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.
ElementsofClassicalElectrodynamics
onlythetransversefields A, E⊥,and B.(B =0inanygauge.)ButtheCoulomb interactionsintheCoulombgaugeareinstantaneous,notretarded(see(1.3.39)).In theLorentzgauge,incontrast,thepotentials(andthereforetheelectricandmagnetic fields)donotpropagateinstantaneouslyandareretardedaslong aswechoosethe retardedGreenfunctionforthewaveequation:
TheexpressionforthesameelectricfieldwhentheCoulombgaugeisusedis
whenweusetheretardedGreenfunctionforthesolutionofthewaveequation(1.3.42).
Ofcourse, E cannotdependonthechoiceofgauge,andsotheexpressions(1.3.45) and(1.3.46)mustbeequivalent,and,inparticular,(1.3.46)mustbea retardedfield, eventhoughthe instantaneous Coulombfieldappearsinthefirstterm.Weshowin AppendixAthatthisisso.
Wecanexpresstheelectricfieldinotherforms.First,write(1.3.45) morecompactly as
bydefining[f ]=
Using
Similarly,
Expressions(1.3.49)and(1.3.50),whichmayberegardedastime-dependentgeneralizationsoftheCoulombandBiot-Savartlaws,arethe Jefimenkoequations forthe electricandmagneticfieldsproducedbyachargedensity ρ(r,t)andacurrentdensity J(r,t).13
1.4DipoleRadiators
Radiationbyacceleratedchargesis,inonewayoranother,responsibleforalllight. Inopticalphysics,weareparticularlyconcernedwithchargeaccelerationintheform ofoscillationsofboundelectrons.Inthecrudestdescription,the radiationfroman excitedatom,forexample,canberegardedasradiationfromanoscillatingelectric dipoleformedbythenegativelychargedelectronsandthepositively chargednucleus. (Thiswillbeclarifiedinthefollowingchapters.)Infact,theradiationresultingfrom anelectricdipoletransitioninanatomisverysimilarinsomewaystothat froma dipoleantenna.Webeginourdiscussionofdipoleradiationbyconsideringthesimple antennasketchedinFigure1.2.
Fig.1.2 Anantennawireoflength L center-fedbyanACcurrent.
Thecurrent I inthewireoscillatesintimeatthefrequency ω andvanishesatthe endpoints z = ±L/2.Ittakestheformofastandingwave:
with k = ω/c,and Im thepeakcurrent.Thevectorpotential(1.3.44)inthisexample is
where,asusual,itisimpliedthatwemusttaketherealpartoftherightside.
13 SeeK.T.McDonald,Am.J.Phys. 65,1074(1997),andreferencestherein.
Fig.1.3 Thevector r fromthemiddleoftheantennawiretothepointofobservation.
Forlargedistancesfromtheantenna,wecanapproximate |r r′| by r inthe denominatoroftheintegrandanduse(seeFigure1.3)
intheexponentinthenumerator:
(1.4.4) and(aftertakingtherealpart)
B(r,t)= ∇× A ∼ = y ˆ x x ˆ y r µ0Im 2πr sin(
φ
where eφ = ˆ x sin φ + ˆ y cos φ istheazimuthal-angleunitvectorat x,y inspherical coordinates.Similarly, E(r,t)= eθ Im 2πr µ0 ǫ0 sin(ωt kr)
where eθ = ˆ x cos θ cos φ + ˆ y cos θ sin φ ˆ z sin θ isthepolar-angleunitvectorat x,y,z insphericalcoordinates.Thecycle-averagedPoyntingvector,
(r)= E × H =
followsbysimplealgebraandtheidentity eθ ×eφ = ˆ r.Theradiatedpoweristherefore
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FASHION IN LETTERS AND THINGS.
Periodicity exists throughout all nature. Day and night, winter and summer, equinox and solstice; years of plenty and years of famine, commerce active and business depressed; volcanoes in state of eruption, then at rest; comets return, eclipses come back, the striae of one glacial period are deepened by those of another, and the leg o’ mutton sleeves that our grandmammas wore in the thirties are again upon us.
When the hounds start game in the mountains, the hunter knowing that the deer moves in a circle, stands still on the run-way, biding his time. So no one need wail and strike his breast if his raiment is out of style: all such should be consoled by the fact that the fashion is surely coming back.
Mode in dress is only an outcrop of a general law. Why does fashion change? Because it is the fashion. The followers of fashion —that is to say, civilized men and women—are not content with being all alike. Esquimaux and Hottentots never vary their styles. But people in the temperate zones are intemperate and desire to excel— to be different from others—distinctive, peculiar, individual. Very seldom is any one strong enough to stand alone, so in certain social circles, by common consent, all overcoats are cut one length—say, to come just above the knee. Then this overcoat is gradually lowered: to the knee, just below the knee, to the ankle—until it conceals the feet. Then an enormous collar is added, which when turned up and viewed from behind completely hides the man. But this thing cannot last; it is not many days before the same men are wearing overcoats so short that the wearers look like matadors ready for the fray.
