AnIntroductiontoQuantumOpticsand QuantumFluctuations
PeterW.Milonni
LosAlamosNationalLaboratoryandUniversityofRochester
GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom
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and Tothememoryofmymother-in-law,Xiu-LanFeng
Tomymother,AntoinetteMarieMilonni
Preface
The quantum theory of light and its fluctuations are applied in areas as diverse as theconceptualfoundationsofquantumtheory,nanotechnology,communications,and gravitationalwavedetection.Theprimarypurposeofthisbookistointroducesomeof themostbasictheoryforscientistswhohavestudiedquantummechanicsandclassical electrodynamicsatagraduateoradvancedundergraduatelevel. Perhapsitmightalso offer some different perspectives and some material that elsewhere.
are not presented in much detail
Anybookpurportingtobeaseriousintroductiontoquantumoptics andfluctuationsshouldincludefieldquantizationandsomeofitsconsequences. Itisnotsoeasyto decidewhichotheraspectsofthisbroadfieldaremostaptorinstructive.Ihaveforthe mostpartwrittenaboutmattersoffundamentalandpresumably long-lastingsignificance.Theseincludespontaneousemissionanditsroleasasourceofquantumnoise; fieldfluctuationsandfluctuation-inducedforces;fluctuation–dissipationrelations;and somedistinctlyquantumaspectsoflight.Ihavetriedtofocusontheessentialphysics, andincalculationshavefavoredtheHeisenbergpicture,asitoften suggestsinterpretationsalongclassicallyfamiliarlines.Somehistoricalnotesthatmight beofinterest tosomereadersareincluded;theseandotherdigressionsappear insmalltype.Also includedareexercisesforreaderswishingtodelvefurtherintosomeofthematerial. IamgratefultomylongtimefriendsPaulR.Berman,RichardJ.Cook,JosephH. Eberly,JamesD.Louck,andG.JordanMaclayfordiscussionsover manyyearsabout muchofthematerialinthisbook.Jordanreadmostofthebookinits nearlyfinalversionandmadeinsightfulcommentsandsuggestions.Thanksalsogo toS¨onkeAdlung andHarrietKonishiofOxfordUniversityPressfortheirpatienceandencouragement.
PeterW.Milonni LosAlamos,NewMexico
2.5Two-StateAtoms
2.6PulsedExcitationandRabiOscillations105
2.7TransitionRatesandtheGoldenRule107
2.8BlackbodyRadiationandFluctuations112
3.3FieldQuantization:EnergyandMomentum135
3.4QuantizedFieldsinDielectricMedia141
3.5PhotonsandInterference
3.6QuantumStatesoftheFieldandTheirStatisticalProperties147
3.7TheDensityOperator
3.8Coherent-StateRepresentationoftheDensityOperator175
3.9CorrelationFunctions
3.10FieldCommutatorsandUncertaintyRelations182
3.11Complementarity:WaveandParticleDescriptionsofLight190
3.12MoreonUncertaintyRelations 197
4InteractionHamiltonianandSpontaneousEmission
4.1Atom–FieldHamiltonian:WhyMinimalCoupling?205
4.2ElectricDipoleHamiltonian 211
4.3TheFieldofanAtom
4.4SpontaneousEmission 222
4.5RadiationReactionandVacuum-FieldFluctuations240
4.6Fluctuations,Dissipation,andCommutators250
4.7SpontaneousEmissionandSemiclassicalTheory253
4.8MultistateAtoms 255
5AtomsandLight:QuantumTheory
5.1OpticalBlochEquationsforExpectationValues270
5.2AbsorptionandStimulatedEmissionasInterferenceEffects271
5.3TheJaynes–CummingsModel 274
5.4CollapsesandRevivals
5.5DressedStates
5.6ResonanceFluorescence
5.7PhotonAnti-BunchinginResonanceFluorescence297
5.8PolarizationCorrelationsofPhotonsfromanAtomicCascade301
5.9Entanglement
6Fluctuations,Dissipation,andNoise
6.1BrownianMotionandEinstein’sRelations325
6.2TheFokker–PlanckEquation
6.3TheLangevinApproach
