HowtoTeachQuantumPhysics

toYourDog

“Charming. A light-hearted and amusing way to learn about one of the mostimportantaspectsofmodernscience.”

WilliamD.Phillips–NobelLaureateinPhysics

“Afast-movingandfunintroductiontosomeofthedeepestmysteriesofmodernphysics.AndEmmyisastar.”

DrSeanCarroll–authorofFromEternitytoHere:

TheQuestfortheUltimateTheoryofTime

“Quantum entanglement, quantum teleportation, and virtual particles are allexplainedwith the author’s characteristic light-hearted touch. Readerswho’veshiedawayfrompopulartreatmentsofphysicsinthepastmayfindhischeerfuldiscussionarealtreat.”

PublishersWeekly

AbouttheAuthor

Chad Orzel is an Associate Professor in the Department of Physics andAstronomy atUnionCollege,NewYork,where he carries out research in thefield of Atomic, Molecular, and Optical Physics. His influential and award-winningblogcanbefoundathttp://scienceblogs.com/principles/.

HOWTOTEACH

QUANTUMPHYSICS

TOYOURDOG

CHADORZEL

AOneworldBook

FirstpublishedinGreatBritainandtheCommonwealth

byOneworldPublications2010

Reprinted2010,2011

ThisebookeditionpublishedbyOneworldPublications2011

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ToKate,whoselaughstartedthewholething

Contents

IntroductionWhyTalktoYourDogaboutPhysics?

AnIntroductiontoQuantumPhysics

Chapter1WhichWay?BothWays:Particle-WaveDuality

Chapter2Where’sMyBone?TheHeisenbergUncertaintyPrinciple

Chapter3Schrödinger’sDog:TheCopenhagenInterpretation

Chapter4ManyWorlds,ManyTreats:TheMany-Worlds

Interpretation

Chapter5AreWeThereYet?TheQuantumZenoEffect

Chapter6NoDiggingRequired:QuantumTunneling

Chapter7SpookyBarkingataDistance:Quantum

Entanglement

Chapter8BeamMeaBunny:QuantumTeleportation

Chapter9BunniesMadeofCheese:VirtualParticlesandQuantumElectrodynamics

Chapter10BewareofEvilSquirrels:MisusesofQuantumPhysics

AcknowledgmentsFurtherReadingGlossaryofImportantTermsIndex

INTRODUCTION

WhyTalktoYourDogaboutPhysics?

AnIntroductiontoQuantumPhysics

TheMohawk-HudsonHumaneSocietyhassetupalittlepaththroughthewoodsnear their facility outside Troy, so you can take a walk with a dog you’rethinkingofadopting.There’sabenchonthesideofthepathinasmallclearing,andIsitdowntolookatthedogI’vetakenout.She sits down next to the bench, and pokes my hand with her nose, so I

scratchbehindherears.MywifeandIhavelookedatloadsofdogstogether,butKatehadtowork,soI’vebeendispatchedtopickoutadogbymyself.Thisoneseemslikeagoodfit.She’s a year-old mixed-breed dog, German shepherd and something else.

She’s got the classic shepherd black and tan coloring, but she’s small for ashepherd, and has floppy ears. The tag on her kennel door gave her name as“Princess,”butthatdoesn’tseemappropriate.“Whatdoyouthink,girl?”Iask.“Whatshouldwecallyou?”“CallmeEmmy!”shesays.“Why’sthat?”“Becauseit’smyname,silly.”Beingcalled“silly”byadogisalittlesurprising,butIguessshehasapoint.

“Okay,Ican’targuewiththat.So,doyouwanttocomeandlivewithus?”“Well,thatdepends,”shesays.“Iliketochasethings.Willtherebecrittersfor

metochase?”“Well, yes.We’ve got a good-sized garden, and there are lots of birds and

squirrels,andtheoccasionalrabbit.”“Ooooh!Ilikebunnies!”Shewagshertailhappily.“Howaboutwalks?WillI

getwalks?”

“Ofcourse.”“Andtreats?Iliketreats.”“You’llgettreatsifyou’reagooddog.”She looks faintlyoffended.“Iamaverygooddog.Youwill giveme treats.

Whatdoyoudoforaliving?”“What?Who’sevaluatingwho,here?”“Ineed toknowifyoudeserveadogasgoodasme.”Thename“Princess”

mayhavebeenmoreaptthanIthought.“Whatdoyoudoforaliving?”“Well,mywife,Kate, is a lawyer, and I’m a professor of physics atUnion

College, over in Schenectady. I teach and do research in atomic physics andquantumoptics.”“Quantumwhat?”“Quantum optics. Broadly defined, it’s the study of the interaction between

light and atoms in situationswhere you have to describe one or both of themusingquantumphysics.”“Thatsoundscomplicated.”“It is, but it’s fascinating stuff.Quantumphysics has all sorts ofweird and

wonderful properties. Particles behave like waves, and waves behave likeparticles. Particle properties are indeterminate until youmeasure them.Emptyspace is full of ‘virtual particles’ popping in and out of existence. It’s reallycool.”“Hmmm.”Shelooksthoughtful,thensays,“Onelasttest.”“What’sthat?”“Rubmybelly.”Sheflopsoveronherback,andIreachdowntorubherbelly.

Afteraminuteofthat,shestandsup,shakesherselfoff,andsays“Okay,you’reprettygood.Let’sgohome.”We head back to the kennel to fill in the adoption paper-work. As we’re

walking, she says, “Quantum physics, eh? I’ll have to learn something aboutthat.”“Well,I’dbehappytoexplainittoyousometime.”

Likemost dog owners, I spend a lot of time talking tomy dog.Most of ourconversationsarefairlytypical—don’teatthat,don’tclimbonthefurniture,let’sgoforawalk.Someofourconversations,though,areaboutquantumphysics.Whydo I talk tomydogaboutquantumphysics?Well, it’swhat Ido fora

living: I’m a university physics professor. As a result, I spend a lot of timethinkingaboutquantumphysics.Whatisquantumphysics?Quantumphysicsisonepartof“modernphysics,”

meaningphysicsbasedonlawsdiscoveredafterabout1900.Lawsandprinciples

of physics that were developed before about 1900 are considered “classical”physics.Classicalphysicsisthephysicsofeverydayobjects—tennisballsandsqueaky

toys, stoves and ice cubes, magnets and electrical wiring. Classical laws ofmotiongovern themotionofanything largeenoughtoseewith thenakedeye.Classical thermodynamics explains thephysicsof heating and coolingobjects,and the operation of engines and refrigerators. Classical electromagnetismexplainsthebehavioroflightbulbs,radios,andmagnets.Modernphysicsdescribesthestrangerworldthatweseewhenwegobeyond

theeveryday.Thisworldwasfirstrevealedinexperimentsdoneinthelate1800sandearly1900s,whichcannotbeexplainedwithclassicallawsofphysics.Newfieldswithdifferentrulesneededtobedeveloped.Modern physics is divided into two parts, each representing a radical

departurefromclassicalrules.Onepart,relativity,dealswithobjectsthatmovevery fast, or are in the presence of strong gravitational forces.AlbertEinsteinintroducedrelativityin1905,andit’safascinatingsubject initsownright,butbeyondthescopeofthisbook.Theother part ofmodernphysics iswhat I talk tomydog about.Quantum

physicsorquantummechanics*isthenamegiventothepartofmodernphysicsdealing with light and things that are very small—molecules, single atoms,subatomic particles. Max Planck coined the word “quantum” in 1900, andEinsteinwontheNobelPrizeforpresentingthefirstquantumtheoryof light.*Thefulltheoryofquantummechanicswasdevelopedoverthenextthirtyyearsorso.Thepeoplewhomade the theory, fromearlypioneers likePlanckandNiels

Bohr, who made the first quantum model of the hydrogen atom, to latervisionarieslikeRichardFeynmanandJulianSchwinger,whoeachindependentlyworked out what we now call “quantum electrodynamics” (QED), are rightlyregarded as titans of physics. Some elements of quantum theory have evenescapedtherealmofphysicsandcapturedthepopularimagination,likeWernerHeisenberg’s uncertainty principle, Erwin Schrödinger’s cat paradox, and theparalleluniversesofHughEverett’smany-worldsinterpretation.Modern life would be impossible without quantum mechanics. Without an

understandingof thequantumnatureof theelectron, itwouldbe impossible tomakethesemiconductorchipsthatrunourcomputers.Withoutanunderstandingof thequantumnatureof light and atoms, itwouldbe impossible tomake thelasersweusetosendmessagesoverfiber-opticcommunicationlines.Quantum theory’s effect on science goes beyond the merely practical—it

forcesphysicists tograpplewith issuesofphilosophy.Quantumphysicsplaces

limitsonwhatwecanknowabouttheuniverseandthepropertiesofobjectsinit.Quantummechanicsevenchangesourunderstandingofwhatitmeanstomakeameasurement. It requires a complete rethinking of the nature of reality at themostfundamentallevel.Quantum mechanics describes an utterly bizarre world, where nothing is

certainandobjectsdon’thavedefinitepropertiesuntilyoumeasurethem.It’saworldwheredistantobjectsareconnectedinstrangeways,wherethereareentireuniverses with different histories right next to our own, and where “virtualparticles”popinandoutofexistenceinotherwiseemptyspace.Quantumphysicsmaysoundlikethestuffoffantasyfiction,butit’sscience.

Theworld described in quantum theory is ourworld, at amicroscopic scale.*The strange effects predicted by quantum physics are real, with realconsequencesandapplications.Quantumtheoryhasbeentestedtoanincrediblelevelofprecision,makingit themostaccurately testedtheoryin thehistoryofscientific theories. Even its strangest predictions have been verifiedexperimentally(aswe’llseeinchapters7,8,and9).So,quantumphysicsisgreatstuff.Butwhatdoesithavetodowithdogs?Dogscometoquantumphysics inabetterposition thanmosthumans.They

approachtheworldwithfewerpreconceptionsthanhumans,andalwaysexpecttheunexpected.Adogcanwalkdownthesamestreeteverydayforayear,anditwillbeanewexperienceeveryday.Everyrock,everybush,everytreewillbesniffedasifithadneverbeensniffedbefore.Ifdogtreatsappearedoutofemptyspaceinthemiddleofakitchen,ahuman

would freak out, but a dogwould take it in stride. Indeed, formost dogs, thespontaneous generation of treats would be vindication—they always expecttreatstoappearatanymoment,fornoobviousreason.Quantum mechanics seems baffling and troubling to humans because it

confoundsourcommonsenseexpectationsabouthowtheworldworks.Dogsareamuchmorereceptiveaudience.Theeverydayworldisastrangeandmarvelousplace toadog, and thepredictionsofquantum theoryareno strangerormoremarvelousthan,say,theoperationofadoorknob.*Discussing quantumphysicswithmy dog is useful because it helpsme see

how to discuss quantum mechanics with humans. Part of learning quantummechanicsislearningtothinklikeadog.Ifyoucanlookattheworldthewayadogdoes,asanendlesssourceofsurpriseandwonder,thenquantummechanicswillseemalotmoreapproachable.Thisbook reproducesa seriesof conversationswithmydogaboutquantum

physics.Each conversation is followedby adetaileddiscussionof thephysicsinvolved,aimedatinterestedhumanreaders.Thetopicsrangefromideasmany

peoplehaveheardof,likeparticle-waveduality(chapter1)andtheuncertaintyprinciple (chapter2), to themoreadvanced ideasofvirtualparticlesandQED(chapter9).Theseexplanationsincludediscussionofboththeweirdpredictionsof the theory (both practical and philosophical), and the experiments thatdemonstrate these predictions. They’re selected for what dogs find mostinterestingandalsoillustratethepartsthathumansfindsurprising.

“Idon’tknow.Ithinkitneeds…more.”“Morewhat?”“Moreme.Youdon’ttalkaboutthefactthatI’manexceptionallycleverdog.”“Fine,but—”“Anddon’tforgetgood.I’mfarbetterthanthoseotherdogs.”“Whatotherdogs?”“Dogswhoaren’tme.”“Look,thisisreallyabookaboutphysics,notabookaboutyou.”“Well,itoughttobemoreaboutme,that’sallI’msaying.”“It’snot,andyou’lljusthavetolivewiththat.”“Okay,fine.Youneedmyhelpwiththephysicsstuff,though.”“Whatdoyoumean?”“Well, sometimes you leave some stuff out, and don’t answer all of my

questions.Youshouldn’tdothat.”“Likewhat?Givemeanexample.”“Ummm…Ican’t thinkofonenow. Ifyou read it tome, though, I’llpoint

themout,andhelpfixthem.”“Okay,thatsoundsfair.Here’swhatwe’lldo.We’llgooverthebooktogether,

and if there are placeswhere you think I’ve left stuff out, we can talk aboutthem,andI’llputyourcommentsinthebook.”“Talkaboutthemlikewe’redoingnow?”“Yes,likewe’redoingnow.”“Andyou’llputtheconversationinthebook?”“Yes,Iwill.”“Inthatcase,weshouldtalkabouthowI’mtheverybest,andI’mcute,andI

shouldgetmoretreats,and—”“Okay,that’saboutenoughofthat.”“Fornow.”

* The terms “quantum physics,” “quantum theory,” and “quantum mechanics” are more or lessinterchangeable.* Inventing relativitydidn’t exactlyhurt,but theofficial reason forEinstein’sNobelwashisquantum

theoryofthephotoelectriceffect(page22).*“Microscopic”foraphysicistmeansanythingtoosmalltobeseenwiththenakedeye.Thiscoversa

range from bacteria to atoms to electrons. It’s a wide range of sizes, but physicists think it would beconfusingtohavemorethanonewordforsmallthings.*Whichunquestionablyfollowsclassicalrules,butdoes,alas,requireopposablethumbstooperate.

CHAPTER1

WhichWay?BothWays:

Particle-WaveDuality

We’reout forawalk,when thedogspotsa squirrelupaheadand takesoff inpursuit.Thesquirrelflees intoagardenanddodgesaroundasmallornamentalmaple tree.Emmydoesn’t alterhercourse in the slightest, and justbefore sheslamsintothetree,Ipullherupshort.“What’dyoudothatfor?”sheasks,indignantly.“Whatdoyoumean?Youwereabouttorunintoatree,andIstoppedyou.”“NoIwasn’t.”Shelooksoffafterthesquirrel,nowsafelyupabiggertreeon

theothersideofthegarden.“Becauseofquantum.”Westartwalkingagain.“Okay,you’regoingtohavetoexplainthat,”Isay.“Well,Ihavethisplan,”shesays.“YouknowhowwhenIchasethebunnies

in the back garden,when I run to the right of the pond, they go left, and getaway?”“Yes.”“AndwhenIruntotheleftofthepond,theygoright,andgetaway?”“Yes.”“Well,I’vethoughtofanewwaytorun,sotheycan’tescape.”“What,throughthemiddleofthepond?”It’sonlyabouteightinchesdeepand

acoupleoffeetacross.“No,silly.I’mgoingtogobothways.I’lltrapthebunniesbetweenme.”“Right…That’san…interestingtheory.”“It’snot a theory, it’squantumphysics.Material particleshavewavenature

and can diffract around objects. If you send a beam of electrons at a barrier,they’llgoaround it to the leftand to the right,at thesame time.”She’s reallygettingintothis,andshedoesn’tevennoticethecatsunningitselfinthegardenacrossthestreet.“So,I’lljustmakeuseofmywavenature,andgoaroundbothsidesofthepond.”

“Andwheredoesrunningheadfirstintoatreecomein?”“Oh, well.” She looks a little sheepish. “I thought I would try it out on

something smaller first. I got a good running start, and Iwas just about togoaroundwhenyoustoppedme.”“Ah.LikeIsaid,aninterestingtheory.Itwon’twork,youknow.”“You’renotgoingtotrytoclaimIdon’thavewavenature,areyou?BecauseI

do.It’sinyourphysicsbooks.”“No,no,you’vegotwavenature,allright.You’vealsogotBuddhanature—”“I’manenlighteneddog!”“—whichwilldoyouaboutasmuchgood.Yousee, a tree isbig, andyour

wavelength is small.Atwalking speed, a twenty-kilogram dog like you has awavelengthofabout10-35meters.Youneedyourwavelengthtobecomparabletothesizeofthetree—maybetencentimeters—inordertodiffractaroundit,andyou’rethirty-fourordersofmagnitudeoff.”“I’ll justchangemywavelengthbychangingmymomentum.Icanrunvery

fast.”“Nice try, but the wavelength gets shorter as you go faster. To get your

wavelengthuptothemillimeterorsoyou’dneedtodiffractaroundatree,you’dhave to be moving at 10-30 meters per second, and that’s impossibly slow. Itwouldtakeabillionyearstocrossthenucleusofanatomatthatspeed,whichisfartooslowtocatchabunny.”“So,you’resayingIneedanewplan?”“Youneedanewplan.”Hertaildroops,andwewalkinsilenceforafewseconds.“Right,”shesays,

“canyouhelpmewithmynewplan?”“Icantry.”“HowdoIusemyBuddhanaturetogoaroundbothsidesofthepondatthe

sametime?”Ireallycan’tthinkofanythingtosaytothat,butaflashofgrayfursavesme.

“Look!Asquirrel!”Isay.Andwe’reoffinpursuit.

Quantum physics hasmany strange and fascinating aspects, but the discoverythatlaunchedthetheorywasparticle-waveduality,orthefactthatbothlightandmatterhaveparticle-like andwavelikeproperties at the same time.Abeamoflight,whichisgenerallythoughtofasawave,turnsouttobehavelikeastreamofparticlesinsomeexperiments.Atthesametime,abeamofelectrons,whichisgenerallythoughtofasastreamofparticles,turnsouttobehavelikeawaveinsome experiments. Particle andwave properties seem to be contradictory, and

yet everything in the universe somehowmanages to be both a particle and awave.The discovery in the early 1900s that light behaves like a particle is the

launchingpointforallofquantummechanics.Inthischapter,we’lldescribethehistoryofhowphysicistsdiscoveredthisstrangeduality.Inorder toappreciatejust what a strange development this is, though, we need to talk about theparticlesandwavesthatweseeineverydaylife.

PARTICLESANDWAVESAROUNDYOU:CLASSICALPHYSICS

Everybodyisfamiliarwiththebehaviorofmaterialparticles.Prettymuchalltheobjectsyouseearoundyou—bones,balls, squeaky toys—behave likeparticlesin the classical sense,with theirmotiondeterminedbyclassicalphysics.Theyhavedifferent shapes, but you canpredict their essentialmotionby imaginingeach as a small, featureless ball with some mass—a particle—and applyingNewton’slawsofmotion.*Atennisballandalongbonetumblingendoverendlookverydifferentinflight,butifthey’rethrowninthesamedirectionwiththesamespeed,they’lllandinthesameplace,andyoucanpredictthatplaceusingclassicalphysics.Aparticle-likeobject has adefinite position (youknow rightwhere it is), a

definitevelocity(youknowhowfastit’smoving,andinwhatdirection),andadefinitemass(youknowhowbigitis).Youcanmultiplythemassandvelocitytogether, to find the momentum. A great big Labrador retriever has moremomentum than a little French poodlewhen they’re bothmoving at the samespeed, and a fast-moving border collie has more momentum than a waddlingbassethoundofthesamemass.Momentumdetermineswhatwillhappenwhentwoparticles collide.When amovingobject hits a stationaryone, themovingobjectwillslowdown,losingmomentum,whilethestationaryobjectwillspeedup,gainingmomentum.The other notable feature of particles is something that seems almost too

obvioustomention:particlescanbecounted.Whenyouhavesomecollectionofobjects,youcanlookatthemanddetermineexactlyhowmanyofthemyouhave—onebone,twosqueakytoys,threesquirrelsunderatreeinthebackgarden.Waves,on theotherhand,areslipperier.Awave isamovingdisturbance in

something,likethepatternsofcrestsandtroughsformedbywatersplashinginagardenpond.Wavesare spreadoutover some regionof spaceby theirnature,formingapatternthatchangesandmovesovertime.Nophysicalobjectsmoveanywhere—the water stays in the pond—but the pattern of the disturbance

changes,andweseethatasthemotionofawave.If youwant to understand awave, there are twoways of looking at it that

provide useful information.One is to imagine taking a snapshot of thewholewave,andlookingatthepatternofthedisturbanceinspace.Forasinglesimplewave,youseeapatternofregularpeaksandvalleys,likethis:

Asyoumovealongthepattern,youseethemediummovingupanddownbyan amount called the “amplitude” of the wave. If you measure the distancebetweentwoneighboringcrestsofthewave(ortwotroughs),you’vemeasuredthe“wavelength,”whichisoneofthenumbersusedtodescribeawave.Theotherthingyoucandois tolookatonelittlepieceofthewavepattern,

andwatchitforalongtime—imaginewatchingaduckbobbingupanddownonalake,say.Ifyouwatchcarefully,you’llseethatthedisturbancegetsbiggerandsmaller in a very regular way—sometimes the duck is higher up, sometimeslowerdown—andmakesapatternin timeverymuchlike thepatterninspace.Youcanmeasurehowoftenthewaverepeatsitselfinagivenamountoftime—howmany times the duck reaches itsmaximumheight in aminute, say—andthat gives you the “frequency” of thewave, which is another critical numberusedtodescribethewave.Wavelengthandfrequencyarerelatedtoeachother—longerwavelengthsmeanlowerfrequency,andviceversa.Youcanalreadyseehowwavesaredifferentfromparticles:theydon’thavea

position.Thewavelengthandthefrequencydescribethepatternasawhole,butthere’snosingleplaceyoucanpointtoandidentifyasthepositionofthewave.Thewaveitselfisadisturbancespreadoverspace,andnotaphysicalthingwithadefinitepositionandvelocity.Youcanassignavelocitytothewavepattern,bylookingathowlongittakesonecrestofthewavetomovefromonepositiontoanother,butagain,thisisapropertyofthepatternasawhole.You also can’t count waves the way you can count particles—you can say

howmanycrestsand troughs thereare inoneparticulararea,but thoseareallpartofasinglewavepattern.Wavesarecontinuouswhereparticlesarediscrete—you can say that you have one, two, or three particles, but you either havewaves, or youdon’t. Individualwavesmayhave larger or smaller amplitudes,buttheydon’tcomeinchunkslikeparticlesdo.Wavesdon’tevenaddtogetherin the same way that particles do—sometimes, when you put two wavestogether, you end upwith a biggerwave, and sometimes you end upwith nowaveatall.Imaginethatyouhavetwodifferentsourcesofwavesinthesamearea—two

rocksthrownintostillwateratthesametime,forexample.Whatyougetwhenyouaddthetwowavestogetherdependsonhowtheylineup.Ifyouaddthetwowaves togethersuchthat thecrestsofonewavefallontopof thecrestsof theother,andthetroughsofonewavefall inthetroughsoftheother(suchwavesarecalled“inphase”),you’llgetalargerwavethaneitherofthetwoyoustartedwith.On theotherhand, ifyouadd twowaves togethersuch that thecrestsofonewavefallinthetroughsoftheotherandviceversa(“outofphase”),thetwowillcancelout,andyou’llendupwithnowaveatall.This phenomenon is called interference, and it’s perhaps themost dramatic

differencebetweenwavesandparticles.

“I don’t know … that’s pretty weird. Do you have any other examples ofinterference?Somethingmore…doggy?”“No, I really don’t. That’s the point—waves are dramatically different than

particles.Nothingthatdogsdealwithonaregularbasisisallthatwavelike.”“Howabout,‘Interferenceis likewhenyouputasquirrel in thegarden,and

thenyouputadoginthegarden,andaminutelater,there’snomoresquirrelinthegarden.’”“That’snotinterference,that’spreypursuit.Interferenceismorelikeputtinga

squirrel in thegarden, thenputtinga secondsquirrel in thegardenone secondlater,andfindingthatyouhavenosquirrelsatall.Butifyouwaittwo secondsbeforeputtinginthesecondsquirrel,youfindfoursquirrels.”“Okay,that’sjustweird.”“That’smypoint.”“Oh.Well,goodjob,then.Anyway,whyarewetalkingaboutthis?”“Well, you need to know a few things about waves in order to understand

quantumphysics.”“Yes,butthisjustsoundslikemaths.Idon’tlikemaths.Whenarewegoingto

talkaboutphysics?”“Wearetalkingaboutphysics.Thewholepointofphysicsistousemathsto

describetheuniverse.”“Idon’twanttodescribetheuniverse,Iwanttocatchsquirrels.”“Well,ifyouknowhowtodescribetheuniversewithmaths,thatcanhelpyou

catchsquirrels.Ifyouhaveamathematicalmodelofwherethesquirrelsarenow,andyouknowtherulesgoverningsquirrelbehavior,youcanuseyourmodeltopredictwherethey’llbelater.Andifyoucanpredictwherethey’llbelater…”“Icancatchsquirrels!”“Exactly.”“Allright,mathsisokay.Istilldon’tseewhatthiswavestuffisfor,though.”“Weneedit toexplainthepropertiesof lightandsoundwaves,whichis the

nextbit.”

WAVESINEVERYDAYLIFE:LIGHTANDSOUND

Wedealwithtwokindsofwavesineverydaylife:lightandsound.Thoughthesearebothexamplesofwavephenomena, theyappear tobehaveverydifferently.Thereasonsforthosedifferenceswillhelpshedsomelight(pardonthepun)onwhyitisthatwedon’tseedogspassingaroundbothsidesofatreeatthesametime.Soundwavesarepressurewavesintheair.Whenadogbarks,sheforcesair

outthroughhermouthandsetsupavibrationthattravelsthroughtheairinalldirections.Whenitreachesanotherdog,thatsoundwavecausesvibrationsinthesecond dog’s eardrums, which are turned into signals in the brain that areprocessedassound,causingtheseconddogtobark,producingmorewaves,untilnearbyhumansgetannoyed.Lightisadifferentkindofwave,anoscillatingelectricandmagneticfieldthat

travelsthroughspace—eventheemptinessofouterspace,whichiswhywecanseedistantstarsandgalaxies.Whenlightwavesstrikethebackofyoureye,theyget turned into signals in thebrain that areprocessed to forman imageof theworldaroundyou.Themoststrikingdifferencebetweenlightandsoundineverydaylifehasto

dowithwhathappenswhentheyencounteranobstacle.Lightwavestravelonlyinstraightlines,whilesoundwavesseemtobendaroundobstacles.Thisiswhyadog in thediningroomcanhearapotatochiphitting thekitchenfloor,eventhoughshecan’tseeit.The apparent bending of sound waves around corners is an example of

diffraction,whichisacharacteristicbehaviorofwavesencounteringanobstacle.Whenawavereachesabarrierwithanopeninginit,likethewallcontainingan

opendoorfromthekitchenintothediningroom,thewavespassingthroughtheopening don’t just keep going straight, but fan out over a range of differentdirections.Howquicklytheyspreaddependsonthewavelengthofthewaveandthesizeoftheopeningthroughwhichtheytravel.Iftheopeningismuchlargerthan the wavelength, there will be very little bending, but if the opening iscomparable to the wavelength, the waves will fan out over the full availablerange.

Ontheleft,awavewithashortwavelengthencountersanopeningmuchlargerthanthewavelength,and thewavescontinuemoreor less straight through.On theright,awavewitha longwavelengthencountersanopeningcomparabletothewavelength,andthewavesdiffractthroughalargerangeofdirections.

Similarly,ifsoundwavesencounteranobstaclelikeachairoratree,theywilldiffract around it, provided the object is not too much larger than thewavelength.This iswhy it takesa largewall tomuffle thesoundofabarkingdog—sound waves bend around smaller obstacles, and reach people or dogsbehindthem.Soundwaves inairhaveawavelengthofameteror so,close to thesizeof

typical obstacles—doors,windows, pieces of furniture.As a result, thewavesdiffractbyalargeamount,whichiswhywecanhearsoundsevenaroundtightcorners.Lightwaves, on the other hand, have a very shortwavelength—less than a

thousandthofamillimeter.Ahundredwavelengthsofvisiblelightwillfitinthethicknessofahair.Whenlightwavesencountereverydayobstacles,theyhardlybendatall, sosolidobjectscastdarkshadows.A tinybitofdiffractionoccursrightat theedgeof theobject,which iswhy theedgesof shadowsarealwaysfuzzy, but for the most part, light travels in a straight line, with no visible

diffraction.Ifwedon’t readily see lightdiffracting likeawave,howdoweknow it’sa

wave? We don’t see diffraction around everyday objects because they’re toolargecomparedtothewavelengthoflight.Ifwelookatasmallenoughobstacle,though,wecanseeunmistakableevidenceofwavebehavior.In 1799 an English physicist named Thomas Young did the definitive

experimenttodemonstratethewavenatureoflight.Youngtookabeamoflightandinsertedacardwithtwoverynarrowslitscut init.Whenhelookedat thelightonthefarsideofthecard,hedidn’tjustseeanimageofthetwoslits,butratheralargepatternofalternatingbrightanddarkspots.Young’sdouble-slitexperimentisacleardemonstrationofthediffractionand

interferenceoflightwaves.Thelightpassingthrougheachoftheslitsdiffractsoutintoarangeofdifferentdirections,andthewavesfromthetwoslitsover-lap.At any given point, the waves from the two slits have traveled differentdistances,andhavegonethroughdifferentnumbersofoscillations.Atthebrightspots,thetwowavesareinphase,andaddtogethertogivelightthatisbrighterthanlightfromeitherslitbyitself.Atthedarkspots,thewavesareoutofphase,andcanceleachotherout.PriortoYoung’sexperiment,therehadbeenalivelydebateaboutthenatureof

light,withsomephysicistsclaimingthatlightwasawave,andothers(includingNewton) arguing that light was a stream of tiny particles. Interference anddiffraction are phenomena that only happen with waves, though, so afterYoung’s experiment (and subsequent experiments by the French physicistAugustin Fresnel), everybody was convinced that light was a wave. Thingsstayedthatwayforaboutahundredyears.

Anillustrationofdouble-slitdiffraction.Ontheleft,thewavesfromtwodifferentslitstravelexactlythesamedistance,andarrive inphase to formabrightspot. In thecenter, thewave fromthe lowerslittravelsanextrahalf-wavelength(darkerline),andarrivesoutofphasewiththewavefromtheupperslit.Thetwocancelout,formingadarkspotinthepattern.Ontheright,thewavefromthelowerslittravelsafullextrawavelength,andagainaddstothewavefromtheupperslittoformabrightspot.

•••

“Howdoesthisrelatetogoingaroundbothsidesofatree?I’mnotinterestedingoingthroughslits,Iwanttocatchbunnies.”“Thesamebasicprocesshappenswhenyouputsmallsolidobstaclesintothe

pathofalightbeam.Youcanthinkofthelightthatgoesaroundtotheleftandthe light that goes around to the right of the obstacle as being like thewavesfromtwodifferentslits.Theytakedifferentpaths to theirdestination,and thuscan be either in phase or out of phasewhen they arrive.You get a pattern ofbrightanddarkspots,justlikewhenyouuseslits.”“Oh.Isupposethatmakessense.So,Ijustneedtogetthebunniestostandat

thespotswhereI’minphasewithme?”“No,becauseofthewavelengththing.We’llgettothatinaminute.Ineedto

talkaboutparticles,first.”“Okay.Icanbepatient.Aslongasitdoesn’ttaketoolong.”

THEBIRTHOFTHEQUANTUM:LIGHTASAPARTICLE

Thefirsthintofaproblemwith thewavemodelof lightcamefromaGermanphysicistnamedMaxPlanckin1900.Planckwasstudyingthethermalradiationemittedbyall objects.Theemissionof lightbyhotobjects is averycommonphenomenon(thebest-knownexampleistheredglowofahotpieceofmetal),andsomethingsocommonseemslike itought tobeeasytoexplain.By1900,though, the problem of explaining how much light of different colors wasemitted(the“spectrum”ofthelight)hadthusfardefeatedthebestphysicistsofthenineteenthcentury.Planckknewthatthespectrumhadaveryparticularshape,withlotsoflight

emittedatlowfrequenciesandverylittleathighfrequencies,andthatthepeakofthespectrum—thefrequencyatwhichthelightemittedisbrightest—dependsonlyontheobject’stemperature.Hehadevendiscoveredaformulatodescribethecharacteristicshapeofthespectrum,butwasstymiedwhenhetriedtofindatheoretical justification for the formula.Everymethodhe triedpredictedmuchmorelightathighfrequenciesthanwasobserved.Indesperation,heresortedtoamathematicaltricktogettherightanswer.Planck’strickwastoimaginethatallobjectscontainedfictitious“oscillators”

that emit light only at certain frequencies. Then he said that the amount ofenergy(E) associatedwith each oscillatorwas related to the frequency of theoscillation(f)byasimpleformula:

E=hf

wherehisaconstant.Whenhefirstmadethisoddassumption,Planckthoughthewoulduseitjusttosetuptheproblem,andthenuseacommonmathematicaltechniquetogetridoftheimaginaryoscillatorsandthisextraconstanth.Muchtohissurprise,though,hefoundthathisresultsmadesenseonlyifhekepttheoscillatorsaround—ifhhadaverysmallbutnonzerovalue.Today,hisknownasPlanck’sconstantinhishonor,andhasthevalue6.626×

10-34 kgm2/s (that’s 0.0000000000000000000000000000000006626 kg m2/s).It’saverysmallnumberindeed,butdefinitelynotzero.Planck’s trick amounts to treating light, which physicists thought of as a

continuous wave, as coming in discrete chunks, like particles. Planck’s“oscillators”couldonlyemitlightindiscreteunitsofbrightness.Thisisalittlelike imaginingapondwherewavescanonlybeone, two,or threecentimetershigh,neveroneandahalfortwoandaquarter.Everydaywavesdon’tworkthatway,butthat’swhatPlanck’smathematicalmodelrequires.These“oscillators”arealsowhatputs the“quantum” in“quantumphysics.”

Planckreferredtothespecificlevelsofenergyinhisoscillatorsas“quanta”(thepluralof“quantum,”fromtheLatinwordfor“howmuch”),soanoscillatoratagivenfrequencymightcontainonequantum(oneunitofenergy,hf),twoquanta,threequanta,andsoon,butneveroneandahalfortwoandaquarter.Thenameforthestepsstuck,andcametobeappliedtotheentiretheorythatgrewoutofPlanck’sdesperatetrick.Thoughhe’softengivencreditfor inventingtheideaof lightquanta,Planck

neverreallybelievedthatlightcameindiscretequanta,andhealwayshopedthatsomebodywould find a cleverway to derive his formulawithout resorting totrickery.ThefirstpersontotalkseriouslyaboutlightasaquantumparticlewasAlbert

Einstein in 1905, who used it to explain the photoelectric effect. Thephotoelectric effect is another physical effect that seems like it ought to besimpletodescribe:whenyoushinelightonapieceofmetal,electronscomeout.Thisformsthebasisforsimplelightsensorsandmotiondetectors:lightfallingonasensorknockselectronsoutofthemetal,whichthenflowthroughacircuit.Whentheamountoflighthittingthesensorchanges,thecircuitperformssomeaction,suchasturninglightsonwhenitgetsdark,oropeningdoorswhenadogpassesinfrontofthesensor.Thephotoelectriceffectoughttobereadilyexplainedbythinkingoflightasa

wave that shakes atoms back and forth until electrons come out, like a dogshakingabagoftreatsuntiltheyflyalloverthekitchen.Unfortunately,thewavemodelcomesoutallwrong: itpredicts that theenergyof theelectrons leavingtheatomsshoulddependontheintensityofthelight—thebrighterthelight,the

harder the shaking, and the faster the bits flying away should move. Inexperiments,though,theenergyoftheelectronsdoesn’tdependontheintensityatall.Instead,theenergydependsonthefrequency,whichthewavemodelsaysshouldn’tmatter.Atlowfrequencies,younevergetanyelectronsnomatterhowhardyoushake,whileathighfrequency,evengentleshakingproduceselectronswithagooddealofenergy.

•••

“Physicistsaresilly.”“Ibegyourpardon?”“Well,anydogknowsthat.Whenyougetabagwithtreatsinit,youalways

shakeitasfastasyoucan,ashardasyoucan.That’showyougetthetreatsout.”“Yes,well,whatcanIsay?Dogshaveanexcellentintuitivegraspofquantum

theory.”“Thankyou.”“Ofcourse,thepointofphysicsistounderstandwhythetreatscomeoutwhen

theydo.”“Maybeforyou.Fordogs,thepointistogetthetreats.”

EinsteinexplainedthephotoelectriceffectbyapplyingPlanck’sformulatolightitself.Einsteindescribedabeamoflightasastreamoflittleparticles,eachwithan energy equal to Planck’s constantmultiplied by the frequency of the lightwave(thesameruleusedforPlanck’s“oscillators”).Eachphoton(thenamenowgiven to these particles of light) has a fixed amount of energy it can provide,dependingonthefrequency;andsomeminimumamountofenergyisrequiredtoknock an electron loose. If the energy of a single photon is more than theminimumneeded, theelectronwillbeknocked loose,andcarry the restof thephoton’senergywith it.Thehigher thefrequency, thehigher thesinglephotonenergyandthemoreenergytheelectronshavewhentheyleave,exactlyas theexperimentsshow.If theenergyofasinglephoton is lower than theminimumenergy for knocking an electron out, nothing happens, explaining the lack ofelectronsatlowfrequencies.*Describing light as a particlewas a hugely controversial idea in 1905, as it

overturnedahundredyears’worthofphysicsandrequiresaverydifferentviewof light. Rather than a continuouswave, likewater poured into a dog’s bowl,lighthastobethoughtofasastreamofdiscreteparticles,likeascoopofkibblepoured into a bowl. And yet each of those particles still has a frequencyassociatedwithit,andsomehowtheyadduptogiveaninterferencepattern,justlikeawave.

Other physicists in 1905 found this deeply troubling, and Einstein’s modeltookawhiletogainacceptance.TheAmericanphysicistRobertMillikanhatedEinstein’sidea,andperformedaseriesofextremelyprecisephotoelectriceffectexperiments in 1916 hoping to prove Einstein wrong.* In fact, his resultsconfirmedEinstein’spredictions,buteventhatwasn’tenoughtogetthephotonidea accepted.Wide acceptance of the photon picture didn’t come until 1923,whenArthurHollyComptondidafamousseriesofexperimentswithX-raysthatdemonstrated unmistakably particle-like behavior from light: he showed thatphotonscarrymomentum,andthismomentumistransferredtootherparticlesincollisions.IfyoutakethePlanckformulafortheenergyofasinglephoton,andcombine

itwithequationsfromEinstein’sspecialrelativity,youfindthatasinglephotonoflightoughttocarryasmallamountofmomentum,givenbytheformula:

p=h/λ

wherepisthesymbolformomentumandλisthewavelengthofthelight.

•••

“Ithoughtyousaidtherewasn’tanyrelativityinthisbook?”“Isaidthebookisn’tabout relativity.That’snotthesamething.Someideas

fromrelativityareimportanttoquantumphysics,aswell.”“What’srelativitygottodowiththis,though?”“Well,whatrelativitysaysisthatbecauseaphotonhassomeenergy,itmust

havesomemomentum,eventhoughitdoesn’thaveanymass.”“So…it’sanE=mc2thing?”“Notexactly,butit’ssimilar.Photonshavemomentumbecauseoftheirenergy

in the sameway thatobjectshaveenergybecauseof theirmass.Andnice jobdroppinganequationinthere.”“Please.EveninferiordogsknowE=mc2.AndIamanexceptionaldog.”

Aphotonwithasmallwavelengthhasalotofmomentum,whileaphotonwithalarge wavelength has very little. That means that the interaction between aphotonoflightandastationaryobjectoughttolookjustlikeacollisionbetweentwoparticles: thestationaryobjectgainssomeenergyandmomentum,and themovingphotonlosessomeenergyandmomentum.Wedon’tnoticethisbecausethemomentuminvolvedistiny—Planck’sconstantisaverysmallnumber—butifwelookatanobjectwithaverysmallmass,likeanelectron,andphotonswithaveryshortwavelength(and thusarelativelyhighmomentum),wecandetect

thechangeinmomentum.In 1923, Compton bounced X-rays with an initial wave-length of 0.0709

nanometers*offasolidtarget(X-raysarejustlightwithanexceptionallyshortwavelength,comparedtoabout500nmforvisiblelight).WhenhelookedattheX-raysthatscatteredoff thetarget,hefoundthat theyhadlongerwavelengths,indicatingthattheyhadlostmomentum(X-raysbouncingoffat90degreesfromtheiroriginaldirectionhadawavelengthof0.0733nm,forexample).Thislossofmomentumisexactlywhatshouldhappeniflightisaparticle:whenanX-rayphotoncomesinandhitsamoreorlessstationaryelectroninatarget,itgivesupsomeofitsmomentumtotheelectron,whichstartsmoving.Afterthecollision,the photon has less momentum, and thus a longer wavelength, exactly asComptonobserved.Theamountofmomentumlostalsodependsontheangleatwhichthephoton

bounces off—a photon that glances off an electron doesn’t lose very muchmomentum,while one that bounces almost straight back loses a lot.Comptonmeasuredthewavelengthatmanydifferentangles,andhisresultsexactlyfitthetheoretical prediction, confirming that the shift was from collisions withelectrons,andnotsomeothereffect.Einstein,Millikan,andComptonallwonNobelprizesfordemonstrating the

particle nature of light. Taken together, Millikan’s photoelectric effectexperiments and Compton’s scattering experiments were enough to get mostphysiciststoaccepttheideaoflightasbeingmadeupofastreamofparticles.*Asstrangeastheideaoflightasaparticlewas,though,whatcamenextwas

evenstranger.

INTERFERINGELECTRONS:PARTICLESASWAVES

Also in 1923, aFrenchPh.D. student namedLouisVictorPierreRaymonddeBroglie†madearadicalsuggestion:hearguedthat thereought tobesymmetrybetweenlightandmatter,andsoamaterialparticlesuchasanelectronoughttohave a wavelength. After all, if light waves behave like particles, shouldn’tparticlesbehavelikewaves?DeBrogliesuggestedthatjustasaphotonhasamomentumdeterminedbyits

wavelength, a material object like an electron should have a wavelengthdeterminedbyitsmomentum:

λ=h/p

whichisjusttheformulaforthemomentumofaphoton(page24)turnedaroundtogivethewavelength.Theideahasacertainmathematicalelegance,whichwasappealing to theoretical physicists even in 1923, but it also seems like patentnonsense—solidobjectsshownosignofbehavinglikewaves.WhendeBrogliepresentedhisideaaspartofhisPh.D.thesisdefense,nobodyknewwhattomakeofit.Hisprofessorsweren’tevensurewhethertogivehimthedegreeornot,andresortedtoshowinghisthesistoEinstein.Einsteinproclaimeditbrilliant,anddeBrogliegothisdegree,buthisideaofelectronsaswaveshadlittlesupportuntiltwoexperiments in the late1920sshowed incontrovertibleproof thatelectronsbehavedlikewaves.In1927,twoAmericanphysicists,ClintonDavissonandLesterGermer,were

bouncingelectronsoffasurfaceofnickel,andrecordinghowmanybouncedoffat different angles. Theywere surprisedwhen their detector picked up a verylargenumberofelectronsbouncingoffatoneparticularangle.Thismysteriousresult was eventually explained as the wavelike diffraction of the electronsbouncingoffdifferentrowsofatomsintheirnickeltarget.Thebeamofelectronspenetratedsomedistance into thenickel,andpartof thebeambouncedoff thefirst row of atoms in the nickel crystal,* while other parts bounced off thesecond, and the third, and so on. Electrons reflecting from all these differentrowsofatomsbehavedlikewaves.Thewavesthatbouncedoffatomsdeeperinthecrystaltraveledfartheronthewayoutthantheonesthatbouncedoffatomsclosertothesurface.Thesewavesinterferedwithoneanother,likelightwavespassing through the different slits in Young’s experiment (though with manyslits,notjusttwo).Mostofthetime,thereflectedwaveswereoutofphaseandcanceledoneanotherout.Forcertainangles,though,theextradistancetraveledwasexactlyrightforthewavestoaddinphaseandproduceabrightspot,whichDavisson andGermer detected as a large increase in the number of electronsreflectedatthatangle.ThedeBroglieformulaforassigningawavelengthtotheelectronpredictstheDavissonandGermerresultperfectly.†

•••

Diffractionofelectronsoffacrystalofnickel.Anincomingelectronbeam(dashedline)passesintoaregularcrystalofatoms,andbitsof thewave(individualelectrons)reflectoffdifferentatoms in thecrystal.Electronsreflectedfromdeeperinthecrystaltravelalongerdistanceonthewayout(darkerline),butforcertainangles,thatdistanceisamultipleofafullwavelength,andthewavesleavingthecrystaladdinphasetogivethebrightspotseenbyDavissonandGermer.

“Wait,howdoes thatwork? If thereare lotsof slits, shouldn’t therebe lotsofspots?”“Notreally.Whenyouaddthewavestogether,youstillgetapatternofbright

and dark spots, but as you use more slits, the bright spots get brighter andnarrower,andthedarkspotsgetdarkerandwider.”“So,ifIrunthroughthepicketfencetotheneighbors’garden,I’llgetbrighter

andnarrowerontheotherside?”“You’dbenarrower,allright,butitwouldn’tbeabrightidea.Thepointhere

is that the ‘slits’ thatDavisson andGermerwere usingwere so close togetherthattheycouldonlyseeonebrightspotintheregionwheretheycouldputtheirdetector.Withadifferentcrystal,orfaster-movingelectrons,theywould’veseenmorespots.”

Ataroundthesametime,GeorgePagetThomsonattheUniversityofAberdeencarriedoutaseriesofexperiments inwhichheshotbeamsofelectronsat thinfilms of metal, and observed diffraction patterns in the transmitted electrons(such patterns are produced in essentially the same way as the pattern in theDavisson-Germerexperiment).DiffractionpatternslikethoseseenbyDavissonandGermer andThomsonare anunmistakable signatureofwavebehavior, asThomas Young showed in 1799, so their experiments provided proof that deBrogliewas right, and electronshavewavenature.DeBrogliewon theNobel

PrizeinPhysicsin1929forhisprediction,andDavissonandThomsonsharedaNobelPrizein1937fordemonstratingthewavenatureoftheelectron.*FollowingtheexperimentsofDavissonandGermerandThomson,scientists

showed that all subatomic particles behave like waves: beams of protons andneutronswilldiffractoffsamplesofatomsinexactlythesamewaythatelectronsdo. In fact, neutron diffraction is now a standard tool for determining thestructureofmaterials at theatomic level: scientists candeducehowatomsarearranged by looking at the interference patterns that result when a beam ofneutrons bounces off their sample. Knowing the structure of materials at theatomic levelallowsmaterialsscientists todesignstrongerand lightermaterialsforuseincars,planes,andspaceprobes.Neutrondiffractioncanalsobeusedtodetermine the structure of biological materials like proteins and enzymes,providingcriticalinformationforscientistssearchingfornewdrugsandmedicaltreatments.