Ladies wear hoops; the hoops expand and expand, until the maximum of possibility in size is reached. Something must be done! The crinoline contracts until these same ladies appear in clinging
skirts, and the pull-back lives its little hour Then the former width of the dress is used to lengthen it. The skirt touches the ground, trails two inches, six, eight, a foot, two feet. Its length becomes too great to drag and so is carried, to the great inconvenience of its owner; or in banquet halls pages are employed. But this is too much, a protest comes and two hundred women in Boston agree to appear on the streets the first rainy day in skirts barely coming to the boot top. “Dress Reform” societies spring up, magazines become the organs of the protestants and the printing presses run over time.
The garb of the Quaker is only a revulsion from a flutter of ribbons and towering headgear. From Beau Brummel lifting his hat with great flourish and uncovering on slight excuse, we have William Penn who uncovers to nobody; the height of Brummel’s hat finds place in the width of Penn’s.
All things move in an orbit.
Even theories have their regular times of incubation. They are hatched, grow lusty, crow in falsetto or else cackle; then they proceed to scratch in the flower beds of conservatism to the hysterical fear of good old ladies, who shoo them away. Or if the damage seems serious, the ladies set dogs—the lap-dogs of war— upon them.
“The sun do move;” Brother Jasper is right. All things move. And when matters get pushed to a point where they fall on t’other side, a Reformer appears. The people proclaim him king, but he modestly calls himself “Protector.” He is spoken of in history as the Savior of the State.
There are only two classes of men who live in history: those who crowd a thing to its extreme limit, and those who then arise and cry “Hold!” A Pharaoh makes a Moses possible. The latter we write down in our books as immortal, the first as infamous.
This is true of all who live in history, whether in the realm of politics, religion or art. History is only a record of ideas (or lack of them) pushed to a point where revulsion occurs. If Rome had been moderate, Luther would have had no excuse.
Literature obeys the law; its orbit is an ellipse. The illustrious names in letters are those of the men who have stood at aphelion or perihelion and waved the flaring comet back.
The so-called great poets are the men stationed by fate at these pivotal points. And as fires burn brightest when the wind is high, so these men facing mob majorities have, through opposition, had their intellects fanned into a flame.
More than thirteen decisive battles have taken place in the world of letters. And the question at issue has always been the same: Radical and Conservative calling themselves Realist, Romanticist, Veritist or What-not struggling for supremacy.
Term it “Veritism” and “Impressionism” if you prefer—juggle the names and put your Union troops in gray, but this does not change the question.
The battle between the two schools of literature is a football game. The extreme goal on one side is tea table chatter, on the other an obscure symbolism. “The difference is this:” said Dion Boucicault, “when Romanticism goes to seed it is ‘rot;’ when Realism reaches a like condition it is only ‘drivel.’”
In literary production why should we hear so much about the dignity of this school and the propriety of that. Men who fail to appreciate the individual excellence of a certain literary output, declare it to be without sense and therefore base. In letters they assume that a style is wholly good or it is wholly bad. They make no allowances for temperament; they would have all men speak in one voice.
Yet liberty need not result in disorder, nor can originality serve as a pretext for boozy inaccuracy. In a literary production the bolder the conception the more irreproachable should be the execution.
There is a tendency for thought to get fixed in set forms, and this form is always that which has been used by some great man. For any one to express thought and feeling in a different way is blasphemy to the eunuchs who guard the tents of Tradition.
Writers of different schools exist because their style fits the mind of a certain style of reader. The sprightly, animated picturesqueness, the play of wit and flights of imagination are only a full expression of what many faintly feel. Thus their mood is mirrored and their thought expressed: hence they are pleased.
In fact the only reason why we like a writer is because he expresses our thought in a way we like. And the reason we dislike a writer is because he deals in that which is not ours. We of course might grow to like him, but the process is slow, for according to Herbert Spencer we must hear a thing six hundred times before we understand. If we comprehend a proposition at once, it is only because it was ours already. If the portrayal of a situation in fiction fascinates us, it is because we (in fancy or fact) have gone before and spied out the land.
There must be more than one school of literature, because there is more than one mood of mind: just as in religion there must be many sects. We worship God not only in sincerity and truth, but according to the temperament our mothers gave us.
The emotional “school of religion” finds its votaries in Methodism: Methodism fits a certain mind. The stately dignity of the Ritualist is a necessity to a certain cast of intellect. And until we get a church that is broad enough, and deep enough, and high enough to allow for temperament in men, “church union” will exist only as an abstract idea.
Until we have a school of literature that will combine all schools and give the liberty to a full expression of every mood, there will be a warfare between the “sects” that give free rein to imagination and the sect that, having no imagination, merely describes. When one school driven by the jibes and jeers of the other tilts to t’other side, a heavy man will start the teeter back, and he is the man we crown.
And let us ever crown the heavy man when we find him.
E H