6.4FourierRepresentation,Stationarity,andPowerSpectrum337
6.5TheQuantumLangevinEquation 341
6.6TheFluctuation–DissipationTheorem351
6.7TheEnergyandFreeEnergyofanOscillatorinaHeatBath366
6.8RadiationReactionRevisited 371
6.9SpontaneousEmissionNoise:TheLaserLinewidth378
6.10AmplificationandAttenuation:TheNoiseFigure391
6.11PhotonStatisticsofAmplificationandAttenuation395
6.12AmplifiedSpontaneousEmission 398
6.13TheBeamSplitter
6.14HomodyneDetection 406
7DipoleInteractionsandFluctuation-InducedForces
7.1VanderWaalsInteractions 409
7.2VanderWaalsInteractioninDielectricMedia423 x
7.3TheCasimirForce 425
7.4Zero-PointEnergyandFluctuations438
7.5TheLifshitzTheory
7.6QuantizedFieldsinDissipativeDielectricMedia459
7.7GreenFunctionsandMany-BodyTheoryofDispersionForces472
7.8DoCasimirForcesImplytheRealityofZero-PointEnergy?480
7.9TheDipole–DipoleResonanceInteraction482
7.10F¨orsterResonanceEnergyTransfer497
7.11SpontaneousEmissionnearReflectors502
7.12SpontaneousEmissioninDielectricMedia509
A.RetardedElectricFieldintheCoulombGauge513
B.TransverseandLongitudinalDeltaFunctions514
C.Photodetection,NormalOrdering,andCausality516 Bibliography
ElementsofClassical Electrodynamics
Thischapterisabriefrefresherinsomeaspectsof(mostly)classicalelectromagnetic theory.Itismainlybackgroundandaccompanimentfortherestof thebook,witha fewsmallconceptualpointsnotalwaysfoundinstandardtreatises.
1.1ElectricandMagneticFields
Maxwell’sequationsfortheelectricfield E andthemagneticinductionfield B are:
where ρ istheelectricchargedensity, J istheelectriccurrentdensity,and c =1/√ǫ0µ0 isthespeedoflightinvacuum.Thefields E and B aredefinedsuchthattheforceon apointcharge q movingwithvelocity v is F = q(E + v × B). (1.1.5)
Newton’ssecondlaw(F = ma)describesthe(non-relativistic)motionofacharge q ofmass m inthe E and B fields.
Equation(1.1.1)isGauss’slaw:thefluxof E throughanyclosedsurface S is proportionaltothenetcharge Q inthevolume V enclosedby S.Equation(1.1.2) impliesthereisnomagneticchargeanalogousto Q.Equation(1.1.3)isFaraday’slaw ofinduction:thelineintegraloftheelectricfieldaroundanyclosedcurve C—the electromotiveforce(emf)inawireloop,forexample,orjustaloopin freespace—is minustherateofchangewithtimeofthemagneticfluxthroughtheloop;theminus signenforcesLenz’slaw,the(experimental)factthattheemfinducedinacoilwhen apoleofamagnetispushedintoitproducesacurrentactingtorepelthemagnet.
AnIntroductiontoQuantumOpticsandQuantumFluctuations.PeterW.Milonni. © PeterW.Milonni2019.Publishedin2019byOxfordUniversityPress. DOI:10.1093/oso/9780199215614.001.0001
ElementsofClassicalElectrodynamics
Equation(1.1.4)relatestheintegralof B aroundaloop C tothecurrent I in C and thefluxof E through C;thefirsttermexpressesOersted’slaw(anelectriccurrent candeflectacompassneedle),whilethesecondtermcorresponds tothe displacement current thatMaxwell,relyingonmechanicalanalogies,addedtothecurrent density J. Withthisadditionalterm(1.1.1)and(1.1.4),togetherwiththeidentity ∇·(∇×B)=0, implythecontinuityequation
(1.1.6) whichsays,inparticular,thatelectricchargeisconserved.(Theadditionaltermalso impliedwaveequationsfortheelectricandmagneticfieldsandthereforethepossibility ofnearlyinstantaneouscommunicationbetweenanytwopointsonEarth!)Maxwell’s equationsexpressallthelawsofelectromagnetismdiscoveredexperimentallybythe pioneers(Amp`ere,Cavendish,Coulomb,Faraday,Lenz,Oersted,etc.)inawonderfully compactform.