EVERYTHINGISMADEOFWAVES:INTERFERENCEOFMOLECULES

So, if allmaterial objects aremade up of particleswithwave properties,whydon’tweseedogsdiffractingaroundtrees?Ifabeamofelectronscandiffractofftwo rows of atoms,why can’t a dog run around both sides of a tree to trap arabbitonthefarside?Theansweristhewavelength:aswiththesoundandlightwaves discussed earlier, the dramatically different behavior of dogs andelectrons encountering obstacles is explained by the difference in theirwavelengths.Thewavelengthisdeterminedbythemomentum,andadoghasalotmoremomentumthananelectron.ThewavelengthofamaterialobjectisgivenbyPlanck’sconstantdividedby

themomentum,whichismassmultipliedbyvelocity.Planck’sconstantisatinynumber, but so is the mass of an electron—about 10-30 kilograms, or0.000000000000000000000000000001 kg. Davisson and Germer’s electrons,movingatthebriskspeedofsixmillionmeterspersecond,hadawavelengthofaboutatenthofananometer(0.0000000001m).That’sextremelysmall,butit’sabouthalfthedistancebetweentwonickelatoms,justrightforseeingdiffraction(justlikesoundwaveswithhalf-meterwavelengthswillreadilydiffractthroughone-meterwidedoors).Thewavelengthofa50-pound(about20kg)dogoutforastroll,ontheother

hand,isabout10-35meters(0.00000000000000000000000000000000001m),or

a millionth of a billionth of a billionth of the wavelength of Davisson andGermer’selectrons.Howdoesthatcomparetothesizeofatree?Well,adog’swavelength compared to the distance between two atoms is like the distancebetween two atoms compared to the diameter of the solar system. There’s nochanceofseeingthewaveassociatedwithadogdiffractoffacrystalofnickel,letalonepassaroundbothsidesofatreeatthesametime.There’salotofroombetweenabeamofelectronsandadog,though,sowhat

is the biggest material object that has been shown to have observable wavenature?In1999, a researchgroup at theUniversity ofViennaheadedbyDr.Anton

Zeilingerobserveddiffractionandinterferencewithmoleculesconsistingof60carbon atoms bound together into a shape like a tiny soccer ball, eachwith amass about a million times that of an electron. They shot these soccer-ball-shapedmoleculestowardadetector,andwhentheylookedatthedistributionofmolecules down-stream, they saw a single narrow beam. Then they sent thebeamthroughasiliconwaferwithacollectionofverysmallslitscutintoit,andlookedatthedistributionofmoleculesonthefarsideoftheslits.Withtheslitsin place, the initial narrow peak grew broader,with distinct “lumps” to eitherside.Thoselumps,likethebrightanddarkspotsseenbyThomasYoungshininglight through a double slit, or the electron diffraction peaks seen byDavissonandGermer,areanunmistakablesignatureofwavebehavior.Moleculespassingthroughtheslitsspreadoutandinterferewithoneanother,justlikelightwaves.

Theinterferencepatternproducedbyabeamofmoleculespassingthroughanarrayofnarrowslits.Theextralumpstoeithersideofthecentralpeakaretheresultofdiffractionandinterferenceofthemolecules passing through the slits. (Reprinted with permission from O. Nairz, M. Arndt, and A.Zeilinger,Am.J.Phys.71,319[2003].Copyright2003,AmericanAssociationofPhysicsTeachers.)

In subsequent experiments, theZeilinger group demonstrated the diffractionofevenlargermolecules,adding48fluorineatomstoeachoftheiroriginal60-carbon-atommolecules.Thesemoleculeshaveamassaboutthreemilliontimesthemassofoneelectron,andstandas thecurrent record for themostmassiveobjectwhosewavenaturehasbeenobserveddirectly.As themass of a particle increases, itswavelength gets shorter and shorter,

anditgetsharderandhardertoseewaveeffectsdirectly.Thisiswhynobodyhaseverseenadogdiffractaroundatree;norarewelikelytoseeitanytimesoon.In terms of physics, though, a dog is nothing but a collection of biologicalmolecules,which theZeilingergrouphasshownhavewaveproperties.So,wecansaywithconfidencethatadoghaswavenature,justthesameaseverythingelse.

“So,whicharetheyreally?”“Whatdoyoumean?”“Well, areelectrons reallyparticlesacting likewaves,or arephotonswaves

actinglikeparticles?”“You’re asking the wrong questions. Or, rather, you’re giving the wrong

answers. The real answer is ‘DoorNumber Three.’ Electrons and photons arebothexamplesofa thirdsortofobject,which isneither justawavenor justaparticle,buthassomewavepropertiesandsomeparticlepropertiesatthesametime.”“So,it’slikeasquirunny?”“Awhat?”“A creature that’s something like a squirrel, and something like a bunny.A

squirunny.”“Iprefer‘quantumparticle,’butIsupposethat’sthebasicidea.Everythingin

theuniverseisbuiltofthesequantumparticles.”“That’sprettyweird.”“Oh,that’sjustthebeginningoftheweirdstuff…”

*SirIsaacNewton,ofthefallingapplestory,setforththreelawsofmotionthatgovernthebehaviorof

movingobjects.Thefirstlawistheprincipleofinertia,thatobjectsatresttendtoremainatrest,andobjectsinmotiontendtoremaininmotionunlessactedonbyanexternalforce.Thesecondlawquantifiesthefirst,andisusuallywrittenastheequationF=ma,forceequalsmasstimesacceleration.Thethirdlawsaysthatforeveryactionthereisanequalandoppositereaction—aforceofequalstrengthintheoppositedirection.These three laws describe themotion ofmacroscopic objects at everyday speeds, and form the core ofclassicalphysics.*Youmightwonderwhyyoucan’tputtogethertwolow-energypho-tonstoprovideenoughenergyto

freeanelectron.Thiswouldrequiretwophotonstohitthesameelectronatthesameinstant,andthatalmostneverhappens.

* Millikan thought the Einstein model lacked “any sort of satisfactory theoretical foundation,” anddescribed its success as “purely empirical,”which is pretty nasty by physics standards. Ironically, thosequotesare fromthe firstparagraphof thepaper inwhichheconclusivelyconfirms thepredictionsof thetheory.*Ananometeris10-9m,oronebillionthofameter(0.000000001m).*A few die-hard theorists still resisted the idea of photons, because even theCompton effect can be

explained without photons, though it’s very complicated. The last resistance collapsed in 1977, whenincontrovertibleproofoftheexistenceofphotonswasprovidedinanexperimentbyKimble,Dagenais,andMandel that looked at the light emitted by single atoms. The seventy-two-year gap between Einstein’sproposalanditsfinalacceptancetellsyousomethingaboutthestubbornnessofphysicistsconfrontedwithanewidea.Itcanbeasdifficulttoseparateaphysicistfromacherishedmodelasit istodragadogawayfromawell-chewedbone.† The proper pronunciation of Louis de Broglie’s surname (his collection of names reflects his

aristocraticbackground—hewasthe7thDucdeBroglie)isthesourceofmuchconfusionamongphysicists.I’ve heard “de-BRO-lee,” “de-BRO-glee,” and “de-BROY-lee,” among others. The correct Frenchpronunciationisapparentlysomethingcloseto“de-BROY,”onlywithagarglysortofsoundtothevowelthatyouneedtobeFrenchtomake.*“Crystal,”toaphysicist,referstoanysolidwitharegularandorderlyarrangementofatomsinit.This

includestheclearandsparklythingsthatwenormallyassociatewiththeword,butalsoalotofmetalsandothersubstances.†Ironically,DavissonandGermersucceededonlybecausetheybrokeapieceof theirapparatus.They

didn’tseeanydiffractioninthefirstexperimentstheydid,becausetheirnickeltargetwasmadeupofmanysmall crystals, each producing a different interference pattern, and the bright spots from the differentpatternsrantogether.Thentheyaccidentallyletairintotheirvacuumsystem.Intheprocessofrepairingthedamage, theymelted the target,which recrystallized into a single large crystal, producing a single, cleardiffractionpattern.Sometimes,theluckiestthingaphysicistcandoistobreaksomethingimportant.*InoneofthegreatbitsofNobeltrivia,Thomson’sfather,J.J.ThomsonofCambridge,wonthe1906

NobelPrizeinPhysicsfordemonstratingtheparticlenatureoftheelectron.Thispresumablyledtosomeinterestingdinner-tableconversationintheThomsonhousehold.

CHAPTER2

Where’sMyBone?TheHeisenberg

UncertaintyPrinciple

I’mmarkingexampapersonthesofawhenEmmycomesintotheroom,lookingconcerned.“What’sthematter?”Iask.“Ican’tfindmybone,”shesays.“Doyouknowwheremyboneis?”“Ihavenoideawhereyourboneis,”Isay,“butIcantellyouexactlyhowfast

it’smoving.”There’ssilenceinresponse,andwhenIlookup,she’sstaringatmeblankly.“It’s a physics joke,” I explain, because that always makes things funnier.

“Youknow,Heisenberg’suncertaintyprinciple?Theuncertaintyinthepositionof an object multiplied by the uncertainty in the momentum is greater thanPlanck’sconstantoverfourpi?Whichmeansthatwhenoneuncertaintyissmall,theothermustbeverylarge.”Nowshe’sglaringatme,almostgrowling.“Stopdoingthat!”shesays.“What?It’snotallthatfunny,butitwasn’tthatbad.”“It’syourfaultthatIcan’tfindmybone.”“Howisitmyfault?”“You went and measured how fast it’s moving, and the position got all

uncertain.AndnowIcan’tfindmybone.”“That’s notwhat happened,” I say. “The uncertainty principle doesn’twork

likethat.”“Yesitdoes.Youjustsaid.Youknowhowfastmyboneismoving,andnowI

can’tfindit.”“Firstofall,thatwasajoke.Ididn’treallymeasurethevelocityofyourbone.

Second,that’saslightlymistakenviewoftheuncertaintyprinciple.It’snotjustthatmeasurementchangesthestateofthesystem,it’sthatwhatwecanmeasureis limited by the fact that position and momentum are undefined until wemeasurethem.”

Shelookspuzzled.“Idon’tseethedifference.”“Well, in the picture where you attribute everything to the effects of

measurement,you implicitlyassume thatwhateveryou’remeasuringhassomedefinite andwell-defined properties, and the uncertainty in those values arisesonlyfromperturbationsthatoccurthroughtheactofmeasuringthem.That’snotwhathappens,though—inquantumtheory,therearenodefinitevaluesforthosequantities.They’renotuncertainbecauseoflimitsonyourmeasurement,they’reuncertain because they are not defined, and they can’t be defined, due to thequantumnatureofreality.”“Oh.”Shelooksthoughtfulforamoment,thenresumesglaring.“Ithinkyou

lostmybone,andyou’rejusttryingtoweaseloutofthisbybeingallconfusing.”“No,that’sreallyhowthetheoryworks.It’samootpoint,though,sinceeven

ifIhadperturbedthepositionofyourbonebymeasuringitsvelocity,there’snowaythatwould’vepreventedyoufromfindingit.”“Really?Whynot?”“Well,becausetheuncertaintyinvolvedwouldbetiny.Imean,yourbonehas

amassofacoupleofhundredgrams,andifImeasureditsvelocitytowithinonemillimeterpersecond,thatwouldgiveanuncertaintyinpositionofonlyabout10-31meters.That’satrillionthofthesizeofaproton—you’dneverevennoticethat.”“Really?Well,where’smybone,smartypants?”“Idon’tknow.DidyoulookundertheTVcabinet?Sometimesitgetskicked

underthere.”She trotsover to theTV, and stickshernoseunder the cabinet. “Here’smy

bone!”Shepawsat it foraminute,andeventuallysucceeds inknockingitoutfrom under the cabinet. “I have a bone!” she announces proudly, and beginschewingitnoisily,theuncertaintyprincipleforgotten.

TheHeisenberguncertaintyprincipleisprobablythesecondmostfamousresultfrommodern physics, after Einstein’sE=mc2 (the most famous result fromrelativity).Mostpeoplewouldn’tknowawavefunctioniftheytrippedoverone,but almost everyone has heard of the uncertainty principle: it is impossible toknowboth the position and themomentumof an object perfectly at the sametime. If youmake a better measurement of the position, you necessarily loseinformationaboutitsmomentum,andviceversa.In this chapter,we’ll describe how the uncertainty principle arises from the

particle-wavedualitywe’vealreadydiscussed.Theuncertaintyprincipleisoftenpresentedasastatementthatameasurementofasystemchangesthestateofthatsystem,andinthisform,referencestoquantumuncertaintyturnupinallsortsof

places, frompolitics to pop culture to sports.*Ultimately, though, uncertaintyhas very little to do with the details of the measurement process. Quantumuncertaintyisafundamentallimitonwhatcanbeknown,arisingfromthefactthatquantumobjectshavebothparticleandwaveproperties.Uncertainty is also the first place where quantum physics collides with

philosophy.Theideaoffundamentallimitstomeasurementrunsdirectlycounterto the goals and foundations of classical physics. Dealing with quantumuncertainty requires a complete rethinking of the basis of physics, and leadsdirectlytotheissuesofmeasurementandinterpretationinchapters3and4.

HEISENBERG’SMICROSCOPE:SEMICLASSICALARGUMENTS

Thetraditionaldescriptionofuncertaintyastheactofmeasurementchangingthestateofthesystemisessentiallybasedinclassicalphysics,andwasdevelopedinthe 1920s and ’30s in order to convince classically trained physicists thatquantumuncertaintyneededtobetakenseriously.Thisiswhatphysicistscallasemiclassicalargument—thephysicsusedisclassical,withafewmodernideasadded on. It’s not the full picture, but it has the advantage of being readilycomprehensible.The ideabehind thesemiclassical treatmentofuncertainty is familiar toany

dog. Imagineyouhavea rabbit in thegardenwhosepositionandvelocityyouwouldliketoknowverywell.Whenyouattempttomakeabetterdeterminationof its position (by getting closer to it), you inevitably change its velocity bymakingitrunaway.Nomatterhowslowlyyoucreepuponit,soonerorlater,italwaystakesoff,andyouneverreallyhaveagoodideaofboththepositionandthevelocity.

Anincomingphotonbouncesoffastationaryelectron,andiscollectedbyamicroscopelensinordertomeasure the electron’s position. In the collision, though, the electron acquires somemomentum,leadingtouncertaintyinitsmomentum.

Anelectron isn’tasentientbeing likea rabbit, so itcan’t runoffof itsownaccord,butasimilarprocesstakesplace.Tomeasurethepositionofanelectron,youneedtodosomethingtomakeitvisible,suchasbouncingaphotonoflightoff it and viewing the scattered light through a microscope. But the photoncarriesmomentum(aswesawinchapter1[page24]),andwhenitbouncesoffthe electron, it changes the momentum of the electron. The electron’smomentumafterthecollisionisuncertain,becausethemicroscopelenscollectsphotonsoversomerangeofangles,soyoucan’ttellexactlywhichwayitwent.Youcanmakethemomentumchangesmallerbyincreasingthewavelengthof

thelight(decreasingthemomentumthatthephotonhasavailabletogivetotheelectron),butwhenyouincreasethewavelength,youdecreasetheresolutionofyourmicroscope,andloseinformationabouttheposition.*Ifyouwanttoknowthepositionwell,youneedtouselightwithashortwavelength,whichhasalotofmomentum, and changes the electron’smomentumby a large amount.Youcan’t determine the position precisely without losing information about themomentum,andviceversa.Therealmeaningof theuncertaintyprinciple isdeeper than that, though. In

themicroscopethoughtexperimentillustratedabove,theelectronhasadefinitepositionandadefinitevelocitybeforeyoustarttryingtomeasureit,andstillhasadefinitepositionandvelocityafterthemeasurement.Youdon’tknowwhattheposition and velocity are, but they have definite values. In quantum theory,however, thesequantities arenotdefined.Uncertainty isnot a statementaboutthelimitsofmeasurement,it’sastatementaboutthelimitsofreality.Askingforthe precise position and momentum of a particle doesn’t even make sense,becausethosequantitiesdonotexist.Thisfundamentaluncertaintyisaconsequenceofthedualnatureofquantum

particles.Aswesawinthepreviouschapter,experimentshaveshownthatlightand matter have both particle-like and wavelike properties. If we’re going todescribe quantum particles mathematically—and physics is all aboutmathematicaldescriptionofreality—weneedtofindsomewayoftalkingabouttheseobjects thatallows themtohavebothparticleandwavepropertiesat thesame time.We’ll find that the only way is to have both the position and themomentumofthequantumparticlesbeuncertain.

BUILDINGAQUANTUMPARTICLE:PROBABILITYWAVES

The usual way of describing particles mathematically, dating from the late1920s, is through quantum wavefunctions. The wavefunction for a particular

object is a mathematical function that has some value at every point in theuniverse,and thatvaluesquaredgives theprobabilityof findingaparticleatagivenpositionatagiventime.Sothequestionweneedtoaskis,Whatsortofwavefunction gives a probability distribution that has both particle and waveproperties?Constructingaprobabilitydistributionforaclassicalparticleiseasy,andthe

resultlookssomethinglikethis:

The probability of finding the object—say, that pesky rabbit in the backgarden—is zero everywhere except right at the well-defined position of theobject. As you look across the garden, you see nothing, nothing, nothing,RABBIT!,nothing,nothing,nothing.This wavefunction doesn’t meet our requirements, though: it has a well-

defined position, but it’s just a single spike, and a spike does not have awavelength. Remember, the wavelength corresponds to themomentum of therabbit,whichisoneofthequantitieswe’retryingtodescribe,soitneedstohavesomevalue.Well, then, how do we draw a probability distribution with an obvious

wavelength?That’salsoeasytodo,anditlookslikethis:

Here,theprobabilityoffindingtherabbitatagivenpositionoscillates:rabbit,Rabbit,RABBIT,Rabbit,rabbit,Rabbit,RABBIT,Rabbit,rabbit,andsoon.Thiswavefunctiondoesn’tmeetour requirements, either.Thewavelength is

easy to define—just measure the distance between two points where theprobability is largest—so we have a well-defined momentum, but we can’tidentifyaspecificpositionfortherabbit.Therabbitisspreadoutovertheentireyard,withagoodprobabilityoffinding itat lotsofdifferentplaces.Thereareplaceswheretheprobabilityofseeingarabbitislow,buttheydon’taccountformuchspace.Whatweneedisa“wavepacket,”awavefunctionthatcombinesparticleand

wavepropertiesinasingleprobabilitydistribution,likethis:

This wavefunction is what we’re after: nothing, nothing, rabbit, Rabbit,RABBIT,Rabbit,rabbit,nothing,nothing.Therabbitisverylikelytobefoundinasmall regionofspace,and theprobabilityof finding itoutside that regiondropsofftozero.Insidethatregion,weseeoscillationsintheprobability,whichallowustomeasureawavelength,andthusthemomentum.

Thiswavepackethastheparticleandwavepropertiesthatwe’relookingfor.As a consequence, it also has some uncertainty in both the position andmomentumoftheparticle.The uncertainty in position is immediately obvious on looking at the wave

packet. The rabbit can’t be pinned down to a specific location, but there areseveraldifferentpositionswithinasmallrangewheretheprobabilityoffindingitisreasonablygood.Therabbitismostlikelytobefoundrightinthecenterofthewavepacket,butthere’sagoodchanceoffindingitalittlebittotheleft,oralittlebittotheright.Thepositionasdescribedbythiswavepacketisnecessarilyuncertain.Theuncertaintyinthewavelengthisnotasobvious,butit’stherebecausethis

wave packet is actually a combination of a great many waves, each with aslightly different momentum. Each of these waves represents a particularpossiblemomentumfortherabbit,sojustasthereareseveraldifferentpositionswheretherabbitmightbefound,therearealsoseveraldifferentpossiblevaluesof themomentum.Themomentumof therabbitdescribedby thiswavepacketthushassomeuncertainty.Howdowe get awave packet by combiningmanywaves?Well, let’s start

with twosimplewaves,onecorresponding toa rabbit casuallyhoppingacrossthegarden,andanotheronewithashorterwavelength(thegraphbelowshows20fulloscillationsofone,inthesamespaceas18oftheother),correspondingtoa rabbitmovingfaster,perhapsbecause itknows there’sadognearby.Nowlet’saddthosetwowavefunctionstogether.

“Waitaminute—nowwehavetwobunnies?”“No,eachwavefunctiondescribesabunnywithaparticularmomentum,but

it’sthesamebunnybothtimes.”“Butdoesn’taddingthemtogethermeanthatyouhavetwobunnies?”“No,inthiscase,itjustmeansthattherearetwodifferentstates*youmight

find the single bunny in. When you look out into the garden, there’s someprobabilityoffindingthebunnymovingslowly,andsomeprobabilityoffindingit moving a little faster. The way we account for that mathematically is byaddingthetwowavestogether.”“Oh.Drat.Iwashopingformorebunnies.”

•••

When we add these two waves together, we find that there are some placeswhere they are in phase, and add up to give a bigger wave. In other places,

they’reoutofphase,andcanceleachotherout.Thewavefunctionwegetfromadding them together (the solid line in the figure) has lumps in it—there areplaceswhereweseewaves,andplaceswhereweseenothing.Whenwesquarethattogettheprobabilitydistribution,wegetthebottomgraph:

Thedashedcurvesinthetopgraphshowthewavefunctionsforthetwodifferentwavelengths(shiftedupsoyoucanseethemclearly).Thesolidcurveshowsthesumofthetwowavefunctions.Thebottomgraphshowstheprobabilitydistributionresultingfromaddingthemtogether(thesquareofthesolidcurveinthetopgraph).

Thecenterpartofthisprobabilitydistributionlooksanawfullotlikethewavepacketwewant.There’saregionwherewehaveagoodprobabilityoffindingthe rabbit, and in that region,we seeawavelengthassociatedwith itsmotion.Outside that region, theprobabilitygoes tozero,meaning that thereareplaceswherewehavenohopeofseeingarabbitatall.Ofcourse, this two-wavewavefunction isn’t exactlywhatwewant,because

theno-rabbit zone is verynarrowand followed immediately by another lump.Butwecanimprovethesituationbyaddingmorewaves:

Thebottomgraph is theprobabilitydistribution fora single frequencywave,with two-, three-, andfive-frequencygraphsaboveit.

Ifwe add together three differentwaves, the regionwherewe have a goodprobabilityofseeingtherabbitgetsnarrower,andwithfivedifferentwaves,it’snarrowerstill.Asweaddmoreandmorewaves,theregionsofhighprobabilitygetnarrower,andthespacesbetweenthembecomewiderandflatter.Whatweendupwithstartstolookslikealongchainofwavepackets.

“So, nowwe’ve got a long chain of bunnies? I thoughtwewere only talkingaboutonebunny.”“Wearetalkingaboutonebunny.Whatwe’vegotisachainofdifferent‘wave

packets,’whicheachdescribeadifferentplacewherewemight find thesinglebunny.Ifwewanttonarrowthatdowntojustonepositionforthebunny,wedoitbyaddingtogetheracontinuousdistributionofwavelengths,notjustasetofregularlyspacedsinglewavelengths.”“Wouldn’t that mean adding together an infinite number of different

wavelengths,though?”“Well, yes, but that’swhatwe have calculus for.” “Oh. I’m not so good at

calculus.”“Very fewdogs are. Just takemyword for it—we canmake a singlewave

packet by summing a continuous distribution of wavelengths, with differentprobabilitiesfordifferentwavelengths.”“Andweendupwithaninfinitenumberofbunnies!”“Sorry, but no. It’s still just one bunny,with an infinite number of possible

velocitiesthatareveryclosetooneanother.”“Drat.Istillwantmorebunnies.”“Well,theinfinitesumwavefunctiondoesdefinethepositionreasonablywell,

soatleastyouknowwheretheonebunnyis.”“True.AndifIknowwhereitis,Icancatchit!”

THELIMITSOFREALITY:THEUNCERTAINTYPRINCIPLE

What does it mean to add together lots of different waves with differentwavelengths in this way? Well, each wave corresponds to a particularmomentum—a different velocity for the (single) rabbit moving through thegarden.Whenweaddthemalltogether,whatwe’redoingissayingthatthere’sachance of finding the rabbit in each of those different states (we’ll talkmoreaboutthisinchapter3).Adding these states together is theoriginof theuncertaintyprinciple. Ifwe

wantanarrowandwell-definedwavepacket,sothatweknowthepositionoftherabbitverywell,weneedtoaddtogetheragreatmanywaves todothat.Eachwavecorrespondstoapossiblemomentumfortherabbit,though,whichgivesalargeuncertainty in themomentum—it couldbemovingat anyoneof a largenumberofdifferentspeeds.Ontheotherhand,ifwewanttoknowthemomentumverywell,wecanusea

small number of different wavelengths, but this gives us a very broad wavepacket,withalargeuncertaintyintheposition.Therabbitcanonlyhaveafewpossiblespeeds,butwecannolongersaywhereitiswithmuchconfidence.Wecan’tproduceawavepacketwithasinglewell-definedpositionwithout

using an infinitely wide distribution of wavelengths, and we can’t produce awavepacketwithasinglewell-definedmomentumwithouthavingitextendoverallofspace.Thebestwecanhopetodoisasinglewavepacketlikewedrewatthebeginning,withasmalluncertaintyinthemomentumandasmalluncertaintyintheposition.Whenwegothroughthemathematicaldetails,wefindthat thesmallest possible product of the uncertainties satisfies the famous Heisenbergrelationship*:

ΔxΔp=h/4π

The uncertainty in the position (Δx) multiplied by the uncertainty in themomentum(Δp)isequaltoPlanck’sconstantdividedby4π.Anyothertypeofwavepacket(andtherearelotsofdifferentshapesfoundinnature)willhavealargeruncertaintyproduct,so therelationship isusuallywrittenwithagreater-than-or-equal-tosign:

ΔxΔp≥h/4π

Theimportantresultremainsthesame,though:nomatterwhatyoudo,there’sno way to make both Δx and Δp zero—as you make one smaller, the othernecessarilygetslarger,andtheproductremainsabovetheHeisenberglimit.Lookedatintermsofwavefunctions,then,wecanseethatthisrelationshipis

muchmore than justapractical limitdue toour inability tomeasurea systemwithoutdisturbingit.Instead,it’sadeepstatementaboutthelimitsofreality.Wesaw in chapter 1 that quantum particles behave like particles—photons havemomentumandcollidewithelectronsintheComptoneffect(page25).Wealsosawthatquantumparticlesbehavelikewaves—electrons,atoms,andmoleculesdiffract around obstacles and form interference patterns.The pricewe pay forhaving both of these sets of properties at the same time is that position andmomentummustalwaysbeuncertain.Themeaningoftheuncertaintyprincipleisnot just that it’s impossible tomeasure thepositionandmomentum, it’s thatthesequantitiesdonotexistinanabsolutesense.

MANIFESTATIONSOFUNCERTAINTY:ZERO-POINTENERGY

Theuncertaintyprinciple forcesus tocompletely rethinkourunderstandingofhow the universe works. Not only does it change the way we look at singlemovingparticles,butithasprofoundconsequencesforthestructureofmatteratthemicroscopiclevel.Mosthumans,andevenmanydogs,pictureatomsastinylittlesolarsystems,

with negatively charged electrons orbiting a positively charged nucleus. ThispictureoriginatedwithNielsBohrin1913,whenheproposedthefirstquantummodelofthehydrogenatom.InBohr’smodel, theoneelectronofahydrogenatomcanorbit thenucleus

onlyincertainveryspecificorbits,withparticularwell-definedvaluesofenergy.Theseorbitsarethe“allowedstates”ofhydrogen,andanelectroninanallowedstatewillhappilyremainthere.Electronscanneverbefoundinanorbitwithanin-betweenenergy.Physicistsoftentalkaboutthesestatesasiftheywerestepsonastaircase,andtheelectronsweredogslookingforaplacetosleep.Thedogcanrestcomfortablyonthegroundfloor,orononeofthesteps,butanyattempttoliedownhalfwaybetweentwostepswillendbadly.Bohr’s model works brilliantly to describe the characteristic colors of light

emitted and absorbed by hydrogen. Electrons can move between the allowedstates by absorbing or emitting photons of light, with the frequency of the

emitted light corresponding to the difference between the energies of the twostates. TheBohrmodel thus solved a problem that had stymied physicists foryears.When Bohr proposed the model, it was a bold break with prior physics.

Unfortunately, it’sacobbled-togethermixofclassicalandquantumideas,withnosolidtheoreticaljustification.LouisdeBroglie’swavemodeloftheelectronprovided themissing theoreticalbasis,butwhileparticle-waveduality justifiestheideaofallowedstates,itrequiresustodiscardtheimageofelectronsorbitingthenucleuslikeplanetsorbitingthesun.The fundamental problem with this picture is the same issue that leads to

uncertainty.Forthesolar-systemmodeltobeaccurate,theelectronmusthaveawell-defined position somewhere along the allowed orbit, and a well-definedmomentummovingitalongthatorbit.Butthiscan’tpossiblywork—ifwetrytodefinetheelectron’spositionwellenoughtolocateitalongaplanetaryorbit,itmusthaveahugeuncertainty inmomentum,meaning thatwecan’t saywhereit’sgoing.Ifwetrytodefinetheelectron’smomentumwellenoughtoplaceitonan orbital track, itmust have a huge uncertainty in position,meaning thatwecan’tevenbesureit’snearthenucleusit’ssupposedtobeorbiting.Whenweaccountforthewavenatureoftheelectron,weareforcedtodiscard

thewhole idea of electrons as planets. Instead, the electron hovers around thenucleusinafuzzysortof“cloud,”withapositionthatisuncertain,butconfinedtoa regionnear thenucleus,andamomentum that isuncertain,but limited tovalues that keep it near the nucleus.Bohr’s idea of allowed energy states stillapplies—the electron will always have one of the limited number of energyvalues predicted by Bohr’s theory—but these states no longer correspond toelectronsmovinginparticularorbits.

“So,wait—theelectronisn’tinaparticularplace,it’sjustsortofneartheatom?”“That’sright.Thedifferentenergystatescorrespondtodifferentprobabilities

offindingtheelectronsatparticularpositions,andhigher-energystateswillgiveyouabetterchanceoffindingtheelectronfartherfromthenucleusthanlower-energy states. But for any of the allowed states, the electron could be at justaboutanypointwithinafewnanometersofthenucleus.”“Butwhathappensifyouhavetwoatomsclosetogether?”“Well,ifyoubringtwoatomscloseenoughtogether,anelectronthatstartsout

attached to one atom can end up on the other atom, because of this quantumuncertaintyintheposition.We’lltalkalittlemoreaboutthisinchapter6,whenwetalkabouttunneling.”“Okay.”

“Youcanalsogetsituationswhereanelectronissortof‘shared’betweentwoatoms. That’s how chemical bonds form. And if you get a bunch of atomstogetherinasolid,oneelectroncanbesharedamongthewholesolid.That’sthebasis for the quantum theory of solids, which lets us understand how metalsconduct electricity and how to make semiconductor computer chips. It’s allbecauseelectronsextendbeyondspecificplanetaryorbits.”“Uncertainelectronsareweird.”“Strictlyspeaking,it’snotjustelectrons.Everythingintheuniverseissubject

totheuncertaintyprinciple,andhasanuncertainpositionandvelocity.”“Thatcan’tberight.Imean,Icanseemybonerightoverthere,andithasa

definiteposition,andavelocityofzero.”“Ah,butthequantumuncertaintyassociatedwithyourboneisdwarfedbythe

practicaluncertaintyinvolvedinmeasuringit.Ifyoulookatitreallycarefully,youmightbeabletospecifyitspositiontowithinamillimeterorso—”“Ialwayslookatmybonecarefully.”“—and with heroic effort, you might bring that down to a hundred

nanometers. In that case, the velocity uncertainty of your hundred-gram bonewould be only 10-27m/s. So, the velocitywould be zero, plus orminus 10-27m/s.”“That’sprettyslow.”“Yes,youcouldsaythat.Atthatspeed,itwouldtaketheageoftheuniverse

tocrossthethicknessofasingleatom.”“Okay,that’sreallyslow.”“Wedon’tseequantumuncertaintyassociatedwitheverydayobjectsbecause

they’rejusttoobig.Weonlyseeuncertaintydirectlywhenwelookatverysmallparticlesconfinedtoverysmallspaces.”“Likeelectronsnearatoms!”“Exactly.”

Uncertainty has another, evenmore profound effect on the structure of atoms.Electronsmust alwayshave uncertainty in both their position andmomentum,andthatmeansthattheenergyofanelectroninanatomcanneverbezero.Tohavezeroenergywhilestillbeingpartofanatom,anelectronwouldneedtobenot moving, sitting right on top of the nucleus. This is impossible, as we’vealreadyseen—theclosestwecancomeistomakeanarrowelectronwavepacketcenteredonthenucleus,whichwillincludelotsofdifferentstateswithnonzeromomentum.Even the lowestenergy-allowed state of hydrogen, then, has someenergy.Thisisageneralphenomenon,andappliestoanyconfinedquantumparticle.

Ifweknowthataparticle is insomeparticular regionofspace, that limits theuncertainty in the position, and increases the uncertainty in the momentum.Confinedquantumparticlesareneverat rest—they’re likepuppies inabasket,alwayssquirmingandwrigglingandshiftingaround,evenwhenthey’reasleep.This tiny residualmotion iscalledzero-pointenergy,which is theminimum

quantum energy associated with a particle due to its confinement. Zero-pointenergy provides an absolute lower limit to the energy a confined particle canhave—no matter how carefully you prepare the system, the particles in thatsystem will always be in motion, with small random fluctuations constantlychangingthemagnitudeanddirectionoftheirvelocity.Zero-point energy is one of the most counterintuitive ideas in quantum

physics,tellingusthatnothingcaneverbeperfectlyatrest.Itmeansthatthereisalwayssomeenergypresentinanysystem,nomatterhowhardyoutrytoextractall the energy. Even empty space has zero-point energy, which leads to somesurprising consequences, including the spontaneous emission of photons fromatoms and tiny forces (called “Casimir forces”) between metal plates in avacuum. The zero-point energy of empty space can even produce short-livedpairsof“virtual”particles,aswe’llseeinchapter9.Zero-point energy is probably the most dramatic manifestation of the

uncertaintyprinciple.Itsexistenceisadirectconsequenceofthequantumnatureofalltheparticlesmakingupouruniverse.

“So, the point of all this is that position andmomentum do not have definitevalues?”“Yes,that’sitexactly.”“Arethosetheonlythingsthishappenswith?”“No, there are lots of different uncertainty relationships between pairs of

physical quantities. There’s uncertainty in angular momentum, for example—youcan’tknowboththedirectionofarotatingobjectandhowfastit’sspinning.There’salsoanuncertaintyrelationshipbetweenthenumberofphotonsinalightbeam and the phase of the wave associated with the beam. Uncertaintyrelationshipsareallovertheplaceinquantumphysics.”“So, basically, nothing is defined in an absolute sense? Isn’t that sort of…

postmodern?”“It’snotallthatbad—it’snotlikedifferentexperimentersgettomakeuptheir

ownresults.Theuncertaintyduetoquantumeffectsisgenerallyverysmall,sofor all practical purposes, we can treat macroscopic objects as if they havedefinite properties. But at the microscopic level, there really isn’t any singledefinedvalueforanyofthesequantities.”

“But you talked earlier aboutmeasuring bunnies in definite positions.Howdoesthatwork,iftheydon’thavedefinitepositions?”“That’s a very good question, and takes us into awhole new area ofweird

stuff,inthenextchapter.”

*Togiveyouanideaofthebreadthofsubjectsinwhichthisshowsup,inJune2008,Googleturnedup

citationsoftheHeisenberguncertaintyprinciplein(amongothers)anarticlefromtheVermontFreePressaboutspeedcameras,aTorontoStararticlecitingtheinfluenceofYouTubeonundergroundartists,andablogarticle about thePhoenixSunsof theUSNationalBasketballAssociation. Incidentally, all of thesearticles also use the uncertainty principle incorrectly—by the end of this chapter, you should hopefullyunderstanditbetterthananyofthem.*Thisiswhyscientistsuseelectronmicroscopestolookatverysmallthings:electronmicroscopesuse

electronsinsteadofvisiblelight,andelectronshavemuchshorterwavelengthsthanvisiblelight.*“State”inphysicsrefers toaparticularcollectionofproperties—position,momentum,energy,etc.A

rabbitinthegardenwithagivenmomentumissaidtobeinamomentumstate.Asecondrabbitinthesamegardenwiththesamemomentumisinthesamestate;asecondrabbitinthesamegardenwithadifferentmomentum is in adifferent state.A third rabbitwith the samemomentumas the first, but in adifferentgarden,isinathirdstate,andsoon.*Thesymbol“Δ”isacapitalGreekletterdelta,asanywell-educateddogcantellyou.Inscience,it’s

used to indicate a change in something or a difference between two things.Δx is the uncertainty in theposition,ortheamountofdifferenceyoucanexpectbetweenthepositionyoueventuallymeasureandthemostlikelypositionfromthewavefunction.Whenyousay“Iburiedanicebonesixteenstepsfromthebigoaktree,giveortakeastep,”the“sixteensteps”isthemostlikelypositionofthebone,andthe“giveortakeastep”isΔx.

CHAPTER3

Schrödinger’sDog:

TheCopenhagenInterpretation

I’minthekitchen,gettingaglassofwater,whenEmmytrots in, tailwagging.“Youshouldgivemeatreat,”shesays.“Ishould?WhyshouldIgiveyouatreat?”“BecauseI’maverygooddog,andIdeserveatreat!”“I’mnotgoingtogiveyouatreatfornoreason,”Isay,“butI’lltellyouwhat

I’lldo.”Ireachintothetreatjar,thenholdoutbothfists.“Guesswherethetreatis,andyoucanhaveit.”Immediately,hernosestartsworking.“Nosniffing,either.”Iputmyhandsbehindmyback.“Justguesswhichhand

hasthetreat.”“Ummm…Okay.Both.”“That’snotoneofthechoices.”“Butit’stherightanswer,”shesays,pouting.“It’slikethatcatinthebox.”“Whatcatinwhatbox?”“Youknow,theoneinthebox.Withthething.It’sdeadandaliveatthesame

time.Inthebox.”“Schrödinger’scat?”“Yes! That’s the one!” Shewags her tail excitedly. “I like that experiment.

Youshoulddothat.”“For one thing, it’s just a thought experiment to show the absurdity of

quantumpredictions.Nobodyeverdiditforreal.Foranother,Idoubtthatpeoplewouldappreciateitifwestartedkillingcats.”“Idon’tcareaboutthekilling.Ijustliketheideaofputtingcatsinboxes.Cats

belonginboxes.”“I’llpass thaton to thescientificcommunity.Butwhatdoes thishave todo

withyourtreat?”

“Well,thetreatcouldbeinyourlefthand,anditcouldbeinyourrighthand.Idon’tknowwhichit’s in,andyouwon’t letmesniff toseewhereit is,sothatmeansthatthetreatisinasuperpositionstateofbothleftandrighthands.UntilImeasure which hand it’s in, the answer is that it’s in both hands at the sametime.”“That’saninterestingargument.Itdoesn’tapplyhere,though.”“Yesitdoes.It’sbasicquantummechanics.”“Well,yes,it’struethatunmeasuredobjectsexistinsuperpositionstatesasa

generalmatter,”Isay,“butthosesuperpositionstatesareextremelyfragile.Anydisturbanceatall—absorbingoremittingevenasinglephoton—willcausethemtocollapseintoclassicalstateswithadefinitevalue.”“Peoplehaveseenthem,though.”“Yes, there have been lots of ‘cat state’ experiments done, but the largest

superpositionanybodyhasmanaged tomake involved something likeabillionelectrons.* That’s nowhere near the size of a dog treat, which would containsomethinglike1022atoms.”“Oh.”“And on top of that, even in the most extreme variant of the Copenhagen

interpretation, the wavefunction is collapsed by the act of observation by aconsciousobserver.Now,youcanargueaboutwhocountsasanobserver—”“Notacat,that’sforcertain.Catsarestupid.”“—butbyanyreasonablestandard,Icountasanobserver.Iknowwhichhand

the treat is in. So you’re dealing with a classical probability distribution, inwhichthetreatisineitheronehandortheother,notaquantumsuperpositioninwhichthetreatisinbothhandsatthesametime.”“Oh.Okay.”Shelooksdisappointed.“So,guesswhichhandthetreatisin.”“Ummm…Istillsayboth.”“Whyisthat?”“BecauseIamanexcellentdog,andIdeservetwotreats!”“Well,yes.Also,I’masucker.”Igiveherbothofthetreats.“Great!Treats!”shesays,crunchinghappily.

One of the most vexing things about studying quantum mechanics is howstubbornly classical the world is. Quantum physics features all sorts ofmarvelousthings—particlesbehavinglikewaves,objectsbeingintwoplacesatthesametime,cats thatarebothaliveanddead—andyet,wedon’tseeanyofthosethingsintheworldaroundus.Whenwelookataneverydayobject,weseeit in a definite classical state—with some particular position, velocity, energy,

and so on—and not in any of the strange combinations of states allowed byquantum mechanics. Particles and waves look completely different, dogs canonly pass on one side or the other of an obstacle, and cats are stubbornly,irritatinglyaliveandnothappyaboutbeingsniffedbystrangedogs.Wedirectlyobserve thestranger featuresofquantummechanicsonlywitha

greatdealofwork,incarefullycontrolledconditions.Quantumstatesturnouttoberemarkablyfragileandeasilydestroyed,andthereasonforthisfragilityisnotimmediately obvious. In fact, determining why quantum rules don’t seem toapply in the macroscopic world of everyday dogs and cats is a surprisinglydifficultproblem.Exactlywhathappensinthetransitionfromthemicroscopictothemacroscopic has troubledmany of the best physicists of the last hundredyears,andthere’sstillnoclearanswer.In this chapter, we’ll lay out the basic principles that are central to

understandingquantumphysics:wavefunctions,allowedstates,probability,andmeasurement.We’ll introduce a key example system, and talk about a simpleexperiment that demonstrates all of the essential features of quantum physics.We’ll talk about the essential randomness of quantum measurement, and thephilosophicalproblemsraisedbythisrandomness,whicharedisturbingenoughthatevensomeofthefoundersofquantumphysicsgaveuponitentirely.

WHATDOESAWAVEFUNCTIONMEAN?INTERPRETATIONOF

QUANTUMMECHANICS

Mostofthephilosophicalproblemswithquantummechanicscenteraroundthe“interpretation”of thetheory.This isaproblemuniquetoquantummechanics,as classical physics doesn’t require interpretation. In classical physics, youpredict the position, velocity, and acceleration of some object, and you knowexactly what those quantities mean and how to measure them. There’s animmediate and intuitive connectionbetween the theory and the reality thatweobserve.Quantummechanics,ontheotherhand,isnotnearlysoobvious.Wehavethe

mathematical equations that govern the theory and allow us to calculatewavefunctions and predict their behavior, but just what those wavefunctionsmean is not immediately clear.We need an “interpretation,” an extra layer ofexplanation, to connect the wavefunctions we calculate to the properties wemeasureinexperiments.The central elements of quantum mechanics can be presented in many

differentways—asmanydifferentwaysastherearebooksonthesubject—butin theend, theyall reston fourbasicprinciples.Youcan thinkof theseas thecoreprinciplesofthetheory,thebasicrulesthatyouhavetoacceptinordertomakeanyprogress.*

CENTRALPRINCIPLESOFQUANTUMMECHANICS

1.Wavefunctions: Every object in the universe is described by a quantumwavefunction.

2.Allowedstates:Aquantumobjectcanonlybeobservedinoneofalimitednumberofallowedstates.

3.Probability:Thewavefunctionofanobjectdetermines theprobabilityofbeingfoundineachoftheallowedstates.

4.Measurement:Measuringthestateofanobjectabsolutelydeterminesthestateofthatobject.

The first principle is the idea of wavefunctions. Every object or system ofobjectsintheuniverseisdescribedbyawavefunction,amathematicalfunctionthat has some value at every point in space. It doesn’t matter what you’redescribing—anelectron,adogtreat,acatinabox—ithasawavefunction,andthatwavefunctionhassomevaluenomatterwhereyoulook.Thevaluecouldbepositive,ornegative,orzero,orevenanimaginarynumber(likethesquarerootof-1),butithasavalueeverywhere.Amathematical formula called the Schrödinger equation (after theAustrian

physicist and noted cad* Erwin Schrödinger, who discovered it) governs thebehavior ofwavefunctions.Given some basic information about the object ofinterest,youcanusetheSchrödingerequationtocalculatethewavefunctionforthatobjectanddeterminehowthatwavefunctionwillchangeovertime,similarto theway you can useNewton’s laws to predict the future position of a doggivenhercurrentpositionandvelocity.Thewavefunction,inturn,determinesalltheobservablepropertiesoftheobject.The second principle is the idea of allowed states. In quantum theory, an

object will only ever be observed in certain states. This principle puts the“quantum”in“quantummechanics”—theenergyinabeamoflightcomesasastreamofphotons,andeachphoton isonequantumof light thatcan’tbesplit.Youcanhaveonephoton,ortwo,orthree,butneveroneandahalforpi.Similarly, an electron orbiting the nucleus of an atom can only be found in

certainveryspecificstates.*Eachofthesestateshasaparticularenergy,andthe

electronwillalwaysbefoundwithoneofthoseenergies,neverinanin-betweenstate.Electronscanmovebetweenthosestatesbyabsorbingoremittinglightofa particular frequency—the red light of a neon lamp, for example, is due to atransition between two states in neon atoms—but they make those jumpsinstantaneously,without passing through the intermediate energies.This is theoriginoftheterm“quantumleap”foradramaticchangebetweentwoconditions—theactualenergyjumpisverysmall,butthechangeinthestatehappensinnotimeatall.The thirdprinciple is the ideaofprobability.Thewavefunctionofanobject

determinestheprobabilitiesofthedifferentallowedstates.Ifyou’reinterestedinthepositionofadog,say,thewavefunctionwilltellyouthatthere’saverygoodprobabilityoffindingthedoginthelivingroom,alowerprobabilityoffindingher in theclosedbedroom,andanextremely lowprobabilityof findingherononeofthemoonsofJupiter.Ifyou’reinterestedintheenergyofthatsamedog,thewavefunctionwilltellyouthatthere’saverygoodprobabilityoffindinghersleeping,agoodprobabilityoffindingherleapingaboutandbarking,andalmostnochanceoffindinghercalmlydoingcalculusproblems.Philosophicalproblemsstarttocreepinatthispoint,becausetheonethingthe

wavefunction won’t give you is certainty. Quantum theory allows you tocalculate only probabilities, not absolute outcomes. You can say that there’ssomeprobabilityoffindingthedoginthelivingroomandsomeprobabilityoffindingthedoginthekitchen,butyoucan’tsayforsurewhereshewillbeuntilyou look. If you repeat the same measurement under the same conditions—asking“Whereisthedog?”atfouro’clockintheafternoon—you’llgetdifferentresultsondifferentdays,butwhenyouputalltheresultstogether,you’llseethattheymatch the probability predicted from thewavefunction.You can’t say inadvance what will happen for any individual measurement, only what willhappenovermanyrepeatedexperiments.Quantumrandomness isa tremendouslydisturbing ideaforpeople raisedon

classicalphysics,whereifyouknowthestartingconditionsofyourexperimentwellenough,youcanpredicttheoutcomewithabsolutecertainty:youknowthatthedogwillbeinthekitchen,andlookingjustconfirmswhatyoualreadyknew.Quantum mechanics doesn’t work that way, though: identically preparedexperiments can give completely different results, and all you can predict areprobabilities. This randomness is the philosophical issue that led Einstein tomakeavarietyofcommentsthathavebeenrenderedas“Goddoesnotplaydicewiththeuniverse.”*

•••

“Physicistsaresilly.”“Whydoyousaythat?”“Well, what’s disturbing about randomness? I never know the outcome of

anythingforcertainbeforeithappens,andI’mfine.”“Well, you’re a dog, not a physicist. But you do make a good point—any

responsiblepracticaltreatmentofclassicalphysicshastoincludesomeelementofprobability in itspredictions, justbecauseyoucanneveraccount forall thelittleperturbationsthatmightaffecttheoutcomeofanexperiment.”“LikethatbutterflyinBrazil,causingallthisweather.”“Exactly.That’stheusualmetaphor:abutterflyflapsitswingsintheAmazon,

and a week later, there’s a storm in Schenectady. It’s the classic example ofchaos theory, which shows that probability is unavoidable even in classicalphysics, because you can never account for every single butterfly that mightaffecttheweather.”“Stupidchaosbutterflies.”“The thing is, quantum probability is a different game altogether. The

probabilitiesweendupwithinclassicalphysicsareapracticallimitation.If,bysomemiracle,youreallycouldkeeptrackofeverybutterfly in theworld, thenyou would be able to predict the weather with certainty, at least for a while.Quantumphysicsdoesn’tallowthat.”“You mean the butterflies are covered by the uncertainty principle, so you

don’tknowwheretheyare?”“Onlypartly—it’sdeeperthanthat.Inquantumphysics,evenifyouperform

the same experiment twice under identical conditions—down to the very lastbutterflywing-flap—youstillwon’tbeabletopredicttheexactoutcomeofthesecond experiment, only the probability of getting various outcomes. Twoidenticalexperimentscanandwillgiveyoudifferentresults.”“Oh.You knowwhat? That is pretty disturbing.Maybe you’re not so silly,

afterall.”“Thanksforthevoteofconfidence.”

