threera- are also the responses of a system to an external field, d diationfieldsinvolveonlythevectorpotentialA.Thisis one also has to give up the temporalcoarse grainingof n o anotherexampledemonstratingtheobservabilityofvec- thecurrentdensitywhichisoftensupposedtobecritical c torpotential. inthekineticapproachofconductivity. [ 1 Copyrightlinewillbeprovidedbythepublisher v 8 8 1 Introduction Thispaperdiscussesthreecloselyre- bound states and coherent radiation from many charged 8 latedproblemsforagroupofnon-relativisticchargedpar- particles. To further exploreradiation phenomenain con- 2 ticlesobeyingquantummechanics(QM):(1)theradiation densedphases,weneedanalternativemethodforcomput- . 0 fields produced by the current density in a pure state of ingfield. 1 the many-bodysystem; (2) derivationof the macroscopic 2 Maxwell equations from the corresponding microscopic 1 equations in a given pure state; and (3) the consequence The “neoclassical” or semi-classical radiation the- : v of the averaging procedure used in (2) to computing the ory (SCRT) of E. T. Jaynes[12] is concerned with the i electricalconductivity. strongradiationofarelativisticornon-relativisticcharged X particle. The motion of the particle is treated quan- r Incondensedphases,thereexistabundantradiationre- tum mechanically while the electromagnetic field is a latedphenomena.Cherenkovradiationandtransitionradi- treated classically[8,9,10,11,12]. The SCRT has suc- ation are two prominent examples[1]. Although quantum ceeded in treating spontaneous emission, absorption, electrodynamics(QED) is believedto be applicableto all induced emission[13,14], photo-dissociation, Raman radiationproblems,itisonlyfeasibletocalculatetheweak scattering[15], radiative level shifts[16] and absorption field(afewphotons)producedbyafewchargedparticles ofradiationbydiffusiveelectron[17].Thusitisnaturalto withperturbationtheory[2,3,4,5,6,7].QEDisdifficultfor extend SCRT to the radiation field caused by a current- thelongtimeevolution[8],thereactionoftheatomonthe carryingpurestateofmanynon-relativisticchargedparti- appliedfield[9,10],theeffectofboundaryconditions[11], cles.Inthispaper,wemergeresultsfromthemicroscopic Copyrightlinewillbeprovidedbythepublisher 2 Firstauthoretal.:Shorttitle response method (MRM)[18,19] with concepts of SCRT whereH′(t) = H +H (t) is thetotalHamiltonianfor fm toextendthelattertomany-particlesystems. S+F,H istheHamiltonianofsystem,H (t)isthefield- fm matterinteraction: Since the pioneering work of Lorentz[20], many schemes have been suggested to derive the macroscopic Maxwell equations from the corresponding microscopic Ne i¯he H (t)= { [A(r ,t)·∇ +∇ ·A(r ,t)] (2) fm j j j j equations[21,22,23]. To obtain the macroscopic fields 2m Xj=1 and sources, usually both temporal coarse graining and spatial coarse graining are taken in addition to ensemble e2A2(r ,t) j average. By assuming that the motion of charged parti- + +eφ(r ,t)} j 2m cles obeys classical mechanics (CM), Russakoff[24] and Robinson[25] (RR) provedthat (i) only the spatialcoarse N i¯hZ e L grainingisrelevant;(ii)thespatialcoarsegrainingiscom- + {− [A(WL,t)·∇L+∇L·A(WL,t)] 2M patible with the ensemble average[26]. Since the motion LX=1 L of conduction electrons in solids is described by QM, it (Z e)2A2(W ,t) L L is desirable to extend Russakoff-Robinson’sprocedure to + −ZLeφ(WL,t)}. 2M suchasituation. L The time dependence of H (t) comes from the ex- Dropping the temporal coarse graining has a pro- fm foundconsequence.Inbothmicroscopicandmacroscopic ternal field. The arguments of Ψ′ are (r1,r2,··· ,rNe; Maxwellequations,currentdensityandchargedensityare W1,W2,··· ,WN;t). In QM, the state of the system at time t is described the sources of electromagnetic field. On the other hand, by the solution Ψ′(t) of Eq.