Ifthechargedensity ρ doesnotchangewithtime,itfollowsthat ∇· J =0and, fromMaxwell’sequations,thattheelectricandmagneticfieldsdonot changewith timeandareuncoupled:
AccordingtoAmp`ere’slaw(∇× B = µ0J),themagneticfieldproducedbyasteady current I inastraightwirehasthemagnitude
atadistance r fromthewireandpointsindirectionsspecifiedbytheright-handrule 1 Itthenfollowsfrom(1.1.5)thatthe(attractive)force f perunitlengthbetweentwo long,parallelwiresseparatedbyadistance r andcarryingcurrents I and I′ is
Untilrecentlythiswasusedtodefinetheampere(A)asthecurrent I = I′ intwolong parallelwiresthatresultsinaforceof2 × 10 7 N/mwhenthewiresareseparatedby 1m.Thisdefinitionoftheampereimpliedthedefinition µ0 =4π × 10 7 Wb/A m, theweber(Wb)beingtheunitofmagneticflux.Withthisdefinitionoftheampere,
1 ThefactthatawirecarryinganelectriccurrentgenerateswhatFaradaywouldlateridentifyas amagneticfieldwasdiscoveredbyOersted.Whilelecturingtostudentsinthespringof1820,Oersted noticedthatwhenthecircuitofa“voltaicpile”wasclosed, therewasadeflectionoftheneedleof amagneticcompassthathappenedtobenearby.Amp`ere,atthetimeamathematicsprofessorin Paris,performedandanalyzedfurtherexperimentsonthemagneticeffectsofelectriccurrents.
thecoulomb(C)wasdefinedasthechargetransportedin1sbyasteadycurrentof1 A.Then,intheCoulomblaw,
fortheforceonapointcharge q2 duetoapointcharge q1,with r12 thevectorpointing from q1 to q2, ǫ0 isinferredfromthe defined valuesof µ0 and c: ǫ0 =8.854 × 10 12 C2/N·m2,or1/4πǫ0 =8.9874 × 109 N·m2/C2 .
IntherevisedInternationalSystemofUnits(SI),theampereisdefined,basedon afixedvaluefortheelectroncharge,asthecurrentcorrespondingto1/(1 602176634 × 10 19)electronspersecond.Thefree-spacepermittivity ǫ0 andpermeability µ0 inthe revisedsystemareexperimentallydeterminedratherthanexactly definedquantities; therelation ǫ0µ0 =1/c2,with c definedas299792458m/s,remainsexact.
Equation(1.1.11)impliesthattheCoulombinteractionenergyoftwoequalcharges q separatedbyadistance r is
)=
Wecanusethisformulatomakeroughestimatesofbindingenergies. Consider,for example,theH+ 2 ion.Thetotalenergyis Etot = Enn + Een + Ekin,where Enn isthe proton–protonCoulombenergy, Een istheCoulombinteractionenergyoftheelectron withthetwoprotons,and Ekin isthekineticenergy.Accordingtothevirialtheorem ofclassicalmechanics, Etot = Ekin,implying
Enn = e2/(4πǫ0r),where e =1.602 × 10 19 Cand r ∼ = 0.106nmistheinternuclear separation.Aroughestimateof Een isobtainedbyassumingthattheelectronsitsat themidpointbetweenthetwoprotons:
Then,
Sincethebinding(ionization)energyofthehydrogenatomis13.6eV, thebinding energyofH+ 2 ,definedasthebindingenergybetweenahydrogenatomandaproton,is estimatedtobe(20.4-13.6)eV=6.8eV.Quantum-mechanicalcalculationsyield2.7 eVforthisbindingenergy.Chemicalbindingenergiesontheorderof afewelectron voltsaretypical.
