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The Modern Revolution in Physics

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The Modern Revolution in Physics
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Science, Physics, Modern Physics, Einstein, Calculating Randomness, Statistical independence, Addition of probabilities, Normalization, Probability Distributions, Exponential Decay, Relativity, Distortion of Time and Space, No simultaneity, Combination of velocities, Momentum, Equivalence of mass and energy, Light as a Particle, Wave Particle Duality, …
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This is an introductory physics textbook designed for use in a typical one year survey course. Contents: 1) Relativity. 2) Rules of Randomness. 3) Light as a Particle. 4) Matter as a Wave. 5) The Atom. This is book 6 in the Light and Matter series of free introductory physics textbooks.
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Benjamin Crowell, Fullerton College, CA
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www.lightandmatter.com
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http://florida.theorangegrove.org/og/file/ae629fb6-ef7d-96d7-9c1f-4879f5dbaaf3/1/ModernPhysics.pdf

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Copyright 1998-2003 Benjamin Crowell. This book is licensed under the Creative Commons Attribution-ShareAlike license, version 1.0, http://creativecommons.org/licenses/by-sa/1.0/ except for those photographs and drawings of which I am not the author, as listed in the photo credits. If you agree to the license, it grants you certain privileges that you …
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Book6intheLightandMatterseriesoffreeintroductoryphysicstextbooks www.lightandmatter.com

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The LightandMatter seriesof introductoryphysicstextbooks: 1NewtonianPhysics 2ConservationLaws 3VibrationsandWaves 4ElectricityandMagnetism 5Optics 6TheModernRevolutioninPhysics

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BenjaminCrowell www.lightandmatter.com

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Fullerton,California www.lightandmatter.com copyright1998-2003BenjaminCrowell edition3.0 rev.May9,2008 ThisbookislicensedundertheCreativeCommonsAttribution-ShareAlikelicense,version1.0, http://creativecommons.org/licenses/by-sa/1.0/,except forthosephotographsanddrawingsofwhichIamnot theauthor,aslistedinthephotocredits.Ifyouagree tothelicense,itgrantsyoucertainprivilegesthatyou wouldnototherwisehave,suchastherighttocopythe book,ordownloadthedigitalversionfreeofchargefrom www.lightandmatter.com.Atyouroption,youmayalso copythisbookundertheGNUFreeDocumentation Licenseversion1.2,http://www.gnu.org/licenses/fdl.txt, withnoinvariantsections,nofront-covertexts,andno back-covertexts. ISBN0-9704670-6-0

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ToGretchen.

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BriefContents 1Relativity13 2RulesofRandomness43 3LightasaParticle67 4MatterasaWave85 5TheAtom111

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Contents 1Relativity 1.1ThePrincipleofRelativity.....14 1.2DistortionofTimeandSpace...18 Time,18.|Space,20.|Nosimultaneity, 20.|Applications,22. 1.3Dynamics...........27 Combinationofvelocities,27.| Momentum,28.|Equivalenceofmassand energy,31. Summary.............37 Problems.............39 2RulesofRandomness 2.1RandomnessIsn'tRandom....45 2.2CalculatingRandomness.....46 Statisticalindependence,46.|Additionof probabilities,47.|Normalization,48.| Averages,48. 2.3ProbabilityDistributions.....50 Averageandwidthofaprobability distribution,51. 2.4ExponentialDecayandHalf-Life..52 Rateofdecay,55. 2.5 R ApplicationsofCalculus....57 Summary.............60 Problems.............62 3LightasaParticle 3.1EvidenceforLightasaParticle..68 3.2HowMuchLightIsOnePhoton?..70 Thephotoelectriceect,70.|Anunexpecteddependenceonfrequency,71.| Numericalrelationshipbetweenenergyand frequency,72. 3.3Wave-ParticleDuality......75 Awronginterpretation:photonsinter10

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feringwitheachother,76.|Theconcept ofaphoton'spathisundened.,76.| Anotherwronginterpretation:thepilotwavehypothesis,77.|Theprobability interpretation,77. 3.4PhotonsinThreeDimensions...80 Summary.............81 Problems.............82 4MatterasaWave 4.1ElectronsasWaves.......86 Whatkindofwaveisit?,89. 4.2 R ? DispersiveWaves......91 4.3BoundStates..........93 4.4TheUncertaintyPrinciple.....96 Theuncertaintyprinciple,96.| MeasurementandSchrodinger'scat,100. 4.5ElectronsinElectricFields....101 Tunneling,102. 4.6 R ? TheSchr odingerEquation...102 Useofcomplexnumbers,105. Summary.............106 Problems.............108 5TheAtom 5.1ClassifyingStates........112 5.2AngularMomentuminThree Dimensions............113 Three-dimensionalangularmomentumin classicalphysics,113.|Three-dimensional angularmomentuminquantumphysics, 114. 5.3TheHydrogenAtom.......115 5.4 ? EnergiesofStatesinHydrogen.117 History,117.|Approximatetreatment, 118. 5.5ElectronSpin..........120 5.6AtomsWithMoreThanOneElectron121 Derivingtheperiodictable,123. Summary.............125 Problems.............127 Appendix1:Exercises 130 Appendix2:PhotoCredits 132 Appendix3:HintsandSolutions 133 11

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a / AlbertEinstein. Chapter1 Relativity Complainingabouttheeducationalsystemisanationalsportamong professorsintheU.S.,andI,likemycolleagues,amoftentempted toimagineagoldenageofeducationinourcountry'spast,orto compareoursystemunfavorablywithforeignones.Realityintrudes, however,whenmyimmigrantstudentsrecounttheoveremphasison rotememorizationintheirnativecountries,andthephilosophythat whattheteachersaysisalwaysright,evenwhenit'swrong. AlbertEinstein'seducationinlate-nineteenth-centuryGermany wasneithermodernnorliberal.Hedidwellintheearlygrades, 1 butinhighschoolandcollegehebegantogetintroubleforwhat today'sedspeakcallscriticalthinking." Indeed,therewasmuchthatdeservedcriticisminthestateof physicsatthattime.Therewasasubtlecontradictionbetweenthe theoryoflightasawaveandGalileo'sprinciplethatallmotion isrelative.Asateenager,Einsteinbeganthinkingaboutthison anintuitivebasis,tryingtoimaginewhatalightbeamwouldlook likeifyoucouldridealongbesideitonamotorcycleatthespeed oflight.Todaywerememberhimmostofallforhisradicaland far-reachingsolutiontothiscontradiction,histheoryofrelativity, butinhisstudentyearshisinsightsweregreetedwithderisionfrom hisprofessors.Onecalledhimalazydog."Einstein'sdistaste forauthoritywastypiedbyhisdecisionasateenagertorenounce hisGermancitizenshipandbecomeastatelessperson,basedpurely onhisoppositiontothemilitarismandrepressivenessofGerman society.HespenthismostproductivescienticyearsinSwitzerland andBerlin,rstasapatentclerkbutlaterasauniversityprofessor. HewasanoutspokenpacistandastubbornopponentofWorld WarI,shieldedfromretributionbyhiseventualacquisitionofSwiss citizenship. Astheepochalnatureofhisworkbecameevident,someliberal GermansbegantopointtohimasamodelofthenewGerman," butaftertheNazicoupd'etat,stagedpublicmeetingsbegan,at whichNaziscientistscriticizedtheworkofthisethnicallyJewish butspirituallynonconformistgiantofscience.WhenHitlerwas appointedchancellor,Einsteinwasonastintasavisitingprofessor atCaltech,andheneverreturnedtotheNazistate.WorldWar 1 Themyththathefailedhiselementary-schoolclassescomesfromamisunderstandingbasedonareversaloftheGermannumericalgradingscale. 13

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b / Therstnuclearexplosiononourplanet,Alamogordo, NewMexico,July16,1945. IIconvincedEinsteintosoftenhisstrictpaciststance,andhe signedasecretlettertoPresidentRoosevelturgingresearchinto thebuildingofanuclearbomb,adevicethatcouldnothavebeen imaginedwithouthistheoryofrelativity.Helaterwrote,however, thatwhenHiroshimaandNagasakiwerebombed,itmadehimwish hecouldburnohisownngersforhavingsignedtheletter. EinsteinhasbecomeakindofscienticSantaClausgurein popularculture,whichispresumablywhythepublicisalwayssotitillatedbyhiswell-documentedcareerasaskirt-chaserandunfaithful husband.Manyarealsosurprisedbyhislifelongcommitmenttosocialism.AfavoritetargetofJ.EdgarHoover'sparanoia,Einstein hadhisphonetapped,hisgarbagesearched,andhismailillegally opened.Acensoredversionofhis1800-pageFBIlewasobtained in1983undertheFreedomofInformationAct,andamorecompleteversionwasdisclosedrecently. 2 Itincludescommentssolicited fromanti-Semiticandpro-Naziinformants,aswellasstatements, fromsourceswhoturnedouttobementalpatients,thatEinstein hadinventedadeathrayandarobotthatcouldcontrolthehuman mind.Eventoday,anFBIwebpage 3 accuseshimofworkingfor orbelongingto34communist-front"organizations,apparentlyincludingtheAmericanCrusadeAgainstLynching.Attheheightof theMcCarthywitchhunt,EinsteinbravelydenouncedMcCarthy, andpubliclyurgeditstargetstorefusetotestifybeforetheHouse UnamericanActivitiesCommittee.Belyinghisother-worldlyand absent-mindedimage,hispoliticalpositionsseeminretrospectnot tohavebeenatallcloudedbynaiveteorthemorefuzzy-minded varietyofidealism.HeworkedagainstracismintheU.S.longbeforethecivilrightsmovementgotunderway.Inanerawhenmany leftistswereonlytooeagertoapologizeforStalinism,heopposedit consistently. ThischapterisspecicallyaboutEinstein'stheoryofrelativity,butEinsteinalsobeganasecond,parallelrevolutioninphysics knownasthequantumtheory,whichstated,amongotherthings, thatcertainprocessesinnatureareinescapablyrandom.Ironically, Einsteinwasanoutspokendoubterofthenewquantumideasthat werebuiltonhisfoundations,beingconvincedthattheOldOne [God]doesnotplaydicewiththeuniverse,"butquantumandrelativisticconceptsarenowthoroughlyintertwinedinphysics. 1.1ThePrincipleofRelativity BythetimeEinsteinwasborn,ithadalreadybeentwocenturies sincephysicistshadacceptedGalileo'sprincipleofinertia.Oneway ofstatingthisprincipleisthatexperimentswithmaterialobjects don'tgivedierentresultsduethestraight-line,constant-speedmo2 FredJerome, TheEinsteinFile ,St.Martin'sPress,2002 3 http://foia.fbi.gov/foiaindex/einstein.htm 14 Chapter1Relativity

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tionoftheapparatus.Forinstance,ifyoutossaballupinthe airwhileridinginajetplane,nothingunusualhappens;theball justfallsbackintoyourhand.Motionisrelative.Fromyourpoint ofview,thejetisstandingstillwhilethefarmsandcitiespassby underneath. TheteenageEinsteinwassuspiciousbecausehisprofessorssaid lightwavesobeyedanentirelydierentsetofrulesthanmaterial objects,andinparticularthatlightdidnotobeytheprincipleof inertia.Theybelievedthatlightwaveswereavibrationofamysterioussubstancecalledtheether,andthatthespeedoflightshould beinterpretedasaspeedrelativetothisether.Thusalthoughthe cornerstoneofthestudyofmatterhadfortwocenturiesbeenthe ideathatmotionisrelative,thescienceoflightseemedtocontain aconceptthatacertainframeofreferencewasinanabsolutestate ofrestwithrespecttotheether,andwasthereforetobepreferred overmovingframes. Experiments,however,failedtodetectthismysteriousether. Apparentlyitsurroundedeverything,andevenpenetratedinside physicalobjects;iflightwasawavevibratingthroughtheether, thenapparentlytherewasetherinsidewindowglassorthehuman eye.Itwasalsosurprisinglydiculttogetagriponthisether. Lightcanalsotravelthroughavacuumaswhensunlightcomesto theearththroughouterspace,soether,itseemed,wasimmuneto vacuumpumps. Einsteindecidedthatnoneofthismadesense.Iftheetherwas impossibletodetectormanipulate,onemightaswellsayitdidn't existatall.Iftheetherdoesn'texist,thenwhatdoesitmeanwhen ourexperimentsshowthatlighthasacertainspeed,3 10 8 meters persecond?Whatisthisspeedrelativeto?Couldwe,atleastin theory,getonthemotorcycleofEinstein'steenagedaydreams,and travelalongsideabeamoflight?Inthisframeofreference,the beam'sspeedwouldbezero,butallexperimentsseemedtoshow thatthespeedoflightalwayscameoutthesame,3 10 8 m/s. Einsteindecidedthatthespeedoflightwasdictatedbythelawsof physicstheonesconcerningelectromagneticinduction,soitmust bethesameinallframesofreference.Thisputbothlightand matteronthesamefooting:bothobeyedlawsofphysicsthatwere thesameinallframesofreference. theprincipleofrelativity Experimentsdon'tcomeoutdierentduetothestraight-line, constant-speedmotionoftheapparatus.Thisincludesbothlight andmatter. ThisisalmostthesameasGalileo'sprincipleofinertia,exceptthat weexplicitlystatethatitappliestolightaswell. Section1.1ThePrincipleofRelativity 15

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Thisishardtoswallow.Ifadogisrunningawayfrommeat5 m/srelativetothesidewalk,andIrunafteritat3m/s,thedog's velocityinmyframeofreferenceis2m/s.Accordingtoeverything wehavelearnedaboutmotion,thedogmusthavedierentspeeds inthetwoframes:5m/sinthesidewalk'sframeand2m/sinmine. How,then,canabeamoflighthavethesamespeedasseenby someonewhoischasingthebeam? Infactthestrangeconstancyofthespeedoflighthadshownup inthenow-famousMichelson-Morleyexperimentof1887.MichelsonandMorleysetupacleverapparatustomeasureanydierence inthespeedoflightbeamstravelingeast-westandnorth-south. Themotionoftheeartharoundthesunat110,000km/hourabout 0.01%ofthespeedoflightistoourwestduringtheday.MichelsonandMorleybelievedintheetherhypothesis,sotheyexpected thatthespeedoflightwouldbeaxedvaluerelativetotheether. Astheearthmovedthroughtheether,theythoughttheywould observeaneectonthevelocityoflightalonganeast-westline. Forinstance,iftheyreleasedabeamoflightinawestwarddirectionduringtheday,theyexpectedthatitwouldmoveawayfrom thematlessthanthenormalspeedbecausetheearthwaschasing itthroughtheether.Theyweresurprisedwhentheyfoundthatthe expected0.01%changeinthespeedoflightdidnotoccur. c / TheMichelson-Morleyexperiment,showninphotographs,and drawingsfromtheoriginal1887 paper.1.Asimplieddrawingoftheapparatus.Abeamof lightfromthesource,s,ispartiallyreectedandpartiallytransmittedbythehalf-silveredmirror h 1 .Thetwohalf-intensitypartsof thebeamarereectedbythemirrorsataandb,reunited,andobservedinthetelescope,t.Ifthe earth'ssurfacewassupposedto bemovingthroughtheether,then thetimestakenbythetwolight wavestopassthroughthemovingetherwouldbeunequal,and theresultingtimelagwouldbe detectablebyobservingtheinterferencebetweenthewaveswhen theywerereunited.2.Inthereal apparatus,thelightbeamswere reectedmultipletimes.Theeffectivelengthofeacharmwas increasedto11meters,which greatlyimproveditssensitivityto thesmallexpecteddifferencein thespeedoflight.3.Inan earlierversionoftheexperiment, theyhadrunintoproblemswith itsextremesensitivenesstovibration,whichwassogreatthat itwasimpossibletoseetheinterferencefringesexceptatbrief intervals...evenattwoo'clock inthemorning.Theytherefore mountedthewholethingona massivestoneoatinginapoolof mercury,whichalsomadeitpossibletorotateiteasily.4.Aphoto oftheapparatus.Notethatitis underground,inaroomwithsolid brickwalls. 16 Chapter1Relativity

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e / AlbertMichelson,in1887, theyearoftheMichelson-Morley experiment. f / GeorgeFitzGerald,18511901. g / HendrikLorentz,1853-1928. AlthoughtheMichelson-MorleyexperimentwasnearlytwodecadesinthepastbythetimeEinsteinpublishedhisrstpaperonrelativityin1905,it'sunclearhowmuchitinuencedEinstein.MichelsonandMorleythemselveswereuncertainaboutwhethertheresult wastobetrusted,orwhethersystematicandrandomerrorswere maskingarealeectfromtheether.Therewereavarietyofcompetingtheories,eachofwhichcouldclaimsomesupportfromtheshaky data.Somephysicistsbelievedthattheethercouldbedraggedalong bymattermovingthroughit,whichinspiredvariationsontheexperimentthatwereconductedonmountaintopsinthin-walledbuildings,d,orwithonearmoftheappartusoutintheopen,andthe othersurroundedbymassiveleadwalls.Inthestandardsanitized textbookversionofthehistoryofscience,everyscientistdoeshis experimentswithoutanypreconceivednotionsaboutthetruth,and anydisagreementisquicklysettledbyadenitiveexperiment.In reality,thisperiodofconfusionabouttheMichelson-Morleyexperimentlastedforfourdecades,andafewreputableskeptics,including Miller,continuedtobelievethatEinsteinwaswrong,andkepttryingdierentvariationsoftheexperimentaslateasthe1920's.Most oftheremainingdoubterswereconvincedbyanextremelyprecise versionoftheexperimentperformedbyJoosin1930,althoughyou canstillndkooksontheinternetwhoinsistthatMillerwasright, andthattherewasavastconspiracytocoveruphisresults. d / DaytonMillerthoughtthattheresultoftheMichelson-Morleyexperimentcouldbeexplainedbecausetheetherhadbeenpulledalongby thedirt,andthewallsofthelaboratory.Thismotivatedhimtocarryouta seriesofexperimentsatthetopofMountWilson,inabuildingwiththin walls. BeforeEinstein,somephysicistswhodidbelievethenegativeresultoftheMichelson-Morleyexperimentcameupwithexplanations thatpreservedtheether.Intheperiodfrom1889to1895,Hendrik LorentzandGeorgeFitzGeraldsuggestedthatthenegativeresult oftheMichelson-Morleyexperimentcouldbeexplainediftheearth, andeveryphysicalobjectonitssurface,wascontractedslightlyby thestrainoftheearth'smotionthroughtheether. 4 4 SeediscussionquestionFonpage26,andhomeworkproblem12 Section1.1ThePrincipleofRelativity 17

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1.2DistortionofTimeandSpace Time Considerthesituationshowningureh.Aboardarocketship wehaveatubewithmirrorsattheends.Ifweletoaashoflightat thebottomofthetube,itwillbereectedbackandforthbetween thetopandbottom.Itcanbeusedasaclock;bycountingthe numberoftimesthelightgoesbackandforthwegetanindication ofhowmuchtimehaspassed:up-downup-down,tick-tockticktock.Thismaynotseemverypractical,butarealatomicclock doesworkonessentiallythesameprinciple.Nowimaginethatthe rocketiscruisingatasignicantfractionofthespeedoflightrelative totheearth.Motionisrelative,soforapersoninsidetherocket, h/1,thereisnodetectablechangeinthebehavioroftheclock,just asapersononajetplanecantossaballupanddownwithout noticinganythingunusual.Buttoanobserverintheearth'sframe ofreference,thelightappearstotakeazigzagpaththroughspace, h/2,increasingthedistancethelighthastotravel. h / Alightbeambouncesbetween twomirrorsinaspaceship. Ifwedidn'tbelieveintheprincipleofrelativity,wecouldsay thatthelightjustgoesfasteraccordingtotheearthboundobserver. Indeed,thiswouldbecorrectifthespeedsweremuchlessthanthe speedoflight,andifthethingtravelingbackandforthwas,say, aping-pongball.Butaccordingtotheprincipleofrelativity,the speedoflightmustbethesameinbothframesofreference.Weare forcedtoconcludethattimeisdistorted,andthelight-clockappears torunmoreslowlythannormalasseenbytheearthboundobserver. Ingeneral,aclockappearstorunmostquicklyforobserverswho areinthesamestateofmotionastheclock,andrunsmoreslowly asperceivedbyobserverswhoaremovingrelativetotheclock. Wecaneasilycalculatethesizeofthistime-distortioneect.In theframeofreferenceshowningureh/1,movingwiththespace18 Chapter1Relativity

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i / Oneobserversaysthelight wentadistance cT ,whilethe othersaysitonlyhadtotravel ct ship,let t bethetimerequiredforthebeamoflighttomovefrom thebottomtothetop.Anobserverontheearth,whoseesthesituationshowningureh/2,disagrees,andsaysthismotiontooka longertime T abiggerletterforthebiggertime.Let v bethe velocityofthespaceshiprelativetotheearth.Inframe2,thelight beamtravelsalongthehypotenuseofarighttriangle,gurei,whose basehaslength base= vT Observersinthetwoframesofreferenceagreeontheverticaldistancetraveledbythebeam,i.e.,theheightofthetriangleperceived inframe2,andanobserverinframe1saysthatthisheightisthe distancecoveredbyalightbeamintime t ,sotheheightis height= ct where c isthespeedoflight.Thehypotenuseofthistriangleisthe distancethelighttravelsinframe2, hypotenuse= cT UsingthePythagoreantheorem,wecanrelatethesethreequantities, cT 2 = vT 2 + ct 2 andsolvingfor T ,wend T = t q 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( v=c 2 Theamountofdistortionisgivenbythefactor1 = q 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( v=c 2 andthisquantityappearssooftenthatwegiveitaspecialname, Greeklettergamma, = 1 q 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( v=c 2 self-checkA Whatis when v =0?Whatdoesthismean? Answer,p.133 Weareusedtothinkingoftimeasabsoluteanduniversal,soit isdisturbingtondthatitcanowatadierentrateforobservers indierentframesofreference.Butconsiderthebehaviorofthe factorshowningurej.Thegraphisextremelyatatlowspeeds, andevenat20%ofthespeedoflight,itisdiculttoseeanything happeningto .Ineverydaylife,weneverexperiencespeedsthat aremorethanatinyfractionofthespeedoflight,sothisstrange strangerelativisticeectinvolvingtimeisextremelysmall.This makessense:Newton'slawshavealreadybeenthoroughlytested byexperimentsatsuchspeeds,soanewtheorylikerelativitymust agreewiththeoldoneintheirrealmofcommonapplicability.This requirementofbackwards-compatibilityisknownasthecorrespondenceprinciple. Section1.2DistortionofTimeandSpace 19

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j / Thebehaviorofthe factor. Space Thespeedoflightissupposedtobethesameinallframesofreference,andaspeedisadistancedividedbyatime.Wecan'tchange timewithoutchangingdistance,sincethenthespeedcouldn'tcome outthesame.Iftimeisdistortedbyafactorof ,thenlengthsmust alsobedistortedaccordingtothesameratio.Anobjectinmotion appearslongesttosomeonewhoisatrestwithrespecttoit,andis shortenedalongthedirectionofmotionasseenbyotherobservers. Nosimultaneity Partoftheconceptofabsolutetimewastheassumptionthatit wasvalidtosaythingslike,IwonderwhatmyuncleinBeijingis doingrightnow."Inthenonrelativisticworld-view,clocksinLos AngelesandBeijingcouldbesynchronizedandstaysynchronized, sowecouldunambiguouslydenetheconceptofthingshappening simultaneouslyindierentplaces.Itiseasytondexamples,however,whereeventsthatseemtobesimultaneousinoneframeof referencearenotsimultaneousinanotherframe.Ingurek,aash oflightissetointhecenteroftherocket'scargohold.According toapassengerontherocket,thepartsofthelighttravelingforwardandbackwardhaveequaldistancestotraveltoreachthefront andbackwalls,sotheygettheresimultaneously.Butanoutside observerwhoseestherocketcruisingbyathighspeedwillseethe 20 Chapter1Relativity

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ashhitthebackwallrst,becausethewallisrushinguptomeet it,andtheforward-goingpartoftheashhitthefrontwalllater, becausethewallwasrunningawayfromit. k / Differentobserversdon'tagree thattheashesoflighthitthe frontandbackoftheshipsimultaneously. Weconcludethatsimultaneityisnotawell-denedconcept. Thisideamaybeeasiertoacceptifwecomparetimewithspace. EveninplainoldGalileanrelativity,pointsinspacehavenoidentityoftheirown:youmaythinkthattwoeventshappenedatthe samepointinspace,butanyoneelseinadierentlymovingframe ofreferencesaystheyhappenedatdierentpointsinspace.For instance,supposeyoutapyourknucklesonyourdeskrightnow, counttove,andthendoitagain.Inyourframeofreference,the tapshappenedatthesamelocationinspace,butaccordingtoan observeronMars,yourdeskwasonthesurfaceofaplanethurtling throughspaceathighspeed,andthesecondtapwashundredsof kilometersawayfromtherst. Relativitysaysthattimeisthesameway|bothsimultaneity andsimulplaceity"aremeaninglessconcepts.Onlywhentherelativevelocityoftwoframesissmallcomparedtothespeedoflight willobserversinthoseframesagreeonthesimultaneityofevents. l / Inthegarage'sframeofreference,1,thebusismoving,and cantinthegarage.Inthebus's frameofreference,thegarageis moving,andcan'tholdthebus. Thegarageparadox Oneofthemostfamousofalltheso-calledrelativityparadoxes hastodowithourincorrectfeelingthatsimultaneityiswelldened. Theideaisthatonecouldtakeaschoolbusanddriveitatrelativistic speedsintoagarageofordinarysize,inwhichitnormallywouldnot t.Becauseofthelengthcontraction,thebuswouldsupposedlyt Section1.2DistortionofTimeandSpace 21

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m / Decayofmuonscreated atrestwithrespecttothe observer. n / Decayofmuonsmoving ataspeedof0.995 c withrespect totheobserver. inthegarage.Theparadoxariseswhenweshutthedoorandthen quicklyslamonthebrakesofthebus.Anobserverinthegarage's frameofreferencewillclaimthatthebustinthegaragebecauseof itscontractedlength.Thedriver,however,willperceivethegarage asbeingcontractedandthusevenlessabletocontainthebus.The paradoxisresolvedwhenwerecognizethattheconceptofttingthe businthegarageallatonce"containsahiddenassumption,the assumptionthatitmakessensetoaskwhetherthefrontandbackof thebuscansimultaneouslybeinthegarage.Observersindierent framesofreferencemovingathighrelativespeedsdonotnecessarily agreeonwhetherthingshappensimultaneously.Thepersoninthe garage'sframecanshutthedoorataninstantheperceivestobe simultaneouswiththefrontbumper'sarrivalatthebackwallofthe garage,butthedriverwouldnotagreeaboutthesimultaneityof thesetwoevents,andwouldperceivethedoorashavingshutlong aftersheplowedthroughthebackwall. Applications Nothingcangofasterthanthespeedoflight. Whathappensifwewanttosendarocketshipoat,say,twice thespeedoflight, v =2 c ?Then willbe1 = p )]TJ/F15 10.9091 Tf 8.484 0 Td [(3.Butyour mathteacherhasalwayscautionedyouabouttheseverepenalties fortakingthesquarerootofanegativenumber.Theresultwould bephysicallymeaningless,soweconcludethatnoobjectcantravel fasterthanthespeedoflight.Eventravelexactlyatthespeedof lightappearstoberuledoutformaterialobjects,since would thenbeinnite. Einsteinhadthereforefoundasolutiontohisoriginalparadox aboutridingonamotorcyclealongsideabeamoflight.Theparadox isresolvedbecauseitisimpossibleforthemotorcycletotravelat thespeedoflight. Mostpeople,whentoldthatnothingcangofasterthanthespeed oflight,immediatelybegintoimaginemethodsofviolatingtherule. Forinstance,itwouldseemthatbyapplyingaconstantforcetoan objectforalongtime,wecouldgiveitaconstantacceleration, whichwouldeventuallymakeitgofasterthanthespeedoflight. We'lltakeuptheseissuesinsection1.3. Cosmic-raymuons Aclassicexperimenttodemonstratetimedistortionusesobservationsofcosmicrays.Cosmicraysareprotonsandotheratomicnucleifromouterspace.Whenacosmicrayhappenstocometheway ofourplanet,therstearth-matteritencountersisanairmolecule intheupperatmosphere.Thiscollisionthencreatesashowerof particlesthatcascadedownwardandcanoftenbedetectedatthe earth'ssurface.OneofthemoreexoticparticlescreatedinthesecosmicrayshowersisthemuonnamedaftertheGreeklettermu, 22 Chapter1Relativity

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Thereasonmuonsarenotanormalpartofourenvironmentisthat amuonisradioactive,lastingonly2.2microsecondsontheaverage beforechangingitselfintoanelectronandtwoneutrinos.Amuon canthereforebeusedasasortofclock,albeitaself-destructingand somewhatrandomone!Figuresmandnshowtheaveragerateat whichasampleofmuonsdecays,rstformuonscreatedatrestand thenforhigh-velocitymuonscreatedincosmic-rayshowers.The secondgraphisfoundexperimentallytobestretchedoutbyafactorofaboutten,whichmatcheswellwiththepredictionofrelativity theory: =1 = p 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( v=c 2 =1 = p 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(.995 2 10 Sinceamuontakesmanymicrosecondstopassthroughtheatmosphere,theresultisamarkedincreaseinthenumberofmuonsthat reachthesurface. Timedilationforobjectslargerthantheatomicscale Ourworldisfortunatelynotfullofhuman-scaleobjectsmovingatsignicantspeedscomparedtothespeedoflight.Forthis reason,ittookover80yearsafterEinstein'stheorywaspublished beforeanyonecouldcomeupwithaconclusiveexampleofdrastic timedilationthatwasn'tconnedtocosmicraysorparticleaccelerators.Recently,however,astronomershavefounddenitiveproof thatentirestarsundergotimedilation.Theuniverseisexpanding intheaftermathoftheBigBang,soingeneraleverythinginthe universeisgettingfartherawayfromeverythingelse.Oneneedonly ndanastronomicalprocessthattakesastandardamountoftime, andthenobservehowlongitappearstotakewhenitoccursina partoftheuniversethatisrecedingfromusrapidly.AtypeofexplodingstarcalledatypeIasupernovallsthebill,andtechnology isnowsucientlyadvancedtoallowthemtobedetectedacrossvast distances.Figureoshowsconvincingevidencefortimedilationin thebrighteninganddimmingoftwodistantsupernovae. Thetwinparadox Anaturalsourceofconfusioninunderstandingthetime-dilation eectissummedupintheso-calledtwinparadox,whichisnotreally aparadox.Supposetherearetwoteenagedtwins,andonestaysat homeonearthwhiletheothergoesonaroundtripinaspaceshipat Section1.2DistortionofTimeandSpace 23

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o / Lightcurvesofsupernovae, showingatime-dilationeffectfor supernovaethatareinmotionrelativetous. relativisticspeedsi.e.,speedscomparabletothespeedoflight,for whichtheeectspredictedbythetheoryofrelativityareimportant. Whenthetravelingtwingetshome,shehasagedonlyafewyears, whilehersisterisnowoldandgray.RobertHeinleinevenwrote asciencectionnovelonthistopic,althoughitisnotoneofhis betterstories. Theparadox"arisesfromanincorrectapplicationoftheprincipleofrelativitytoadescriptionofthestoryfromthetraveling twin'spointofview.Fromherpointofview,theargumentgoes, herhomebodysisteristheonewhotravelsbackwardonthereceding earth,andthenreturnsastheearthapproachesthespaceshipagain, whileintheframeofreferencexedtothespaceship,theastronaut twinisnotmovingatall.Itwouldthenseemthatthetwinonearth istheonewhosebiologicalclockshouldtickmoreslowly,notthe oneonthespaceship.Theawinthereasoningisthattheprinciple ofrelativityonlyappliestoframesthatareinmotionatconstant velocityrelativetooneanother,i.e.,inertialframesofreference. Theastronauttwin'sframeofreference,however,isnoninertial,becauseherspaceshipmustacceleratewhenitleaves,deceleratewhen itreachesitsdestination,andthenrepeatthewholeprocessagain onthewayhome.Theirexperiencesarenotequivalent,because theastronauttwinfeelsaccelerationsanddecelerations.Acorrect treatmentrequiressomemathematicalcomplicationtodealwiththe changingvelocityoftheastronauttwin,buttheresultisindeedthat it'sthetravelingtwinwhoisyoungerwhentheyarereunited. 5 5 Readersfrequentlywonderwhytheeectsofthedecelerationsdon'tcancel outtheeectsoftheaccelerations.Thereareacoupleofsubtleissueshere.First, there'snoclearcutwaytodecidewhetherthetimedistortionhappensduring theaccelerationsanddecelerations,orduringthelongperiodsofconstant-speed cruisinginbetween.Thisisbecausesimultaneityisn'twelldened,sothere'sno well-denedanswerifEarth-boundEmmaasks,Ismysister'stimedistorted rightnow ?"DuringthelongperiodwhenspacefaringSarahiscruisingaway fromEarthatconstantspeed,Emmamayobservethathersister'svoiceon 24 Chapter1Relativity