The finalprincipleofquantum theory is the ideaofmeasurement. Inquantummechanics,measurement isanactiveprocess.Theactofmeasuringsomethingcreatestherealitythatweobserve.*Togiveaconcreteexample,let’simaginethatyouhaveadogtreatinoneof

two boxes. The boxes are sealed, soundproof (so you can’t hear the treatrattling),andairtight(soyoucan’tsniffitout):youcan’ttellwhichboxthetreatisinwithoutopeningoneoftheboxes.Ifwewanttodescribethisasaquantummechanicalobject,weneedtowrite

down a wavefunction with two parts, one part describing the probability offindingthetreatintheboxontheleft,andtheotherdescribingtheprobabilityoffindingthetreat intheboxontheright.Wecandothisbyaddingtogetherthewavefunctions for the treat being in the left-handboxonly and the right-handbox only, just aswe did in the preceding chapter (page 42)whenwemade awavepacketbyaddingtogetherrabbitstates.Now,imaginethatyouopenoneoftheboxes,andfindthetreat,thenclosethe

boxbackup.Youstillhaveone treatand twoboxes,butyou’vemeasured thepositionofthetreat.Whatdoesthewavefunctionlooklike?Thewavefunctionnowhasonlyonepart—thepiecedescribingatreatinthe

left-handbox—becauseweknowexactlywherethetreatis.Ifyoufounditintheleft-handbox,thenexttimeyouopenthatbox,there’sa100%chancethatitwillbethere,andthere’sa0%chanceoffindingthetreatintheright-handbox.Theother part thatwas there before youopened the box, giving the probability ofbeingintheright-handbox,isgone,duetothemeasurementyoumade.Now throw away those boxes, take two new boxes prepared in the same

manner as the first pair, and you’ll have a two-part wavefunction again. Theresultofopeningthefirstboxwon’tnecessarilybethesame,though.Youmightvery well find the treat in the right-hand box this time. If you do, and keepclosingandreopeningthatsetofboxes,you’llalwaysfindthetreatintheright-hand box. Again, you go from a two-part wavefunction to a one-partwavefunction.So,what’sthebigdeal?Afterall,that’sjusthowprobabilitieswork,right?In

the first experiment, the treatwas in the left-hand box all along, but you justdidn’tknowit,andinthesecondexperiment,thetreatwasintheright-handbox.Thestateof the treatdidn’tchange,butyourknowledgeabout thestateof thetreatdid.Quantum probabilities don’t work that way. When we have a two-part

wavefunction(a“superpositionstate”),itdoesn’tmeanthattheobjectisinoneofthetwostates,itmeansthattheobjectisinbothstatesatthesametime.Thedogtreatisn’tintheleft-handboxallalong,it’ssimultaneouslyinbothleftandrightboxesuntilafteryouopenthebox,andfinditinoneortheother.

“That’sprettystrange.Whyshouldwebelieveit?”“Well,wecandemonstratetheweirdfeaturesofquantummechanicswithan

experimentcalledaquantumeraser.”“Great!Ilikethat!Let’serasesomecats!”“Itdoesn’tworkonmacroscopicobjects.Itusespolarizedlight,whichIhave

toexplainfirst.”

“Ohdear…Whycan’twejusterasestuff?”“I’llkeepitasshortasIcan,butthisisimportantstuff.Polarizedlightisthe

bestsystemaroundforgivingconcreteexamplesofquantumeffects.We’llneeditforthischapter,andalsochapters7and8.”“Oh,allright.AslongasIcanerasestufflater.”“We’llseewhatwecando.”

SUPERPOSITIONANDPOLARIZATION:ANEXAMPLESYSTEM

We can show both the existence of superposition states and the effects ofmeasurement using the polarization of light. Polarized photons are extremelyusefulfortestingthepredictionsofquantummechanics,andwillshowupagainand again in coming chapters, so we need to take a little time to discusspolarizationoflight,whichcomesfromtheideaoflightasawave.A wave, such as a beam of light, is defined by five properties. We have

alreadytalkedaboutfourofthese:thewavelength(distancebetweencrestsinthewavepattern), frequency (howmany times thewaveoscillatesper secondatagivenpoint),amplitude(thedistancebetweenthetopofacrestandthebottomof a trough), and the direction in which the wave moves. The fifth is thepolarization,whichisbasicallythedirectionalongwhichthewaveoscillates.Animpatient dog owner out for awalk can attempt to get his dog’s attention byshaking the leadupanddown,whichmakesaverticallypolarizedwave in thelead, or by shaking the lead from side to side, which makes a horizontallypolarizedwave.Like a shaken lead, a classical light wave has a direction of oscillation

associatedwith it. The oscillation is always at right angles to the direction ofmotion, but can point in any direction around that (that is, left, right, up, ordown,relativetothedirectionthelightismoving).Physiciststypicallyrepresentthepolarizationstateofabeamoflightbyanarrowpointingalongthedirectionof oscillation—avertically polarizedbeamof light is representedby an arrowpointing up, and a horizontally polarized beam of light is represented by anarrowpointingtotheright,asseeninthefigurebelow.

Left:verticalpolarization,representedasanuparrow.Middle:horizontalpolarization,representedasarightarrow.Right:polarizationbetweenverticalandhorizontal,whichcanbethoughtofasasumofhorizontalandverticalcomponents.

•••

“Wait,whatarethesepictures,again?”“Imagine that you’re right behind the beam of light, and looking down the

direction ofmotion.The arrow indicates the direction of the oscillation of thewave.Anuparrowmeansthatyou’llseethewavemovingupanddown;arightarrowmeansthatyou’llseeitmovingsidetoside.”“So…anuparrowislikechasingabunnythatboundsupanddown,whilea

rightarrowislikechasingasquirrelthatzigzagsbackandforth?”“Okay,thatworks.”“Areupandtotherighttheonlyoptions?”“You can have arrows in other directions, too. An arrow to the left also

indicates a side-to-side oscillation, but it’s out of phasewith the arrow to theright.”“So,arightarrowisasquirrelthatzigstotherightfirst,andaleftarrowisa

squirrelthatzagstotheleftfirst?”“Yes. If you insist on examples involving prey animals.” “I like prey

animals!”

•••

Thepolarizationofawavecanbehorizontalorvertical,butalsoanyangle inbetween.Wecanthinkofthein-betweenanglesasbeingmadeupofahorizontalpartandaverticalpart,asshowninthefigureabove.Eachofthesecomponentsislessintense(thatis,ithasasmalleramplitude,indicatedinthefigurebythelengthofthearrow)thanthetotalwave,buttheyaddtogethertogivethesamefinalintensityatsomeangle.Youcanthinkofthisadditionasacombinationofsteps,justlikethewaythatwecangetfromonepointtoanotherbyeithertakingthree steps east followed by four steps north, or by taking five steps in adirectionabout37°eastofduenorth.

“So,anin-betweenangleis likeabunnythat’szigzaggingleftandright,whilealsohoppingupanddown?”“Yes,that’sright.”“Orasquirrelthat’sjumpingupanddownwhileitzigzagsleftandright?”“Ithinkthat’saboutenoughpreyexamplesfornow.”

“You’renofun.”

Thinking of in-between polarizations as a sum of horizontal and verticalcomponentsisausefultrickbecauseitmakesiteasytoseewhathappenswhenlightencountersapolarizingfilter.Polarizingfiltersaredevices thatwillallowlightpolarizedataparticularangle—vertical,say—topassthroughunimpeded,while light polarized at an angle 90° away—horizontal—will be completelyabsorbed.Youcanunderstandtheeffectbyimaginingadogonaleadthatpassesthroughapicket fence. Ifyoushake the leadupanddown, thewavewillpassright through, but side-to-side shaking will be blocked by the boards of thefence.When light at an angle between vertical and horizontal strikes a vertically

oriented polarizing filter, only the vertical component of the light will passthrough.This lowers the intensityof the lighton theother side,byanamountthatdependsontheangle.Forsmallangles,mostofthelightmakesitthrough—atanangleof30°,thebeamonthefarsideisthree-fourthsasbrightastheinitialbeam—while for larger angles, most of the beam is blocked—at 60° fromvertical,thebeamonthefarsideisonlyone-fourthasbrightastheinitialbeam.Atanangleof45°midwaybetweenhorizontalandvertical,exactlyhalfofthelightwillpassthroughthefilter.Thelightonthefarsideofthefilterispolarizedattheangleofthefilter,no

matterwhatangleitstartedat.Forthisreason,polarizingfiltersarecommonlycalledpolarizers:lightpassingthroughaverticallyorientedpolarizingfilterwillemergeasverticallypolarizedlight,whetheritstartedwithverticalpolarizationor at some other angle. The overall amount of light will be different, but thepolarization will be the same. All of the light passing through a verticallyoriented filter will pass through a second vertical filter, and all of it will beblockedbyahorizontallyorientedfilter.

“Whatisallthisgoodfor,anyway?”“Otherthanhelpingdemonstratequantumphysics?Plenty.Lightpolarization

is an extremely useful thing.Digital displays onwatches,mobile phones, andtelevisionsuseapolarizerinfrontofalightsourcetovarytheamountoflightthatgetsthrough.Andpolarizingfiltersarealsousedtomakesunglasses.”“Sunglasses?”“Yes, those sunglasses that I wear when I take you for walks are actually

polarizing filters. The light from the sun is unpolarized—it’s as likely to behorizontalasvertical—butwhenlightreflectsoffasurface, it tendstobecomeslightly polarized. Light reflecting off the road out in front of uswhenwe’re

walking hasmore horizontal polarization than vertical, so bywearing verticalpolarizersassunglasses,Icanblockmostofthatlight.”“What’sthepointofthat?Doesn’titmakeithardertosee?”“Actually,itreducestheglareofftheroad,andmakesiteasiertoseethingsup

ahead.”“Things…Likebunniesintheroad?”“Forexample,yes.”“CanIhavesomepolarizedsunglassessoIcanseebunnies?”“TheonesIhavewon’tfitonyourears,butwe’ll lookintoit.Later.First,I

havetotalkaboutquantummeasurementwithpolarizedlight.”“Oh,okay.Quantumphysics.Right.”

Howdoesallthisapplytolightasaparticle,though?Wespentagoodchunkofchapter 1 describing how a beam of light is both a stream of photons and asmoothwave.Thelastfewpageshavebeendiscussingpolarizationinclassicalterms.Howdowehandlelightpolarizationinquantumphysics?Whenwe’re dealingwith classical lightwaves, it’s easy to understand how

partofawavecanpassthroughthefilter.Whenwetalkaboutlightintermsofphotons, though, the filter is an all-or-nothing proposition. Any given photoneithermakes it through,orgets absorbedby the filter.Thereareno“parts”ofphotons.We handle the interaction between photons and polarizing filters by saying

thateachphotonhasaprobabilityofpassingthroughthefilterthatisequaltothefractionofthetotalwavethatmakesitthroughintheclassicalmodel.Ifabeamoflightwithapolarizationat60°fromverticalencountersaverticalpolarizingfilter,thebeamonthefarsidewillbeone-fourthasbright,meaningthatithasone-fourth asmanyphotons.Thatmeans that each individual photonhas onlyonechanceinfourofmakingitthroughthepolarizingfilter.Each photon making it through the filter will also have its polarization

determined by the filter. Only one photon in four may make it through avertically oriented polarizing filter, but every one of those photons will passthroughasecondverticalfilter,andnoneofthemwillpassthroughahorizontalfilter. “Vertical” and “horizontal” are then the allowed states of the singlephoton’spolarization—whenwemeasurethepolarizationusingafilter,wewillfind the photon in one of those two states (either passing through the verticalfilter,orbeingabsorbedbyit),andnotanywhereinbetween.Polarized photons thus provide an excellent system for looking at the core

principles of quantummechanics. Each individual photon can be described intermsofawavefunction,withtwopartscorrespondingtothetwoallowedstates,

horizontalandverticalpolarization.Thatwavefunctiongivesyoutheprobabilityof the photon passing through a polarizing filter, and after you make ameasurementofthepolarizationwithafilter,thephotonsareinonlyoneoftheallowedstates.Asinglephotonpassingthroughapolarizingfilterdemonstratesalltheessentialfeaturesofquantumphysics.Asaresult,polarizedphotonshavebeenusedinmanyexperimentsdemonstratingquantumphenomena.

“So,letmegetthisstraight.Aphotonatananglebetweenhorizontalandverticalisinasuperpositionstate?Andsendingitthroughapolarizingfilteristhesameasmeasuringit?”“Yes.Yougetall the featuresofquantumsuperpositionandmeasurement—

wavefunctions, allowed states, probability, and measurement—using singlepolarizedphotons.”“ButI thoughtyousaidall thisstuffworkedthesamewaywhenyoutalked

aboutlightasaclassicalwave?”“Well, yes. The end result is the same as the classical polarized wave

description.”“What’sthebigdeal,then?Imean,yourbigexampleofquantumweirdnessis

somethingthatjustreproducesclassicalresults?”“Well,no. Imean, that’snotmybigexample.Thebigexampleofquantum

weirdnessisinthenextsection.”“Oh.Well,carryon,then.”

(UN)MEASURINGAPHOTON:THEQUANTUMERASER

Oneofthebestdemonstrationsoftheweirdnessofquantumsuperpositionsisanexperimentcalledaquantumeraser.Thequantumeraserencapsulateseverythingthat’s strange about single-particle quantum physics in a single experiment:particle-waveduality,superpositionstates,andtheactivenatureofmeasurement.If you can understand the quantum eraser, you’ve understood the essentialelementsofquantumphysics.Manydifferentvariantsofquantum-eraserexperimentshavebeendoneover

the years,* but the simplest starts with a variant of Young’s double-slitexperiment(page18).Ifwesendabeamofphotonsatapairofnarrowslits,wewillseeaninterferencepatternonthefarsideoftheslits,builtupoutofsinglephotonsdetectedatparticularpoints(asshowninthefigureonthenextpage).Wecanseethepatternonlybecauselightpassesthroughbothslitsat thesametime.Ifweblockoneslit, theinterferencepatternwilldisappear,andwe’llsee

onlyabroadscatteringofphotonsduetothelightpassingthroughtheunblockedslit.The interference pattern that we see indicates that the photons are in a

superpositionstate:thewavefunctiondescribingeachphotonhastwoparts,oneforthephotonpassingthroughtheleftslit,andtheotherforthephotonpassingthrough the right slit. Each photon has passed through both slits, at the sametime,andthe interferencebetweenthose twocomponents iswhatproduces thepatternwesee. Interferencepatternsalways turnupwhenyouhavea two-partwavefunction.Whenweblockoneslit,weonlyhaveaone-partwavefunction,destroyingthesuperposition,andthereisnointerferencepattern.

•••

“Wait, I thought the interferencewasbetween twodifferentphotons—one thatwentthroughtheleftslit,andonethatwentthroughtherightslit?”

Interferencepatternbuiltup fromsinglephotons.Left toright,1/30second,1second,100seconds.ImagesbyLymanPageatPrincetonUniversity,reprintedwithpermission.

“That’saneasythingtothink,sinceweusuallysendinlotsofphotonsatthesametime.Wecanshowthatthat’snotthecase,though,bysendinglightattheslitsonephotonatatime.”“Howdoesasinglephotongiveyouaninterferencepattern?”“Itdoesn’t.Eachindividualphotonshowsupasasinglespot,ataparticular

positiononthescreen,andwhereanyindividualphotonturnsupisrandom.”“There’stheprobabilitythingagain.”“Exactly.Theindividualphotonsarerandom,butifyourepeattheexperiment

overandoveragain,andkeeptrackofallthephotons,you’llseethemadduptoformaninterferencepattern.Therearesomeplaceswhereyou’reverylikelytofind a photon, and other placeswhere there’s absolutely no chance of findingone. The overall pattern is determined by the probability distribution you get

fromthewavefunctionforeachindividualphotoninterferingwithitself.”“So it’soneparticle,but it goes throughboth slits, and thenendsupatone

placeontheotherside?”“Exactly.”“That’sjustweird.”“That’squantumphysics.”

Insteadofblockingoneslit,though,let’simaginecoveringthetwoslitswithtwodifferentpolarizingfilters,oneverticalandonehorizontal.Weputafilterontheleftslitthatwillpassonlyhorizontallypolarizedlight,andweputafilterontheright slit that will pass only vertically polarized light. If we send in lightpolarizedatanangleof45ºtothevertical,ithasa50%chanceofgoingthrougha horizontal polarizing filter, and a 50% chance of going through a verticalpolarizingfilter,sowegetsomelightthrougheachslit.Thisarrangementof filtersgivesusawayofmeasuringwhich slit the light

wentthrough.Ifweputaverticalpolarizerinfrontofourdetector,wewillonlysee light that went through the right-hand slit, and if we put a horizontalpolarizer in front of our detector,wewill only see light thatwent through theleft-hand slit. The polarizer in front of the detector lets us tell which slit thephotonwent through, justas ifwehadputadetector rightnext to theslitandmeasuredthepositiondirectly.What happenswhenwe do this?Whenwe look at the lightwith the filters

over the two slits,we don’t see any sign of an interference pattern.Whenwemeasure the polarization of the light, we measure which slit the light wentthrough, and that takes us from a two-part wavefunction, which produces aninterference pattern, to a one-part wavefunction, which does not. The act ofmeasuring which slit the photon went through destroys the component of thewavefunctiondescribingthephotongoingthroughtheotherslit,justastheactofopening one of the boxes destroyed the component for the treat being in theotherbox.Wedon’tevenneedtoputapolarizingfilteronthedetector—byputtingthe

polarizersover theslits,wehave“tagged”eachphoton,and themerefact thatwecanmeasurewhichslit itwent through isenough todestroy thepattern. Inthetreats-in-boxesexample,thisislikesomeonewriting“Treat”ontheoutsideof theboxcontaining the treat—weno longerneed toopen thebox todestroythesuperposition.The disappearing pattern is pretty weird in its own right, but things get

weirder:we can undo themeasurement after the fact by using a 45º polarizerinsteadofahorizontalorverticalpolarizertolookat thelightaftertheslits.If

wedothis,weseeaninterferencepatternagain!A45ºpolarizerwillpasseitherhorizontal or vertical polarization, eachwith a 50% probability, whichmeansanylightwedetectafterthepolarizercouldhavegonethrougheitherofthetwoslits, or even both at once. The third filter “erases” the information we hadgainedby tagging thephoton, like somebody removing the label from theboxcontaining the treat. Inserting the extra polarizermakes it as if we had nevermade the measurement at all. The second part of the wavefunction isn’tdestroyedafterall,andwecanseeinterference.Thequantum-eraserexperimentencapsulateseverythingthatisstrangeabout

the core principles of quantummechanics. The appearance of the interferencepatternshowsthesuperpositionofquantumstates,aseachphotongoesthroughboth slits at the same time, and the disappearance and reappearance of thepattern when we add polarizing filters shows the active nature of quantummeasurement. Just the fact that it ispossible tomeasurewhichslit theparticlewentthroughisenoughtocompletelychangetheresultsoftheexperiment.

WHATYOUSEEISALLTHEREIS:

THECOPENHAGENINTERPRETATION

Thesefourideas—wavefunctions,allowedstates,probability,andmeasurement—arethecentralelementsofquantumtheory.Theinterferencepatternsweseeinexperiments with photons* confirm that quantum particles really do occupymultiple states at the same time. The disappearance and reappearance of thepatterninthequantum-eraserexperimentconfirmsthatmeasurementisanactiveprocessanddetermineswhathappensinsubsequentexperiments.Westillhaveaproblem,though,becausethereisnomathematicalprocessfor

describinghowtogetfromaprobabilitytotheresultofameasurement.WeusetheSchrödingerequationtocalculatethewavefunctionsfortheallowedstatesofa physical object, and we use the wavefunction to calculate the probabilitydistribution, butwe cannot use theprobabilitydistribution topredict the exactresult of an individual measurement. Something mysterious happens in theprocessofmakingameasurement.This“measurementproblem”istheoriginofthecompetinginterpretationsof

quantum mechanics, and the point where physics is forced to becomephilosophy.All interpretations use the samemethods to calculate probabilitiesfor the outcomes of repeated measurements. They differ only in how theyexplain the step from a quantum superposition state, where the wavefunction

consists of two (or more) states at the same time, to the classical result of asinglemeasurement,wheretheobjectisfoundinoneandonlyonestate.The first interpretation put forward for quantum theory was developed by

NielsBohrandcoworkersathisinstituteinDenmark,andisthusknownastheCopenhagen interpretation. The Copenhagen interpretation is a very ad hocapproach to the problem of measurement in quantummechanics (which is insomewaystypicalofBohr’sapproach*).The Copenhagen interpretation tries to avoid the problems of superposition

andmeasurementbydrawingastrictlinebetweenmicroscopicandmacroscopicphysics. Microscopic objects—photons, electrons, atoms, and molecules—aregoverned by the rules of quantummechanics, butmacroscopic objects—dogs,physicists, and measurement apparatus—are governed by classical physics.There’s an absolute separation between the two, and you will never see amacroscopicobjectbehavinginaquantummanner.Quantum measurement involves the interaction of a macroscopic

measurementapparatuswithamicroscopicobject,andthat interactionchangesthe state of the microscopic object. The usual description is that thewavefunction“collapses”intoasinglestate.This“collapse,”intheCopenhageninterpretation, is an actual change of the wavefunction from a spread-outquantum statewithmultiple possiblemeasurement outcomes to a statewith asinglemeasuredvalue.*In themost extreme variants of theCopenhagen interpretation, the collapse

requires not only amacroscopicmeasurement apparatus, but also a consciousobserver to note the measurement. In this view, a tree that falls in the foresthasn’treallyfallenuntilsomeperson(ordog)comesalongandobservesit.

“So,wait, doesn’t thatmean rejecting the entire idea of an objective physicalreality?”“In itsmost extreme forms, yes.WernerHeisenbergwas probably themost

radical of theCopenhagen crowd, and he insisted quite strongly that it was amistake to talk about electrons having an independent reality. InHeisenberg’sview, the only things we can really talk about are the outcomes of specificmeasurements.Herejectedthewholenotionoftalkingaboutwhattheelectronsweredoingbetweenmeasurements.”“That’s…prettyradical.Idon’tthinkIlikethat.”“You’renotalone,believeme.”

TheCopenhageninterpretationraisesagreatmanyproblems,amongthemthatthere’snoobviousreasonfor theabsolutedistinctionbetweenmicroscopicand

macroscopic physics. As we’ve already seen, while it gets more difficult todetect quantumbehavior as objects get larger andmore complicated, it is stillpossible to see wave behavior in rather largemolecules.Macroscopic objectsoughttobedescribedbyquantumwavefunctionsandquantumrules.Anotherproblemiswhoorwhatcountsasan“observer”forthepurposesof

collapsing the wavefunction. The requirement that somebody observe theoutcome of a measurement before the measurement really “counts” seems toassign some sort of mystical quality to “consciousness,” and that idea makesmanyphysicistsuncomfortable.Eventheideaofthe“collapse”itselfisproblematic.Nomathematicalformula

exists to describe the collapse—you can use the Schrödinger equation todescribe how awavefunction changes betweenmeasurements, but there is noway todescribe the“collapse”process.Allyoucando is choosea result, andstartoverwithanewwavefunctionafterthemeasurement.Manyphysicistsfindthisalittletoomagicalforcomfort.The most famous illustrations of the problems with the Copenhagen

interpretationaretheinfamous“Schrödinger’scat”thoughtexperiment,andthefollow-upthoughtexperimentof“Wigner’sfriend.”Despitehisroleincreatingquantum theory, Erwin Schrödinger, like Einstein, had deep philosophicalproblemswith its interpretation,andbecamedisillusionedwith theentirefield.Schrödinger’s cat, which is arguably more famous than his equation, is adiabolicalthoughtexperimentthroughwhichSchrödingerattemptedtoillustratetheabsurdityof theCopenhagen interpretation.He imaginedplacingacat inasealedboxwitharadioactiveatomthathasa50%chanceofdecayingwithinonehour,andadevicethatwillreleasepoisongasiftheatomdecays,killingthecat.What,heasked,isthestateofthecatattheendofthehour?As Schrödinger noted, according to the Copenhagen interpretation the

wavefunctiondescribing thecatwouldbeequalparts“alive”and“dead.”Thiswouldlastuntiltheexperimenteropensthebox,atwhichpointitwouldcollapseintooneofthetwostates.*Thisseemscompletelyabsurd,though—theideaofacat that is both dead and alive at the same time is outlandish.And yet this isexactlywhatseemstohappenwithphotons.TheCopenhagen interpretation also seems to be saying that physical reality

doesnotexistuntilameasurement ismade,whichposes itsownphilosophicalproblems. EugeneWigner brought this out by adding another layer to the catexperiment,imaginingthattheentirethingwasconductedbyafriend,andonlyreportedtohimlater.Wigneraskedwhenthewavefunctioncollapsed:Whenthefriendopened thebox,or later,whenWignerheard the result?Hasa tree inaforestreallyfallenbeforeyourdogtellsyouthatit’sontheground?

Noneof theCopenhageninterpretation’sanswerstothesequestionsareverysatisfying,philosophically.Whilequantummechanicsdoesanoutstandingjobofdescribing the behavior of microscopic objects and collections of objects, theworldweseeremainsstubbornly, infuriatinglyclassical.Somethingmysterioushappensinthetransitionfromtheweirdworldofsimplequantumobjectstothemuch largerworldofeverydayobjects.TheCopenhagenapproachof insistingon an absolute division between microscopic and macroscopic strikes manyphysicistsassimplydodgingthequestion:itsayswhathappens,butnotwhy.Howbest to handle the transition between quantum and classical remains a

subject of active debate. Some future theory may lead to a detailedunderstanding of what, exactly, happens when we make a measurement of aquantumobject.Untilthen,we’restuckwithoneofthevariousinterpretationsofquantummechanics.

“Idon’tthinkIlikethisinterpretation.It’sawfullysolipsistic,isn’tit?”“You’re not alone. There aren’t very many physicists these days who are

reallyhappywiththeCopenhageninterpretation.”“So,whatinterpretationdoyoulike?”“Me?Itendtogowiththe‘shutupandcalculate’interpretation.Thenameis

sometimesattributedtoRichardFeynman,*buttheideaisjusttoavoidthinkingaboutit.Quantummechanicsgivesusverygoodtoolsforcalculatingtheresultsof experiments, and the question of what goes on during measurement isprobablybetterlefttophilosophy.”“I don’t think I like that one, either. It’s hard to work a calculator without

opposablethumbs.”“Well,thereareallsortsofdifferentinterpretations—there’sthemany-worlds

interpretation, David Bohm’s non-local mechanics, and something called the‘transactional interpretation.’ There are almost as many interpretations ofquantummechanicsastherearepeoplewhohavethoughtdeeplyaboutquantummechanics.”“Ilikethemany-worldsinterpretation.Youshouldtalkaboutthat.”“Goodidea.That’sthenextchapter.”“Iknewthat.”

*Seechapter4,page101.*Sortoflike“Thoushaltnotclimbonthefurniture”fordogslivingwithhumans.*Schrödingerwasalmostasnotoriousforhiswomanizingasforhiscontributionstophysics.Hecame

upwith the equation that bears his namewhile on a ski holidaywith one of hismany girlfriends, andfathereddaughterswith threedifferentwomen,noneof themhiswife (who, incidentally,knewabouthis

affairs).HisunconventionalpersonallifecosthimapositionatOxfordafterheleftGermanyin1933,buthecarriedonlivingmoreorlessopenlywithtwowomen(onethewifeofacolleague)formanyyears.*Aswediscussedattheendofchapter2.*Einsteinhadmanynegativethingstosayabouttheprobabilisticnatureofquantummechanics,butthe

originoftheusualformulationisalettertoMaxBornin1926,inwhichhewrote,“Thetheorydeliversalot,buthardlybringsusclosertothesecretoftheOldOne.IforoneamconvincedthatHedoesnotthrowdice”(quotedinDavidLindley’sUncertainty,p.137).*WernerHeisenbergwentsofarastosaythattheresultsofmeasurementsweretheonlyreality—thatit

madenosensetotalkaboutwhereanelectronwasorwhatitwasdoingbetweenmeasurements.*TheMay2007issueofScientificAmericanevendescribesaquantum-eraserexperimentthatyoucan

doathome,usingalaserpointer,tinfoil,wire,andafewpiecesofcheappolarizingfilm.*Andelectrons,andatoms,andmolecules…* As we said in the last chapter (page 49), Bohr’s first great contribution to physics was a simple

quantummodelofhydrogen. Itwasacobbled-togethermixofquantumandclassical ideaswithnoclearjustification that happened to give the right result, and it’s unclearwhat ledBohr to put it forth. It did,however,pointthewaytowardthemodernquantumtheorythatwe’rediscussinginthisbook.*Theword“collapse”hascometobestronglyassociatedwithCopenhagen-typeinterpretations.There

are other approaches to the problem of the projection of a multicomponent wavefunction onto a singlemeasurementresultthatdon’tinvolveaphysicalchangeinthewavefunction.We’lllookatthebest-knownexampleofthese“no-collapse”interpretationsinchapter4.*Or,asTerryPratchettdescribed it inhisnovelLordsandLadies, applied toaparticularlynastycat:

“Technically,acatlockedinaboxmaybealiveoritmaybedead.Youneverknowuntilyoulook.Infact,themereactofopeningtheboxwilldeterminethestateofthecat,althoughinthiscasetherewerethreedeterminate states the cat could be in: these being Alive, Dead, and Bloody Furious” (p. 226, Harperpaperback).*Feynmantends togetcredit foranythingcleversaidbyaphysicist in the latterhalfof the twentieth

century.“Shutupandcalculate”probablyisn’tFeynman,though—itsfirstappearanceinprintseemstobeaDavidMermincolumninPhysicsToday(April1989,p.4),asheexplainsintheMay2004issue(p.10).

CHAPTER4

ManyWorlds,ManyTreats:

TheMany-WorldsInterpretation

I’m sitting at the computer typing,when Emmy bumps up againstmy legs. Ilookdown,andshe’ssniffingtheflooraroundmyfeetintently.“Whatareyoudoingdownthere?”“I’mlookingforsteak!”shesays,wagginghertailhopefully.“I’mprettycertainthatthere’snosteakdownthere,”

Isay.“I’venevereatensteakatthecomputer,andI’vecertainlyneverdroppedanyonthefloor.”“Youdidinsomeuniverse,”shesays,stillsniffing.Isigh.“Allright,whatridiculoustheoryhasyoursillylittledoggybraincome

upwithnow?”“Well,it’spossiblethatyouwouldeatsteakatthecomputer,yes?”“Idoeatsteak,yes,andIsometimeseatatthecomputer,sookay.”“Andifyouweretoeatsteakatthecomputer,you’dprobablydropsomeon

thefloor.”“Idon’tknowaboutthat…”“Seriously,pal,I’veseenyoueat.”Yes,thedogcallsme“pal.”Theremaybe

obedienceclassesinherfuture.“Allright,we’llallowthepossibility.”“Therefore,it’spossiblethatyoudroppedsteakonthefloor.Andaccordingto

Everett’smany-worldsinterpretationofquantummechanics,thatmeansthatyoudiddropsteakonthefloor.WhichmeansIjustneedtofindit.”“Well, technically, what the many-worlds interpretation says is that there’s

somebranchof theunitarilyevolvingwavefunctionof theuniverse inwhich Idroppedsteakonthefloor.”“Ummm…right.Anyway,Ijustneedtofindtheunitarywhatsit.”“Thethingis,though,wecanonlyperceiveonebranchofthewavefunction.”

“Maybeyoucanonlyperceiveonebranch.Ihaveaverygoodnose.Icansniffintoextradimensions.They’refullofevilsquirrels.Withgoatees.”“That’sStarTrek,notscience,andanyway,extradimensionsareacompletely

differentthing.Inthemany-worldsinterpretation,oncetherehasbeensufficientdecoherencebetweenthebranchesofthewavefunctionthatthere’snopossibilityof interference between the different parts, they’re effectively separate andinaccessibleuniverses.”“Whatdoyoumean,decoherence?”“Well, say I did have a piece of steak here—stop wagging your tail, it’s

hypothetical—quantummechanics says that if I dropped it on the floor, thenpicked it back up, there could be an interference between the wavefunctiondescribing the bit of steak that fell and thewavefunction describing the bit ofsteakthatdidn’tfall.Because,ofcourse,there’sonlyaprobabilitythatI’ddropit,soyouneedbothbits.”“Whatwouldthatmean?”“I’mnotreallysurewhatthatwouldlooklike.Thepointis,though,itdoesn’t

reallymatter.Thesteak isconstantly interactingwith itsenvironment—theair,thedesk,thefloor—”“Thedog!”“Whatever.Thoseinteractionsareessentiallyrandom,andunmeasured.They

leadtoshiftsinthewavefunctionsofthedifferentbitsofsteak,andthoseshiftsmake it so thewavefunctions don’t interfere cleanly anymore.That process iscalleddecoherence,andithappensveryquickly.”“Howquickly?”sheasks,lookinghopeful.“It depends on the exact situation, but as a rough guess, probably 10-30

secondsorless.”“Oh.”Shedeflatesalittle.“That’sfast.”“Yes.Andoncethatdecoherencehashappened,thedifferentbranchesofthe

wavefunction can’t interact with one another any more. Which means,essentially, that the different branches become separate universes that arecompletely inaccessible to one another. Things that happen in these other‘universes’haveabsolutelynoeffectonwhathappensinouruniverse.”“Whydoweonlyseeonebranchofthewhatchamacallit?”“Ah, now that’s the big question.Nobody really knows. Somepeople think

thismeans thatquantummechanics is fundamentally incomplete, and there’sawholecommunityof scientistsdoing research into the foundationsofquantumtheory and its interpretations. The important thing is, there’s no way you’regoingtofindsteakundermydeskinthisuniverse,sopleasegetoutofthere.”“Oh. Okay.” She mopes out from under the desk, head down and tail

drooping.“Well,lookonthebrightside,”Isay.“Intheuniversewhereaversionofme

droppedapieceofsteakonthefloor,there’salsoaversionofyou.”“Really?”Herheadpicksup.“Yes.Andyou’reamightyhunter,soyouprobablygot to thesteakbeforeI

couldpickitup.”“Really?”Hertailstartswagging.“Yes.So,intheuniversewhereIdroppedsteak,yougottoeatsteak.”Thetailwagsfuriously.“Ilikesteak!”“Iknowyoudo.”IsavewhatIwasworkingon.“Tellyouwhat,howaboutwe

goforawalk?”“Goodplan!”Andshe’soff,clatteringdownthestairsforthebackdoorand

thelead.

Few physicists have ever been entirely happy with the Copenhageninterpretation discussed in the previous chapter. Numerous alternatives havebeenproposed,eachattempting to findamore satisfyingway todealwith theproblem of quantum measurement. The most famous of these is commonlyknown as the many-worlds interpretation, which has achieved a dominantposition in pop culture, if not among physicists, thanks to its prediction of anearly infinite number of alternate universes in which events took a differentpaththantheonewesee.It’sawonderfulsciencefictionconceit,turningupinbooks, films,and the famousStarTrek episode featuringanevilSpockwithagoatee.In this chapter,we’ll talk about themany-worlds interpretation, and how it

addressessomeoftheproblemsraisedbytheCopenhageninterpretation.We’llalsodiscussthephysicalprocessknownas“decoherence,”inwhichfluctuatinginteractions with the environment obscure the effects of interference betweendifferent parts of the wavefunction. Decoherence is central to the modernunderstanding of quantum mechanics, and may be the critical factor forunderstandingthemovefromthemicroscopicworldofquantumphysicstotheclassicalworldofeverydayobjects.

THENAMEASUREMENTOCCURS:PROBLEMSWITHCOPENHAGEN

The most disturbing element of the Copenhagen interpretation by far, for aphysicist at least, is the lack of amathematical procedure for describingwhat

happens when you make a measurement of some quantity. The Schrödingerequation allows you to calculate what happens to the wavefunction betweenmeasurements, but at the instant of a measurement the Copenhageninterpretationsaysthatnormalphysicsstops,andsomethinghappenstoselectasingle outcome in a way that does not involve any known mathematicalequation.TheadhocnatureoftheCopenhageninterpretation,withitsarbitrarydivision

betweenmicroscopicandmacroscopicphysicsanditsmysterious“wavefunctioncollapse,” is tremendously disturbing, because the whole project of moderntheoreticalphysicsistofindasingleconsistentmathematicaldescriptionoftheworld. The unexplained process of wavefunction collapse is like the famousSidneyHarriscartoonofascientistwhohaswritten“Thenamiracleoccurs”asthesecondstepofaproblem.Normalsciencehasnoroomformiracles,andtheCopenhagencollapseideaisalittletoomiraculousforcomfort.Mostphysicists (particularlyexperimentalists) are content touse the ideaof

wavefunctioncollapse as a calculational shortcut andgoabout thebusinessofpredictingandmeasuringthephysicalworld,forwhichregularquantumtheoryworks astoundingly well. In this “shut up and calculate” interpretation, theproblemoffindingaconsistentexplanationforquantummeasurementispushedasidetobedealtwithbyphilosophers.Somebettertheorymayeventuallycomealong, but until then,we should dowhatwe canwithwhatwe’ve got (whichturnsouttobeanawfullot).Thenatureofmeasurementhasbeenaproblemfromthefirstdaysofquantum

theory, though,anda fewphysicistshavealwayschosen to thinkdeeplyabouttheseissues.Manyofthesephysiciststhinkthatthelackofaclearexplanationfor the “collapse” of the wavefunction indicates that the Copenhageninterpretation is fundamentally flawed. Thus, they have always searched forsomealternativeinterpretation.

THEREISNOCOLLAPSE:HUGHEVERETT’SMANY-WORLDSINTERPRETATION

In 1957, a graduate student at Princeton University named Hugh Everett IIIsuggested a solution to the “collapse” problem that’s breathtaking in itssimplicity.Thereasonthereisnomathematicalmethodtodescribethecollapseof the wavefunction, Everett said, is because there is no such thing as thecollapseofthewavefunction.Thewavefunctionalwaysandeverywhereevolvesaccording to the Schrödinger equation, but we only see a small piece of the

largerwavefunctionoftheuniverse.Let’sreturntothepreviouschapter’sexampleofdogtreatsinsealedboxesto

seehowthisworks.Ifweimaginethatwehaveonedogtreatintwoboxes,theCopenhagenpicturesaysthatweinitiallyhaveawavefunctionforthetreatthatconsists of two pieces at the same time. This wavefunction changes in timeaccordingtotheSchrödingerequation.Whenweopenaboxandletthedoglookinside, the wavefunction instantaneously collapses into only one of those twostates,withthetreatineithertheleft-handboxortheright-handbox.Wepredictfuture changes by starting over againwith theSchrödinger equation using thenewone-partwavefunction.In theEverettpicture, there isnocollapse.Thewavefunction startsout ina

superposition,atwo-partwavefunctionwithpiecescorrespondingtothetreatinbothleft-handandright-handboxes,andwhenweopenaboxthatsuperpositionjust becomes a little bigger. Now the superposition includes not just theapparatusbutalsothedogmeasuringthepositionofthetreat.Onepieceis“treatintheleft-handboxplusadogwhoknowsthetreatisintheleft-handbox,”andtheotheris“treatintheright-handboxplusadogwhoknowsthetreatisintheright-handbox.”Thisprocesscontinuesasyoumoveintothefuture.Ifthenextstep in the experiment involves the dog either eating the treat or not (a lowprobabilityoutcome,butit’spossible),thewavefunctioncontainsfourpieces:adogwhoatethetreatfromtheleft-handbox;adogwhodidn’teatthetreatfromtheleft-handbox;adogwhodidn’teatthetreatfromtheright-handbox,andadogwhoatethetreatfromtheright-handbox.The increase in complexity is evenmore striking inmathematical notation.

Westartwithatwo-componentwavefunctionforjustthetreat:

Ψtotal=|L>treat+|R>treat

wheretheangledbracketsrepresentwavefunctionsforthetreatbeingintheleftorrightboxes.Thenwebringinthedog:

Ψtotal=|L>treat|L>dog+|R>treat|R>dog

andfinally,thedecisiontoeatthetreatornot:

Ψtotal=|L>treat|L>dog|eat>+|L>treat|L>dog|not>

+|R>treat|R>dog|eat>+|R>treat|R>dog|not>

Asyoucansee, thisgetsverycomplicatedveryquickly,but itsevolution isalwaysdescribedbytheSchrödingerequation.

“Youknow,I’mnotgettingalotoutoftheseequations.”“You’re not supposed to understand them in detail. They’re just there to

illustrate the increasing complexity of the wavefunction in a more compactmanner.”“So,basically,they’rejustsupposedtolookscary?”“Prettymuch.”“Oh.Goodjob,then.”

•••

The Everett picture doesn’t immediately appear to be an improvement. Themysterious “collapse” is removed, but at the price of a wavefunction that’sexpanding exponentially. At first glance, it also seems to defy reality, as wenever see systems in more than one state. If all these extra pieces of thewavefunction are running around, why don’t we perceive objects as being inmultiplestatesatthesametime?Theanswer,accordingtoEverett,isthatwecan’tseparatetheobserverfrom

the wavefunction. The observer is included with the rest of the system—thecomponentsarethingslike“treatintheleft-handboxplusadogwhoknowsthetreatisintheleft-handbox”—andasaconsequence,weonlyperceiveourownsmall part of the overall wavefunction. This branching is the origin of thequantum randomness that Einstein and others found so troubling: thewavefunction always evolves in a smooth and continuous way, but we onlyexperienceonebranchofthewavefunctionatatime,andwhichbranchweseeisa random choice. Other versions of ourselves exist in the other branches,experiencingdifferentoutcomes(forthisreason,theinterpretationissometimesreferredtoasthe“many-minds”interpretation).None of the myriad other branches have any detectable influence over the

eventsinourbranch,andourbranchhasnodetectableinfluenceovertheeventsinanyoftheothers.Forallintentsandpurposes,thoseotherbranchesareself-containedparalleluniverses,completelyinaccessiblefromouruniverse.Thisistheoriginofthename“many-worlds”forthetheory:it’sasiftheuniverseforksevery time a measurement is made, and is constantly spawning new paralleluniverseswithslightlydifferenthistories.

WAVEFUNCTIONSFALLAPART:DECOHERENCE

These noninteracting branches present a serious but subtle problem for themany-worlds interpretation.Everyother two-partwavefunctionwe’veseenhasledtosomesortofinterferencephenomenon.So,whydon’tweseeinterferencearoundusallthetimeiftherearealltheseextrabranchestothewavefunction?Whatisitthatsealsthese“paralleluniverses”offfromus?The answer is a process called decoherence, which prevents the different

branchesofthewavefunctionfrominteractingwithoneanother.Decoherenceisthe resultof random, fluctuating interactionswitha larger environment,whichdestroy the possibility of interference between different branches of thewavefunction, andmake theworldwe experience look classical.Decoherencedoesn’tjustoccurinthemany-worldsinterpretationofquantummechanics—it’sarealphysicalprocess,compatiblewithanyinterpretation*—butit’sparticularlyimportant in the modern view of many-worlds (which is sometimes called“decoherenthistories”asaresult—theinterpretationhasalmostasmanynamesasuniverses†).Decoherenceisabsolutelycriticaltothemodernviewofquantummechanics

and quantum interpretations. Themost common semiclassical explanations ofdecoherenceleavealottobedesired,though,astheyareinaccurate,andoftensomewhat circular. The real theory of decoherence is subtle and difficult tounderstand.Aswith theuncertaintyprinciple (chapter2,page40), though, it’sworthsomeefforttounpackit,becauseitprovidesamuchricherunderstandingofthewaytheuniverseworks.Tounderstandtheideaofdecoherence,let’sthinkabouttheconcreteexample

ofasimpleinterferometer,adeviceconsistingoftwobeamsplittersthatsplitabeamoflightinhalfandacoupleofmirrorsthatbringitbacktogetheragain.‡Interferometers like this are extremely important in physics, not only fordemonstratingquantumeffects,butbecausetheyformthebasisfortheworld’smost sensitive detectors of rotation, acceleration, and gravity.These allow themeasurementoftinyforcesinphysicsexperiments,andalsofindapplicationinsubmarinenavigation.Light enters the interferometerwhen it strikes abeamsplitter,whichpasses

halfofthelightthroughwithoutperturbingitandreflectstheotherhalfoffata90°angle.Thesetwobeamsseparatefromeachother,andthenaresteeredbacktogether using two mirrors, and recombined on a second beam splitter. Thesecondbeamsplitterislinedupsothatthetransmittedlightfromonebeamandthereflectedlightfromtheotherfollowexactlythesamepathandinterferewitheachotherbeforefallingononeoftwodetectors.Youmightthinkthateachofthetwodetectorswoulddetectexactlyhalfofthe

light,becauseeachdetectorreceivesone-quarteroftheoriginalbeamfromeach

ofthetwopaths(¼+¼=½).Eachdetectorcanactuallyseeanythingfromnolightatalluptothefullintensityoftheinitialbeam,though,becausethewavesthattookdifferentpathsinterferewitheachother,likethewavesinthedouble-slitexperimentinchapter1(page18).Ifthepathsfollowedbythetwobeamshaveexactlythesamelength,thetwo

lightwavesundergothesamenumberofoscillationsenroutetoDetector2,andinterfere constructively, giving a bright spot—all of the light entering theinterferometerhitsthatdetector.*On theotherhand, ifonepath is longer thanthe other by one half of thewavelength of the light, the light along that pathundergoes an extra half-oscillation, and the twowaves interfere destructively:the crests of the wave hittingMirror 1 line up with the troughs of the wavehittingMirror2,andtheycancelout,givingnolightatDetector2.Ifweincreasethe lengthdifference toonefullwavelength, thepeaksalignagain,andwegetanother bright spot, and so on. Between those two extremes we get someintermediate amount of light. We can generate an interference pattern byrepeating the experiment many times, and changing the length of one pathslightly.ThiswillgiveusapatternofalternatinglightanddarkspotsonDetector2.