ElementsofClassicalElectrodynamics
ConsiderasanotherexampletheenergyreleasedinthefissionofaU235 nucleus. Sincethereare92protons,theCoulombinteractionenergyofthe protonsis
where R isthenuclearradius.Ifthenucleusissplitintwo,thevolumedecreases beafactorof2,andtheradiusthereforedecreasesto(1/2)1/3R,sincethevolumeis proportionaltotheradiuscubed.ThesumoftheCoulombinteractionenergiesofthe daughternucleiistherefore
Theenergyreleasedinfissionis Uf = U1 U2 =0.37U1.Taking R =10 14 mforthe nuclearradius,weobtain Uf =4.8 × 108 eV=480MeV,comparedwiththeactual valueofabout170MeVpernucleus.Thusweobtainthecorrectorderofmagnitude withonlyelectrostaticinteractions,withoutaccountingforthestrongforcebetween nucleonsandwithouthavingtoknowthat E = mc2 2 Thephysicaloriginoftheenergy releasedinthissimplemodelistheCoulombinteractionofchargedparticles,justasin achemicalcombustionreaction.Buttheenergyreleasedinchemicalreactionstypically amountstojustafewelectronvoltsperatom;theenormouslylargerenergyreleased pernucleusinthefissionofU235 isduetothesmallsizeofthenucleuscomparedwith anatomandtothelargenumberofcharges(protons)involved.
1.2Earnshaw’sTheorem
Electrostaticsisbasedon(1.1.7).Weintroduceascalarpotential φ(r)suchthat E(r)= −∇φ(r),sothat ∇× E =0issatisfiedidentically.Then, ∇· E = ρ/ǫ0 implies thePoissonequation
ortheLaplaceequation
inaregionfreeofcharges.Thereaderhasprobablyenjoyedsolvingtheseequationsin homeworkproblemsforvarioussymmetricalconfigurationsofchargedistributionsand conductorssubjecttoboundaryconditions.Here,wewillrecallonlyoneimplication oftheelectrostaticMaxwellequations, Earnshaw’stheorem:achargedparticlecannot beheldatapointofstableequilibriumbyanyelectrostaticfield.Thisfollowssimply fromGauss’slaw(seeFigure1.1).Thetheoremiseasilygeneralizedto anynumberof charges:noarrangementofpositiveandnegativechargesinfree spacecanbeinstable equilibriumunderelectrostaticforcesalone.
2 “SomehowthepopularnotiontookholdlongagothatEinstein’stheoryofrelativity,inparticular hisfamousequation E = mc2,playssomeessentialroleinthetheoryoffission...butrelativity isnotrequiredindiscussingfission.”—R.Serber, TheLosAlamosPrimer,UniversityofCalifornia Press,Berkeley,1992,p.7.
Fig.1.1 Apointinsidesomeimaginedclosedsurfaceinfreespace.Forthatpointtobeone ofstableequilibriumforapositivepointcharge,forexample,theelectricfieldmustpoint everywheretowardit,whichwouldimplyanegativefluxofelectricfieldthroughthesurface. ThiswouldviolateGauss’slaw,because ∇· E =0infreespace.
AmoreformalproofofEarnshaw’stheoremstartsfromtheforce
onapointcharge q,or,equivalently,thepotentialenergy U (r)= qφ(r); ∇· E =0in freespaceimpliesLaplace’sequation,
whichmeansthatthepotentialenergyhasnolocalmaximumorminimuminside thesurfaceofFigure1.1;alocalmaximumorminimumwouldrequirethatallthree secondderivativesinLaplace’sequationhavethesamesign,whichwouldcontradict theequation.Itisonlypossibleatanypointtohaveamaximumalongonedirection andaminimumalonganother(saddlepoints).Inparticular,nocombinationofforces involving1/r potentialenergies,suchas,forexample,electrostaticplusgravitational interactions,canresultinpointsofstableequilibrium,sincethesumoftheLaplacians overallthepotentialsiszero.