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Thetwinparadox"reallyisn'taparadoxatall.Itmayevenbe apartofyourordinarylife.Theeectwasrstveriedexperimentallybysynchronizingtwoatomicclocksinthesameroom,andthen sendingoneforaroundtriponapassengerjet.Theyboughtthe clockitsownticketandputitinitsownseat.Theclocksdisagreed whenthetravelingonegotback,andthediscrepancywasexactly theamountpredictedbyrelativity.Theeectsarestrongenough tobeimportantformakingtheglobalpositioningsystemGPS workcorrectly.Ifyou'veevertakenaGPSreceiverwithyouona hikingtrip,thenyou'veusedadevicethathasthetwinparadox" programmedintoitscalculations.YourhandheldGPSboxgetssignalsfromasatellite,andthesatelliteismovingfastenoughthatits timedilationisanimportanteect.Sofarnoastronautshavegone fastenoughtomaketimedilationadramaticeectintermsofthe humanlifetime.TheeectontheApolloastronauts,forinstance, wasonlyafractionofasecond,sincetheirspeedswerestillfairly smallcomparedtothespeedoflight.AsfarasIknow,noneofthe astronautshadtwinsiblingsbackonearth! Anexampleoflengthcontraction Figurepshowsanartist'srenderingofthelengthcontractionfor thecollisionoftwogoldnucleiatrelativisticspeedsintheRHICacceleratorinLongIsland,NewYork,whichwentonlinein2000.The goldnucleiwouldappearnearlysphericalorjustslightlylengthened likeanAmericanfootballinframesmovingalongwiththem,butin thelaboratory'sframe,theybothappeardrasticallyforeshortened astheyapproachthepointofcollision.Thelaterpicturesshowthe nucleimergingtoformahotsoup,inwhichexperimentershopeto observeanewformofmatter. theradiosoundsabnormallyslow,andconcludethatthetimedistortionisin progress.Sarah,however,saysthatsheherselfisnormal,andthatEmmais theonewhosoundsslow.Eachtwinexplainstheother'sperceptionsasbeing duetotheincreasingseparationbetweenthem,whichcausestheradiosignals tobedelayedmoreandmore.Theotherthingtounderstandisthat,evenif wedodecidetoattributethetimedistortiontotheperiodsofaccelerationand deceleration,weshouldexpectthetime-distortingeectsofaccelerationsand decelerationstoreinforce,notcancel.Thisisbecausethereisnocleardistinction betweenaccelerationanddecelerationthatcanbeagreeduponbyobserversin dierentinertialframes.ThisisafactaboutplainoldGalileanrelativity,not Einstein'srelativity.Supposeacarisinitiallydrivingwestwardat100km/hr relativetotheasphalt,thenslamsonthebrakesandstopscompletely.Inthe asphalt'sframeofreference,thisisadeceleration.Butfromthepointofview ofanobserverwhoiswatchingtheearthrotatetotheeast,theasphaltmaybe movingeastwardataspeedof1000km/hr.Thisobserverseesthebrakescause an acceleration ,from900km/hrto1000km/hr:theasphalthaspulledthecar forward,forcingcartomatchitsvelocity. Section1.2DistortionofTimeandSpace 25

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DiscussionquestionB p / Collidingnucleishowrelativisticlengthcontraction. DiscussionQuestions A Apersoninaspaceshipmovingat99.99999999%ofthespeed oflightrelativetoEarthshinesaashlightforwardthroughdustyair,so thebeamisvisible.Whatdoesshesee?Whatwoulditlookliketoan observeronEarth? B Aquestionthatstudentsoftenstrugglewithiswhethertimeand spacecanreallybedistorted,orwhetheritjustseemsthatway.Compare withopticalillusionsormagictricks.Howcouldyouverify,forinstance, thatthelinesinthegureareactuallyparallel?Arerelativisticeffectsthe sameornot? C Onaspaceshipmovingatrelativisticspeeds,wouldalectureseem evenlongerandmoreboringthannormal? D Mechanicalclockscanbeaffectedbymotion.Forexample,itwas asignicanttechnologicalachievementtobuildaclockthatcouldsail aboardashipandstillkeepaccuratetime,allowinglongitudetobedetermined.Howisthissimilartoordifferentfromrelativistictimedilation? E WhatwouldtheshapesofthetwonucleiintheRHICexperiment lookliketoamicroscopicobserverridingontheleft-handnucleus?To anobserverridingontheright-handone?Cantheyagreeonwhatis happening?Ifnot,whynotafterall,shouldn'ttheyseethesamething iftheybothcomparethetwonucleiside-by-sideatthesameinstantin time? F Ifyoustickapieceoffoamrubberoutthewindowofyourcarwhile drivingdownthefreeway,thewindmaycompressitalittle.Doesitmake sensetointerprettherelativisticlengthcontractionasatypeofstrain thatpushesanobject'satomstogetherlikethis?Howdoesthisrelateto discussionquestionE? 26 Chapter1Relativity

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1.3Dynamics Sofarwehavesaidnothingabouthowtopredictmotioninrelativity.DoNewton'slawsstillwork?Doconservationlawsstillapply? Theanswerisyes,butmanyofthedenitionsneedtobemodied, andcertainentirelynewphenomenaoccur,suchastheconversion ofmasstoenergyandenergytomass,asdescribedbythefamous equation E = mc 2 Combinationofvelocities Theimpossibilityofmotionfasterthanlightisaradicaldierencebetweenrelativisticandnonrelativisticphysics,andwecanget atmostoftheissuesinthissectionbyconsideringtheawsinvariousplansforgoingfasterthanlight.Thesimplestargumentofthis kindisasfollows.SupposeJanettakesatripinaspaceship,and acceleratesuntilsheismovingat0.8 c %ofthespeedoflight relativetotheearth.Shethenlaunchesaspaceprobeintheforward directionataspeedrelativetohershipof0.4 c .Isn'ttheprobethen movingatavelocityof1.2timesthespeedoflightrelativetothe earth? TheproblemwiththislineofreasoningisthatalthoughJanet saystheprobeismovingat0.4 c relativetoher,earthboundobservers disagreewithherperceptionoftimeandspace.Velocitiestherefore don'taddthesamewaytheydoinGalileanrelativity.Supposewe expressallvelocitiesasfractionsofthespeedoflight.TheGalilean additionofvelocitiescanbesummarizedinthisadditiontable: q / Galileanadditionofvelocities. Thederivationofthecorrectrelativisticresultrequiressometedious algebra,whichyoucanndinmybook SimpleNature ifyou're curious.I'lljuststatethenumericalresultshere: Janet'sprobe,forexample,ismovingnotat1.2 c butat0.91 c whichisadrasticallydierentresult.Thedierencebetweenthe twotablesismostevidentaroundtheedges,wherealltheresults areequaltothespeedoflight.Thisisrequiredbytheprincipleof relativity.Forexample,ifJanetsendsoutabeamoflightinstead Section1.3Dynamics 27

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r / Relativisticadditionofvelocities.Thegreenovalnearthecenterofthetabledescribesvelocitiesthatarerelativelysmallcomparedtothespeedoflight,and theresultsareapproximatelythe sameastheGalileanones.The edgesofthetable,highlightedin blue,showthateveryoneagrees onthespeedoflight. ofaprobe,bothsheandtheearthboundobserversmustagreethat itmovesat1.00timesthespeedoflight,not0.8+1=1.8.On theotherhand,thecorrespondenceprinciplerequiresthattherelativisticresultshouldcorrespondtoordinaryadditionforlowenough velocities,andyoucanseethatthetablesarenearlyidenticalinthe center. Momentum Here'sanotherawedschemefortravelingfasterthanthespeed oflight.Thebasicideacanbedemonstratedbydroppingapingpongballandabaseballstackedontopofeachotherlikeasnowman. Theyseparateslightlyinmid-air,andthebaseballthereforehastime tohittheoorandreboundbeforeitcollideswiththeping-pong ball,whichisstillonthewaydown.Theresultisasurpriseifyou haven'tseenitbefore:theping-pongballiesoathighspeedand hitstheceiling!Asimilarfactisknowntopeoplewhoinvestigate thescenesofaccidentsinvolvingpedestrians.Ifacarmovingat 90kilometersperhourhitsapedestrian,thepedestrianiesoat nearlydoublethatspeed,180kilometersperhour.Nowsuppose thecarwasmovingat90percentofthespeedoflight.Wouldthe pedestrianyoat180%of c ? Toseewhynot,wehavetobackupalittleandthinkabout wherethisspeed-doublingresultcomesfrom.Foranycollision,there isaspecialframeofreference,thecenter-of-massframe,inwhich thetwocollidingobjectsapproacheachother,collide,andrebound withtheirvelocitiesreversed.Inthecenter-of-massframe,thetotal momentumoftheobjectsiszerobothbeforeandafterthecollision. Figures/1showssuchaframeofreferenceforobjectsofvery unequalmass.Beforethecollision,thelargeballismovingrelatively slowlytowardthetopofthepage,butbecauseofitsgreatermass, itsmomentumcancelsthemomentumofthesmallerball,whichis movingrapidlyintheoppositedirection.Thetotalmomentumis zero.Afterthecollision,thetwoballsjustreversetheirdirectionsof motion.Weknowthatthisistherightresultfortheoutcomeofthe collisionbecauseitconservesbothmomentumandkineticenergy, 28 Chapter1Relativity

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s / Anunequalcollision,viewedinthecenter-of-massframe,1,and intheframewherethesmallballisinitiallyatrest,2.Themotionis shownasitwouldappearonthelmofanold-fashionedmoviecamera, withanequalamountoftimeseparatingeachframefromthenext.Film 1wasmadebyacamerathattrackedthecenterofmass,lm2byone thatwasinitiallytrackingthesmallball,andkeptonmovingatthesame speedafterthecollision. andeverythingnotforbiddenismandatory,i.e.,inanyexperiment, thereisonlyonepossibleoutcome,whichistheonethatobeysall theconservationlaws. self-checkB Howdoweknowthatmomentumandkineticenergyareconservedin gures/1? Answer,p.133 Let'smakeupsomenumbersasanexample.Saythesmallball hasamassof1kg,thebigone8kg.Inframe1,let'smakethe velocitiesasfollows: beforethecollision afterthecollision -0.8 0.8 0.1 -0.1 Figures/2showsthesamecollisioninaframeofreferencewhere thesmallballwasinitiallyatrest.Tondallthevelocitiesinthis frame,wejustadd0.8toalltheonesintheprevioustable. beforethecollision afterthecollision 0 1.6 0.9 0.7 Inthisframe,asexpected,thesmallballiesowithavelocity, Section1.3Dynamics 29

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1.6,thatisalmosttwicetheinitialvelocityofthebigball,0.9. Ifallthosevelocitieswereinmeterspersecond,thenthat'sexactlywhathappened.Butwhatifallthesevelocitieswereinunits ofthespeedoflight?Nowit'snolongeragoodapproximation justtoaddvelocities.Weneedtocombinethemaccordingtothe relativisticrules.Forinstance,thetableonpage28tellsusthat combiningavelocityof0.8timesthespeedoflightwithanother velocityof0.8resultsin0.98,not1.6.Theresultsareverydierent: beforethecollision afterthecollision 0 0.98 0.83 0.76 t / An8-kgballmovingat83%of thespeedoflighthitsa1-kgball. Theballsappearforeshortened duetotherelativisticdistortionof space. Wecaninterpretthisasfollows.Figures/1isoneinwhichthe bigballismovingfairlyslowly.Thisisverynearlythewaythe scenewouldbeseenbyanantstandingonthebigball.According toanobserverinframet,however,bothballsaremovingatnearly thespeedoflightafterthecollision.Becauseofthis,theballs appearforeshortened,butthedistancebetweenthetwoballsisalso shortened.Tothisobserver,itseemsthatthesmallballisn'tpulling awayfromthebigballveryfast. Nowhere'swhat'sinterestingaboutallthis.Theoutcomeshown ingures/2wassupposedtobetheonlyonepossible,theonly onethatsatisedbothconservationofenergyandconservationof momentum.Sohowcanthe dierent resultshowninguretbe possible?Theansweristhatrelativistically,momentummustnot equal mv .Theold,familiardenitionisonlyanapproximation that'svalidatlowspeeds.Ifweobservethebehaviorofthesmall ballinguret,itlooksasthoughitsomehowhadsomeextrainertia. It'sasthoughafootballplayertriedtoknockanotherplayerdown withoutrealizingthattheotherguyhadathree-hundred-poundbag fullofleadshothiddenunderhisuniform|hejustdoesn'tseem toreacttothecollisionasmuchasheshould.Thisextrainertiais describedbyredeningmomentumas p = mv Atverylowvelocities, iscloseto1,andtheresultisverynearly mv ,asdemandedbythecorrespondenceprinciple.Butatvery 30 Chapter1Relativity

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highvelocities, getsverybig|thesmallballingurethasa of5.0,andthereforehasvetimesmoreinertiathanwewould expectnonrelativistically. Thisalsoexplainstheanswertoanotherparadoxoftenposed bybeginnersatrelativity.Supposeyoukeeponapplyingasteady forcetoanobjectthat'salreadymovingat0.9999 c .Whydoesn't itjustkeeponspeedinguppast c ?Theansweristhatforceisthe rateofchangeofmomentum.At0.9999 c ,anobjectalreadyhasa of71,andthereforehasalreadysuckedup71timesthemomentum you'dexpectatthatspeed.Asitsvelocitygetscloserandcloserto c ,its approachesinnity.Tomoveat c ,itwouldneedaninnite momentum,whichcouldonlybecausedbyaninniteforce. Equivalenceofmassandenergy Nowwe'rereadytoseewhymassandenergymustbeequivalent asclaimedinthefamous E = mc 2 .Sofarwe'veonlyconsidered collisionsinwhichnoneofthekineticenergyisconvertedintoany otherformofenergy,suchasheatorsound.Let'sconsiderwhat happensifablobofputtymovingatvelocity v hitsanotherblob thatisinitiallyatrest,stickingtoit.Thenonrelativisticresultis thattoobeyconservationofmomentumthetwoblobsmustyo togetherat v= 2.Halfoftheinitialkineticenergyhasbeenconverted toheat. 6 Relativistically,however,aninterestingthinghappens.Ahot objecthasmoremomentumthanacoldobject!Thisisbecause therelativisticallycorrectexpressionformomentumis mv ,and themorerapidlymovingatomsinthehotobjecthavehighervalues of .Inourcollision,thenalcombinedblobmustthereforebe movingalittlemoreslowlythantheexpected v= 2,sinceotherwise thenalmomentumwouldhavebeenalittlegreaterthantheinitial momentum.Toanobserverwhobelievesinconservationofmomentumandknowsonlyabouttheoverallmotionoftheobjectsandnot abouttheirheatcontent,thelowvelocityafterthecollisionwould seemtobetheresultofamagicalchangeinthemass,asifthemass oftwocombined,hotblobsofputtywasmorethanthesumoftheir individualmasses. Nowweknowthatthemassesofalltheatomsintheblobsmust bethesameastheyalwayswere.Thechangeisduetothechangein withheating,nottoachangeinmass.Theheatenergy,however, seemstobeactingasifitwasequivalenttosomeextramass. Butthiswholeargumentwasbasedonthefactthatheatisa formofkineticenergyattheatomiclevel.Would E = mc 2 applyto 6 Adouble-massobjectmovingathalfthespeeddoesnothavethesame kineticenergy.Kineticenergydependsonthesquareofthevelocity,socutting thevelocityinhalfreducestheenergybyafactorof1/4,which,multipliedby thedoubledmass,makes1/2theoriginalenergy. Section1.3Dynamics 31

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u / ANewYorkTimesheadlinefromNovember10,1919, describingtheobservations discussedinexample1. otherformsofenergyaswell?Supposearocketshipcontainssome electricalenergystoredinabattery.Ifwebelievedthat E = mc 2 appliedtoformsofkineticenergybutnottoelectricalenergy,then wewouldhavetobelievethatthepilotoftherocketcouldslow theshipdownbyusingthebatterytorunaheater!Thiswould notonlybestrange,butitwouldviolatetheprincipleofrelativity, becausetheresultoftheexperimentwouldbedierentdepending onwhethertheshipwasatrestornot.Theonlylogicalconclusionis thatallformsofenergyareequivalenttomass.Runningtheheater thenhasnoeectonthemotionoftheship,becausethetotal energyintheshipwasunchanged;oneformofenergyelectrical wassimplyconvertedtoanotherheat. Theequation E = mc 2 tellsushowmuchenergyisequivalent tohowmuchmass:theconversionfactoristhesquareofthespeed oflight, c .Since c abignumber,yougetareallyreallybignumber whenyoumultiplyitbyitselftoget c 2 .Thismeansthatevenasmall amountofmassisequivalenttoaverylargeamountofenergy. v / example1 Gravitybendinglightexample1 Gravityisauniversalattractionbetweenthingsthathavemass, andsincetheenergyinabeamoflightisequivalenttoasome verysmallamountofmass,weexpectthatlightwillbeaffected bygravity,althoughtheeffectshouldbeverysmall.Therstimportantexperimentalconrmationofrelativitycamein1919when starsnexttothesunduringasolareclipsewereobservedtohave 32 Chapter1Relativity

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shiftedalittlefromtheirordinaryposition.Iftherewasnoeclipse, theglareofthesunwouldpreventthestarsfrombeingobserved. Starlighthadbeendeectedbythesun'sgravity.Figurevisa photographicnegative,sothecirclethatappearsbrightisactuallythedarkfaceofthemoon,andthedarkareaisreallythe brightcoronaofthesun.Thestars,markedbylinesaboveand belowthen,appearedatpositionsslightlydifferentthantheirnormalones. Blackholesexample2 Astarwithsufcientlystronggravitycanpreventlightfromleaving.Quiteafewblackholeshavebeendetectedviatheirgravitationalforcesonneighboringstarsorcloudsofgasanddust. You'velearnedaboutconservationofmassandconservationof energy,butnowweseethatthey'renotevenseparateconservation laws.Asaconsequenceofthetheoryofrelativity,massandenergyareequivalent,andarenotseparatelyconserved|onecan beconvertedintotheother.Imaginethatamagicianwaveshis wand,andchangesabowlofdirtintoabowloflettuce.You'dbe impressed,becauseyouwereexpectingthatbothdirtandlettuce wouldbeconservedquantities.Neitheronecanbemadetovanish, ortoappearoutofthinair.However,thereareprocessesthatcan changeoneintotheother.Afarmerchangesdirtintolettuce,and acompostheapchangeslettuceintodirt.Atthemostfundamentallevel,lettuceanddirtaren'treallydierentthingsatall;they're justcollectionsofthesamekindsofatoms|carbon,hydrogen,and soon.Becausemassandenergyareliketwodierentsidesofthe samecoin,wemayspeakofmass-energy,asingleconservedquantity, foundbyaddingupallthemassandenergy,withtheappropriate conversionfactor: E + mc 2 Arustingnailexample3 Anironnailisleftinacupofwateruntilitturnsentirelytorust. Theenergyreleasedisabout0.5MJ.Intheory,wouldasufcientlyprecisescaleregisterachangeinmass?Ifso,howmuch? Theenergywillappearasheat,whichwillbelosttotheenvironment.Thetotalmass-energyofthecup,water,andironwill indeedbelessenedby0.5MJ.Ifithadbeenperfectlyinsulated, therewouldhavebeennochange,sincetheheatenergywould havebeentrappedinthecup.Thespeedoflightis c =3 10 8 meterspersecond,soconvertingtomassunits,wehave m = E c 2 = 0.5 10 6 J )]TJ/F39 10.9091 Tf 5 -8.836 Td [(3 10 8 m = s 2 =6 10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(12 kilograms. Thechangeinmassistoosmalltomeasurewithanypractical Section1.3Dynamics 33

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technique.Thisisbecausethesquareofthespeedoflightis suchalargenumber. Electron-positronannihilationexample4 Naturalradioactivityintheearthproducespositrons,whichare likeelectronsbuthavetheoppositecharge.Aformofantimatter,positronsannihilatewithelectronstoproducegammarays,a formofhigh-frequencylight.Suchaprocesswouldhavebeen consideredimpossiblebeforeEinstein,becauseconservationof massandenergywerebelievedtobeseparateprinciples,and thisprocesseliminates100%oftheoriginalmass.Theamount ofenergyproducedbyannihilating1kgofmatterwith1kgof antimatteris E = mc 2 =kg 3.0 10 8 m = s 2 =2 10 17 J, whichisonthesameorderofmagnitudeasaday'senergyconsumptionfortheentireworld'spopulation! PositronannihilationformsthebasisforthemedicalimagingtechniquecalledaPETpositronemissiontomographyscan,inwhich apositron-emittingchemicalisinjectedintothepatientandmapped bytheemissionofgammaraysfromthepartsofthebodywhere itaccumulates. Onecommonlyhearssomemisinterpretationsof E = mc 2 ,one beingthattheequationtellsushowmuchkineticenergyanobject wouldhaveifitwasmovingatthespeedoflight.Thiswouldn't makemuchsense,bothbecausetheequationforkineticenergyhas 1 = 2init, KE = = 2 mv 2 ,andbecauseamaterialobjectcan'tbe madetomoveatthespeedoflight.However,thisnaturallyleads tothequestionofjusthowmuchmass-energyamovingobjecthas. Weknowthatwhentheobjectisatrest,ithasnokineticenergy,so itsmass-energyissimplyequaltotheenergy-equivalentofitsmass, mc 2 E = mc 2 when v =0, wherethesymbol E standsformass-energy.Youcanwritethis symbolyourselfbywritinganE,andthenaddinganextralineto it.Havefun!Thepointofusingthenewsymbolissimplyto remindourselvesthatwe'retalkingaboutrelativity,soanobjectat resthas E = mc 2 ,not E =0aswe'dassumeinclassicalphysics. Supposewestartacceleratingtheobjectwithaconstantforce. Aconstantforcemeansaconstantrateoftransferofmomentum, but p = mv approachesinnityas v approaches c ,sotheobject willonlygetcloserandclosertothespeedoflight,butneverreach it.Nowwhatabouttheworkbeingdonebytheforce?Theforce 34 Chapter1Relativity

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keepsdoingworkanddoingwork,whichmeansthatwekeepon usingupenergy.Mass-energyisconserved,sotheenergybeing expendedmustequaltheincreaseintheobject'smass-energy.We cancontinuethisprocessforaslongaswelike,andtheamountof mass-energywillincreasewithoutlimit.Wethereforeconcludethat anobject'smass-energyapproachesinnityasitsspeedapproaches thespeedoflight, E !1 when v c Nowthatwehavesomeideawhattoexpect,whatistheactual equationforthemass-energy?Asprovedinmybook SimpleNature itis E = mc 2 self-checkC Verifythatthisequationhasthetwopropertieswewanted. Answer,p.133 KEcomparedto mc 2 atlowspeedsexample5 Anobjectismovingatordinarynonrelativisticspeeds.Compare itskineticenergytotheenergy mc 2 ithaspurelybecauseofits mass. Thespeedoflightisaverybignumber,so mc 2 isahugenumberofjoules.Theobjecthasagiganticamountofenergybecauseofitsmass,andonlyarelativelysmallamountofadditional kineticenergybecauseofitsmotion. Anotherwayofseeingthisisthatatlowspeeds, isonlyatiny bitgreaterthan1,so E isonlyatinybitgreaterthan mc 2 Thecorrespondenceprincipleformass-energyexample6 Showthattheequation E = m c 2 obeysthecorrespondence principle. Asweaccelerateanobjectfromrest,itsmass-energybecomes greaterthanitsrestingvalue.Classically,weinterpretthisexcess mass-energyastheobject'skineticenergy, KE = E v )]TJ/F32 10.9091 Tf 10.909 0 Td [(E v =0 = m c 2 )]TJ/F116 10.9091 Tf 10.909 0 Td [(mc 2 = m )]TJ/F39 10.9091 Tf 10.909 0 Td [(1 c 2 Expressing as )]TJ/F39 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F116 10.9091 Tf 10.909 0 Td [(v 2 = c 2 )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 = 2 andmakinguseoftheapproximation+ p 1+ p forsmall ,wehave 1+ v 2 = 2 c 2 so KE m + v 2 2 c 2 )]TJ/F39 10.9091 Tf 10.91 0 Td [(1 c 2 = 1 2 mv 2 Section1.3Dynamics 35

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whichistheclassicalexpression.Asdemandedbythecorrespondenceprinciple,relativityagreeswithclassicalphysicsat speedsthataresmallcomparedtothespeedoflight. 36 Chapter1Relativity

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Summary Notation ..........anabbreviationfor1 = p 1 )]TJ/F20 10.9091 Tf 10.909 0 Td [(v 2 =c 2 E ..........mass-energy Summary Theprincipleofrelativitystatesthatexperimentsdon'tcome outdierentduetothestraight-line,constant-speedmotionofthe apparatus.UnlikehispredecessorsgoingbacktoGalileoandNewton,Einsteinclaimedthatthisprincipleappliednotjusttomatter buttolightaswell.Thisimpliesthatthespeedoflightisthesame, regardlessofthemotionoftheapparatususedtomeasureit.This seemsimpossible,becauseweexpectvelocitiestoaddinrelative motion;thestrangeconstancyofthespeedoflightwas,however, observedexperimentallyinthe1887Michelson-Morleyexperiment. Basedonlyonthisprincipleofrelativity,Einsteinshowedthat timeandspaceasseenbyoneobserverwouldbedistortedcompared toanotherobserver'sperceptionsiftheyweremovingrelativeto eachother.Thisdistortionisquantiedbythefactor = 1 q 1 )]TJ/F21 7.9701 Tf 12.105 4.295 Td [(v 2 c 2 where v istherelativevelocityofthetwoobservers.Aclockappearstorunfastesttoanobserverwhoisnotinmotionrelativeto it,andappearstoruntooslowlybyafactorof toanobserverwho hasavelocity v relativetotheclock.Similarly,ameter-stickappearslongesttoanobserverwhoseesitatrest,andappearsshorter tootherobservers.Timeandspacearerelative,notabsolute.In particular,thereisnowell-denedconceptofsimultaneity. Allofthesestrangeeects,however,areverysmallwhentherelativevelocitiesaresmallrelativetothespeedoflight.Thismakes sense,becauseNewton'slawshavealreadybeenthoroughlytested byexperimentsatsuchspeeds,soanewtheorylikerelativitymust agreewiththeoldoneintheirrealmofcommonapplicability.This requirementofbackwards-compatibilityisknownasthecorrespondenceprinciple. Relativityhasimplicationsnotjustfortimeandspacebutalso fortheobjectsthatinhabittimeandspace.Thecorrectrelativistic equationformomentumis p = mv whichissimilartotheclassical p = mv atlowvelocities,where 1,butdivergesfromitmoreandmoreatvelocitiesthatapproachthespeedoflight.Since becomesinniteat v = c ,an inniteforcewouldberequiredinordertogiveamaterialobject Summary 37

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enoughmomentumtomoveatthespeedoflight.Inotherwords, materialobjectscanonlymoveatspeedslowerthanthespeedof light.Relativistically,massandenergyarenotseparatelyconserved. Massandenergyaretwoaspectsofthesamephenomenon,known asmass-energy,andtheycanbeconvertedtooneanotheraccording totheequation E = mc 2 Themass-energyofamovingobjectis E = mc 2 .Whenanobjectis atrest, =1,andthemass-energyissimplytheenergy-equivalent ofitsmass, mc 2 .Whenanobjectisinmotion,theexcessmassenergy,inadditiontothe mc 2 ,canbeinterpretedasitskinetic energy. ExploringFurther RelativitySimplyExplained ,MartinGardner.Abeatifully clear,nonmathematicalintroductiontothesubject,withentertainingillustrations.Apostscript,writtenin1996,followsuponrecent developmentsinsomeofthemorespeculativeideasfromthe1967 edition. WasEinsteinRight?|PuttingGeneralRelativitytothe Test ,CliordM.Will.Thisbookmakesitclearthatgeneral relativityisneitherafantasynorholyscripture,butascientic theorylikeanyother. 38 Chapter1Relativity

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Astronautsinthreedierentspaceshipsarecommunicating witheachother.ThoseaboardshipsAandBagreeontherateat whichtimeispassing,buttheydisagreewiththeonesonshipC. aDescribethemotionoftheothertwoshipsaccordingtoAlice, whoisaboardshipA. bGivethedescriptionaccordingtoBetty,whoseframeofreference isshipB. cDothesameforCathy,aboardshipC. 2 aFigureionpage19isbasedonalightclockmovingata certainspeed, v .Bymeasuringwitharuleronthegure,determine v=c bBysimilarmeasurements,ndthetimecontractionfactor whichequals T=t cLocateyournumbersfrompartsaandbasapointonthegraph ingurejonpage20,andcheckthatitactuallyliesonthecurve. Makeasketchshowingwherethepointisonthecurve. p 3 Thisproblemisacontinuationofproblem2.Nowimagine thatthespaceshipspeedsuptotwicethevelocity.Drawanew triangleonthesamescale,usingarulertomakethelengthsofthe sidesaccurate.Repeatpartsbandcforthisnewdiagram. p 4 Whathappensintheequationfor whenyouputinanegative numberfor v ?Explainwhatthismeansphysically,andwhyitmakes sense. 5 aBymeasuringwitharuleronthegraphingureoonpage 24,estimatethe valuesofthetwosupernovae. p bFigureogivesthevaluesof v=c .Fromthese,compute values andcomparewiththeresultsfromparta. p cLocatethesetwopointsonthegraphingurej,andmakea sketchshowingwheretheylie. 6 TheVoyager1spaceprobe,launchedin1977,ismovingfaster relativetotheearththananyotherhuman-madeobject,at17,000 meterspersecond. aCalculatetheprobe's p bOverthecourseofoneyearonearth,slightlylessthanoneyear passesontheprobe.Howmuchless?Thereare31millionseconds inayear. p 7 aAfreeneutronasopposedtoaneutronboundintoan atomicnucleusisunstable,andundergoesbetadecaywhichyou maywanttoreview.Themassesoftheparticlesinvolvedareas Problems 39

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follows: neutron1.67495 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(27 kg proton1.67265 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(27 kg electron0.00091 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(27 kg antineutrino < 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(35 kg Findtheenergyreleasedinthedecayofafreeneutron. p bNeutronsandprotonsmakeupessentiallyallofthemassofthe ordinarymatteraroundus.Weobservethattheuniversearoundus hasnofreeneutrons,butlotsoffreeprotonsthenucleiofhydrogen, whichistheelementthat90%oftheuniverseismadeof.Wend neutronsonlyinsidenucleialongwithotherneutronsandprotons, notontheirown. Ifthereareprocessesthatcanconvertneutronsintoprotons,we mightimaginethattherecouldalsobeproton-to-neutronconversions,andindeedsuchaprocessdoesoccursometimesinnuclei thatcontainbothneutronsandprotons:aprotoncandecayintoa neutron,apositron,andaneutrino.Apositronisaparticlewith thesamepropertiesasanelectron,exceptthatitselectricalcharge ispositiveseechapter7.Aneutrino,likeanantineutrino,has negligiblemass. Althoughsuchaprocesscanoccurwithinanucleus,explainwhy itcannothappentoafreeproton.Ifitcould,hydrogenwouldbe radioactive,andyouwouldn'texist! 8 aFindarelativisticequationforthevelocityofanobject intermsofitsmassandmomentumeliminating p bShowthatyourresultisapproximatelythesameastheclassical value, p=m ,atlowvelocities. cShowthatverylargemomentaresultinspeedsclosetothespeed oflight. ? 9 aShowthatfor v = = 5 c comesouttobeasimple fraction. bFindanothervalueof v forwhich isasimplefraction. 10 Anobjectmovingataspeedveryclosetothespeedoflight isreferredtoasultrarelativistic.Ordinarilyluckilytheonlyultrarelativisticobjectsinouruniversearesubatomicparticles,such ascosmicraysorparticlesthathavebeenacceleratedinaparticle accelerator. aWhatkindofnumberis foranultrarelativisticparticle? bRepeatexample5onpage35,butinsteadofverylow,nonrelativisticspeeds,considerultrarelativisticspeeds. cFindanequationfortheratio E =p .Thespeedmayberelativistic,butdon'tassumethatit'sultrarelativistic. p dSimplifyyouranswertopartcforthecasewherethespeedis ultrarelativistic. p eWecanthinkofabeamoflightasanultrarelativisticobject| 40 Chapter1Relativity

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itcertainlymovesataspeedthat'ssucientlyclosetothespeed oflight!Supposeyouturnonaone-wattashlight,leaveitonfor onesecond,andthenturnito.Computethemomentumofthe recoilingashlight,inunitsofkg m = s. p fDiscusshowyouranswerinparterelatestothecorrespondence principle. 11 Asdiscussedinbook3ofthisseries,thespeedatwhichadisturbancetravelsalongastringundertensionisgivenby v = p T= where isthemassperunitlength,and T isthetension. aSupposeastringhasadensity ,andacross-sectionalarea A Findanexpressionforthemaximumtensionthatcouldpossibly existinthestringwithoutproducing v>c ,whichisimpossible accordingtorelativity.Expressyouranswerintermsof A ,and c .Theinterpretationisthatrelativityputsalimitonhowstrong anymaterialcanbe. bEverysubstancehasatensilestrength,denedastheforce perunitarearequiredtobreakitbypullingitapart.ThetensilestrengthismeasuredinunitsofN = m 2 ,whichisthesameasthe pascalPa,themksunitofpressure.Makeanumericalestimate ofthemaximumtensilestrengthallowedbyrelativityinthecase wheretheropeismadeoutofordinarymatter,withadensityon thesameorderofmagnitudeasthatofwater.Forcomparison, kevlarhasatensilestrengthofabout4 10 9 Pa,andthereisspeculationthatbersmadefromcarbonnanotubescouldhavevalues ashighas6 10 10 Pa. cAblackholeisastarthathascollapsedandbecomeverydense, sothatitsgravityistoostrongforanythingevertoescapefromit. Forinstance,theescapevelocityfromablackholeisgreaterthan c ,soaprojectilecan'tbeshotoutofit.Manypeople,whenthey hearthisdescriptionofablackholeintermsofanescapevelocity greaterthan c ,wonderwhyitstillwouldn'tbepossibletoextract anobjectfromablackholebyothermeansthanlaunchingitout asaprojectile.Forexample,supposeweloweranastronautintoa blackholeonarope,andthenpullhimbackoutagain.Whymight thisnotwork? 12 Theearthisorbitingthesun,andthereforeiscontracted relativisticallyinthedirectionofitsmotion.Computetheamount bywhichitsdiametershrinksinthisdirection. Problems 41