(1), the microscopic charge currentdensity is also the response of a system to an ex- ternal field. Therefore Russakoff-Robinson’s procedure density ρΨ′ and current density jΨ′ in state Ψ′ are com- means that the temporal coarse-grained average is irrele- pletelydeterminedbyΨ′(t).Themicroscopicchargeden- vant to derive the irreversibility caused by a conduction sity ρΨ′ in state Ψ′ is ρΨ′(r,t) = dτΨ′∗ρ(r)Ψ′, where process. ρ(r) = eδ(r − r ) − ZReδ(r − W ) is the j j L L L InSec.2,wewriteoutthemicroscopicMaxwellequa- charge dePnsity operator of thPe system, dτ b= dr1dτ1 is b tionfor a pure state. In Sec.3, we applySCRT to analyze the volume element in the whole configurational space, theradiationfieldsfromthecurrentdensitycausedbyone dτ1 = dr2···drNedW1···dWN. Carrying out the in- and many charged particles. For a quantal charged parti- tegrals and using the antisymmetry of Ψ′ for exchanging cle,threeradiationfieldsinvolveonlythevectorpotential. thecoordinatesoftwoelectrons, Herewehaveonemoreexampledemonstratingthephys- ical reality of vector potential. In contrastto the classical ρΨ′(r,t)=N e dτ1Ψ′Ψ′∗− Z e dτLΨ′Ψ′∗, radiationfield,fiveradiationfieldsareperpendiculartothe e Z X L Z L canonical momentum of particle. For a group of charged (3) particlesobeyingquantummechanics,oneofitsradiation wheredτL =dr1···drNedW1···dWL−1dWL+1···dWN, fieldsisperpendiculartoA(xj,t)×[∇×(∇jΨ′)],where theargumentsoftheΨ′ inthefirsttermare(r,r2···rNe; Ψ′ is the many-body wave function, A(xj,t) is the vec- W1···WN),theargumentsoftheΨ′ inthesecondterm torpotentialfeltbythejth particleattimet.Thiskindof are(r1,··· ,rNe;W1,··· ,WL−1,r,WL+1··· ,WN;t). radiation does not exist for a single charged particle. In 2.1 Microscopic Maxwell equations in a pure Sec.4, we apply the RR procedure to derive the macro- state If the system is in a state Ψ′(t), the microscopic scopic Maxwell equations for a group of charged parti- Maxwellequationsare cles obeying QM. We show that without temporal coarse grained average, the irreversibility of conduction process ∇·bΨ′ =0, ∇×eΨ′ =−∂bΨ′/∂t, (4) isstillincludedinthepresenttheory. and 2 MicroscopicMaxwellequationsforapurestate Consider a system (S) with Ne electrons and N nuclei ∇·eΨ′ =ρΨ′/ǫ0, c2∇×bΨ′ =jΨ′/ǫ0+∂eΨ′/∂t, (5) inanexternalelectromagneticfield(F)describedbyvec- torandscalarpotentials(A,φ).Denotethecoordinatesof where eΨ′ and bΨ′ are the microscopic electric field and Ne electronsasr1,r2,··· ,rNe,thecoordinatesofN nu- magneticinductionat(r,t)inpurestateΨ′. cleiasW1,W2,··· ,WN.Intheexternalfield(A,φ),the Applying∂/∂tonthefirstinhomogeneousequationin stateΨ′(t)ofthesystemisdeterminedbythemany-body (5),and∇·onthesecond,onehasthecontinuityequation Schro¨dingerequation forapurestateΨ′[27]: i¯h∂Ψ′/∂t=H′Ψ′, (1) ∂ρΨ′(r,t)/∂t+∇·jΨ′(r,t)=0. (6) Copyrightlinewillbeprovidedbythepublisher pssheaderwillbeprovidedbythepublisher 3 N (Z e)2 Eq.(6)helpsusfindthemicroscopiccurrentdensity[18,19] − L A (r,t) dτLΨ′∗Ψ′, jΨ′ fromρΨ′: LX=1 ML α Z jΨ′(r,t)= i¯h2emNe Z dτ1(Ψ′∇Ψ′∗−Ψ′∗∇Ψ′) awrheere(r,thre2,·a·rg·u,mrNene;tsWo1f,·Ψ·′· ,iWn tNhe;t)1,stthaned a2rngdumliennetss of the Ψ′ in the 3rd and 4th lines is (r1,··· ,rNe; − Nmee2A(r,t)Z dτ1Ψ′∗Ψ′ (7) Wwit1h,·E·q·.(,7W), wL−e1fi,nrd,WthaLt+t1he··c·u,rWrenNtd;etn).sCityominpastraintegΨEq′.i(s12) relatedtothetimederivativeofpolarizationinstateΨ′by: N i¯hZ e −LX=1 2MLL Z dτL(Ψ′∇Ψ′∗−Ψ′∗∇Ψ′) jαΨ′(r,t)= ∂PαΨ∂′t(r,t), α=x,y,z. (13) N (Z e)2 − L A(r,t) dτLΨ′∗Ψ′, Since both the free carriers and the bound electrons are LX=1 ML Z includedin Eq.(8), the polarizationPαΨ′ definedin Eq.(9) contains the contributions from both carriers and bound where the arguments of the Ψ′ in the 1st and 2nd lines electrons. aorfeth(er,rΨ2′,·i·n· ,trhNee;3rdWa1n,d···4t,hWlNine;ts),isth(er1,a·rg··um,reNnets; by t2h.e3cRoaudplieadtioeqnufiateioldnsT(h1e,4s,5ta)t[e14o]f.SIn+Fordiisndareytemrmaitnereid- W1,··· ,WL−1,r,WL+1··· ,WN;t).Thecurrentden- als,theinducedmotionofthechargedparticlesbyaweak sity(7)isvalidforanarbitrarygauge.InEq.(7),thecontri- butiontojΨ′ fromnucleiissimilartothatfromelectrons; external field is non-relativistic: the energy radiated by a particleismuchsmallerthanthemechanicalenergyofthat wewillnotkeepthelasttwoterms. particle.Usingdirectproduct|Ψ′i⊗|e,bitorepresentthe 2.2 Currentdensityanddipolemomentdensity stateofS+F isthenallowed. The αth (α = x,y,z) component of the dipole density The radiation fields produced by a current-carrying operatorisdefinedas purestateΨ′(t)are[28]: Ne N ρbdα(r)=Xj=1erjαδ(r−rj)−LX=1ZLeWLαδ(r−WL()8.) bΨra′d(x,t)= 4πǫ10c3 Z d3x′[∂∂jtΨ′′(x′,|tx′)−]rext′×|2(x−x′), WemayextractthepolarizationPΨ′(r,t)instateΨ′from (14) α and dα(t)|Ψ′ =Z drPαΨ′(r,t), (9) eΨra′d(x,t)= 4πǫ10c2 Z d3x′ (15) wheredα(t)|Ψ′ istheαthcomponentoftheinduceddipole {[∂∂jtΨ′′(x′,t′)]ret×(x−x′)}×(x−x′), instateΨ′: |x−x′|3 dα(t)|Ψ′ =Z drZ dτΨ′∗ρdαΨ′, α=x,y,z. (10) w∂jhΨe′r(ex′t,′t′)=/∂tt′ −oft|hxe−curxr′e|n/tc.deIntsiistytthheattidmeteerdmeirnivesattihvee b radiationfield. The time dependenceof the induceddipolein state Ψ′(t) Denote x′ as the origin of the source distribution, x′ resultsfromthetimedependenceofΨ′(t).Thetimederiva- 0 an arbitrarypointin a localized chargedistribution,x the tiveofthepolarizationcanbefoundfrom∂dα(t)|Ψ′/∂t: observationpoint.TheintegrandsofEqs.