TheReverendSamuelEarnshawpresentedhistheoremin1842inthecontextofthe “luminiferousether”andelasticitytheory.Heshowedthat forcesvaryingastheinversesquare ofthedistancebetweenparticlescouldnotproduceastable equilibrium,andconcludedthat theethermustbeheldtogetherbynon-inverse-squareforces.Maxwellstatedthetheorem as“achargedbodyplacedinafieldofelectricforcecannotbe instableequilibrium,”and proveditforelectrostatics.3
Earnshaw’stheoreminelectrostaticsonlysaysthatstable equilibriumcannotoccurwith electrostaticforces alone.Ifotherforcesacttoholdnegativechargesinplace,apositive chargecan,ofcourse,bekeptinstableequilibriumbyasuitabledistributionofthenegative charges.Similarly,achargecanbeinstableequilibriumin electricfieldsthatvaryintime,or inadielectricmedium(heldtogetherbynon-electrostatic forces!)inwhichanydisplacement ofthechargeresultsinarestoringforceactingbackonit,asoccursforachargeatthecenter ofadielectricspherewithpermittivity ǫ < ǫ0 4
Thingsarealittlemorecomplicatedinmagnetostatics.Therearenomagneticmonopoles, andthepotentialenergyofinterestis U(r)= m B foramagneticdipole m inamagnetic field B.Forinducedmagneticdipoles, m = αmB,where αm > 0foraparamagneticmaterial
3 J.C.Maxwell, TreatiseonElectricityandMagnetism,Volume1,DoverPublications,NewYork, 1954,p.174.
4 See,forinstance,D.F.V.James,P.W.Milonni,andH.Fearn, Phys.Rev.Lett. 75,3194(1995).
ElementsofClassicalElectrodynamics
(thedipoletendstoalignwiththe B field), αm < 0foradiamagneticmaterial(thedipole tendsto“anti-align”withthefield),thepotentialenergyis
and ∇2U = (1/2)αm ∇2B2.Fortheretobeapointofstableequilibriumthefluxoftheforce F throughanysurfacesurroundingthepointinfreespacemust benegative,which,fromthe divergencetheorem,requiresthat ∇· F = −∇2U< 0,or αm∇2B2 < 0atthatpoint.Nowin freespace ∇× B =0,and,consequently, ∇× (∇× B)= ∇(∇· B) −∇2B =0,so ∇2B =0 and
Therefore,wecannothave αm∇2B2 < 0intheparamagneticcase,thatis,aparamagnetic particlecannotbeheldinstableequilibriuminamagnetostaticfield.Butitispossiblefora diamagneticparticletobeinstableequilibriuminamagnetostaticfield:thisissimplybecause B2,unlikeanyofthethreecomponentsof B itself,does not satisfyLaplace’sequationand can havealocalminimum.Ordinarydiamagneticmaterials(wood,water,proteins,etc.)areonly veryweaklydiamagnetic,butlevitationispossibleinsufficientlystrongmagneticfields.The mostspectacularpracticalapplicationatpresentofmagneticlevitation—“maglev”trains—is basedonthelevitationofsuperconductors(αm →−∞)inmagneticfields.
1.3GaugesandtheRelativityofFields
Theelectricandmagneticfieldsofinterestinopticalphysicsarefar fromstaticand must,ofcourse,bedescribedbythecoupled,time-dependentMaxwellequations.In thissection,webrieflyreviewsomegaugeandLorentztransformationpropertiesimpliedbytheseequations.
