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Chapter 4 Duality（第1页）

Chapter4Duality

&wodifferentviewsoflight，asapartidasawave，bothsightaheyhaveeaabledbothandiuralworldaanddesigeologies。Yettheyappeartobevastlydifferentintheirofwhatlightactuallyis。Oheparticlemodelviewslightasalotity，abuhatmovesalongawell-defiory。Ohewavemhtasadiffuseeingthroughspaotothemotionofsolidthings。Howthesetwopicturespossiblyrefertothesamething?ThisdilemmawasreizedearlyonbyHuygensandhisporaries，butthetwoviewsremaiensioivedessoflight，uhtury。

WhenMaxwelldevelopedhistheoryicfields，hewasabletousethistoexplaiiesoflightaswavemotionofthosefields，aster3。ThistriumphappearedtotheexperimentsofThomasYoungandAugusteFresnel（desChapter3）byprovidiionoftheerferenddiffra，thatdidwithiiclemodel。Yettheceptoftrajeaiillremairaordinarilypowerfuloheanalysisanddesignofopticalsystems。Sothere’saruceofthesetwopictures—adualismhysics—thatrequiressomesideration。Howtheybereciled?

Lookingattrajectain

IhturytheFrenPierredeFermatproposedaningeniousformulatioionthatwasverydifferentfromthatofSSnell’slawdealswiththegeofdirehtataweentareheray，defiioninwhichitistravellingtowardstheihepointatwhichithitstheisdirealteredbyanamountproportioiooftherefradicesofthetwomaterials。Itisonlythelocalpropertiesoftherayahatareimportant。Snell’slaliesateataloory，asiftherayis‘feeling’itswayalong，adjustiioentersaerface。

&’swasradicallydifferehatoneshoulddefioryiartingandendingpoints，asshowninFigure21。Hesuggestedthatthequestiontoaskis:whatisthepaththatthelighttakestotraversethespathetwopoints?Heproposedthatitshouldtakethepaththatmiimeofflightbetweewopoints。ThatthisgivesthesameanswerasSnellisremarkableandprofou’s‘pritime’suggeststhatthelightsiderstheoverallpictureofthesituation，andthatthenotionofarayisoakesintoatboththeinitialandfinalpositionsaionsaswellaseverythihetrastwiththelocalmodelofaparticlereaediateeelling。

21。Fermat’sofalightrayasapathofleasttimegthestartahetrajectory。Therayiweentwoopticalmediainwhichlightmovesatdiferentspeeds。

ThisideabytheGermannaturalphilottfriedWilhelmvonLeibniz，ton’sporaryandantagonist。LeibnizressedbytheholisticpictureoftheprocessdescribedbyFermat，aof‘optimization’thatitimplied:arayexploresthewholeofspadpicksjustthatpaththatwillmiraweenthespecifiedbegins。Hedevelopedthemathematicaltoolsforanalysingthisidea—thecalculusofvariations—bywhichtheeffectssmallatrajectorywouldhaveootraversethemodifiedtrajectorycouldbecalculated。LeibnizreizedtheimportahatFermat’sprincipleprovided:themovementoflightfromooanotherdefiimal’trajectory。

&akenwasLeibnizbythisizatioedittoateleologiciple:thattheworlditself，inallitsaspects，wasoimaltrajectorybetointandafinishingpoiradiherentinsuchaposition，liedoutsideoftherealmofsce，ooaireinhisnoveldide，whereLeibniz’sideasareputihloss，whoinsistsdisastersbothnaturalandmahelessevidehisisthe‘bestofallpossibleworlds’。

&ingwavesandrays

&heless，Leibniz’smathematicalideasprovedtobeveryfruitful。TheybytherenownedIrishmathematiWilliamRowanHamiltohtury。Heshowedformallyhowtheideaofawavebealliedtothatofaofparticles。Wavesbedefiheirwavelength，amplitude，andphase（seeFigure15）。Particlesaredefiioionoftravel（seeFigure5），aionofparticlesbytheirdehehematagivenpositieofdireediainwhichthelightmovesarecharacterizedbytheirrefradices。Thisvaryacrossspaple，attheinterfaihereisasteptherefradexacrossthebouhet>