(14,15)canbeex- ∂∂tdα(t)|Ψ′ =Z dr∂∂tPαΨ′(r,t). (11) pcaaunsdeedthienraadsimataiollnpfiaerladmreepterresεen=ts|oxu′t−goxin′0g|/e|nxer−gyx,′0it|.mBues-t have asymptotic form proportionalto |x−x′|−1[27,28]. 0 CombiningEqs.(1,10,11)andintegratingbyparts,onehas: Theelectricfieldandthemagneticinductioninfarregion are ∂PΨ′(r,t) N i¯he ∂Ψ′∗ ∂Ψ′ α∂t = 2em Z dτ1(Ψ′ ∂r −Ψ′∗∂r ) (12) bΨ′ =−n×pΨ′, eΨ′ =cn×(n×pΨ′), (16) α α rad rad −Nmee2Aα(r,t)Z dτ1Ψ′∗Ψ′ awnhderpeΨn′ a=re((xx−,tx)[′02)9/,|3x0−].xT′0h|e,tvheecatorgrufimelednptsΨo′fiesΨrga′idv,ebnΨrba′yd N i¯hZ e ∂Ψ′∗ ∂Ψ′ −LX=1 2MLL Z dτL(Ψ′ ∂rα −Ψ′∗∂rα) pΨ′(x,t)= 4πǫ10c3 Z d3x′ (17) Copyrightlinewillbeprovidedbythepublisher 4 Firstauthoretal.:Shorttitle ∞ (|x′−cx′0|cosθ)q ∂q+1jΨ′(x′,t−s0), FromEqs.(17,22)wecanseethatinthedipoleapprox- q! ∂tq+1 Xq=0 imation the quantal acceleration a(t)|ψ′ is the source of radiation. The radiation fields produced by the first two whereθistheanglebetween(x−x′)and(x′−x′).The 0 0 terms in Eq.(19) are already knownin CM. The last term q =0termgivesthedipoleapproximation,theq =1term in Eq.(19) is the new feature of a quantal charged parti- gives the quadrupole and magnetic dipole approximation cle. The corresponding radiation fields are perpendicular etc[29,30]. to(∇×B),theyshouldbedetectableforachargedparti- Ψ′ ′ ′ ′ clemovinginanon-uniformmagneticfield.Bytestingthe 3 Radiationfieldscomputedfrom∂j (x ,t )/∂t polarization of scattered fields, it can also be observedin thescatteringphenomenon. 3.1 A moving electron Let us consider the motion Letusconsidermoregeneralradiationfieldsproduced of an electron in an electromagnetic field (A,φ) and an- by a quantal charged particle moving in an external field otherexternalfieldU′(x).Analkaliatominadilutealkali (φ,A).Sinceweconsideronlynon-relativisticmotion,the gasisanexample.Denotethesingleparticlewavefunction effectofself-fieldsoftheparticleonitsmotioncanbene- odf[ thde3xpψar′t∗icrlψe′a]/sdψt′(inx,stt)a,tetheψa′(vxe,rat)gecvanelobceityfovu(ntd)|ψfr′o=m glected.TakingtimederivativeinEq.(20),onefinds ScRhrodingerequation[14,18]: ∂jψ′(x,t)/∂t= e2Eψ′ψ′∗− e ∇U′(x)ψ′ψ′∗ v(t)|ψ′ = 2im¯h Z d3x(ψ′∇ψ′∗−ψ′∗∇ψ′) m m e2 e3 e + [ψ′(i¯h∇ψ′∗)+ψ′∗(−i¯h∇ψ′)]×B− (A×B)ψ′ψ′∗ − d3xψ′∗A(x,t)ψ′. (18) 2m2 m2 mZ ψTh′(exa,vt)eriasg[1e4a,1cc8e]leration a(t)|ψ′ = dv(t)|ψ′/dt in state +me A(∇·jψ′)− me32ψ′ψ′∗(A·∇)A i¯he2 a(t)|ψ′ =m−1Z d3xψ′∗[eE−∇U′]ψ′ +2m2{ψ′[(∇ψ′∗)·∇]A−ψ′∗[(∇ψ′)·∇]A} 2 ¯h e +me2 Z d3xψ′∗(−i¯h∇−eA)ψ′×B +4m2[ψ′∇∇2ψ′∗+ψ′∗∇∇2ψ′−(∇ψ′∗)∇2ψ′−(∇ψ′)∇2ψ′∗] i¯he i¯he2 − d3xψ′∗ψ′(∇×B), (19) + {ψ′[A·∇](∇ψ′∗)−ψ′∗[A·∇](∇ψ′)]} 2m2 Z 2m2 wheretheargumentsofthefieldsare(x,t).Thefirstterm i¯he2 + [(∇ψ′∗)A·∇ψ′−(∇ψ′)A·∇ψ′∗] in Eq.(19) is the acceleration caused by the electric field 2m2 andexternalfieldU′,thesecondtermisthemagneticforce. i¯he2 TheyareexpectedfromCM.Thethirdtermisaquantum + (ψ′∇ψ′∗−ψ′∗∇ψ′)∇·A, (23) effect: an additional small component along the average 2m2 directionof∇×B.Theratioofthethirdtermtothesec- whereE(x,t)=−∇φ(x,t)−∂A(x,t)/∂tandB(x,t)= ondis∼a/L,whereaisthecharacteristiclengthofwave ∇×A(x,t)aretheelectricfieldandmagneticinduction, function ψ′, L is the characteristic length scale in which the arguments of A, B, E and ψ′ are (x,t). Integrating magneticfieldchanges. Eq.(23)overspace d3x,onecanchecksumrule(22)us- Forasingleelectron,Eq.(7)reducesto: ing integration by Rparts. However one observes that, ac- cordingtoEqs.(14,15),theradiationfieldisdeterminedby jψ′(x,t)= i¯he(ψ′∇ψ′∗−ψ′∗∇ψ′) the time derivativeof ∂jψ′(x,t)/∂t currentdensityrather 2m thanbytheaverageaccelerationa(t). − e2A(x,t)ψ′∗ψ′. (20) The radiation fields produced by the first three terms m areperpendiculartotheelectricforce,externalforceofU′, ComparingEq.(18)andEq.(20),wehave: and magnetic force respectively. They are well known in theclassicaldescription[27,28,29]. v(t)|ψ′ =e−1Z d3xjψ′(x,t). (21) tumTfheeatruermesaionfintghetemrmicsroosfcEopqi.