Theinterferometerdescribedinthetext.Lightentersfromtheleft,issplitbyabeamsplitterintotwodifferentpaths,andthenrecombinedbythesecondbeamsplitter.IfthebeamhittingMirror1travelsexactlythesamedistanceasthebeamhittingMirror2,interferencecausesallthelighttohittheupperdetector(Detector1),andnonetohitthedetectoratright(Detector2).

ThefractionofthelightreachingDetector2dependsonlyonthetimeittakesforthelighttotraveleachpath.Wecanimaginethelightmovingdownthetwoarms as two identical dogs setting out around a single block in oppositedirections. They agree in advance that the first dog to arrive at the oppositecornerchoosesthepathfortherestofthewalk.Ifthetwodogswalkatthesamespeed, and the two paths are the same length, they will always arrive at the

oppositecorneratthesametime.Ifonepathisslightlylongerthantheother,thedogtakingthatpathwillalwaysbelatearriving,andtheywillalwaysfollowthesamerouteafterwards.“Wait,what’squantumaboutthis?You’rejusttalkingaboutwavesanddogs.”“Youcandescribethebasicoperationoftheinterferometerclassically,butit

alsoworksperfectlywell ifyousend inonephotonata time.That’sasystemyoucanonlyexplainwithquantumphysics.”“Don’tyouneedlotsofphotonstoseethepattern?”“Yes,butyoucanrepeattheexperimentmanytimes,andbuildupthepattern.

Yousetitupsothetwopathshaveequallength,andrepeatit1,000times,andyou’llsee1,000photonsatonedetector.Thenyoumoveonemirroralittlebit,andrepeattheexperimentanother1,000times,andsee700photons,andsoon.Ifyoukeepdoingthisoverandoveragain,you’lltraceoutthesamepatternyouseewithabrightbeam.”“LikethewayImeasuredthewavefunctionofabunnyinthebackgardenby

markingitspositioneverynightwhenIwentouttochaseit?”“Technically, you measured a probability distribution, the square of a

wavefunction,notthewavefunctionitself,butyes,that’sthebasicidea.”“Thebunnyismostlikelytobeunderneaththebirdfeeder,youknow.”“Yes,becauseiteatsthespilledseed.Trytofocus,please.”“Okay.”

Whenwemovetoaquantumdescriptionoftheinterferometer,talkingaboutitintermsofsinglephotonsandwavefunctions,wesaythatthephotonwavefunctionsplits into two parts at the first beam splitter. In the popular view of many-worlds,youmightbetemptedtosaythatthisisalsowhentheuniversesplitsintwo,asthatiswhenwefirstacquireasecondbranchofthewavefunction.Thisistempting,butwrong—thewavefunctionhastwobranches,buttheyare

not “separate universes” at this point. We know this because when we bringthem back together, at the second beam splitter, we see interference. Theprobabilityofaphotonreachingourdetectorchangesaswechangethelengthofone of the paths in the interferometer, indi cating that the two separate pathsinfluenceeachother.Thephotongoesbothwaysatthesametime,andinterfereswithitselfwhenwerecombinethebranches.We see this interference pattern because the two parts of the wavefunction

haveapropertycalledcoherence.“Coherence”isaslipperyword,butwhenwesaythattwowavefunctionsare“coherent,”wemeanthattheybehaveasiftheycame from a single source.* In the case of the interferometer, they did comefrom a single source, and each of the pieces experiences essentially the same

interactionsasitgoesthroughtheinterferometer,sotheyremaincoherentallthewaythrough.Theonlyfactordeterminingwhethertheinterferenceisinphaseoroutofphaseisthelengthdifferencebetweenthetwopaths.Toturnthetwobranchesofthewavefunctionintotwoseparateuniverses,we

need to destroy that coherence. Without coherence, the two pieces of thewavefunctionwillnot interferetogiveadetectablepattern,andwewillnotbeable toseeanyinfluenceofonepathontheother.Thephotonwill looklikeaclassical particle in two different universes, passing straight through the beamsplitterinoneuniverse,andreflectingoffitintheother.Thecoherencebetweenthetwobranchesofthewavefunctionisdestroyedby

interaction with a larger environment. To see how this works, let’s imaginemakingtheinterferometerverylong,sothatthebeamshavetopassthroughalotofairbetweenthetwobeamsplitters:

Aninterferometerwithaverylongdistancebetweenthebeamsplitters.

In our dog-walking example, we can imagine this longer interferometer assending our two identical dogs aroundHyde Park in opposite directions. Thelonger path allowsmore time for distractions to build up—squirrels to chase,droppedfoodtoeat,horsedroppingstorollin.Nowthedogsnolongermoveatthe same speed—sometimes they speed up, and sometimes they slow down,dependingonwhattheysee.Theorderinwhichthetwodogsarrivedependsnotonlyonthelengthofthepaththeyfollow,butonwhattheyencounteredalongtheway.Thesamethinghappenswithlight.Aslighttravelsthroughtheinterferometer,

itoccasionallyinteractswiththeatomsandmoleculesmakinguptheair.Lightpassing along one arm might encounter a region with more molecules thanaverage, and slowdown a little, or itmight encounter a regionwith very fewmolecules,andspeedupabit.Theseeffectsareverysmall,buttheyaddupoverthelengthofthepath,justlikethedistractionsencounteredbyourwalkingdogs.

Theinterferencepattern isdeterminednotonlyby the lengthof the twopaths,butalsobyinteractionswiththeenvironmentintheformoftheairthatthelightpassesthrough.Interactions with the environment, like distractions around Hyde Park, are

essentiallyrandom,andfluctuating—thatis,they’redifferentfromoneplacetoanother, andone time to another.Adogwon’t finddropped foodon the samepatheveryday,andaphotonwon’tinteractwithamoleculeinthesamepartofthe interferometer every time. Interactions shift the pattern, and the shift isdifferenteachtimeweruntheexperiment.Theendresult—whetherweseelightat Detector 2, or which dog wins the race—will be completely random,determinedbyfactorsoutsideourcontrol.Thankstotheserandominteractions,thelightwavesalongthetwopathsare

no longer coherent, soweno longer see a clean interference patternwhenwebring themback together. Insteadwe see a pattern that’s constantly changing,shifting position millions of times a second. The bright and dark spots blurtogether,wipingout thepattern.Thissameeffectcarriesoverwhenwedo theexperiment with single photons—the two pieces of the wavefunction are nolonger coherent. When the photons undergo random, fluctuating interactionswithalargeenvironment,theinterferencepatternisdestroyed.Thisprocessof randominteractionsdestroyingquantumeffects isknownas

decoherence,becauseit’sdestroyingthecoherencebetweenthedifferentpartsofthe photon wavefunction. Decoherence prevents the various branches of thewavefunction from affecting one another in any detectable way—while wewould expect the different branches to come back together and produceinterferencepatterns,wedon’tseethat.Thankstodecoherence,theinterferencebetweendifferentbranchesisrandom,andthuscan’tbedetected.

“So,nowwehavetwodifferentphotonsthatdon’tinterfere?”“No,it’smoresubtlethanthat.Westillhaveonlyonephoton,butitappearsin

two different branches of the wavefunction. We don’t see any interferencebecauseofdecoherence.”“It’sonephotonthatdoesn’tinterferewithitself?”“It interferes with itself, but the interaction with the environment leads to

randomshifts thatmake the interferencepatterndifferentevery time.Wecan’tbuildupapattern throughrepeatedmeasurements,because it’salwaysmovingaround.Thepatternshiftsmillionsoftimesasecond,andyou’reaslikelytogetabrightspotasadarkspot,smearingthewholethingoutinto—”“Anin-betweenspot?”“Exactly.”

“So, it’s like if I tried to measure the bunny wavefunction by marking itsposition when I went out to chase it, but sometimes I was chasing squirrelsinsteadofbunnies?”“Well, thesquirrelsandbunniesbothtendtobefoundunderthebirdfeeder,

so that wouldn’t make much difference. It’s more like trying to measure thebunnywavefunctionbyrecordingthepositioneverynight,whileIkeepmovingthebirdfeedertodifferentspotsintheyard.”“Oh.Thatwouldbemean.Don’tdothat.”“I’mnotgoingto.It’sjustananalogyforthewaythatdecoherencewipesout

the interference pattern. Even though each individual photon interferes withitself,thetwopartsofthewavefunctionaren’tcoherent,sowecan’tbuildupapatternfromrepeatedmeasurements.”“But if we don’t see a pattern, how do we know that there’s interference?

What’s the difference between quantum particles that don’t produce aninterferencepatternandordinaryclassicalparticles?”“That’s exactly the point: there is no difference. Because of the random

interactions,thequantumeffectsaresmearedout,andwe’releftwithsomethingthatdoesn’t look likean interferencepattern,even though there is interferencehappeningallthetime.Youendupdetectingaphoton50%ofthetime,exactlyasyouwouldifitwereaclassicalparticlewithnowaveproperties.”“Idon’t know. It’skindofZen, isn’t it? ‘What is thepatternofonephoton

interfering?’”“Actually,that’sprettygood.”“Thanks.IhaveBuddhanature,youknow.”

THEINFLUENCEOFTHEENVIRONMENT:DECOHERENCEANDMEASUREMENT

You will sometimes see explanations of decoherence describing it as ameasurement process, saying things like “when a photon interactswith an airmolecule, that’s the same as measuring the position of the photon, whichdestroys the interference pattern.” This is almost exactly backwards—decoherenceisn’taresultofmeasuringthephotonthroughinteractions,it’stheresultofnotmeasuringtheinteractionbetweenthephotonanditsenvironment.This is a subtle point, but it’s critical to the modern understanding of

decoherence. As we send individual photons through the interferometer, eachone interferes with itself, according to a wavefunction that shows someinterferencepattern.Ifwecouldsendinthousandsofphotons,andgetthesame

interactioneachtime,wecouldrepeatthemeasurementoverandover,andtraceoutaninterferencepatternlikethefirstdashedcurveshowninthefigureonthenextpage.Wecan’tguaranteeexactly thesame interactionbetween thephotonand the

environment every time, though, anymore thanwecanguarantee that aHydeParktouristwilldropfoodinthesameplaceeverytimeourdogsgoforawalk.As a result, the second photon sent inwill interferewith itself according to adifferentwavefunction. Ifwecould repeat this experiment thousandsof times,wewouldtraceoutapatternliketheseconddashedcurveinthefigure.Thethirdphotonhasyetanotherdifferentwavefunction,whichwouldtraceoutapatternlikethethirdcurve,andsoon.In the end, we get one photon drawn from each of these patterns, and

thousandsofothers,eachwithpeaksindifferentpositions.Thecumulativeeffectis to traceoutapattern that is thesumofmanydifferent interferencepatterns.Thisisshownasthesolidlineinthefigure,whichbarelyshowsanyinterferenceatall.Whatdoesthishavetodowithmeasuringtheenvironment?Well,wheneach

ofthephotonswesendininteractswiththeair,itmakesasmallchangeinthestateoftheenvironment—anairmoleculeismovingalittlebitfaster,oralittlebitslower,ortheinternalstateofthatmoleculeischangedinsomeway.Exactlywhat happens to the environment depends on exactly what sort of interactionwenton,andthat,inturn,determineswhathappenstothephoton.

Thedashedlinesshowtheinterferencepatternsforphotonswiththreedifferentphases.Thesolidlinerepresentsthesumofseveralsuchpatterns,showingthattheinterferencepatternisalmostcompletelywipedout.

Ifwecouldkeep trackofeverything thathappened to theenvironment—theexactstateofeveryairmolecule in thetwopaths throughtheinterferometer—we could use that information towork outwhat happened to the photon, andchoosetolookonlyatphotonswhosewavefunctionsproduceidenticalpatterns.Only a tiny number of photons will have exactly the same result, but if yourepeattheexperimentoftenenough,you’llfindsome—the159thphotonthroughmightproduceexactlythesamepatternasthefirst,andthenthe1022nd,andthe5,674th,andsoon.Ifyoulookonlyatthosephotons,you’llseethemtraceoutaninterferencepattern,justasiftherewerenodecoherence.

“So,evenifyouweremovingthefeedereverynight,Icouldstillfindthebunnywavefunction?”“Ifyoukepttrackofwherethefeederwaseachtime,yes.”“Because the bunny would always be most likely to be underneath the

feeder?”“Right.Ifyouputallthepositionstogether,you’dfindamess,butifyouonly

usedcaseswherethefeederwasundertheoaktree,you’dfindthebunnymostlyunder the oak. If you only used caseswhere the feederwas between the twomapletrees,you’dmostlyfindthebunnythere,andsoon.”“Okay.You’renotgoingtodothat,though,right?”“Iwouldn’tupsetyoulikethat.”“See,thisiswhyyou’remyfavoritehuman.”

Wesee,then,thatthemany-worldspictureremovesbothofthemajorproblemswith theCopenhagen interpretation.Not only does it eliminate themysterious“collapse” of the wavefunction, it gets rid of the arbitrary division betweenmicroscopic and macroscopic in the Copenhagen picture. According to themany-worlds interpretation, macroscopic objects like cats in boxes do obeyquantum rules, and show superposition and interference effects.We don’t seethose effects because of decoherence caused by interactions with theenvironment.Ifwecouldkeeptrackofeveryinteractionbetweenthecatanditsenvironment, though, we could reconstruct the wavefunction, and verify thatquantummechanicsworksoneveryscale.Of course, it’s impossible to keep track of the exact state of every particle

makingupeventhesimplestofscientificexperiments,letalonealltheparticlesmaking up amacroscopic object like a piece of steak or a hungry dog. As aresult, theprocessofdecoherencehappensall thetime,andhappensextremelyquickly. You see decoherence any time you have an object interactingwith alarger environment, and all objects are always interacting with their

environments. Themore atoms you have, themore chances you have for thesystem to interact with the environment, and the faster decoherence will takeplace.Apieceofsteakcontainssomethinglike1023atoms,sodecoherencewilltakeplaceextremelyquickly,soquicklythatwe’llnevergetachancetoseeanyquantumeffects.*Real objects are extremely difficult to isolate from the environment well

enough to be able to see quantum interference, but it can be done.Experimentalists have been able to demonstrate quantum effects using smallnumbers of particles inside ultra-high vacuum chambers, often involvingcomponents cooled to temperatures close to absolute zero. The largest suchexperiment involved about a billion electrons in a loop of superconductor, attemperatureswithina fewdegreesofabsolutezero.Theelectronswereplacedintoasuperpositionofstatescorrespondingtoclockwiseandcounterclockwiseflow around the loop (like a dog walking clockwise and counterclockwisearoundHydeParkatthesametime).Abillionelectronssoundslikealot,butit’sstillprettytinycomparedtoeverydayobjects.Itservestodemonstrate,though,thatwith sufficiently careful control of the environment,we can see quantumbehaviorwithlargenumbersofparticles.

GETTINGTOREALITY:DECOHERENCEANDINTERPRETATIONS

The idea of decoherence is by no means exclusive to the many-worldsinterpretationofquantummechanics.Decoherenceisarealphysicalprocessthathappensinallinterpretations.IntheCopenhageninterpretation,itservesasthefirst step in the process ofmeasurement, selecting the states you can possiblyend up in. Decoherence turns a coherent superposition of two or more states(bothAandB)intoanincoherentmixtureofdefinitestates(eitherAorB).Thensomeother,unknown,mechanismcausesthewavefunctiontocollapseintooneofthosestates,givingthemeasurementresult.Inthemany-worldsview,decoherenceiswhatpreventsthedifferentbranchesofthewavefunctionfrominteractingwith one another,while each branch contains an observerwho only perceivesthatonebranch.In either case, decoherence is an essential step in getting from quantum

superpositions to classical reality. All of the concrete predictions of quantumtheory are absolutely identical, regardless of interpretation. Whicheverinterpretationyou favor,youuse the sameequations to find thewavefunction,and the wavefunction gives you the probabilities of the different possibleoutcomesofanymeasurement.Noknownexperimentwilldistinguishbetween

the Copenhagen interpretation and the many-worlds interpretation,* so whichyouuseisessentiallyamatterofpersonaltaste.Theyarejusttwodifferentwaysofthinkingaboutwhathappensasyoumovefromtheprobabilitiespredictedbythewavefunctiontotheresultofanactualmeasurement.

“SohowdoIpickmyuniverse?”“Pardon?”“I want to be in the universe where I eat steak. What determines which

universeIendupin,andhowdoIchangeittogivemetheonethatIwant?”“Well, there’s nothing you can do to affect the outcome of a quantum

measurement, that we know of. It’s completely random,whether youwant tothinkofitasacollapsingwavefunction,orjustperceivingasinglebranchoftheentire wavefunction of the universe. Either way, you’re stuck with randomoutcomes.”“But I thought you said that in many-worlds the wavefunction always

followedtheSchrödingerequation?Can’tyouusethattopredictwhichbranchistherealone?”“Youcanuseittopredicttheprobabilitiesofthedifferentbranches,buteach

branch has its own version of you, perceiving its own set of measurementoutcomes.Eachof them think they’re the ‘real’ branch, andwonderwhy theydidn’tendupinadifferentbranch.Thetheorydoesn’tsayanythingaboutoneofthembeing‘real.’”“So…it’sapunt,basically?”“Sorry,butyes. It’smathematicallymoreelegant, butnot reallymuchmore

satisfyingthantheCopenhageninterpretation,whenyougetrightdowntoit.Itjustsortofpushesthequestionbackalevel.”“Interpretationsarestupid.”“You’renotaloneinthinkingso,butwe’restuckwiththemforthemoment.”“Well,Idon’t likethem.Iwanttoeatsteak.Ifquantummeasurementwon’t

helpmeeatsteak,thenIdon’twantanymoretodowithit.”“Interpretationsaren’tthewholeofquantumtheory,byanystretch.Thereare

onlya fewpeoplewhospend their timeworkingon thatstuff.Mostphysicistsdon’tbotherworryingabout interpretations, andusequantummechanics todousefulstuffinstead.”“Ifmeasurementisrandom,howcanyoudoanythingusefulwithit?”“Well,that’sthenextchapter.”

*Infact,alloftheviablecandidatestoreplacetheCopenhageninterpretationincludedecoherenceasan

importantpartofthemeasurementprocess.†“Many-worlds,”“many-histories,”“many-minds,”“decoherenthistories,”“relativestateformulation,”

and“theoryoftheuniversalwavefunction,”amongothers.‡Thisiscalleda“Mach-Zehnderinterferometer”aftertheGermanandSwissphysicistswhoinventedit.*Thereisasmallshiftinthe“phase”ofawavewhenitreflectsoffabeamsplitter,asifthereflected

beamhad traveled a small extra distance.As a result,when the twopaths have equal length, thewaveshittingDetector2areinphase,andaddtogiveabrightspot,whilethewavesreachingDetector1areoutofphase,andcanceleachotherout.Forpathswithdifferent lengths, the twodetectorsgivecomplementarysignals—whenoneseesnolight,theotherdetectsthefullinitialintensity,andviceversa.*Thisisaveryroughdefinition,anddoesn’tcaptureeverything—wavesfromoppositeendsofasingle

largesourcemaynotbecoherent,forexample—butitdoesgetthebasicideaacross.*Thisisontopofthefactthatthewavelengthofa10-grampieceofsteakwouldbesomethinglike10-

29meters,asdiscussedinchapter1,makingitnearlyimpossibletomeasureasteakyinterferencepatternevenifdecoherenceweren’tanissue.*Oranyof theother interpretations, for thatmatter. Interpretationsofquantummechanics are sortof

“metatheories,”eachgivingadifferentglossontheresultsofanexperiment,butnotchangingtheresults.Every now and then, you will run across somebody claiming to have experimentally “proved” someinterpretationoranother,butthey’reinevitablyconfused.

CHAPTER5

AreWeThereYet?

TheQuantumZenoEffect

I’vehadalong,annoyingdayatwork,runningfromonecommitteemeetingtoanother,andIcomehomewithapoundingheadache.IhaveanhourorsountilKategetshome,andallIwantisanap.Emmyisecstatictoseeme,anddoestheHappyDancealloverthelivingroom.“Hooray!You’rehome!Yippeee!”She’swagginghertailsohardshealmost

losesherbalance.ThishappenseveryafternoonwhenIcomehome.“It’sgoodtoseeyou,too.”“Let’sdosomethingfun!Let’splayfetch!Let’sgoforawalk!Let’splayfetch

onawalk!”“Let’s let me take a nap.” She stops bounding immediately, and looks

crestfallen.Herearsandtaildroop.“Nowalk?”“Notrightnow,”Isay,lyingdownonthesofa.“Letmesleepforhalfanhour,

andthenwe’lldosomethingfun.”“Promise?”“I promise. Now be quiet. The sooner I get to sleep, the sooner we’ll do

somethingfun.”“Oh.Okay.”Iliedownandgetcomfortableonthesofa.I’mjuststartingtosettleinformy

nap,whenacold,wetnosepokesmeintheface.“Areyouasleep?”“No,I’mnotasleep.”“Oh.”Aminutepasses.Poke.“Areyouasleep?”“No.”Anotherminutepasses.Poke.“Areyouasleep?”

“No!” I sitbackup. “And I’mnevergoing toget to sleep ifyoudon’t stoppokingmewithyournoseandaskingthatquestion.”“Whynot?”“Every time you pokeme, youwakeme back up, and I have to start over

again.Ifyoukeepwakingmeup,Idon’tgetallthewayasleep,andyoudon’tgettodoanythingfun.”“Oh.”Shebrightensup.“It’sjustliketheZeroEffect!”“Thewhat?”“Youknow.Theparadoxwiththeblokewhocan’tcatchtheturtlebecausehe

hastogohalfthewaythere,andthenanotherhalf,andsoon,sohenevergetsanywhere.”“YoumeanZeno’sparadox.Zeno,withannasin‘nap.’TheZeroEffect isa

filmwithBillPullmanandBenStiller.”“Whatever.Idon’tspellsowell.”“Anyway,whatyou’rethinkingofisthequantumZenoeffect,andyes,thisis

sortof like that. Ifyouhaveasystemthat’smovingfromonestate toanother,with theprobabilityofbeing in the secondstate increasingover time,youcanpreventthestatechangebyrepeatedmeasurements.Everytimeyoumeasureittobeinthefirststate,yourestarttheprocess.”“Right.SowhenIaskifyou’reasleep,Icollapseyourwavefunctionbackto

the‘awake’state,andyouneedtostartnappingagain.”“OryoufindyourselfperceivingthebranchofthewavefunctioninwhichI’m

awake again, in themany-worlds picture.But yes, that’s the basic idea, and agoodanalogy.”“I’maphilosophicaldog!”“Yes,you’reveryclever.Nowshutupandletmesleep.”“Okay.Iwon’taskifyou’reasleepanymore.”“Thankyou.”I settle back down onto the sofa, and start to feelwarm and cozy, and feel

myselfdriftingoff…Poke.“Areyouawake?”

Whetheryouprefer theCopenhageninterpretation,many-worlds,oroneof themanyothers,somethinghappenswhenyoumakeameasurement.Whetheryouthinkthisinvolvesthephysicalcollapseofawavefunction,orjustlimitingyourperception to a single branch of an expanding and evolving wavefunction,measurementisanactiveprocess.Beforeyoumeasureanobject’sstate,itexistsin a quantum superposition of all possible states, while immediately after themeasurement,youobserveoneandonlyonestate.In this chapter, we’ll look at the most dramatic consequence of active

measurement,the“quantumZenoeffect.”We’llseethatmakingrepeatedmeasurementsofaquantumparticlecanpreventitfromchangingitsstate.WecanalsousethequantumZenoeffecttodetectthepresenceofobjectswithouthittingthemwithevenasinglephotonoflight.

YOUCAN’TGETANYWHEREFROMHERE:ZENO’SPARADOX

The name of the effect is a reference to the famous paradoxes of the GreekphilosopherZenoofElea,wholivedinthefifthcenturyB.C.E.Thereareseveraldifferentversionsoftheparadoxes,butallofthempurporttoshowthatmotionisimpossible.Here’samoderncanineversionoftheargument:inordertoreachatreaton

the farsideof the room,adogfirstneeds tocrosshalf thewidthof the room,which takes a finite time. Then, she needs to cross half of the remainingdistance,whichtakesafinitetime,andthenhalfoftheremainingdistance,andsoon.Thedistance across the room isdivided into an infinitenumberof halfsteps,eachrequiringafinitetimetocross.Ifyouaddtogetheraninfinitenumberof steps,each takinga finite time, it should takean infiniteamountof time tocrosstheroom.Thus,it’simpossibleforthepoordogtoevergetallthewaytothetastytreat.Happily for hungrydogs everywhere, there’s amathematical solution to the

apparentparadox:asthedistancegetssmaller,thetimerequiredtocrossitalsogetssmaller.Ifit takesonesecondtocrosshalfthewidthoftheroom,it takeshalfasecondtocrossthenextquarter,andaquarterofasecondtocrossthenexteighth,andsoon.Addingtogetherallthosetimes,wefindthat:

The total time is the sum of an infinite number of terms, but the terms getsmallerasyougo.Mathematicians learnedhowtoaddthissortofserieswhencalculuswas invented in the seventeenthandeighteenthcenturies.The infinitesumgives a finite result: the dog crosses the room in two seconds.Motion ispossibleafterall,andagooddogcanalwaysreachhertreats.*

WATCHEDPOTSANDMEASUREDATOMS:THEQUANTUMZENOEFFECT

The quantum Zeno effect uses the active nature of quantum measurement topreventaquantumobject(likeanatom)frommovingfromonestatetoanother,bymaking repeatedmeasurements. Ifwemeasure the atom a very short timeafterthetransitionstarts,itwillmostlikelybefoundintheinitialstate.Theactofmeasuringtheatomprojectsitbackintotheinitialstate,aswesawinchapter3,andthetransitionstartsover.Ifwekeepmeasuringthestateoftheatom,wekeepputtingitbackwhereit

started.TheatomisinapredicamentreminiscentofZeno’sparadox—takinganinfinitenumberofstepstowardsomegoal,butnevergettingthere.*Astheoldsayinghasit,awatchedpotneverboils,atleastaslongasit’saquantumpot.Thisisdramaticallydifferentfromclassicalphysics.Measuringthestateofa

classicalobjectdoesnotchangethestate—ifapotofwateris50%ofthewaytoboilingwhenthemeasurementismade,it’sstill50%ofthewaytoboilingafterthemeasurement. The quantum Zeno effect works only because of the activenatureofquantummeasurement—thewaterinaquantumpotiseitherboilingornotboiling.Ifyoufindthatitisn’tboiling,youneedtostartoveragain,asifyouhadneverheatedit.ThedefinitivequantumZenoeffectexperimentwasdonein1990byWayne

Itano inDaveWineland’s group at theUSNational Institute ofStandards andTechnology(NIST)inColorado,usingberylliumions.Ionsarejustatomswithone electron removed, and like all atoms, they have a collection of allowedenergystates,whichtheymovebetweenbyabsorbingoremittinglight.Itano’sexperiment collected a few thousand beryllium ions, and made them moveslowlyfromonestatetoanotherbyexposingthemtomicrowaves.Left unmeasured, the ions took 256milliseconds to complete the transition

from State 1 to State 2.* Their state during this process was described by awavefunction with two parts, corresponding to the probability of finding theatominState1andState2.Atthestartoftheexperiment,theatomswere100%inState1,andattheend,theywere100%inState2.Inbetween,theprobabilityofState2steadilyincreased,whiletheprobabilityofState1steadilydecreased.The experimentersmeasured the state of the ions using an ultraviolet laser

with its frequencychosenso thatan ion inState1wouldhappilyabsorb light,while ions in State 2 would not absorb any light. Ions in State 1 absorbedphotons fromthe laserand re-emitted thema fewnanoseconds later,makingabrightspotonacamerapointedat the ions.Ions inState2,ontheotherhand,produced no light when illuminated by the laser. The total amount of lightreaching thecamera, then,wasadirectmeasurementof thenumberof ions inState1.To demonstrate the quantum Zeno effect, the NIST group trapped a large

numberofions,allinState1.Thentheyturnedonthemicrowaves,waited256milliseconds, and pulsed on the laser. None of the ions produced any light,indicatingthat100%ofthesamplehadmovedtoState2,asexpected.Thentheyrepeated the experiment, with two laser pulses: one after 128 ms (halfwaythroughthemovetoState2),andoneafter256ms.Inthiscase,theysawhalfasmuchlightafter256ms, indicating thatonly50%of thesamplehadmade thetransitiontoState2.ThedecreasedprobabilityisexplainedbythequantumZenoeffect.Thelaser

pulsehalfwaythroughmeasuredthestateoftheions.ManyofthemwerefoundinState1,andthemeasurementdestroyedtheState2partofthewavefunction.Theseatomswerenow100%inState1,sothetransitionhadtostartoveragain,with the probability of State 2 increasing slowly. After another 128 ms, theprobabilityoffindingtheionsinState2wasonly50%.TheprobabilityofmovingfromState1toState2decreasedfurtherwithmore

measurements.Withfourpulses(at64,128,192,and256ms),only35%oftheatomsmadethetransition.Witheightpulses,only19%madethetransition.Witha totalof64 laserpulsesover the full experimental interval (oneevery4ms),fewerthan1%oftheatomsmadethetransition.Alloftheseprobabilitieswerein excellent agreement with the theoretical predictions of the quantum Zenoeffect,asshowninthefigurebelow.

“So,whenyoumakeameasurement,theionabsorbsaphoton,andthatcollapsesthewavefunction?”“Actually,theiondoesn’tneedtoabsorbaphotonatall.TheWinelandgroup

repeated the experiment starting with the ion in State 2. In that case, the ionstarts out in the ‘dark’ state, and doesn’t absorb any photons during themeasurements. They still got the same result—the probability of making thetransitionfromState2toState1decreasedwithmoremeasurements,exactlyaspredicted.”“Wait,notabsorbingaphotonisthesameasabsorbingaphoton?”

TheprobabilityofmakingatransitionfromonestatetoanotherinthequantumZenoeffectexperimentdoneby theWinelandgroup (W.M. Itano,D.J.Heinzen,J.J.Bollinger,andD.J.Wineland, Phys.Rev.A41,2295–2300[1990],modifiedandreprintedwithpermission).Blackbarsarethetheoreticalprediction, gray bars are the experimental result, with error bars showing the experimentaluncertainty.Theprobabilityof changing statesdecreasesas thenumberofmeasurements increases,whethertheionsstartinState1orState2.

“Whenitcomestothinkingofthephotonsasmeasurementtools,yes.It’sjustlikethetreatintwoboxes—ifyouopenoneoftheboxes,andfinditempty,youknowthetreathastobeintheotherbox.Thatdeterminesthestateofthetreatjustasifyouopenedtheboxandfoundatreatthere.”“It’snotasmuchfun,though,becauseIdon’tgetthetreat.”“Yes,well,yourlifeisverydifficult.”

The quantum Zeno effect does not depend on a particular interpretation ofquantummechanics.It’seasiertodiscusswhat’sgoingonusingtheCopenhagenlanguageofwavefunctioncollapse,butwecanequallywelldescribeitintermsofthemany-worldsinterpretation.Inthemany-worldspicture,newbranchesofthewavefunction appear at eachmeasurement step, butwe aremore likely toperceivethehigherprobabilitybranch.Theprobabilityofseeingastatechangeisthesameinbothinterpretations.Wecanuse thequantumZenoeffect todramatically reduce thechanceofa

systemchangingstates,simplybymeasuringitmanytimes.Wecannevermaketheprobabilityof transitionexactlyzero—there’salwaysasmallchance that itwill change in spite of the measurements—but we can make it very small,demonstratingthepowerofquantummeasurement.

“Humansaresosilly.Ifyouwanttostopthetransition,wouldn’titbeeasiertojustturnoffthemicrowaves?”

“Well,yes,butthepointistodemonstratethatthequantumZenoeffectisreal.It’snotinterestingbecauseitcanstopionsfromchangingstates;it’sinterestingbecauseofwhatittellsusaboutquantumphysics.”“Yes,butwhatgoodisit?Canitdoanythinguseful?”“Well,youcanuseitto

detectobjectswithouthavingthemabsorbanylight.”“Objects…likebunnies?”“Yes,hypothetically.”“Ilikethesoundofthat!”

MEASURINGWITHOUTLOOKING:QUANTUMINTERROGATION

The quantum Zeno effect can be exploited to do some amazing things. AcollaborationbetweentheUniversityof

Westartwithaphotonontheleft-handsideoftheapparatus,bouncingbackand forthbetween twomirrors.There is a small chanceof thephoton leaking through the centralmirror, soover time thephotonwill shift into theright-handsideof theapparatus. If there isanabsorbingobject (arabbit,say)ontheright-handside,though,itwillpreventthephotonfrommoving,throughthequantumZenoeffect.

Innsbruck and Los Alamos National Laboratory has demonstrated that it’s

possibletouselighttodetectthepresenceofanabsorbingobjectwithouthavingit absorb any photons, by using the quantum Zeno effect to stop a photonmovingfromoneplacetoanother.Inthefuture,thistechniquemaybeusedtostudythepropertiesofquantumsystemsthataretoofragiletosurviveabsorbingevenasinglephoton.Here’sasimplifiedversionofthisquantuminterrogationexperiment:imagine

that we have a single photon bouncing back and forth between two perfectmirrors. Halfway between those two, we insert a thirdmirror that’s not quiteperfect.The wavefunction for this system has two pieces, one corresponding to

findingthephotoninthelefthalfoftheapparatus,andtheothercorrespondingtofinding thephoton in therighthalf. Ifwestart theexperimentwithasinglephotoninthelefthalf,wefindthatovertime,itwillslowlymoveintotherighthalf. Each time the photon hits the imperfect central mirror, there’s a smallchancethatitgoesthrough,sotheleft-sidepieceofthewavefunctiongetsalittlesmaller, and the right-side piece gets bigger. Eventually, the left-side piece isreducedtozero,andthereisa100%chanceoffindingthephotonontherightside.Thentheprocessreversesitself.Thephotonwillslowly“slosh”backandforth between the two sides of the apparatus, just as the ions in the NISTexperimentmovedbetweenState1andState2.We can trigger the quantumZeno effect by adding a device tomeasure the

positionof thephoton,suchasarabbit in therighthalfof theapparatus.Eachtimethephotonhits thecentralmirror, therabbitmeasureswhetherthephotonpassed through the mirror: being very skittish, the rabbit will run away if itdetectsevenasinglephotonontherightside.The“sloshing”thathappensintheno-rabbitcaseisblockedbythequantum

Zeno effect when the rabbit is present. If the photon does pass through themirror, the rabbit absorbs it and flees. The photon no longer exists, so itswavefunction is zero, and nothing changes after that. If it doesn’t make itthrough, thephoton isdefinitelyon the left-handside,and thewavefunction isputbackintheinitialphoton-on-the-leftstate,andeverythingstartsoveragain.ThequantumZenoeffectletsusdowhatanydogwantsto:determinewhether

there’sarabbitintheapparatuswithoutscaringitoff.Westartwithaphotonontheleftside,waitlongenoughforittomoveovertotherightside,andthenlookat theleftsideof theapparatus.If there’snophotonthere, there’snorabbitontheright,eitherbecausetherabbitabsorbedthephotonandranoff,orbecausethereneverwasa rabbit and thephotonhas“sloshed”over to the right. If thephotonisstillintheleft-handsideoftheapparatus,weknowthatnotonlywastherearabbit,butitisstillthere,andhasnotabsorbedevenasinglephotonof

light.There isalwaysachance that thephotonwillmake it throughandscare the

rabbitaway,butwecanmakethischanceaslowaswelike,bydecreasingtheprobability that the photon will leak through the mirror. We’ll have to waitlonger to complete the measurement, as the time required for the photon to“slosh”intotherightsidewillincrease,butthechancesofsuccessfullydetectingtherabbitimprovedramatically.Ifthephotonneedstobouncebackandforthonthe left-hand side 100 times before it “sloshes” to the right, the probability ofdetectingarabbitwithoutscaringitoffis98.8%.Ifyourepeatedtheexperiment1,000times,only12rabbitswouldbescaredoff.

“Great!So,allIneedtodoisgetsomebigmirrors…”“No.Youarenotsettingthisexperimentupinthebackgarden.”“ButIcanusethequantumZenoeffecttosneakuponthebunnies…”“No.Just…No.Youarenotputtinggreatbigmirrorsacrossthegarden,and

that’sfinal.”“Awww…”

Quantum interrogation hasn’t been used to catch rabbits, but it has beendemonstrated experimentally using polarized photons, by physicists inInnsbruck, LosAlamos, and Illinois. Quantum interrogation allows you to dosome incredible things—takingpicturesofobjectswithouteverbouncing lightoff them, forexample.Thisprobably isn’tuseful for spypurposes (unlessyoucan somehow get your enemies to obligingly store their secrets between twomirrors),butitmightbeessentialforprobingfragilequantumsystemslikelargecollectionsofatomsinsuperpositionstatesthatcan’tsurvivetheabsorptionofaphoton.Whether you think of it in terms of collapsing wavefunctions, or a single

expandingwavefunctionundergoingdecoherence,thequantumZenoeffectisadramaticdemonstrationof thestrangenatureofquantummeasurement.Unlikeclassicalmeasurement,theactofmeasuringaquantumsystemchangesthestateof that system, leaving it in only one of the allowed states, which is verydifferent than what we expect classically. With a clever arrangement of theexperimentalsituation,thiscanbeexploitedtopreventasystemfromchangingstates,oreven toextract information fromasystemwithout interactingwith itdirectly.“That’sreallyinteresting.Weird,butinteresting.”“Thanks.”“Now,ifyou’llexcuseme,Ineedtogoandlookinmybowl.”

“Whyisthat?”“Well,I’mgoingtousetheZenoeffecttogetmorefood.Ireckon,ifIkeep

measuringmybowltobefullofdogfood,I’llalwayshavefood,nomatterhowmuchIeat.Thatwillbefun.”“Of course, if you keep measuring your bowl to be empty, it’ll always be

empty,andyou’llneverhavefood.”“Oh.Thatwouldbebad.Ididn’tthinkofthat.”“Anyway,you’dneedtohavesomenaturalquantumprocessthatcausedfood

to appear in the bowl for that to work. Things aren’t going to appear for noreason,justbecauseyouwanttomeasurethem.”“Well, you sometimes put food in my bowl, right? And you’re a natural

process.”“Inamannerofspeaking.”“So,howaboutputtingsomefoodinmybowl?”“Oh,allright.It’salmostdinnertime.Comeon.”“Oooh!Dinner!”

*WhilethesummingofinfiniteseriesisacceptedastheresolutionofZeno’sparadoxbyphysicistsand

engineers and most mathematicians, some philosophers do not accept this as a sufficient resolution ofZeno’sparadox(StanfordEncyclopediaofPhilosophy).Thisjustprovesthatphilosophersaremadderthanmathematicians,orevencats.*AbetterGreekliteraryallusionmightbethemythofSisyphus,whowascondemnedtospendeternity

pushingaboulderupahill,onlytohaveitslipfreeandrollbacktothebottomagain.Thename“Sisyphuseffect”wasusedforsomethingelse,though,sothisiscalledthequantumZenoeffect.*Aquarter of a second seemspretty fast to humansor dogs, but it’s really slow for an atom.Atoms

usuallychangestatesinafewbillionthsofasecond.

CHAPTER6

NoDiggingRequired:

QuantumTunneling

We’resittinginthegarden,enjoyingabeautifulsunnyafternoon.I’mlyingonadeckchairreadingabook,andEmmyissprawledoutonthegrass,baskinginthesunandkeepinganeyeoutforsquirrelincursions.“CanIaskaquestion?”sheasks.“Hmm?Okay,goahead.”“Whatdoyouknowabouttunneling?”“Tunneling, eh?” I put my book down. “Well, it’s a process by which a

particlecangettotheothersideofabarrierdespitenothavingenoughenergytopassoverthebarrier.”“Barrier?Likeafence?”“Well,metaphorically,atleast.”“Likethefencebetweenthisgardenandthenext?”Shelooksreallyhopeful.“Oh.Isthatwhatthisisabout?”“There are bunnies over there!” Shewags her tail for aminute, then looks

crestfallen.“ButIcan’tgettothem.”“True,butIdon’t thinktunnelingis theanswer.Itworksforsmallparticles,

butwouldn’tworkforadog.”“Whynot?”“Well,youcanthinkofabarrierintermsofpotentialandkineticenergy.For

example, right now, all your energy is potential energy, because you’re notmoving.But you could startmoving, say, if you took off after a squirrel, andturnedthatpotentialintokineticenergy.”“I’mveryfast.Ihavelotsofenergy.”“Yes,Iknow.You’reagreattrialtous.Anyway,whetheryou’resittingstill,

ormoving,youhavethesametotalamountofenergy.It’sjustaquestionofwhatformit’sin.”

“Okay,butwhatdoesthishavetodowiththefence?”“Well,youcanthinkof thefenceasbeingaplacewhereyoucanonlygoif

youhaveenoughenergy.Foryoutobeatthespotwherethefenceis,youwouldhave to jumpveryhighorelseoccupy thesamespaceas thefence,andeitherwouldtakeanawfullotofenergy.”“Ican’tjumpthathigh.That’swhyIcan’tgetthebunnies.”“Right.Youdon’thaveenoughenergytogetoverthefence.Andbecauseyou

don’t have enough energy, you can’t end up in the neighbors’ garden, andeverybodyismuchhappierthatway,believeme.”“Exceptme.”Shepouts.“Yes, well, except you.” I scratch behind her ears by way of apology.

“Anyway,quantummechanicspredictsthateventhoughyoudon’thaveenoughenergytogooverthefence,there’sstillachancethatyoucouldendupontheother side. You could just sort of… pass through the fence, as if it weren’tthere.”“Likethebunniesdo!”“Pardon?”“Thebunnies.Theygobackandforththroughthefenceallthetime.”“Yes,well,that’sbecausetheyfitbetweenthebarsofthefence.Ithasnothing

todowithquantumtunneling.”Istopforamoment.“Ofcourse, it’snotabadanalogy.Thebunniesdon’thaveenoughenergytogooverthefence,either,butthey can go through it, and end up on the other side. Which is sort of liketunneling.”“SohowdoItunnelthroughthefence?”“Well,youcouldeatfewertreats,andbecomethinenoughtopassbetweenthe

barslikethebunniesdo.”“Idon’tlikethatplan.I’magooddog.IdeservethetreatsIget.”“And you get the treats you deserve. The other option would be quantum

tunnelingthroughthefence,butquantumtunnelingisn’tsomethingyoudo,it’ssomethingthatjusthappens.Ifyousendawholebunchofparticlesatthebarrier,a small number of them will show up on the other side. Butwhich ones gothroughiscompletelyrandom.It’sallaboutprobability.”“So,Ijustneedtorunatthefenceenoughtimes,andI’llendupontheother

side?”“I wouldn’t try it. The probability of a particle tunneling through a barrier

depends on the thickness of the barrier and the quantum wavelength of theparticle. The probability of a fifty-pound dog passing through a half-inchaluminiumbarrierwouldbesomethinglikeoneoveretothepoweroftentothethirty-six.Doyouknowwhatthatis?”

“What?”“Zero.Ornearenoughtomakenodifference.Sodon’tgothrowingyourself

atthefence.”She’squietforaminute.“Anyway,Ihopethatanswersyourquestion.”Ipickmybookbackup.“Sortof.”“Sortof?”“Well, the quantum stuff was interesting, and all, but I was thinking of

classicaltunneling.”“Classicaltunneling?”“Iwasgoingtodigaholeunderthefence.”“Oh.”“It’sagoodplan!”Shewagshertailenthusiastically,andlooksverypleased

withherself.“No,it’snot.Onlybaddogsdigholes.”“Oh.”Hertailstops,andherheaddroops.“ButI’magooddog,right?”“Yes,you’reaverygooddog.You’rethebest.”“Rubmybelly?”Sheflipsoveronherback,andlookshopeful.“Oh,okay…”Iputmybookbackdown,andleanovertorubherbelly.

“Tunneling” is one of the most unexpected quantum phenomena, where aparticleheadedatsomesortofobstacle—say,adog running towarda fence—will pass right through it as if it weren’t there. This odd behavior is a directconsequenceoftheunderlyingwavenatureofquantumparticlesseeninchapter2.In thischapter,we’ll talkabout theessentialphysicsconceptofenergy, and

how energy determineswhere particles can be found.We’ll see that thewavenature of matter allows quantum particles to turn up in places that classicalphysicssaystheycan’treach,passingintooreventhroughsolidobjects.We’llalso see how tunneling lets scientists build microscopes that can study thestructureofmatter,makingpossiblerevolutionarydevelopmentsinbiochemistryandnanotechnology.