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42 Chapter1Relativity

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ThecontinentalU.S.gotitsrsttasteofvolcanisminrecentmemorywith theeruptionofMountSt.Helensin1980. Chapter2 RulesofRandomness Givenforoneinstantanintelligencewhichcouldcomprehend alltheforcesbywhichnatureisanimatedandtherespective positionsofthethingswhichcomposeit...nothingwouldbe uncertain,andthefutureasthepastwouldbelaidoutbefore itseyes. PierreSimondeLaplace,1776 TheQuantumMechanicsisveryimposing.Butaninnervoice tellsmethatitisstillnotthenaltruth.Thetheoryyields much,butithardlybringsusnearertothesecretoftheOld 43

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One.Inanycase,IamconvincedthatHedoesnotplaydice. AlbertEinstein HoweverradicalNewton'sclockworkuniverseseemedtohiscontemporaries,bytheearlytwentiethcenturyithadbecomeasortof smuglyaccepteddogma.Luckilyforus,thisdeterministicpictureof theuniversebreaksdownattheatomiclevel.Theclearestdemonstrationthatthelawsofphysicscontainelementsofrandomness isinthebehaviorofradioactiveatoms.Picktwoidenticalatoms ofaradioactiveisotope,saythenaturallyoccurringuranium238, andwatchthemcarefully.Theywilldecayatdierenttimes,even thoughtherewasnodierenceintheirinitialbehavior. Wewouldbeinbigtroubleiftheseatoms'behaviorwasaspredictableasexpectedintheNewtonianworld-view,becauseradioactivityisanimportantsourceofheatforourplanet.Inreality,each atomchoosesarandommomentatwhichtoreleaseitsenergy,resultinginanicesteadyheatingeect.Theearthwouldbeamuch colderplanetifonlysunlightheateditandnotradioactivity.Probablytherewouldbenovolcanoes,andtheoceanswouldneverhave beenliquid.Thedeep-seageothermalventsinwhichliferstevolved wouldneverhaveexisted.Buttherewouldbeanevenworseconsequenceifradioactivitywasdeterministic:afterafewbillionyearsof peace,alltheuranium238atomsinourplanetwouldpresumably pickthesamemomenttodecay.Thehugeamountofstorednuclear energy,insteadofbeingspreadoutovereons,wouldallbereleased atoneinstant,blowingourwholeplanettoKingdomCome. 1 Thenewversionofphysics,incorporatingcertainkindsofrandomness,iscalledquantumphysicsforreasonsthatwillbecome clearlater.Itrepresentedsuchadramaticbreakwiththeprevious,deterministictraditionthateverythingthatcamebeforeis consideredclassical,"eventhetheoryofrelativity.Theremainder ofthisbookisabasicintroductiontoquantumphysics. DiscussionQuestion A IsaidPicktwoidenticalatomsofaradioactiveisotope.Aretwo atomsreallyidentical?Iftheirelectronsareorbitingthenucleus,canwe distinguisheachatombytheparticulararrangementofitselectronsat someinstantintime? 1 Thisisundertheassumptionthatalltheradioactiveheatingcomesfrom uraniumatoms,andthatalltheatomswerecreatedatthesametime.In reality,bothuraniumandthoriumatomscontribute,andtheymaynothaveall beencreatedatthesametime.Wehaveonlyageneralideaoftheprocesses thatcreatedtheseheavyelementsinthegascloudfromwhichoursolarsystem condensed.Someportionofthemmayhavecomefromnuclearreactionsin supernovaexplosionsinthatparticularnebula,butsomemayhavecomefrom previoussupernovaexplosionsthroughoutourgalaxy,orfromexoticeventslike collisionsofwhitedwarfstars. 44 Chapter2RulesofRandomness

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2.1RandomnessIsn'tRandom Einstein'sdistasteforrandomness,andhisassociationofdeterminismwithdivinity,goesbacktotheEnlightenmentconceptionofthe universeasagiganticpieceofclockworkthatonlyhadtobeset inmotioninitiallybytheBuilder.Manyofthefoundersofquantummechanicswereinterestedinpossiblelinksbetweenphysicsand EasternandWesternreligiousandphilosophicalthought,butevery educatedpersonhasadierentconceptofreligionandphilosophy. BertrandRussellremarked,SirArthurEddingtondeducesreligion fromthefactthatatomsdonotobeythelawsofmathematics.Sir JamesJeansdeducesitfromthefactthattheydo." Russell'switticism,whichimpliesincorrectlythatmathematics cannotdescriberandomness,remindsushowimportantitisnot tooversimplifythisquestionofrandomness.Youshouldnotsimplysurmise,Well,it'sallrandom,anythingcanhappen."For onething,certainthingssimplycannothappen,eitherinclassical physicsorquantumphysics.Theconservationlawsofmass,energy, momentum,andangularmomentumarestillvalid,soforinstance processesthatcreateenergyoutofnothingarenotjustunlikely accordingtoquantumphysics,theyareimpossible. Ausefulanalogycanbemadewiththeroleofrandomnessin evolution.Darwinwasnottherstbiologisttosuggestthatspecies changedoverlongperiodsoftime.Histwonewfundamentalideas werethatthechangesarosethroughrandomgeneticvariation, andchangesthatenhancedtheorganism'sabilitytosurviveand reproducewouldbepreserved,whilemaladaptivechangeswouldbe eliminatedbynaturalselection.Doubtersofevolutionoftenconsider onlytherstpoint,abouttherandomnessofnaturalvariation,but notthesecondpoint,aboutthesystematicactionofnaturalselection.Theymakestatementssuchas,thedevelopmentofacomplex organismlikeHomosapiensviarandomchancewouldbelikeawhirlwindblowingthroughajunkyardandspontaneouslyassemblinga jumbojetoutofthescrapmetal."Theawinthistypeofreasoningisthatitignoresthedeterministicconstraintsontheresultsof randomprocesses.Foranatomtoviolateconservationofenergyis nomorelikelythantheconquestoftheworldbychimpanzeesnext year. DiscussionQuestion A Economistsoftenbehavelikewannabephysicists,probablybecause itseemsprestigioustomakenumericalcalculationsinsteadoftalking abouthumanrelationshipsandorganizationslikeothersocialscientists. TheirstrivingtomakeeconomicsworklikeNewtonianphysicsextends toaparalleluseofmechanicalmetaphors,asintheconceptofamarket'ssupplyanddemandactinglikeaself-adjustingmachine,andthe idealizationofpeopleaseconomicautomatonswhoconsistentlystriveto maximizetheirownwealth.Whatevidenceisthereforrandomnessrather thanmechanicaldeterminismineconomics? Section2.1RandomnessIsn'tRandom 45

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a / Theprobabilitythatone wheelwillgiveacherryis1/10. Theprobabilitythatallthree wheelswillgivecherriesis 1 = 10 1 = 10 1 = 10. 2.2CalculatingRandomness Youshouldalsorealizethatevenifsomethingisrandom,wecanstill understandit,andwecanstillcalculateprobabilitiesnumerically. Inotherwords,physicistsaregoodbookmakers.Agoodbookiecan calculatetheoddsthatahorsewillwinaracemuchmoreaccurately thataninexperiencedone,butneverthelesscannotpredictwhatwill happeninanyparticularrace. Statisticalindependence Asanillustrationofageneraltechniqueforcalculatingodds, supposeyouareplayinga25-centslotmachine.Eachofthethree wheelshasonechanceintenofcomingupwithacherry.Ifall threewheelscomeupcherries,youwin$100.Eventhoughthe resultsofanyparticulartrialarerandom,youcanmakecertain quantitativepredictions.First,youcancalculatethatyourodds ofwinningonanygiventrialare1 = 10 1 = 10 1 = 10=1 = 1000= 0.001.Here,Iamrepresentingtheprobabilitiesasnumbersfrom 0to1,whichisclearerthanstatementslikeTheoddsare999to 1,"andmakesthecalculationseasier.Aprobabilityof0represents somethingimpossible,andaprobabilityof1representssomething thatwilldenitelyhappen. Also,youcansaythatanygiventrialisequallylikelytoresultin awin,anditdoesn'tmatterwhetheryouhavewonorlostinprior games.Mathematically,wesaythateachtrialisstatisticallyindependent,orthatseparategamesareuncorrelated.Mostgamblers aremistakenlyconvincedthat,tothecontrary,gamesofchanceare correlated.Iftheyhavebeenplayingaslotmachineallday,they areconvincedthatitisgettingreadytopay,"andtheydonot wantanyoneelseplayingthemachineandusingup"thejackpot thattheyhavecoming."Inotherwords,theyareclaimingthat aseriesoftrialsattheslotmachineisnegativelycorrelated,that losingnowmakesyoumorelikelytowinlater.Crapsplayersclaim thatyoushouldgotoatablewherethepersonrollingthediceis hot,"becausesheislikelytokeeponrollinggoodnumbers.Craps players,then,believethatrollsofthedicearepositivelycorrelated, thatwinningnowmakesyoumorelikelytowinlater. Mymethodofcalculatingtheprobabilityofwinningontheslot machinewasanexampleofthefollowingimportantruleforcalculationsbasedonindependentprobabilities: thelawofindependentprobabilities Iftheprobabilityofoneeventhappeningis P A ,andtheprobabilityofasecondstatisticallyindependenteventhappening is P B ,thentheprobabilitythattheywillbothoccuristhe productoftheprobabilities, P A P B .Iftherearemorethan twoeventsinvolved,yousimplykeeponmultiplying. 46 Chapter2RulesofRandomness

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Notethatthisonlyappliestoindependentprobabilities.For instance,ifyouhaveanickelandadimeinyourpocket,andyou randomlypulloneout,thereisaprobabilityof0.5thatitwillbe thenickel.Ifyouthenreplacethecoinandagainpulloneout randomly,thereisagainaprobabilityof0.5ofcomingupwiththe nickel,becausetheprobabilitiesareindependent.Thus,thereisa probabilityof0.25thatyouwillgetthenickelbothtimes. Supposeinsteadthatyoudonotreplacetherstcoinbefore pullingoutthesecondone.Thenyouareboundtopulloutthe othercointhesecondtime,andthereisnowayyoucouldpullthe nickelouttwice.Inthissituation,thetwotrialsarenotindependent,becausetheresultofthersttrialhasaneectonthesecond trial.Thelawofindependentprobabilitiesdoesnotapply,andthe probabilityofgettingthenickeltwiceiszero,not0.25. Experimentshaveshownthatinthecaseofradioactivedecay, theprobabilitythatanynucleuswilldecayduringagiventimeintervalisunaectedbywhatishappeningtotheothernuclei,and isalsounrelatedtohowlongithasgonewithoutdecaying.The rstobservationmakessense,becausenucleiareisolatedfromeach otheratthecentersoftheirrespectiveatoms,andthereforehaveno physicalwayofinuencingeachother.Thesecondfactisalsoreasonable,sinceallatomsareidentical.Supposewewantedtobelieve thatcertainatomswereextratough,"asdemonstratedbytheir historyofgoinganunusuallylongtimewithoutdecaying.Those atomswouldhavetobedierentinsomephysicalway,butnobody haseversucceededindetectingdierencesamongatoms.Thereis nowayforanatomtobechangedbytheexperiencesithasinits lifetime. Additionofprobabilities ThelawofindependentprobabilitiestellsustousemultiplicationtocalculatetheprobabilitythatbothAandBwillhappen, assumingtheprobabilitiesareindependent.Whatabouttheprobabilityofanor"ratherthananand?"IftwoeventsAandB aremutuallyexclusive,thentheprobabilityofoneortheotheroccurringisthesum P A + P B .Forinstance,abowlermighthavea 30%chanceofgettingastrikeknockingdownalltenpinsanda 20%chanceofknockingdownnineofthem.Thebowler'schanceof knockingdowneitherninepinsortenpinsistherefore50%. Itdoesnotmakesensetoaddprobabilitiesofthingsthatare notmutuallyexclusive,i.e.,thatcouldbothhappen.SayIhavea 90%chanceofeatinglunchonanygivenday,anda90%chanceof eatingdinner.TheprobabilitythatIwilleateitherlunchordinner isnot180%. Section2.2CalculatingRandomness 47

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b / Normalization:theprobabilityofpickinglandplusthe probabilityofpickingwateradds upto1. Normalization IfIspinaglobeandrandomlypickapointonit,Ihaveabouta 70%chanceofpickingapointthat'sinanoceananda30%chance ofpickingapointonland.Theprobabilityofpickingeitherwaterorlandis70%+30%=100%.Waterandlandaremutually exclusive,andtherearenootherpossibilities,sotheprobabilities hadtoaddupto100%.Itworksthesameiftherearemorethan twopossibilities|ifyoucanclassifyallpossibleoutcomesintoa listofmutuallyexclusiveresults,thenalltheprobabilitieshaveto addupto1,or100%.Thispropertyofprobabilitiesisknownas normalization. Averages Anotherwayofdealingwithrandomnessistotakeaverages. Thecasinoknowsthatinthelongrun,thenumberoftimesyouwin willapproximatelyequalthenumberoftimesyouplaymultiplied bytheprobabilityofwinning.Inthegamementionedabove,where theprobabilityofwinningis0.001,ifyouspendaweekplaying, andpay$2500toplay10,000times,youarelikelytowinabout10 times,000 0.001=10,andcollect$1000.Ontheaverage,the casinowillmakeaprotof$1500fromyou.Thisisanexampleof thefollowingrule. ruleforcalculatingaverages Ifyouconduct N identical,statisticallyindependenttrials, andtheprobabilityofsuccessineachtrialis P ,thenonthe average,thetotalnumberofsuccessfultrialswillbe NP .If N islargeenough,therelativeerrorinthisestimatewillbecome small. Thestatementthattheruleforcalculatingaveragesgetsmore andmoreaccurateforlargerandlarger N knownpopularlyasthe lawofaverages"oftenprovidesacorrespondenceprinciplethat connectsclassicalandquantumphysics.Forinstance,theamount ofpowerproducedbyanuclearpowerplantisnotrandomatany detectablelevel,becausethenumberofatomsinthereactorisso large.Ingeneral,randombehaviorattheatomicleveltendsto averageoutwhenweconsiderlargenumbersofatoms,whichiswhy physicsseemeddeterministicbeforephysicistslearnedtechniquesfor studyingatomsindividually. Wecanachievegreatprecisionwithaveragesinquantumphysics becausewecanuseidenticalatomstoreproduceexactlythesame situationmanytimes.Ifwewerebettingonhorsesordice,wewould bemuchmorelimitedinourprecision.Afterathousandraces,the horsewouldbereadytoretire.Afteramillionrolls,thedicewould bewornout. 48 Chapter2RulesofRandomness

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c / Whyaredicerandom? self-checkA Whichofthefollowingthings must beindependent,which could beindependent,andwhichdenitelyare not independent? theprobabilityofsuccessfullymakingtwofree-throwsinarowin basketball theprobabilitythatitwillraininLondontomorrowandtheprobability thatitwillrainonthesamedayinacertaincityinadistantgalaxy yourprobabilityofdyingtodayandofdyingtomorrow Answer,p.133 DiscussionQuestions A Newtonianphysicsisanessentiallyperfectapproximationfordescribingthemotionofapairofdice.IfNewtonianphysicsisdeterministic, whydoweconsidertheresultofrollingdicetoberandom? B Whyisn'titvalidtodenerandomnessbysayingthatrandomness iswhenalltheoutcomesareequallylikely? C Thesequenceofdigits121212121212121212seemsclearlynonrandom,and41592653589793seemsrandom.Thelattersequence,however,isthedecimalformofpi,startingwiththethirddigit.Thereisastory abouttheIndianmathematicianRamanujan,aself-taughtprodigy,thata friendcametovisithiminacab,andremarkedthatthenumberofthe cab,1729,seemedrelativelyuninteresting.Ramanujanrepliedthaton thecontrary,itwasveryinterestingbecauseitwasthesmallestnumber thatcouldberepresentedintwodifferentwaysasthesumoftwocubes. TheArgentineauthorJorgeLuisBorgeswroteashortstorycalledThe LibraryofBabel,inwhichheimaginedalibrarycontainingeverybook thatcouldpossiblybewrittenusingthelettersofthealphabet.Itwouldincludeabookcontainingonlytherepeatedlettera;alltheancientGreek tragediesknowntoday,allthelostGreektragedies,andmillionsofGreek tragediesthatwereneveractuallywritten;yourownlifestory,andvarious incorrectversionsofyourownlifestory;andcountlessanthologiescontainingashortstorycalledTheLibraryofBabel.Ofcourse,ifyoupicked abookfromtheshelvesofthelibrary,itwouldalmostcertainlylooklikea nonsensicalsequenceoflettersandpunctuation,butit'salwayspossible thattheseeminglymeaninglessbookwouldbeascience-ctionscreenplaywritteninthelanguageofaNeanderthaltribe,orthelyricstoaset ofincomparablybeautifullovesongswritteninalanguagethatneverexisted.Inviewoftheseexamples,whatdoesitreallymeantosaythat somethingisrandom? Section2.2CalculatingRandomness 49

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d / Probabilitydistributionfor theresultofrollingasingledie. e / Rollingtwodiceandadding themup. f / Aprobabilitydistribution forheightofhumanadults.Not realdata. 2.3ProbabilityDistributions Sofarwe'vediscussedrandomprocesseshavingonlytwopossible outcomes:yesorno,winorlose,onoro.Moregenerally,arandom processcouldhavearesultthatisanumber.Someprocessesyield integers,aswhenyourolladieandgetaresultfromonetosix,but somearenotrestrictedtowholenumbers,forexamplethenumber ofsecondsthatauranium-238atomwillexistbeforeundergoing radioactivedecay. Considerathrowofadie.Ifthedieishonest,"thenweexpect allsixvaluestobeequallylikely.Sinceallsixprobabilitiesmust addupto1,thenprobabilityofanyparticularvaluecomingup mustbe1/6.Wecansummarizethisinagraph,d.Areasunder thecurvecanbeinterpretedastotalprobabilities.Forinstance, theareaunderthecurvefrom1to3is1 = 6+1 = 6+1 = 6=1 = 2,so theprobabilityofgettingaresultfrom1to3is1/2.Thefunction shownonthegraphiscalledtheprobabilitydistribution. Figureeshowstheprobabilitiesofvariousresultsobtainedby rollingtwodiceandaddingthemtogether,asinthegameofcraps. Theprobabilitiesarenotallthesame.Thereisasmallprobability ofgettingatwo,forexample,becausethereisonlyonewaytodo it,byrollingaoneandthenanotherone.Theprobabilityofrolling asevenishighbecausetherearesixdierentwaystodoit:1+6, 2+5,etc. Ifthenumberofpossibleoutcomesislargebutnite,forexample thenumberofhairsonadog,thegraphwouldstarttolooklikea smoothcurveratherthanaziggurat. Whataboutprobabilitydistributionsforrandomnumbersthat arenotintegers?Wecannolongermakeagraphwithprobabilityonthe y axis,becausetheprobabilityofgettingagivenexact numberistypicallyzero.Forinstance,thereiszeroprobabilitythat aradioactiveatomwilllastfor exactly 3seconds,sincethereare innitelymanypossibleresultsthatarecloseto3butnotexactly three,forexample2.999999999999999996876876587658465436.It doesn'tusuallymakesense,therefore,totalkabouttheprobability ofasinglenumericalresult,butitdoesmakesensetotalkabout theprobabilityofacertainrangeofresults.Forinstance,theprobabilitythatanatomwilllastmorethan3andlessthan4secondsis aperfectlyreasonablethingtodiscuss.Wecanstillsummarizethe probabilityinformationonagraph,andwecanstillinterpretareas underthecurveasprobabilities. Butthe y axiscannolongerbeaunitlessprobabilityscale.In radioactivedecay,forexample,wewantthe x axistohaveunitsof time,andwewantareasunderthecurvetobeunitlessprobabilities. 50 Chapter2RulesofRandomness

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g / Aclose-upoftherighthandtailofthedistributionshown intheguref. h / Theaverageofaprobabilitydistribution. i / ThefullwidthathalfmaximumFWHMofaprobability distribution. Theareaofasinglesquareonthegraphpaperisthen unitlessareaofasquare =widthofsquarewithtimeunits heightofsquare. Iftheunitsaretocancelout,thentheheightofthesquaremust evidentlybeaquantitywithunitsofinversetime.Inotherwords, the y axisofthegraphistobeinterpretedasprobabilityperunit time,notprobability. Figurefshowsanotherexample,aprobabilitydistributionfor people'sheight.Thiskindofbell-shapedcurveisquitecommon. self-checkB Comparethenumberofpeoplewithheightsintherangeof130-135cm tothenumberintherange135-140. Answer,p.133 Lookingfortallbasketballplayersexample1 Acertaincountrywithalargepopulationwantstondverytall peopletobeonitsOlympicbasketballteamandstrikeablow againstwesternimperialism.Outofapoolof10 8 peoplewhoare therightageandgender,howmanyaretheylikelytondwhoare over225cmfeet4inchesinheight?Figureggivesaclose-up ofthetailofthedistributionshownpreviouslyinguref. Theshadedareaunderthecurverepresentstheprobabilitythat agivenpersonistallenough.Eachrectanglerepresentsaprobabilityof0.2 10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(7 cm )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 1cm=2 10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(8 .Thereareabout35 rectanglescoveredbytheshadedarea,sotheprobabilityofhavingaheightgreaterthan225cmis7 10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(7 ,orjustunderonein amillion.Usingtheruleforcalculatingaverages,theaverage,or expectednumberofpeoplethistallis 8 10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(7 =70. Averageandwidthofaprobabilitydistribution IfthenextMartianyoumeetasksyou,Howtallisanadulthuman?,"youwillprobablyreplywithastatementabouttheaverage humanheight,suchasOh,about5feet6inches."Ifyouwanted toexplainalittlemore,youcouldsay,Butthat'sonlyanaverage. Mostpeoplearesomewherebetween5feetand6feettall."Without botheringtodrawtherelevantbellcurveforyournewextraterrestrialacquaintance,you'vesummarizedtherelevantinformationby givinganaverageandatypicalrangeofvariation. Theaverageofaprobabilitydistributioncanbedenedgeometricallyasthehorizontalpositionatwhichitcouldbebalanced ifitwasconstructedoutofcardboard,h.Aconvenientnumerical measureoftheamountofvariationabouttheaverage,oramountof uncertainty,isthefullwidthathalfmaximum,orFWHM,dened ingurei.TheFWHMwasintroducedinchapter2of Vibrations andWaves Section2.3ProbabilityDistributions 51

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Agreatdealmorecouldbesaidaboutthistopic,andindeed anintroductorystatisticscoursecouldspendmonthsonwaysof deningthecenterandwidthofadistribution.Ratherthanforcefeedingyouonmathematicaldetailortechniquesforcalculating thesethings,itisperhapsmorerelevanttopointoutsimplythat therearevariouswaysofdeningthem,andtoinoculateyouagainst themisuseofcertaindenitions. Theaverageisnottheonlypossiblewaytosaywhatisatypical valueforaquantitythatcanvaryrandomly;anotherpossibledenitionisthemedian,denedasthevaluethatisexceededwith50% probability.Whendiscussingincomesofpeoplelivinginacertain town,theaveragecouldbeverymisleading,sinceitcanbeaected massivelyifasingleresidentofthetownisBillGates.Noristhe FWHMtheonlypossiblewayofstatingtheamountofrandomvariation;anotherpossiblewayofmeasuringitisthestandarddeviation denedasthesquarerootoftheaveragesquareddeviationfrom theaveragevalue. 2.4ExponentialDecayandHalf-Life Mostpeopleknowthatradioactivitylastsacertainamountof time,"butthatsimplestatementleavesoutalot.Asanexample, considerthefollowingmedicalprocedureusedtodiagnosethyroid function.Averysmallquantityoftheisotope 131 I,producedina nuclearreactor,isfedtoorinjectedintothepatient.Thebody's biochemicalsystemstreatthisarticial,radioactiveisotopeexactly thesameas 127 I,whichistheonlynaturallyoccurringtype.Nutritionally,iodineisanecessarytraceelement.Iodinetakeninto thebodyispartlyexcreted,buttherestbecomesconcentratedin thethyroidgland.Iodizedsalthashadiodineaddedtoittopreventthenutritionaldeciencyknownasgoiters,inwhichtheiodinestarvedthyroidbecomesswollen.Asthe 131 Iundergoesbetadecay, itemitselectrons,neutrinos,andgammarays.Thegammarayscan bemeasuredbyadetectorpassedoverthepatient'sbody.Asthe radioactiveiodinebecomesconcentratedinthethyroid,theamount ofgammaradiationcomingfromthethyroidbecomesgreater,and thatemittedbytherestofthebodyisreduced.Therateatwhich theiodineconcentratesinthethyroidtellsthedoctoraboutthe healthofthethyroid. Ifyoueverundergothisprocedure,someonewillpresumably explainalittleaboutradioactivitytoyou,toallayyourfearsthat youwillturnintotheIncredibleHulk,orthatyournextchildwill haveanunusualnumberoflimbs.Sinceiodinestaysinyourthyroid foralongtimeonceitgetsthere,onethingyou'llwanttoknowis whetheryourthyroidisgoingtobecomeradioactiveforever.They mayjusttellyouthattheradioactivityonlylastsacertainamount oftime,"butwecannowcarryoutaquantitativederivationofhow 52 Chapter2RulesofRandomness

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theradioactivityreallywilldieout. Let P surv t betheprobabilitythataniodineatomwillsurvive withoutdecayingforaperiodofatleast t .Ithasbeenexperimentallymeasuredthathalfall 131 Iatomsdecayin8hours,sowehave P surv hr=0.5. Nowusingthelawofindependentprobabilities,theprobability ofsurvivingfor16hoursequalstheprobabilityofsurvivingforthe rst8hoursmultipliedbytheprobabilityofsurvivingforthesecond 8hours, P surv hr=0.50 0.50 =0.25. Similarlywehave P surv hr=0.50 0.5 0.5 =0.125. Generalizingfromthispattern,theprobabilityofsurvivingforany time t thatisamultipleof8hoursis P surv t =0.5 t= 8hr Wenowknowhowtondtheprobabilityofsurvivalatintervals of8hours,butwhataboutthepointsintimeinbetween?What wouldbetheprobabilityofsurvivingfor4hours?Well,usingthe lawofindependentprobabilitiesagain,wehave P surv hr= P surv hr P surv hr, whichcanberearrangedtogive P surv hr= p P surv hr = p 0.5 =0.707. Thisisexactlywhatwewouldhavefoundsimplybypluggingin P surv t =0.5 t= 8hr andignoringtherestrictiontomultiplesof8 hours.Since8hoursistheamountoftimerequiredforhalfofthe atomstodecay,itisknownasthehalf-life,written t 1 = 2 .Thegeneral ruleisasfollows: exponentialdecayequation P surv t =0.5 t=t 1 = 2 Usingtheruleforcalculatingaverages,wecanalsondthenumberofatoms, N t ,remaininginasampleattime t : N t = N 0.5 t=t 1 = 2 Bothoftheseequationshavegraphsthatlooklikedying-outexponentials,asintheexamplebelow. Section2.4ExponentialDecayandHalf-Life 53

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j / Calibrationofthe 14 Cdatingmethodusingtreeringsandartifactswhoseageswereknownfromothermethods.RedrawnfromEmilio Segr e, NucleiandParticles ,1965. 14 C Datingexample2 AlmostallthecarbononEarthis 12 C,butnotquite.Theisotope 14 C,withahalf-lifeof5600years,isproducedbycosmicraysin theatmosphere.Itdecaysnaturally,butisreplenishedatsucha ratethatthefractionof 14 Cintheatmosphereremainsconstant, at1.3 10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(12 .Livingplantsandanimalstakeinboth 12 Cand 14 Cfromtheatmosphereandincorporatebothintotheirbodies. Oncethelivingorganismdies,itnolongertakesinCatomsfrom theatmosphere,andtheproportionof 14 Cgraduallyfallsoffas itundergoesradioactivedecay.Thiseffectcanbeusedtond theageofdeadorganisms,orhumanartifactsmadefromplants oranimals.Figurejshowstheexponentialdecaycurveof 14 C invariousobjects.Similarmethods,usinglonger-livedisotopes, provetheearthwasbillionsofyearsold,notafewthousandas somehadclaimedonreligiousgrounds. 54 Chapter2RulesofRandomness

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RadioactivecontaminationatChernobylexample3 Oneofthemostdangerousradioactiveisotopesreleasedbythe Chernobyldisasterin1986was 90 Sr,whosehalf-lifeis28years. aHowlongwillitbebeforethecontaminationisreducedtoone tenthofitsoriginallevel?bIfatotalof10 27 atomswasreleased, abouthowlongwoulditbebeforenotasingleatomwasleft? aWewanttoknowtheamountoftimethata 90 Srnucleus hasaprobabilityof0.1ofsurviving.Startingwiththeexponential decayformula, P surv =0.5 t = t 1 = 2 wewanttosolvefor t .Takingnaturallogarithmsofbothsides, ln P = t t 1 = 2 ln0.5, so t = t 1 = 2 ln0.5 ln P Pluggingin P =0.1and t 1 = 2 =28years,weget t =93years. bThisisjustliketherstpart,but P =10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(27 .Theresultis about2500years. Rateofdecay Ifyouwanttondhowmanyradioactivedecaysoccurwithina timeintervallastingfromtime t totime t + t ,themoststraightforwardapproachistocalculateitlikethis: numberofdecaysbetween t and t + t = N t )]TJ/F20 10.9091 Tf 10.909 0 Td [(N t + t = N [ P surv t )]TJ/F20 10.9091 Tf 10.909 0 Td [(P surv t + t ] = N h 0.5 t=t 1 = 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(0.5 t + t =t 1 = 2 i = N h 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(0.5 t=t 1 = 2 i 0.5 t=t 1 = 2 Aproblemariseswhen t issmallcomparedto t 1 = 2 .Forinstance, supposeyouhaveahunkof10 22 atomsof 235 U,withahalf-lifeof 700millionyears,whichis2.2 10 16 s.Youwanttoknowhowmany decayswilloccurin t =1s.Sincewe'respecifyingthecurrent numberofatoms, t =0.Asyouplugintotheformulaaboveon yourcalculator,thequantity0.5 t=t 1 = 2 comesoutonyourcalculator toequalone,sothenalresultiszero.That'sincorrect,though. Inreality,0.5 t=t 1 = 2 shouldequal0.999999999999999968,butyour calculatoronlygiveseightdigitsofprecision,soitroundeditoto one.Inotherwords,theprobabilitythata 235 Uatomwillsurvive for1sisveryclosetoone,butnotequaltoone.Thenumberof decaysinonesecondistherefore3.2 10 5 ,notzero. Section2.4ExponentialDecayandHalf-Life 55

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Well,mycalculatoronlydoeseightdigitsofprecision,justlike yours,sohowdidIknowtherightanswer?Thewaytodoitisto usethefollowingapproximation: a b 1+ b ln a ,if b 1 Thesymbol meansismuchlessthan."Usingit,wecannd thefollowingapproximation: numberofdecaysbetween t and t + t = N h 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(0.5 t=t 1 = 2 i 0.5 t=t 1 = 2 N 1 )]TJ/F26 10.9091 Tf 10.909 15.382 Td [( 1+ t t 1 = 2 ln0.5 0.5 t=t 1 = 2 ln2 N .5 t=t 1 = 2 t t 1 = 2 Thisalsogivesusawaytocalculatetherateofdecay,i.e.,the numberofdecaysperunittime.Dividingby t onbothsides,we have decaysperunittime ln2 N t 1 = 2 0.5 t=t 1 = 2 ,if t t 1 = 2 Thehotpotatoexample4 Anuclearphysicistwithadementedsenseofhumortosses youacigarbox,yellinghotpotato.Thelabelontheboxsays contains10 20 atomsof 17 F,half-lifeof66s,producedtodayin ourreactorat1p.m.Ittakesyoutwosecondstoreadthelabel, afterwhichyoutossitbehindsomeleadbricksandrunaway.The timeis1:40p.m.Willyoudie? Thetimeelapsedsincetheradioactiveuorinewasproduced inthereactorwas40minutes,or2400s.Thenumberofelapsed half-livesistherefore t = t 1 = 2 =36.Theinitialnumberofatoms was N =10 20 .Thenumberofdecayspersecondisnow about10 7 s )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 ,soitproducedabout2 10 7 high-energyelectrons whileyouhelditinyourhands.Althoughtwentymillionelectrons soundslikealot,itisnotreallyenoughtobedangerous. Bytheway,noneoftheequationswe'vederivedsofarwasthe actualprobabilitydistributionforthetimeatwhichaparticular radioactiveatomwilldecay.Thatprobabilitydistributionwouldbe foundbysubstituting N =1intotheequationfortherateof decay. Ifthesheernumberofequationsisstartingtoseemformidable, let'spauseandthinkforasecond.Thesimpleequationfor P surv is somethingyoucanderiveeasilyfromthelawofindependentprobabilitiesanytimeyouneedit.Fromthat,youcanquicklyndthe 56 Chapter2RulesofRandomness