(c2c3u)rrreesnutltdfernosmitythjeψ′quinana- Theaverageaccelerationandthetimederivative∂jψ′(x,t)/∂t purestateψ′.AccordingtoEqs.(14,15),theradiationfields producedbythefourthtermareperpendicularto(A×B), ofcurrentdensityarerelatedby: the radiationfieldsproducedbythe fifthtermareperpen- ∂jψ′(x,t) dicular to A. The radiation fields produced by the sixth a(t)|ψ′ =e−1Z d3x ∂t . (22) termsareperpendicularto(A·∇)A.Theratioofthefourth Copyrightlinewillbeprovidedbythepublisher pssheaderwillbeprovidedbythepublisher 5 term to the second term is λ/a, where λ is the character- − e3 [A×B]n′+ e A(r,t)(∇·jΨ′)− e3 [(A·∇)A]n′ m2 m m2 isticlengthofA.Thefifthtermisthesameorderofmag- nitude as the third term. The ratio of the fifth term to the i¯he2N + e dτ1{Ψ′[(∇Ψ′∗)·∇]A(r,t) secondtermisA/(h¯/ea),forordinarymagneticfieldthis 2m2 Z isasmallnumber.Thesixthtermisthesameorderofmag- nitudeasthefourthterm. −Ψ′∗[(∇Ψ′)·∇]A(r,t)} The radiation fields producedby the seventh term are 2 ¯h eN perpendicular to [(∇ψ′) · ∇]A. The ratio of the seventh + e dτ1[Ψ′∇∇2Ψ′∗+Ψ′∗∇∇2Ψ′ term to the second term is ∼ (h¯/ea2)/B. The radiation 4m2 Z fieldsproducedbytheeighthtermareperpendiculartothe −(∇Ψ′∗)∇2Ψ′−(∇Ψ′)∇2Ψ′∗] canonicalmomentum∇ψ′.Theratiooftheeighthtermto the second term is ∼1. In CM, no such a radiation field +i¯he2Ne dτ1{Ψ′[A(x ,t)·∇](∇ Ψ′∗) exist. The radiation field producedby the ninth terms are 2m2 Z j j X perpendicularto[A·∇](∇ψ′).Theratiooftheeighthand j theninthtermstothesecondtermisλ/a. −Ψ′∗[A(x ,t)·∇](∇ Ψ′)]} j j The radiation fields produced by the tenth term of Eq.(23) are also perpendicularto (∇ψ′)A·∇ψ′∗, where +i¯he2Ne dτ1[(∇Ψ′∗)A(x ,t)·∇ Ψ′ −i¯h∇ = mv+eAisthecanonicalmomentumoperator. 2m2 XZ j j j In contrast to the eighth term, which does not depend on vectorpotential,the tenth term dependson A·∇ψ′. The −(∇Ψ′)A(xj,t)·∇jΨ′∗] ratioofthetenthtermtothefifthtermis∼1.Boththefifth i¯hN e2 and the tenth terms exist for a constant vector potential. + e dτ1(Ψ′∇Ψ′∗−Ψ′∗∇Ψ′) ∇ ·A(x ,t) Theradiationfieldsproducedbytheeleventhtermareper- 2m2 Z X j j j pendicular to (ψ′∇ψ′∗)∇ · A. The eleventh term is the i¯he2N sameorderofmagnitudeastheseventhterm. + e dτ1 A(x ,t)×{[∇×(∇ Ψ′∗)]Ψ′ Before the Bohm-Aharonov effect was discovered, it 2m2 Z X j j ′ ′ j had been assumed that any (φ ,A ) is indistinguishable from (φ,A) if they satisfy A′ = A + ∇Λ and φ′ = −[∇×(∇jΨ′)]Ψ′∗}, φ+∂Λ/∂t, where is Λ is an arbitrary scalar function of where ∇ is abbreviatedas ∇. n′(r,t) = N dτ′Ψ′Ψ′∗ r e timeandcoordinate.TheBohm-Aharonoveffectgivesthe is the number density of electrons in state Ψ′.RU and U′ first example that beside B = ∇ × A, vector potential are the interactionbetweentwo electronsand the interac- canbedetectedinanotherway:thephaseshift e A·dx. h¯ tion betweenan electronanda nucleus.Because the con- The radiation fields produced by the fourth, fifHth, sixth, tributions from nuclei are less important and do not have ninth,tenthandeleventhtermsinvolveeitherAor∇·A. anynewfeaturesforvisiblelight,theyarenotincludedin Theydemonstratetheobservabilityofvectorpotential:one Eq.(24).Ofcoursethenuclearcontributionsareimportant maydetecttheseradiationfieldsoutsidealongsolenoidfor forinfraredradiation. whichB=0butA6=0. There are 12 terms in Eq.(24). The first to eleventh Eq.(23) is a resolution of ∂jΨ′(x′,t′)/∂t′. It directly termsareamany-bodygeneralizationofthecorresponding resultsfromthetimedependentSchrodingerequationand single-particletermsinEq.(23).Thecoherentscatteringin Eq.(20).Onemaywonderwhetheraregroupingtheterms CM is produced by several particles in which their posi- inEq.(23)leadstoadifferentexplanationfortheradiation tionsarecorrelatedintherangeofonewavelenth[27,28]. fields.Theanswerisno:becauseEqs.(14,15)arelinearin Thisfeatureisalsoinheritedbythefirsttoeleventhterms. ∂jΨ′(x′,t′)/∂t′, a recombinationof terms only results to Wedon’thavesumrulesforthesystemwithmorethanone a differentchoice of the basis of fields. The new basis of chargedparticles. fieldisasuperpositionoftheoldfields. ThetwelfthtermofEq.(24)exitsonlyforasystemwith 3.2 A group of charged particles Taking the time morethanonechargedparticlewhichobeyQM,andrep- derivativeinEq.