Weintroduceavectorpotential A suchthat B = ∇× A,consistentwith ∇· B =0. From(1.1.3),itfollowsthatwecanwrite E = −∇φ ∂A/∂t,and,from(1.1.4)and theidentity ∇× (∇× A)= ∇(∇· A) −∇2A,
Intermsof φ and A,(1.1.1)becomes
2φ + ∂ ∂t (∇· A)= ρ/ǫ0. (1.3.2)
Theselasttwoequationsforthepotentials φ and A areequivalenttotheMaxwell equations(1.1.3)and(1.1.1),andthedefinitionsof φ and A ensurethattheremaining twoMaxwellequationsaresatisfied.But φ and A arenotuniquelyspecifiedby B = ∇× A and E = −∇φ ∂A/∂t:wecansatisfyMaxwell’sequationswithdifferent potentials A′ and φ′ obtainedfromthe gaugetransformations A = A′ + ∇χ,and φ = φ′ ∂χ/∂t with B = ∇× A = ∇× A′,and E = −∇φ ∂A/∂t = −∇φ′ ∂A′/∂t 5
5 Theword“gauge”inthiscontextwasintroducedbyHermannWeylin1929.
1.3.1LorentzGauge
Wecan,forexample,choosetheLorentzgaugeinwhichthescalarandvectorpotentials arechosensuchthatweobtainthefollowingequation:6
Then,from(1.3.1)and(1.3.2),
TheadvantageoftheLorentzgauge,asthenamesuggests,comeswhentheequations ofelectrodynamicsareformulatedsoastobe“manifestly”invariantundertheLorentz transformationsofrelativitytheory,asdiscussedbelow.
Recallasolutionofthescalarwaveequation
usingtheGreenfunction G satisfying
Fromastandardrepresentationforthedeltafunction,
with R = r r′,and T = t t′.ThecorrespondingFourierdecompositionofthe Greenfunction,
anditsdefiningequation(1.3.7),implythat
6 Recallthatthe“Lorentzgauge”isreallyaclassofgauges,aswecanreplace A by A + ∇ψ,and φ by φ ∂ψ/∂t,andstillsatisfy(1.3.3)aslongas ∇2ψ (1/c2) ∂2ψ/∂t2 =0.TheCoulombgauge condition,similarly,remainssatisfiedundersuch“restricted”gaugetransformationswith ∇2ψ =0, butforpotentialsthatfalloffatleastasfastas1/r, r beingthedistancefromthecenterofalocalized chargedistribution, ψ =0.WhatisgenerallycalledtheLorentzgaugeconditionwas actuallyproposed aquarter-centurybeforeH.A.LorentzbyL.V.Lorenz,whoalsoformulatedequationsequivalentto Maxwell’s,independentlyofMaxwellbutafewyearslater.SeeJ.D.JacksonandL.B.Okun,Rev. Mod.Phys. 73,663(2001).
Howcanwedealwiththesingularitiesat ω = ±kc intheintegrationover ω?A physicallyreasonableassumptionisthat G(r,t; r′,t′)is0for T = t t′ < 0,thatis, fortimesbeforethedeltafunction“source”isturnedon.Wecansatisfythiscondition byintroducingthepositiveinfinitesimal ǫ anddefiningthe retarded Greenfunction:
(1.3.11)
Nowthepoleslienotontherealaxisbutinthelowerhalfofthecomplex plane.Since e iωT → 0for T< 0andlarge,positiveimaginarypartsof ω,wecanreplacethe integrationpathin(1.3.11)byonealongtherealaxisandclosedinalarge(radius →∞)semicircleintheupperhalf-plane.Andsincetherearenopolesinside this closedpath,wehavethedesiredpropertythat G(r,t; r ′,t′)=0(t<t
For T = t t′ > 0,similarly,wecanclosetheintegrationpathwithaninfinitelylarge semicircleinthelowerhalfofthecomplexplane.Theintegrationpathnowencloses thepolesat ω = ±kc iǫ,andtheresiduetheoremgives
G(r,t; r ′,t′)= 1 2π 3 c 2iR ∞ −∞ dkeikR( 2πi
Thesolutionof(1.3.5),forexample,isthen
,t
undertheassumptionthatitistheretardedGreenfunctionthatis physicallymeaningful,ratherthanthe“advanced”Greenfunctionorsomelinearcombinationofadvanced
andretardedGreenfunctions.7 Thecontributionofthechargedensityat r′ tothe scalarpotentialat r attime t dependsonthevalueofthechargedensityattheretardedtime t −|r r′|/c,andlikewiseforthevectorpotential.Evaluationofthese potentialsgivesexpressionsthataremorecomplicatedthantrivially retardedversions oftheirstaticforms,aswenowrecallforasimplebutimportantexample.