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exactequationfortherateofdecay.Thederivationoftheapproximateequationsfor t t isalittlehairier,butnotethatexcept forthefactorsofln2,everythingintheseequationscanbefound simplyfromconsiderationsoflogicandunits.Forinstance,alonger half-lifewillobviouslyleadtoaslowerrateofdecays,soitmakes sensethatwedividebyit.Asfortheln2factors,theyareexactly thekindofthingthatonelooksupinabookwhenoneneedsto knowthem. DiscussionQuestions A Inthemedicalprocedureinvolving 131 I,whyisitthegammarays thataredetected,nottheelectronsorneutrinosthatarealsoemitted? B For1s,Fredholdsinhishands1kgofradioactivestuffwitha half-lifeof1000years.Gingerholds1kgofadifferentsubstance,witha half-lifeof1min,forthesameamountoftime.Didtheyplacethemselves inequaldanger,ornot? C Howwouldyouinterpretitifyoucalculated N t ,andfounditwas lessthanone? D Doesthehalf-lifedependonhowmuchofthesubstanceyouhave? Doestheexpectedtimeuntilthesampledecayscompletelydependon howmuchofthesubstanceyouhave? 2.5 R ApplicationsofCalculus Theareaundertheprobabilitydistributionisofcourseanintegral. Ifwecalltherandomnumber x andtheprobabilitydistribution D x ,thentheprobabilitythat x liesinacertainrangeisgivenby probabilityof a x b = Z b a D x d x Whataboutaverages?If x hadanitenumberofequallyprobable values,wewouldsimplyaddthemupanddividebyhowmanywe had.Iftheyweren'tequallylikely,we'dmaketheweightedaverage x 1 P 1 + x 2 P 2 +...Butweneedtogeneralizethistoavariable x that cantakeonanyofacontinuumofvalues.Thecontinuousversion ofasumisanintegral,sotheaverageis averagevalueof x = Z xD x d x wheretheintegralisoverallpossiblevaluesof x Section2.5 R ApplicationsofCalculus 57

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Probabilitydistributionforradioactivedecayexample5 Hereisarigorousjusticationforthestatementinsection2.4 thattheprobabilitydistributionforradioactivedecayisfoundby substituting N =1intotheequationfortherateofdecay.We knowthattheprobabilitydistributionmustbeoftheform D t = k 0.5 t = t 1 = 2 where k isaconstantthatweneedtodetermine.Theatomis guaranteedtodecayeventually,sonormalizationgivesus probabilityof0 t < 1 =1 = Z 1 0 D t d t Theintegralismosteasilyevaluatedbyconvertingthefunction intoanexponentialwith e asthebase D t = k exp h ln 0.5 t = t 1 = 2 i = k exp t t 1 = 2 ln0.5 = k exp )]TJ/F39 10.9091 Tf 9.809 7.38 Td [(ln2 t 1 = 2 t whichgivesanintegralofthefamiliarform R e cx d x = = c e cx Wethushave 1= )]TJ/F116 10.9091 Tf 9.68 8.467 Td [(kt 1 = 2 ln2 exp )]TJ/F39 10.9091 Tf 9.809 7.38 Td [(ln2 t 1 = 2 t whichgivesthedesiredresult: k = ln2 t 1 = 2 Averagelifetimeexample6 Youmightthinkthatthehalf-lifewouldalsobetheaveragelifetimeofanatom,sincehalftheatoms'livesareshorterandhalf longer.Butthehalfwhoselivesarelongerincludesomethatsurviveformanyhalf-lives,andtheserarelong-livedatomsskewthe average.Wecancalculatetheaveragelifetimeasfollows: averagelifetime= Z 1 0 tD t d t Usingtheconvenientbasee formagain,wehave averagelifetime= ln2 t 1 = 2 Z 1 0 t exp )]TJ/F39 10.9091 Tf 9.808 7.38 Td [(ln2 t 1 = 2 t d t 58 Chapter2RulesofRandomness

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Thisintegralisofaformthatcaneitherbeattackedwithintegrationbypartsorbylookingitupinatable.Theresultis R xe cx d x = x c e cx )]TJ/F39 7.9701 Tf 12.938 4.296 Td [(1 c 2 e cx ,andthersttermcanbeignoredforour purposesbecauseitequalszeroatbothlimitsofintegration.We endupwith averagelifetime= ln2 t 1 = 2 t 1 = 2 ln2 2 = t 1 = 2 ln2 =1.443 t 1 = 2 whichis,asexpected,longerthanonehalf-life. Section2.5 R ApplicationsofCalculus 59

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Summary SelectedVocabulary probability....thelikelihoodthatsomethingwillhappen,expressedasanumberbetweenzeroandone normalization..thepropertyofprobabilitiesthatthesumof theprobabilitiesofallpossibleoutcomesmust equalone independence..thelackofanyrelationshipbetweentworandomevents probabilitydistribution...... acurvethatspeciestheprobabilitiesofvariousrandomvaluesofavariable;areasunder thecurvecorrespondtoprobabilities FWHM......thefullwidthathalf-maximumofaprobabilitydistribution;ameasureofthewidthofthe distribution half-life......theamountoftimethataradioactiveatom willsurvivewithprobability1/2withoutdecaying Notation P .........probability t 1 = 2 ........half-life D .........aprobabilitydistributionusedonlyinoptionalsection2.5;notastandardizednotation Summary Quantumphysicsdiersfromclassicalphysicsinmanyways,the mostdramaticofwhichisthatcertainprocessesattheatomiclevel, suchasradioactivedecay,arerandomratherthandeterministic. Thereisamethodtothemadness,however:quantumphysicsstill rulesoutanyprocessthatviolatesconservationlaws,anditalso oersmethodsforcalculatingprobabilitiesnumerically. Inthischapterwefocusedoncertaingenericmethodsofworking withprobabilities,withoutconcerningourselveswithanyphysical details.Withoutknowinganyofthedetailsofradioactivedecay, forexample,wewerestillabletogiveafairlycompletetreatment oftherelevantprobabilities.Themostimportantofthesegeneric methodsisthelawofindependentprobabilities,whichstatesthatif tworandomeventsarenotrelatedinanyway,thentheprobability thattheywillbothoccurequalstheproductofthetwoprobabilities, probabilityofAandB = P A P B [ifAandBareindependent]. Themostimportantapplicationistoradioactivedecay.Thetime thataradioactiveatomhasa50%chanceofsurvivingiscalled thehalf-life, t 1 = 2 .Theprobabilityofsurvivingfortwohalf-livesis = 2 = 2=1 = 4,andsoon.Ingeneral,theprobabilityofsurviving 60 Chapter2RulesofRandomness

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atime t isgivenby P surv t =0.5 t=t 1 = 2 Relatedquantitiessuchastherateofdecayandprobabilitydistributionforthetimeofdecayaregivenbythesametypeofexponential function,butmultipliedbycertainconstantfactors. Summary 61

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Ifaradioactivesubstancehasahalf-lifeofoneyear,doesthis meanthatitwillbecompletelydecayedaftertwoyears?Explain. 2 Whatistheprobabilityofrollingapairofdiceandgetting snakeeyes,"i.e.,bothdicecomeupwithones? 3 aExtracthalf-livesdirectlyfromthegraphsshowningures mandnonpage22. p bCheckthattheratiobetweenthesetwonumbersreallyisabout 10,ascalculatedinthetextbasedonrelativity. 4 Useacalculatortochecktheapproximationthat a b 1+ b ln a if b 1,usingsomearbitrarynumbers.Thenseehowgoodthe approximationisforvaluesof b thatarenotquiteassmallcompared toone. 5 Makeupanexampleofanumericalprobleminvolvingarateof decaywhere t t 1 = 2 ,buttheexactexpressionfortherateofdecay onpage55canstillbeevaluatedonacalculatorwithoutgetting somethingthatroundsotozero.Checkthatyougetapproximately thesameresultusingbothmethodsonpage55tocalculatethe numberofdecaysbetween t and t + t .Keepplentyofsignicant guresinyourresults,inordertoshowthedierencebetweenthem. 6 Deviseamethodfortestingexperimentallythehypothesis thatagambler'schanceofwinningatcrapsisindependentofher previousrecordofwinsandlosses. 7 Refertotheprobabilitydistributionforpeople'sheightsin gurefonpage50. aShowthatthegraphisproperlynormalized. bEstimatethefractionofthepopulationhavingheightsbetween 140and150cm. p 62 Chapter2RulesofRandomness

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Problem8. 8 aAnuclearphysicistisstudyinganuclearreactioncausedin anacceleratorexperiment,withabeamofionsfromtheaccelerator strikingathinmetalfoilandcausingnuclearreactionswhenanucleusfromoneofthebeamionshappenstohitoneofthenucleiin thetarget.Aftertheexperimenthasbeenrunningforafewhours, afewbillionradioactiveatomshavebeenproduced,embeddedin thetarget.Shedoesnotknowwhatnucleiarebeingproduced,but shesuspectstheyareanisotopeofsomeheavyelementsuchasPb, Bi,FrorU.Followingonesuchexperiment,shetakesthetargetfoil outoftheaccelerator,sticksitinfrontofadetector,measuresthe activityevery5min,andmakesagraphgure.Theisotopesshe thinksmayhavebeenproducedare: isotopehalf-lifeminutes 211 Pb36.1 214 Pb26.8 214 Bi19.7 223 Fr21.8 239 U23.5 Whichoneisit? bHavingdecidedthattheoriginalexperimentalconditionsproducedonespecicisotope,shenowtriesusingbeamsofionstravelingatseveraldierentspeeds,whichmaycausedierentreactions. Thefollowingtablegivestheactivityofthetarget10,20and30minutesaftertheendoftheexperiment,forthreedierentionspeeds. activitymillionsofdecays/safter... 10min20min30min rstionspeed1.9330.8320.382 secondionspeed1.2000.5450.248 thirdionspeed6.5441.2960.248 Sincesuchalargenumberofdecaysisbeingcounted,assumethat thedataareonlyinaccurateduetoroundingowhenwritingdown thetable.Whichareconsistentwiththeproductionofasingle isotope,andwhichimplythatmorethanoneisotopewasbeing created? Problems 63

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9 Allheliumonearthisfromthedecayofnaturallyoccurring heavyradioactiveelementssuchasuranium.Eachalphaparticle thatisemittedendsupclaimingtwoelectrons,whichmakesita heliumatom.Iftheoriginal 238 Uatomisinsolidrockasopposed totheearth'smoltenregions,theHeatomsareunabletodiuse outoftherock.Thisprobleminvolvesdatingarockusingthe knowndecaypropertiesofuranium238.Supposeageologistnds asampleofhardenedlava,meltsitinafurnace,andndsthatit contains1230mgofuraniumand2.3mgofhelium. 238 Udecaysby alphaemission,withahalf-lifeof4.5 10 9 years.Thesubsequent chainofalphaandelectronbetadecaysinvolvesmuchshorterhalflives,andterminatesinthestablenucleus 206 Pb.Youmaywantto reviewalphaandbetadecay.Almostallnaturaluraniumis 238 U, andthechemicalcompositionofthisrockindicatesthattherewere nodecaychainsinvolvedotherthanthatof 238 U. aHowmanyalphasareemittedindecaychainofasingle 238 U atom? [Hint:Useconservationofmass.] bHowmanyelectronsareemittedperdecaychain? [Hint:Useconservationofcharge.] cHowlonghasitbeensincethelavaoriginallyhardened? p 10 Physiciststhoughtforalongtimethatbismuth-209wasthe heavieststableisotope.Veryheavyelementsdecaybyalphaemissionbecauseofthestrongelectricalrepulsionofalltheirprotons. However,a2003paperbyMarcillacetal.describesanexperiment inwhichbismuth-209lostitsclaimtofame|itactuallyundergoes alphadecaywithahalf-lifeof1.9 10 19 years. aAfterthealphaparticleisemitted,whatistheisotopeleftover? bComparethehalf-lifetotheageoftheuniverse,whichisabout 14billionyears. cAtablespoonofPepto-Bismolcontainsabout4 10 20 bismuth209atoms.Onceyou'veswallowedit,howmuchtimewillittake, ontheaverage,beforetherstatomicdecay? p 11 Ablindfoldedpersonresagunatacirculartargetofradius b ,andisallowedtocontinueringuntilashotactuallyhitsit. Anypartofthetargetisequallylikelytogethit.Wemeasurethe randomdistance r fromthecenterofthecircletowherethebullet wentin. aShowthattheprobabilitydistributionof r mustbeoftheform D r = kr ,where k issomeconstant.Ofcoursewehave D r =0 for r>b bDetermine k byrequiring D tobeproperlynormalized. p cFindtheaveragevalueof r p 64 Chapter2RulesofRandomness

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dInterpretingyourresultfrompartc,howdoesitcomparewith b= 2?Doesthismakesense?Explain. R 12 Wearegivensomeatomsofacertainradioactiveisotope, withhalf-life t 1 = 2 .Wepickoneatomatrandom,andobserveitfor onehalf-life,startingattimezero.Ifitdecaysduringthatone-halflifeperiod,werecordthetime t atwhichthedecayoccurred.Ifit doesn't,weresetourclocktozeroandkeeptryinguntilwegetan atomthatcooperates.Thenalresultisatime0 t t 1 = 2 ,witha distributionthatlooksliketheusualexponentialdecaycurve,but withitstailchoppedo. aFindthedistribution D t ,withthepropernormalization. p bFindtheaveragevalueof t p cInterpretingyourresultfrompartb,howdoesitcomparewith t 1 = 2 = 2?Doesthismakesense?Explain. R 13 Thespeed, v ,ofanatominanidealgashasaprobability distributionoftheform D v = bve )]TJ/F21 7.9701 Tf 6.586 0 Td [(cv 2 where0 v< 1 c relatestothetemperature,and b isdetermined bynormalization. aSketchthedistribution. bFind b intermsof c p cFindtheaveragespeedintermsof c ,eliminating b .Don't trytodotheindeniteintegral,becauseitcan'tbedoneinclosed form.Therelevantdeniteintegralcanbefoundintablesordone withcomputersoftware. p R 14 Neutrinosinteractsoweaklywithnormalmatterthat,of theneutrinosfromthesunthatentertheearthfromthedayside, onlyabout10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(10 ofthemfailtoreemergeonthenightside.From thisfact,estimatethethicknessofmatter,inunitsoflight-years, thatwouldberequiredinordertoblockhalfofthem.Thishalfdistance"isanalogoustoahalf-lifeforradioactivedecay. Problems 65

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66 Chapter2RulesofRandomness

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Inrecentdecades,ahugeholeintheozonelayerhasspreadoutfrom Antarctica. Chapter3 LightasaParticle Theonlythingthatinterfereswithmylearningismyeducation. AlbertEinstein Radioactivityisrandom,butdothelawsofphysicsexhibitrandomnessinothercontextsbesidesradioactivity?Yes.Radioactive decaywasjustagoodplaypentogetusstartedwithconceptsof randomness,becauseallatomsofagivenisotopeareidentical.By stockingtheplaypenwithanunlimitedsupplyofidenticalatomtoys,naturehelpedustorealizethattheirfuturebehaviorcouldbe dierentregardlessoftheiroriginalidenticality.Wearenowready toleavetheplaypen,andseehowrandomnesstsintothestructure ofphysicsatthemostfundamentallevel. Thelawsofphysicsdescribelightandmatter,andthequantum revolutionrewrotebothdescriptions.Radioactivitywasagoodexampleofmatter'sbehavinginawaythatwasinconsistentwith classicalphysics,butifwewanttogetunderthehoodandunderstandhownonclassicalthingshappen,itwillbeeasiertofocuson lightratherthanmatter.Aradioactiveatomsuchasuranium-235 isafterallanextremelycomplexsystem,consistingof92protons, 143neutrons,and92electrons.Light,however,canbeasimplesine wave. Howeversuccessfultheclassicalwavetheoryoflighthadbeen |allowingthecreationofradioandradar,forexample|itstill 67

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failedtodescribemanyimportantphenomena.Anexamplethat iscurrentlyofgreatinterestisthewaytheozonelayerprotectsus fromthedangerousshort-wavelengthultravioletpartofthesun's spectrum.Intheclassicaldescription,lightisawave.Whenawave passesintoandbackoutofamedium,itsfrequencyisunchanged, andalthoughitswavelengthisalteredwhileitisinthemedium, itreturnstoitsoriginalvaluewhenthewavereemerges.Luckily forus,thisisnotatallwhatultravioletlightdoeswhenitpasses throughtheozonelayer,orthelayerwouldoernoprotectionat all! 3.1EvidenceforLightasaParticle a / Imagesmadebyadigitalcamera.Ineachsuccessiveimage, thedimspotoflighthasbeen madeevendimmer. Foralongtime,physiciststriedtoexplainawaytheproblems withtheclassicaltheoryoflightasarisingfromanimperfectunderstandingofatomsandtheinteractionoflightwithindividualatoms andmolecules.Theozoneparadox,forexample,couldhavebeen attributedtotheincorrectassumptionthattheozonelayerwasa smooth,continuoussubstance,wheninrealityitwasmadeofindividualozonemolecules.Itwasn'tuntil1905thatAlbertEinstein threwdownthegauntlet,proposingthattheproblemhadnothingto dowiththedetailsoflight'sinteractionwithatomsandeverything todowiththefundamentalnatureoflightitself. Inthosedaysthedataweresketchy,theideasvague,andthe experimentsdiculttointerpret;ittookageniuslikeEinsteintocut throughthethicketofconfusionandndasimplesolution.Today, however,wecangetrighttotheheartofthematterwithapieceof ordinaryconsumerelectronics,thedigitalcamera.Insteadoflm,a digitalcamerahasacomputerchipwithitssurfacedividedupintoa gridoflight-sensitivesquares,calledpixels."Comparedtoagrain ofthesilvercompoundusedtomakeregularphotographiclm,a digitalcamerapixelisactivatedbyanamountoflightenergyorders ofmagnitudesmaller.Wecanlearnsomethingnewaboutlightby usingadigitalcameratodetectsmallerandsmalleramountsof light,asshowninguresa/1througha/3.Figure1isfake,but2 and3arerealdigital-cameraimagesmadebyProf.LymanPage ofPrincetonUniversityasaclassroomdemonstration.Figure1is 68 Chapter3LightasaParticle

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b / Awaterwaveispartially absorbed. c / Astreamofbulletsispartiallyabsorbed. whatwewouldseeifweusedthedigitalcameratotakeapictureof afairlydimsourceoflight.Ingures2and3,theintensityofthe lightwasdrasticallyreducedbyinsertingsemitransparentabsorbers likethetintedplasticusedinsunglasses.Goingfrom1to2to3, moreandmorelightenergyisbeingthrownawaybytheabsorbers. Theresultsaredramaticallydierentfromwhatwewouldexpect basedonthewavetheoryoflight.Iflightwasawaveandnothing butawave,b,thentheabsorberswouldsimplycutdownthewave's amplitudeacrossthewholewavefront.Thedigitalcamera'sentire chipwouldbeilluminateduniformly,andweakeningthewavewith anabsorberwouldjustmeanthateverypixelwouldtakealongtime tosoakupenoughenergytoregisterasignal. Butguresa/2anda/3showthatsomepixelstakestronghits whileotherspickupnoenergyatall.Insteadofthewavepicture, theimagethatisnaturallyevokedbythedataissomethingmore likeahailofbulletsfromamachinegun,c.Eachbullet"oflight apparentlycarriesonlyatinyamountofenergy,whichiswhydetectingthemindividuallyrequiresasensitivedigitalcamerarather thananeyeorapieceoflm. AlthoughEinsteinwasinterpretingdierentobservations,this istheconclusionhereachedinhis1905paper:thatthepurewave theoryoflightisanoversimplication,andthattheenergyofabeam oflightcomesinnitechunksratherthanbeingspreadsmoothly throughoutaregionofspace. d / EinsteinandSeurat:twins separatedatbirth?Detailfrom SeineGrandeJatte byGeorges Seurat,1886. Wenowthinkofthesechunksasparticlesoflight,andcallthem Section3.1EvidenceforLightasaParticle 69

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e / Apparatusforobserving thephotoelectriceffect.Abeam oflightstrikesacapacitorplate insideavacuumtube,andelectronsareejectedblackarrows. photons,"althoughEinsteinavoidedthewordparticle,"andthe wordphoton"wasinventedlater.Regardlessofwords,thetroublewasthatwavesandparticlesseemedlikeinconsistentcategories. ThereactiontoEinstein'spapercouldbekindlydescribedasvigorouslyskeptical.Eventwentyyearslater,Einsteinwrote,There arethereforenowtwotheoriesoflight,bothindispensable,and| asonemustadmittodaydespitetwentyyearsoftremendouseort onthepartoftheoreticalphysicists|withoutanylogicalconnection."Intheremainderofthischapterwewilllearnhowtheseeming paradoxwaseventuallyresolved. DiscussionQuestions A Supposesomeonerebutsthedigitalcameradataingurea,claimingthattherandompatternofdotsoccursnotbecauseofanythingfundamentalaboutthenatureoflightbutsimplybecausethecamera'spixels arenotallexactlythesamesomearejustmoresensitivethanothers. Howcouldwetestthisinterpretation? B Discusshowthecorrespondenceprincipleappliestotheobservationsandconceptsdiscussedinthissection. 3.2HowMuchLightIsOnePhoton? Thephotoelectriceffect Wehaveseenevidencethatlightenergycomesinlittlechunks, sothenextquestiontobeaskedisnaturallyhowmuchenergyis inonechunk.Themoststraightforwardexperimentalavenuefor addressingthisquestionisaphenomenonknownasthephotoelectriceect.Thephotoelectriceectoccurswhenaphotonstrikes thesurfaceofasolidobjectandknocksoutanelectron.Itoccurs continuallyallaroundyou.Itishappeningrightnowatthesurface ofyourskinandonthepaperorcomputerscreenfromwhichyou arereadingthesewords.Itdoesnotordinarilyleadtoanyobservableelectricaleect,however,becauseontheaverage,freeelectrons arewanderingbackinjustasfrequentlyastheyarebeingejected. Ifanobjectdidsomehowloseasignicantnumberofelectrons, itsgrowingnetpositivechargewouldbeginattractingtheelectrons backmoreandmorestrongly. Figureeshowsapracticalmethodfordetectingthephotoelectriceect.Twoverycleanparallelmetalplatestheelectrodesofa capacitoraresealedinsideavacuumtube,andonlyoneplateisexposedtolight.Becausethereisagoodvacuumbetweentheplates, anyejectedelectronthathappenstobeheadedintherightdirectionwillalmostcertainlyreachtheothercapacitorplatewithout collidingwithanyairmolecules. Theilluminatedbottomplateisleftwithanetpositivecharge, andtheunilluminatedtopplateacquiresanegativechargefrom theelectronsdepositedonit.Thereisthusanelectriceldbetween 70 Chapter3LightasaParticle

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f / Thehamsterinherhamster ballislikeanelectronemerging fromthemetaltiledkitchenoor intothesurroundingvacuum woodoor.Thewoodooris higherthanthetiledoor,soas sherollsupthestep,thehamster willloseacertainamountof kineticenergy,analogousto E s Ifherkineticenergyistoosmall, shewon'tevenmakeitupthe step. theplates,anditisbecauseofthiseldthattheelectrons'pathsare curved,asshowninthediagram.However,sincevacuumisagood insulator,anyelectronsthatreachthetopplatearepreventedfrom respondingtotheelectricalattractionbyjumpingbackacrossthe gap.Insteadtheyareforcedtomaketheirwayaroundthecircuit, passingthroughanammeter.Theammetermeasuresthestrength ofthephotoelectriceect. Anunexpecteddependenceonfrequency ThephotoelectriceectwasdiscoveredserendipitouslybyHeinrichHertzin1887,ashewasexperimentingwithradiowaves.He wasnotparticularlyinterestedinthephenomenon,buthedidnotice thattheeectwasproducedstronglybyultravioletlightandmore weaklybylowerfrequencies.Lightwhosefrequencywaslowerthana certaincriticalvaluedidnotejectanyelectronsatall. 1 Thisdependenceonfrequencydidn'tmakeanysenseintermsoftheclassical wavetheoryoflight.Alightwaveconsistsofelectricandmagnetic elds.Thestrongertheelds,i.e.,thegreaterthewave'samplitude,thegreatertheforcesthatwouldbeexertedonelectronsthat foundthemselvesbathedinthelight.Itshouldhavebeenamplitude brightnessthatwasrelevant,notfrequency.Thedependenceon frequencynotonlyprovesthatthewavemodeloflightneedsmodifying,butwiththeproperinterpretationitallowsustodetermine howmuchenergyisinonephoton,anditalsoleadstoaconnectionbetweenthewaveandparticlemodelsthatweneedinorderto reconcilethem. Tomakeanyprogress,weneedtoconsiderthephysicalprocess bywhichaphotonwouldejectanelectronfromthemetalelectrode. Ametalcontainselectronsthatarefreetomovearound.Ordinarily, intheinteriorofthemetal,suchanelectronfeelsattractiveforces fromatomsineverydirectionaroundit.Theforcescancelout.But iftheelectronhappenstonditselfatthesurfaceofthemetal, theattractionfromtheinteriorsideisnotbalancedoutbyany attractionfromoutside.Inpoppingoutthroughthesurfacethe electronthereforelosessomeamountofenergy E s ,whichdepends onthetypeofmetalused. Supposeaphotonstrikesanelectron,annihilatingitselfandgivingupallitsenergytotheelectron. 2 Theelectronwilllose kineticenergythroughcollisionswithotherelectronsasitplows throughthemetalonitswaytothesurface;loseanamountof kineticenergyequalto E s asitemergesthroughthesurface;and losemoreenergyonitswayacrossthegapbetweentheplates, 1 InfactthiswasallpriortoThomson'sdiscoveryoftheelectron,soHertz wouldnothavedescribedtheeectintermsofelectrons|wearediscussing everythingwiththebenetofhindsight. 2 Wenowknowthatthisiswhatalwayshappensinthephotoelectriceect, althoughithadnotyetbeenestablishedin1905whetherornotthephotonwas completelyannihilated. Section3.2HowMuchLightIsOnePhoton? 71

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g / Adifferentwayofstudyingthephotoelectriceffect. h / Thequantity E s + e V indicatestheenergyofonephoton. Itisfoundtobeproportionalto thefrequencyofthelight. duetotheelectriceldbetweentheplates.Eveniftheelectron happenstoberightatthesurfaceofthemetalwhenitabsorbsthe photon,andeveniftheelectriceldbetweentheplateshasnotyet builtupverymuch, E s isthebareminimumamountofenergythat theelectronmustreceivefromthephotonifitistocontributeto ameasurablecurrent.Thereasonforusingverycleanelectrodesis tominimize E s andmakeithaveadenitevaluecharacteristicof themetalsurface,notamixtureofvaluesduetothevarioustypes ofdirtandcrudthatarepresentintinyamountsonallsurfacesin everydaylife. Wecannowinterpretthefrequencydependenceofthephotoelectriceectinasimpleway:apparentlytheamountofenergy possessedbyaphotonisrelatedtoitsfrequency.Alow-frequency redorinfraredphotonhasanenergylessthan E s ,soabeamof themwillnotproduceanycurrent.Ahigh-frequencyblueorviolet photon,ontheotherhand,packsenoughofapunchtoallowan electrontogetoutoftheelectrode.Atfrequencieshigherthanthe minimum,thephotoelectriccurrentcontinuestoincreasewiththe frequencyofthelightbecauseofeectsand. Numericalrelationshipbetweenenergyandfrequency PromptedbyEinstein'sphotonpaper,RobertMillikanwhom weencounteredinbook4ofthisseriesguredouthowtousethe photoelectriceecttoprobepreciselythelinkbetweenfrequency andphotonenergy.Ratherthangoingintothehistoricaldetailsof Millikan'sactualexperimentsalengthyexperimentalprogramthat occupiedalargepartofhisprofessionalcareerwewilldescribea simpleversion,showningureg,thatisusedsometimesincollege laboratorycourses.Theideaissimplytoilluminateoneplateof thevacuumtubewithlightofasinglewavelengthandmonitorthe voltagedierencebetweenthetwoplatesastheychargeup.Since theresistanceofavoltmeterisveryhighmuchhigherthanthe resistanceofanammeter,wecanassumetoagoodapproximation thatelectronsreachingthetopplatearestucktherepermanently, sothevoltagewillkeeponincreasingforaslongaselectronsare makingitacrossthevacuumtube. Atamomentwhenthevoltagedierencehasareachedavalue V,theminimumenergyrequiredbyanelectrontomakeitoutof thebottomplateandacrossthegaptotheotherplateis E s + e V. As V increases,weeventuallyreachapointatwhich E s + e V equalstheenergyofonephoton.Nomoreelectronscancrossthe gap,andthereadingonthevoltmeterstopsrising.Thequantity E s + e V nowtellsustheenergyofonephoton.Ifwedeterminethis energyforavarietyoffrequencies,h,wendthefollowingsimple relationshipbetweentheenergyofaphotonandthefrequencyof thelight: E = hf 72 Chapter3LightasaParticle

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where h isaconstantwithanumericalvalueof6.63 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(34 J s. Notehowtheequationbringsthewaveandparticlemodelsoflight underthesameroof:theleftsideistheenergyofone particle of light,whiletherightsideisthefrequencyofthesamelight,interpretedasa wave .Theconstant h isknownasPlanck'sconstantsee historicalnoteonpage73. self-checkA Howwouldyouextract h fromthegraphingureh?Whatifyoudidn't evenknow E s inadvance,andcouldonlygraph e V versus f ? Answer,p.133 Sincetheenergyofaphotonis hf ,abeamoflightcanonlyhave energiesof hf ,2 hf ,3 hf ,etc.Itsenergyisquantized|thereisno suchthingasafractionofaphoton.Quantumphysicsgetsitsname fromthefactthatitquantizesthingslikeenergy,momentum,and angularmomentumthathadpreviouslybeenthoughttobesmooth, continuousandinnitelydivisible. HistoricalNote WhatI'mpresentinginthischapterisasimpliedexplanationofhow thephotoncouldhavebeendiscovered.Theactualhistoryismore complex.MaxPlanck-1947beganthephotonsagawithatheoreticalinvestigationofthespectrumoflightemittedbyahot,glowing object.Heintroducedquantizationoftheenergyoflightwaves,inmultiplesof hf ,purelyasamathematicaltrickthathappenedtoproducethe rightresults.Planckdidnotbelievethathisprocedurecouldhaveany physicalsignicance.Inhis1905paperEinsteintookPlanck'squantizationasadescriptionofreality,andappliedittovarioustheoreticaland experimentalpuzzles,includingthephotoelectriceffect.Millikanthen subjectedEinstein'sideastoaseriesofrigorousexperimentaltests.AlthoughhisresultsmatchedEinstein'spredictionsperfectly,Millikanwas skepticalaboutphotons,andhispapersconspicuouslyomitanyreferencetothem.OnlyinhisautobiographydidMillikanrewritehistoryand claimthathehadgivenexperimentalproofforphotons. Numberofphotonsemittedbyalightbulbpersecondexample1 Roughlyhowmanyphotonsareemittedbya100-Wlightbulbin 1second? Peopletendtorememberwavelengthsratherthanfrequencies forvisiblelight.Thebulbemitsphotonswitharangeoffrequenciesandwavelengths,butlet'stake600nmasatypicalwavelengthforpurposesofestimation.Theenergyofasinglephoton Section3.2HowMuchLightIsOnePhoton? 73

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is E photon = hf = hc Apowerof100Wmeans100joulespersecond,sothenumber ofphotonsis 100J E photon = 100J hc = 3 10 20 Momentumofaphotonexample2 Accordingtothetheoryofrelativity,themomentumofabeam oflightisgivenby p = E = c seehomeworkproblem10onpage 40.Applythistondthemomentumofasinglephotoninterms ofitsfrequency,andintermsofitswavelength. Combiningtheequations p = E = c and E = hf ,wend p = E c = hf c Toreexpressthisintermsofwavelength,weuse c = f : p = hf f = h Thesecondformturnsouttobesimpler. DiscussionQuestions A Thephotoelectriceffectonlyeverejectsaverytinypercentageof theelectronsavailablenearthesurfaceofanobject.Howwelldoesthis agreewiththewavemodeloflight,andhowwellwiththeparticlemodel? Considerthetwodifferentdistancescalesinvolved:thewavelengthofthe light,andthesizeofanatom,whichisontheorderof10 )]TJ/F39 6.9738 Tf 6.226 0 Td [(10 or10 )]TJ/F39 6.9738 Tf 6.227 0 Td [(9 m. B WhatisthesignicanceofthefactthatPlanck'sconstantisnumericallyverysmall?Howwouldoureverydayexperienceoflightbedifferent ifitwasnotsosmall? C Howwouldtheexperimentsdescribedabovebeaffectedifasingle electronwaslikelytogethitbymorethanonephoton? D Drawsomerepresentativetrajectoriesofelectronsfor V =0, V lessthanthemaximumvalue,and V greaterthanthemaximumvalue. E Explainbasedonthephotontheoryoflightwhyultravioletlightwould bemorelikelythanvisibleorinfraredlighttocausecancerbydamaging DNAmolecules.HowdoesthisrelatetodiscussionquestionC? F Does E = hf implythataphotonchangesitsenergywhenitpasses fromonetransparentmaterialintoanothersubstancewithadifferentindexofrefraction? 74 Chapter3LightasaParticle