THEABILITYTOGETTHINGSDONE:ENERGY

Inorder toexplainquantumtunneling,weneed tofirst talkabout theclassicalphysicsofenergy.While theterm“energy”haspassedfromphysics intomoregeneral use, its physics meaning is slightly different from its everyday,

conversationaluse.A one-sentence definition of the term “energy” in physics might be: “The

energycontentofanobjectisameasureofitsabilitytochangeitsownmotionor the motion of another object.” An object can have energy because it ismoving,orbecauseitisheldstationaryinaplacewhereitmightstartmoving.Everyobjecthassomeenergysimplybecauseithasmass(Einstein’sE=mc2)andbecauseitstemperatureisaboveabsolutezero.*Alloftheseformsofenergycanbeusedtosetastationaryobjectintomotion,ortostopordeflectanobjectthatismoving.Themostobviousformofenergyiskineticenergy,theenergyassociatedwith

amovingobject.Thekineticenergyofanobjectmovingataneverydaysortofspeed is equal to half its mass times the velocity squared, or as it’s usuallywritten:

KE=½mv2

Kinetic energy is always a positive number, and increases as you increaseeitherthemassorthespeed.AGreatDanehasmorekineticenergythanalittleChihuahuamoving at the same speed,while a hyperactiveSiberian husky hasmore kinetic energy than a sleepy old bloodhound of the samemass. Kineticenergy is similar to momentum, but it increases faster as you increase thevelocity,andunlikemomentum,itdoesn’tdependonthedirectionofmotion.Objectsthatarenotalreadymovinghavethepotentialtostartmovingdueto

interactions with other objects.We describe this as potential energy. A heavyobjectonatablehaspotentialenergy:it’snotmoving,butitcanacquirekineticenergyifahyperactivedogbumpsintothetableanditfallsonher.Twomagnetsheldclosetoeachotherhavepotentialenergy:whenreleased,they’lleitherrushtogetherorflyapart.Adogalwayshaspotentialenergy,evenwhensleeping:attheslightestsound,shecanleapupandstartbarkingatnothing.Energyisessentialtophysicsbecauseit’sa“conservedquantity”:thelawof

conservationofenergysaysthatwhileenergymaybeconvertedfromoneformtoanother, thetotalamountofenergyinagivensystemdoesnotchange.Thisturns some difficult problems into bookkeeping exercises: the total energy(kinetic plus potential) has to be the same at the endof the problemas at thebeginning,sowhateverenergyisleftoverwhenyousubtractthefinalpotentialfromthetotalhastobekineticenergy.*To get a better feel for how energy works, let’s think about a concrete

example:aballthrownupintheair.Asanydogknows,whatgoesupmustcomedown, and a ball that’s thrown upwith some initial velocitywill slow down,

stop,andthenfallbackdown.Youcansee this in thefigureonthenextpage,whichshowstheheightofaballataseriesofregularlyspacedinstants.Atlowheights, theball ismovingfast,andcoversalotofgroundfromonepicturetothe next.Near the top of its flight, the ballmoves very little, and at the verypeak,it’sperfectlystillforasplitsecond.Wecandescribe this flight in termsof energy.An instant after the toss, the

ballismoving,soithaslotsofkineticenergy,butit’sneartheground,andhasnopotentialenergy.Thetotalenergyisthusequaltothekineticenergy.Wecanthinkofthisasbeingakindofenergysupply,likeajarfulloftreats,shownbythe black bar in the figure. As the ball moves upwards, its kinetic energydecreases (because it’s notmoving as fast), and its potential energy increases(becauseit’shigherofftheground).Thekineticenergyleveldrops,replacedbypotentialenergy(showningray),butthetotalenergyremainsthesame.

Aballthrownupintheairstartsoutmovingrapidlyupwards,slowsduetogravity,andturnsaroundandfallsback.Thepicturesshowthepositionoftheballatregularintervals.Thebarsshowtheenergyoftheball,withblackindicatingkineticenergyandgrayindicatingpotential.Neargroundlevel,alloftheenergyiskineticenergy,whileatthepeakofitsflight,alloftheenergyispotentialenergy.

Atthepeakof its flight, theballhaspotentialenergy,butnokineticenergy,becauseforasplitsecond,it’snotmovingatall.Onthewaybackdown,itgoesthroughthesameprocessinreverse:itstartswithpotentialenergybutnokineticenergy,andendsupwithkineticenergy(thesameamountitstartedwith)butnopotentialenergy.

“You’vegotthisbackwards,youknow.”“Ido?”“Yes,thejarfulloftreatsshouldbethepotentialenergy,becausetreatshave

thepotentialtogivemeenergy,whenIeatthem.Theemptyjarshouldbekinetic

energy,becauseIrunallovertheplaceafterIeattreats.”“Youmayhaveapointthere.Ofcourse,noanalogycomparingenergytodog

treatsisevergoingtobeperfect.”“Whyisthat?”“Becausewhileyou can convert potential energy tokinetic energy, you can

also convert kinetic energy back to potential energy. Which would be likeputtingtreatsbackinthejar.”“Oh.Ineverdothat.”“Believeme,we’venoticed.”

Wealsoseefromlookingattheenergyofaballinflightthatenergylimitsthemotionof theball.Theball startswithsome totalenergy,allkinetic,andas itgoesup, itconverts that topotentialenergy.Oncetheinitialkineticenergyhasbeen turned into potential, the ball stopsmoving. The ball can’t go beyond acertainmaximumheight,becausethatwouldrequireitstotalenergytoincrease,and that can’t happen.* Themaximum height the ball can reachwith a givenamount of energy is called the “turning point,” because the ball reversesdirectionatthatpoint.Heightsabovetheturningpointare“forbidden,”becausetheballdoesn’thaveenoughenergytoreachthem.

FOLLOWTHEBOUNCINGWAVEFUNCTION:

AQUANTUMBALL

Thethrownballisasimpleexampleofenergyinaction,andthinkingaboutitinenergytermsmaynotseemthathelpful.Energyanalysiscanbeappliedtomanymore complicated systems, though, including situations that can only bedescribedmathematically using energy.As a result, energy is one of themostimportanttoolsthatphysicistshaveforunderstandingtheworld.Energy is especially important in talking about quantummechanics. Aswe

saw in chapter 2, quantum particles do not have a well-defined position orvelocity, so there’s noway to keep track of those properties aswe dowith aclassical system. Conservation of energy still applies, though, so we canunderstandquantumsystemsbylookingattheirenergy.Infact,theSchrödingerequation uses the potential energy of a quantum object to predict what willhappentothewavefunctionofthatobject,soeverycalculationdoneinquantummechanicsisfundamentallyaboutenergy.Wecanseehowenergyrelatestowavefunctionsbyimaginingaquantumball

thrownintheair.Wecanpredictsomefeaturesofitswavefunctionjustbyusing

whatweknowaboutitsenergy.Kineticenergyissimilartomomentum,andweknowfromchapter1(page10)thatthemomentumdeterminesthewavelength.Near the ground, where the kinetic energy is high, the ball should have highmomentumand thus thewavefunctionshouldhaveashortwavelength.Higherup, where the ball is moving slowly, the ball has low momentum, and thewavefunctionshouldhavea longerwavelength.Wealsoexpect theprobabilityof finding theball above the turningpoint to be zero, because theball shouldnevergohigherthanallowedbyitsinitialenergy.Wecancalculatethewavefunctionforthissystem,andwefindaprobability

distributionthatlookslikethis:

Thesolidcurveshowstheprobabilitydistributionforfindingaquantumparticleatagivenheight.Thedashedcurveistheprobabilitydistributionforaclassicalparticle.

Lookingat thisgraph,weseemoreor lesswhatweexpect.Theprobabilitydistributionoscillatesmorerapidlyatlowelevation(ontheleft),thanathigherelevation. On closer inspection, though, we notice something strange: theprobabilitydoesnotgotozeroexactlyattheclassicalturningpoint(atabout17unitsofheight,where thedashedcurvegoes tozero). Itdropsoff tozero,butthere’sa rangeofheightsabove the turningpointwhere theprobability is stillsignificant.There’ssomeprobabilityoffindingtheballatheightsitshouldneverbeabletoreach!Whydoesn’t theprobabilitygo tozero rightat the turningpoint?Well, if it

did,therewouldbeasuddenchangeinthewavefunctionatthatpoint.Weknowfrom chapter 2 (page 47), though, that making a sudden change in thewavefunction requires adding together a huge number of wavefunctions withdifferent wavelengths. Many wavelengths means a large uncertainty in themomentum, and therefore a large uncertainty in kinetic energy. But we don’t

havealargeuncertaintyintheenergy—weknowhowhardwethrewtheball.Asmallenergyuncertaintyleavesuswithalargeuncertaintyinthepositionoftheturning point, which means no sharp changes in the wavefunction and awavefunctionthatextendsintotheforbiddenregion.Theballcan’tbothhaveawell-definedenergyandturnaroundexactlywhere

classicalphysicssaysitshould.Ifwewantasmalluncertaintyintheenergy,wehavetoacceptmoreuncertaintyintheposition,andthatmeansthattherewillbesomechanceoffindingtheballathigherelevationthanclassicalphysicsallows.

“Whydothewigglesgetbiggernearthetop?”“Ialreadysaidthat.Theballismovingslower,sothewavelengthgetslonger.”“Yes,buttheygettaller,too.”“Oh,that.That’salsobecausetheballisslowingdown.Theballspendsmore

time near the top of its flight where it’smoving slowly than it does near thebottomwhereit’smovingfast,sothere’sahigherprobabilityoffindingitnearthe top. You see the same thing with a classical ball, if you work out theprobabilitydistribution—that’sthedashedcurve.”“So,wait,themostlikelypositionfortheballiswayupintheair?”“Yes.Youcanseethatbylookingatthefigureonpage125showingtheball

inflight.Therearealotmorepicturesoftheballathighelevationsthanatlowones.That’swhywhenweplayfetch,youusuallycatchtheballwhenit’satthetopofitsflight—it’snotmovingveryfast,soit’seasiertoget.”“I’mverygoodatcatchingthings.Weshouldgoandplayfetch!That’sfun!”“Afterwefinishthischapter,okay?Ihaven’ttalkedabouttunnelingyet.”“Oh, right.Let’s talk about tunneling. Passing through solid objects is even

morefunthanfetch.”“Don’tgetyourhopesuptoomuch…”

LIKEIT’SNOTEVENTHERE:

BARRIERPENETRATIONANDTUNNELING

Howdoesuncertaintyintheturningpointleadtoparticlespassingthroughsolidobjects?Well, the inside of a solid object is a forbidden region—interactionsbetween the atoms making up the two objects make the potential energyenormousforoneobjectinsideanother.It’salittleliketryingtostickaseconddogintoakennelthatalreadycontainsoneunfriendlydog—you’llhaveahardtime getting the second dog in there, and if you do, you’ll see a lot of extra

energy,intheformofgrowlingandbarkingandsnapping.In quantum mechanics, though, wavefunctions can extend into forbidden

regions, and that works even for solid objects—there’s a tiny probability offinding one object inside another. Better yet, if the forbidden region is verynarrow, quantummechanics predicts a small probability of one object passingthroughtheother,eventhoughitdoesn’thaveenoughenergytomakeitintotheforbiddenregion,letalonetotheotherside.Thesimplestexampleofthisisanelectronhittingathinpieceofmetal,where

the potential energy ismuch higher. Classical physics tells us that the kineticenergyofanelectronoutsidethemetaldetermineswhathappenswhenitreachestheedgeofthemetal.Iftheelectron’skineticenergyislarge,itcanconvertmostofitsenergytopotential,andstillhavekineticenergylefttomovethroughthemetal. If the kinetic energy outside is less than the potential inside themetal,though,there’snowaytheelectroncanenterwithoutincreasingitstotalenergy.The edge of themetal becomes a turning point, and themetal is a forbiddenregion:anelectroncominginfromtheleftbouncesoffthesurfaceandgoesbackwhereitcamefrom.Anelectroncominginfromtherightbouncesofftheothersurfaceinthesameway.Accordingtoquantummechanics,though,wecan’thaveasharpturningpoint

at the edgeof themetal.Aswith the thrownball, the electron’swavefunctionextendsintotheforbiddenregionwherethepotentialenergyisgreaterthantheenergy of the incoming particles. There’s some probability of finding theelectronsinsidethemetal,eventhoughclassicalphysicssaysit’sforbidden.Theprobabilityoffindingaparticleintheforbiddenregionishighestneartheedge,and decreases rapidly as youmove farther in. If the forbidden region extendsoveralongenoughdistance,theprobabilitydropstozero,*andthat’stheendofit.For a very narrow barrier, though, there is some probability of finding the

electronsattheoppositeedgeoftheforbiddenregionfromwheretheyentered.Beyondthatpoint,they’renolongerforbiddentobethere—they’rebackoutinemptyspace,andmoveoffwiththesameenergytheyhadatthestart.Somebodywatching theexperimentwould seea tiny fractionof the incomingparticles—oneinamillion,say—simplypassthroughthebarrierasifitweren’teventhere.Thisiscalledtunneling,becausetheelectronshavepassedtheforbiddenregioneventhoughit’simpossibleforthemtobeinsideit.Inasense,they’veduckedunderthebarrier,likeabaddogwhotunnelsunderafence.Thewavefunctionfor thissituation isshownabove.Onthe left,wehavean

incomingelectronwithsomemomentumandenergy,representedbyawavewithawell-definedwavelength.†Whentheelectronreachestheedgeofthemetal,it

enterstheforbiddenregion,andtheprobabilitydecreasesrapidly.Itdoesn’tgetto zero before it reaches the right edge of the forbidden region, though, so itemergesthereasanotherwavewiththesamewavelengthasontheleft.

The probability distribution for an electron coming in from the left, hitting a barrier where thepotentialenergyofanelectronwouldincreaseabovethetotalenergyoftheelectron.Theprobabilitydropsoffrapidlyinsidetheforbiddenregion,butdoesnotreachzero,sothereissomeprobabilityoffindingtheelectrontotherightofthebarrier.

The smaller height of thewave to the right of the barrier indicates that theprobabilityoffindingtheelectronontherightismuchlowerthantheprobabilityoffindingitontheleft.Theprobabilityoftunnelingdecreasesexponentiallyasthe barrier thickness increases—if you double the thickness, the probability ismuchlessthanhalfoftheoriginalprobability.Ontheotherhand,astheenergyof the incoming electrons increases, they penetrate farther into the forbiddenregion,andtheprobabilityofonemakingitallthewaythroughincreases.

“Sotheelectronsjustdrillholesthroughthebarrier?”“No,theypassthroughitasifitweren’tthereatall.Theydon’thaveenough

energytopunchthrough.”“Buthowdoyouknowthat?”“Well, the electrons show up on the far side of the barrierwith exactly the

same energy as before they hit it. If theywere boring little holes through thebarrier,theywouldlosesomeenergyintheprocess,andwe’dbeabletodetectthat.”“Maybethey’rejustreallytinyholes?”“No,wecanlookatthatwithascanningprobemicroscope,andtherearen’t

holes.”“What’sascanningprobemicroscope?”“That’sanexcellentandveryconvenientquestion…”

FEELINGSINGLEATOMS:

SCANNINGTUNNELINGMICROSCOPY

Tunneling has a more direct technological application than most of the otherweirdquantumphenomenawe’vediscussed.Tunnelingisthebasisforadevicecalledascanningtunnelingmicroscope(STM),whichuseselectrontunnelingtomake images of objects as small as a single atom. The STMwas invented in1981at IBMZurich,andhasbecomeanessential tool forpeoplestudying theatomic structure of solid materials. Its inventors, Gerd Binnig and HeinrichRohrer,wontheNobelPrizeinPhysicsin1986.AnSTMconsistsofasampleofelectricallyconductingmaterialandavery

sharpmetal tipbroughtwithina fewnanometersof the surfaceof the sample.Thetipisheldataslightlydifferentvoltagethanthesample,soelectronsinthetipwanttomovefromthetipintothesample.Theelectronscan’tflowdirectlyfromthetipintothesample,though,becausethesmallgapbetweenthetipandthesampleactsasabarrierpreventingthemovementofelectrons.*If the gap between the tip and the sample is small enough, though—a

nanometerorso—there’ssomechancethatelectronswilltunnelfromthetiptothe sample. That produces a small current, which can be measured. Thetunnelingprobability(andthusthecurrent)increasesdramaticallyasthetipgetsclosertothesurface,sochangesinthecurrentcanbeusedtodetecttinychangesinthedistancebetweenthetwo—changessmallerthanthediameterofasingleatom.Makingan imagewithanSTMis like runningyour fingeracrossa surface,

and feeling thebumpsandscratches.Youscan the tipbackand forthover thesurfaceof thesample,keeping theheightof the tipconstant,andasyoumovethe tip, youmonitor the current flowing between the tip and the sample. Thecurrent increaseswhenever there’s a small bump sticking up from the surfacemakingiteasierforelectronstotunnelacross,anddecreaseswheneverthere’sasmalldip in thesurface.Ifyoutakea largenumberofheightmeasurementsatpointsonagrid,youcanputthemtogethertocreateanimageoftheindividualatomsmakingupthesurfaceofyoursample.

Aschematicofa scanning tunnelingmicroscope.A sharp tip ispositionedclose to the surfaceofamaterial,andmovedbackandforthinaregularpattern.Electronsfromthetipwilltunnelacrossthegapbetweenthetipandthesurface,producingasmallelectriccurrentthatisamplifiedandmeasured.The amount of current depends very sensitively on the distance between the tip and the surface,allowingareconstructionofthesurfacesensitiveenoughtodetectsingleatoms.

Notonlycanyouseesingleatoms,butifyoubringthetipintodirectcontactwiththesurface,youcanpushindividualatomsaround.Scientistshaveusedthisability to make a number of incredible structures, such as the oval-shaped“corral”showninthepictureonthenextpage,madeatIBM’sAlmadenresearchlaboratory. The bumpsmaking up the “corral” are individual iron atoms on acoppersurface,whichhavebeendraggedintoplacebytheSTM.Suchstructurescan be used to study the quantum behavior of electrons inside the “corral,”whichaccountsforthewavyfeaturesseenonthecoppersurface.Scanning tunnelingmicroscopeshave revolutionized the studyof solids and

surfaces,andthetechnologymayleadtonewmanufacturingtechniquesfortinydevices. Other scientists have used STMs to study and manipulate individualstrands of DNA, providing a more detailed understanding of the behavior ofgeneticmaterialandpossiblynewdrugsortreatmentsforgeneticdamage.Allofthisismadepossiblebytheunderlyingwavenatureofmatter.

Iron atoms on a copper surface, arranged in a “corral” pattern using an STM. The wave patterninsidethecorralisduetothewavenatureofelectronsonthecoppersurface.ImagecourtesyofIBM.

“That’sniceandeverything,butI’mnotinterestedinmicroscopicbunnies.Whathasquantumtunnelingeverdoneforme?”“Well,foronething,youwouldn’tbeabletoenjoyanicesunnydayifnotfor

tunneling.”“Whatdoyoumean?”“Well,theSunshinesbecauseoffusionreactionsinthecore,right?”“Everybodyknowsthat.Eventhebeagledowntheroadknowsthat,andthat

dogisreallystupid.”“Yes. Well. Anyway, fusion works by sticking protons together to make

helium fromhydrogen.Becauseprotons arepositively charged, they repeloneanother,andthatrepulsionsetsupabarrier.AndashotastheSunis,theprotonsintheSunstilldon’thaveenoughenergytogetoverthatbarrierdirectly.”“Sotheytunnelthrough?”“Exactly.Theprobabilityofanygivenprotontunnelingthroughthebarrieris

prettylow,buttherearelotsandlotsofprotonsintheSun,andenoughofthemdotunnelthroughtokeepthereactiongoing.Soit’sreallytunnelingthatletstheSunshine.”“Hmm.Isupposethatisprettyimpressive.”“I’msogladyouapprove…”“Canwegoandplayfetch,now?”

*Temperaturemeasurestheenergyduetothemotionoftheindividualatomsmakingupanobject,and

“absolute zero” is the imaginary temperature at which that motion would cease. No real object can be

cooledallthewaytoabsolutezero,though,andevenifonecoulditwouldstillhavezero-pointenergy,asdiscussedinchapter2(page49).* Potential energy is generally much easier to calculate than kinetic energy. Potential energy usually

dependsonlyonthepositionsoftheinteractingobjects,whilethekineticenergydependsonthevelocity,whichdependsonwhathashappened in the recentpast.Theeasiestway to tackleanenergyproblem isusually to calculate the potential energy using the position, and find the kinetic energy by process ofelimination.Forexample,whenarollercoasterpausesatthetopofabighill,weknowthatallofitsenergyispotentialenergy.Lateron,wecaneasilycalculatethepotentialenergyfromtheheightofthetrack,andthatletsusfindthekineticenergy(andthusthespeed)withoutneedingtoknowwhathappenedinbetween.*Thetotalenergyofanobjectcanbeincreased,byaddingenergyfromsomeothersource,inthesame

way that adog’s treat jar canbe refilledby a friendlyhuman.Theextra energydoesnot come for free,though—theenergyoftheoutsideobjecthastodecrease,inthesamewaythatahuman’sbankbalancewilldecrease in order to supply the treats. The total energy of the entire universe—balls, dogs, treats, andhumans—isaconstant,andhasnotincreasedordecreasedinthefourteenbillionyearssincetheBigBang.* Strictly speaking, the probability is never exactly zero—the mathematical function describing the

probabilityisanexponential,andwhileitgetsclosertozeroastheelectronmovesintothebarrier,itnevergetsallthewaythere.Quantumphysicspredictsatinyprobabilitythataballthrownintheairwilltunnelthroughtheforbiddenregionthatstartsatitsclassicalmaximumheight,andendupontheMoon.That’snota good bet, though—the probability is so small that it’s indistinguishable from zero, for all practicalpurposes.†Noticethattheuncertaintyinthepositionisverylarge—theelectroncouldbejustaboutanywhereto

theleftoftheforbiddenregion.*Apotentialenergybarrierdoesn’thavetobeasolidphysicalobject.Anairgapwilldojustfine,which

iswhyyoucan’tmakealightbulblightupbyjustholdingitclosetothesocket.

CHAPTER7

SpookyBarkingataDistance:

QuantumEntanglement

Emmy is napping in the living room, but wakes up as I pass through. Shestretcheshugely,thenfollowsmeintothekitchenlookingpleasedwithherself.“I’mgoingtomeasureabunny,”sheannounces.“Begyourpardon?”She’salwaysmakingtheseweirdannouncements.“I’ve figuredouthow tomeasureboth thepositionand themomentumofa

bunny.”“Youhave,haveyou?Howareyougoingtodothat?”“I’mgoing toputabiggridof lines in thebackgarden, and thenwhen the

bunny is righton topof agridmark, all Ihave todo ismeasurehow fast it’sgoing.”Shewagshertailproudly.“Right…Andhowareyougoing tomeasurewhen thebunny is rightona

gridmark?”“Whatdoyoumean?I’mjustgoingtolook.”“Okay,whichmeans you’ll see the bunny, and the bunnywill see you, and

thenitwillchangeitsvelocitytorunaway.”“Oh.”Hertaildroops.“Ididn’tthinkofthat.”“Look, we’ve been through this. There’s no way around the uncertainty

principle.Reallycleverhumanshavetriedtofindawayaroundit,anditcan’tbedone.EinsteinspentyearsarguingaboutitwithNielsBohr.”“Didhecomeupwithanything?”“He tried lotsofdifferentarguments,butnoneof themactuallyworked.He

even had a really clever argument that quantum mechanics was incomplete,involvingtwoentangledparticles,preparedsothattheirstatesarecorrelated.”“Correlatedhow?”“Well, let’s say I have two treats inmy hand—stop drooling, it’s a thought

experiment—andoneofthemissteak,andtheotherischicken.”

“Ilikesteak.Ilikechicken.”She’sdroolingalloverthefloor.“Yes,Iknow.Thoughtexperiment,remember?”Igrabsomepapertowelsto

mopupthefloor.“Now,imagineIthrowthesetwotreatsinoppositedirections,onetoyou,andonetosomeotherdog.”“Don’tdothat.Otherdogsdon’tdeservetreats.”“It’sahypothetical,trytokeepup.Now,ifyougotthesteaktreat,youwould

knowimmediately that theotherdoggot thechicken treat.And—whyareyoulookingallsad?”“Ilikehypotheticalchickentreats.”“Yougotahypotheticalsteaktreat.”“Oooh!Ilikehypotheticalsteak.”“The point is, bymeasuringwhat sort of treat you got, you knowwhat the

othertreatis,withoutevermeasuringit.”“Yes,so?What’sweirdaboutthat?”“Well,inthequantumversion,thestateoftheparticlesisindeterminateuntil

oneofthemismeasured.WhenIthrowthetreats,untilyougetoneandfindoutwhetherit’ssteakorchicken,it’snoteither.Insomesense,it’sboth.”“Chickensteak!Steakchicken!Sticken!”“You’reridiculous.Anyway,Einsteinthoughtthiswasaproblem,andthatthe

fact that you could predict the state of one particle by measuring the otherparticlemeantthatbothofthemhadtohavedefinitestatesthewholetime.”“Thatmakessense.”“In a classical world, yes. Einstein’s argument fails, though, because he’s

assumedwhat’scalled‘locality’—thatmeasuringoneparticledoesnotaffecttheother. In fact, measuring the state of one determines the state of the other,absolutelyandinstantaneously.”Shelooksreallybotheredbythis.“Idon’tlikethatidea.Wouldn’tthatrequire

amessagetotravelfromonetreattotheother?”“That’swhatbotheredEinstein,andhecalleditspukhafteFernwirkung.”“‘Spookyactionatadistance’?”shetranslates.“SincewhendoyouknowGerman?”“Justlookatme.”Sheturnssidewaysforasecond,showingoffherblackand

tancoloringandpointednose.“Germanshepherd,remember?”“Of course, how silly of me. Anyway, yes, this bothered Einstein because

informationcannotpassbetweenseparatedobjectsfasterthanthespeedoflight.Butquantummechanicsisnonlocal,andtheentangledparticlesactlikeasingleobject.AmannamedJohnBellshowedthat it’spossible toput limitsonwhatyoucanmeasureintheorieswheretheparticleshavedefinitestates,andshowedthat those limits are different than the limits for entangled quantum particles.

Peoplehavedone the experiments and found that thequantum theory is right.Thestateoftheparticlesreallyisindeterminateuntilthey’remeasured.”“SoEinsteinwaswrong?”“Aboutthis,yes.Andgenerally,aboutthebasisofquantumtheory.”“Buthewasreallyclever,wasn’the?”“Yes.Einsteinwas arguably cleverer thanBohr.Bohrwonall their debates,

though, because he had the advantage of being right.” I bend over to scratchbehindherears.“You’reprettyclever,butyou’renoEinstein.”“I’m,like,thecanineEinstein,though,right?”“Allrightthen.AsfarasIknow,you’retheEinsteinofthedogworld.”“CanIhavesomesteak,then?Orchicken?”“Maybe.” Igraba treatoutof the jaron thecounter. “You’ll findoutwhen

youmeasureit.”I throwthetreatoutof thebackdoor,andshegoesboundingafterit.“Great!Indeterminatetreats!”

Everythingwe have talked about so far has been a one-particle phenomenon.Mostoftheexperimentsneedtoberepeatedmanytimestoseetheeffects,usingdifferentindividualparticlespreparedthesameway,butatafundamentallevel,all the interference, diffraction, and measurement effects we’ve talked aboutworkwithoneparticleatatime.Eachparticleinaninterferenceexperimentcanbe thought of as interferingwith itself, andmeasurement phenomena like thequantumZenoeffectinvolvethestateofasingleparticle.*Ofcourse,theworldweliveininvolvesagreatmanyparticles,soweneedto

look at what happens when we apply quantum physics to systems involvingmore thanoneparticle.Whenwedo, it’sno surprise thatwe find someweirdthingsgoingon,startingwiththeideaof“entangledstates.”Inthischapter,we’lllookattheideaof“entangled”particles,whosestatesare

correlatedsothatmeasuringoneparticledeterminestheexactstateoftheother.EntangledparticlesarethebasisforthemostseriouschallengeEinsteinmountedagainst quantum theory, known as the Einstein, Podolsky, and Rosen (EPR)paradox. We’ll talk about John Bell’s famous theorem resolving the EPRparadox, and its disturbing implications for the commonsense view of reality.Finally,we’lltalkabouttheexperimentsthatproveBell’stheorem,andshowthelengthsthatphysicistsgotoinchallengingnewideas.

SLEEPINGDOGSLETEACHOTHERLIE:ENTANGLEMENTANDCORRELATIONS

Entanglement is fundamentally about correlations between the states of twoobjects.Toillustratetheidea,let’sthinkabouttwodogs—we’llusemyparents’Labradorretriever,RD,andmyin-laws’Bostonterrier,Truman—whocaneachbeinoneoftwostates:“awake”or“asleep.”Ifthedogsarecompletelyseparatefromeachother,therearefourstateswecouldfindourtwo-dogsystemin:wecanfindbothdogsawake,bothdogsasleep,TrumanawakewhileRDisasleep,orTrumanasleepwhileRDisawake.If we bring the two dogs together and allow them to interact, though, a

correlationdevelopsbetweenthestateofthetwodogs.IfTrumanisasleepwhileRDisawake,RDwillwakeTrumanuptoplay,andviceversa.Youwilleitherfindbothdogs awakeorbothdogs asleep, but neverone awake and theotherasleep.Wegofromfourpossiblestatestoonlytwo.Moreover, this correlation allows us to know the state of one of the dogs

withoutmeasuringit.IfTrumanisawake,weknowthatRDmustbeawake,andifTrumanisasleep,weknowthatRDmustbeasleep.WecanlookatRDifwewant,butwe’lljustconfirmwhatwealreadyknow.Measuringthestateofoneofthetwodogsimmediatelyandabsolutelytellsusthestateoftheotherdog.

ISQUANTUMMECHANICSINCOMPLETE?THEEPRARGUMENT

What does this have to dowith Einstein? Einsteinwas a strong believer in adeterministicuniverse,inwhichwecanalwaystraceaclearpathfromcausetoeffect. He had major philosophical problems with quantum mechanics. Inparticular,hewasbotheredby the idea thatpropertiesofquantumparticlesareundefineduntiltheyaremeasured,andthentakeonrandomvalues.Fromthelate1920sthroughthemid-1930s,Einsteinhadaseriesofarguments

withNielsBohr,whowasalsophilosophicallyinclined*butwasachampionofthequantumtheory.Einsteinfirstattackedtheideaofuncertaintywithanumberof different ingenious thought experiments that would perform measurementsforbidden by the uncertainty principle—measuring both the position andmomentum of an electron, for example. Every time he did, Bohr found asemiclassicalcounterargumentshowingthatEinstein’sproposedexperimenthadsomehiddenflaw.†Intheearly1930s,Einsteinreconciledhimselftouncertainty,butheremained

troubledbyquantumtheory,andfoundanewproblemtoattack.Hearguedthatthe existing quantum theory did not contain all the information needed todescribe a particle’s properties. In a 1935 paper titled “Can Quantum-MechanicalDescriptionofPhysicalRealityBeConsideredComplete?”Einstein

and colleagues Boris Podolsky and Nathan Rosen presented an ingeniousargumentforthisclaim,usingtheideaofanentangledstate.Theyproposedanexperimenttodemonstratethissupposedincompletenessbyentanglingthestatesof twoparticles, and then separating them so that they no longer interact (buttheir states do not change). You can then measure the two in separateexperiments that have no possible influence on each other, and see whathappens.In the EPR scheme, measuring the position of one of the two particles

(Particle A) allows you to predict the position of the other (Particle B) withabsolutecertainty.Atthesametime,ifyoumeasuredthemomentumofParticleB,youwouldknowwith certainty themomentumofParticleA.According toEinstein, Podolsky, and Rosen, since there’s no way for measurements ofParticleA to affect theoutcomeofmeasurements ofParticleB, or viceversa,boththepositionandthemomentumofeachparticlemusthavedefinitevaluesthe whole time. This suggests that quantum mechanics is incomplete: theinformationneededtodescribetheprecisestateoftheparticlesexists,butisnotcapturedbyquantumtheory.

“That’sjustwhatIwassaying!”“Whatwas?”“Abunnydoeshaveadefinitepositionandmomentum.All thatuncertainty

businesswasjustyoubeingallconfusingandstuff.”“Itsoundslikeaconvincingargument,butifyouremember,Ialsosaiditwas

wrong.It’sbrilliantlywrong,butthere’sstillaflawinoneoftheirassumptions,namelytheideathatit’simpossibleforameasurementofoneparticletoaffecttheoutcomeofthestateoftheotherparticle.”“Oh,really?Proveit.”“I’llgetthere.Justgivemeaminute…”

“DON’TKNOW”VS.“CAN’TKNOW”:LOCALHIDDENVARIABLES

Bohr’s initial response to the EPR argument was rushed and nearlyincomprehensible.*Herefinedthislater,buthewasneverabletocomeupwitha convincing semiclassical counterargument, in theway that he had in all hisother debateswith Einstein. The reason is simple: there is no such argument.Quantum mechanics is a “nonlocal” theory, meaning that measurementsseparated by a large distance can affect one another inways thatwouldn’t beallowedbyclassicalphysics.

ThesortoftheorypreferredbyEinstein,Podolsky,andRoseniscalledalocalhiddenvariable(LHV)theory,aftertheunderlyingassumptionsthatmakeupthemodel.“Hiddenvariable”meansthatallquantitiesthatmightbemeasuredhavedefinite values, but those values are not known to the people doing theexperiment. “Local”means thatmeasurements and interactions at onepoint inspacecanonly instantaneouslyaffect things in the immediateneighborhoodofthat point. Long-distance interactions are possible, but those interactionsmusttakesome time tobecommunicatedfromoneplace toanother,ataspeed lessthanorequaltothespeedoflight.*Locality is so central to classical physics that it may seem too obvious to

challenge.Locality says that some timemustpassbetweencausesandeffects.Whenahumancallstoadogoutinthegarden,thedogwon’tcomerunninguntilenoughtimehaspassedforthesoundofthecalltotravelfromthehumantothedog.† Nothing the human does can have any influence on the dog’s actionsbeforethattime.LocalityiswhatmakestheEPRargumentaparadox.Nothingintheproposed

experimentlimitsthetimebetweenthetwomeasurements.YoucankeepParticleA at Princeton, and send ParticleB toCopenhagen, and agree tomeasure thepositionofAandthemomentumofBat,say,onenanosecondpastnoon,EasternStandard Time. There is no possible way for any message to travel fromPrinceton to Copenhagen in time to influence the outcome of the secondmeasurement. Hence, assuming locality is true, the two measurements arecompletely independent of each other, and eachmust reflect some underlyingreality.Asobviousastheassumptionoflocalityseems,thisisexactlythepointwhere

theargumentfails.Quantummechanicsisanonlocaltheory,andameasurementmade on one of two entangled objects will affectmeasurementsmade on theother instantaneously, nomatter how far apart the two are.Ameasurement inPrincetoncandetermine the resultofameasurement inCopenhagen,providedtheobjectsbeingmeasuredareentangled.Becausequantummechanicsisnonlocal, thestateof twoentangledparticles

remains indeterminate until one of the two ismeasured.Not only do you notknowthestateoftheparticles,youcan’tknowit.Intermsofourdogexample(page143),untilsomebodymeasuresthestateofoneofthetwodogs,bothdogsare simultaneously asleep and awake—thewavefunction for the system has apartcorrespondingto“TrumanasleepandRDasleep”andapartcorrespondingto“TrumanawakeandRDawake,”butneitherdogisdefinitelyasleeporawake.Thedogsexistinasuperposition,likeafriendlierversionofSchrödinger’scat.Thestateofagivendogtakesonadefinitevalueonlywhenit ismeasured,

andwhenthathappens,thestateoftheotherdogissimultaneouslydetermined.The instant that youmeasure one, you determine the state of both, nomatterwheretheyare.IfTrumanisawake,soisRD,andifTrumanisasleep,soisRD.Ifyou take themintodifferent roomsbeforemeasuring their states,you’ll stillfindthemcorrelated,despitethefactthatmeasuringTruman’sconditiondoesnotdirectlyaffectRD,andnoinformationpassesbetweenthem.Thetwoseparateddogsareasinglequantumsystem,andameasurementofanypartofthatsystemaffectsthewhole.Nonlocalityprevents theEPRexperiment frombeingable tocircumvent the

uncertainty principle. A measurement of Particle A does perturb the state ofParticle B, exactly as if themeasurement had beenmade on Particle B. Thisholdstruenomatterhowcarefullythetwoareseparatedbeforethemeasurement—theentangledparticlesareasingle,nonlocalquantumsystem.Nonlocalitypresentsaphilosophicalchallengetothebasisofclassicalscience

as profound and disturbing as the issues of probability and measurementdiscussed in chapters 3 and 4. The instantaneous projection of the entangledobjects onto definite states* is a conclusion that we’re forced to by quantumtheory,andthere’snothinglikeitinclassicalmechanics.With the EPR paper, quantum physics reached a philosophical impasse.

Supporters ofBohr’s orthodox quantum theorywere unconvinced by theEPRargument, but could not present a compelling counterargument. Meanwhile,people likeEinsteinwhowerebotheredby the implicationsofquantumtheorypointedtotheEPRargumentassuggestingsomedeepertheorythatwouldmakesenseofthisweirdandunpleasantquantumbusiness.MorepeopletookBohr’ssidethanEinstein’s,becausequantumtheoryprovidedsuchaccuratepredictionsofatomicproperties,butneithersidecouldthinkofadefinitiveexperiment.

SETTLINGTHEDEBATE:BELL’STHEOREM

This impasse lasted for almost thirty years, until the Irish physicist JohnBellcameupwith away todistinguishbetween thepredictionsofquantum theoryandthoseofthelocalhiddenvariablemodelspreferredbyEinstein.BellrealizedthatLHV theorieshavedefinite particle states andonly local interactions, andarethuslimitedinwaysthatquantumtheoryisnot.Heprovedamathematicaltheoremstating thatentangledquantumparticleshave their statescorrelated inwaysthatnopossiblelocalhiddenvariabletheorycanmatch.Thesecorrelationscan bemeasured experimentally; ameasurement showing correlations beyondtheLHVlimitswouldconclusivelyprovethatBohrwasright,andEinsteinwas

wrong.Bell’stheoremiscriticaltothemodernunderstandingofquantummechanics,

soit’sworthexploringinsomedetail.Itcan’tbedemonstratedwithdogs,butit’snottoohardtodousingthepolarizedphotonswetalkedaboutinchapter3(page65).Tobe concrete, let’s think about twophotonswhosepolarizations are thesame—ifoneismeasuredtobehorizontal,theotherisalsohorizontal;ifoneismeasuredata45ºangle,theotherisatthesame45ºangle.Thenwelookatthreedifferentpossiblemeasurements.The traditional arrangement calls for two experimenters—they’re usually

called “Alice” and “Bob,” but we’ll stick with “Truman” and “RD,” becausethey’regooddogs—toeachreceiveoneofthetwophotons.TrumanandRDareeachgivenapolarizerandaphotondetector,whichcombinetomakedetectorsthat register either a “1” or a “0,” depending onwhether the photonmakes itthrough the polarizer or not. For example, if the polarizer is set to vertical, avertically polarized photon will be transmitted and give a “1,” while ahorizontallypolarizedphotonwillbeblocked,andgivea“0.”Ifthepolarizerisset at 45° to the vertical, a vertically polarized photon has a 50% chance ofmakingitthroughandbeingrecordedasa“1,”otherwiseitwillbeblockedandrecordedasa“0.”Theexperimentissimple:eachdogsetshispolarizeratoneofthreeangles,a,

b,orc.Hethenrecordsthedetectorreading(“0”or“1”)foronephoton.Thentheychangethedetectorsettings,anddoitagain.Afterrepeatingthisoverandover,theywillhavetriedallthepossiblecombinationsofdetectorsettingsmanytimes,andthentheycomparetheirresults.

Schematic of ameasurement to test Bell’s theorem. Truman andRD each take one photon from anentangledphoton source,andmeasure itspolarizationalongoneof threeanglesusingapolarizingfilter and a photon detector. They can distinguish between quantummechanics and a local hiddenvariable theory by measuring how often they detect the same thing when their polarizers are atdifferentangles.

When theycompare results, they’llnotice two things.When theirpolarizersaresettothesameangle,they’llseethatbothgetthesameanswer(either“1”or

“0”),everytime.They’llalsoseethatnomatterwhatangletheychoose,theygetequalnumbersof“0”and“1”results—iftheyrepeattheexperiment1,000timesatagivenangle,theywillget500“0”sand500“1”s.Thesetwoobservationsaretruewhetherthey’redealingwithaquantumentangledstate,orastategovernedbyanLHVtheory.

“Wait,shouldn’titdependontheangles?”“Whatangles?”“Your a, b, and c angles.Whydo they get equal numbers of ‘0’s and ‘1’s?

Shouldn’tthemeasurementresultsdependonwhichangletheychoose?Like,iftheyhavetheirpolarizerssetvertically,theyalwaysdetecta‘1’?”“No,thestateswe’redealingwitharestatesofindeterminatepolarization.In

thequantumpicture,thepolarizationisundefined,whileintheLHVpicture,it’sequallylikelytobeeitherhorizontalorvertical.”“Doesn’t that mean they’re at 45º? Then shouldn’t they get ‘1’ every time

whentheyputthepolarizersat45º?”“No,theygetthesameresultat45º.Thephotonsareequallylikelytobe45º

counterclockwise fromvertical, or 45º clockwise, or any other angle. It reallydoesn’t matter what angles they choose for a, b, and c—even ‘vertical’ and‘horizontal’arekindofarbitrary.”“Nothey’renot.”“Yestheyare.WhenIsaysomethingweird,andyoulookatmesideways—

likeyou’redoingrightnow—thatchangeswhat‘vertical’lookslike,right?”“I guess. Everything looks different from an angle, and sometimes weird

humanstuffmakesmoresense.”“It’sthesamethinghere.Theanglestheysetforthepolarizersdeterminewhat

‘0’ and ‘1’ will mean, in the same way that tilting your head changes yourperception of ‘horizontal’ and ‘vertical.’ They still have an equal chance ofgettingeither result.Whatyou seedependsonwhatyou’re looking for.Togobacktothetreatanalogyfrompage140,it’slikeatreatwhereifyou’relookingfor‘meat,’yougeteithersteakorchicken,butifyou’relookingfor‘notmeat,’yougeteitherpeanutbutterorcheese.”“Thosetreatssoundgood.Youshouldbuymesomeofthose.”“Idon’tthinktheyhavethematthepetshop,butI’lllook.”

To testBell’s theorem,we askhowoften theyget the same answerwith theirdetectorsatdifferentsettings.Thatis,howmanytimesdidTrumanrecorda“0”withthedetectorinposition“a,”whileRDgota“0”inposition“c,”orTrumana“1”inposition“b”andRDa“1”inposition“a,”andsoon.Theprobabilityof

bothdogsgettingthesameresultwithdifferentdetectorsettingsisverydifferentforLHVtheoriesandquantummechanics.

THEEPROPTION:LOCALHIDDENVARIABLEPREDICTION

Thekey toBell’s theorem is that all thepredictionsof a localhiddenvariabletheorycanbewrittendowninadvance,solet’sdothat.Eachphotonhasawell-defined state, andwe can represent that state by a set of three numbers, eachgivingthedefiniteoutcomeofameasurementinpolarizerpositiona,b,orc.Thetwo-photonsystemoffersatotalofeightpossiblestates,whichwecanrepresentinatable:

TotestBell’stheorem,weneedtheprobabilityofbothdogsgettingthesameanswer with different detector settings. Looking at the table, we see that nomatter what angles we pick, four of the eight possible states give the sameanswer.Forexample,ifTrumansetshisdetectortopositiona,andRDsetshistob,states1and2willgivethemeacha“1”andstates7and8willgivethemeacha“0.” IfTrumanchoosescandRDchoosesa, the four statesgiving thesameanswerare1,3,6,and8,andsoon.We’renotstuckwithexactly50%probabilityofgettingthesameanswerfor

different settings, though.We’re free to adjust the probability of the photonsbeing in a particular state—say, making state 1 more likely, and state 6 lesslikely—though any changewemakehas to endupwith equal probabilities offinding“0”or“1”foreachdetectorsetting.Ifweplayaroundwiththeprobabilitiesoftheindividualstates,wefindthat

we can cover a limited range of possible probabilities. We can make the

maximum probability of both dogs getting the same result 100%, but theminimumprobabilityis33%,not0%.Nomatterwhatwedo,wecannevermaketheprobabilitylowerthan33%.*Notice thatwehaven’tsaidanythingaboutwhatcauses thosestates,orhow

they are chosen.We don’t need to themere fact that we can write down thelimited number of possible results places restrictions on the experiment. Nomodel inwhich the twophotonshavewell-defined stateswhen they leave thesourcecangiveaprobabilityoflessthan33%forthetwomeasurementstogivethe same outcome. The probability must be less than or equal to 100%, andgreaterthanorequalto33%.†SimilarlimitsholdtrueforanyLHVtheoryyoucandreamup.

THEBOHROPTION:QUANTUMMECHANICALPREDICTION

Toprovequantummechanicscorrect,then,weneedtofindsomedetectoranglesforwhich the probability of both dogs getting the same answerwith differentsettings is less than 33%. Bell showed that this can be done, thanks toentanglement: measuring the polarization of one of the two photonsinstantaneouslydeterminesthepolarizationoftheother.Inthequantumpicture,thestateofthetwophotonsisindeterminateuntilthe

instantwhenoneofthetwoismeasured,whenithasa50%chanceofendingupas a 0or 1.At that instant, thepolarizationof the secondphoton is set to thesame angle as the first, whatever that is. If the first photon passed through averticalpolarizer,recordinga“1,”thesecondphotonisnowverticallypolarized.If the first photonwas blocked by the vertical polarizer, recording a “0,” thesecond photon is now horizontally polarized. The possible outcomes of thesecondmeasurementarethendeterminedbythefirstpolarizerangle.To prove Bell’s theorem, let’s imagine Truman sets his detector to vertical

polarization(whichwe’llcall“a”).RDsetshisdetectortoeither60°clockwisefromvertical (“b”), or 60° counterclockwise fromvertical (“c”).What are thepossibleways to get the same answer for both dogswhen they have differentpolarizersettings?Well, half of the time, Truman will detect a “1” with his detector, which

means thatwewant theprobabilityofRDalsogettinga“1.”*SinceTruman’spolarizerisvertical,theentangledphotonhittingRD’sdetectorisalsoverticallypolarized. If his detector is set to position “b,” then the angle between thevertical photon and RD’s polarizer is 60°, and the probability of the photonpassingthroughthepolarizeris25%.Thesameholdsforposition“c,”whichis

60°from“a”intheotherdirection.The other half of the time, Truman measures a “0,” and both entangled

photonsarehorizontal.RD’sphotonagainhasa25%chanceofbeingblockedandgivinga“0,”†foreitherangle.Nomatterwhat value Trumanmeasures, then, quantum theory tells us that

thereisonlya25%chancethatRDwillgetthesamevaluewithhisdetectoratadifferent polarizer setting. This directly contradicts the prediction of the localhiddenvariabletheory,whichgaveaminimumchanceof33%.OnlyoneinfourofRD’smeasurements is the sameasTruman’s,whereLHVsays thatat leastoneinthreeshouldbethesame.Youmight think that the two theories shouldgive the same results, because

they’re describing the same system, in the same way that the differentinterpretationsofquantummechanicsallgivethesamepredictions.That’swhatmostphysicists thought,untilBellshowedotherwise.Thecoreassumptionsofthelocalhiddenvariabletheoriesmeanthattheyaresubjecttostrictlimits—youcanwritedownatableliketheoneaboveshowingallpossibleresults.Quantumtheoriesdonothavethesamelimitations,soacleverexperimentcandistinguishbetweenthem.*The results are different because quantum mechanics is nonlocal—the

polarization of RD’s photon is not set in advance, but is determined by theoutcomeofTruman’smeasurement.The probability of getting the same resultwithdifferentsettingsislowerbecausethetwomeasurementsaffecteachother,no matter how far apart they are, or when they’re made. Einstein called this“spooky,”andit’shardtoarguewithhim.