Forapointcharge q movingsuchthatitspositionattime t is u(t), ρ(r′,t′)= qδ3[r′ u(t′)],andthescalarpotentialis
φ(r,t)= q 4πǫ0 d3
Toperformtheintegration,wechangevariablesfrom x′,y′,z′,t′ to y1 = x′ ux(t′), y2 =
φ(r,t)= q
(y4), (1.3.16)
wherenow r′ = u(t′), t′ = t −|r r′|/c,and J isthe4 × 4Jacobiandeterminant, J = ∂(y1,y2,y3,y4) ∂(x′,y′,z′,t′) , (1.3.17)
whichisfoundbystraightforwardalgebratobe J =1 [u(t′)/c] · r r′ |r r′| (1.3.18) Therefore, φ(r,t)= q 4πǫ
or,inmorecompactnotation, φ(r,t)= 1 4
(1.3.20)
where R isthedistancefromthechargetotheobservationpoint r, ˆ n istheunitvector pointingfromthepointchargetothepointofobservation, v = ˙ u isthevelocityofthe charge,andthesubscript“ret”meansthatallthequantitiesinbracketsareevaluated attheretardedtime t′ = t −|r r′|/c.Likewise,thesolutionof(1.3.4)fortheretarded vectorpotentialis
(1.3.21) sincethecurrentdensityassociatedwiththepointchargeis J = qvδ3[r u(t)].
7 Wefollowherethenearlyuniversalpracticeinclassicalelectrodynamicsofsettingto0the(zerotemperature)solutionsofthehomogeneousMaxwellequations,thatis,wepresumethereareno “source-free”fields.In quantum electrodynamics,however,therearefluctuatingfields,withobservable physicalconsequences,evenatzerotemperature.NontrivialsolutionsofthehomogeneousMaxwell equationsalsoappearintheclassicaltheorycalled stochasticelectrodynamics.SeeSection7.4.1.
ElementsofClassicalElectrodynamics
These Li´enard–Wiechertpotentials arecomplicated.Foronething, φ(r,t),forinstance,is not simply q/4πǫ0[R]ret,which“almosteveryonewould,atfirst,think.”8 Instead, φ(r,t)dependsnotonlyonthepositionofthechargeattheretardedtime t′ , butalsoonwhatthevelocitywasat t′.Forachargemovingwithconstantvelocity v alongthe x axis,forexample,
ifwedefineourcoordinatessuchthat,at t =0,thechargeisat(x =0,y =0,z =0). Thesolutionofthisequationfor t′ (<t)is
Since R = c(t t′)andthecomponentofvelocityalong r′ attheretardedtime t′ is v × (x vt′)/|r′|,itfollowsfrom(1.3.22)and(1.3.23)that [R Rv ˆ
andthereforethat
(1.3.25) and Ax(x,y,z,t)=
forachargedparticlemovingwithconstantvelocity v alongthe x direction. Wecanderivetheseresultsmoresimplyusingthefactthat,inspecialrelativity theory, φ and A transformasthecomponentsofafour-vector(φ/c, A).Inaspacetimecoordinatesystem(x′,y′,z′,t′)inwhichacharge q isatrest,
′(x ′ ,y ′ ,z ′,t′)= q
Thecoordinates(x,y,z,t)inthe“lab”frame,inwhichthechargeismovinginthe positive x directionwithconstantvelocity v,arerelatedtotherest-framecoordinates bytheLorentztransformations:
8 Feynman,Leighton,andSands,VolumeII,p.21–9.(WerefertobooksintheBibliographyusing theirauthors’italicizedsurnames.)
Thepotential φ(x,y,x,t),forinstance,isobtainedbytransforming
,z
,t
) fromtherestframeofthechargetoaframemovingwithvelocity v alongthe x axis:
whichisjust(1.3.25).Thatweobtained(1.3.25)directlyfromthesolutionofthe waveequationfor φ withoutmakinganyLorentztransformationsisnotsurprising, ofcourse,becausetheMaxwellequationsarethecorrectequationsofelectromagnetic theoryinspecialrelativity;theyarecorrectinanyinertialframe. Indeed,theLi´enard–Wiechertpotentialswereobtainedbeforethedevelopmentofthetheoryofspecial relativity.Whatspecialrelativityshowsisthat v canberegardedastherelative velocitybetweenthecoordinatesysteminwhichthechargeisatrestandthesystem inwhichitismovingwithvelocity v.