(7),weobtain resentsanewfeatureoftheradiationfieldproducedbythe motionofmanychargedparticles.Ifthereisonlyonepar- ∂jΨ′(r,t)/∂t= e2E(r,t)n′(r,t) (24) ticle,∇×(∇jΨ′)=0,Eq.(24)isreducedtoEq.(23).The m radiationfieldsemittedbythetwelfthtermofEq.(24) are perpendiculartoA(x ,t)×[∇×(∇ Ψ′)],andaredifferent j j +eNe dτ1Ψ′Ψ′∗[− ∇U(r,x )− ∇U′(r,W )] tothoseemittedbythefirsttotheeleventhterms.Thera- m Z k α tioofthetwelfthtermtothemagneticforceisλ/a′,where X X k α a′isthecharacteristiclengthofΨ′.Itshouldbedetectable e2N forthescatteringofvisiblelightbyparticlessmallerthan + e{ dτ1[Ψ′(i¯h∇Ψ′∗)+Ψ′∗(−i¯h∇Ψ′)]}×B(r,t) 2m2 Z 100nm. Copyrightlinewillbeprovidedbythepublisher 6 Firstauthoretal.:Shorttitle In QED, the coupling between electromagnetic field TheargumentsofeΨ′,bΨ′,jΨ′ andρΨ′ are(R,t),thespa- andchargedparticlesistreatedasaperturbation.Theob- tialdifferential∇ isrespecttothespatiallycoarsegained R servablephotonsareexpressedbyexternallines.Tocalcu- coordinate R. Because the spatial coarse graining (25) is lateaprocesswithnphotons,oneneedsanth ordertran- alinearmap,andEqs.(4,5)arelinear,Eqs.(26,27)arealso sitionamplitude[2,3].Thisisimpracticalforastrongfield linearaboutalltruncatedquantitiesappearinthem. (manyphotons)producedbyalargenumberofinteracting Ifonedistinguishesthefreechargesandboundcharges chargedparticles. In SCRT, H is treated at zero order. fm Becausenohighenergyphotonappearsintheusualstateof in ρΨ′ and jΨ′, various order electric and magnetic mo- a condensed phase and nanoscale system, Eqs.(14,15,24) ments of the bound chargesalso appear as the sources of aresuitablefortheemissionandscatteringoflight[12,13, fields[24,25].Suchadescriptionappearsintextbooks[27, 14,15,16]. 28], and is convenient for the crystalline metals, alloys (thereisacleardistinctionbetweenfreecarriersandbound 4 MacroscopicMaxwellequations Forananoscale charges) and insulators at low frequency. In amorphous systemoramacroscopicsystem,weusuallycannotspecify semiconductors,thehoppingprobabilityforlocalizedcar- theinitialconditionofthesystempreciselyinthesensethat riers is[34] ∼ 1012 −1013sec−1. On a time scale shorter thestateofthesysteminthefuturecanbepredictedtothe than 10−12sec, one cannot distinguish a localized carrier maximumextentallowedbyQM[36].Theresultsobtained from a boundcharge.Thereforefor (i) metals, alloys and in Sec.3 are not directly applicable. To explore radiation insulatorsathighfrequency(inter-bandtransition);and(ii) fieldsproducedbyamacroscopicornanoscalesystem,we plasma[35] and amorphoussemiconductors[18,19,32,33] havetoaveragetheaboveresultsoverarepresentativeen- atanyfrequency,itisconvenientnottodistinguishthecon- semble.Inotherwords,weneedthemacroscopicMaxwell tributions from carriers and from bound electrons in ρΨ′ equationsforagroupofchargedparticlesobeyingQM. andjΨ′,i.e.nottomakethemultipoleexpansion. α Suppose that the system is in a good thermal contact with a reservoir (B) such that after a short equilibration For the following three reasons, we do not need[24, time,thesystemreachesasteadystate:thethermodynamic 25,27] to take an additional temporal coarse grained av- state of the system is specified by the intensive parame- erage over fields and sources at any stage. First of all, ters of B (temperature, chemical potential etc.). The heat Eqs.(4,5) are linear in all quantities appearing in them: evolved is transferred to the bath[31,32,33]; The motion allthequantitiesareadditive.Thereforefortypicalcut-off ofsystemhasthesamefrequencyastheexternalfield. wave vectork0 = 106cm−1 and numberdensityof parti- 4.1 Spatial coarse grained average To describe clesn = 1022cm−3,therelativefluctuationofatruncated thefinitespatialresolutioninamacroscopicmeasurement, quantityissmall[27]∝n−1/2k3/2 ∼10−2.Thespatialav- 0 for each microscopic quantity ξΨ′(r,t) defined in pure erage(25)isenough.Secondly,thetimev−1k−1 spentby 0 stateΨ′(t),oneintroduces[24,25]atruncatedquantityξΨ′ aparticlewithspeedv ∼106m·sec−1traversingadistance : k0−1is10−14sec,isstillintherangeofatomicormolecular ξΨ′(R,t)= ∞ d3rξΨ′(R−r,t)f(r), (25) motions[27],averagingoversuch a time periodafter spa- Z tial average(25) is pointless.Third,beforethespatial av- −∞ erage(25),ifoneaveragesoveratimeperiodshorterthan where the scalar weight function f(r) satisfies two the time scale ofatomicandmolecularmotions,onecan- conditions: (i) ∞ d3rf(r) = 