Hamiltoortantidlytherefradexspaparedwiththelengthofanopticalwave。Thatis，iftheiookplaascaleofclosetoawavelehewavecharacteroflightwasevident。Ifitvariedmoresmoothlyandveryslowlyiheparticlepictureprovidedaiohesimplerraypictureemergesfromthemoreplexioeredsituations。Theappearanceofhenomena，suchasdiffradinterference，othesizescalesofthewavelengthoflightauresinwhichitpropagatesaresimilar。Thusyouseediffrapatternsarisiheobjectthatthelighthitsisafewmidiameter，orhasaverysharpedge，suchasthedelicatestruabird’sfeather，orabutterflywing。Otherwise，asintheeraleoryprovidesasuffitdes，siiveindexisunifhouttheglassofthelensitself。

22。Hamilton’sideaofraysasgwavefronts-thusjoitionsoflight。

Further，HamiltoFermat’strajectoriesrelateddirectlytoapropertyofthewave—thewavefrohelotsatwhichthewavehasthesamephaseateatinspastanyouseetheripplesonthesurfaceofapoointoit，thecircularpatterhesewavefroheplathesurfacewheretheeaks’（hs）atagiveninstantoftime。Now，whatHamiltohatrayscouldbesideredasliihewavefrles，asshownihusgadjatwavefrontsbyawell-defiory。

Hamilton’s‘optialogy’

Thisremarkableresultsuggestedanotherprofoundilton’sso-called‘optialogy’。Whathehatthewell-knownformulatiohemotionandpositionofsolidbodiesofmatter—wasbasedorajectories。Theideathatthesemayalsobeiimal’hadbeensideredbyPierreLouisMaupertuisihtury。

&uishadformulatedawaytoevaluatetheoptimalvalueofaquantitycalledthe‘a’—essentiallythevelocityofthebodymultipliedbythedistamoves（asmass）—alorajectory。

&hattheainimalforanactualtrajectorybetweentwopoints，justasi’sargumentthatthetimetakenthtrayshouldbeminimal。Maupertuis’‘pria’isverysimilarioFermat’s‘pritime’。Indeed，LeonhardEuler，aSwissmathemati，showedhowtouseLeibniz’scalculustoderiveon’sfamousequationsofmotionfromMaupertuis’prihusEulerectedadesofatrajetermsofapartigitswaythroughitseooneinwhichthewholeofspathespecifiedstartingandfinishingpoihepath。

WhatHamiltondidwastofiioedthevariationsinasofasimpledesofthespeviroinwhioving。Aurnsouttohaveaverysimilarformtotheonehefoundfthetrajectoriesoflightrays（forwhiviroionisjusthowtherefradexgeswithpositioninthemedium）。Sothereisahintofalatentahetrajectoriesofsolidobjedafictivewavefront:perhapsallbodiesmighthavebothparticle-liketrajedroperties?Ioion，andhiseponymousfun，turnsouttobeveryimportantinthinkingabouttheepiandinglight—quantummeics。

Unsolvedpuzzles

Thiswasbyheonlyhiunityforsce。Aboutthistime，towardstheehtury，lightstillofferedafewpuzzlesthatwereunexplaiermsmodelsofitsproperties，evenwiththerethatHamiltonhadprovided。Twoofthemostimportantofthesewere:thecolourofhotobjegtheSun），andthecolourofdifferentatomsinaflame。

&hiedup，theygecolour。Takealumpofmetal。Asitgetshotterafirstglowsred，thehenwhite。Whydoesthishappen?Thisquestionstumpedmastistsofthetime，ingMaxwellhimself。TheproblemwasthatMaxwell’stheht，liedtothisproblem，ihecetbluerahetemperatureihoutalimit，eventuallymovingouteofhumanvisioraviolet—beyiorum。Butthisdoesnothappeninpractice。