Oncewehave φ and A,wecanobtaintheelectricandmagneticfieldsusing E = −∇φ ∂A/∂t and B = ∇× A.From(1.3.25)andthecorrespondingformulas for A,
and
Moregenerally,theelectricandmagneticfieldstransformas
whentheprimedframemoveswithrespecttotheunprimedframeat aconstant velocity v inthe x direction.9
Theresult(1.3.31),forexample,canbeobtainedfromtheCoulombfieldinaframe inwhichthechargeisatrest,usingthesetransformationlawstorelatethefieldsin thetwoinertialframes.Inparticular,apurelyelectricfieldinoneframeimpliesa magneticfieldinanother,andviceversa.10
Inthecaseofachargedparticlemovingwithavelocitythatvariesintime,the electricandmagneticfieldscanbecalculatedfromtheLi´enard–Wiechertpotentials, asisdoneinstandardtexts.Here,weonlyrecalltheformulasforthe(retarded)fields intheradiationzonewhentheparticlemotionisnon-relativistic(v ≪ c):
ThepowerradiatedpersolidangleiscalculatedusingthesefieldsandthePoynting vector:
where θ istheanglebetween r andtheacceleration v.Integrationoverallsolidangles resultsinthe(non-relativistic)Larmorformulafortheradiatedpower:
9 ThesetransformationsareappliedinSection2.8toblackbodyradiationfields.
10 “Whatledmemoreorlessdirectlytothespecialtheoryofrelativitywastheconvictionthatthe electromotiveforceactingonabodyinmotioninamagneticfieldwasnothingelsebutanelectric field.”—Einstein,quotedinR.S.Shankland,Am.J.Phys. 32,16(1964),p.35.
1.3.2CoulombGauge
IntheCoulombgaugewechoose χ suchthat ∇· A =0.11 Inthisgauge,
and
ThescalarpotentialsatisfiesthePoissonequation(1.3.37)andisgivenintermsofthe chargedensity ρ(r,t)bytheinstantaneousCoulombpotential,
ifthechargedistributionisspecifiedthroughoutallspace.(Ofcourse,thisisnot alwaysthecase;inmanyexamplesinelectrostatics,forexample,thepotentialsare specifiedonconductors,andsurfacechargedistributionsarededuced after solving Laplace’sequationwithboundaryconditions.)Equation(1.3.38)canberewritten using Helmholtz’stheorem:anyvectorfield F(r,t)canbeuniquelydecomposedin transverseandlongitudinalpartsdefinedrespectivelyby12
Inotherwords, F = F⊥ + F ,with ∇· F⊥ = ∇× F =0.IntheCoulombgaugethe vectorpotential A isatransversevectorfield(∇· A =0);writing J = J⊥ + J in (1.3.38),wehave
wherewehaveusedthechargeconservationcondition(1.1.6).
AlthoughtheLorentzgaugeisperfectlysuitedforrelativistictheory,theCoulomb gaugealsoofferssomeadvantages,andisalmostalwaysusedinquantumoptics.Inthe Coulombgauge,thelongitudinalfield E = −∇φ iseffectivelyeliminatedandreplaced byCoulombinteractionsofthecharges,andquantizationofthefieldtheninvolves
11 Forexplicitformsofthe χ’sthateffectthegaugetransformations,seeJ.D.Jackson,Am.J. Phys. 70,917(2002).
12 AproofofHelmholtz’stheoremisoutlinedinAppendixB.