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j / Bulletspassthroughadouble slit. k / Awaterwavepassesthrough adoubleslit. 3.3Wave-ParticleDuality i / Waveinterferencepatterns photographedbyProf.Lyman Pagewithadigitalcamera.Laser lightwithasinglewell-dened wavelengthpassedthrougha seriesofabsorberstocutdown itsintensity,thenthroughasetof slitstoproduceinterference,and nallyintoadigitalcamerachip. Atripleslitwasactuallyused, butforconceptualsimplicitywe discusstheresultsinthemain textasifitwasadoubleslit.In panel2theintensityhasbeen reducedrelativeto1,andeven moresoforpanel3. Howcanlightbebothaparticleandawave?Wearenow readytoresolvethisseemingcontradiction.Ofteninsciencewhen somethingseemsparadoxical,it'sbecauseweeitherdon'tdeneour termscarefully,ordon'ttestourideasagainstanyspecicreal-world situation.Let'sdeneparticlesandwavesasfollows: Wavesexhibitsuperposition,andspecicallyinterferencephenomena. Particlescanonlyexistinwholenumbers,notfractions Asareal-worldcheckonourphilosophizing,thereisoneparticularexperimentthatworksperfectly.Wesetupadouble-slitinterferenceexperimentthatweknowwillproduceadiractionpattern iflightisanhonest-to-goodnesswave,butwedetectthelightwith adetectorthatiscapableofsensingindividualphotons,e.g.,adigitalcamera.Tomakeitpossibletopickoutindividualdotsfrom individualphotons,wemustuselterstocutdowntheintensityof thelighttoaverylowlevel,justasinthephotosbyProf.Pagein section3.1.Thewholethingissealedinsidealight-tightbox.The resultsareshowningurei.Infact,thesimilarguresinsection 3.1aresimplycutoutsfromthesegures. Neitherthepurewavetheorynorthepureparticletheorycan explaintheresults.Iflightwasonlyaparticleandnotawave,there wouldbenointerferenceeect.Theresultoftheexperimentwould belikeringahailofbulletsthroughadoubleslit,j.Onlytwo spotsdirectlybehindtheslitswouldbehit. If,ontheotherhand,lightwasonlyawaveandnotaparticle, wewouldgetthesamekindofdiractionpatternthatwouldhappen Section3.3Wave-ParticleDuality 75

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l / Asinglephotoncango throughbothslits. withawaterwave,k.Therewouldbenodiscretedotsinthephoto, onlyadiractionpatternthatshadedsmoothlybetweenlightand dark. Applyingthedenitionstothisexperiment,lightmustbeboth aparticleandawave.Itisawavebecauseitexhibitsinterference eects.Atthesametime,thefactthatthephotographscontain discretedotsisadirectdemonstrationthatlightrefusestobesplit intounitsoflessthanasinglephoton.Therecanonlybewhole numbersofphotons:fourphotonsingurei/3,forexample. Awronginterpretation:photonsinterferingwitheachother Onepossibleinterpretationofwave-particledualitythatoccurred tophysicistsearlyinthegamewasthatperhapstheinterferenceeffectscamefromphotonsinteractingwitheachother.Byanalogy,a waterwaveconsistsofmovingwatermolecules,andinterferenceof waterwavesresultsultimatelyfromallthemutualpushesandpulls ofthemolecules.Thisinterpretationwasconclusivelydisprovedby G.I.Taylor,astudentatCambridge.ThedemonstrationbyProf. Pagethatwe'vejustbeendiscussingisessentiallyamodernized versionofTaylor'swork.Taylorreasonedthatifinterferenceeects camefromphotonsinteractingwitheachother,abareminimumof twophotonswouldhavetobepresentatthesametimetoproduce interference.Bymakingthelightsourceextremelydim,wecanbe virtuallycertainthattherearenevertwophotonsintheboxatthe sametime.Ingurei,theintensityofthelighthasbeencutdown somuchbytheabsorbersthatifitwasintheopen,theaverage separationbetweenphotonswouldbeontheorderofakilometer! Atanygivenmoment,thenumberofphotonsintheboxismost likelytobezero.Itisvirtuallycertainthattherewerenevertwo photonsintheboxatonce. Theconceptofaphoton'spathisundened. Ifasinglephotoncandemonstratedouble-slitinterference,then whichslitdiditpassthrough?Theunavoidableanswermustbethat itpassesthroughboth!Thismightnotseemsostrangeifwethink ofthephotonasawave,butitishighlycounterintuitiveifwetry tovisualizeitasaparticle.Themoralisthatweshouldnotthink intermsofthe path ofaphoton.Likethefullyhumanandfully divineJesusofChristiantheology,aphotonissupposedtobe100% waveand100%particle.Ifaphotonhadawelldenedpath,thenit wouldnotdemonstratewavesuperpositionandinterferenceeects, contradictingitswavenature.Inthenextchapterwewilldiscuss theHeisenberguncertaintyprinciple,whichgivesanumericalway ofapproachingthisissue. 76 Chapter3LightasaParticle

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Anotherwronginterpretation:thepilotwavehypothesis Asecondpossibleexplanationofwave-particledualitywastaken seriouslyintheearlyhistoryofquantummechanics.Whatifthe photon particle islikeasurferridingontopofitsaccompanying wave ?Asthewavetravelsalong,theparticleispushed,orpiloted" byit.Imaginingtheparticleandthewaveastwoseparateentities allowsustoavoidtheseeminglyparadoxicalideathataphotonis bothatonce.Thewavehappilydoesitswavetricks,likesuperpositionandinterference,andtheparticleactslikearespectable particle,resolutelyrefusingtobeintwodierentplacesatonce.If thewave,forinstance,undergoesdestructiveinterference,becoming nearlyzeroinaparticularregionofspace,thentheparticlesimply isnotguidedintothatregion. Theproblemwiththepilotwaveinterpretationisthattheonly wayitcanbeexperimentallytestedorveriedisifsomeonemanages todetachtheparticlefromthewave,andshowthattherereallyare twoentitiesinvolved,notjustone.Partofthescienticmethodis thathypothesesaresupposedtobeexperimentallytestable.Since nobodyhasevermanagedtoseparatethewavelikepartofaphoton fromtheparticlepart,theinterpretationisnotusefulormeaningful inascienticsense. Theprobabilityinterpretation Thecorrectinterpretationofwave-particledualityissuggested bytherandomnatureoftheexperimentwe'vebeendiscussing:even thougheveryphotonwave/particleispreparedandreleasedinthe sameway,thelocationatwhichitiseventuallydetectedbythe digitalcameraisdierenteverytime.Theideaoftheprobability interpretationofwave-particledualityisthatthelocationofthe photon-particleisrandom,buttheprobabilitythatitisinacertain locationishigherwherethephoton-wave'samplitudeisgreater. Morespecically,theprobabilitydistributionoftheparticlemust beproportionaltothe square ofthewave'samplitude, probabilitydistribution / amplitude 2 Thisfollowsfromthecorrespondenceprincipleandfromthefact thatawave'senergydensityisproportionaltothesquareofitsamplitude.Ifwerunthedouble-slitexperimentforalongenoughtime, thepatternofdotsllsinandbecomesverysmoothaswouldhave beenexpectedinclassicalphysics.Topreservethecorrespondence betweenclassicalandquantumphysics,theamountofenergydepositedinagivenregionofthepictureoverthelongrunmustbe proportionaltothesquareofthewave'samplitude.Theamountof energydepositedinacertainareadependsonthenumberofphoSection3.3Wave-ParticleDuality 77

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m / Example3. tonspickedup,whichisproportionaltotheprobabilityofnding anygivenphotonthere. Amicrowaveovenexample3 Thegureshowstwo-dimensionaltopandone-dimensional bottomrepresentationsofthestandingwaveinsideamicrowave oven.Grayrepresentszeroeld,andwhiteandblacksignifythe strongestelds,withwhitebeingaeldthatisintheoppositedirectioncomparedtoblack.Comparetheprobabilitiesofdetecting amicrowavephotonatpointsA,B,andC. AandCarebothextremesofthewave,sotheprobabilitiesof detectingaphotonatAandCareequal.Itdoesn'tmatterthatwe haverepresentedCasnegativeandAaspositive,becauseitis thesquareoftheamplitudethatisrelevant.TheamplitudeatBis about1/2asmuchastheothers,sotheprobabilityofdetectinga photonthereisabout1/4asmuch. Theprobabilityinterpretationwasdisturbingtophysicistswho hadspenttheirpreviouscareersworkinginthedeterministicworld ofclassicalphysics,andironicallythemoststrenuousobjections againstitwereraisedbyEinstein,whohadinventedthephoton conceptintherstplace.Theprobabilityinterpretationhasneverthelesspassedeveryexperimentaltest,andisnowaswellestablished asanypartofphysics. Anaspectoftheprobabilityinterpretationthathasmademany peopleuneasyisthattheprocessofdetectingandrecordingthe photon'spositionseemstohaveamagicalabilitytogetridofthe wavelikesideofthephoton'spersonalityandforceittodecidefor onceandforallwhereitreallywantstobe.Butdetectionormeasurementisafterallonlyaphysicalprocesslikeanyother,governed bythesamelawsofphysics.Wewillpostponeadetaileddiscussion ofthisissueuntilthefollowingchapter,sinceameasuringdevice likeadigitalcameraismadeofmatter,butwehavesofaronly discussedhowquantummechanicsrelatestolight. Whatistheproportionalityconstant?example4 Whatistheproportionalityconstantthatwouldmakeanactual equationoutofprobabilitydistribution / amplitude 2 ? Theprobabilitythatthephotonisinacertainsmallregionof volume v shouldequalthefractionofthewave'senergythatis withinthatvolume: P = energyinvolume v energyofphoton = energyinvolume v hf Weassume v issmallenoughsothattheelectricandmagnetic 78 Chapter3LightasaParticle

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eldsarenearlyconstantthroughoutit.Wethenhave P = 1 8 k j E j 2 + 1 2 o j B j 2 v hf Wecansimplifythisformidablelookingexpressionbyrecognizingthatinanelectromagneticwave, j E j and j B j arerelatedby j E j = c j B j .Withsomealgebra,itturnsoutthattheelectricand magneticeldseachcontributehalfthetotalenergy,sowecan simplifythisto P =2 )]TJ/F39 7.9701 Tf 11.209 -4.541 Td [(1 8 k j E j 2 v hf = v 4 khf j E j 2 Asadvertised,theprobabilityisproportionaltothesquareofthe wave'samplitude. DiscussionQuestions A Inexample3onpage78,aboutthecarrotinthemicrowaveoven, showthatitwouldbenonsensicaltohaveprobabilitybeproportionalto theelditself,ratherthanthesquareoftheeld. B Einsteindidnottrytoreconcilethewaveandparticletheoriesof light,anddidnotsaymuchabouttheirapparentinconsistency.Einstein basicallyvisualizedabeamoflightasastreamofbulletscomingfrom amachinegun.Inthephotoelectriceffect,aphotonbulletwouldonly hitoneatom,justasarealbulletwouldonlyhitoneperson.Suppose someonereadinghis1905paperwantedtointerpretitbysayingthat Einstein'sso-calledparticlesoflightaresimplyshortwave-trainsthatonly occupyasmallregionofspace.Comparingthewavelengthofvisiblelight afewhundrednmtothesizeofanatomontheorderof0.1nm,explain whythisposesadifcultyforreconcilingtheparticleandwavetheories. C Canawhitephotonexist? D Indouble-slitdiffractionofphotons,wouldyougetthesamepattern ofdotsonthedigitalcameraimageifyoucoveredoneslit?Whyshouldit matterwhetheryougivethephotontwochoicesoronlyone? Section3.3Wave-ParticleDuality 79

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n / Thevolumeunderasurface. 3.4PhotonsinThreeDimensions UpuntilnowI'vebeensneakyandavoidedafulldiscussionofthe three-dimensionalaspectsoftheprobabilityinterpretation.Theexampleofthecarrotinthemicrowaveoven,forexample,reduced toaone-dimensionalsituationbecausewewereconsideringthree pointsalongthesamelineandbecausewewereonlycomparingratiosofprobabilities.Thepurposeofbringingitupnowistohead oanyfeelingthatyou'vebeencheatedconceptuallyratherthanto prepareyouformathematicalproblemsolvinginthreedimensions, whichwouldnotbeappropriateforthelevelofthiscourse. Atypicalexampleofaprobabilitydistributioninsection2.3 wasthedistributionofheightsofhumanbeings.Thethingthat variedrandomly,height, h ,hadunitsofmeters,andtheprobabilitydistributionwasagraphofafunction D h .Theunitsofthe probabilitydistributionhadtobem )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 inversemeterssothatareasunderthecurve,interpretedasprobabilities,wouldbeunitless: area=heightwidth=m )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 m. Nowsupposewehaveatwo-dimensionalproblem,e.g.,theprobabilitydistributionfortheplaceonthesurfaceofadigitalcamera chipwhereaphotonwillbedetected.Thepointwhereitisdetected wouldbedescribedwithtwovariables, x and y ,eachhavingunits ofmeters.Theprobabilitydistributionwillbeafunctionofboth variables, D x y .Aprobabilityisnowvisualizedasthevolume underthesurfacedescribedbythefunction D x y ,asshownin guren.Theunitsof D mustbem )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 sothatprobabilitieswillbe unitless:probability=depthlengthwidth=m )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 m m. Generalizingnallytothreedimensions,wendbyanalogythat theprobabilitydistributionwillbeafunctionofallthreecoordinates, D x y z ,andwillhaveunitsofm )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 .Itis,unfortunately, impossibletovisualizethegraphunlessyouareamutantwithanaturalfeelforlifeinfourdimensions.Iftheprobabilitydistribution isnearlyconstantwithinacertainvolumeofspace v ,theprobabilitythatthephotonisinthatvolumeissimply vD .Ifyouknow enoughcalculus,itshouldbeclearthatthiscanbegeneralizedto P = R D d x d y d z if D isnotconstant. 80 Chapter3LightasaParticle

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Summary SelectedVocabulary photon...........aparticleoflight photoelectriceect....theejection,byaphoton,ofanelectronfromthesurfaceofanobject wave-particleduality...theideathatlightisbothawaveand aparticle Summary Aroundtheturnofthetwentiethcentury,experimentsbeganto showproblemswiththeclassicalwavetheoryoflight.Inanyexperimentsensitiveenoughtodetectverysmallamountsoflightenergy, itbecomesclearthatlightenergycannotbedividedintochunks smallerthanacertainamount.Measurementsinvolvingthephotoelectriceectdemonstratethatthissmallestunitoflightenergy equals hf ,where f isthefrequencyofthelightand h isanumber knownasPlanck'sconstant.Wesaythatlightenergyisquantized inunitsof hf ,andweinterpretthisquantizationasevidencethat lighthasparticlepropertiesaswellaswaveproperties.Particlesof lightarecalledphotons. Theonlymethodofreconcilingthewaveandparticlenatures oflightthathasstoodthetestofexperimentistheprobability interpretation:theprobabilitythattheparticleisatagivenlocation isproportionaltothesquareoftheamplitudeofthewaveatthat location. Oneimportantconsequenceofwave-particledualityisthatwe mustabandontheconceptofthepaththeparticletakesthrough space.Toholdontothisconcept,wewouldhavetocontradictthe wellestablishedwavenatureoflight,sinceawavecanspreadoutin everydirectionsimultaneously. Summary 81

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. Forsomeofthesehomeworkproblems,youmaynditconvenient torefertothediagramoftheelectromagneticspectrumshownin section6.4of ElectricityandMagnetism 1 Giveanumericalcomparisonofthenumberofphotonsper secondemittedbyahundred-wattFMradiotransmitteranda hundred-wattlightbulb. p 2 Twodierentashesoflighteachhavethesameenergy.One consistsofphotonswithawavelengthof600nm,theother400nm. Ifthenumberofphotonsinthe600-nmashis3.0 10 18 ,howmany photonsareinthe400-nmash? p 3 Whenlightisreectedfromamirror,perhapsonly80%of theenergycomesback.Therestisconvertedtoheat.Onecould trytoexplainthisintwodierentways:80%ofthephotonsare reected,orallthephotonsarereected,buteachloses20%of itsenergy.Basedonyoureverydayknowledgeaboutmirrors,how canyoutellwhichinterpretationiscorrect?[Basedonaproblem fromPSSCPhysics.] 4 Supposewewanttobuildanelectroniclightsensorusingan apparatusliketheonedescribedinthesectiononthephotoelectric eect.Howwoulditsabilitytodetectdierentpartsofthespectrum dependonthetypeofmetalusedinthecapacitorplates? 5 Thephotoelectriceectcanoccurnotjustformetalcathodes butforanysubstance,includinglivingtissue.IonizationofDNA moleculescancausecancerorbirthdefects.Iftheenergyrequiredto ionizeDNAisonthesameorderofmagnitudeastheenergyrequired toproducethephotoelectriceectinametal,whichofthesetypes ofelectromagneticwavesmightposesuchahazard?Explain. 60Hzwavesfrompowerlines 100MHzFMradio microwavesfromamicrowaveoven visiblelight ultravioletlight x-rays 6 Thebeamofa100-Woverheadprojectorcoversanareaof 1m 1mwhenithitsthescreen3maway.Estimatethenumber ofphotonsthatareinightatanygiventime.Sincethisisonly 82 Chapter3LightasaParticle

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Problem7. Problem8. anestimate,wecanignorethefactthatthebeamisnotparallel. 7 Thetwodiractionpatternsweremadebysendingaashof lightthroughthesamedoubleslit.Giveanumericalcomparisonof theamountsofenergyinthetwoashes. p Problems8and9wereswitchedafteredition3.0,becausedoing8 rstmakesiteasiertodo9. 8 Threeofthefourgraphsareproperlynormalizedtorepresent singlephotons.Whichoneisn't?Explain. 9 PhotonFredhasagreaterenergythanphotonGinger.For eachofthefollowingquantities,explainwhetherFred'svalueof thatquantityisgreaterthanGinger's,lessthanGinger's,orequal toGinger's.Ifthereisnowaytotell,explainwhy. frequency speed wavelength period electriceldstrength magneticeldstrength 10 Giveexperimentalevidencetodisprovethefollowinginterpretationofwave-particleduality: Aphotonisreallyaparticle,but ittravelsalongawavypath,likeazigzagwithroundedcorners. Cite aspecic,realexperiment. 11 Inthephotoelectriceect,electronsareobservedwithvirtuallynotimedelay 10ns,evenwhenthelightsourceisveryweak. Aweaklightsourcedoeshoweveronlyproduceasmallnumberof ejectedelectrons.Thepurposeofthisproblemistoshowthatthe lackofasignicanttimedelaycontradictedtheclassicalwavetheoryoflight,sothroughoutthisproblemyoushouldputyourselfin theshoesofaclassicalphysicistandpretendyoudon'tknowabout photonsatall.Atthattime,itwasthoughtthattheelectronmight havearadiusontheorderof10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(15 m.Recentexperimentshave shownthatiftheelectronhasanynitesizeatall,itisfarsmaller. aEstimatethepowerthatwouldbesoakedupbyasingleelectron inabeamoflightwithanintensityof1mW = m 2 p bTheenergy, E s ,requiredfortheelectrontoescapethroughthe surfaceofthecathodeisontheorderof10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(19 J.Findhowlongit wouldtaketheelectrontoabsorbthisamountofenergy,andexplain whyyourresultconstitutesstrongevidencethatthereissomething wrongwiththeclassicaltheory. p Problems 83

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84 Chapter3LightasaParticle

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DorothymeltstheWickedWitch oftheWest. Chapter4 MatterasaWave [In]afewminutesIshallbeallmelted...Ihavebeenwicked inmyday,butIneverthoughtalittlegirllikeyouwouldever beabletomeltmeandendmywickeddeeds.Lookout hereIgo! TheWickedWitchoftheWest AstheWickedWitchlearnedthehardway,losingmolecular cohesioncanbeunpleasant.That'swhyweshouldbeverygratefulthattheconceptsofquantumphysicsapplytomatteraswell aslight.Ifmatterobeyedthelawsofclassicalphysics,molecules wouldn'texist. Consider,forexample,thesimplestatom,hydrogen.Whydoes onehydrogenatomformachemicalbondwithanotherhydrogen atom?Roughlyspeaking,we'dexpectaneighboringpairofhydrogenatoms,AandB,toexertnoforceoneachotheratall, attractiveorrepulsive:therearetworepulsiveinteractionsproton AwithprotonBandelectronAwithelectronBandtwoattractive interactionsprotonAwithelectronBandelectronAwithproton B.Thinkingalittlemoreprecisely,weshouldevenexpectthatonce thetwoatomsgotcloseenough,theinteractionwouldberepulsive. Forinstance,ifyousqueezedthemsoclosetogetherthatthetwo protonswerealmostontopofeachother,therewouldbeatremendouslystrongrepulsionbetweenthemduetothe1 =r 2 natureofthe 85

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electricalforce.Therepulsionbetweentheelectronswouldnotbe asstrong,becauseeachelectronrangesoveralargearea,andisnot likelytobefoundrightontopoftheotherelectron.Thiswasonly aroughargumentbasedonaverages,buttheconclusionisvalidated byamorecompleteclassicalanalysis:hydrogenmoleculesshould notexistaccordingtoclassicalphysics. Quantumphysicstotherescue!Aswe'llseeshortly,thewhole problemissolvedbyapplyingthesamequantumconceptstoelectronsthatwehavealreadyusedforphotons. 4.1ElectronsasWaves Westartedourjourneyintoquantumphysicsbystudyingtherandombehaviorof matter inradioactivedecay,andthenaskedhow randomnesscouldbelinkedtothebasiclawsofnaturegoverning light .Theprobabilityinterpretationofwave-particledualitywas strangeandhardtoaccept,butitprovidedsuchalink.Itisnow naturaltoaskwhetherthesameexplanationcouldbeappliedto matter.Ifthefundamentalbuildingblockoflight,thephoton,is aparticleaswellasawave,isitpossiblethatthebasicunitsof matter,suchaselectrons,arewavesaswellasparticles? AyoungFrencharistocratstudyingphysics,LouisdeBroglie pronouncedbroylee",madeexactlythissuggestioninhis1923 Ph.D.thesis.Hisideahadseemedsofarfetchedthattherewas seriousdoubtaboutwhethertogranthimthedegree.Einsteinwas askedforhisopinion,andwithhisstrongsupport,deBrogliegot hisdegree. Onlytwoyearslater,AmericanphysicistsC.J.DavissonandL. GermerconrmeddeBroglie'sideabyaccident.Theyhadbeen studyingthescatteringofelectronsfromthesurfaceofasample ofnickel,madeofmanysmallcrystals.Onecanoftenseesucha crystallinepatternonabrassdoorknobthathasbeenpolishedby repeatedhandling.Anaccidentalexplosionoccurred,andwhen theyputtheirapparatusbacktogethertheyobservedsomething entirelydierent:thescatteredelectronswerenowcreatinganinterferencepattern!Thisdramaticproofofthewavenatureofmatter cameaboutbecausethenickelsamplehadbeenmeltedbytheexplosionandthenresolidiedasasinglecrystal.Thenickelatoms, nownicelyarrangedintheregularrowsandcolumnsofacrystalline lattice,wereactingasthelinesofadiractiongrating.Thenew crystalwasanalogoustothetypeofordinarydiractiongratingin whichthelinesareetchedonthesurfaceofamirrorareection gratingratherthanthekindinwhichthelightpassesthroughthe transparentgapsbetweenthelinesatransmissiongrating. Althoughwewillconcentrateonthewave-particledualityofelectronsbecauseitisimportantinchemistryandthephysicsofatoms, 86 Chapter4MatterasaWave

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alltheotherparticles"ofmatteryou'velearnedaboutshowwave propertiesaswell.Figurea,forinstance,showsawaveinterference patternofneutrons. Itmightseemasthoughallourworkwasalreadydoneforus, andtherewouldbenothingnewtounderstandaboutelectrons: theyhavethesamekindoffunnywave-particledualityasphotons. That'salmosttrue,butnotquite.Therearesomeimportantways inwhichelectronsdiersignicantlyfromphotons: 1.Electronshavemass,andphotonsdon't. 2.Photonsalwaysmoveatthespeedoflight,butelectronscan moveatanyspeedlessthan c 3.Photonsdon'thaveelectriccharge,butelectronsdo,soelectric forcescanactonthem.Themostimportantexampleisthe atom,inwhichtheelectronsareheldbytheelectricforceof thenucleus. 4.Electronscannotbeabsorbedoremittedasphotonsare.Destroyinganelectron,orcreatingoneoutofnothing,would violateconservationofcharge. Inchapter5wewilllearnofonemorefundamentalwayinwhich electronsdierfromphotons,foratotalofve. a / Adouble-slitinterferencepatternmadewithneutrons.A. Zeilinger,R.G ahler,C.G.Shull, W.Treimer,andW.Mampe,ReviewsofModernPhysics,Vol.60, 1988. Becauseelectronsaredierentfromphotons,itisnotimmediatelyobviouswhichofthephotonequationsfromchapter3canbe appliedtoelectronsaswell.Aparticleproperty,theenergyofone photon,isrelatedtoitswavepropertiesvia E = hf or,equivalently, E = hc= .Themomentumofaphotonwasgivenby p = hf=c or Section4.1ElectronsasWaves 87

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p = h= example2onpage74.Ultimatelyitwasamatterofexperimenttodeterminewhichoftheseequations,ifany,wouldwork forelectrons,butwecanmakeaquickanddirtyguesssimplyby notingthatsomeoftheequationsinvolve c ,thespeedoflight,and somedonot.Since c isirrelevantinthecaseofanelectron,we mightguessthattheequationsofgeneralvalidityarethosethatdo nothave c inthem: E = hf p = h ThisisessentiallythereasoningthatdeBrogliewentthrough,and experimentshaveconrmedthesetwoequationsforallthefundamentalbuildingblocksoflightandmatter,notjustforphotonsand electrons. Thesecondequation,whichIsoft-pedaledinchapter3,takes onagreaterimportantforelectrons.Thisisrstofallbecausethe momentumofmatterismorelikelytobesignicantthanthemomentumoflightunderordinaryconditions,andalsobecauseforce isthetransferofmomentum,andelectronsareaectedbyelectrical forces. Thewavelengthofanelephantexample1 Whatisthewavelengthofatrottingelephant? Onemaydoubtwhethertheequationshouldbeappliedtoan elephant,whichisnotjustasingleparticlebutaratherlargecollectionofthem.Throwingcautiontothewind,however,weestimatetheelephant'smassat10 3 kganditstrottingspeedat10 m/s.Itswavelengthisthereforeroughly = h p = h mv = 6.63 10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(34 J s 3 kgm = s 10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(37 )]TJ/F39 10.9091 Tf 5 -8.837 Td [(kg m 2 = s 2 s kg m = s =10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(37 m. Thewavelengthfoundinthisexampleissofantasticallysmall thatwecanbesurewewillneverobserveanymeasurablewave phenomenawithelephants.Theresultisnumericallysmallbecause Planck'sconstantissosmall,andasinsomeexamplesencountered previously,thissmallnessisinaccordwiththecorrespondenceprinciple. 88 Chapter4MatterasaWave

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Althoughasmallermassintheequation = h=mv doesresultinalongerwavelength,thewavelengthisstillquiteshorteven forindividualelectronsundertypicalconditions,asshowninthe followingexample. Thetypicalwavelengthofanelectronexample2 Electronsincircuitsandinatomsaretypicallymovingthrough voltagedifferencesontheorderof1V,sothatatypicalenergyis e V,whichisontheorderof10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(19 J.Whatisthewavelength ofanelectronwiththisamountofkineticenergy? Thisenergyisnonrelativistic,sinceitismuchlessthan mc 2 Momentumandenergyarethereforerelatedbythenonrelativistic equation KE = p 2 = 2 m .Solvingfor p andsubstitutingintothe equationforthewavelength,wend = h p 2 m KE =1.6 10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(9 m. Thisisonthesameorderofmagnitudeasthesizeofanatom, whichisnoaccident:aswewilldiscussinthenextchapterin moredetail,anelectroninanatomcanbeinterpretedasastandingwave.Thesmallnessofthewavelengthofatypicalelectron alsohelpstoexplainwhythewavenatureofelectronswasn'tdiscovereduntilahundredyearsafterthewavenatureoflight.To scaletheusualwave-opticsdevicessuchasdiffractiongratings downtothesizeneededtoworkwithelectronsatordinaryenergies,weneedtomakethemsosmallthattheirpartsarecomparableinsizetoindividualatoms.ThisisessentiallywhatDavisson andGermerdidwiththeirnickelcrystal. self-checkA Theseremarksabouttheinconvenientsmallnessofelectronwavelengths applyonlyundertheassumptionthattheelectronshavetypicalenergies.Whatkindofenergywouldanelectronhavetohaveinorderto havealongerwavelengththatmightbemoreconvenienttoworkwith? Answer,p.133 Whatkindofwaveisit? Ifasoundwaveisavibrationofmatter,andaphotonisa vibrationofelectricandmagneticelds,whatkindofawaveis anelectronmadeof?Thedisconcertingansweristhatthereis noexperimentalobservable,"i.e.,directlymeasurablequantity,to correspondtotheelectronwaveitself.Inotherwords,thereare deviceslikemicrophonesthatdetecttheoscillationsofairpressure inasoundwave,anddevicessuchasradioreceiversthatmeasure theoscillationoftheelectricandmagneticeldsinalightwave, butnobodyhaseverfoundanywaytomeasureanelectronwave directly. Section4.1ElectronsasWaves 89

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b / Thesetwoelectronwaves arenotdistinguishablebyany measuringdevice. Wecanofcoursedetecttheenergyormomentumpossessedby anelectronjustaswecoulddetecttheenergyofaphotonusinga digitalcamera.InfactI'dimaginethatanunmodieddigitalcamerachipplacedinavacuumchamberwoulddetectelectronsjustas handilyasphotons.Butthisonlyallowsustodeterminewherethe wavecarrieshighprobabilityandwhereitcarrieslowprobability. Probabilityisproportionaltothesquareofthewave'samplitude, butmeasuringitssquareisnotthesameasmeasuringthewave itself.Inparticular,wegetthesameresultbysquaringeithera positivenumberoritsnegative,sothereisnowaytodeterminethe positiveornegativesignofanelectronwave. Mostphysiciststendtowardtheschoolofphilosophyknownas operationalism,whichsaysthataconceptisonlymeaningfulifwe candenesomesetofoperationsforobserving,measuring,ortestingit.Accordingtoastrictoperationalist,then,theelectronwave itselfisameaninglessconcept.Nevertheless,itturnsouttobeone ofthoseconceptslikeloveorhumorthatisimpossibletomeasure andyetveryusefultohavearound.Wethereforegiveitasymbol, thecapitalGreekletterpsi,andaspecialname,theelectron wavefunction becauseitisafunctionofthecoordinates x y ,and z thatspecifywhereyouareinspace.Itwouldbeimpossible,for example,tocalculatetheshapeoftheelectronwaveinahydrogenatomwithouthavingsomesymbolforthewave.Butwhenthe calculationproducesaresultthatcanbecompareddirectlytoexperiment,thenalalgebraicresultwillturnouttoinvolveonly 2 whichiswhatisobservable,notitself. Since,unlike E and B ,isnotdirectlymeasurable,wearefree tomaketheprobabilityequationshaveasimpleform:insteadof havingtheprobabilitydensityequaltosomefunnyconstantmultipliedby 2 ,wesimplydenesothattheconstantofproportionalityisone: probabilitydensity= 2 Sincetheprobabilitydensityhasunitsofm )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 ,theunitsofmust bem )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 = 2 DiscussionQuestion A Frequencyisoscillationspersecond,whereaswavelengthismeters peroscillation.Howcouldtheequations E = hf and p = h = bemade tolookmorealikebyusingquantitiesthatweremorecloselyanalogous? Thismoresymmetrictreatmentmakesiteasiertoincorporaterelativity intoquantummechanics,sincerelativitysaysthatspaceandtimearenot entirelyseparate. 90 Chapter4MatterasaWave

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c / Partofaninnitesinewave. 4.2 R ? DispersiveWaves Acolleagueofminewhoteacheschemistrylovestotellthestory aboutanexceptionallybrightstudentwho,whentoldoftheequation p = h= ,protested,ButwhenIderivedit,ithadafactorof 2!"Theissuethat'sinvolvedisarealone,albeitonethatcouldbe glossedoverandis,inmosttextbookswithoutraisinganyalarms inthemindoftheaveragestudent.Thepresentoptionalsection addressesthispoint;itisintendedforthestudentwhowishesto delvealittledeeper. Here'showthenow-legendarystudentwaspresumablyreasoning.Westartwiththeequation v = f ,whichisvalidforanysine wave,whetherit'squantumorclassical.Let'sassumewealready know E = hf ,andaretryingtoderivetherelationshipbetween wavelengthandmomentum: = v f = vh E = vh 1 2 mv 2 = 2 h mv = 2 h p Thereasoningseemsvalid,buttheresultdoescontradicttheacceptedone,whichisafterallsolidlybasedonexperiment. Themistakenassumptionisthatwecangureeverythingoutin termsofpuresinewaves.Mathematically,theonlywavethathasa perfectlywelldenedwavelengthandfrequencyisasinewave,and notjustanysinewavebutaninnitelylongone,c.Theunphysical thingaboutsuchawaveisthatithasnoleadingortrailingedge,so itcanneverbesaidtoenterorleaveanyparticularregionofspace. Ourderivationmadeuseofthevelocity, v ,andifvelocityistobea meaningfulconcept,itmusttellushowquicklystumass,energy, momentum,...istransportedfromoneregionofspacetoanother. Sinceaninnitelylongsinewavedoesn'tremoveanystufromone regionandtakeittoanother,thevelocityofitsstu"isnotawell denedconcept. Ofcoursetheindividualwavepeaksdotravelthroughspace,and onemightthinkthatitwouldmakesensetoassociatetheirspeed withthespeedofstu,"butaswewillsee,thetwovelocitiesare ingeneralunequalwhenawave'svelocitydependsonwavelength. Suchawaveiscalleda dispersive wave,becauseawavepulseconsistingofasuperpositionofwavesofdierentwavelengthswillseparate disperseintoitsseparatewavelengthsasthewavesmovethrough spaceatdierentspeeds.Nearlyallthewaveswehaveencountered Section4.2 R ? DispersiveWaves 91