“Can’tyoujustmakeabettertheory?”“Whatkindofbettertheory?”“Abetterhiddenvariabletheory.Thatmatchesthepredictionsbetter.”“That’s the whole point. Bell didn’t look at a particular theory—what he

showedisthatthere’snopossiblelocalhiddenvariabletheorythatcanreproduceall the predictions of quantum mechanics. If the two measurements areindependent of each other, there’s no way to arrange things so that themeasurementsshowthesamecorrelationthatyouseewithquantummechanics.”“Somakethemeasurementsdependoneachother.”“Thatworks, but that isn’t a local hidden variable theory anymore. In fact,

David Bohm worked out a version of quantummechanics that uses nonlocalhidden variables, and reproduces all the predictions of quantum theory usingparticleswithdefinitepositionsandvelocities.”“Thatsoundsnice.Whydon’tpeopleusethat?”

“Well,Bohm’stheoryintroducesanextra‘quantumpotential,’afunctionthatextends through the entire universe and changes instantaneously when youchange some property of the experiment. It’s a really weird object, and it’s aheadache when doing calculations. It’s also easier to extend regular quantummechanics to be compatiblewith relativity, inwhat’s known as quantum fieldtheory.”“It’snotwrong,though?”“No,itpredictsthesamethingsasregularquantumtheory.Youcanlookatit

as an extreme version of a quantum inter pretation, like the Copenhageninterpretationormany-worldspicturesthatwetalkedaboutearlier.Itaddsalittlemore maths to the theory, but doesn’t predict anything different in practicalterms.”“Hmmm.”“The important thing for this discussion is that Bohm’s theory is nonlocal,

whichiswhattheEPRparadoxandBell’stheoremarereallyabout.Fromthose,we know that quantum theory can’t be a strictly local theory, wheremeasurementsintwodifferentplaceshavenoeffectoneachother.”“Thatstillannoysme.Howdoweknowthatthat’sreallytrue?”“I’mgladyouaskedthat…”

Thisexample isa specificdemonstrationofBell’s theorem,but it captures theflavorofthegeneraltheorem.WhatBellshowedisthattherearelimitsonwhatcanbeachievedwithLHVtheoriesingeneral,andthatundercertainconditions,quantummechanicswillexceedthoselimits.Acleverexperimentcandetermineonce and for all whether quantummechanics is right, or whether it could bereplacedbyalocalhiddenvariabletheoryasEinsteinhoped.

LABORATORYTESTSANDLOOPHOLES:THEASPECTEXPERIMENTS

Bell published his famous theorem in 1964. In 1981 and 1982, the FrenchphysicistAlainAspect and colleagues testedBell’s predictionwith a series ofthree experiments that are generally considered to conclusively rule out localhiddenvariabletheories.*Theyneededallthreeexperimentstocloseaseriesof“loopholes,”gapsintheirresultsthatsomelocalhiddenvariablemodelsmightslipthrough.We’ll describe all three experiments here, because they’re outstanding

examples of the art of experimental physics. More than that, though, they

demonstratethelengthsyouneedtogotoifyouwanttoconvincephysicistsofsomething. You need to answer not only the obvious objections, but alsoobjectionsthatareimprobableenoughtoseemalittleridiculous.The first experiment, published in 1981, was essentially the same as our

thoughtexperimentwithTrumanandRD.Aspect’sgroupmadecalciumatomsemit twophotonswithinafewnanosecondsofeachother,heading inoppositedirections. These photons are guaranteed to have the same polarization—it’sequallylikelytobeeitherhorizontalorvertical(oranyotherpairofangles),butifonephotonishorizontal,theothermustalsobehorizontal.ThisisexactlytheentangledstateyouneedinordertotestBell’stheorem.In the first experiment, they placed two detectors on opposite sides of their

entangled photon source, with a polarizer in front of each detector. Thepolarizersweresettovariousdifferentangles,andtheymeasuredthenumberoftimes they counted a photon at both detectors—that is, both detectors reading“1,”intermsofourexampleabove.Physicists like todealwithnumbers, and for the specificconfiguration they

used, a local hidden variable treatment predicts that their results should boildown to a number between -1 and 0. When they did the experiment, theymeasured a value of 0.126,with an uncertainty of plus orminus 0.014.* Thedifference between the maximum LHV value and their measurement is ninetimeslargerthantheuncertaintyinthemeasurement,meaningthatthere’saonein1036probabilityofthishappeningbychance.†

ThefirstAspectexperiment.Anexcitedcalciumatomemitstwophotonswithentangledpolarizations.Eachphotonheadstowardasingledetectorwithapolarizingfilterinfrontofit,settoanappropriateangle.

So, that’s the end of LHV theories, right? It looks just like our imaginaryexperiment above, and that’s an astonishingly small probability of thishappeningbyaccident.Whydidtheyneedtodoasecondexperiment,letaloneathird?Unfortunately,there’saloopholeintheirresultthatallowssomeLHVtheories

tosurvive.Inourthoughtexperiment,weimaginedTrumanandRDwithphotondetectors thatwereabsolutelyperfect,because they’reverygooddogs.Aspect

andhis coworkers are only human, though, and sowere stuck using detectorswith limited efficiency. On rare occasions, a detector would fail to record aphotonthatwasreallythere.This is a problem, because their experiment recorded a “0” when they

expected a photon and didn’t see one—they assumed that those photonswereblockedbythepolarizers.Butbecausetheirdetectorssometimesfailedtodetectphotons, it’s conceivable that the first Aspect experiment just looked like itviolated theLHVprediction. If someof their “0”s really should’vebeen“1”s,thatcouldconfusetheirresults.ForLHVtheoriestoslipthroughthisloopholetheuniversewouldneedtobe

somewhatperverse,butit’spossible,sotheydidasecondexperiment,publishedin1982,usingtwodetectorsforeachphoton.They closed the detector efficiency loophole by directly detecting both

possiblepolarizations, andonlycountingexperimentswhere theydetectedonephoton on each side of the apparatus. They replaced the polarizers withpolarizing beam splitters that directed each polarization to its own detector. Ifone of the detectors failed to record a photon, that run of the experimentwasdiscarded.TheirmeasuredvalueinthesecondexperimentexceededtheLHVlimitbyan

astonishing40timestheuncertainty,andtheoddsofthathappeningbychanceare so small it’s ridiculous. So, why did they do the third experiment? Asimpressive as the second experiment was, it still left a loophole, becausesomethingcouldhavepassedmessagesbetweentheirdetectorsandtheirsource.

The secondAspect experiment. The entangled photons leave the source and head toward a pair ofdetectors with a polarizing beam splitter in front of them. These beam splitters direct the “0”polarizationtoonedetectorandthe“1”polarizationtoanother,ensuringthatnophotonsaremissedintheexperiment.

TotestBell’s theorem,itneedstobeimpossiblefor themeasurementatonedetector todependonwhathappens at theotherdetectorwithout some faster-than-lightinteraction.Ifthere’sawaytosendmessagesbetweenthedetectorsatspeedslessthanthatoflight,allbetsareoff.Inthefirsttwoexperiments,theychosethedetectorsettingsinadvance,andleftthemsetformuchlongerthanit

took light to pass between the source and the detector. Somethingmight havecommunicatedthepolarizersettingsfromthedetectorstothesource,whichthensentoutphotonswithdefinitepolarizationvalueschosentomatchthequantumpredictions.Whentheexperimenterschangedtheangles,thenewvalueswouldbe sent to the source, which would change the polarizations sent out. Theirresultsseemedtoprovequantumtheory,buttheymighthavebeenthevictimsofacosmicconspiracy.Thethirdexperimentfoundaningeniouswaytoclosethatloophole,aswell.

Aspect and his colleagues ruled out any possibility of some sort of universalconspiracymimicking the quantum results by changing their detector settingsfasterthanlightcouldgofromthesourcetothedetector.They replaced thebeamsplitterswith fast optical switches that coulddirect

the photons to one of two detectors, each set for a different polarization. Theswitchesflippedbetweenthedetectorsevery10nanoseconds,whileit tookthephotons 40 ns to reach the detector. In effect, which detector a given photonwouldhitwasnotdecideduntilafterthephotonhadalreadyleftthesource.The third experiment’s results exceeded the LHV limit by five times the

uncertainty.Thechancesofsucharesulthappeningbyaccidentwereaboutoneinahundredbillion—betterthanthechancesfortheothertwoexperiments,butstilllowenoughtobeconvincing.

ThethirdAspectexperiment.Thetwoentangledphotonsleavethesource,andheadtowardfastopticalswitchesthatsendeachphotontowardoneoftwodifferentpolarizers,withthechoicenotbeingmadeuntilafterthephotonshaveleftthesource.

Eventhethirdexperimentdoesn’tcloseeveryloophole,*butAspectstoppedthere, because the experiments were extraordinarily difficult. A number ofpeople have repeated these experiments, using more modern sources ofentangledphotons,†anda2008experimenthaseventestedBell’stheoremusingentangled states of ions insteadof photons, but no loophole-free test has beendone.Asaresult,therearestillafewpeoplewhoarguethatLHVtheorieshaveneverbeencompletelyruledout.

These fewdie-hard theoristsaside, thevastmajorityofphysicistsagree thattheBell’stheoremexperimentsdonebyAspectandcompanyhaveconclusivelyshownthatquantummechanicsisnonlocal.Ouruniversecannotbedescribedbyanytheoryinwhichparticleshavedefinitepropertiesatalltimes,andinwhichmeasurements made in one place are not affected by measurements in otherplaces.Aspect’sexperimentsrepresentaresoundingdefeatfortheviewoftheworld

favoredbyEinsteinandpresentedintheEinstein,Podolsky,andRosenpaperin1935.ButwhiletheEPRpaperiswrong,it’sbrilliantlywrong,forcingphysiciststograpplewiththephilosophicalimplicationsofnonlocality.Exploringtheideasraisedinthepaperhasdeepenedourunderstandingofthebizarrenatureofourquantum universe. The idea of quantum entanglement exploited in the EPRpaperalso turnsout toallowus todosomeamazing thingsusing thenonlocalnatureofquantumreality.

“Physicistsarereallyweird.”“Yes,nonlocalityisstrange.”“Notthat,theloopholes.Dophysicistsreallybelievethattherearemessages

being passed back and forth between different bits of their apparatus? Whatwouldcarrythemessages?”“I’m not sure anybody ever suggested a plausible mechanism, but it really

doesn’tmatter.Theycouldbecarriedbyinvisiblequantumbunnies,forall thedifferenceitmakes.”“Quantumbunnies?”“Invisible quantum bunnies. Moving at the speed of light. Don’t get your

hopesup.”“Awww…”“Anyway, the third Aspect experiment prettymuch rules out anymeans of

carryingmessagesbetweenpartsoftheapparatus,involvingbunniesoranythingelse.Thepointis,priortothat,itwasatleastpossibleinprinciplefortheretobeanother explanation. And in science, you have to rule out all possibleexplanations, even theones that seem really unlikely, if youwant to convinceanybodyofanextraordinaryclaim.”“Eventheonesinvolvingbunnies?”“Eventheonesinvolvingbunnies.Andanyway,theideathatdistantparticles

canbecorrelatedinanonlocalfashionisn’tallthatmuchweirderthanquantumbunnieswouldbe.”“Goodpoint.So,what’sthisgoodfor?”“Whatdoyoumean?”

“Youdroppedareallyheavyhintinthatlastparagraphthatthisentanglementstuff is good for something.What’s it good for, sendingmessages faster thanlight?”“No, you can’t use it for faster-than-light communication, because the

detections are random. There are correlations between particles, but thepolarizationof eachpairwill be random. I can’t sendamessage to somebodyelseusingEPRcorrelations—allIcansendisarandomstringofnumbers.”“Sowhatgoodisit?”“Well, random strings of numbers can be useful for quantum cryptography,

makingunbreakablecodes.Andtheideaofentanglementiscentraltoquantumcomputing, which could solve problems no normal computer can tackle. Andthere’squantumteleportation,usingentanglementtomovestatesfromoneplacetoanother.There’sallsortsofstuffoutthere,ifyoulookforit.”“Teleportationsoundscool!Talkaboutthat.”“Well,that’snext…”

* The process of decoherence (described in chapter 4) involves the interaction of a single quantum

particlewithamuchlargerenvironment,butwecareonlyaboutthestateofthesingleparticle.*WernerHeisenberg,whodevelopedtheuncertaintyprinciplewhileworkingwithBohr,oncedescribed

Bohras“primarilyaphilosopher,notaphysicist.”†Inalmostallof thosecases,Bohr’sargumentdependedontheeffectofmeasurementonthesystem.

SomethingintheprocessbywhichEinsteinproposedtomeasurethepositionwouldcauseachangeinthemomentum(asinthecaseoftheHeisenbergmicroscopethoughtexperimentdiscussedinchapter2 [page38]),orviceversa.Measuringthesystemrequiresaninteraction,andthatinteractionchangesthestateofthesysteminawaythatintroducessomeuncertaintyinthequantitiesbeingmeasured.* Bohrwas somewhat famous for the opacity of his writing, but he outdid himself in this case. The

crucialparagraphofhispaperreferstothequantumconnectionbetweendistantobjectsas“aninfluenceonthe very conditions which define the possible types of predictions regarding the future behavior of thesystem” (italics in original), and declares that the quantum view “may be characterized as a rationalutilizationofallpossibilitiesofunambiguousinterpretationofmeasurement,compatiblewiththefiniteanduncontrollableinteractionbetweentheobjectsandthemeasuringinstrumentsofquantumtheory.”*Thislight-speedlimitisoneofthemainconsequencesofEinstein’stheoryofrelativity,andthusvery

importanttohisconceptionofphysics.†Orevenlonger,dependingonwhatthedogisdoingwhencalled.*IfyouprefertheCopenhagenview,thisprojectioninvolvesarealcollapseofthewavefunctionintoa

single state. If you prefer many-worlds, the apparent projection onto a single state comes because weperceiveonlyasinglebranchofthewavefunction.Ineithercase,theresultingcorrelationisthesame,andtheeffectisinstantaneous.*Weget themaximumvalueof100%if thesystemhasa50%chanceofbeing instate1anda50%

chanceofbeinginstate8.Wegettheminimumvalueof33%byneverlettingthesystembeinstate1orstate8,andmakingtheothersixstatesequallylikely.Ifyoulookatstates2through7,you’llseethatnomatterwhattwodifferentanglesyouchoose,therearealwaystwostatesthatgiveyouthesameanswerforbothdetectors.†AsaresultthepredictionsofBell’stheoremareoftencalled“Bellinequalities.”*We’reassumingthatTruman’sphotonismeasuredfirst,forthesakeofclarity.Theresultisthesameif

weassumeRD’sphotonisthefirstonemeasured.

† The horizontal photon has a 75% chance of passing through the polarizer to the detector in eitherposition.A“0”tomatchTruman’sresulthappensonlywhenRD’sphotonisblocked,a25%probability.*Thisraisesthequestionofwhetherasufficientlycleverexperimentmightdistinguishbetween,say,the

Copenhagen interpretation and the many-worlds interpretation. This is a much harder problem thandistinguishingbetweenquantumandLHVtheories.SomefutureJohnBellmayyetcomealongandfindtherighttest,butnoonehasmanagedyet.* JohnClauser and a couple of other people had done earlier tests, but theAspect (pronounced “As-

PAY”)experimentshadbetterprecision,andsoareregardedasthedefinitivetests.*Thisuncertaintyisatechnicallimitationbasedonthedetailsoftheirexperiment,andnotanythingto

dowiththeHeisenberguncertaintyprinciple.†1036isabillionbillionbillionbillion,anumbersolargethatitmighthavemadeevenCarl“Billions

andBillions”Saganblink.*The third experiment actually reopens thedetector efficiency loop-hole, because theyusedonlyone

detectorforeachpolarizer.† One experiment by Paul Kwiat (who was part of the Innsbruck–Los Alamos team doing quantum

interrogationexperimentsinchapter5)andcolleaguesatLosAlamossawaneffectamind-boggling100timeslargerthantheuncertainty.

CHAPTER8

BeamMeaBunny:

QuantumTeleportation

Emmy trots intomyoffice, lookingpleasedwithherself.This isnever agoodsign.“Ihaveaplan!”sheannounces.“Really.Whatsortofplanisthis?”“Aplantogetthosepeskysquirrels.”Theykeepescapingupthetreesinthe

garden,andshe’sgettingfrustrated.“Is this a better plan than the onewhere youwere going to learn to fly by

eatingthespilledseedfromthebirdfeeder?”“Thatwasgoingtowork,”shesays,indignantly.“Andforyourinformation,

yes,it’samuchbetterplanthanthat.”“Well,then,I’mallears.What’sthisbrilliantplan?”“Teleportation.”Shelookssmug,andwagshertailvigorously.“Teleportation?”“Yes.”“Okay,you’regoingtohavetoexplainthatalittle.”“Well,I’mthinking,theproblemis,theycanseemecomingfromthehouse,

andtheygettothetreesbeforeIdo.IfIcouldgetbetweenthemandthetrees,though,Icouldgetthembeforetheygetaway.”“Okay,I’mwithyousofar.”“So, I justneed to teleportout into thegarden, insteadofgoing through the

door.”Herwholebackendiswaggingnow.“Right…Andhow,exactly,didyouplantoaccomplishthisfeat?”“Well…Iwashopingyouwouldhelpme.”“Me?”“Yes. I read where some physicists have done quantum teleportation, and

you’reaphysicist,andyou’rereallyclever,andyouknowaboutquantum,soI

washopingyouwouldhelpmebuildateleporter.”Sheputsherheadinmylap.“Pleeeeease?I’magooddog.”Iscratchbehindherears.“Youareagooddog,butIreallycan’thelp.Forone

thing, I don’t do teleportation experiments in my lab. But even if I did, Iwouldn’tbeabletohelpyouuseteleportationtocatchsquirrels.”“Whynot?”“Well, the existing teleportation experiments all deal with single particles,

usually photons. You’re made up of probably 1026 atoms—a hundred trilliontrillion—whichisfarmorethananybodyhaseverteleported.”“Yes,butyou’rereallyclever.Youcanjust…makeitbigger.”“Iappreciateyourconfidence,butno.Thebiggerproblemisthatthequantum

teleportationpeopledointherealworldisn’tliketheteleportationyouseewiththetransportersonStarTrek.”“Howso?”“Well, all that quantum teleportation does is transmit the state of a particle

fromoneplacetoanother.IfIhaveanatomhere,forexample,Ican‘teleport’itto thegarden,andendupwithanatomtherethat’s intheexactsamequantumstate as the atom I startedwithhere.At the endof theprocess, though, I stillhavetheoriginalatomherewhereitstarted—itdoesn’tmovefromoneplacetoanother.”“That’sprettyrubbish.What’sthepointofthat?”“Well, quantum mechanics won’t let you make an exact copy of a state

withoutchangingtheoriginalstate,andquantumstatesofthingslikeatomsarepretty fragile. If you really needed to get a particular quantum state from oneplacetoanother,yourbestbetmightbetoteleportit.”Shelooksalittledubious.“YoucoulduseittomakeaquantumversionoftheInternet,ifyouhadacoupleofquantumcomputersthatyouneededtoconnecttogether.”“Well,okay.Sojustteleportmystateintothegarden,andI’lluseittocatch

squirrels.”“EvenifIknewhowtoentangleyourstatewithawholebunchofphotons—

which I don’t—I would need to have raw material out in the garden. Therewouldneedtobeanotherdogoutthere,onethatlookedjustlikeyou.”Hertailstopsdead.“Wedon’tlikethosedogs,”shesays.“Dogsthatlookjust

likeme.Inmygarden.Wedon’tlikethosedogsatall.”Shelooksdistressed.“No,wedon’t.Oneofyouisall thedogweneed.”Sheperksupabit.“So,

yousee,teleportationisn’tagoodplan,afterall.”“No,Isupposenot.”She’squietforamoment,andlooksthoughtful.“Well,”

shesays,“Isupposeit’sbacktoplanA.”“PlanA?”

“CanIhavesomebirdseed?”

“Quantumteleportation”isprobablythebest-knownapplicationofthenonlocalcorrelations discussed in the previous chapter. The name certainly fires theimagination, conjuring up images of Star Trek and other fictional settings inwhich people, either through fictional science or just plain magic, caninstantaneouslytransportobjectsfromoneplacetoanother.Theobjectstartsatpoint A, disappears with a soft *whoosh*, and reappears at point B, somedistanceaway.Thehighexpectationscreatedbysciencefictionmaketherealityofquantum

teleportation seemdisappointing.Realquantum teleportation involvesonly thetransferofaquantumstatefromonelocationtoanother,andnotthemovementofcompleteobjects.Thetransferisalsoslowerthanthespeedoflight,becauseinformation needs to be sent from one place to another. This is a greatdisappointment to dogs hoping to beam themselves out into places whereunsuspectingsquirrelsarewaiting.Nevertheless,it’samarvelouslycleveruseofquantumtheory,tyingtogether

severalof the topics thatwe’vealready talkedabout. In thischapter,we’ll seehow indeterminacy and quantum measurement make it difficult to transmitinformationaboutquantumstatesfromoneplacetoanother.We’llseehowthe“quantum teleportation” scheme makes ingenious use of nonlocality andentangledstatestoavoidtheseproblems,andwhyyoumightwantto.Quantum teleportation is a complex and subtle subject, probably the most

difficult topicdiscussedinthisbook.It’salsothebestexamplewehaveof thestrangenessandpowerofquantumphysics.

DUPLICATIONATADISTANCE:CLASSICAL“TELEPORTATION”

Wecan’t teleport in thewayenvisioned in science fiction and fantasy,but theessenceof teleportation is justduplicationatadistance—youtakeanobjectatone place, and replace it with an exact copy at some other location. By thatdefinition,wedohaveanapproximationofteleportationusingclassicalphysics:afaxmachine.If you have a document that youwant to send instantly from one place to

another—for example, if Truman has just been given a really nice bone, andwants to tauntRDby sendinghimapictureof it—youcando thiswith a faxmachine. The machine works by scanning the document, converting it toelectronic instructions for creating an identical document, and sending that

information over telephone lines to another faxmachine at a distant location,which prints a copy.What’s transmitted is not the document itself, but ratherinformationabouthowtomakethatdocument.The operation of a fax machine is different from the fictional idea of

teleportation, but the differences are not all that significant.When you fax adocument from one place to another, you end upwith two copies in differentlocations,butifyouregardthisasaproblem,youcouldalwaysattachashreddertothesender’sfaxmachinetodestroytheoriginal.Thecopyproducedbyafaxmachineisn’tperfect,butthat’sjustamatteroftheresolutionofthescannerandprinter, and you can always imagine getting a better scanner and printer. Thetransmissionislimitedbythetimeittakestotransmittheinformationfromoneplace to another, so it’s not perfectly instantaneous, but that’s not a majorproblemformosttransactionsinvolvingafaxmachine.Ifyouwantedtoapproximatethefictionalidealofteleportationinaclassical

world, the best you could do would be to upgrade the concept of the faxmachine.Trumanwould take a bone, and place it in amachine,whichwouldscanthebonetodeterminethearrangementofatomsandmoleculesmakingupthebone.ThenhewouldsendthisinformationtoRD’s“teleportation”machine,whichwouldassembleanidenticalboneoutofmaterialsathandandpresentittohimtochew.

NOCLONINGALLOWED:QUANTUMLIMITATIONS

When we turn to quantum teleportation, we’re talking about “teleporting” aquantumobject.Thismeans not just getting the right physical arrangement oftheatomsandmoleculesmakinguptheobject,butalsogettingallthoseparticlesintherightquantumstates,includingsuperpositionstates.TrumancoulduseanupgradedfaxmachinetosendRDacatinabox,buthewouldneedaquantumteleportationdevice to sendacat inabox thatwas30%alive,30%dead,and40%bloodyfurious.Thisturnsouttobevastlymoredifficultthantheclassicalanalogue,duetotheactivenatureofquantummeasurement.Whileintheoryitispossibletodoquantumteleportationwithanyobject,in

practice,alloftheexperimentsdonetodatehaveusedphotons,sowe’llimaginethatTrumanistryingtosendasinglephotonofaparticularpolarizationtoRD.*Aswesawbackinchapter3(page65),apolarizedphotoncanbethoughtofasasuperpositionofhorizontalandverticalpolarizations,with someprobabilityoffindingeitherofthosetwoallowedstates.When we describe a photon with a polarization between vertical and

horizontal,wewriteawavefunctionforthatphotonthatisasuperpositionstate:it’sapartsvertical,andbpartshorizontal:

a|V>+b|H>

The numbers a and b tell us the probability of finding vertical or horizontalpolarization.† Infact,anyobject inasuperpositionstatewillbedescribedbyawavefunction exactly like this one. Ifwe can find away to teleport a photonpolarizationfromTrumantoRD,wecanusethesametechniquetoteleportthestateofacat inabox—it’s justamatterof increasing thenumberofparticlesinvolved.So,Trumanhas a photon that hewants to send toRD.The classical recipe

tellshimtosimplymeasurethepolarizationofthephoton,thencallRDonthephone,andtellhimhowtoprepareanidenticalstate.ButtheonlywayTrumancanmeasure thepolarization is ifhealreadyknowssomethingabout thestate,andcansethispolarizationdetectorappropriately.Forexample,ifheknowsthatthephotoniseitherverticalorhorizontal,hecansenditataverticallyorientedpolarizer.Ifitpassesthrough,heknowsthatthepolarizationwasvertical,andifitgetsabsorbed,heknowsitwashorizontal.HecanthensendthatinformationtoRD,whocanprepareaphotonintheappropriatestate.Unfortunately, if the polarization is at some intermediate angle—a parts

vertical and b parts horizontal—it’s impossible for Truman to make thenecessary measurement. The numbers a and b tell us the probability of thephotonpassingthroughaverticalorhorizontalpolarizer,butthere’snowayofmeasuringbothaandbforasinglephoton—eitheritpassesthroughapolarizeror itdoesn’t.Evenif thephotonpasses through, thesuperposition isdestroyedandit’sleftinoneoftheallowedstates.You can only determine both probabilities by repeating the measurement

manytimesusingidenticallypreparedphotons.Thatdoesn’thelpustotransmitthepolarizationofasinglephoton,though,whichisourgoal.This polarization measurement problem is a specific example of the no-

cloningtheorem.WilliamWoottersandWojchiechZurekprovedin1982thatitisimpossibletomakeaperfectcopyofanunknownquantumstate.Unlessyoualreadyhavesomeideawhatthestateis,youchangethestatewhenyoutrytomeasure it, and can never be sure that your copy is faithful. If Truman reallyneeds to send RD a perfect copy of a single photon, without knowing itspolarizationinadvance,he’llneedtofindamorecleverwayofdoingit.

“Whynotjustsendthephoton?”

“Pardon?”“Imean, it’s aphoton.They travelplacesat the speedof light—that’swhat

theydo. If IhadaphotonandIwantedtosendit tosomeotherdog—whichIdon’t, by the way. Other dogs don’t deserve my photons. If I did, though, Iwouldjustpointthephotonattheotherdog,andletitgo.”“Oh.Well, therearea lotof things that canhappen toaphotonon theway

fromoneplacetoanotherthatwouldchangethepolarization.Ifyouwanttobesurethatthedogontheotherendgetsexactlythepolarizationyoustartedwith,teleportationisasurewayofdoingthat.”“That’sasillythingtowanttodo,anyway.”“Not really, but you’ll have towait until the end of the chapter to find out

why.”

AMAGICCOMPASS:CLASSICALANALOGUEOFQUANTUMTELEPORTATION

It’s hard to find a classical analogue for quantum teleportation, because theissuesinvolvedareinherentlynonclassical.Butwecangetalittleoftheflavorofwhat’s involvedby thinkingof thephoton teleportationprocess ingraphicalterms.Wecanalsogetahintofwhatquantumteleportationwillreallyrequire.Aswesawinchapter3(page66),wecanrepresentaphotonpolarizationby

anarrowindicatingthedirectionofpolarization.Wecanthinkofthehorizontalandverticalcomponentsintermsofthenumberofstepswetakeinthedifferentdirections:youtakeastepsintheverticaldirection,andbstepsinthehorizontaldirection.Inthisgraphicalpicture,teleportationisaproblemofaligningarrows.Truman

hasanarrowpointinginsomedirection,andbothdogswillgetsteakifRDcanmakehisarrowpointinthesamedirection.Howdotheymanagethis?Theonlywaythetwodogscangettheirarrowsalignedisiftheyhavesome

sharedreference.Iftheyeachhaveacompass,Trumancancomparehisarrowtothedirectionofthecompassneedle,andtellRDtopointhisarrow,say,17°eastof due north.The compass provides a reference that they both share, and anyschemeforphotonteleportationwillneedasimilarreference.

Representing light polarization as a sum of horizontal and vertical components. The larger arrowsrepresent two different photon states, while the smaller arrows are the vertical and horizontalcomponents.

The problem of teleporting a photon is much harder than simply aligningarrows, though, becauseof theno-cloning theorem.Truman can’tmeasure thedirectionofhisarrowwithoutdisturbingit.Somehow,heneedstocommunicatethe direction of his arrow to RD without measuring it. What he needs is anonlocalreference,akindofmagiccompassthatcancommunicateadirectiontoRD’s compass without making a measurement. Quantum teleportation ispossible because the quantum entanglement that we discussed in chapter 7providesthiskindofnonlocalreference.

BEAMMEAPHOTON:QUANTUMTELEPORTATION

Quantumteleportationwasdevelopedin1993byateamofphysicistsworkingatIBM(includingWilliamWoottersoftheno-cloningtheorem).Itusesafour-stepprocesstotransferanunknownstatefromoneplacetoanother:

FOURSTEPSFORQUANTUMTELEPORTATION

1.Shareapairofentangledparticleswithyourpartner.2.Makean“entanglingmeasurement”betweenoneoftheentangledparticlesandtheparticlewhosestateyouwanttoteleport.

3.Sendtheresultofyourmeasurementtoyourpartnerbyclassicalmeans.

4. Tell your partner how to adjust the state of his particle according to themeasurementresult.

This recipe for teleportation exploits quantum entanglement to generate acopyofanarbitrarystateatadistant location throughonemeasurementandaphonecall.Itusestheactivenatureofquantummeasurementtoalignoneofthetwo entangled photons with the state to be “teleported.” In the process, thesecondentangledphotonisinstantlyconvertedtoapolarizationthatdependsonthe original state. The no-cloning theorem still applies, so the state of theoriginalparticleisalteredbythemeasurement,butattheendoftheprocess,thesecond entangled photon is in the same state as the original photon before“teleportation.”Here’s how it works: let’s imagine that Truman has a single photon in a

particular polarization state, and he wants to get exactly that state to his oldfriendRD(buthecan’tjustsenditstraightthere).Anticipatingthatthissituationmightcomeup,TrumanandRDhavepreviouslysharedapairofphotonsinanentangled state, each taking one. The polarizations of these photons areindeterminateuntilmeasured,buttheyareguaranteedtobeoppositeeachother.So,thetwodogshaveatotalofthreephotons:Photon1isthestatethatTrumanwants to convey toRD (at some randomlychosenangle,describedbya|V>+b|H>), Photon 2 is Truman’s photon from the entangled pair, and Photon 3 isRD’sphotonfromtheentangledpair.TheteleportationprocedureoutlinedabovewillallowRDtoturnhisPhoton3intoanexactcopyofPhoton1.

Acartoonversionofquantumteleportation.Atthebeginningoftheprocess,Trumanhastwophotons,Photon 1 in a definite (though unknown) state that he wants to send to RD, and Photon 2 in anindeterminate state that is entangled with RD’s Photon 3. After the teleportation procedure is

completed,Trumanhastwophotonsinanindeterminatestateentangledwitheachother,andRDhasaphotonwhosepolarizationisidenticaltotheoriginalpolarizationofPhoton1.

Teleportationworksbecausequantumphysicsisnonlocal.Wesawinchapter7 that any measurement Truman makes on Photon 2 will instantaneouslydetermine the polarization of RD’s Photon 3. Of course, it’s not as simple asmeasuring the individual polarizations of Photon 1 andPhoton 2—we alreadysaw that that won’t work. Instead, what Truman does is to make a jointmeasurement of the two photons together. He measures whether the twopolarizationsarethesameordifferent—notwhattheyare,justwhetherthey’rethesame.If Truman measured the two photons individually, asking whether they’re

horizontal or vertical, there are four possible outcomes. Both photons can bevertical(wewritethisasV1V2,wherethefirstletterindicatesthepolarizationofPhoton 1 and the second that of Photon 2), both can be horizontal (H1H2),Photon 1 can be vertical andPhoton 2 horizontal (V1H2), or Photon 1 can behorizontal andPhoton2vertical (H1V2).These four outcomeswill occurwithdifferentprobabilities, dependingonwhat thepolarizationof theoriginal statewas.For teleportation, Truman doesn’t measure the individual polarizations, but

insteadaskswhether they’re the same.This stillgives fourpossibleoutcomes,twowiththesamepolarization,andtwowithoppositepolarizations.These“Bellstates”aretheallowedstatesforapairofentangledphotons,andwhenTrumanmakeshismeasurement,he’llfindPhotons1and2inoneofthesefourstates:

These states are superpositions of the four possible outcomes from theindependent measurements, just like Schrödinger’s famous cat is in asuperposition of “alive” and “dead.”* Each of the polarizations is stillindeterminate—ifyougoontomeasuretheindividualpolarizationofPhoton1,youareequallylikelytogethorizontalorvertical.WhenyoudomeasurePhoton1, you determine the state of Photon 2 to be either the same or opposite,dependingonwhichofthefourstatesyou’rein.

“Wait a minute—why are there four outcomes? Shouldn’t there just be two?What’swiththeplusesandminuses?Eitherthey’rethesame,orthey’renot.”“That’s true,but inquantummechanics, thereare twodifferent stateswhere

theyhave thesamepolarization,State IandState II,and twowhere theyhaveoppositepolarizations,StateIIIandStateIV.Thatgivesfourstates.”

“Butwhat’sthedifferencebetweenStateIandStateII?”

“They’redifferentstates, in thesamewaythat |V>+ |H>and |V>– |H>aredifferentstatesforasinglephoton.”“Wait—theyare?”“Yes.You can see it by thinking of how they add together to give a single

polarization at a different angle. You can imagine the |H> as being one stepeitherleftorright,andthe |V>beingonestepeitherupordown. |V>+ |H>isthenonestepup,andonetotheright,while|V>–|H>isonestepup,andonetotheleft.”“So,|V>+|H>is45ºtotherightofvertical,and|V>–|H>is45ºtotheleftof

vertical?”“Exactly.Theybothgiveafifty-fiftychanceofbeingmeasuredashorizontal

or vertical, but they’re different states. If you rotated your polarizer 45ºclockwise,the|V>+|H>photonswouldallmakeitthrough,whilethe|V>–|H>photonswouldallbeblocked.”“So,StateIisupandtotheright,whileStateIIisupandtotheleft?”“Well, it’dbemorecomplicated than that.Thereare twoparticles, soyou’d

needtodoitinfourdimensions,orsomething,butthat’sthebasicidea.”“Okay, I suppose I’mconvinced.Wait—yousaid theoriginal twoentangled

photonsneedtohaveoppositepolarizations.Shouldn’ttheybeinStateIIIorIV,then?”“You’reabsolutelyright.Intheusualteleportationprocedure,Photons2and3

needtobeinStateIV.Ididn’tmentionthatearlier,becauseI thoughtitwouldcomplicatethingsneedlessly.Wellspotted.”“I’maverycleverdog.Youcan’tgetanythingpastme.”

WhenTrumanmakeshismeasurementaskingwhetherPhotons1and2havethesamepolarization,Photons1and2areprojectedintooneofthesefourstates.Atthatinstant,theentanglementbetweenPhotons2and3meansthatRD’sPhoton3 is put into a definite polarization state that depends onwhich state Trumanmeasured.TherearefourpossibleresultsforthepolarizationofRD’sPhoton3,whose horizontal and vertical components are related to the horizontal and

verticalcomponentsofTruman’soriginalPhoton1.Eachresultisasimplerotationoftheoriginalpolarizationstate—thearrows

pointinadifferentdirection,butstillinvolveastepsinonedirection(up,down,left, or right), and b steps in another. Given the outcome of Truman’smeasurement,RDknowshowtorecovertheoriginalstateofTruman’sphoton,eventhoughhedoesn’tknowwhatthatstatewas.All Truman has to do, then, is phone RD and tell him the result of the

measurement. At that point, RD knows exactly what he needs to do to getPhoton3intotherightstate.

ThestateofRD’sPhoton3after“teleportation,”foreachofthefourpossibleoutcomesofTruman’sentanglingmeasurement.EachstateisasimplerotationoftheinitialpolarizationofPhoton1(dottedarrow).

BasedontheresultofTruman’smeasurement,RDcanrotatethepolarizationofPhoton3,andknowthathe’sgotexactlythestatethatTrumanstartedwith.This scheme transfers the polarization state of Photon 1 to Photon 3,

transformingitintoaperfectcopyoftheinitialstateofPhoton1.Intheprocess,though,theentanglingmeasurementmadeonPhotons1and2haschangedthestateofPhoton1sothatitisnolongerinthesamestateaswhenitstarted—it’sinanindeterminateentangledstatewithPhoton2.It’simpossibleforbothdogstoendupwithexactlythesamestate,satisfyingtheno-cloningtheorem.Wealsoseethatteleportationisnotinstantaneous.ThepolarizationofPhoton

3 is instantaneously determined when Truman makes the measurement onPhotons 1 and 2, but there’s one more step, because Photon 3 is notinstantaneously put into the correct state. Instead, it goes into one of four

possible states, depending on the outcome of Truman’s measurement. TheteleportationisnotcompleteuntilRDmakesthefinalrotationofPhoton3.RDcan’t do that until he receives the message containing the outcome of themeasurement,andthatmessagehastotravelfromonedogtotheotherataspeedlessthanorequaltothespeedoflight.

TELEPORTINGACROSSTHEDANUBE:EXPERIMENTALDEMONSTRATION

Theideaofteleportationwasfirstproposedin1993,anditwasdemonstratedin1997byagroupinInnsbruckheadedbyAntonZeilinger.*Theyproducedtheirentangledphotonsby sendingaphoton fromanultraviolet laser into a specialcrystal that produces two infrared photons, each having half the energyof theoriginalphoton.Theysentthelaserthroughthecrystaltwice,toproduceatotaloffourphotons.Onepairwasusedastheentangledpairneededforteleportation(Photons2 and3),while oneof theother twowas sent through apolarizer toprovide thestate tobe teleported (Photon1).The fourthphotonwasusedasatriggertolettheexperimentersknowwhentocollectdata.Photons 1 and 2 were brought together on a beam splitter in a way that

performedtheentanglingmeasurement.TheycouldonlydetectoneofthefourBell states, butwhen they did, they knew that Photon 3was projected into aparticularpolarization.WhentheydetectedPhotons1and2inStateIV(25%ofthe time), they sent a signal to their analyzer to measure the polarization ofPhoton3.Because theyset thepolarizationofPhoton1 themselves, theywereable to repeat the experiment many times, and confirm that Photon 3 waspolarizedatexactlytheanglepredictedbytheteleportationprotocol.TheinitialdemonstrationusedonlyoneofthefourpossibleBellstatesinthe

measurementstep, for reasonsofexperimentalconvenience,and teleported thepolarizationstateallofhalfameter.Subsequentexperimentshaveexpandedthemeasurements to include all four outcomes and extended the distanceconsiderably. In 2004 theZeilinger group teleported photons fromone side oftheDanubeRiver*totheother(adistanceofabout600meters)overanopticalfiber,showingthatteleportationispracticaloverlongerdistances.*

Schematic of the Zeilinger group teleportation experiment. An ultraviolet laser passes through adownconversioncrystal,whereitproducestwoinfraredphotons(Photons2and3),whichserveastheentangledpairforteleportation.Theultravioletlaserthenhitsamirror,andpassesthroughthecrystalagain,producinganotherpair(Photons1and4),oneofwhichservesasthephotontobeteleported,while theother isa trigger to let theexperimentersknowthatall fourphotonshavebeenproduced.Photons1and2arebroughttogetherforanentanglingmeasurement,andwhentheyarefoundintheappropriateBellstate,thepolarizationofPhoton3ismeasuredtoconfirmthe“teleportation.”

“Okay,butwhat’sthepoint?”“Whatdoyoumean?”“Well,whocaresifyoucanteleportphotonstates?”“Photonsaren’t theonlythingswhosestatescanbeteleported.Themathsis

exactly thesameforany two-statesystem,soyoucanuse thesameschemetoteleport thestateofasingleelectronspin, forexample,or transferaparticularsuperpositionoftwoenergylevelsfromoneatomtoanother.”“Yes,butifyoucanexchangetheentangledatomsorelectrons,whydon’tyou

justsendthem,insteadofteleportingthem?”“Atomicstatesandelectronspinsaresortoffragile,andit’shardtosendthem

longdistanceswithoutthestategettingmessedup.Whatyoucandoistotakeanatom,say,andentangle itwithonephotonfromanentangledpair,andusetheotherphotontoteleportthestateofthefirstatomontoanotheratominadistantplace.”“Okay,that’salittlebetter,butit’sstilljustoneatom.”“It doesn’t have to be. In 2006, a group at the Niels Bohr Institute in

Copenhagenused teleportation to transfer a collective state fromonegroupofatoms toanother.Therewereabouta trillionatoms ineachof the twogroups,which is stillpretty small compared todogsandpeople,but it shows thatyou

canapplythetechniquetoalargersystem.”“Thatstillsoundsprettyuseless,butIsupposeit’sgettingbetter.”“Thankyou.You’reverykind.”

WHATISITALLFOR?APPLICATIONSOFTELEPORTATION

Thequantumteleportationprotocolletsususeentanglementtofaithfullymoveaparticularquantumstatefromonelocationtoanother,withoutphysicallymovingtheinitialobject.Itcanbeusedtoreproducephotonstatesatdistantlocations,ortotransferasuperpositionstatefromoneatomorgroupofatomstoanother.Ofcourse,it’sstillalongwayfromthesciencefictionideal.Aswith theclassical faxmachine, theonly thing transmitted is information.

Quantumteleportationallowsustotransferaparticularstateorsuperpositionofstatesfromoneplacetoanother,inthesamewaythatthefaxmachineallowsustosendafacsimileofwhat’sprintedonapaperdocumentovertelephonelines.Ifthestatebeing“teleported”isthestateofanatom,however,therehavetobeappropriate atomswaiting at the other end of the teleportation scheme, in thesamewaythatthereceivingfaxmachineneedstobeloadedwithpaperandink.Ifthegoalistotransferanobjectfromoneplacetoanother,though,it’snot

obvious that you need quantum teleportation. Quantum teleportation moves aparticular state from one place to another, but if you’re sending an inanimateobjectlikeadogtreatfromoneplacetoanother,youmaynotneedtopreservethe exact state. As long as you have the right molecules in the right placesrelativetooneanother,itdoesn’tmakemuchdifferencetothetasteortextureofthetreatiftheatomsinthefacsimiletreatarenotinpreciselythesamestatesasthe original.All you really need is a faxmachine thatworks at themolecularlevel,andthere’snothinginherentlyquantumaboutthat.Sowhyshouldwecareaboutquantumteleportation?Quantum teleportation

maynotbeneededtomoveinanimateobjects,butitmaybecrucialformovingconscious entities. Some scientists believe that consciousness is essentially aquantumphenomenon—RogerPenrose,forexample,promotesthis ideainTheEmperor’sNewMind.Ifthey’reright,wewouldneedaquantumteleporter,notjustafaxmachine,tomovepeopleordogs,inordertoproperlyreproducetheirbrainstate.QuantumteleportationmaybethekeytoensuringthatwhenScottybeamsyouuptotheEnterprise,youarrivethinkingthesamethoughtsaswhenyouleft.We’renotevenclose to teleportingpeople, though,so thecurrent interest in

quantum teleportation involvesmuchsmallerobjects.Quantum teleportation is

useful and important for situationswhere state information is the critical itemthatneedstobemovedfromoneplacetoanother.Theprimaryapplicationforthissortofthingtodayisinquantumcomputing.Aquantumcomputer,liketheclassicalcomputersweusetoday,isessentially

alargecollectionofobjectsthatcantakeontwostates,called“0”and“1.”*Youcanstringthese“bits”togethertorepresentnumbers.Forexample, thenumber“229”wouldberepresentedbyeightbitsinthepattern“11100101.”Inaquantumcomputer,however, the “qubits”† canbe foundnot just in the

“0”and“1”states,butinsuperpositionsof“0”and“1”atthesametime.Theycanalsobeinentangledstates,withthestateofonequbitdependingonthestateof another qubit in a different location. These extra elements let a quantumcomputer solve certain kinds of problems much faster than any classicalcomputer—factoring large numbers, for example. The modern cryptographyschemes used to encode messages—whether they’re government secrets orcreditcardtransactionsontheInternet—relyonfactoringbeingaslowprocess.A working quantum computer might be able to crack these codes quickly,leadingtointenseinterestinquantumcomputingfromgovernmentsandbanks.‡Theprecisequantumstateofanindividualqubitiscriticaltothefunctioning

ofaquantumcomputer,andit’sherethatquantumteleportationmayfindusefulapplications.Acalculation involvinga largenumberofqubitsmay require theentanglement of two qubits that are separated by a significant distance in thecomputer. Teleportation might be useful as a way of doing the necessaryoperations.Furtherdowntheroad,ifwewanttoconnecttogethertwoormorequantum

computersindifferentlocations,tomakewhatJeffKimbleofCaltechcallsthe“Quantum Internet,” schemesbasedonentanglement and teleportationmaybeessential.Thiswouldallowstillgreaterimprovementsincomputing,inthesamewaythattheclassicalInternetdoesforeverydaycomputers.*Whatever its eventual applications, quantum teleportation is a fascinating

topic. It shows us that the nonlocal effects of quantum entanglement and the“spooky action at a distance” explored in the EPR paper can be put to use,moving information around in a way that can’t be done by more traditionalmeans.Itmaynothelpdogstocatchsquirrels(notyet,anyway),butit’sanothersourceofinsightregardingthedeepandbizarrequantumnatureoftheuniverse.

“Idon’tknow.Istillthinkit’srubbish.”“Why’sthat?”“Well,Imean,ifyoucallsomething‘teleportation,’Iexpectittobegoodfor

morethanjustmovingstateinformation.”

“Thatissortofunfortunate,Iagree.I’mnottheonewhomadeupthename,though.”“So, that’s it for entanglement, then? Just Aspect experiments and

teleportation?”“No,notatall.Therearelotsofthingsyoucanusequantumentanglementfor.