1; and (ii) F(k) = −∞ noteliminatefastfluctuation[27].Ontheotherhand,ifone −∞∞d3re−ik·rfR(r) → 0 for k > k0. k0 is solely de- averages over a time period longer than the time scale of tRerminedby the type of problemand calculationwe have atomicandmolecularmotions,onesmearedorevenelim- in mind[25]. Condition (ii) means that the spatial Fourier inatesthescatteringphenomenon[1]. components of the field variables are irrelevant above a cut-off wave vector k0. ξΨ′ is ρΨ′ or any Cartesian To obtain a macroscopic observable from the corre- componentof e, b and jΨ′. Using the truncated quantity sponding microscopic quantity in the pure state, we first truncatethemicroscopicquantityinagivenpurestateand ξΨ′(R,t) is stricter than the simple coarse grained aver- thentakeanensembleaverage.Foragroupofchargedpar- ageΩR−1 r∈ΩRd3rξΨ′(r,t)forsolids[25],whereΩR isa ticlesinteractingwithanelectromagneticfield,wecanre- physicaliRnfinitesimalvolumearoundpointR. placetheensembleaveragewithanaverageoverallpossi- TheMaxwellequationsforthetruncatedfieldsare ble initial purestates. Since fora givenexternalfield, the couplingbetweenfieldandthesystemcanbeincludedwith ∇ ·bΨ′ =0, ∇ ×eΨ′ =−∂b/∂t, (26) R R additionalterms[representedbyH (t)inEq.(2)]to the fm systemHamiltonianH,thestateΨ′(t)ofsystemattimet and iscompletelydeterminedbythevalueofΨ′ ataprevious moment through Eq.(1). If we adiabatically introduce an ∇R·eΨ′ =ρΨ′/ǫ0, c2∇R×bΨ′ =jΨ′/ǫ0+∂eΨ′/∂t. external field, then the ensemble average is changed into (27) anaverageovervariousinitialvaluesΨ′(−∞)ofstateΨ′. Copyrightlinewillbeprovidedbythepublisher pssheaderwillbeprovidedbythepublisher 7 Denote derivativeofpolarization,andthesourceofradiationfield istimederivativeofcurrentdensity. ξ(R,t)= W[Ψ′(−∞)]ξΨ′(R,t), (28) Eqs.(13,14,15) have shown that the time derivative XΨ′ relation is correct for the microscopic quantities in a pure state. It is easy to see that the time derivative re- where ξΨ′ is any truncated quantity: ρΨ′ or any Carte- lation is also correct for the macroscopic quantities. For sian component of eΨ′, bΨ′ and jΨ′. W[Ψ′(−∞)] is the the truncated quantity PΨ′(R,t) = ∞ d3rPΨ′(R − probability that the system is initially in state Ψ′(−∞). r,t)f(r) and the macroscopic polarizRa−ti∞on P(R,t) = W[Ψ′(−∞)]dependsonlyontheenergyofΨ′(−∞),can W[Ψ′(−∞)]PΨ′(R,t),Eq.(13)implies betakenaseitheracanonicaloragrandcanonicaldistribu- Ψ′ P tion(p678of[31]).Thenξ(R,t)istheusualmacroscopic ∂PΨ′(R,t) ∂P (R,t) observable[32]. jΨ′(R,t)= α andj (R,t)= α . Because Eqs.(26,27) are linear, we can average the α ∂t α ∂t (31) fieldsandsourcesoverallpossibleinitialpurestatesofthe Sometimes,thesecondequationin(31)istakenasaredef- system. Then we obtain the macroscopic Maxwell equa- inition of polarization[35,38]. Since Eqs.(29,30) have the tions: samestructureasEqs.(4,5),themacroscopicradiationfield ∇·B=0, ∇×E=−∂B/∂t, (29) isdeterminedby∂j (R,t)/∂t. α and 4.3 Current density as the response of system to an external field In Sec.4.1, we have seen that (1)if ∇·E=ρ/ǫ0, c2∇×B=j/ǫ0+∂E/∂t. (30) we average the motion of particles over the microscopic time scale v−1k−1, the averaged charge and currentden- 0 HereweusedthefamiliarsymbolsEandBtodenotethe sitystilloscillateinfrequencyk0v;(ii)ifweaveragefields ensembleaverageofeΨ′ andbΨ′.TheargumentsofE,B, andsourcesoverlongertimescale(e.g.severalperiodsof jandρare(R,t),∇representstheoperatorrespecttoR. externalfield), the scattering phenomenaare smeared[27, Themacroscopicsources(ρandj)andfieldsdependonthe 1].Thereforewe shouldnottake temporalcoarse-grained intensiveparametersofachosenensemble. average even for describing the electromagneticphenom- In view of facts that (i) the microscopic Eqs.(4,5) are enainamixedstateofamacroscopicsystem. linear; and (ii) both the ensemble average and truncation On the other hand, the induced charge density ρ and (25)arelinearmapsofthemicroscopicfieldsandsources, the current density j are also the macroscopic responses toobtainEqs.