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d / Anite-lengthsinewave. e / Abeatpatterncreatedby superimposingtwosinewaves withslightlydifferentwavelengths. havebeennondispersive.Forinstance,soundwavesandlightwaves inavacuumhavespeedsindependentofwavelength.Awaterwave isonegoodexampleofadispersivewave.Long-wavelengthwater wavestravelfaster,soashipatseathatencountersastormtypicallyseesthelong-wavelengthpartsofthewaverst.Whendealing withdispersivewaves,weneedsymbolsandwordstodistinguish thetwospeeds.Thespeedatwhichwavepeaksmoveiscalledthe phasevelocity, v p ,andthespeedatwhichstu"movesiscalled thegroupvelocity, v g Aninnitesinewavecanonlytellusaboutthephasevelocity, notthegroupvelocity,whichisreallywhatwewouldbetalking aboutwhenwereferredtothespeedofanelectron.Ifaninnite sinewaveisthesimplestpossiblewave,what'sthenextbestthing? Wemightthinktherunnerupinsimplicitywouldbeawavetrain consistingofachopped-osegmentofasinewave,d.However,this kindofwavehaskinksinitattheend.Asimplewaveshouldbe onethatwecanbuildbysuperposingasmallnumberofinnite sinewaves,butakinkcanneverbeproducedbysuperposingany numberofinnitelylongsinewaves. Actuallythesimplestwavethattransportsstufromplaceto placeisthepatternshowninguree.Calledabeatpattern,itis formedbysuperposingtwosinewaveswhosewavelengthsaresimilar butnotquitethesame.Ifyouhaveeverheardthepulsatinghowling soundofmusiciansintheprocessoftuningtheirinstrumentstoeach other,youhaveheardabeatpattern.Thebeatpatterngetsstronger andweakerasthetwosinewavesgoinandoutofphasewitheach other.Thebeatpatternhasmorestu"energy,forexample intheareaswhereconstructiveinterferenceoccurs,andlessinthe regionsofcancellation.Asthewholepatternmovesthroughspace, stuistransportedfromsomeregionsandintootherones. Ifthefrequencyofthetwosinewavesdiersby10%,forinstance,thentenperiodswillbeoccurbetweentimeswhentheyare inphase.Anotherwayofsayingitisthatthesinusoidalenvelope" thedashedlinesingureehasafrequencyequaltothedierence infrequencybetweenthetwowaves.Forinstance,ifthewaveshad frequenciesof100Hzand110Hz,thefrequencyoftheenvelope wouldbe10Hz. Toapplysimilarreasoningtothewavelength,wemustdenea quantity z =1 = thatrelatestowavelengthinthesamewaythat frequencyrelatestoperiod.Intermsofthisnewvariable,the z of theenvelopeequalsthedierencebetweenthe z 'softhetwosine waves. Thegroupvelocityisthespeedatwhichtheenvelopemoves throughspace.Let f and z bethedierencesbetweenthe frequenciesand z 'softhetwosinewaves,whichmeansthatthey equalthefrequencyand z oftheenvelope.Thegroupvelocityis 92 Chapter4MatterasaWave

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f / Threepossiblestandingwavepatternsforaparticleina box. v g = f envelope envelope = f= z.If f and z aresuciently small,wecanapproximatethisexpressionasaderivative, v g = d f d z Thisexpressionisusuallytakenasthedenitionofthegroupvelocityforwavepatternsthatconsistofasuperpositionofsinewaves havinganarrowrangeoffrequenciesandwavelengths.Inquantummechanics,with f = E=h and z = p=h ,wehave v g =d E= d p Inthecaseofanonrelativisticelectrontherelationshipbetween energyandmomentumis E = p 2 = 2 m ,sothegroupvelocityis d E= d p = p=m = v ,exactlywhatitshouldbe.Itisonlythephase velocitythatdiersbyafactoroftwofromwhatwewouldhave expected,butthephasevelocityisnotthephysicallyimportant thing. 4.3BoundStates Electronsareattheirmostinterestingwhenthey'reinatoms,that is,whentheyareboundwithinasmallregionofspace.Wecan understandagreatdealaboutatomsandmoleculesbasedonsimple argumentsaboutsuchboundstates,withoutgoingintoanyofthe realisticdetailsofatom.Thesimplestmodelofaboundstateis knownastheparticleinabox:likeaballonapooltable,the electronfeelszeroforcewhileintheinterior,butwhenitreaches anedgeitencountersawallthatpushesbackinwardonitwith alargeforce.Inparticlelanguage,wewoulddescribetheelectron asbouncingoofthewall,butthisincorrectlyassumesthatthe electronhasacertainpaththroughspace.Itismorecorrectto describetheelectronasawavethatundergoes100%reectionat theboundariesofthebox. Likeagenerationofphysicsstudentsbeforeme,Irolledmy eyeswheninitiallyintroducedtotheunrealisticideaofputtinga particleinabox.Itseemedcompletelyimpractical,anarticial textbookinvention.Today,however,ithasbecomeroutinetostudy electronsinrectangularboxesinactuallaboratoryexperiments.The box"isactuallyjustanemptycavitywithinasolidpieceofsilicon, amountinginvolumetoafewhundredatoms.Themethodsfor creatingtheseelectron-in-a-boxsetupsknownasquantumdots" wereaby-productofthedevelopmentoftechnologiesforfabricating computerchips. Forsimplicitylet'simagineaone-dimensionalelectroninabox, i.e.,weassumethattheelectronisonlyfreetomovealongaline. Theresultingstandingwavepatterns,ofwhichtherstthreeare showninguref,arejustlikesomeofthepatternsweencountered withsoundwavesinmusicalinstruments.Thewavepatternsmust bezeroattheendsofthebox,becauseweareassumingthewalls areimpenetrable,andthereshouldthereforebezeroprobabilityof Section4.3BoundStates 93

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g / Thespectrumofthelight fromthestarSirius.Photograph bytheauthor. ndingtheelectronoutsidethebox.Eachwavepatternislabeled accordingto n ,thenumberofpeaksandvalleysithas.Inquantumphysics,thesewavepatternsarereferredtoasstates"ofthe particle-in-the-boxsystem. Thefollowingseeminglyinnocuousobservationsabouttheparticleintheboxleadusdirectlytothesolutionstosomeofthemost vexingfailuresofclassicalphysics: Theparticle'senergyisquantizedcanonlyhavecertainvalues. Eachwavelengthcorrespondstoacertainmomentum,andagiven momentumimpliesadenitekineticenergy, E = p 2 = 2 m .Thisis thesecondtypeofenergyquantizationwehaveencountered.The typewestudiedpreviouslyhadtodowithrestrictingthenumber ofparticlestoawholenumber,whileassumingsomespecicwavelengthandenergyforeachparticle.Thistypeofquantizationrefers totheenergiesthatasingleparticlecanhave.Bothphotonsand matterparticlesdemonstratebothtypesofquantizationunderthe appropriatecircumstances. Theparticlehasaminimumkineticenergy. Longwavelengthscorrespondtolowmomentaandlowenergies.Therecanbenostate withanenergylowerthanthatofthe n =1state,calledtheground state. Thesmallerthespaceinwhichtheparticleisconned,thehigher itskineticenergymustbe. Again,thisisbecauselongwavelengths givelowerenergies. Spectraofthingasesexample3 Afactthatwasinexplicablebyclassicalphysicswasthatthin gasesabsorbandemitlightonlyatcertainwavelengths.This wasobservedbothinearthboundlaboratoriesandinthespectra ofstars.Figuregshowstheexampleofthespectrumofthestar Sirius,inwhichtherearegapteethatcertainwavelengths.Takingthisspectrumasanexample,wecangiveastraightforward explanationusingquantumphysics. Energyisreleasedinthedenseinteriorofthestar,buttheouter layersofthestararethin,sotheatomsarefarapartandelectrons areconnedwithinindividualatoms.Althoughtheirstandingwavepatternsarenotassimpleasthoseoftheparticleinthe box,theirenergiesarequantized. Whenaphotonisonitswayoutthroughtheouterlayers,itcanbe absorbedbyanelectroninanatom,butonlyiftheamountofenergyitcarrieshappenstobetherightamounttokicktheelectron fromoneoftheallowedenergylevelstooneofthehigherlevels.Thephotonenergiesthataremissingfromthespectrumare theonesthatequalthedifferenceinenergybetweentwoelectronenergylevels.Themostprominentoftheabsorptionlinesin Sirius'sspectrumareabsorptionlinesofthehydrogenatom. 94 Chapter4MatterasaWave

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h / Example5:TwohydrogenatomsbondtoformanH 2 molecule.Inthemolecule,the twoelectrons'wavepatterns overlap,andareabouttwiceas wide. Thestabilityofatomsexample4 InmanyStarTrekepisodestheEnterprise,inorbitaroundaplanet, suddenlylostenginepowerandbeganspiralingdowntowardthe planet'ssurface.Thiswasutternonsense,ofcourse,duetoconservationofenergy:theshiphadnowayofgettingridofenergy, soitdidnotneedtheenginestoreplenishit. Consider,however,theelectroninanatomasitorbitsthenucleus.Theelectron does haveawaytoreleaseenergy:ithasan accelerationduetoitscontinuouslychangingdirectionofmotion, andaccordingtoclassicalphysics,anyacceleratingchargedparticleemitselectromagneticwaves.Accordingtoclassicalphysics, atomsshouldcollapse! Thesolutionliesintheobservationthataboundstatehasaminimumenergy.Anelectroninoneofthehigher-energyatomic statescananddoesemitphotonsandhopdownstepbystepin energy.Butonceitisinthegroundstate,itcannotemitaphoton becausethereisnolower-energystateforittogoto. Chemicalbondsinhydrogenmoleculesexample5 Ibeganthischapterwithaclassicalargumentthatchemical bonds,asinanH 2 molecule,shouldnotexist.Quantumphysics explainswhythistypeofbondingdoesinfactoccur.Whenthe atomsarenexttoeachother,theelectronsaresharedbetween them.Theboxisabouttwiceaswide,andalargerboxallowsa smallerenergy.Energyisrequiredinordertoseparatetheatoms. Aqualitativelydifferenttypeofbondingisdiscussedinonpage 123. DiscussionQuestions A Neutronsattracteachotherviathestrongnuclearforce,soaccording toclassicalphysicsitshouldbepossibletoformnucleioutofclustersof twoormoreneutrons,withnoprotonsatall.Experimentalsearches, however,havefailedtoturnupevidenceofastabletwo-neutronsystem dineutronorlargerstableclusters.Thesesystemsareapparentlynot justunstableinthesenseofbeingabletobetadecaybutunstablein thesensethattheydon'tholdtogetheratall.Explainbasedonquantum physicswhyadineutronmightspontaneouslyyapart. B Thefollowingtableshowstheenergygapbetweenthegroundstate andtherstexcitedstateforfournuclei,inunitsofpicojoules.Thenuclei werechosentobeonesthathavesimilarstructures,e.g.,theyareall sphericalinshape. nucleusenergygappicojoules 4 He3.234 16 O0.968 40 Ca0.536 208 Pb0.418 Explainthetrendinthedata. Section4.3BoundStates 95

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4.4TheUncertaintyPrinciple Theuncertaintyprinciple Eliminatingrandomnessthroughmeasurement? Acommonreactiontoquantumphysics,amongbothearlytwentieth-centuryphysicistsandmodernstudents,isthatweshould beabletogetridofrandomnessthroughaccuratemeasurement.If Isay,forexample,thatitismeaninglesstodiscussthepathofa photonoranelectron,youmightsuggestthatwesimplymeasure theparticle'spositionandvelocitymanytimesinarow.Thisseries ofsnapshotswouldamounttoadescriptionofitspath. Apracticalobjectiontothisplanisthattheprocessofmeasurementwillhaveaneectonthethingwearetryingtomeasure.This maynotbeofmuchconcern,forexample,whenatraccopmeasuresyourcar'smotionwitharadargun,becausetheenergyand momentumoftheradarpulsesaren'tenoughtochangethecar's motionsignicantly.Butonthesubatomicscaleitisaveryreal problem.Makingavideotapeofanelectronorbitinganucleusis notjustdicult,itistheoreticallyimpossible,evenwiththevideo camerahookeduptothebestimaginablemicroscope.Thevideo cameramakespicturesofthingsusinglightthathasbouncedo themandcomeintothecamera.Ifevenasinglephotonofthe rightwavelengthwastobounceooftheelectronweweretryingto study,theelectron'srecoilwouldbeenoughtochangeitsbehavior signicantlyseehomeworkproblem4. TheHeisenberguncertaintyprinciple Thisinsight,thatmeasurementchangesthethingbeingmeasured,isthekindofideathatclove-cigarette-smokingintellectuals outsideofthephysicalsciencesliketoclaimtheyknewallalong.If only,theysay,thephysicistshadmademoreofahabitofreading literaryjournals,theycouldhavesavedalotofwork.TheanthropologistMargaretMeadhasrecentlybeenaccusedofinadvertently encouragingherteenagedSamoaninformantstoexaggeratethefreedomofyouthfulsexualexperimentationintheirsociety.Ifthisis consideredadamningcritiqueofherwork,itisbecauseshecould havedonebetter:otheranthropologistsclaimtohavebeenableto eliminatetheobserver-as-participantproblemandcollectuntainted data. TheGermanphysicistWernerHeisenberg,however,showedthat inquantumphysics, any measuringtechniquerunsintoabrickwall whenwetrytoimproveitsaccuracybeyondacertainpoint.Heisenbergshowedthatthelimitationisaquestionof whatthereistobe known ,eveninprinciple,aboutthesystemitself,notoftheinabilityofaparticularmeasuringdevicetoferretoutinformationthat isknowable. 96 Chapter4MatterasaWave

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Suppose,forexample,thatwehaveconstructedanelectronina boxquantumdotsetupinourlaboratory,andweareabletoadjust thelength L oftheboxasdesired.Allthestandingwavepatterns prettymuchllthebox,soourknowledgeoftheelectron'sposition isoflimitedaccuracy.Ifwewrite x fortherangeofuncertainty inourknowledgeofitsposition,then x isroughlythesameasthe lengthofthebox: x L Ifwewishtoknowitspositionmoreaccurately,wecancertainly squeezeitintoasmallerspacebyreducing L ,butthishasanunintendedside-eect.Astandingwaveisreallyasuperpositionoftwo travelingwavesgoinginoppositedirections.Theequation p = h= onlygivesthemagnitudeofthemomentumvector,notitsdirection,soweshouldreallyinterpretthewaveasa50/50mixtureof aright-goingwavewithmomentum p = h= andaleft-goingone withmomentum p = )]TJ/F20 10.9091 Tf 8.485 0 Td [(h= .Theuncertaintyinourknowledgeof theelectron'smomentumis p =2 h= ,coveringtherangebetween thesetwovalues.Evenifwemakesuretheelectronisintheground state,whosewavelength =2 L isthelongestpossible,wehavean uncertaintyinmomentumof p = h=L .Ingeneral,wend p & h=L withequalityforthegroundstateandinequalityforthehigherenergystates.Thusifwereduce L toimproveourknowledgeofthe electron'sposition,wedosoatthecostofknowinglessaboutits momentum.Thistrade-oisneatlysummarizedbymultiplyingthe twoequationstogive p x & h Althoughwehavederivedthisinthespecialcaseofaparticleina box,itisanexampleofaprincipleofmoregeneralvalidity: theHeisenberguncertaintyprinciple Itisnotpossible,eveninprinciple,toknowthemomentumandthe positionofaparticlesimultaneouslyandwithperfectaccuracy.The uncertaintiesinthesetwoquantitiesarealwayssuchthat p x & h Thisapproximationcanbemadeintoastrictinequality, p x> h= 4 ,butonlywithmorecarefuldenitions,whichwewillnot botherwith. 1 NotethatalthoughIencouragedyoutothinkofthisderivation intermsofaspecicreal-worldsystem,thequantumdot,Inever 1 Seehomeworkproblems6and7. Section4.4TheUncertaintyPrinciple 97

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madeanyreferencetospecicmeasuringequipment.Theargument issimplythatwecannot know theparticle'spositionveryaccurately unlessit has averywelldenedposition,itcannothaveaverywell denedpositionunlessitswave-patterncoversonlyaverysmall amountofspace,anditswave-patterncannotbethuscompressed withoutgivingitashortwavelengthandacorrespondinglyuncertainmomentum.Theuncertaintyprincipleisthereforearestriction onhowmuchthereistoknowaboutaparticle,notjustonwhatwe canknowaboutitwithacertaintechnique. Anestimateforelectronsinatomsexample6 Atypicalenergyforanelectroninanatomisontheorderof volt e ,whichcorrespondstoaspeedofabout1%ofthespeed oflight.Ifatypicalatomhasasizeontheorderof0.1nm,how closearetheelectronstothelimitimposedbytheuncertainty principle? Ifweassumetheelectronmovesinalldirectionswithequal probability,theuncertaintyinitsmomentumisroughlytwiceits typicalmomentum.Thisonlyanorder-of-magnitudeestimate,so wetake p tobethesameasatypicalmomentum: p x = p typical x = m electron .01 c .1 10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(9 m =3 10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(34 J s ThisisonthesameorderofmagnitudeasPlanck'sconstant,so evidentlytheelectronisrightupagainstthewall.Thefactthat itissomewhatlessthan h isofnoconcernsincethiswasonlyan estimate,andwehavenotstatedtheuncertaintyprincipleinits mostexactform. self-checkB Ifweweretoapplytheuncertaintyprincipletohuman-scaleobjects, whatwouldbethesignicanceofthesmallnumericalvalueofPlanck's constant? Answer,p.133 self-checkC Supposerainisfallingonyourroof,andthereisatinyholethatlets raindropsintoyourlivingroomnowandthen.Allthesedropshitthe samespotontheoor,sotheyhavethesamevalueof x .Notonly that,butiftherainisfallingstraightdown,theyallhavezerohorizontal momentum.Thusitseemsthattheraindropshave p =0, x =0, and p x =0,violatingtheuncertaintyprinciple.Tolookfortheholein thisargument,considerhowitwouldbeactedoutonthemicroscopic scale:anelectronwavecomesalongandhitsanarrowslit.Whatreally happens? Answer,p.133 HistoricalNote ThetruenatureofHeisenberg'sroleintheNaziatomicbombeffortis afascinatingquestion,anddramaticenoughtohaveinspiredawellreceived1998theatricalplay,Copenhagen.Therealstory,however, 98 Chapter4MatterasaWave

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i / WernerHeisenberg. mayneverbecompletelyunraveled.Heisenbergwasthescienticleader oftheGermanbombprogramupuntilitscancellationin1942,whenthe Germanmilitarydecidedthatitwastooambitiousaprojecttoundertake inwartime,andtoounlikelytoproduceresults. SomehistoriansbelievethatHeisenbergintentionallydelayedand obstructedtheprojectbecausehesecretlydidnotwanttheNazisto getthebomb.Heisenberg'sapologistspointoutthatheneverjoined theNaziparty,andwasnotanti-Semitic.Heactivelyresistedthegovernment's Deutsche-Physik policyofeliminatingsupposedJewishinuencesfromphysics,andasaresultwasdenouncedbytheS.S.asa traitor,escapingpunishmentonlybecauseHimmlerpersonallydeclared himinnocent.Onestrongpieceofevidenceisasecretmessagecarried totheU.S.in1941,byoneofthelastJewstoescapefromBerlin,and eventuallydeliveredtothechairmanoftheUraniumCommittee,which wasthenstudyingthefeasibilityofabomb.Themessagestated...that alargenumberofGermanphysicistsareworkingintensivelyonthe problemoftheuraniumbombunderthedirectionofHeisenberg,[and] thatHeisenberghimselftriestodelaytheworkasmuchaspossible, fearingthecatastrophicresultsofsuccess.Buthecannothelpfulllingtheordersgiventohim,andiftheproblemcanbesolved,itwillbe solvedprobablyinthenearfuture.Sohegavetheadvicetoustohurry upifU.S.A.willnotcometoolate.Themessagesupportstheviewthat Heisenbergintentionallymisledhisgovernmentaboutthebomb'stechnicalfeasibility;GermanMinisterofArmamentsAlbertSpeerwrotethat hewasconvincedtodroptheprojectaftera1942meetingwithHeisenbergbecausethephysiciststhemselvesdidn'twanttoputtoomuch intoit.HeisenbergalsomayhavewarnedDanishphysicistNielsBohr personallyinSeptember1941abouttheexistenceoftheNazibomb effort. Ontheothersideofthedebate,criticsofHeisenbergsaythathe clearlywantedGermanytowinthewar,thathevisitedGerman-occupied territoriesinasemi-ofcialrole,andthathesimplymaynothavebeen verygoodathisjobdirectingthebombproject.OnavisittotheoccupiedNetherlandsin1943,hetoldacolleague,Democracycannot developsufcientenergytoruleEurope.Thereare,therefore,onlytwo alternatives:GermanyandRussia.AndthenaEuropeunderGerman leadershipwouldbethelesserevil.Somehistorians 2 arguethatthe realpointofHeisenberg'smeetingwithBohrwastotrytoconvincethe U.S.nottotrytobuildabomb,sothatGermany,possessinganuclear monopoly,woulddefeattheSovietsthiswasaftertheJune1941entryoftheU.S.S.R.intothewar,butbeforetheDecember1941Pearl HarborattackbroughttheU.S.in.BohrapparentlyconsideredHeisenberg'saccountofthemeeting,publishedafterthewarwasover,tobe inaccurate. 3 Thesecret1941messagealsohasacuriousmoralpassivitytoit,asifHeisenbergwassayingIhopeyoustopmebeforeIdo somethingbad,butweshouldalsoconsiderthegreatriskHeisenberg wouldhavebeenrunningifheactuallyoriginatedthemessage. 2 AHistoricalPerspectiveonCopenhagen,DavidC.Cassidy,PhysicsToday, July2000,p.28,http://www.aip.org/pt/vol-53/iss-7/p28.html 3 Bohrdraftedseveralreplies,butneverpublishedthemforfearofhurting Heisenbergandhisfamily.Bohr'spapersweretobesealedfor50yearsafterhis death,buthisfamilyreleasedthemearly,inFebruary2002. Section4.4TheUncertaintyPrinciple 99

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j / Schr odinger'scat. MeasurementandSchr odinger'scat Inchapter3Ibrieymentionedanissueconcerningmeasurementthatwearenowreadytoaddresscarefully.Ifyouhangaround alaboratorywherequantum-physicsexperimentsarebeingdone andsecretlyrecordthephysicists'conversations,you'llhearthem saymanythingsthatassumetheprobabilityinterpretationofquantummechanics.Usuallytheywillspeakasthoughtherandomness ofquantummechanicsentersthepicturewhensomethingismeasured.Inthedigitalcameraexperimentsofchapter3,forexample, theywouldcasuallydescribethedetectionofaphotonatoneofthe pixelsasifthemomentofdetectionwaswhenthephotonwasforced tomakeupitsmind."Althoughthismentalcartoonusuallyworks fairlywellasadescriptionofthingsoneexperiencesinthelab,it cannotultimatelybecorrect,becauseitattributesaspecialroleto measurement,whichisreallyjustaphysicalprocesslikeallother physicalprocesses. Ifwearetondaninterpretationthatavoidsgivinganyspecialroletomeasurementprocesses,thenwemustthinkoftheentire laboratory,includingthemeasuringdevicesandthephysiciststhemselves,asonebigquantum-mechanicalsystemmadeoutofprotons, neutrons,electrons,andphotons.Inotherwords,weshouldtake quantumphysicsseriouslyasadescriptionnotjustofmicroscopic objectslikeatomsbutofhuman-scalemacroscopic"thingslike theapparatus,thefurniture,andthepeople. ThemostcelebratedexampleiscalledtheSchrodinger'scatexperiment.Luckilyforthecat,thereprobablywasnoactualexperiment|itwassimplyathoughtexperiment"thatthephysicisttheGermantheoristSchrodingerdiscussedwithhiscolleagues. Schrodingerwrote: Onecanevenconstructquiteburlesquecases.Acatisshutupinasteel container,togetherwiththefollowingdiabolicalapparatuswhichone mustkeepoutofthedirectclutchesofthecat:Ina[radiationdetector] thereisatinymassofradioactivesubstance,solittlethatinthecourse ofanhourperhapsoneatomofitdisintegrates,butalsowithequal probabilitynotevenone;ifitdoeshappen,the[detector]respondsand ...activatesahammerthatshattersalittleaskofprussicacid[lling thechamberwithpoisongas].Ifonehasleftthisentiresystemtoitself foranhour,thenonewillsaytohimselfthatthecatisstillliving,if inthattimenoatomhasdisintegrated.Therstatomicdisintegration wouldhavepoisonedit. Nowcomesthestrangepart.Quantummechanicssaysthatthe particlesthecatismadeofhavewaveproperties,includingthe propertyofsuperposition.Schrodingerdescribesthewavefunction ofthebox'scontentsattheendofthehour: Thewavefunctionoftheentiresystemwouldexpressthissituationby havingthelivingandthedeadcatmixed...inequalparts[50/50proportions].Theuncertaintyoriginallyrestrictedtotheatomicdomain 100 Chapter4MatterasaWave

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k / Anelectroninagentle electriceldgraduallyshortens itswavelengthasitgainsenergy. hasbeentransformedintoamacroscopicuncertainty... AtrstSchrodinger'sdescriptionseemslikenonsense.Whenyou openedthebox,wouldyouseetwoghostlikecats,asinadoubly exposedphotograph,onedeadandonealive?Obviouslynot.You wouldhaveasingle,fullymaterialcat,whichwouldeitherbedead orvery,veryupset.ButSchrodingerhasanequallystrangeand logicalanswerforthatobjection.Inthesamewaythatthequantum randomnessoftheradioactiveatomspreadtothecatandmadeits wavefunctionarandommixtureoflifeanddeath,therandomness spreadswideronceyouopenthebox,andyourownwavefunction becomesamixtureofapersonwhohasjustkilledacatandaperson whohasn't. DiscussionQuestions A Compare p and x forthetwolowestenergylevelsoftheonedimensionalparticleinabox,anddiscusshowthisrelatestotheuncertaintyprinciple. B Onagraphof p versus x,sketchtheregionsthatareallowedand forbiddenbytheHeisenberguncertaintyprinciple.Interpretthegraph: Wheredoesanatomlieonit?Anelephant?Caneither p or x bemeasuredwithperfectaccuracyifwedon'tcareabouttheother? 4.5ElectronsinElectricFields Sofartheonlyelectronwavepatternswe'veconsideredhave beensimplesinewaves,butwheneveranelectronndsitselfinan electriceld,itmusthaveamorecomplicatedwavepattern.Let's considertheexampleofanelectronbeingacceleratedbytheelectrongunatthebackofaTVtube.Theelectronismovingfrom aregionoflowvoltageintoaregionofhighervoltage.Sinceits chargeisnegative,itlosesPEbymovingtoahighervoltage,soits KEincreases.Asitspotentialenergygoesdown,itskineticenergy goesupbyanequalamount,keepingthetotalenergyconstant.Increasingkineticenergyimpliesagrowingmomentum,andtherefore ashorteningwavelength,k. Thewavefunctionasawholedoesnothaveasinglewell-dened wavelength,butthewavechangessograduallythatifyouonlylook atasmallpartofityoucanstillpickoutawavelengthandrelate ittothemomentumandenergy.Thepictureactuallyexaggeratesbymanyordersofmagnitudetherateatwhichthewavelength changes. Butwhatiftheelectriceldwasstronger?Theelectriceldin aTVisonly 10 5 N/C,buttheelectriceldwithinanatomis morelike10 12 N/C.Ingurel,thewavelengthchangessorapidly thatthereisnothingthatlookslikeasinewaveatall.Wecould getageneralideaofthewavelengthinagivenregionbymeasuring thedistancebetweentwopeaks,butthatwouldonlybearough Section4.5ElectronsinElectricFields 101

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m / 1.Kinkslikethisdon't happen.2.Thewaveactually penetratesintotheclassically forbiddenregion. approximation.Supposewewanttoknowthewavelengthatpoint P.Thetrickistoconstructasinewave,liketheoneshownwiththe dashedline,whichmatchesthecurvatureoftheactualwavefunction ascloselyaspossiblenearP.Thesinewavethatmatchesaswellas possibleiscalledtheosculating"curve,fromaLatinwordmeaning tokiss."Thewavelengthoftheosculatingcurveisthewavelength thatwillrelatecorrectlytoconservationofenergy. l / Atypicalwavefunctionofanelectroninanatomheavycurve andtheosculatingsinewavedashedcurvethatmatchesitscurvature atpointP. Tunneling Weimplicitlyassumedthattheparticle-in-a-boxwavefunction wouldcutoabruptlyatthesidesofthebox,m/1,butthatwould beunphysical.Akinkhasinnitecurvature,andcurvatureisrelated toenergy,soitcan'tbeinnite.Aphysicallyrealisticwavefunction mustalwaystailo"gradually,m/2.Inclassicalphysics,aparticle canneverenteraregioninwhichitspotentialenergywouldbe greaterthantheamountofenergyithasavailable.Butinquantum physicsthewavefunctionwillalwayshaveatailthatreachesinto theclassicallyforbiddenregion.Ifitwasnotforthiseect,called tunneling,thefusionreactionsthatpowerthesunwouldnotoccur duetothehighpotentialenergythatnucleineedinordertoget closetogether!Tunnelingisdiscussedinmoredetailinthenext section. 4.6 R ? TheSchr odingerEquation Insection4.5wewereabletoapplyconservationofenergytoanelectron'swavefunction,butonlybyusingtheclumsygraphicaltechniqueofosculatingsinewavesasameasureofthewave'scurvature. Youhavelearnedamoreconvenientmeasureofcurvatureincalculus:thesecondderivative.Torelatethetwoapproaches,wetake thesecondderivativeofasinewave: d 2 d x 2 sin 2 x = d d x 2 cos 2 x = )]TJ/F26 10.9091 Tf 10.303 15.382 Td [( 2 2 sin 2 x 102 Chapter4MatterasaWave

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Takingthesecondderivativegivesusbackthesamefunction,but withaminussignandaconstantoutinfrontthatisrelatedto thewavelength.Wecanthusrelatethesecondderivativetothe osculatingwavelength: [1] d 2 d x 2 = )]TJ/F26 10.9091 Tf 10.303 15.382 Td [( 2 2 Thiscouldbesolvedfor intermsof,butitwillturnouttobe moreconvenienttoleaveitinthisform. Usingconservationofenergy,wehave E = KE + PE = p 2 2 m + PE = h= 2 2 m + PE [2] Notethatbothequation[1]andequation[2]have 2 inthedenominator.Wecansimplifyouralgebrabymultiplyingbothsidesof equation[2]bytomakeitlookmorelikeequation[1]: E = h= 2 2 m + PE = 1 2 m h 2 2 2 2 + PE = )]TJ/F15 10.9091 Tf 14.469 7.38 Td [(1 2 m h 2 2 d 2 d x 2 + PE Furthersimplicationisachievedbyusingthesymbol ~ h witha slashthroughit,readh-bar"asanabbreviationfor h= 2 .Wethen havetheimportantresultknownasthe Schrodingerequation : E = )]TJ/F32 10.9091 Tf 11.883 7.38 Td [(~ 2 2 m d 2 d x 2 + PE ActuallythisisasimpliedversionoftheSchrodingerequation, applyingonlytostandingwavesinonedimension.Physicallyitis astatementofconservationofenergy.Thetotalenergy E mustbe constant,sotheequationtellsusthatachangeinpotentialenergy mustbeaccompaniedbyachangeinthecurvatureofthewavefunction.Thischangeincurvaturerelatestoachangeinwavelength, whichcorrespondstoachangeinmomentumandkineticenergy. Section4.6 R ? TheSchr odingerEquation 103