It’s the key to quantum computing, as I said, and you can use it for ‘densecoding,’sendingtwobitsofinformationforeveryonebittransmitted.”“That’sstilljustmovinginformationaround.”“There’salsoquantumcryptography,whereyouuseentanglementtotransmit

astringofrandomnumbersfromonepersontoanother,numbersthattheycanthenusetoencodemessagesinacompletelysecureway.There’snopossibilityof anyone eavesdroppingon theirmessages, because the eavesdroppingwouldchangetheparticlestates,andmessupthecodeinawaythatcanbedetected.”“Stilljustinformation.”“Well,okay,buttherearepeoplewhothinkthattheproperwaytothinkabout

quantumphysicsisintermsofinformation.Insomesense,thewholescienceofphysicsisreallyallaboutinformation.”“Really?Well,I’madog,andI’mallaboutgettingsquirrels.”“Okay,butthat’sreallyaboutinformation,too.”“Howso?”“Well,foryourinformation,there’sabigfatsquirrelsittingrightinthemiddle

ofthelawn.”“Brilliant!Fat,squeakysquirrels!”

*We’lltalkaboutwhyhemightwanttodosuchanoddthingattheendofthechapter.† Strictly speaking, a2 is the probability of finding vertical polarization, and b2 the probability of

horizontalpolarizationanda2+b2=1.So foraphotonat30° from thevertical,witha75%chanceofpassingaverticalpolarizer,a= andb=½.*Strictlyspeaking,then,beforeitsstateismeasured,Schrödinger’scatcanbeinoneoftwostates:“alive

plusdead,”or“aliveminusdead.”*The sameAntonZeilingerwas seen in chapter1heading thegroup that demonstrateddiffractionof

molecules,andchapter5doingquantuminterrogation.Hehasmadealonganddistinguishedcareerdoingexperimentstodemonstratetheweirdandwonderfulfeaturesofquantummechanics.*Inthesevenyearsbetweenthetwoexperiments,ProfessorZeilingermovedfromInnsbrucktoVienna.*There’snoinherentproblemwithsendingphotonsoververylongdistances—lightmanagestoreachus

fromdistantgalaxies,afterall—butinteractionswiththeenvironmentcandestroyentangledstatesthroughtheprocessofdecoherencediscussedinchapter4.TheViennaexperimentshowsthatdecoherencecanbeavoidedlongenoughtosendentangledphotonsoverusefuldistances.*Aclassicalcomputerusesmillionsof tiny transistorsonsiliconchips;quantumcomputerscoulduse

anythingwithatleasttwostates—atoms,molecules,electrons.†The“qu”isfor“quantum.”Physicistsarenotwidelyadmiredfortheirabilitytothinkupclevernames.‡Infact,theNationalSecurityAgencyisoneofthelargestfundersofquantumcomputingresearchin

theUnitedStates.*Ofcourse,itmayalsoleadtoquantume-mailfromdogsinNigeriaofferingninebillionkilosofkibble

ifwe’lljustprovidethebankaccountinformationtohelpwithasimpletransaction…

CHAPTER9

BunniesMadeofCheese:

VirtualParticlesandQuantum

Electrodynamics

Emmyisstandingatthewindow,wagginghertailexcitedly.Ilookoutside,andthegardenisempty.“Whatareyoulookingat?”Iask.“Bunnies made of cheese!” she says. I look again, and the garden is still

empty.“There are no bunnies out there,” I say, “and there are certainly not any

bunniesmadeofcheese.Thegardenisempty.”“Butparticlesarecreatedoutofemptyspaceallthetime,aren’tthey?”“You’restillreadingmyquantumphysicsbooks?”“It’sboringherewhenyou’renothome.Anyway,answerthequestion.”“Well, yes, in a sense. They’re called virtual particles, and under the right

conditions, the zero-point energy of the vacuum can occasionally manifest aspairsofparticles,onenormalmatterandoneantimatter.”“See?”shesays,wagginghertailharder.“Bunniesmadeofcheese!”“I’mnotsurehowthathelpsyou,”Isay.“Virtualparticleshavetoannihilate

oneanotherinaveryshorttime,inordertosatisfytheenergy-timeuncertaintyprinciple. A virtual electron-positron pair lasts something like 10-21 secondsbeforeitdisappears.They’renotaroundlongenoughtoberealparticles.”“Buttheycanbecomereal,can’tthey?”Shelooksalittleconcerned.“Imean,

whataboutHawkingradiation?”“Well,yes,inasense.Theideaisthatavirtualpaircreatednearablackhole

canhaveoneofitsmemberssuckedintotheblackhole,atwhichpointtheotherparticlezipsoffandbecomesreal.”

Thetail-waggingpicksupagain.“Bunniesmadeofcheese!”“What?”Shegivesanexasperatedsigh.“Look,virtualparticlesarecreatedallthetime,

aren’tthey?Includinginourgarden?”“Yes,that’sright.”“Includingbunnies,yes?”“Well,technically,itwouldhavetobeabunny-antibunnypair…”“Andthesebunnies,theycouldbemadeofcheese.”“It’s not very likely, but I suppose in aMax Tegmark* sort of ‘everything

possiblemustexist’kindofuniverse,thenyes,there’sapossibilitythatabunny-antibunnypairmadeofcheese(andanticheese)mightbecreatedinthegarden,but—”“AndifIeatone,theotherbecomesreal.”She’swagginghertailsohardthat

herwholerearendisshaking.“Yes,buttheywouldn’tlastverylongbeforetheyannihilatedeachother.”“I’mveryfast.”“Giventhemassofabunny,they’donlylast10-52seconds.Ifthat.”“Inthatcase,you’dbetterletmeoutside.SoIcancatchthebunniesmadeof

cheese.”Isigh.“Ifyouwantedtogooutside,whydidn’tyoujustsaythat?”“Whatfunwouldthatbe?Anyway,bunniesmadeofcheese!”I look out the window again. “I still don’t see any bunnies, but there is a

squirrelbythebirdfeeder.”“Hurray!Squirrels!”Iopenthedoor,andshegoeschargingoutsideafterthe

squirrel,whomakesitupatreejustintime.

Backinchapter2,wesawthatthewavenatureofmattergivesrisetozero-pointenergy,meaningthatnoquantumparticlecaneverbecompletelyatrest,butwillalways have at least some energy. Incredibly, this idea applies even to emptyspace.Inquantumphysics,evenaperfectvacuumisaconstantstormofactivity,with“virtualparticles”poppingintoexistenceforafleetingmoment, thankstozero-pointenergy,thendisappearingagain.Theideaof“virtualparticles”poppinginandoutofexistenceinthemiddleof

emptyspaceisoneofthemostcompellingandbizarreideasinmodernphysics.Inthischapter,we’lltalkaboutquantumelectrodynamics(“QED”forshort),theunderlying theory thatgivesrise to the ideaofvirtualparticles.We’llalso talkabouttheexperimentsthatmakeQEDarguablythemostpreciselytestedtheoryin the history of science. Ironically, though, our discussionof this ultraprecisetheoryneedstostartwiththeHeisenberguncertaintyprinciple.

COUNTINGTAKESTIME:ENERGY-TIMEUNCERTAINTY

The best-known version of the uncertainty principle is the one thatwe talkedabout in chapter 2 (page 48), which puts a limit on the uncertainties in thepositionandmomentumofaparticle.Ataveryfundamentallevel,themoreweknowaboutthepositionofaparticle,thelesswecanknowabouthowfastitismoving,andviceversa.Slightly lesswell known is the uncertainty relationship between energy and

time,whichsaysthattheuncertaintyinenergymultipliedbytheuncertaintyintimehastobelargerthanPlanck’sconstantdividedbyfourpi:

ΔEΔt≥h/4π

As with position-momentum uncertainty, this means that the more we knowaboutoneofthesetwoquantities,thelesswecanknowabouttheother.Theideathatenergyissomehowrelatedtotimemayseemstrangeatfirst,but

wecanunderstanditbythinkingaboutlight.Aswesawinchapter1(page21),theenergyofaphotonisdeterminedbythefrequencyassociatedwiththatcolorof light. A low-uncertainty measurement of energy, then, requires a precisemeasurementoffrequency.So,howdoyoumakeaprecisemeasurementoffrequency?Imaginethatyou

wanttomeasuretherateatwhichanexciteddogiswagginghertail,whichisafairlyregularoscillationwithsmallfluctuationsinthefrequencyandamplitude:sometimesshewagsa littlefaster,sometimesa littleslower,sometimesfartherto theright,sometimesfarther to the left.What is thebestwaytomeasure thewaggingfrequency?Frequency ismeasured in oscillations per second, so you need to count the

number of wags that take place in some fixed time interval. If you wait fiveseconds, and count ten tail wags, that’s a frequency of two oscillations persecond. Any such measurement will always have some uncertainty, though—whenyoucountedtenoscillations,wasitreallytenfulloscillations,orten-and-a-little-bit?Hadher tail gotten all theway to the right, orwas shewagging itfartherthattimearound?Tominimizethatuncertainty,youneedtolookoveramuchlongertime.The

uncertaintyinyourcountofwagswilltendtobeaconstant—onetenthofatailwag,say—sothemoreoscillationsyoucount,thebetteryoudointermsoftherelativeuncertainty.Ifyouwatchatailwaggingforfiveseconds,andcounttenoscillations,plus

orminusonetenth,thefrequencyyoumeasureis

f=(10±0.1oscillations)/(5seconds)=2.00±0.02Hz*

That is, the frequency is somewhere between 1.98 and 2.02 oscillations persecond.If you watch for fifty seconds (assuming the dog doesn’t explode from

impatience), you’ll count a hundred wags, plus or minus one tenth, and thefrequencyisthen

f=(100±0.1oscillations)/(50seconds)=2.000±0.002Hz

Increasingthenumberofoscillationsthatyoumeasureleadstoadecreaseintheuncertaintyofthefrequency,andamoreprecisedeterminationofjusthowhappythedogis.Thecostof thatdecrease in frequencyuncertainty isan increase in the time

uncertainty.Togetonetenththeenergyuncertainty,youspenttentimesaslongmaking the measurement, which means you can’t say exactly when youmeasured the frequency. You know the average frequency over those fiftyseconds,butyoucan’tpointtoaspecificinstantandsaythatthefrequencyrightthenwas2.000Hz.All you can say is that over that fifty-second interval, thedog’stailwaswaggingatabout2Hz,butatanygiveninstant,itmayhavebeenfasterorslower.You could bemore specific about themeasurement time, bymeasuring for

only half a second, say, and counting one tail wag, but then the frequencyuncertainty becomes much larger: f = 2.0 ± 0.2 Hz. You can’t have a smalluncertainty in both the frequency of an oscillation and the time when it wasmeasured.The same logic applies to measuring the frequency of light, though the

oscillationsaremuchtoofasttocountbyhand.Wedirectlyseethisuncertaintyprinciple in action in the interaction between light and atoms.We know fromchapters2and3thatatomswillbefoundonlyincertainallowedenergystates,andthattheymovebetweenthesestatesbyabsorbingoremittingphotons.When an atommoves from a high-energy state to a lower-energy state, the

frequencyoftheemittedphotonisdeterminedbytheenergydifferencebetweenthetwostates.Thatenergydifferencehassomeuncertainty,though,thatdependson how much time the atom spent in the higher-energy state. Two identicalatoms placed in the same high-energy state can thus emit photons with veryslightlydifferentfrequencies.Thedifferenceistiny—fortypicalatoms,it’saboutonehundred-millionthof

thefrequencyofthephotons.Thiscanbemeasuredusinglasers,though,andthis

tiny frequency uncertainty limits our ability tomake certainmeasurements ofatomicproperties.

“So it takes you awhile to count things.Bigdeal.What does that have to dowithbunniesmadeofcheese?”“The frequency-counting example is just one example of a more general

principle.Energy-timeuncertaintyholdsnomatterwhatformtheenergyisin.”“Why?”“Well,allformsofenergyhavetobeequivalent,becauseyoucanconvertone

formofenergytoanother.So,ifyouhaveaphotonofuncertainenergy,andyouuse it to start an electronmoving, the kinetic energy of that electronmust beuncertain.”“Istilldon’tseewhatthishastodowithbunnies.”“We know from Einstein’s theory of relativity that mass and energy are

equivalent—”“E=mc2!”“Exactly.Sincemass is justanother formofenergy,youcanconvertenergy

intomassandmassintoenergy.Masshasuncertaintyjustlikealltheotherformsofenergy,andthatuncertaintyisrelatedtohowlongthemassstuckaround.”“Soabunnymadeofcheesewouldhaveanuncertainmass?”“Right. If itwasaroundforonlyashort time, theuncertaintycouldbevery

large—for something like a top quark, which sticks around for only 10-25seconds,thequantumuncertaintyduetothatlifetimeiscloseto1%ofthetotalmass.”“Butifitwasaroundforalongtime,themassuncertaintywouldbesmall?I

wantabunnymadeofcheesewithasmallmassuncertainty!”“Goodluckwiththat.Abunnymadeofcheesewouldhaveanawfullyshort

lifetimearoundyou.”“Okay.Goodpoint.”

WHENTHEHUMANSAREAWAY…:VIRTUALPARTICLES

Howdoes thisgetus rabbitsmadeof cheese?Well, let’s thinkabout applyingthisuncertaintyprincipletoemptyspace.Ifwelookatsomesmallregionoveralong period of time, we can be quite confident that it is empty. Over a shortinterval, though, we can’t say for certain that it isn’t empty. The spacemightcontainsomeparticles,andinquantummechanics,thatmeansitwill.Uncertainty about the emptiness of space isn’t as strange as itmay seemat

first.Ifaphysicistorastagemagiciangivesadogaboxtoinspectatleisure,shecan conclusively state that the box is empty. She can sniff in all the corners,checkforfalsebottoms,andmakeabsolutelysurethatthere’snothinghidinginsomelittlerecess.Ifshe’sallowedonlyabriefpeekoraquicksniff insidethebox, though, she can’t be as confident that the box is empty. Theremight besomethingtuckedintoacornerthatshewasn’tabletodetectinthatshorttime.The amount of time needed to determine whether the box is empty also

dependsonthesizeofthethingyoumightexpecttofind.Youdon’tneedtolookfor very long to determine whether the box contains Professor Schrödinger’sfamouscat,butifyou’reattemptingtoruleoutthepresenceofamuchsmallerobject—acrumbofadogtreat,say—amorethoroughinspectionisrequired,andthattakestime.*Thesameideaappliestoemptyspaceinquantumphysics,viatheenergy-time

uncertainty relationship.Whenwe lookatanemptyboxovera longperiodoftime,wecanmeasureitsenergycontentwithasmalluncertainty,andknowthatthereisonlyzero-pointenergy—noparticlesareinthebox.Ifwelookoveronlyashortinterval,however,theuncertaintyintheenergycanbequitelarge.SinceenergyisequivalenttomassthroughEinstein’sfamousE=mc2,thismeansthatwe can’t be certain that the box doesn’t contain any particles. And as withSchrödinger’scat,ifwedon’tknowtheexactstateofwhat’sinthebox,it’sinasuperpositionofall theallowedstates.Thecat isbothaliveanddead,and theboxisbothemptyandfullofallmannerofparticles,atthesametime.To put it another way, the box can contain some particles, as long as they

appearanddisappearfastenoughthatwedon’tseethemdirectly.Buthowcanwearrange forparticles todisappearsoconveniently?Ordinaryparticlesdon’tdisappearwhenwe’renotlooking,unlessthey’reedibleandthere’sadoginthearea.Particlescandisappearfromtheboxprovidedtheycomeinpairs,onematter

andone antimatter.Everyparticle in theuniversehas an antimatter equivalentwiththesamemassandtheoppositecharge—theantiparticleforanelectronisapositron,theantiparticleforaprotonisanantiproton,andsoon.Whenaparticleof ordinarymatter comes into contact with its antiparticle, the two annihilateeachother,convertingtheirmassintoenergy.Inpractical terms, thismeansthataparticleanditsantiparticlecanpopinto

existencefromnothinginsidethebox.This temporarily increases theenergyinside theboxbyasmallamount—an

electron-positronpairincreasestheenergybytwoelectronmassesmultipliedbythe speed of light squared—but as long as theymutually annihilate in a shortenoughtime,there’snoproblem,becausetheuncertaintyintheenergyis large

enoughtocovertwoextraparticles.*Howlongdoweneedtolooktoruleoutanypossibilityoftheboxcontaining

aparticular typeofparticle?Well,weneed theuncertainty in theenergy tobesmallerthantheenergythattheparticlehasfromEinstein’sE=mc2,sothetimewilldependon themass.Tomake theenergyuncertaintysmallenoughtoruleoutthepossibilityoftheboxcontainingoneelectronandonepositron,weneedonly to lookat thebox for10-21 seconds.That’sabillionthofa trillionthofasecond—slightlylessthanthetimerequiredforlighttotravelfromonesideofanatomtotheother.Aprotonandantiprotonhaveagreatermass,andwouldlasteven less time—10-24 seconds or less. A rabbit with a mass of a kilogram(whetherit’smadeofcheeseorsomethingelse)wouldlastfor10-52seconds,†or0.0000000000000000000000000000000000000000000000000001s.“Ithinkyou’remakingthismuchmorecomplicatedthanitneedstobe.How

hardisittomeasurezeroparticles?”“This,fromadogwhohastosnifftheentirekitchenflooreverynight,onthe

offchancethatwedroppedamoleculeoffoodwhilemakingdinner?”“Butyoudodropfoodsometimes,andIlikeyourfoodbetterthanmine.”“Thepoint is, it’s really difficult tomeasure zero.You cannever really say

thatsomethinghasadefinitevalueofzero,onlythatthevalueisnobiggerthanthe uncertainty associated with your measurement. Some people spend theirwhole scientific careers measuring things to be zero, with better and betterprecision.”“Thatsoundsprettydepressing.”“Itdoesrequireacertainpersonalitytype.Anyway,that’ssortofasideissue,

becauseweknowthatvirtualparticlesexist.Wecandetecttheireffects.”“Wait, if these things only stick around for such a short time, how do you

detectthem?”“Wecan’tdetect them,notdirectly.Weknowtheyexistbecausewecansee

theeffects theyhaveonotherparticles.Virtualparticlespopping inandoutofexistencechangethewaythatrealparticlesinteractwithoneanother.”“Howdoesthatwork?”“Well…”

EVERYPICTURETELLSASTORY:

FEYNMANDIAGRAMSANDQED

Virtual particles, particles that appear and disappear too quickly to be seendirectly,mayseem too fanciful fora serious scientific theory. In fact, theyareabsolutelycriticaltothetheoryknownasquantumelectrodynamicsorQED,whichdescribestheinteractionbetweenlightandmatteratthemostfundamentallevel. QED describes all interactions between electrons, or between electronsand electric or magnetic fields, in terms of electrons absorbing and emittingphotons.Thebest-knownformofQEDreliesonpicturescalledFeynmandiagrams,

afterthenotedphysicistandcolorfulcharacterRichardFeynman,whoinventedthem as a sort of calculational shortcut. These diagrams represent complexcalculations in the form of pictures that tell a story about what happens asparticles interact.* The simplest Feynman diagram for an electron interactingwithanelectricormagneticfieldlookslikethis:

Thestraightlinesrepresentanelectronmovingthroughsomeregionofspace,andthesquigglylinerepresentsaphotonfromtheelectromagneticfield.Inthisdiagram, time flows from the bottom of the picture to the top of the picture,while the horizontal direction indicatesmotion through space, so the diagramitself is a story in pictures: “Once upon a time, therewas an electron, whichinteractedwithaphoton,andchangeditsdirectionofmotion.”Youmightthinkthatthislittlestory,boringasitis,isthecompletepictureof

what happens when an electron interacts with light, but it turns out there aremyriadotherpossibilities,thankstovirtualparticles.Forexample,wecanhavediagramslikethese:

These tell much more eventful stories: on the left, our electron is movingalong through space, and just before it interacts with the photon from the

electromagnetic field, itemitsavirtualphoton*andchangesdirection.Then itabsorbs the realphoton from the field,changesdirectionagain, then reabsorbsthe virtual photon it emitted earlier. The right diagram is even stranger: thevirtual photon spontaneously turns into an electron-positron pair,† which thenmutuallyannihilate,turningbackintoaphoton,whichgetsreabsorbed.These diagrams show the odd relationship between virtual particles and the

normallawsofphysics.Thevirtualparticlesseemtodisobeysomebasicrulesofphysics—arealelectroncouldnevermovefastenoughtogetbackinfrontofaphotontoreabsorbit—butvirtualparticlescanbreaktherulesaslongasthey’renot around long enough to violate energy-time uncertainty. It’s similar to therelationshipbetweenadogandfurniture:whenhumansarelooking,beingonthesofaisstrictlyforbidden,butaslongastherearenohumansaround,andshegetsoffbeforetheyseeher,it’sagreatplacetonap.Eachof theseFeynmandiagramsrepresentsa tinysliceof thestoryofwhat

might happen to an electron interacting with a photon from an electric ormagnetic field. Each of these diagrams also stands for a calculation in QEDgivingthefinalenergyoftheelectron,andthelikelihoodoftheeventsdepicted.Asyouincreasethenumberofvirtualparticles,thediagramsbecomemuchlesslikely,buttheyremainpossible,aslongastheyhappenveryquickly.The diagrams on the previous page only involve electrons, positrons, and

photons,butanytypeofparticlecanturnupasavirtualparticle.Thelikelihoodof a particle appearing decreases as its mass increases, so a virtual proton-antiprotonpairismuchlesslikelythanavirtualelectron-positronpair(aprotonhas almost 2,000 times themass of an electron), but in principle, there is nolimit.Ifyouwaitlongenough,youshouldexpectanyandeverytypeofobjectshowingupasavirtualparticle—evenrabbitsmadeofcheese.

“Okay,sovirtualparticlesaffectthewayelectronsinteractwithphotons.Bigdeal.WhyshouldIcare?”“ThephotonsinthoseFeynmandiagramscouldrepresentanysortofelectric

or magnetic interaction. Those diagrams could be describing an electroninteracting with another electron, or with a proton, and that happens all thetime.”“Okay,Ihavenoideawhatyou’retalkingabout.”“Well,whenyou’redealingwithQED,youtalkaboutinteractionsintermsof

theexchangeofparticles.Twoelectronsthatrepeleachotherdosobypassingaphoton from one to the other—one emits a photon, which travels over to theother, andgets absorbed.Theabsorptionandemissionchange themomentum,andweseethatasaforcepushingthetwoparticlesapart.”

“Thatsoundscomplicated.Whywouldyouthinkaboutitthatway?”“Itturnsouttobemoreconvenient,mathematically.Youcanalsolookatitas

anaturalwaytoincorporatethefactthatnothingcantravelfasterthanlight.Inthe classical picture of electric forces developed before Einstein, when youchange the position of one electron, the force on the other should changeinstantaneously,nomatterhowfarawayitis.Thatgoesagainstrelativity,whichsaysthatthere’snowaytotransmitanythingfasterthanlight.”“Oh.That’saproblem.”“That’s right. If we think of the forces as arising from photons which are

passedfromoneparticletoanother,though,thattakescareoftheproblem.Thephotons travel at the speed of light, and the force doesn’t change until thephotonsgetthere.”“So,thephotonsinthosediagrams…”“Theycouldberealphotons,duetotheelectroninteractingwithamagnetor

an electric field, or they could be photons emitted by another electron. Eitherway, the effect is the same—the presence of virtual particles changes theinteraction,andwecandetectthat.”“Youstillhaven’texplainedhow todetect it, though.”“I’mgetting there. If

you’dstopinterrupting…”

THEMOSTPRECISELYTESTEDTHEORYINHISTORY:

EXPERIMENTALVERIFICATIONOFQED

Virtual particles come and go in an extremely short time, faster than we candirectlyobserve.Weknowtheyexist, though,because interactionswithvirtualparticles change the way electrons interact. The effect is tiny, but we canmeasure it, and it agrees with experimental observations to fourteen decimalplaces.Not only do these experiments confirm thatQED is correct, theymayprovideawaytodetecttheexistenceofnewsubatomicparticleswithoutusingparticleacceleratorscostingbillionsofpounds.Howdoesthiswork?Well,becausethevirtualparticlesarearoundforsucha

short time,we don’t knowwhich of themany possibilities took place for anygivenelectron.Quantummechanicstellsusthatifwedon’tknowtheexactstateofaparticle,itexistsinacombinationofallthepossiblestates—asuperpositionstate like those we’ve discussed in previous chapters. So, when physicistscalculatetheeffectofanelectroninteractingwithanelectricormagneticfield,theyneedtoincludeallthepossibleFeynmandiagramsdescribingtheprocess.*

Inacertainsense,astheelectronpassesfrompointAtopointB,itfollowsallthepossiblepathsfromAtoBatthesametime.Itabsorbsasinglerealphoton,butalsoemitsandreabsorbsavirtualphoton,andthatvirtualphotondoesanddoesn’tturnintoanelectron-positronpair,andsoon.Alloftheseprocessesarepossible,sotheyallcontributetothesuperpositionofFeynmandiagrams.Anotherwayofthinkingaboutthisistonotethatweneverseearealelectron

interactingwithasinglephoton;rather,whatweseeisthecumulativeeffectofagreatmanyrepeatedinteractions.Ifwecouldwatcheachoftheseindetail,wewouldseethatmostofthetime,theelectronsimplyabsorbsthephoton,withnovirtual funny business.Maybe one time in ten thousand, itwill emit a virtualphoton.Onevirtualphotonintenthousandwillcreateavirtualelectron-positronpair,andsoon.Eachtimeyougetoneofthesedifferentprocesses,itchangesthetotalenergybyasmallamount,andthatneedstobetakenintoaccount.Inthisview,theelectronislikeadogwalkingdownthestreet.Thechancesof

thedogstoppingtosniffanygivenplantareverysmall,*buttherearedozensofplantsalongthesideofanysuburbanstreet,andoneofthemisboundtosmellfascinating,anddemandfurtherinvestigation.Thehumanholdingtheleadwillneedtotakethatdelayintoaccountwhenplanningthewalk.Wedon’thavetheabilitytowatcheachoftheinteractionsofarealelectronin

sufficient detail to keep track of just how many times virtual particles wereinvolved in the interaction,anymore thanwecanpredictexactlywhichplantsthe dog will stop to sniff, but we can model the effect that this has on theinteraction.AddingupFeynmandiagramscanbe thoughtofascalculating theaveragechangeintheenergyoftheelectronfromabsorbingonerealphoton.Wethendescribe thecumulativeeffectofmanyphotonabsorptions,whichmayormay not involve virtual particles, as a series of absorptions with the sameaverageenergy,inthesamewaythatwecandescribetheprogressofadogbysaying thathewill stop three timesonevery streetonaverage. Itmaybe fivetimesonsomestreets,andonlyonetimeonothers,butovertheentirewalk,theeffectisthesameasifhestoppedthreetimesoneverystreet.Whetheryou thinkof it as a superpositionof all possiblepaths at the same

time,oranaverageinteractionbeingusedtocoverthedetailsofmanyindividualinteractions,theeffectofvirtualparticlesshowsupwhenwelookatanelectroninteractingwithamagneticfield.Electrons(andallothermaterialparticles)haveapropertycalled“spin”thatmakesthemactliketinylittlemagnets,withnorthand southpoles.*The energy of an electronwhose poles are alignedwith themagnetic field is very slightly different than the energy of an electronwhosepolespointintheoppositedirection.Theenergydifferencebetweenthesestates inamagneticfielddependsona

number called the gyromagnetic ratio, or g-factor, of the electron, whichbasicallytellsyouhowbigamagnetyougetforagivenamountof“spin.”Thesimplestquantumtheoryoftheelectronsaysthatthevalueofthisratioshouldbeexactly2,whichiswhatyouwouldgetiftherewerenovirtualparticles.Thankstothecontributionsofvirtualparticles,theactualvalueisveryslightlyhigher.Thisvaluecanbemeasuredwithextraordinaryprecision.In2008,ateamof

physicists led by Gerald Gabrielse of Harvard made the most accuratemeasurement to date of the electron g-factor in experiments involving singleelectronsheldinaPenningtrap.Theexperimentalvaluetheyobtainedwas:

g=2.00231930436146±0.00000000000056

They compared their result to the result of aQEDcalculation that involvedsumming almost a thousand individual Feynman diagrams*—includingprocesses much more complex than those described above—and theory andexperimentwere inperfectagreement, to fourteendecimalplaces.This iswhywecansaywithconfidencethatvirtualparticlesexist,nomatterhowstrangetheideamayseematfirstglance.Whilethisagreementisextremelyimpressive,adisagreementwouldbeeven

moreinteresting.Theg-factorfortheelectronincludesonlytheeffectsofvirtualphotons and virtual electron-positron pairs, but the analogous quantity for themuon(atypeofparticlesimilartoanelectron,butwithalargermass)showstheeffects of a number of other, more exotic, virtual particles.† The latestmeasurements of theg-factor of themuon showa tinydifferencebetween theexperimental value and the theoretical prediction. This difference might bemerelyacalculationormeasurementerror,oritmighthintattheexistenceofanew particle thatwasn’t included in the calculations—some particle that isn’tpartofthestandardmodelofparticlephysics.It’sstillmuchtooearlytosayforsure,butifthisdifferenceholdsup,itcouldbethefirstexperimentaltestofthemanytheoriesforphysicsbeyondthestandardmodel.Experimentalistshavealongwaytogobeforetheystarttoseetheeffectsof

virtualrabbitsmadeofcheese,ofcourse.Butifitispossibletomakerabbitsoutofcheese,weknowthattheymustbeouttheresomewhere.

“So that’s what you mean when you say ‘Quantum mechanics is the mostaccuratelytestedtheoryinthehistoryoftheories.’”“Well,Ididn’tsayitlikethat…”“Whatever.Yousayitallthetime,anditsoundsreallypompous.”“Fine.Anyway,yes, that’swhat Imean.Quantumelectrodynamicshasbeen

used to predict the g-factor of the electron to fourteen decimal places, and itagrees perfectly with experimental measurements. And QED is a relativelystraightforward extension of ordinary quantum mechanics to situations whereyouneedtothinkaboutrelativity.”“So,whatelseisitgoodfor?”“Well,asIsaid,youcanhaveallsortsofdifferentthingsasvirtualparticles.

This includes particles that have been predicted by theorists, but not observedyet—iftheyreallyexist,they’llshowupasvirtualparticles.”“Howdoesthathelpanything?”“Some of those hypothetical particles would allow interactions that aren’t

possiblewith any of the particleswe know about. In addition to themuon g-factor, there’s the ‘electric dipolemoment’ of the electron. If the right sort ofparticles exist, they would change the way an electron interacts with electricfieldsinsideatomsandmolecules.Itmightbepossibletodetectthetinyshiftintheallowedstatesinexperimentswithlasers.”“So,ifyousawthisdipolemomentthing,youwouldknowthatnewtypesof

particlesexist?”“Andifyoudon’tseeit,youcanruleoutsometypesofparticles.Thefactthat

nobodyhasobservedanelectricdipolemomentinanelectronyethasruledoutsome of the simpler models that theorists like to use. If some of the newexperimentsdon’tseeanything,itcouldreallymakethetheoristssquirm.”“That’sprettycool.”“It’s especially cool because these are tabletop experiments, not expensive

particle accelerators. There are loads of groups doing these experiments—atBerkeley, Yale, Washington, Colorado, and other places—together with theLarge Hadron Collider, they’re our best chance of learning something reallynew.”“Ofcourse,noneofthishelpswithwhat’sreallyimportant.”“Thatbeing?”“Providingmewithbunniesmadeofcheese.”“Well,whatcanIsay?Insomeareas,physicsstillhasalongwaytogo.”“I’llsay.”

*MaxTegmarkisacosmologistatMITknownforproposingthatouruniverseisoneofavastnumberof

universesina larger“multiverse.”AccordingtoTegmark, thismultiversecontainseverypossiblekindofuniversethatcanbedescribedmathematically,eventhosethatwouldmakenosensetous.Tegmark’sworkissomewhatsimilartothe“modalrealism”ofthephilosopherDavidKelloggLewis.*Theunitsforfrequencyare“Hertz,”abbreviatedHz,aftertheGermanphysicistHeinrichHertz,who

wasthefirstphysicisttodemonstrateexperimentallythatlightisanelectromagneticwave.*It’sevenharderforhumans,whosenosesdon’tworkwellenoughtosniffoutthereallytinycrumbsin

thecorners.*Or,toputityetanotherway,iftheparticle-antiparticlepairisaroundforonlyashorttime,theirmasses

havealargeuncertainty.Ifthetimeisshortenough,theuncertaintyinthemasscanbelargerthanthemass,inwhich casewe can’t say that themasswasn’t zero. Inwhich case, they can exist for that short time,becausetwoparticleswithzeromassdon’tincreasetheenergyinsidetheboxatall.†This is an interval so short itmay literallyhavenomeaning—insomeexotic theories, time itself is

quantized,andcomes instepsof10-44s (the“Planck time”). Insuchmodels, it’s impossible tohaveanintervalshorterthanthat.*Feynmanwasfamousforhisveryintuitiveapproachtophysics,andthediagram-basedtechniquehe

developed gainedwide acceptance because it provides a convenientway of thinking about the complexcalculationsrequiredforQED.AtthesametimeFeynmanwasdoinghiswork,JulianSchwingerdevelopedamuchmoreformalapproachtothesameproblems,givingthesameresultsastheFeynmanapproachinamore mathematically rigorous way. Both approaches are widely used in theoretical physics today, andFeynman and Schwinger shared the 1965 Nobel Prize with Shin-Ichiro Tomonaga, who independentlydevelopedsomeofthesametechniquesasSchwinger.* In a Feynman diagram, any particle that appears and disappears before the end of the “story” is a

“virtualparticle.”†Apositron canbemathematicallydescribed as an electronmovingbackwards in time, another trick

inventedbyFeynman,hencethedownwardsarrow.*Of course, there are an infinite number of things thatmight happen, and thus an infinite numberof

possiblediagrams. Inpractice, though, themorecomplicated thediagram, the smaller thecontribution itmakestotheanswer,sotheoreticalphysicistsneedtoadduponlyasmanydiagramsastheyneedtomatchtheprecisionofanexperiment.*Unlessthedogisabassethound,inwhichcasehe’llwanttothoroughlysniffeverything.*Thename“spin”isbecausethisissimilartowhatyouwouldseeiftheelectronwereaspinningballof

charge.Theelectronsarenotliterallyspinning,butthemathsisthesame.*Animpressiveachievementinitsownright,donebythegroupofToichiroKinoshitaatCornell.†Themuong-factorcalculationsincludevirtualmuons,tauparticles,quarks,andgluons,whichaccount

formostoftheknowntypesofsubatomicparticles.

CHAPTER10

BewareofEvilSquirrels:

MisusesofQuantumPhysics

I’mputtingseedinthesquirrelproofbirdfeederwhenIhearalittlevoiceabovemyhead.“Pssst!Oi,human!”There’sasquirrelperchedonthebranchofatree,staringatme.Ilookaround,

butEmmyisonthefarsideof thegarden, intentlysniffingthebaseof thebigoaktree.“Whatdoyouwant?”Iask.“Howaboutyougivemesomeofthatseed?”“Idon’tthink—what’swithyourface?Isthatagoatee?”“It’safake.Wewear’emtoconfusethedog—shethinkswe’refromanother

dimension.Look,howaboutsellingmesomebirdseed,then?”“What’sasquirrelgoingtousetobuybirdseed?”“How does free energy sound, hmm?” Somehow, despite the fake goatee,

fuzzytail,andprotrudingteeth,hemanagestolooksmug.“Freeenergy?”“Yes,wecantellyouhowtoextractanearlyinfiniteamountofenergyfrom

ordinarywater.Thatoughttobeworthsomebirdseed,eh?”“Really.Freeenergy.”“Youbet.Weextractthezero-pointvibrationalenergyfromwatermolecules,

leavingtheminalowerenergystatethanordinarymolecules.Youcanturnthatenergy directly into electricity, and use it to power lights, or computers, orbirdseed-making machines.” Looking closely, I see that the fake goatee isobviouslyheldonbystring.“Soundstoogoodtobetrue.What’sthecatch?Toxicwasteproducts?”“No, no—the only waste is still water. In fact, it’s better than water. It’s

superwater!”“What’ssosuperaboutit?”“Well, it’s in a different quantum state, right? So it’s got, like, special

propertiesandstuff.Youcandrinkit,andit’llcurediseases.”“Howdoesthatwork?”“Well,youdrinkit,andconcentrateonmeasuringyourwavefunctiontobein

ahealthystate.Ifyoudoitright,youcanthinkyourwaytoperfecthealth.”“Youdon’tsay.”“Yes,we runsomeworkshopsandclassesandstuff. I’mahundredandsix,

andinperfecthealth.Forjustahandfulofbirdseedaweek,we’llletyouinonthesecret.”“Really?”Thedogisstillonthefarsideofthegarden,carefullycheckingthe

bushesforrabbits.“Youcanalsouseittopoweraquantumcomputer,butthat’llcostmorethan

birdseed.For a jar of peanut butter aweek, you can have the schematics of aquantum computer that you can use to crack the encryption on credit cardtransactions.”“Crikey.”“Iknow.Prettycool,eh?”“Thedogwasright.Youareevilsquirrels.”“Asifyouwouldn’tdoitifyouknewhow.So,how’boutthatbirdseed?”“Okay,I’llgiveyousomebirdseed…”Iputalittlepileofseeddownonthe

ground,aboutsixfeetfromthetrunkofthetree.“Thanks,mate,”saysthesquirrel,scamperingdownthetrunk.“You’reastar.”“Don’tmentionit,”Isay,steppingbetweenthesquirrelandthetree.“Emmy!”

Acrosstheyard,herheadsnapsaround.“Look!Evilsquirrel!”“Ooooh!”Shecomeschargingacrosstheyard,teethbared.Thesquirreltries

to runbackup the tree, but I blockhis path, so he turns and flees toward themapletreesatthebackofthegarden,withthedogsnappingathistail.Ispotsomethinginthegrass,andbenddowntoretrieveatinyfakegoatee.I

dropitinthebinonmywaybackintothehouse.

In the preceding chapters,we have talked about a lot ofweird andwonderfulfeatures of quantummechanics.Wave-particle duality, quantummeasurement,EPRcorrelations,virtualparticles—somanyaspectsofquantumtheorydefyoureverydayexperiencethatquantummechanicsstartstoseemlikemagic.Noneofthenormalrulesseemtoapply,anditmaylookasthoughabsolutelyanythingispossible.This is a commonmisconception regarding quantummechanics, and you’ll

finditrepeatedinlotsofplaces.AlittletimewithGooglewillturnupdozensofsitesoffering“quantum”methodstoproduceenergyfornothing, improveyourhealthandwell-being,orevenamasswealthandpower:lotsofpeopleoutthere

aremakingmoneybypeddlingquantummechanicsasmagic.Quantummechanicsisnotmagic,though.Nomatterhowunlikelyoramazing

it seems, quantummechanics is a scientific theory that has to conform to thegeneral principles of physics. The word “quantum” in the description of aphenomenonordevicedoesnotallowittocreateenergyoutofnothingorsendmessagesfasterthanthespeedoflight.Theseprinciplesarebuiltintothedeepstructureoftheuniverse.Quantummechanicsisnotonlycompatiblewiththoserules,butinsomecases,theserulesarisefromquantumbehavior.While many of the predictions of quantum mechanics seem to defy our

everydayintuitionforhowtheworldworks,theydonotsuspendalltherulesofcommon sense. In particular, they do not supersede the most importantcommonsenserulefordealingwiththeworld:ifsomethingsoundstoogoodtobetrue,italmostcertainlyis.We’ve spent the bulk of this book talking about the wonderful features of

quantum theory, but I want to close on a cautionary note. There are a lot ofpeoplepeddlingafalseversionofquantummechanicsasmagic,offeringresultsbeyondyourmostwildlyunrealisticdreams.Someofthemareconartists,andsomeof them are sincere but deluded, but they’re allwrong.The conmen arehard to separate from themerely confused, but it is not difficult to spot falseversionsofquantumtheory,andinthischapter,I’llpointoutafewofthemostcommonproblems.

A“QUANTUM”FREELUNCH:FREEENERGY

Oneofthetwomainareasworkedbyconartistsabusingquantumtheoryisthefieldof“freeenergy.”Freeenergyfraudstersalwaysclaimtohavedevelopedaschemethatwillproducehugeamountsofenergyforatrivialamountofwork.Thistakeslotsofdifferentforms,butthebasicappealisalwaysthesame:youputasmallamountofworkin,andgetalargeamountofelectricityout.All itwilltakeisasmallinvestmentofcashtogettheprototypesystemworking,andsoonyou’llbeoutfromunderthethumbofthepowercompanyforever…This is nothing more than a claim to have invented a perpetual motion

machine,andscientistshaveknownforhundredsofyearsthatperpetualmotionmachinesareimpossible.Quantummechanicsdoesnotchangethatconclusion.The most common fake explanation for a “quantum” perpetual motion

machine is that it is tapping the zero-point energy of some systemor another.Thiszero-pointenergyistheenergythatquantumphysicstellsusispresentevenin a system in its lowest possible energy state.When you’ve extracted all the

energy thatyoucanfromaquantumsystem—reduced thekineticenergy to itslowest possible value and removed outside interactions that would raise thepotentialenergy—thereisstillsomeresidualenergyleftinthesystem.Conartists liketopoint tothiszero-pointenergyasaresourcetobetapped.

“There’sstillenergy there,” theysay,“andourdevice taps thatenergy tokeeptheperpetualmotionmachinemoving.”Wesawinchapter2(page52),though,thatzero-pointenergyexistsbecause

matterisfundamentallywavelike,andquantumparticlesmustalwayshavesomewavelength. For the energy of a system to truly be zero, it would have to beperfectly still at a specific position, and that is impossible for any systemdescribed by a wave. Zero-point energy, like the Heisenberg uncertaintyprinciple, is a consequence of the fundamental wave nature ofmatter. Just asthere isnoway toevade the limits imposedbyuncertainty, there isnoway toextract the zero-point energy to do useful work. Trying to use the zero-pointenergyislikeaskingforhalfaphoton—it’sarequestthatmakesnosense.Probably the most successful proponent of free energy via bogus quantum

theory to date is a company called Black Light Power. Bob Park of theUniversity of Maryland and the American Physical Society has spent nearlytwenty years debunking the claims of the company’s founder, Randell Mills,which is well covered in Park’s book Voodoo Science: The Path fromFoolishness to Fraud (Oxford, 2000). Despite Park’s efforts, though, BlackLightPowerisstillaround,peddlingaremarkableenergy-generatingprocessinwhich (according to their website) “energy is released as the electrons ofhydrogen atoms are induced by a catalyst to transition to lower-energy levels(i.e., drop to lower base orbits around each atom’s nucleus) corresponding tofractionalquantumnumbers.”Thesemysterious lower-energyhydrogenatoms,called“hydrinos,”areclaimedtohaveallsortsofmagicalproperties,supposedlyenablingnewhigh-voltagebatteriesandmiraculouslightsources(noneofwhichareavailableyet,butthey’repromisedanydaynow).Thissoundsimpressivelyscience-y,butevenadogcantellthatit’snonsense.

Hydrogen is the simplest atom in the universe, consisting of a single protonorbited by a single electron. The first quantum model of hydrogen was putforward by Niels Bohr in 1913, and a full quantum treatment was developedusingtheSchrödingerequationinthe1920s.Quantumelectrodynamicswasfirstappliedtohydrogenin1947,andmodernQEDmodelsofhydrogenagreewithexperimentalresults to thesamephenomenalprecisionas themeasurementsoftheelectrong-factor.Thehydrogenatomisoneofthebestunderstoodandmostpreciselytestedsystemsintheuniverse.Modern physics leaves no room for states “below the ground state” in

hydrogen. For such states to exist, our understanding of fundamental physicswould need to be so far wrong that it would be impossible to achieve thefourteen-decimal-place agreement between experiment and theory that we seewithQED.Another commonly cited source of “quantum” free energy is the “vacuum

energy.”This is justavariantof thezero-pointenergyschemethatpurports totap the zero-point energy of empty space—the constantly appearing anddisappearingseaofvirtualparticlesthatQEDshowsmustexist.“Vacuumenergy”schemesarenomorepossiblethan“hydrino”power.Empty

space does contain energy in the form of virtual particles, but those particlesappearatrandom,anddisappearagaininatinyfractionofasecond.Wehavenowayofmakingelectronsappearondemand,nordowehaveanywayofmakingthem stick around to do useful work. Vacuum energy exerts a small but realinfluenceonelectronsandotherparticles,but it isnotanenergy resource thatcanbetapped.Anyclaimofperpetualmotionor“freeenergy”isessentiallyaclaimthatyou

can get something for nothing. Any dog knows, deep down, that that’s notpossible—youdon’tgettreatswithoutdoingsomesortofatrick.Throwingtheword“quantum”arounddoesn’tchangethatbasicfact:youcan’tgetsomethingfromnothing.Anyonewhoclaimsotherwiseissellingsomething.

“Ithoughtyousaidvirtualparticlesdidbecomereal,asinHawkingradiation?”“Sort of.The idea is that an electron and a positron can appear right at the

edgeofablackhole, insuchaway thatoneof themfalls into theblackhole,whiletheotherescapes.”“Right,soelectronsarecreatedoutofnothing!”“Wrong.WhenavirtualelectronbecomesrealthroughtheHawkingprocess,

theblackholeactuallylosesabitofmass,tomakeupforit.Theenergytomaketherealelectrondoesn’tcomefromthevacuumenergy,itcomesfromtheblackhole,whichgetswhittledawaytonothing,onevirtualparticleatatime.”“So,Isupposeit’snotmuchuseasapowersource,then?”“No,notreally.Evenassumingyoucouldcontainandcontrolablackhole,it

would be consumed through the power generation process, just like anythingelse.There’snosuchthingasafreelunch.”“Yes there is. You never charge me for lunch. Or breakfast, or dinner, or

snacks…”“Youearnyourfoodbyprotectingthehousefromevilsquirrels!Speakingof

which…”

MEASURINGYOURWAYTOHEALTH:“QUANTUMHEALING”

Theothermainsourceofmisusedquantummechanicsis“alternative”medicine.BookshopsandtheInternetaboundwithpeoplepitchingquantummechanicsasthekeytohealth,wealth,andlonglife.The most common form of these claims involves quantum measurement.