(29,30),wecanactuallyfirsttaketheensem- ofthesystemto anexternalfield.Thereforeboththepro- bleaverageoverthemicroscopicquantities,andthentrun- cedureinSec.4.1andthetimederivativerelationrequires catetheensembleaveragedquantities. that temporal coarse graining should not be taken in the The derivation suggested in this section differs from macroscopic charge density and current density either as previousones[21,22,23,35]intwoaspects:(i)todescribe thesourcesoffieldsorastheresponsestoexternalfields. the electromagnetic phenomena in macroscopic media, The microscopic response method (MRM) used this temporal coarse graining is unnecessary[27]. Only a spa- factasthestartingpointtocalculateconductivityandHall tial coarse-grainedand ensemble average are needed; (ii) mobility[18,32,19].Theentropyproductionisreflectedin Because for a given external field, the interaction of the the existence of a steady state of the system, which is in system and field can be written with additional terms good contact with a heat and material bath. Because the in the Hamiltonian of system[37]. Then for each wave system is in a good thermal and chemical contact with functionΨ′ which satisfies Eq.(1), one may introducethe a bath in an external field, after a short transient period, microscopicresponsejΨ′ forthatstate.Thustheensemble the system will eventually reach a steady state which os- average Eq.(28) can be delayed to the final stage[18,32] cillates at the same frequencyas the externalfield. In the ratherthantakingattheoutset[37,21,22,23]. steadystate,theparameterscharacterizingtheensembledo SinceEqs.(4,5)havethesamestructureasEqs.(29,30), notchange:theevolvedorabsorbedenergyaretransferred toobtaintheradiationfieldsproducedbyamixedstate,we to or taken fromthe bath. For a classical chargedoscilla- onlyneedtoreplace∂jΨ′(x′,t′)/∂t′with∂j(x′,t′)/∂t′. tor undergoingforced oscillation, it is well-known that if 4.2 Polarization, current density and the source the inputenergyis completelydissipated, the system will ofradiationfield Foragroupofchargedparticlesobey- eventually reach a steady state. With perturbation theory, ingCM, the induceddipolemomentisdeterminedbythe one can show that a solid in good thermal contact with a displacement of charges. The current density is propor- bathwillreachasteadystateinanexternalfield[39,33]. tional to the velocities of charges. The radiation field is Temporal coarse graining is a key step in the ki- causedbytheaccelerationsofcharges.We willshowthat netic descriptions of the irreversibility[31,37]. For the for a group of charged particles obeying QM, there exist processes caused by mechanical perturbations, temporal similar relations among polarization, current density and coarse grained average can be avoided. Faber[40] and the source of radiation field: the current density is time Mott[41]conjecturedanexpressionfortheacconductivity Copyrightlinewillbeprovidedbythepublisher 8 Firstauthoretal.:Shorttitle for strong scattering based upon assumptions: (i) in the [2]W.Heitler,TheQuantumTheoryofRadiation,Thirdedition, Kramers-Heisenberg dispersion relation for Raman scat- ClarendonPress,Oxford(1954). tering, put the final state and initial state the same; (ii) [3]V.B.Berestetskii,L.P.PitaevskiiandE.M.Lifshitz,Quan- requirethezerofrequencylimitconsistentwiththeGreen- tumElectrodynamics,2ndedition,Butterworth-Heinemann, wood’s dc conductivity formula[42], this was realized by Oxford(1982). takinglongtimelimit(i.e.temporalcoarsegraining)inthe [4]C.Cohen-Tannoudji,J.Dupont-RocandG.Grynberg,Pho- contractedKramers-Heisenbergrelation.Ifthesecondstep tonsandAtoms,JohnWiley,NewYork,(1997). [5]C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, wasnotused,thefirststepwouldendupwiththeconduc- Atoms-PhotonInteractions,JohnWiley,NewYork,(1998). tivity formula (expressed by the single electronic states) [6]D.P.Craig,T.Thirunamachandran,Acc.Chem.Res.19,10- derivedbytheMRMinwhichnotemporalcoarsegraining 16(1986);MolecularQuantumElectrodynamics,Academic is taken[18,19,32,39]. For crystalline metals, such a for- Press,London(1998). mulawas written downwithoutderivation,cf. Eq.(13.37) [7]A. Salam, Molecular Quantum Electrodynamics: Long- of[43]. RangeIntermolecularInteractions,JohnWiley(2010). [8]C.R.StroudJr.andE.T.Jaynes,Phys.Rev.A1,106(1970). 