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n / Tunnelingthroughabarrier. self-checkD ConsideringtheassumptionsthatweremadeinderivingtheSchr odinger equation,woulditbecorrecttoapplyittoaphoton?Toanelectronmovingatrelativisticspeeds? Answer,p. 134 Usuallyweknowrightothebathowthepotentialenergydependson x ,sothebasicmathematicalproblemofquantumphysics istondafunction x thatsatisestheSchrodingerequation foragivenfunction PE x .Anequation,suchastheSchrodinger equation,thatspeciesarelationshipbetweenafunctionandits derivativesisknownasadierentialequation. Thestudyofdierentialequationsingeneralisbeyondthemathematicallevelofthisbook,butwecangainsomeimportantinsights byconsideringtheeasiestversionoftheSchrodingerequation,in whichthepotentialenergyisconstant.Wecanthenrearrangethe Schrodingerequationasfollows: d 2 d x 2 = 2 m PE )]TJ/F20 10.9091 Tf 10.909 0 Td [(E ~ 2 whichboilsdownto d 2 d x 2 = a where,accordingtoourassumptions, a isindependentof x .Weneed tondafunctionwhosesecondderivativeisthesameastheoriginal functionexceptforamultiplicativeconstant.Theonlyfunctions withthispropertyaresinewavesandexponentials: d 2 d x 2 [ q sin rx + s ]= )]TJ/F20 10.9091 Tf 8.485 0 Td [(qr 2 sin rx + s d 2 d x 2 qe rx + s = qr 2 e rx + s Thesinewavegivesnegativevaluesof a a = )]TJ/F20 10.9091 Tf 8.485 0 Td [(r 2 ,andtheexponentialgivespositiveones, a = r 2 .Theformerappliestotheclassically allowedregionwith PE
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wavefunctionontheleftsideofthebarriertothewavefunctionon therightis qe rx + s qe r x + w + s = e )]TJ/F21 7.9701 Tf 6.587 0 Td [(rw Probabilitiesareproportionaltothesquaresofwavefunctions,so theprobabilityofmakingitthroughthebarrieris P = e )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 rw =exp )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(2 w ~ p 2 m PE )]TJ/F20 10.9091 Tf 10.909 0 Td [(E self-checkE Ifweweretoapplythisequationtondtheprobabilitythatapersoncan walkthroughawall,whatwouldthesmallvalueofPlanck'sconstant imply? Answer,p.134 Useofcomplexnumbers Inaclassicallyforbiddenregion,aparticle'stotalenergy, PE + KE ,islessthanits PE ,soits KE mustbenegative.Ifwewantto keepbelievingintheequation KE = p 2 = 2 m ,thenapparentlythe momentumoftheparticleisthesquarerootofanegativenumber. ThisisasymptomofthefactthattheSchrodingerequationfails todescribeallofnatureunlessthewavefunctionandvariousother quantitiesareallowedtobecomplexnumbers.Inparticularitisnot possibletodescribetravelingwavescorrectlywithoutusingcomplex wavefunctions. Thismayseemlikenonsense,sincerealnumbersaretheonly onesthatare,well,real!Quantummechanicscanalwaysberelatedtotherealworld,however,becauseitsstructureissuchthat theresultsofmeasurementsalwayscomeouttoberealnumbers. Forexample,wemaydescribeanelectronashavingnon-realmomentuminclassicallyforbiddenregions,butitsaveragemomentum willalwayscomeouttoberealtheimaginarypartsaverageoutto zero,anditcannevertransferanon-realquantityofmomentum toanotherparticle. Acompleteinvestigationoftheseissuesisbeyondthescopeof thisbook,andthisiswhywehavenormallylimitedourselvesto standingwaves,whichcanbedescribedwithreal-valuedwavefunctions. Section4.6 R ? TheSchr odingerEquation 105

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Summary SelectedVocabulary wavefunction..thenumericalmeasureofanelectronwave,or ingeneralofthewavecorrespondingtoany quantummechanicalparticle Notation ~ ..........Planck'sconstantdividedby2 usedonlyin optionalsection4.6 .........thewavefunctionofanelectron Summary Lightisbothaparticleandawave.Matterisbothaparticleand awave.Theequationsthatconnecttheparticleandwaveproperties arethesameinallcases: E = hf p = h= Unliketheelectricandmagneticeldsthatmakeupaphotonwave,theelectronwavefunctionisnotdirectlymeasurable.Only thesquareofthewavefunction,whichrelatestoprobability,has directphysicalsignicance. Aparticlethatisboundwithinacertainregionofspaceisa standingwaveintermsofquantumphysics.Thetwoequations abovecanthenbeappliedtothestandingwavetoyieldsomeimportantgeneralobservationsaboutboundparticles: 1.Theparticle'senergyisquantizedcanonlyhavecertainvalues. 2.Theparticlehasaminimumenergy. 3.Thesmallerthespaceinwhichtheparticleisconned,the higheritskineticenergymustbe. Theseimmediatelyresolvethedicultiesthatclassicalphysicshad encounteredinexplainingobservationssuchasthediscretespectra ofatoms,thefactthatatomsdon'tcollapsebyradiatingawaytheir energy,andtheformationofchemicalbonds. Astandingwaveconnedtoasmallspacemusthaveashort wavelength,whichcorrespondstoalargemomentuminquantum physics.Sinceastandingwaveconsistsofasuperpositionoftwo travelingwavesmovinginoppositedirections,thislargemomentum shouldactuallybeinterpretedasanequalmixtureoftwopossible momenta:alargemomentumtotheleft,oralargemomentumto theright.Thusitisnotpossibleforaquantumwave-particleto beconnedtoasmallspacewithoutmakingitsmomentumvery 106 Chapter4MatterasaWave

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uncertain.Ingeneral,theHeisenberguncertaintyprinciplestates thatitisnotpossibletoknowthepositionandmomentumofa particlesimultaneouslywithperfectaccuracy.Theuncertaintiesin thesetwoquantitiesmustsatisfytheapproximateinequality p x & h Whenanelectronissubjectedtoelectricforces,itswavelength cannotbeconstant.Thewavelength"tobeusedintheequation p = h= shouldbethoughtofasthewavelengthofthesinewave thatmostcloselyapproximatesthecurvatureofthewavefunction ataspecicpoint. Innitecurvatureisnotphysicallypossible,sorealisticwavefunctionscannothavekinksinthem,andcannotjustcutoabruptly attheedgeofaregionwheretheparticle'senergywouldbeinsucienttopenetrateaccordingtoclassicalphysics.Instead,the wavefunctiontailso"intheclassicallyforbiddenregion,andasa consequenceitispossibleforparticlestotunnel"throughregions whereaccordingtoclassicalphysicstheyshouldnotbeabletopenetrate.Ifthisquantumtunnelingeectdidnotexist,therewould benofusionreactionstopoweroursun,becausetheenergiesof thenucleiwouldbeinsucienttoovercometheelectricalrepulsion betweenthem. ExploringFurther TheNewWorldofMr.Tompkins:GeorgeGamow'sClassicMr.TompkinsinPaperback ,GeorgeGamow.Mr.Tompkinsndshimselfinaworldwherethespeedoflightisonly30miles perhour,makingrelativisticeectsobvious.Laterpartsofthebook playsimilargameswithPlanck'sconstant. TheFirstThreeMinutes:AModernViewoftheOriginof theUniverse ,StevenWeinberg.Surprisinglysimpleideasallow ustounderstandtheinfancyoftheuniversesurprisinglywell. ThreeRoadstoQuantumGravity ,LeeSmolin.Thegreatestembarrassmentofphysicstodayisthatweareunabletofully reconcilegeneralrelativitythetheoryofgravitywithquantum mechanics.Thisbookdoesagoodjobofintroducingthelayreader toadicult,speculativesubject,andshowingthateventhough wedon'thaveafulltheoryofquantumgravity,wedohaveaclear outlineofwhatsuchatheorymustlooklike. Summary 107

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Inatelevision,supposetheelectronsareacceleratedfromrest throughavoltagedierenceof10 4 V.Whatistheirnalwavelength? p 2 UsetheHeisenberguncertaintyprincipletoestimatethe minimumvelocityofaprotonorneutronina 208 Pbnucleus,which hasadiameterofabout13fmfm=10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(15 m.Assumethat thespeedisnonrelativistic,andthencheckattheendwhetherthis assumptionwaswarranted. p 3 Afreeelectronthatcontributestothecurrentinanohmic materialtypicallyhasaspeedof10 5 m/smuchgreaterthanthe driftvelocity. aEstimateitsdeBrogliewavelength,innm. p bIfacomputermemorychipcontains10 8 electriccircuitsina 1cm 2 area,estimatethelinearsize,innm,ofonesuchcircuit. p cBasedonyouranswersfrompartsaandb,doesanelectrical engineerdesigningsuchachipneedtoworryaboutwaveeects suchasdiraction? dEstimatethemaximumnumberofelectriccircuitsthatcanton a1cm 2 computerchipbeforequantum-mechanicaleectsbecome important. 4 Onpage96,Idiscussedtheideaofhookingupavideocameratoavisible-lightmicroscopeandrecordingthetrajectoryofan electronorbitinganucleus.Anelectroninanatomtypicallyhasa speedofabout1%ofthespeedoflight. aCalculatethemomentumoftheelectron. p bWhenwemakeimageswithphotons,wecan'tresolvedetails thataresmallerthanthephotons'wavelength.Supposewewanted tomapoutthetrajectoryoftheelectronwithanaccuracyof0.01 nm.Whatpartoftheelectromagneticspectrumwouldwehaveto use? cAsfoundinhomeworkproblem10onpage40,themomentumof aphotonisgivenby p = E=c .Estimatethemomentumofaphoton ofhavingthenecessarywavelength. p dComparingyouranswersfrompartsaandc,whatwouldbethe eectontheelectronifthephotonbouncedoofit?Whatdoes thistellyouaboutthepossibilityofmappingoutanelectron'sorbit aroundanucleus? 5 Findtheenergyofaparticleinaone-dimensionalboxoflength 108 Chapter4MatterasaWave

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L ,expressingyourresultintermsof L ,theparticle'smass m ,the numberofpeaksandvalleys n inthewavefunction,andfundamental constants. p 6 TheHeisenberguncertaintyprinciple, p x & h ,canonlybe madeintoastrictinequalityifweagreeonarigorousmathematical denitionof x and p .Supposewedenethedeltasintermsofthe fullwidthathalfmaximumFWHM,whichwerstencounteredin VibrationsandWaves ,andrevisitedonpage51ofthisbook.Now considerthelowest-energystateoftheone-dimensionalparticlein abox.Asarguedonpage97,themomentumhasequalprobability ofbeing h=L or )]TJ/F20 10.9091 Tf 8.485 0 Td [(h=L ,sotheFWHMdenitiongives p =2 h=L aFind x usingtheFWHMdenition.Keepinmindthatthe probabilitydistributiondependsonthesquareofthewavefunction. bFind x p p 7 If x hasanaveragevalueofzero,thenthestandarddeviation oftheprobabilitydistribution D x isdenedby 2 = s Z D x x 2 d x wheretheintegralrangesoverallpossiblevaluesof x Interpretation:if x onlyhasahighprobabilityofhavingvaluesclose totheaveragei.e.,smallpositiveandnegativevalues,thething beingintegratedwillalwaysbesmall,because x 2 isalwaysasmall number;thestandarddeviationwillthereforebesmall.Squaring x makessurethateitheranumberbelowtheaverage x< 0ora numberabovetheaverage x> 0willcontributeapositiveamount tothestandarddeviation.Wetakethesquarerootofthewhole thingsothatitwillhavethesameunitsas x ,ratherthanhaving unitsof x 2 Redoproblem6usingthestandarddeviationratherthantheFWHM. Hints:Youneedtodeterminetheamplitudeofthewavebased onnormalization.You'llneedthefollowingdeniteintegral: R = 2 )]TJ/F21 7.9701 Tf 6.586 0 Td [(= 2 u 2 cos 2 u d u = 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 = 24. p R 8 Insection4.6wederivedanexpressionfortheprobability thataparticlewouldtunnelthrougharectangularpotentialbarrier.Generalizethistoabarrierofanyshape.[Hints:Firsttry generalizingtotworectangularbarriersinarow,andthenusea seriesofrectangularbarrierstoapproximatetheactualcurveofan arbitrarypotential.Notethatthewidthandheightofthebarrierin theoriginalequationoccurinsuchawaythatallthatmattersisthe areaunderthe PE -versusx curve.Showthatthisisstilltruefor aseriesofrectangularbarriers,andgeneralizeusinganintegral.]If youhaddonethiscalculationinthe1930'syoucouldhavebecome afamousphysicist. R ? Problems 109

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110 Chapter4MatterasaWave

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Awavefunctionofanelectronina hydrogenatom. Chapter5 TheAtom Youcanlearnalotbytakingacarengineapart,butyouwillhave learnedalotmoreifyoucanputitallbacktogetheragainandmake itrun.Halfthejobofreductionismistobreaknaturedowninto itssmallestpartsandunderstandtherulesthosepartsobey.The secondhalfistoshowhowthosepartsgotogether,andthatisour goalinthischapter.Wehaveseenhowcertainfeaturesofallatoms canbeexplainedonagenericbasisintermsofthepropertiesof boundstates,butthiskindofargumentclearlycannottellusany detailsofthebehaviorofanatomorexplainwhyoneatomacts dierentlyfromanother. Thebiggestembarrassmentforreductionistsisthatthejobof puttingthingsbacktogetherisusuallymuchharderthanthetaking themapart.Seventyyearsafterthefundamentalsofatomicphysics weresolved,itisonlybeginningtobepossibletocalculateaccuratelythepropertiesofatomsthathavemanyelectrons.Systems consistingofmanyatomsareevenharder.Supercomputermanufacturerspointtothefoldingoflargeproteinmoleculesasaprocess whosecalculationisjustbarelyfeasiblewiththeirfastestmachines. Thegoalofthischapteristogiveagentleandvisuallyoriented guidetosomeofthesimplerresultsaboutatoms. 111

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a / Eightwavelengthstaround thiscircle: ` =8. 5.1ClassifyingStates We'llfocusourattentionrstonthesimplestatom,hydrogen,with oneprotonandoneelectron.Weknowinadvancealittleofwhat weshouldexpectforthestructureofthisatom.Sincetheelectron isboundtotheprotonbyelectricalforces,itshoulddisplayaset ofdiscreteenergystates,eachcorrespondingtoacertainstanding wavepattern.Weneedtounderstandwhatstatesthereareand whattheirpropertiesare. Whatpropertiesshouldweusetoclassifythestates?Themost sensibleapproachistousedconservedquantities.Energyisone conservedquantity,andwealreadyknowtoexpecteachstateto haveaspecicenergy.Itturnsout,however,thatenergyaloneis notsucient.Dierentstandingwavepatternsoftheatomcan havethesameenergy. Momentumisalsoaconservedquantity,butitisnotparticularly appropriateforclassifyingthestatesoftheelectroninahydrogen atom.Thereasonisthattheforcebetweentheelectronandtheprotonresultsinthecontinualexchangeofmomentumbetweenthem. Whywasn'tthisaproblemforenergyaswell?Kineticenergyand momentumarerelatedby KE = p 2 = 2 m ,sothemuchmoremassiveprotonneverhasverymuchkineticenergy.Wearemakingan approximationbyassumingallthekineticenergyisintheelectron, butitisquiteagoodapproximation. Angularmomentumdoeshelpwithclassication.Thereisno transferofangularmomentumbetweentheprotonandtheelectron, sincetheforcebetweenthemisacenter-to-centerforce,producing notorque. Likeenergy,angularmomentumisquantizedinquantumphysics. Asanexample,consideraquantumwave-particleconnedtoacircle,likeawaveinacircularmoatsurroundingacastle.Asine waveinsuchaquantummoat"cannothaveanyoldwavelength, becauseanintegernumberofwavelengthsmusttaroundthecircumference, C ,ofthemoat.Thelargerthisintegeris,theshorter thewavelength,andashorterwavelengthrelatestogreatermomentumandangularmomentum.Sincethisintegerisrelatedtoangular momentum,weusethesymbol ` forit: = C ` Theangularmomentumis L = rp Here, r = C= 2 ,and p = h= = h`=C ,so L = C 2 h` C = h 2 ` 112 Chapter5TheAtom

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b / Theangularmomentum vectorofaspinningtop. Intheexampleofthequantummoat,angularmomentumis quantizedinunitsof h= 2 ,andthisturnsouttobeacompletely generalfactaboutquantumphysics.Thatmakes h= 2 apretty importantnumber,sowedenetheabbreviation ~ = h= 2 .This symbolisreadh-bar." quantizationofangularmomentum Theangularmomentumofaparticleduetoitsmotionthrough spaceisquantizedinunitsof ~ self-checkA Whatistheangularmomentumofthewavefunctionshownonpage 111? Answer,p.134 5.2AngularMomentuminThreeDimensions Upuntilnowwe'veonlyworkedwithangularmomentuminthe contextofrotationinaplane,forwhichwecouldsimplyusepositiveandnegativesignstoindicateclockwiseandcounterclockwise directionsofrotation.Ahydrogenatom,however,isunavoidably three-dimensional.Let'srstconsiderthegeneralizationofangularmomentumtothreedimensionsintheclassicalcase,andthen considerhowitcarriesoverintoquantumphysics. Three-dimensionalangularmomentuminclassicalphysics Ifwearetocompletelyspecifytheangularmomentumofaclassicalobjectlikeatop,b,inthreedimensions,it'snotenoughto saywhethertherotationisclockwiseorcounterclockwise.Wemust alsogivetheorientationoftheplaneofrotationor,equivalently,the directionofthetop'saxis.Theconventionistospecifythedirection oftheaxis.Therearetwopossibledirectionsalongtheaxis,and asamatterofconventionweusethedirectionsuchthatifwesight alongit,therotationappearsclockwise. Angularmomentumcan,infact,bedenedasavectorpointing alongthisdirection.Thismightseemlikeastrangedenition,since nothingactuallymovesinthatdirection,butitwouldn'tmakesense todenetheangularmomentumvectorasbeinginthedirectionof motion,becauseeverypartofthetophasadierentdirectionof motion.Ultimatelyit'snotjustamatterofpickingadenition thatisconvenientandunambiguous:thedenitionwe'reusingis theonlyonethatmakesthetotalangularmomentumofasystema conservedquantityifwelettotal"meanthevectorsum. Aswithrotationinonedimension,wecannotdenewhatwe meanbyangularmomentuminaparticularsituationunlesswepick apointasanaxis.Thisisreallyadierentuseofthewordaxis" thantheoneinthepreviousparagraphs.Herewesimplymeana pointfromwhichwemeasurethedistance r .Inthehydrogenatom, Section5.2AngularMomentuminThreeDimensions 113

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c / 1.Thisparticleismoving directlyawayfromtheaxis,and hasnoangularmomentum. 2.Thisparticlehasangular momentum. thenearlyimmobileprotonprovidesanaturalchoiceofaxis. Three-dimensionalangularmomentuminquantumphysics Oncewestarttothinkmorecarefullyabouttheroleofangular momentuminquantumphysics,itmayseemthatthereisabasic problem:theangularmomentumoftheelectroninahydrogenatom dependsonbothitsdistancefromtheprotonanditsmomentum, soinordertoknowitsangularmomentumpreciselyitwouldseem wewouldneedtoknowbothitspositionanditsmomentumsimultaneouslywithgoodaccuracy.This,however,mightseemtobe forbiddenbytheHeisenberguncertaintyprinciple. Actuallytheuncertaintyprincipledoesplacelimitsonwhatcan beknownaboutaparticle'sangularmomentumvector,butitdoes notpreventusfromknowingitsmagnitudeasanexactintegermultipleof ~ .Thereasonisthatinthreedimensions,therearereally threeseparateuncertaintyprinciples: p x x & h p y y & h p z z & h Nowconsideraparticle,c/1,thatismovingalongthe x axisat position x andwithmomentum p x .Wemaynotbeabletoknow both x and p x withunlimitedaccuracy,butwecanstillknowthe particle'sangularmomentumabouttheoriginexactly.Classically, itiszero,becausetheparticleismovingdirectlyawayfromthe origin:ifitwastobenonzero,wewouldneedbothanonzero x and anonzero p y .Inquantumterms,theuncertaintyprincipledoesnot placeanyconstrainton x p y Suppose,ontheotherhand,aparticlendsitself,asingure c/2,ataposition x alongthe x axis,anditismovingparalleltothe y axiswithmomentum p y .Ithasangularmomentum xp y aboutthe z axis,andagainwecanknowitsangularmomentumwithunlimited accuracy,becausetheuncertaintyprincipleonrelates x to p x and y to p y .Itdoesnotrelate x to p y Asshownbytheseexamples,theuncertaintyprincipledoes notrestricttheaccuracyofourknowledgeofangularmomentaas severelyasmightbeimagined.However,itdoespreventusfrom knowingallthreecomponentsofanangularmomentumvectorsimultaneously.Themostgeneralstatementaboutthisisthefollowingtheorem,whichwepresentwithoutproof: theangularmomentumvectorinquantumphysics Themostthatcanbeknownaboutanangularmomentumvector isitsmagnitudeandoneofitsthreevectorcomponents.Bothare quantizedinunitsof ~ 114 Chapter5TheAtom

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e / Theenergyofastatein thehydrogenatomdependsonly onits n quantumnumber. 5.3TheHydrogenAtom Derivingthewavefunctionsofthestatesofthehydrogenatomfrom rstprincipleswouldbemathematicallytoocomplexforthisbook, butit'snothardtounderstandthelogicbehindsuchawavefunction invisualterms.Considerthewavefunctionfromthebeginningof thechapter,whichisreproducedbelow.Althoughthegraphlooks three-dimensional,itisreallyonlyarepresentationofthepartofthe wavefunctionlyingwithinatwo-dimensionalplane.Thethirdupdowndimensionoftheplotrepresentsthevalueofthewavefunction atagivenpoint,notthethirddimensionofspace.Theplanechosen forthegraphistheoneperpendiculartotheangularmomentum vector. d / Awavefunctionofahydrogen atom. Eachringofpeaksandvalleyshaseightwavelengthsgoingaround inacircle,sothisstatehas L =8 ~ ,i.e.,welabelit ` =8.Thewavelengthisshorternearthecenter,andthismakessensebecausewhen theelectronisclosetothenucleusithasalowerPE,ahigherKE, andahighermomentum. Betweeneachringofpeaksinthiswavefunctionisanodalcircle,i.e.,acircleonwhichthewavefunctioniszero.Thefullthreedimensionalwavefunctionhasnodalspheres:aseriesofnestedsphericalsurfacesonwhichitiszero.Thenumberofradiiatwhichnodes occur,including r = 1 ,iscalled n ,and n turnsouttobeclosely relatedtoenergy.Thegroundstatehas n =1asinglenodeonly at r = 1 ,andhigher-energystateshavehigher n values.There isasimpleequationrelating n toenergy,whichwewilldiscussin section5.4. Thenumbers n and ` ,whichidentifythestate,arecalledits quantumnumbers.Astateofagiven n and ` canbeoriented inavarietyofdirectionsinspace.Wemighttrytoindicatethe Section5.3TheHydrogenAtom 115

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orientationusingthethreequantumnumbers ` x = L x = ~ ` y = L y = ~ and ` z = L z = ~ .Butwehavealreadyseenthatitisimpossibleto knowallthreeofthesesimultaneously.Togivethemostcomplete possibledescriptionofastate,wechooseanarbitraryaxis,saythe z axis,andlabelthestateaccordingto n ` ,and ` z f / Thethreelowest-energystatesofhydrogen. Angularmomentumrequiresmotion,andmotionimplieskinetic energy.Thusitisnotpossibletohaveagivenamountofangular momentumwithouthavingacertainamountofkineticenergyas well.Sinceenergyrelatestothe n quantumnumber,thismeans thatforagiven n valuetherewillbeamaximumpossible ` .It 116 Chapter5TheAtom

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turnsoutthatthismaximumvalueof ` equals n )]TJ/F15 10.9091 Tf 10.909 0 Td [(1. Ingeneral,wecanlistthepossiblecombinationsofquantum numbersasfollows: n canequal1,2,3,... ` canrangefrom0to n )]TJ/F15 10.9091 Tf 10.909 0 Td [(1,instepsof1 ` z canrangefrom )]TJ/F20 10.9091 Tf 8.485 0 Td [(` to ` ,instepsof1 Applyingtheserules,wehavethefollowinglistofstates: n =1, ` =0, ` z =0 onestate n =2, ` =0, ` z =0 onestate n =2, ` =1, ` z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,0,or1 threestates ... ... self-checkB Continuethelistfor n =3. Answer,p.134 Figurefshowsthelowest-energystatesofthehydrogenatom. Theleft-handcolumnofgraphsdisplaysthewavefunctionsinthe x )]TJ/F20 10.9091 Tf 11.27 0 Td [(y plane,andtheright-handcolumnshowstheprobabilitydistributioninathree-dimensionalrepresentation. DiscussionQuestions A Thequantumnumber n isdenedasthenumberofradiiatwhich thewavefunctioniszero,including r = 1 .Relatethistothefeaturesof theguresonthefacingpage. B Basedonthedenitionof n ,whycan'ttherebeanysuchthingas an n =0state? C Relatethefeaturesofthewavefunctionplotsingureftothecorrespondingfeaturesoftheprobabilitydistributionpictures. D Howcanyoutellfromthewavefunctionplotsingurefwhichones havewhichangularmomenta? E Criticizethefollowingincorrectstatement:The ` =8wavefunction inguredhasashorterwavelengthinthecenterbecauseinthecenter theelectronisinahigherenergylevel. F Discusstheimplicationsofthefactthattheprobabilitycloudinofthe n =2, ` =1stateissplitintotwoparts. 5.4 ? EnergiesofStatesinHydrogen History Theexperimentaltechniqueformeasuringtheenergylevelsof anatomaccuratelyisspectroscopy:thestudyofthespectrumof lightemittedorabsorbedbytheatom.Onlyphotonswithcertainenergiescanbeemittedorabsorbedbyahydrogenatom,for example,sincetheamountofenergygainedorlostbytheatom mustequalthedierenceinenergybetweentheatom'sinitialand nalstates.Spectroscopyhadactuallybecomeahighlydeveloped artseveraldecadesbeforeEinsteinevenproposedthephoton,and Section5.4 ? EnergiesofStatesinHydrogen 117

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theSwissspectroscopistJohannBalmerdeterminedin1885that therewasasimpleequationthatgaveallthewavelengthsemitted byhydrogen.Inmodernterms,wethinkofthephotonwavelengths merelyasindirectevidenceabouttheunderlyingenergylevelsof theatom,andwereworkBalmer'sresultintoanequationforthese atomicenergylevels: E n = )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(2.2 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(18 J n 2 Thisenergyincludesboththekineticenergyoftheelectronand theelectricalenergy.Thezero-leveloftheelectricalenergyscale ischosentobetheenergyofanelectronandaprotonthatare innitelyfarapart.Withthischoice,negativeenergiescorrespond toboundstatesandpositiveenergiestounboundones. Wheredoesthemysteriousnumericalfactorof2.2 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(18 Jcome from?In1913theDanishtheoristNielsBohrrealizedthatitwas exactlynumericallyequaltoacertaincombinationoffundamental physicalconstants: E n = )]TJ/F20 10.9091 Tf 9.681 7.38 Td [(mk 2 e 4 2 ~ 2 1 n 2 where m isthemassoftheelectron,and k istheCoulombforce constantforelectricforces. Bohrwasabletocookupaderivationofthisequationbasedon theincompleteversionofquantumphysicsthathadbeendeveloped bythattime,buthisderivationistodaymainlyofhistoricalinterest. Itassumesthattheelectronfollowsacircularpath,whereasthe wholeconceptofapathforaparticleisconsideredmeaninglessin ourmorecompletemodernversionofquantumphysics.Although Bohrwasabletoproducetherightequationfortheenergylevels, hismodelalsogavevariouswrongresults,suchaspredictingthat theatomwouldbeat,andthatthegroundstatewouldhave ` =1 ratherthanthecorrect ` =0. Approximatetreatment Afullandcorrecttreatmentisimpossibleatthemathematical levelofthisbook,butwecanprovideastraightforwardexplanation fortheformoftheequationusingapproximatearguments. Atypicalstanding-wavepatternfortheelectronconsistsofa centraloscillatingareasurroundedbyaregioninwhichthewavefunctiontailso.Asdiscussedinsection4.6,theoscillatingtype ofpatternistypicallyencounteredintheclassicallyallowedregion, whilethetailingooccursintheclassicallyforbiddenregionwhere theelectronhasinsucientkineticenergytopenetrateaccordingto classicalphysics.Weusethesymbol r fortheradiusofthespherical boundarybetweentheclassicallyallowedandclassicallyforbidden regions.Classically, r wouldbethedistancefromtheprotonat 118 Chapter5TheAtom

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g / Theenergylevelsofa particleinabox,contrastedwith thoseofthehydrogenatom. whichtheelectronwouldhavetostop,turnaround,andheadback in. If r hadthesamevalueforeverystanding-wavepattern,then we'dessentiallybesolvingtheparticle-in-a-boxprobleminthree dimensions,withtheboxbeingasphericalcavity.Considerthe energylevelsoftheparticleinaboxcomparedtothoseofthehydrogenatom,g.They'requalitativelydierent.Theenergylevels oftheparticleinaboxgetfartherandfartherapartaswegohigher inenergy,andthisfeaturedoesn'tevendependonthedetailsof whethertheboxistwo-dimensionalorthree-dimensional,oritsexactshape.Thereasonforthespreadingisthattheboxistakento becompletelyimpenetrable,soitssize, r ,isxed.Awavepattern with n humpshasawavelengthproportionalto r=n ,andthereforea momentumproportionalto n ,andanenergyproportionalto n 2 .In thehydrogenatom,however,theforcekeepingtheelectronbound isn'taninniteforceencounteredwhenitbouncesoofawall,it's theattractiveelectricalforcefromthenucleus.Ifweputmoreenergyintotheelectron,it'slikethrowingaballupwardwithahigher energy|itwillgetfartheroutbeforecomingbackdown.This meansthatinthehydrogenatom,weexpect r toincreaseaswego tostatesofhigherenergy.Thistendstokeepthewavelengthsof thehighenergystatesfromgettingtooshort,reducingtheirkinetic energy.Thecloserandclosercrowdingoftheenergylevelsinhydrogenalsomakessensebecauseweknowthatthereisacertainenergy thatwouldbeenoughtomaketheelectronescapecompletely,and thereforethesequenceofboundstatescannotextendabovethat energy. Whentheelectronisatthemaximumclassicallyalloweddistance r fromtheproton,ithaszerokineticenergy.Thuswhentheelectron isatdistance r ,itsenergyispurelyelectrical: [1] E = )]TJ/F20 10.9091 Tf 9.681 7.38 Td [(ke 2 r Nowcomestheapproximation.Inreality,theelectron'swavelength cannotbeconstantintheclassicallyallowedregion,butwepretend thatitis.Since n isthenumberofnodesinthewavefunction,we caninterpretitapproximatelyasthenumberofwavelengthsthat tacrossthediameter2 r .Wearenotevenattemptingaderivation thatwouldproduceallthecorrectnumericalfactorslike2and andsoon,sowesimplymaketheapproximation [2] r n Finallyweassumethatthetypicalkineticenergyoftheelectronis onthesameorderofmagnitudeastheabsolutevalueofitstotal energy.Thisistruetowithinafactoroftwoforatypicalclassical systemlikeaplanetinacircularorbitaroundthesun.Wethen Section5.4 ? EnergiesofStatesinHydrogen 119

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h / Thetophasangularmomentumbothbecauseofthe motionofitscenterofmass throughspaceandduetoits internalrotation.Electronspinis roughlyanalogoustotheintrinsic spinofthetop. have absolutevalueoftotalenergy[3] = ke 2 r K = p 2 = 2 m = h= 2 = 2 m h 2 n 2 = 2 mr 2 Wenowsolvetheequation ke 2 =r h 2 n 2 = 2 mr 2 for r andthrow awaynumericalfactorswecan'thopetohavegottenright,yielding [4] r h 2 n 2 mke 2 Plugging n =1intothisequationgives r =2nm,whichisindeed ontherightorderofmagnitude.Finallywecombineequations[4] and[1]tond E )]TJ/F20 10.9091 Tf 21.196 7.38 Td [(mk 2 e 4 h 2 n 2 whichiscorrectexceptforthenumericalfactorsweneveraimedto nd. DiscussionQuestions A Statesofhydrogenwith n greaterthanabout10areneverobserved inthesun.Whymightthisbe? B Sketchgraphsof r and E versus n forthehydrogen,andcompare withanalogousgraphsfortheone-dimensionalparticleinabox. 5.5ElectronSpin It'sdisconcertingtothenoviceping-pongplayertoencounterfor thersttimeamoreskilledplayerwhocanputspinontheball. Eventhoughyoucan'tseethattheballisspinning,youcantell somethingisgoingonbythewayitinteractswithotherobjectsin itsenvironment.Inthesameway,wecantellfromthewayelectrons interactwithotherthingsthattheyhaveanintrinsicspinoftheir own.Experimentsshowthatevenwhenanelectronisnotmoving throughspace,itstillhasangularmomentumamountingto ~ = 2. Thismayseemparadoxicalbecausethequantummoat,forinstance,gaveonlyangularmomentathatwereintegermultiplesof ~ ,nothalf-units,andIclaimedthatangularmomentumwasalwaysquantizedinunitsof ~ ,notjustinthecaseofthequantum moat.Thatwholediscussion,however,assumedthattheangular momentumwouldcomefromthemotionofaparticlethroughspace. The ~ = 2angularmomentumoftheelectronissimplyapropertyof theparticle,likeitschargeoritsmass.Ithasnothingtodowith 120 Chapter5TheAtom

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whethertheelectronismovingornot,anditdoesnotcomefromany internalmotionwithintheelectron.Nobodyhaseversucceededin ndinganyinternalstructureinsidetheelectron,andevenifthere wasinternalstructure,itwouldbemathematicallyimpossibleforit toresultinahalf-unitofangularmomentum. Wesimplyhavetoacceptthis ~ = 2angularmomentum,called thespin"oftheelectron|MotherNaturerubsournosesinitas anobservedfact.Protonsandneutronshavethesame ~ = 2spin, whilephotonshaveanintrinsicspinof ~ .Ingeneral,half-integer spinsaretypicalofmaterialparticles.Integralvaluesarefoundfor theparticlesthatcarryforces:photons,whichembodytheelectric andmagneticeldsofforce,aswellasthemoreexoticmessengers ofthenuclearandgravitationalforces. Aswasthecasewithordinaryangularmomentum,wecandescribespinangularmomentumintermsofitsmagnitude,andits componentalongagivenaxis.Wenotatethesequantities,inunits of ~ ,as s and s z ,soanelectronhas s =1 = 2and s z =+1 = 2or-1/2. Takingelectronspinintoaccount,weneedatotaloffourquantumnumberstolabelastateofanelectroninthehydrogenatom: n ` ` z ,and s z .Weomit s becauseitalwayshasthesamevalue.The symbols ` and ` z includeonlytheangularmomentumtheelectron hasbecauseitismovingthroughspace,notitsspinangularmomentum.Theavailabilityoftwopossiblespinstatesoftheelectron leadstoadoublingofthenumbersofstates: n =1, ` =0, ` z =0, s z =+1 = 2or )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 = 2 twostates n =2, ` =0, ` z =0, s z =+1 = 2or )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 = 2 twostates n =2, ` =1, ` z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,0,or1, s z =+1 = 2or )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 = 2 sixstates ... ... 5.6AtomsWithMoreThanOneElectron Whataboutotheratomsbesideshydrogen?Itwouldseemthat thingswouldgetmuchmorecomplexwiththeadditionofasecond electron.Ahydrogenatomonlyhasoneparticlethatmovesaround much,sincethenucleusissoheavyandnearlyimmobile.Helium, withtwo,wouldbeamess.Insteadofawavefunctionwhosesquare tellsustheprobabilityofndingasingleelectronatanygivenlocationinspace,aheliumatomwouldneedtohaveawavefunction whosesquarewouldtellustheprobabilityofndingtwoelectrons atanygivencombinationofpoints.Ouch!Inaddition,wewould havetheextracomplicationoftheelectricalinteractionbetweenthe twoelectrons,ratherthanbeingabletoimagineeverythinginterms ofanelectronmovinginastaticeldofforcecreatedbythenucleus alone. Despiteallthis,itturnsoutthatwecangetasurprisinglygood descriptionofmany-electronatomssimplybyassumingtheelecSection5.6AtomsWithMoreThanOneElectron 121

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tronscanoccupythesamestanding-wavepatternsthatexistina hydrogenatom.Thegroundstateofhelium,forexample,would havebothelectronsinstatesthatareverysimilartothe n =1 statesofhydrogen.Thesecond-lowest-energystateofheliumwould haveoneelectroninan n =1state,andtheotherinan n =2states. Therelativelycomplexspectraofelementsheavierthanhydrogen canbeunderstoodasarisingfromthegreatnumberofpossiblecombinationsofstatesfortheelectrons. Asurprisingthinghappens,however,withlithium,thethreeelectronatom.Wewouldexpectthegroundstateofthisatomto beoneinwhichallthreeelectronssettledowninto n =1states. Whatreallyhappensisthattwoelectronsgointo n =1states,but thethirdstaysupinan n =2state.Thisisaconsequenceofanew principleofphysics: thePauliexclusionprinciple Onlyoneelectroncaneveroccupyagivenstate. Therearetwo n =1states,onewith s z =+1 = 2andonewith s z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 = 2,butthereisnothird n =1stateforlithium'sthird electrontooccupy,soitisforcedtogointoan n =2state. ItcanbeprovedmathematicallythatthePauliexclusionprincipleappliestoanytypeofparticlethathashalf-integerspin.Thus twoneutronscanneveroccupythesamestate,andlikewisefortwo protons.Photons,however,areimmunetotheexclusionprinciple becausetheirspinisaninteger.Materialobjectscan'tpassthrough eachother,butbeamsoflightcan,andthebasicreasonisthatthe exclusionprincipleappliestoonebutnottotheother. 1 1 Therearealsoelectricalforcesbetweenatoms,butasarguedonpage86, theattractionsandrepulsionstendtocancelout. 122 Chapter5TheAtom

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j / Hydrogenishighlyreactive. i / Thebeginningoftheperiodictable. Derivingtheperiodictable Wecannowaccountforthestructureoftheperiodictable,which seemedsomysteriouseventoitsinventorMendeleev.Therstrow consistsofatomswithelectronsonlyinthe n =1states: H1electroninan n =1state He2electronsinthetwo n =1states Thenextrowisbuiltbyllingthe n =2energylevels: Li2electronsin n =1states,1electroninan n =2state Be2electronsin n =1states,2electronsin n =2states ... O2electronsin n =1states,6electronsin n =2states F2electronsin n =1states,7electronsin n =2states Ne2electronsin n =1states,8electronsin n =2states Inthethirdrowwestartinonthe n =3levels: Na2electronsin n =1states,8electronsin n =2states,1 electroninan n =3state ... Wecannowseealogicallinkbetweenthellingoftheenergylevels andthestructureoftheperiodictable.Column0,forexample, consistsofatomswiththerightnumberofelectronstollallthe availablestatesuptoacertainvalueof n .ColumnIcontainsatoms likelithiumthathavejustoneelectronmorethanthat. Thisshowsthatthecolumnsrelatetothellingofenergylevels, butwhydoesthathaveanythingtodowithchemistry?Why,for example,aretheelementsincolumnsIandVIIdangerouslyreactive?Consider,forexample,theelementsodiumNa,whichisso reactivethatitmayburstintoameswhenexposedtoair.The electroninthe n =3statehasanunusuallyhighenergy.Ifwelet asodiumatomcomeincontactwithanoxygenatom,energycan bereleasedbytransferringthe n =3electronfromthesodiumto oneofthevacantlower-energy n =2statesintheoxygen.This energyistransformedintoheat.AnyatomincolumnIishighly reactiveforthesamereason:itcanreleaseenergybygivingaway theelectronthathasanunusuallyhighenergy. ColumnVIIisspectacularlyreactivefortheoppositereason: theseatomshaveasinglevacancyinalow-energystate,soenergy isreleasedwhentheseatomsstealanelectronfromanotheratom. Itmightseemasthoughtheseargumentswouldonlyexplain reactionsofatomsthatareindierentrowsoftheperiodictable, becauseonlyinthesereactionscanatransferredelectronmovefrom ahighern statetoalowern state.Thisisincorrect.An n =2electroninuorineF,forexample,wouldhaveadierentenergythan an n =2electroninlithiumLi,duetothedierentnumberof protonsandelectronswithwhichitisinteracting.Roughlyspeaking,the n =2electroninuorineismoretightlyboundlowerin Section5.6AtomsWithMoreThanOneElectron 123

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energybecauseofthelargernumberofprotonsattractingit.The eectoftheincreasednumberofattractingprotonsisonlypartly counteractedbytheincreaseinthenumberofrepellingelectrons, becausetheforcesexertedonanelectronbytheotherelectronsare inmanydierentdirectionsandcanceloutpartially. Neutronstarsexample1 Here'sanexoticexamplethatdoesn'teveninvolveatoms.When astarrunsoutoffuelforitsnuclearreactions,itbeginstocollapse underitsownweight.SinceNewton'slawofgravitydependson theinversesquareofthedistance,thegravitationalforcesbecomestrongerasthestarcollapses,whichencouragesittocollapseevenfurther.Thenalresultdependsonthemassofthe star,butlet'sconsiderastarthat'sonlyalittlemoremassivethan ourownsun.Suchastarwillcollapsetothepointwherethegravitationalenergybeingreleasedissufcienttocausethereaction p+e )]TJ/F23 10.9091 Tf 10.115 -3.959 Td [(! n+ tooccur.Asyoufoundinhomeworkproblem7on page39,thisreactioncanonlyoccurwhenthereissomesource ofenergy,becausethemass-energyoftheproductsisgreater thanthemass-energyofthethingsbeingconsumed.Theneutrinosyoffandareneverheardfromagain,sowe'releftwitha starconsistingonlyofneutrons! Nowtheexclusionprinciplecomesintoplay.Thecollapsecan't continueindenitely.Thesituationisinfactcloselyanalogous tothatofanatom.Aleadatom'scloudof82electronscan't shrinkdowntothesizeofahydrogenatom,becauseonlytwo electronscanhavethelowest-energywavepattern.Thesame happenswiththeneutronstar.Nophysicalrepulsionkeepsthe neutronsapart.They'reelectricallyneutral,sotheydon'trepelor attractoneanotherelectrically.Thegravitationalforceisattractive,andasthecollapseproceedstothepointwheretheneutrons comewithinrangeofthestrongnuclearforce 10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(15 m,we evenstartgettingnuclearattraction.Theonlythingthatstopsthe wholeprocessistheexclusionprinciple.Thestarendsupbeing onlyafewkilometersacross,andhasthesamebillion-ton-perteaspoondensityasanatomicnucleus.Indeed,wecanthinkofit asonebignucleuswithatomicnumberzero,becausethereare noprotons. Aswithaspinninggureskaterpullingherarmsin,conservation ofangularmomentummakesthestarspinfasterandfaster.The wholeobjectmayendupwitharotationalperiodofafractionofa second!Suchastarsendsoutradiopulseswitheachrevolution, likeasortoflighthouse.Thersttimesuchasignalwasdetected, radioastronomersthoughtthatitwasasignalfromaliens. 124 Chapter5TheAtom

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Summary SelectedVocabulary quantumnumberanumericallabelusedtoclassifyaquantum state spin........thebuilt-inangularmomentumpossessedby aparticleevenwhenatrest Notation n ..........thenumberofradialnodesinthewavefunction,includingtheoneat r = 1 ~ .......... h= 2 L ..........theangularmomentumvectorofaparticle, notincludingitsspin ` ..........themagnitudeofthe L vector,dividedby ~ ` z .........the z componentofthe L vector,dividedby ~ ;thisisthestandardnotationinnuclear physics,butnotinatomicphysics s ..........themagnitudeofthespinangularmomentum vector,dividedby ~ s z .........the z componentofthespinangularmomentumvector,dividedby ~ ;thisisthestandard notationinnuclearphysics,butnotinatomic physics OtherTerminologyandNotation m ` .........alessobviousnotationfor ` z ,standardin atomicphysics m s .........alessobviousnotationfor s z ,standardin atomicphysics Summary Hydrogen,withoneprotonandoneelectron,isthesimplest atom,andmorecomplexatomscanoftenbeanalyzedtoareasonablygoodapproximationbyassumingtheirelectronsoccupystates thathavethesamestructureasthehydrogenatom's.Theelectron inahydrogenatomexchangesverylittleenergyorangularmomentumwiththeproton,soitsenergyandangularmomentumare nearlyconstant,andcanbeusedtoclassifyitsstates.Theenergy ofahydrogenstatedependsonlyonits n quantumnumber. Inquantumphysics,theangularmomentumofaparticlemoving inaplaneisquantizedinunitsof ~ .Atomsarethree-dimensional, however,sothequestionnaturallyarisesofhowtodealwithangularmomentuminthreedimensions.Inthreedimensions,angular momentumisavectorinthedirectionperpendiculartotheplane ofmotion,suchthatthemotionappearsclockwiseifviewedalong thedirectionofthevector.Sinceangularmomentumdependson bothpositionandmomentum,theHeisenberguncertaintyprinciple limitstheaccuracywithwhichonecanknowit.Themostthecan Summary 125

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beknownaboutanangularmomentumvectorisitsmagnitudeand oneofitsthreevectorcomponents,bothofwhicharequantizedin unitsof ~ Inadditiontotheangularmomentumthatanelectroncarriesby virtueofitsmotionthroughspace,itpossessesanintrinsicangular momentumwithamagnitudeof ~ = 2.Protonsandneutronsalso havespinsof ~ = 2,whilethephotonhasaspinequalto ~ Particleswithhalf-integerspinobeythePauliexclusionprinciple:onlyonesuchparticlecanexistisagivenstate,i.e.,witha givencombinationofquantumnumbers. Wecanenumeratethelowest-energystatesofhydrogenasfollows: n =1, ` =0, ` z =0, s z =+1 = 2or )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 = 2 twostates n =2, ` =0, ` z =0, s z =+1 = 2or )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 = 2 twostates n =2, ` =1, ` z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,0,or1, s z =+1 = 2or )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 = 2 sixstates ... ... Theperiodictablecanbeunderstoodintermsofthellingofthese states.Thenonreactivenoblegasesarethoseatomsinwhichthe electronsareexactlysucienttollallthestatesuptoagiven n value.Themostreactiveelementsarethosewithonemoreelectron thananoblegaselement,whichcanreleaseagreatdealofenergy bygivingawaytheirhigh-energyelectron,andthosewithoneelectronfewerthananoblegas,whichreleaseenergybyacceptingan electron. 126 Chapter5TheAtom

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Problem2. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 aAdistancescaleisshownbelowthewavefunctionsand probabilitydensitiesillustratedinsection5.3.Comparethiswith theorder-of-magnitudeestimatederivedinsection5.4fortheradius r atwhichthewavefunctionbeginstailingo.Wastheestimatein section5.4ontherightorderofmagnitude? bAlthoughwenormallysaythemoonorbitstheearth,actually theybothorbitaroundtheircommoncenterofmass,whichisbelow theearth'ssurfacebutnotatitscenter.Thesameistrueofthe hydrogenatom.Doesthecenterofmasslieinsidetheprotonor outsideit? 2 Thegureshowseightofthepossiblewaysinwhichanelectron inahydrogenatomcoulddropfromahigherenergystatetoastate oflowerenergy,releasingthedierenceinenergyasaphoton.Of theseeighttransitions,only D E ,and F producephotonswith wavelengthsinthevisiblespectrum. aWhichofthevisibletransitionswouldbeclosesttotheviolet endofthespectrum,andwhichwouldbeclosesttotheredend? Explain. bInwhatpartoftheelectromagneticspectrumwouldthephotons fromtransitionsA, B ,and C lie?Whatabout G andH?Explain. cIsthereanupperlimittothewavelengthsthatcouldbeemitted byahydrogenatomgoingfromoneboundstatetoanotherbound state?Istherealowerlimit?Explain. 3 Beforethequantumtheory,experimentalistsnotedthatin manycases,theywouldndthreelinesinthespectrumofthesame atomthatsatisedthefollowingmysteriousrule:1 = 1 =1 = 2 + 1 = 3 .Explainwhythiswouldoccur.Donotusereasoningthat onlyworksforhydrogen|suchcombinationsoccurinthespectra ofallelements.[Hint:Restatetheequationintermsoftheenergies ofphotons.] 4 Findanequationforthewavelengthofthephotonemitted whentheelectroninahydrogenatommakesatransitionfromenergylevel n 1 tolevel n 2 .[Youwillneedtohavereadoptionalsection 5.4.] p 5 aVerifythatPlanck'sconstanthasthesameunitsasangular momentum. bEstimatetheangularmomentumofaspinningbasketball,in unitsof ~ Problems 127

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6 Assumethatthekineticenergyofanelectroninthe n =1 stateofahydrogenatomisonthesameorderofmagnitudeasthe absolutevalueofitstotalenergy,andestimateatypicalspeedat whichitwouldbemoving.Itcannotreallyhaveasingle,denite speed,becauseitskineticandpotentialenergytradeoatdierent distancesfromtheproton,butthisisjustaroughestimateofa typicalspeed.Basedonthisspeed,werewejustiedinassuming thattheelectroncouldbedescribednonrelativistically? 7 Thewavefunctionoftheelectroninthegroundstateofa hydrogenatomis = )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 = 2 a )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 = 2 e )]TJ/F21 7.9701 Tf 6.587 0 Td [(r=a where r isthedistancefromtheproton,and a = ~ 2 =kme 2 = 5.3 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(11 misaconstantthatsetsthesizeofthewave. aCalculatesymbolically,withoutplugginginnumbers,theprobabilitythatatanymoment,theelectronisinsidetheproton.Assumetheprotonisaspherewitharadiusof b =0.5fm.[Hint: Doesitmatterifyouplugin r =0or r = b intheequationforthe wavefunction?] p bCalculatetheprobabilitynumerically. p cBasedontheequationforthewavefunction,isitvalidtothink ofahydrogenatomashavinganitesize?Can a beinterpreted asthesizeoftheatom,beyondwhichthereisnothing?Oristhere anylimitonhowfartheelectroncanbefromtheproton? 8 Usephysicalreasoningtoexplainhowtheequationforthe energylevelsofhydrogen, E n = )]TJ/F20 10.9091 Tf 9.681 7.38 Td [(mk 2 e 4 2 ~ 2 1 n 2 shouldbegeneralizedtothecaseofaheavieratomwithatomic number Z thathashadallitselectronsstrippedawayexceptfor one. ? 9 Thisquestionrequiresthatyoureadoptionalsection5.4. Amuonisasubatomicparticlethatactsexactlylikeanelectron exceptthatitsmassis207timesgreater.Muonscanbecreatedby cosmicrays,anditcanhappenthatoneofanatom'selectronsis displacedbyamuon,formingamuonicatom.Ifthishappensto ahydrogenatom,theresultingsystemconsistssimplyofaproton plusamuon. aHowwouldthesizeofamuonichydrogenatominitsground statecomparewiththesizeofthenormalatom? bIfyouweresearchingformuonicatomsinthesunorinthe earth'satmospherebyspectroscopy,inwhatpartoftheelectromagneticspectrumwouldyouexpecttondtheabsorptionlines? 128 Chapter5TheAtom

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10 Consideraclassicalmodelofthehydrogenatominwhich theelectronorbitstheprotoninacircleatconstantspeed.Inthis model,theelectronandprotoncanhavenointrinsicspin.Using theresultofproblem17frombook4,ch.6,showthatinthis model,theatom'smagneticdipolemoment D m isrelatedtoitsangularmomentumby D m = )]TJ/F20 10.9091 Tf 8.485 0 Td [(e= 2 m L ,regardlessofthedetailsof theorbitalmotion.Assumethatthemagneticeldisthesameas wouldbeproducedbyacircularcurrentloop,eventhoughthere isreallyonlyasinglechargedparticle.[Althoughthemodelis quantum-mechanicallyincorrect,theresultturnsouttogivethecorrectquantummechanicalvalueforthecontributiontotheatom's dipolemomentcomingfromtheelectron'sorbitalmotion.There areothercontributions,however,arisingfromtheintrinsicspinsof theelectronandproton.] Problems 129

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Appendix1:Exercises Exercise1A:SportsinSlowlightland InSlowlightland,thespeedoflightis20mi/hr=32km/hr=9m/s.Thinkofanexampleof howrelativisticeectswouldworkinsports.Thingscangetverycomplexveryquickly,sotry tothinkofasimpleexamplethatfocusesonjustoneofthefollowingeects: relativisticmomentum relativistickineticenergy relativisticadditionofvelocities timedilationandlengthcontraction Dopplershiftsoflight equivalenceofmassandenergy timeittakesforlighttogettoanathlete'seye deectionoflightraysbygravity

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Exercise5A:QuantumVersusClassicalRandomness 1.Imaginetheclassicalversionoftheparticleinaone-dimensionalbox.Supposeyouinsert theparticleintheboxandgiveitaknown,predeterminedenergy,butarandominitialposition andarandomdirectionofmotion.Youthenpickarandomlatermomentintimetoseewhere itis.Sketchtheresultingprobabilitydistributionbyshadingontopofalinesegment.Does theprobabilitydistributiondependonenergy? 2.Dosimilarsketchesfortherstfewenergylevelsofthequantummechanicalparticleina box,andcomparewith1. 3.Dothesamethingasin1,butforaclassicalhydrogenatomintwodimensions,whichacts justlikeaminiaturesolarsystem.Assumeyou'realwaysstartingoutwiththesamexedvalues ofenergyandangularmomentum,butapositionanddirectionofmotionthatareotherwise random.Dothisfor L =0,andcomparewithareal L =0probabilitydistributionforthe hydrogenatom. 4.Repeat3foranonzerovalueof L ,sayL= ~ 5.Summarize:Aretheclassicalprobabilitydistributionsaccurate?Whatqualitativefeatures arepossessedbytheclassicaldiagramsbutnotbythequantummechanicalones,orvice-versa? 131

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Appendix2:PhotoCredits Exceptasspecicallynotedbeloworinaparentheticalcreditinthecaptionofagure,alltheillustrationsin thisbookareundermyowncopyright,andarecopyleftlicensedunderthesamelicenseastherestofthebook. Insomecasesit'sclearfromthedatethatthegureispublicdomain,butIdon'tknowthenameofthe artistorphotographer;Iwouldbegratefultoanyonewhocouldhelpmetogivepropercredit.Ihaveassumed thatimagesthatcomefromU.S.governmentwebpagesarecopyright-free,sinceproductsoffederalagenciesfall intothepublicdomain.I'veincludedsomepublic-domainpaintings;photographicreproductionsofthemarenot copyrightableintheU.S.BridgemanArtLibrary,Ltd.v.CorelCorp.,36F.Supp.2d191,S.D.N.Y.1999. WhenPSSCPhysics"isgivenasacredit,itindicatesthatthegureisfromthersteditionofthetextbook entitledPhysics,bythePhysicalScienceStudyCommittee.Theearlyeditionsofthesebooksneverhadtheir copyrightsrenewed,andarenowthereforeinthepublicdomain.Thereisalsoablanketpermissiongivenin thelaterPSSCCollegePhysicsedition,whichstatesonthecopyrightpagethatThematerialstakenfromthe originalandsecondeditionsandtheAdvancedTopicsofPSSCPHYSICSincludedinthistextwillbeavailable toallpublishersforuseinEnglishafterDecember31,1970,andintranslationsafterDecember31,1975." CreditstoMillikanandGalerefertothetextbooksPracticalPhysicsandElementsofPhysics. Botharepublicdomain.The1927versiondidnothaveitscopyrightrenewed.Sinceispossiblethatsomeof theillustrationsinthe1927versionhadtheircopyrightsrenewedandarestillundercopyright,Ihaveonlyused themwhenitwasclearthattheywereoriginallytakenfrompublicdomainsources. Inafewcases,Ihavemadeuseofimagesunderthefairusedoctrine.However,Iamnotalawyer,andthe lawsonfairusearevague,soyoushouldnotassumethatit'slegalforyoutousetheseimages.Inparticular, fairuselawmaygiveyoulessleewaythanitgivesme,becauseI'musingtheimagesforeducationalpurposes, andgivingthebookawayforfree.Likewise,ifthephotocreditsayscourtesyof...,"thatmeansthecopyright ownergavemepermissiontouseit,butthatdoesn'tmeanyouhavepermissiontouseit. Cover Collidingnuclei: courtesyofRHIC. 13 Einstein: ProfessorEinstein'sVisittotheUnitedStates,"The ScienticMonthly12:5,p.483,publicdomain. 13 Trinitytest: U.S.military,publicdomain. 17 Michelson: 1887,publicdomain. 17 Lorentz: paintingbyArnhemensis,publicdomainWikimediaCommons. 17 FitzGerald: before1901,publicdomain. 16 Michelson-Morleyexperiment: Lineart:redrawnfromthe1887 paperbyMichelsonandMorley.Photoofthe1887apparatus:publicdomain.PhotosoftheMillerexperiment: ca.1930,believedtobepublicdomain. 26 Collidingnuclei: courtesyofRHIC. 32 Eclipse: 1919,public domain. 32 Newspaperheadline: 1919,publicdomain. 43 MountSt.Helens: public-domainimageby AustinPost,USGS. 67 Ozonemaps: NASA/GSFCTOMSTeam. 68 Digitalcameraimage: courtesyof LymanPage. 75 Diractedphotons: courtesyofLymanPage. 99 WernerHeisenberg: ca.1927,believedto bepublicdomain. 85 WickedWitch: artbyW.W.Denslow,1900.Quotefrom TheWizardofOz ,L.Frank Baum,1900. 123 Hindenburg: PublicdomainproductoftheU.S.Navy.

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Appendix3:HintsandSolutions AnswerstoSelf-Checks AnswerstoSelf-ChecksforChapter1 Page19,self-checkA: At v =0,weget =1,so t = T .Thereisnotimedistortionunless thetwoframesofreferenceareinrelativemotion. Page29,self-checkB: Thetotalmomentumiszerobeforethecollision.Afterthecollision, thetwomomentahavereversedtheirdirections,buttheystillcancel.Neitherobjecthaschanged itskineticenergy,sothetotalenergybeforeandafterthecollisionisalsothesame. Page35,self-checkC: At v =0,wehave =1,sothemass-energyis mc 2 asclaimed.As v approaches c approachesinnity,sothemassenergybecomesinniteaswell. AnswerstoSelf-ChecksforChapter2 Page49,self-checkA: Mostpeoplewouldthinktheywerepositivelycorrelated,but theycouldbeindependent.Thesemustbeindependent,sincethereisnopossiblephysical mechanismthatcouldmakeonehaveanyeectontheother.Thesecannotbeindependent, sincedyingtodayguaranteesthatyouwon'tdietomorrow. Page51,self-checkB: Theareaunderthecurvefrom130to135cmisabout3/4ofa rectangle.Theareafrom135to140cmisabout1.5rectangles.Thenumberofpeopleinthe secondrangeisabouttwiceasmuch.Wecouldhaveconvertedthesetoactualprobabilities rectangle=5cm 0.005cm )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 =0.025,butthatwouldhavebeenpointless,becausewe werejustgoingtocomparethetwoareas. AnswerstoSelf-ChecksforChapter3 Page73,self-checkA: Theaxesofthegrapharefrequencyandphotonenergy,soitsslope isPlanck'sconstant.Itdoesn'tmatterifyougraph e V ratherthan W + e V ,becausethat onlychangesthey-intercept,nottheslope. AnswerstoSelf-ChecksforChapter4 Page89,self-checkA: Wavelengthisinverselyproportionaltomomentum,sotoproducea largewavelengthwewouldneedtouseelectronswithverysmallmomentaandenergies.In practicalterms,thisisn'tveryeasytodo,sincerippinganelectronoutofanobjectisaviolent process,andit'snotsoeasytocalmtheelectrondownafterward. Page98,self-checkB: Undertheordinarycircumstancesoflife,theaccuracywithwhich wecanmeasurethepositionandmomentumofanobjectdoesn'tresultinavalueof p x thatisanywherenearthetinyorderofmagnitudeofPlanck'sconstant.Werunupagainstthe ordinarylimitationsontheaccuracyofourmeasuringtechniqueslongbeforetheuncertainty principlebecomesanissue. Page98,self-checkC: Theelectronwavewillsuersingle-slitdiraction,andspreadoutto

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thesidesafterpassingthroughtheslit.Neither p nor x iszeroforthediractedwave. Page104,self-checkD: No.Theequation KE = p 2 = 2 m isnonrelativistic,soitcan'tbe appliedtoanelectronmovingatrelativisticspeeds.Photonsalwaysmoveatrelativisticspeeds, soitcan'tbeappliedtothem,either. Page105,self-checkE: DividingbyPlanck'sconstant,asmallnumber,givesalargenegative resultinsidetheexponential,sotheprobabilitywillbeverysmall. AnswerstoSelf-ChecksforChapter5 Page113,self-checkA: Ifyoutraceacirclegoingaroundthecenter,yourunintoaseriesof eightcompletewavelengths.Itsangularmomentumis8 ~ Page117,self-checkB: n =3, ` =0, ` z =0:onestate n =3, ` =1, ` z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,0,or1:threestates n =3, ` =2, ` z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,-1,0,1,or2:vestates 134 Appendix3:HintsandSolutions

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Index absorptionspectrum,94 angularmomentum andtheuncertaintyprinciple,113 inthreedimensions,113 quantizationof,112 atoms helium,121 lithium,122 sodium,123 withmanyelectrons,121 averages,48 ruleforcalculating,48 Balmer,Johann,118 blackhole,33 Bohr,Niels,118 boundstates,93 box particleina,93 carbon-14dating,54 cat Schrodinger's,100 chemicalbonds quantumexplanationfor,95 classicalphysics,44 complexnumbers useinquantumphysics,105 correspondenceprinciple dened,19 formass-energy,35 forrelativisticadditionofvelocities,28 forrelativisticmomentum,31 fortimedilation,19 cosmicrays,22 Darwin,Charles,45 Davisson,C.J.,86 deBroglie,Louis,86 decay exponential,52 digitalcamera,68 duality wave-particle,75 Eddington,Arthur,45 Einstein,Albert,44,67 andrandomness,45 biography,13 electron asawave,86 spinof,120 wavefunction,89 emissionspectrum,94 energy equivalencetomass,31 quantizationofforboundstates,94 Enlightenment,45 ether,14,16 evolution randomnessin,45 exclusionprinciple,122 exponentialdecay,52 rateof,55 gammafactor,19 garageparadox,21 gas spectrumof,94 geothermalvents,44 Germer,L.,86 goiters,52 groupvelocity,92 half-life,52 Heisenberguncertaintyprinciple,96 inthreedimensions,114 Heisenberg,Werner Nazibombeort,98 uncertaintyprinciple,96 helium,121 Hertz,Heinrich,71 Hindenburg,123 hydrogenatom,115 angularmomentumin,112 classicationofstates,112 energiesofstatesin,117 energyin,112 Lquantumnumber,115

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momentumin,112 nquantumnumber,115 independence statistical,46 independentprobabilities lawof,46 inertia principleofrelativity,15 iodine,52 Jeans,James,45 Laplace,PierreSimonde,43 light momentumof,40 speedof,15 mass equivalencetoenergy,31 mass-energy conservationof,33 correspondenceprinciple,35 ofamovingparticle,35 matter asawave,85 measurementinquantumphysics,100 Michelson-Morleyexperiment,16 Millikan,Robert,72 molecules nonexistenceinclassicalphysics,85 momentum oflight,40 relativistic,28 muons,22 neutron spinof,121 neutronstars,124 Newton,Isaac,44 normalization,48 ozonelayer,68 particle denitionof,75 particleinabox,93 pathofaphotonundened,76 Pauliexclusionprinciple,122 periodictable,123 phasevelocity,92 photoelectriceect,70 photon Einstein'searlytheory,70 energyof,72 inthreedimensions,80 spinof,121 pilotwavehypothesis,77 Planck'sconstant,73 Planck,Max,73 positron,34 probabilities additionof,47 normalizationof,48 probabilitydistributions averagesof,51 widthsof,51 probabilitydistributions,50 probabilityinterpretation,77 proteinmolecules,111 proton spinof,121 quantumdot,93 quantummoat,112 quantumnumbers,115 ` ,115 ` z ,115 m ` ,125 m s ,125 n ,115 s ,121 s z ,121 quantumphysics,44 radar,67 radio,67 radioactivity,52 randomness,45 relativity principleof,15 RHICaccelerator,25 Russell,Bertrand,45 Schrodingerequation,102 Schrodinger'scat,100 Schrodinger,Erwin,100 simultaneity,20 Sirius,94 136 Index

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sodium,123 space relativisticeects,20 spectrum absorption,94 emission,94 spin,120 electron's,120 neutron's,121 photon's,121 proton's,121 StarTrek,95 states bound,93 supernovae,23 Taylor,G.I.,76 time relativisticeects,18 tunneling,102 twinparadox,23 ultravioletlight,68 uncertaintyprinciple,96 inthreedimensions,114 velocity additionof,16 relativistic,27 group,92 phase,92 Voyagerspaceprobe,39 wave denitionof,75 dispersive,91 wave-particleduality,75 pilot-waveinterpretationof,77 probabilityinterpretationof,77 wavefunction complexnumbersin,105 oftheelectron,89 WickedWitchoftheWest,85 Index 137

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UsefulData MetricPrexes M-mega-10 6 k-kilo-10 3 m-milli-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(3 -Greekmumicro-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(6 n-nano-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(9 p-pico-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(12 f-femto-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(15 Centi-,10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(2 ,isusedonlyinthecentimeter. TheGreekAlphabet Aalpha Nnu Bbeta xi )-1633(gamma oOomicron delta pi Eepsilon Prho Zzeta sigma Heta Ttau theta Yupsilon Iiota phi Kkappa Xchi lambda psi Mmu omega SubatomicParticles particlemasskgradiusfm electron9.109 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(31 0.01 proton1.673 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(27 1.1 neutron1.675 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(27 1.1 Theradiiofprotonsandneutronscanonlybegivenapproximately,sincetheyhavefuzzysurfaces.Forcomparison,a typicalatomisaboutamillionfminradius. NotationandUnits quantityunitsymbol distancemeter,m x x timesecond,s t t masskilogram,kg m densitykg = m 3 velocitym/s v accelerationm = s 2 a forceN=kg m = s 2 F pressurePa=1N = m 2 P energyJ=kg m 2 = s 2 E powerW=1J = s P momentumkg m = s p angularmomentumkg m 2 = sorJ s L periods T wavelengthm frequencys )]TJ/F19 5.9776 Tf 5.757 0 Td [(1 orHz f gammafactorunitless probabilityunitless P prob.distributionvarious D electronwavefunctionm )]TJ/F19 5.9776 Tf 5.756 0 Td [(3 = 2 Earth,Moon,andSun bodymasskgradiuskmradiusoforbitkm earth5.97 10 24 6.4 10 3 1.49 10 8 moon7.35 10 22 1.7 10 3 3.84 10 5 sun1.99 10 30 7.0 10 5 | FundamentalConstants gravitationalconstant G =6.67 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(11 N m 2 = kg 2 Coulombconstant k =8.99 10 9 N m 2 = C 2 quantumofcharge e =1.60 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(19 C speedoflight c =3.00 10 8 m/s Planck'sconstant h =6.63 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(34 J s 138 Index