5 Summary According to the semi-classical radia- [9]M.D.CrispandE.T.Jaynes,Phys.Rev.179,1253(1969). tiontheory,theradiationfieldsaredeterminedbythetime [10]P.W.Milonni,PhysicsReports25,1(1976). derivativeofthecurrentdensity.Forapurestateofamany- [11]A.I. Akhiezer and N.F. Shul’ga, Physics Reports 234, 297 body system, a strict current density expression has been (1993). obtained from the microscopic response method. Thus [12]E.T.JaynesandF.W.Cummings,Proc.IEEE51,89(1963). wehaveestablishedmicroscopicMaxwellequationsfora [13]R.K.Nesbet,Phys.Rev.Lett.27,553(1971). pure state. For a charged quantal particle, three radiation [14]L.I.Schiff,QuantumMechanics,3rdedition,McGraw-Hill, fieldsinvolvevectorpotentialonly.Thisisanewexample NewYork(1968). demonstrating the observability of vector potential. Five [15]S.-Y.Lee,J.Chem.Phys.76,3064(1982). radiationfieldsareperpendicularthecanonicalmomentum [16]R.K.Nesbet,Phys.Rev.A4,259(1971). [17]B. Melig and M. Wilkinson, J. Phys. Condens. Matter 9, oftheparticle.Inamany-bodysystem,oneofitsradiation fieldsisperpendiculartoA(x ,t)×[∇×(∇ Ψ′)],which 3277(1997). j j [18]M.-L. Zhang and D. A. Drabold, Phys. Rev. Lett. 105, doesnotexistforasinglechargedparticle.Wepredictthat 186602(2010). thisformofradiationcanbedetected. [19]M.-L.ZhangandD.A.Drabold,Phys.Stat.Sol.B248,2015 We have extended Russakoff-Robinson’s ansatz to a (2011). group of charged particles obeying QM to derive macro- [20]H.A.Lorentz,Thetheoryofelectrons,G.E.Stechert,New scopic Maxwell equations. Only the spatial coarse grain- York(1923). ing is relevant. The charge density and current density [21]P.MazurandB.R.A.Nijboer,Physica19,971(1953). as sources or responses is well-defined for a pure state. [22]K.Schram,Physica26,1080(1960). Everymacroscopicquantitycanbeobtainedfromthecor- [23]S.R.DeGrootandJ.Vlieger,Physica31,254(1965). responding microscopic quantity by taking spatial coarse [24]G.Russakoff,Am.J.Phys.38,1188(1970). grainingandensembleaverage,cf.Eqs.(25,28).Temporal [25]F.N.H.Robinson,Physica54,329(1971). coarse graining is inadequate for the macroscopic fields, [26]F. N. H. Robinson, Macroscopic Electromagnetism, Perga- sources and the responses of system to field. The consis- monPress,NewYork(1973). tency among the induced displacement, the current and [27]J. D.Jackson, Classical Electrodynamics, 2nd edition, Wi- the acceleration requires that one should not take tem- ley,NewYork(1975). poral coarse graining in the current density (transport [28]W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, Second Edition, Addison-Wesley, London coefficients). This agrees with the microscopic response (1962). method[18,32,33,19].Theentropyproductionisreflected [29]L. D. Landau and E.M. Lifshitz, The Classical Theory of bythe existenceof a steady state of the system in a good Fields,Butterworth-Heinmann,Oxford(1975). thermalconnectionwith a bath. If temporalcoarse grain- [30]M.P.Davison,AnnalesFondationLouisdeBroglie30,259 ing is taken in the transport coefficients obtained by the (2005). microscopic response method, one can recover the corre- [31]C.M.VanVliet,EquilibrimandNon-equilibriumStatistical spondingresultsobtainedbythekineticapproaches. Mechanics,WorldScientific,Singapore(2008). [32]M.-L. Zhang and D. A. Drabold, Phys. Rev. E.83, 012103 Acknowledgements Thisworkissupported bytheArmy (2011). ResearchLaboratoryandArmyResearchOfficeunderGrantNo. [33]M.-L.ZhangandD.A.Drabold,J.Phys.:CondensedMat- W911NF1110358andtheNSFunderGrantDMR09-03225. ter,23,085801(2011). [34]M.-L.ZhangandD.A.Drabold,Eur.Phys.J.B77,7(2010). References [35]L. P. Pitaevskii and E.M. Lifshitz, Physical Kinetics, [1]L. D. Landau, E. M. Lifshitz and L. P. Pitaevski˘ı, Eletro- Butterworth-Heinemann,Oxford(1981). dynamics of Continuous Media, 2nd edition, Butterworth [36]R. C. Tolman, The Principles of Statistical Mechanics, HeinemannLtd,Oxford(1984). DoverPublications,Inc.NewYork(1979). Copyrightlinewillbeprovidedbythepublisher pssheaderwillbeprovidedbythepublisher 9 [37]R.Kubo,M.TodaandN.Hashitsume,StatistialPhysicsII, 2ndedition,Springer-Verlag,Berlin(1991). [38]A.Aldea,J.Phys.C:SolidStatePhys.5,1649(1972). [39]M.-L.Zhang and D. A. Drabold, Phys. Rev. B 81, 085210 (2010). [40]T.E. Faber,inOptical Propertiesand ElectronicStructure of Metals and Alloys, edited by F. Abele`s, p. 259, North- HollandPublishingCompany,Amsterdam(1966). [41]N.F.Mott,AdvancesinPhysics16,49(1967). [42]D.A.Greenwood,Proc.Phys.Soc.(London)71,585(1958). [43]N.W.AshcroftandN.D.Mermin,SolidStatePhysics,Saun- dersCollegePublishing,FortWorth(1976). Copyrightlinewillbeprovidedbythepublisher