Fraudsters note that in quantum theory states aren’t determined until they aremeasured.Theythenclaimthatthekeytohealthissimplytomeasureyourselfashealthy.Youcanliveforever, inthislineofthinking,byrunningaquantumZenoeffectexperimentonyourself—ifyou’realwaysmeasuringyourselftobeinfinehealth,quantummeasurementwillseetoitthatyounevergetsick.Thebest exampleof this lineof quantumquackery isDeepakChopra,who

evenhasabooktitledQuantumHealing(Bantam,1990),thefirstofastringofbestselling alternative medicine books. What is “quantum healing,” you ask?Chopraoffersaone-paragraphexplanationina1995interview:

Quantumhealingishealingthebodymindfromaquantumlevel.Thatmeansfromalevelwhichisnotmanifestatasensorylevel.Ourbodiesultimatelyarefieldsofinformation,intelligenceandenergy.Quantumhealinginvolvesashiftinthefieldsofenergyinformation,soastobringaboutacorrectionin an idea that has gonewrong. So quantum healing involves healing onemode of consciousness,mind,tobringaboutchangesinanothermodeofconsciousness,body.*

Heusesalotofscientific-soundingterms,butthisisjustwordsalad.IthasallthescientificvalidityofthetechnobabbleonoldepisodesofStarTrek—allhe’smissingisacallto“reversethepolarity”ofsomething.HeexpandsontheseideasinAgelessBody,TimelessMind(Harmony,1994)

whose subtitle promises a “Quantum Alternative to Getting Old.” Hisexplanationofthephysicalbasisofhisideasshowsanimpressiveignoranceofhistory,boldlydeclaringthat“Einsteintaughtusthatthephysicalbody,likeallmaterialobjects, isanillusion,andtryingtomanipulateitcanbelikegraspingthe shadow and missing the substance” (p. 10). This is almost exactly theopposite of Einstein’s view, as we saw in chapter 7. Einstein was profoundlydisturbed by the idea of quantum indeterminacy, which Chopra takes to anentirelynewlevel,arguingthatnothingactuallyexists:

Becausetherearenoabsolutequantitiesinthematerialworld,itisfalsetosaythatthereevenisanindependentworld“out there.”Theworldisareflectionof thesensoryapparatus thatregisters it…Allthatisreally“outthere”israw,unformeddatawaitingtobeinterpretedbyyou,theperceiver.Youtake “a radically ambiguous flowing quantum soup” as physicists call it,* and use your senses tocongealthesoupintothesolidthree-dimensionalworld.[p.11]

While he acknowledges that this descent into solipsism may sound“disturbing,”perhapsbecausecongealedsoupsoundsunappealing,hesees thisas a feature, not a drawback, writing that “there is incredible liberation inrealizing that you can change your world—including your body—simply bychanging your perception.” (pp. 11–12) In other words, since nothing reallyexists,youmightaswellbehealthy,wealthy,andyouthful.It’salljustamatterof“perception,”whichistosay,measurement.The idea of “quantum healing,” using the active nature of quantum

measurementtoensuregoodhealth,suffersfromtwomajorproblems.ThefirstproblemisthatChopraandotherauthorsareapplyingquantumideastosystemsthatare far too large toshowquantumeffects.Aswe’veseenagainandagainthroughout this book, quantumeffects are extremelydifficult to teaseout, andthelargerthesystembeingstudied,theharderitistoseequantumeffects.Thelargestobjecteverseeninaquantumsuperpositionstateisacollectionofabouta billion electrons (see chapter 4, page 101), while the quantum Zeno effect(chapter5)hasonlybeendemonstratedwithsingleparticles.Thebiggerproblem,though,isthatquantummeasurementsarefundamentally

random.Thestateofaquantumsystemisindeterminateuntilameasurementismade, and the specific outcome of an individual measurement cannot bepredicted. It doesn’t matter whether you subscribe to a Copenhagen-likeinterpretationinwhichthewavefunctioncollapsestoasinglevalue,oramany-worlds-like interpretation inwhich you simply perceive a single branch of anever-expandingwavefunction, or evenChopra’s congealed soup interpretation:thereisnowaytoknowinadvancehowagivenquantummeasurementwillturnout.Theremayverywellbeabranchofthewavefunctionoftheuniverseinwhich

everydogenjoysperfecthealthandan infinite supplyof fat, slow rabbits,butthere is no way to influence measurement outcomes to reach that universe.Meditationdoesn’thelp,positivethinkingwon’tgetthejobdone,drugsdon’tdoany good—there is no known way to influence the quantum structure of theuniverse to generate a particular outcome of a quantummeasurement, and noscientific study has ever detected a hint of one. If itwere possible to achievegreat things simply by wanting them badly enough, physicists would have amuch easier time demonstrating quantumeffects,* and dogswould never lackforsteak,cheese,andrabbits.Meditationmay loweryour stress level, and thinkingpositive thoughtsmay

improveyourmood,andeitherofthosethingsmaymakeyoufeelbetteraboutyour lot in life and thus help you find the energy to catch that rabbit.There’snothing quantum about that, though, and you’re not tapping into the deep

structureoftheuniverseinanymeaningfulway.

“You’re rightabout themeditation thing.Meditatingreallyhelpsme lowermystresslevel.”“Sincewhendoyoumeditate?”“Since always. I have Buddha nature. I like to meditate in the sun in the

garden.”“That’snotmeditating,that’ssleeping.Youreyesareclosed,andyousnore.”“That’snotsnoring,that’s…amantra.”“You’reridiculous.Whatstressdoyouhave,anyway?”“Oh,mylifeisveryhard.Ihavetoworryaboutwhethertosleepintheliving

room,orthediningroom,ortheoffice.Iworryaboutwhyyou’renotpattingme,whyyou’renotgivingmetreats…”“Okay,stopit.You’regivingmeaheadache.”“Youshouldtrymeditating!”

SPOOKYHEALINGTHROUGHENTANGLEMENT:“DISTANTHEALING”

Anothercommonformofquantumquackeryisclaimingquantumnonlocalityasa basis for “alternative” or “traditional” medicine. The claim is that thecorrelationsseeninBell’stheoremandtheAspectexperimentsshowthatthereissome deeper, “transcendent,” level of reality. This supposedly produces aconnectionbetweenalllivingthings,andpractitionerscanusethisconnectiontodiagnoseproblemsorevenhealpeoplewithoutactuallytouchingthem.It’salsotheallegedbasisforallsortsofESPphenomena.ThemostextremevariantofthisideaisfoundinbookslikeDistantHealing,

byJackAngelo(SoundsTrue,2008),whichproclaims:

Quantum mechanics scientists believe that the unified field theory connects everything in theuniverse including gravity, nuclear reactions, electromagnetic fields, and human consciousness.Modernphysics, then,supports thefindingofDistantHealing that thought forms,suchas ideasandinformation, are able to travel from one part of the human family to another via a network ofconsciousness.[pp.180–81]

Again, this is Star Trek–level stuff. There is no “unified field theory” inphysics—indeed,thelackofaunifiedfieldtheoryisoneofthegreatchallengesof modern physics. Even if there were a unified field theory, “humanconsciousness”isnotoneoftheelementsitwouldinclude.

Theidea is fleshedout inmoredetail,withmoreerroneous justifications, inTiffanySnow’sForward from theMind:DistantHealing,Bilocation,MedicalIntuition&PrayerinaQuantumWorld(SpiritJourneyBooks,2006):

Entanglement (sometimes called nonlocality) is where a faster-than-the-speed-of-light signal isinstantaneouslycommunicatedbetweentwoparticleswhichsomehowremainintouchwitheachothernomatterhowfarapart theyare.Whathappens toone instantlyhappens to theother,evenacrossagalaxy,throughtheentangledwaveformconnection.Andifwelookatthebeginningoftheuniversethrough a “Big Bang” theory, we see all energy (which includes us) was entangled in the verybeginning,soweallhaveanindent[sic]witheachothernow,eventhoughwemaybefarapart.Verysimply,whatoneofusdoes,doesaffectallothers.[p.31]

This starts off with an explanation that is only slightly wrong, but it goescompletelyofftherailsbytheendoftheparagraph.As we saw in chapter 7, entanglement does, indeed, allow for nonlocal

correlations between the states of entangled particles. However, thesecorrelationsmustfirstbeestablishedthroughlocalinteractions—thephotonsintheAspectexperimentswereinitiallyproducedbythesameatom,forexample.The quantum connection between these entangled particles is extremely

fragile, and it’s easily broken by interactions with the rest of the universe,leading to decoherence. Physicists have to work very hard to produce anentangled state that lasts even a tenth of a second. No entanglement-basedconnectioncouldpossiblysurvivethefourteenbillionyearssincetheBigBang.Since there is no residual entanglement from the Big Bang, there is no

inherentconnectionbetweenseparatedobjects.Icaneasilyarrangeforthestatesoftwodogstobecomecorrelated,asdiscussedinchapter7(page143),butonlybybringingthosedogstogetherandallowingthemtointeractwitheachother.Ifthe two dogs are always separated, there is no way for them to becomeentangled.Similarly, even leavingaside the fact thathumanorgansare far toolargetoshowquantumeffects,thereissimplynowaytoestablishaconnectionbetween, say, the liver of a patient and the hands of a “healer,”without somecontactbetweenthem.Entanglement is also invoked as an explanation for homeo pathy. In

homeopathictreatments,minutequantitiesofherbsortoxinsareplacedinwaterand thendiluted toapointwhere there shouldnotbea singlemoleculeof theoriginalherbortoxininagivenwatersample.Homeopathsclaimthatthewater“remembers”thepresenceoftheoriginalsubstance,though,andacquiressomeofitsproperties,whichsupposedlyenablesthewatertohealpatientswhodrinkit.Entanglementissometimescitedasanexplanationforthis“memory”effect,withtheclaimbeingthattheinteractionbetweenthewaterandtheherbortoxinestablishes a connection along the lines of the correlations seen in theAspect

experiments.The absolute pinnacle of the quantum-entanglement explanation of

homeopathy ispresented in theworkofLionelMilgrom,who theorizes that itinvolves not merely entanglement between water and toxin, but a three-wayentanglementbetweenthepatient,thepractitioner,andtheremedy(hecallsthis“PPR” entanglement).Milgrom has a wonderful ability to ape the jargon andnotationofquantumphysics,andwritesina2006paper:

[I]t should be possible to use notions of quantum entanglement (and by implication, informationprocessing) to illustratecertainfeaturesof the therapeuticprocess inhomeopathyandotherCAMs.*Consequently,theeffectsofinvestigatinghomeopathyandotherCAMsusingblindedtrialproceduresshouldalsobeamenabletosuchillustration.Thus,indouble-blindedprovings,eachofthecomponentsin the PPR entangled state may be thought of as two-state “macro-qubits” … and, therefore, byimplication,thehomeopathicprocessmightbeconsideredtoinvolvemacro-quantum“teleportation.”…However,itisonlytheentangledstatewhichcontainsinformationaboutthewholesystem.Thus,anything which breaks the entangled state will necessarily lead to loss of information about theintegrationoffunctionofthesystemsasawholesystem.Clearly,thiscouldhappenin[double-blindrandomized control trials†] of homeopathic efficacy, where either the remedy or patient andpractitionerareremovedfromtheirentangledtherapeuticcontext.*

Thisisatrulybreathtakingapplicationof“quantum”reasoning.NotonlydoesMilgromattributethecurativeeffectsofhomeopathicremediestothis“macro-quantum ‘teleportation,’ ” but in a lovely bit of logical judo, he uses thisargument to explain away the failure of homeopathic remedies to outperformplacebos in properly run clinical trials. It’s all quantum, you see, and thusattempting tomeasure the performance in a manner consistent with scientificprinciples ruins everything. Milgrom concludes that standard medical testingprotocols simplycan’tbeused tomeasurehomeopathybecause“they seem todestroy the very effects they were purportedly designed to investigate,” animpressiveattempttousequantummechanicstoavoid(possiblyviatunnelingorteleportation) meeting the exacting standards applied to conventional medicaltreatments.Theclaimthatentanglementexplainshomeopathyispatentnonsense.Patients

and practitioners are much too large to exhibit quantum behavior, and eventhough there is the slight possibility of an entangling interaction betweenmoleculesinasolution,entanglementisjustacorrelationbetweenthestatesoftwo systems—when an atom is in a particular state, a photon is verticallypolarized,orwhenonedogisawake,anotherdogisalsoawake.Thisdoesnotmean that one system has acquired characteristics of the other. A quantuminteractioncansetupacorrelationbetweenanatomandaphoton,but itcan’tturn an atom into a photon or water into healing elixir, any more than an

interactionbetweentwodogscanturnaLabradorretrieverintoaBostonterrier.

BEWAREOFEVILSQUIRRELS:QUANTUMPHYSICSISNOTMAGIC

Quantum mechanics is a weird and wonderful theory, and it makes someamazing things possible. Most modern technology depends on quantummechanics in one way or another—modern electronic devices and computerchips rely on quantumeffects in order to operate, and optical devices like thelasers and LEDs used in modern telecommunications are fundamentallyquantumdevices.Quantumtheorymayalsoprovideforfuturetechnologieswithamazingpotential—quantumcomputersthatcansolveproblemsfasterthananyclassical computer, or quantum cryptography systems that protect messagesusingunbreakablecodes.Asastoundingasitsresultsare,though,quantummechanicsdoesnotprovide

abasisformiracles.Itspredictionsdefyeverydayintuition,butthetheorydoesnotcompletelyoverridecommonsense.Ifsomebodypromisesresultsthatsoundtoogood to be true, chances are they’re lying, either to youor to themselves.Dropping a few quantum buzzwords into the explanation doesn’t make freeenergyoreternalyouthanymoreplausible.Therearemanyfascinatingaspectsofquantumtheorythatwehaven’ttalked

abouthere.There are also a lot of evil squirrels in theworld,withorwithoutgoatees.Thinkcarefullyaboutclaimsmaderegardingquantumeffects,andkeepinmindthatwhileitmaybeweird,quantummechanicsisnotmagic.Ifyoudothat, you’ll have no trouble finding the wonderful aspects of our quantumuniverse,andavoidingquacksandloonies.

“Yikes,that’sprettydepressing.”“What?Quantumtheorydoesn’tneedtobemagictobecool.”“No,notthat.I’mtalkingabouttheconartists.Iknewsquirrelswereevil,but

Ididn’tknowtherewerehumanswhowerethatbad.”“Yes, it’sa littledepressing toseeagoodtheoryabused in thisway.Buton

somelevel,it’sasignofprogress.”“Howdoyoucometothatconclusion?”“Well,ifyougobacktothe1800s,youcanfindpeoplemakingthesamesorts

of magical claims about electricity. All sorts of nonsensical devices wereproposedthatweresupposedtodomagicalthingsbecausetheyusedelectricity.”“Really?”“And in the mid-1900s, it was atomic or nuclear power. People suggested

using nuclear power for the most absurd things, and any number of scamsclaimedatomicpowerasabasis.”“So?What’syourpoint?”“Well, nobody really falls for either of those anymore.We’ve got used to

electricity and nuclear power, and people no longer believe ridiculous claimsmadeaboutthem.”“So‘quantum’isthenew‘atomic’?”“Prettymuch.Conartistshavehadtomoveonto‘quantum’asanexplanation,

because theoldexplanationsdon’tworkanymore.So, ina sense, the fact thatpeople are peddling ‘quantum’ nonsense means that the general public hasbecomea tinybit lesscredulousover theyears. ‘Quantum’stillworksbecausemostpeopledon’tknowwhatitmeans.”“Soyouneedtoteachmorepeopleaboutquantum.”“Exactly.Hencethisbook.”“AndI’mhelping!I’mapublic-servicedog!”“You’reaverygooddog.”“So,isthatitforthebook,then?”“Prettymuch.Why?”“Well,ifyou’refinishedwiththebook,canwegoforawalk?”“Okay.”“Andifweseeanyevilsquirrels…”“Ifweseeevilsquirrels,youcanbitethem.”“Ooooh!”

*The interview is online atwww.healthy.net/scr/interview.asp?ID=167, andwas retrieved in summer

2008.*Theplural“physicists”hereisprobablynotjustified—“radicallyambiguousflowingquantumsoup”is

notacommonphraseinphysics.TheonlyactualphysicistwhohaseveruseditappearstobeNickHerbert,apromoterof“QuantumTantra,”whichisaboutwhatyouwouldexpect.*Nottomentionsecuringfundingfortheirexperiments.*“CAM”=“complementaryandalternativemedicine.”Acronymsmakeanythingsoundmorescientific.†Double-blind randomized control trials aremedical tests inwhich patients are randomly selected to

receiveeitherthetreatmentbeingtestedoraplacebo,andneitherthepatientnorthedoctordispensingthetreatmentknowswhichiswhich.Thesearethegoldstandardformodernmedicalresearch.*LionelR.Milgrom,Evidence-BasedComplementaryandAlternativeMedicine4,7–16(2006).Quotes

fromp.14.

Acknowledgments

I learned about the physics described in this book from a large number ofmentorsandcolleaguesoveraperiodofalmost twentyyears.Manythanksaredue to Bill Phillips, Steve Rolston, Paul Lett, Kris Helmerson, Ivan Deutsch,AephraimSteinberg,LuisOrozco,PaulKwiat,MarkKasevich,DaveDeMille,SeyffieMaleki,KevinJones,JeffStrait,StuartCrampton,andBillWootters.Allthegoodpartsoftheexplanationsareultimatelyduetothem;anymistakesareoriginaltome.Igotmanyhelpfulcommentsonanearlydraftofthisbookfrommyintrepid

beta readers:JaneAcheson,LisaBao,AaronBergman,SeanCarroll,YoonHaLee,MattMcIrvin,andFrancesMoffet.MichaelNielsenandDavidKaiseralsomadehelpfulcommentsondraftcopies.Allof themhelpedmake thisabetterbookthanitwould’vebeenotherwise.Thisbookgrewoutofacoupleofpostsonmyweblog,“UncertainPrinciples”

(http://scienceblogs.com/principles/), which eventually became the openingdialoguesofchapters4and9.Thanksaredue to the folksatScience-Blogs—ChristopherMims, Katherine Sharpe, Erin Johnson, and ArikiaMillikan—forprovidingmewith a platform, and toCoryDoctorowofBoingBoing and thepeople at Digg for promoting those posts. Barrett Garese, Erin Hosier, andPatrickNielsenHaydendeservethanksforconvincingmethatwritingaphysicsbookwithmydogwasagood idea.Andof course, thanks tomyeditor,BethWareham,andagent,ErinHosier,foralltheirhelpgettingthebookintoshape,andhelpingmenavigate thepublishingprocess,which is completelydifferentthananythinginphysics.Emmy was adopted from the Mohawk & Hudson River Humane Society

shelter inMenands, New York (http://www.mohawkhumanesociety.org/). Likemostanimalshelters,theyareanexcellentsourceofwonderfuldogs(andotherpets),and Iwouldencourageanyone thinkingofgettingadog to lookat theirlocalshelter.I’vebeenluckyenoughtoknowalotofdogsovertheyears—Patches,Rory,

Truman,thelategreatRD,Bodie,andevenTinker—andthere’salittlebitofallof them in this book. Most of the credit goes to Emmy, though, who isunquestionablythebestEmmyever,andtheQueenofNiskayuna.

Manythanksareduetomyfriendsandfamily,whohavebeentremendouslysupportivedespitefindingthewholethingalittleweird.Andlast,butfarfromleast,thankstomywife,KateNepveu,forreadinginnumerabledraftsandgentlycorrectingmygrammar;forpatientlylisteningtomerantandkickideasaround;andforbabyClaire,whocomplicatedthingsinthebestwaypossible,andmostofallforinspiringthewholethingbylaughingwhenIhavesillyconversationswiththedog.Thisquiteliterallywouldnothavehappenedwithouther.

FurtherReading

DavidLindley’sUncertainty:Einstein,Heisenberg,Bohr, and the Struggle fortheSoulofScience (Doubleday,2007)providesaveryreadableintroductiontotheearlyhistoryofquantumtheory,aswellasadetailedaccountofthedebatesaboutthetheoryandthemeaningoftheuncertaintyprinciple.Louisa Gilder’s The Age of Entanglement: When Quantum Physics Was

Reborn(Knopf,2008)coverssomeofthesameterritoryasUncertainty,butwithmoreofanemphasisonentanglement,andgoesontodescribeBohm’snon-localhidden variable theory, Bell’s theorem, and the first experimental tests ofnonlocality.Thebookisbuiltaroundseveralreconstructedconversationsamongtheimportantfiguresinthestory,withdialoguepiecedtogetherfromlettersandmemoirs.TheTestsofTime:ReadingsintheDevelopmentofPhysicalTheory,editedby

LisaM.Dolling,ArthurF.Gianelli,andGlennN.Statile(PrincetonUniversityPress, 2003), reproduces many of the classic papers in early quantum theory,includingBohr’soriginalmodelofhydrogen;theEinstein,Podolsky,andRosenpaper;Bohr’sresponsetoEPR;andJohnBell’sfamoustheorem.TheTheoryofAlmostEverything:TheStandardModel,theUnsungTriumph

ofModernPhysicsbyRobertOerter(Plume,2006)givesanexcellentoverviewofthestateofmodernphysics,includingbasicquantumtheory,QED,andissuesinparticlephysicsthatarenotdiscussedinthisbook.RichardFeynman’sQED:TheStrangeTheoryofLightandMatter(Princeton

UniversityPress,2006)isawonderfullyaccessibleexplanationofthedetailsofquantum electrodynamics. His autobiographical books (Surely You’re Joking,Mr. Feynman and What Do You Care What Other People Think?) have lessphysics,butaregreatfun.Formoremathematically inclined readers,TheQuantumChallenge, Second

Edition: Modern Research on the Foundations of Quantum Mechanics byGeorge Greenstein and Arthur G. Zajonc (Jones and Bartlett, 2005) gives anexcellent overview of many of the experiments that have demonstrated thestrangefeaturesofquantummechanics.JamesGleick’sGenius:TheLifeandScienceofRichardFeynmanincludesa

descriptionof thedevelopmentofQED,and therivalrybetweenFeynmanand

JulianSchwinger.Onanartisticnote,MichaelFrayn’splayCopenhagenmakespowerfuluseof

quantum ideas in exploring the famous falling-out between Niels Bohr andWernerHeisenberg.And finally, George Gamow’s “Mr. Tompkins” stories (collected in Mr.

TompkinsinPaperback[CambridgeUniversityPress,CantoImprint,1993])aretheoriginalwhimsicalexplorationofmodernphysics,throughthedaydreamsofan unassuming bank clerk. No physics book involving a talking dog couldpossiblyfailtomentionthem.

GlossaryofImportantTerms

allowed state: One of a limited number of states in which an objectmay bemeasuredinquantummechanics.Forexample,adogatrestcaneitherbefoundonthefloor,oronthesofa,butneverhalfwaybetweenthefloorandthesofa.

antimatter:Everyparticleintheuniversehasanantimatterequivalent,withthesame mass and the opposite charge. When a particle of ordinary matterencountersitsantiparticle,thetwoannihilate,convertingtheirmassintoenergy.

Bell’s theorem: A mathematical theorem proved by John Bell, showing thatentangled quantum particles have their states correlated in ways that no localhiddenvariable(LHV)theorycanmatch.

classicalphysics:Physicstheoriesdevelopedbeforeabout1900,describingthebehavior of everyday objects.Core components areNewton’s laws ofmotion,Maxwell’s equations for electricity and magnetism, and the laws ofthermodynamics.

coherence:Apropertyofwavesorwavefunctions,roughlydefinedasbehavingasifthewavescamefromasinglesource.Addingtogethertwocoherentwavesgivesaclearinterferencepattern;addingtogethertwoincoherentwavesgivesarapidlyshiftingpattern that smearsoutandbecomes indistinct.Theprocessof“decoherence”destroysthecoherencebetweentwowavesfromasinglesourcethroughrandomandfluctuatinginteractionswithalargerenvironment.

conservationofenergy:The lawof conservationof energy states that energycanbechangedfromoneformtoanother,butthetotalenergyofagivensystemisalwaysthesame.Forexample,adogcanconvertthepotentialenergystoredinfood into kinetic energy as she chases a squirrel, but she cannot gain morekineticenergythanthetotalenergyavailableinherfood.

Copenhagen interpretation: The philosophical framework for quantummechanicsdevelopedbyNeilsBohrandcolleaguesathisinstituteinDenmark.TheCopenhageninterpretationinsistsonanabsolutesplitbetweenmicroscopic

systems,whicharedescribedbyquantummechanics,andmacroscopicsystems,whicharedescribedbyclassicalphysics.Theinteractionbetweenamicroscopicquantum system and a macroscopic measuring apparatus causes thewavefunctionto“collapse”intooneoftheallowedstatesforthatsystem.

decoherence: A process by which random, fluctuating interactions with theenvironment destroy our ability to see an interference pattern for a quantumparticle. Decoherence is particularly important for the many-worldsinterpretation,whereitensuresthatdifferentbranchesofthewavefunctionoftheuniversewillnotaffectoneanother.

diffraction:Acharacteristicbehaviorofwaves,inwhichwavespassingthroughanarrowopeningor aroundanobstacle spreadouton the far side.Adogcanhearapotatocrisphittingthekitchenfloorfromthelivingroombecausesoundwavesdiffractthroughthekitchendoorandaroundcorners.

Einstein, Podolsky, and Rosen (EPR) paradox: A famous paper by AlbertEinstein, Boris Podolsky, and Nathan Rosen that used entangled particles toarguethatquantummechanicswasincomplete.TheirargumenthasbeenprovenwrongbyexperimentstestingBell’stheorem,buthasledtothedevelopmentofquantumteleportationandotherquantuminformationtechnologies.

energy:Ameasureofanobject’sabilitytochangeitsownmotionorthemotionofanotherobject.Energycomesinmanyforms,suchaskineticenergy,potentialenergy,andmassenergy(Einstein’sE=mc2).Energymaybeconvertedfromoneformtoanother,butcannotbecreatedordestroyed.

energy-time uncertainty: A variation of theHeisenberg uncertainty principlestatingthatitisimpossibletoknowboththeexactenergyofsomeobjectandtheexacttimeatwhichitwasmeasured.Thislimitsthelifetimeofvirtualparticlesinquantumelectrodynamics.

entanglement:Aquantum“connection”between twoobjectswhose statesarecorrelated in such a way that measuring one also determines the state of theother. A classical analogy is two dogs in the same room: either both will beawake, or both will be asleep. If you measure one dog to be awake, youimmediately know that the other is also awake. Similar correlations exist forquantum particles, but their states are indeterminate until one of the two ismeasured, at which time the state of both is instantaneously determined, no

matterhowfaraparttheyare.

Feynmandiagram:Apicturerepresentingapossiblesequenceofeventsforachargedparticleinteractingwithlight.EachdiagramstandsforacalculationinQED,and theenergyof the interactingparticle is foundbyadding togetherallthe possible diagrams for that particle. The diagrams are named afterRichardFeynman,whoinventedthemasacalculationalshortcut.

gyromagneticratio/“g-factor”:Anumber,giventhesymbolg,thatdetermineshowanelectroninteractswithamagneticfield.Thesimplesttheoryofquantummechanics predicts that g = 2, but QED predicts a value that is very slightlylarger.TheexperimentallymeasuredvalueofgagreeswiththeQEDpredictiontofourteendecimalplaces.

Hawkingradiation:Aprocessbywhich“virtualparticles”causeblackholestoevaporate.Whenaparticle-antiparticlepairappearsnearablackhole,oneofthetwo can fall into theblackhole,while theother escapes. In order to conserveenergy,theblackholemustloseatinybitofmass.Overtime,theblackholeiswhittleddowntonothing,oneparticlemassatatime.

interference: A phenomenon that occurswhen two ormorewaves are addedtogether. If the peaks of one wave line up with the peaks of the other (“inphase”),theresultisamuchlargerwave.Ifthepeaksofonewavelineupwiththevalleysoftheother(“outofphase”),theresultisnowaveatall.Interferencepatterns involving single particles are the clearest demonstration of quantumbehavior.

kineticenergy:Energyassociatedwithamovingobject.Foreverydayobjects,thekineticenergyisequaltohalfthemasstimesthespeedsquared(½mv2).AGreatDanehasmorekineticenergythanaChihuahuamovingatthesamespeed,while a hyperactive Siberian husky has more kinetic energy than a sleepybloodhoundofthesamemass.

localhiddenvariable(LHV)theory:AtheoryofthesortpreferredbyEinstein,Podolsky,andRosen.InanLHVtheory,measurementsmadeinonepositionareindependentofmeasurementsmadeatotherpositions(“local”),andparticlesarealways in definite states, though the exact values are unknown (“hiddenvariables”). LHV theories cannot duplicate all the predictions of quantummechanics (according to Bell’s theorem), and have been disproven in

experimentsbyAlainAspect,amongothers.

many-worlds interpretation: The philosophical frame-work for quantummechanicsdevelopedbyHughEverettIIIatPrincetoninthe1950s.Themany-worlds interpretation avoids the “wavefunction collapse” problem of theCopenhagen interpretation by saying that all possible measurement outcomestake place in different branches of the wavefunction—in some part of thewavefunction,everydogeatssteak.Sadly,weonlyperceiveasinglebranch.Theother branches of the wavefunction are effectively separate universes, due todecoherence,which prevents the different branches from having ameasurableeffectononeanother.

measurement: Inquantummechanics,anactiveprocess thatchanges thestateofthesystembeingmeasured.Beforeameasurementismade,aquantumobjectwill be in a superpositionof all the allowed states; after themeasurement, theobjectwillbeinoneandonlyonestate.TheCopenhageninterpretationandthemany-worldsinterpretationoffertwodifferentwaysofdescribingwhathappensduringameasurement.

modern physics: Physics theories developed after about 1900, consistingprincipallyofrelativityandquantummechanics.

momentum: A quantity associated with motion that determines what willhappentoanobjectduringacollision.Inclassicalphysics,momentumismasstimesvelocity(p=mv);asmallChihuahuamustbemovingmuchfasterthanaGreatDanetohavethesamemomentum.Inquantummechanics,themomentumofaparticledeterminesitswavelength,throughthedeBroglierelationλ=h/p.

no-cloningtheorem:Amathematical theoremshowing that it is impossible tomakeaperfectcopyofaquantumobjectwithoutknowingitsstateinadvance.

particle-waveduality:Afeatureofquantummechanics,inwhichobjectshavebothparticleandwaveproperties.Classicalphysicssaysthatlightisawave,butquantumphysicssaysthatabeamoflightisalsoastreamofphotons.Classicalphysics says that an electron is a particle, but quantum physics says that anelectron also has a wavelength that depends on itsmomentum. Electrons andphotons are each “quantum particles,” a third class of object that is neitherparticlenorwave,buthaspropertiesofeach.

photoelectriceffect:Aneffectdiscoveredinthelate1800s,wherelightfalling

on metal knocks out electrons. Einstein explained the photoelectric effect in1905 by applying Planck’s quantum hypothesis to light directly, describing abeamoflightasastreamofphotons.

photon:A“particle”oflight.Abeamoflightmaybethoughtofasastreamofparticles,likekibblebeingpouredintoadog’sbowl.EachphotonhasanenergygivenbyPlanck’sconstantmultipliedbythefrequencyassociatedwiththatcoloroflight(E=hf).

Planck’s Constant (h): The constant that relates energy to frequency ormomentum to wavelength in quantum physics.The measured value ish=6.6261×10-34 J-s, or 0.0000000 000000000000000000000000006261 J-s,whichisaverysmallnumberindeed.

polarization:Apropertyoflight,correspondingtothedirectionofoscillationofthe light wave in classical physics. Any polarization can be described as acombination of vertical and horizontal polarizations, and will determine theprobabilityofthatlightpassingthroughaverticalorhorizontalpolarizer.

polarizer/ polarizing filter: Amaterial that allows through light polarized atsomeangle,andblockslightpolarized90°awayfromthatangle.Lightpolarizedatsomeintermediateanglehasaprobabilityofpassingthroughthatdependsontheangle.

potentialenergy:Energyassociatedwithanobjectthatisnotcurrentlymoving,buthas thepotential to startmoving.Adogalwayshaspotential energy, evenwhen sleeping: at the slightest sound, she can leap up and start barking atnothing.

probability:Thewavefunctionforaquantumobjectdescribestheprobabilityoffindingtheobjectinanyofitsallowedstateswhenameasurementismade.Forexample, there is a high probability of finding the dog in the kitchen, a highprobabilityoffindingthedoginthelivingroom,andaverylowprobabilityoffindingthedogonthesofa,ifsheknowswhat’sgoodforher.

quantumcomputer: A computermade up of “qubits” that not only can takevalues of “0” and “1” like the bits in a classical computer, but alsosuperpositions of “0” and “1.” Such a computer could solve certain types ofproblems, such as the factoringof largenumbers,much faster than a classical

computer. The difficulty of factoring large numbers is the basis for moderncryptography, so aquantumcomputerwould let an evil squirrel decipher yourcreditcardtransactions,andcleanoutyourbankaccounttobuybirdseed.

quantum electrodynamics: “QED” for short. The theory describing theinteractions between charged particles and light, developed by RichardFeynman,JulianSchwinger,andShin-IchiroTomonagaaround1950.Feynman’sformulationisthebestknownversion,whichdescribesinteractionsintermsoftheexchangeof“virtualparticles.”

quantum eraser: A demonstration of quantum measurement in which aninterferencepatternisdestroyedbymakingitpossibletomeasuretheexactpathtaken by a particle, but recovered by doing something to confuse thatmeasurement.

quantum field theory: A theory that combines quantum mechanics withEinstein’s relativity, to cover particlesmoving at speeds close to the speed oflight, and the interactions between such particles. The simplest quantum fieldtheoryisquantumelectrodynamics.

quantuminterrogation:A techniqueusing thequantumZeno effect todetectthe presence of an object without letting it absorb even a single photon. Itsapplicationstorabbitstalkingshouldbeobvioustoanydog.

quantummechanics/quantumphysics/quantumtheory:Thesubjectofthisbook, quantum mechanics was developed in the first half of the twentiethcentury,anddescribesthebehaviorofandinteractionsamongatoms,molecules,subatomicparticles,andlight.

quantum teleportation: A procedure for transferring the exact state of aquantumparticle fromoneplace toanotherwithoutmeasuring itormoving it,using entangled particles as a resource. Sadly, it does not allowdogs to beamthemselvesintoplaceswheretheycaneasilycatchsquirrels.

quantumZenoeffect:Ademonstrationofquantummeasurement inwhichanobjectcanbepreventedfromchangingstatesbyrepeatedlymeasuringitsstate.The classical equivalent is a dog who prevents her owner from napping byconstantlyasking“Areyouasleep?”

relativity:ThetheorydevelopedbyAlbertEinsteintodescribegravityandthe

behaviorofobjectsmovingatspeedsclosetothespeedoflight.

Schrödingerequation:Themathematicalformulathatphysicistsusetofindthewavefunction for a particular quantum system, and predict how it changes intime.

Schrödinger’s cat: A thought experiment proposed by Erwin Schrödinger,intended to show the absurdity of quantum superpositions.He imagined a catenclosedinaboxwithadevicethathada50%chanceofkillingthecatwithinonehour;quantumphysicssaysthatattheendofthehourthecatisequalpartsaliveanddead,untilitsstateismeasured.Thisexperimenthasmadehimaherotocaninephysicists.

semiclassicalargument:Adescriptionofaphysicalsystemthatismostlybasedon classical physics, with a few modern ideas added in an ad hoc manner.Examples of semiclassicalmodels include the “Heisenbergmicroscope” (page38)andtheBohrmodelofhydrogen(page49).

state/ quantum state: A particular collection of properties (position,momentum, energy, etc.) describing an object. For example, “sleeping in theliving room,” “sleeping in the kitchen,” and “running around the house” arethreedifferentpossiblestatesforadog.

superposition state: In quantum mechanics, an object can exist in asuperposition of two or more allowed states at the same time, until ameasurement is made. Such superposition states give rise to interferencepatterns,whichcanbedetectedexperimentally,eventhoughthesystemcanonlybemeasuredinoneallowedstate.

thermalradiation:Alsocalled“black-bodyradiation,”thelightthatisemittedbyahotobject, suchas thecharacteristic redglowofahotburneronastove.The spectrum of this light depends only on the temperature of the object.ExplainingthisspectrumledMaxPlancktointroducequantummechanics.

tunneling: A quantum phenomenon in which a particle that does not haveenough energy to pass over a barrier passes through the barrier anyway,appearingontheotherside,likeabaddogdiggingaholeunderafence.

uncertainty principle/ Heisenberg uncertainty principle: One of a set ofmathematical relationships limiting the precision with which complementary

properties can bemeasured. The best known uncertainty principle is betweenmomentum and position, and says that it is impossible to know both exactlywherearabbitis,andexactlyhowfastitismoving.Anyattempttospecifythepositionmorepreciselywill lead to increasedmomentumuncertainty,andviceversa. The energy-time uncertainty relationship is also important, as itdeterminesthelengthoftimethatvirtualparticlescanexistinQED.

virtualparticle:Aparticle inaFeynmandiagramthatappearsanddisappearstooquicklytobemeasureddirectly.Theseusuallyappearaspairsofonenormalparticleandoneantimatterparticle,mostoftenoneelectronandonepositron.Inprinciple, anything can show up as a virtual particle, even a rabbit made ofcheese.

wavefunction:Amathematical functionwhose squaregives theprobabilityoffindinganobjectinanyofitsallowedstates.Inquantummechanics,allobjectsaredescribedbywave-functions.

zero-point energy: The tiny amount of energy that is always present in aquantumobject,thankstothewavenatureofmatter.Confinedquantumparticlesareneverperfectly at rest—they’re likepuppies in abasket, always squirmingandwrigglingandshiftingaround,evenwhenthey’reasleep.

Index

Note:Pagenumbersinitalicsrefertodiagrams;numbersinboldtyperefertoGlossaryentries.

allowedstates72,91,140,227,271,272andelectrons61,63,64,73andions133andphotons73,85,208

Angelo,Jack258antimatter221,231,271Aspect,Alain187–93,258,259–60,275atoms:emissionsofphotons65,227,236

solar-systemmodel60–62

Bell,John168,169,177Bell’stheorem177–186,181,187,258,271,273,275andAspectexperiments187–194

Binnig,Gerd160blackholes,andHawkingradiation222,252,274BlackLightPower250–1Bohm,David97,186–7Bohr,Niels5,272andEinstein166–168,171,173,176andHeisenberg171n.,270andquantummodelofhydrogen60–62,251,281seealsoCopenhageninterpretation

Casimirforces65chaostheory75Chopra,Deepak253–6classicalphysics:andBohr’smodel60–62andCopenhageninterpretation92definition3–4,280andenergy147–151,156–7andinterpretation71andlocality174andmeasurement132andmomentum13,276–7andparticlesandwaves12–20andprobability76

andteleportation199,203–5anduncertainty46–49Clauser,John187n..coding,dense220coherence114–17,271–2Compton,ArthurHolly29,31–32computer,quantum194,217–8,246–7,263,279consciousness,andquantumteleportation217conservationofenergy149–81,152–3,272Copenhageninterpretation69,91,272anddecoherence123,125,276andmeasurement92–97,103,123,256andwavefunctioncollapse92–97,104,105,122,134–5,256corralpattern162,163cryptography194,218–9,247,263

Davisson,Clinton33,35,36,37deBroglie,LouisVictorPierreRaymond32,36,61deBroglierelation34,277decoherence100–03,108–18,272,272–3,276andentangledstates169–70andenvironment116,118–23,169n.,259andreality123–6

diffraction12,273ofelectrons34,35,36andlightwaves21–22,23ofmolecules38–9,39ofneutrons36–7andsoundwaves20–21,21

duplicationatadistanceseequantumteleportation

Einstein,Albert4,273,275andBohr166–168,171,173,176anddeBroglie32andphotoelectriceffect27–29,31,277andrandomness75,107,170andrelativity29,174n.,228,280anduncertainty170,254

Einstein,Podolsky,andRosen(EPR)paradox169,171–2,173,186,193,219,273electrodynamicsseequantumelectrodynamicselectrons:andallowedstates61,63,64,73diffraction34,35,36electricdipolemoment243mass38momentum33,37,47,48–49

asparticles36n.,40–41andphotoelectriceffect27,28,31–2andpositrons230,234,235,238–9,241,252andsolar-systemmodel60–62andspin240andtunneling156–60,159,161–2anduncertaintyprinciple47,48–9,60–3wavelength33,34,37–8aswaves33–7,40–1,61–2,163andzero-pointenergy64–6seealsoquantumelectrodynamics

energy273inclassicalphysics147–51conservation149–51,152–3,272free245–9kinetic144,148–52,150,153,155,156–7,250,275andmass31,148,195,229–30potential144,148–52,150,153,156–7,159,250,278vacuum252andwavefunctions153zero-point64–6,148n.,221–2,230,245–6,249–51,283

energy–timeuncertainty221–2,224–8,230–1,235–6,273,282entanglement165–94,274andAspectexperiments187–93,188,190,192,258,259–60andBell’stheorem177–86,213andcorrelation166–7,169–70,177–86,194,198,258,259–60andEPRparadox171–6andhomeopathy260–2andnonlocality173–6,186,187–94,198–9,207–8,219,259andteleportation205–15,216

environment,andinterference116–7,118–23,260EPRseeEinstein,Podolsky,andRosen(EPR)paradoxEverett,HughIII,andmany-worldsinterpretation5,100,105–7,276

faxmachines199–200Feynman,Richard5,96,233,270,279Feynmandiagram233–5,238–40,274,282Frayn,Michael,Copenhagen270frequency,measurement225–7Fresnel,Augustin24g-factor240–2,274Gabrielse,Gerald240–1Gamow,George270Germer,Lester33,35,36,37–40Gilder,Louisa269Gleick,James270

Greenstein,GeorgeandZajonc,ArthurG.270gyromagneticratio240–42,274

Harris,Sidney104Hawkingradiation222,252,274–5health:distanthealing258–62andquantumphysics253–57

Heisenberg,WernerandBohr171n.,270andmeasurement76n.,93

uncertaintyprinciple5,43–66,224,273Heisenbergmicroscope46–9,171n.,271Heisenbergrelationship58–9Herbert,Nick255n.Hertz,Heinrich226n.homeopathy,andentanglement260–2hydrogen,Bohrmodel60–2,251,281

IBM:Almadenresearchlaboratory162andquantumteleportation205andscanningtunnelingmicroscope160

indeterminacy:andentanglement212andmeasurement3,167–8,175,206,255–6andpolarization179,183,206,206,209,212andteleportation198

information220,254interference34,168,275,281anddecoherence108–18andenvironment116–7,118–23,120,260andlightwaves23,29,86–90,87,110–17,118–23andmolecules38–9,39andneutrondiffraction37andsoundwaves18

interferometer109–10,111,114–17,115,119–21interpretationsseeCopenhageninterpretation;many-worldsinterpretation;‘shutupandcalculate’

interpretation;transactionalinterpretationinterrogation,quantum136–41,192n.,280ions,andquantumZenoeffect132–5,135Itano,W.M.132,135n.Kimble,Dagenais,andMandel32n.kineticenergy144,148–52,150,153,155,156–7,250,275Kinoshita,Toichiro241n.Kwiat,Paul192n.

LargeHadronCollider243Lewis,DavidKellogg223n.LHVtheoryseelocalhiddenvariabletheorylight:anddiffraction21–2,23andinterference23,29,86–90,87,110–17,118–23asparticle14,24–32,84–5polarization78–85,89–90,203–5,204,278quantumtheory4–5spectrum25aswave20–4,27,79,84,85wavelength21,29,30–1,48seealsophotons

Lindley,David269localhiddenvariabletheory173–6,271,275andAspectexperiments187–93,188,190andBell’stheorem177–86

locality167,174

Mach–Zehnderinterferometer109n.magic,andquantummechanics245–9,251,263–4many-worldsinterpretation5,97,99–126,272,276anddecoherence108–123andwavefunctionbranches100–1,108,109–14,117,124–5,129,136,256andwavefunctioncollapse105–8,122,124–5

mass:andenergy31,148,195,229–30ofmolecules39

mathematics:andCopenhageninterpretation93–4,103–4,105andwavefunctions48–9,58–9,71,72–3,94–5,106seealsoSchrödingerequation

measurement276inclassicalphysics132andcorrelation169,177–85,194,258anddecoherence118–23,169n.anddeterminationofstate43–6,53,72,76–8,85–90,91,94–7,129,131–4,140offrequency225–7andhealthandhealing254–6limitations46,59macroscopic92ofphotons85,86–90,201–2andquantumeraser78–9,86–90,279andquantuminterrogation136–41,192n.andquantumZenoeffect128–9,131–40,135,168–9,253,256

andteleportation206,207–8,212–15anduncertaintyprinciple43–9,47,50–7,65,171seealsoCopenhageninterpretation;many-worldsinterpretation

medicine,alternative253–61meditation256–7Mermin,David96n.microscopes:electron48n.scanningtunneling(STM)160–4,162,163

Milgrom,LionelR.261–2Millikan,Robert29,32Mills,Randell250modernphysics4,276molecules,wavenature38–9,39,94momentum276–7angular65inclassicalphysics13,276–7andelectrons33,37,47,48–9andkineticenergy153,155andphotons30–1,47,48–9andposition43–9,51–3,57–60,61,155,225,282anduncertaintyprinciple43–9,52–3,61–4,170–2andwavelength13,48,50–2,57–8,152–5

motion:Newtonianlaws3,14,73

andZeno’sparadox130,132muon241–2

NationalInstituteofStandardsandTechnology(NIST)132–3neutrons,aswaves37Newton,SirIsaac:andlight24andmotion3,14,73

NielsBohrInstitute,Copenhagen215–6NISTseeNationalInstituteofStandardsandTechnologyno-cloningtheorem203,204,205–6,212,277nonlocality:anddistanthealing258–62andentanglement168,173–6,186,187–94,198–9,259andteleportation205,207–8,219

observation,andwavefunctioncollapse69,92–3,107–8,102Oerter,Robert270oscillation:andlightwaves23,25–6,79–80,110–11,227

measurement225–7

Park,Bob250particle–waveduality11–42,277andallowedstates61,63andclassicalphysics13–20andlightandsound20–32,84andquantumeraser86anduncertaintyprinciple43–9,60andwavefunctionprobability50–7

particles:andantiparticles271inclassicalphysics12–20collision15,29,47,48andelectrons33–37,40–1entangled166–94,274,280andfrequency29andlight24–32mass39andquantumtunneling143–164aswaves32–41,147,164seealsoantimatter;photons;virtualparticles

Penningtrap241Penrose,Roger217perpetualmotionmachines249–50,252philosophy,andquantumphysics5–6,46,74,91,176,272photoelectriceffect,quantumtheory5n.,27–9,32,277photons40–1,277andallowedstates73,85,208emissions53,194,202andenvironment116–7,118–23infrared180–2,181measurement85,86–90andmomentum30–31,47,48–9andphotoelectriceffect28–9andquantumZenoeffect136–40,137andteleportation201–5,212–15virtual238–9,241andwavefunction85,86,113,121,201aswaves40–1seealsolight;polarization

physicsseeclassicalphysics;modernphysics;quantumphysicsPlanck,Max4,24–6,277,281–2Planck’sconstant(h)277–8andlight25–6,28–30

anduncertaintyprinciple59,224andwavelength38

Podolsky,Boris169,273,275polarization78–85,80,88–9,177–84,188–91,278andindeterminacy179,183,207,207,209,212andteleportation201–3,208–10,212–15

polarizer/polarizingfilter82–5,88–9,177–9,184,188–91,212–15,278position:andmomentum43–9,51–3,57–60,61,155,224,282anduncertaintyprinciple43–9,52–3,61–4,170–2andwavefunctionprobability49–57

potentialenergy144,148–52,150,153,156–7,159,250,278barrier158n.

Pratchett,Terry,LordsandLadies95n.probability278classicaldistribution70andtunneling156–60,161,164andwavefunction49–57,72,73–4,76–7,85,87–8,91,113,124,154

protons:andantiprotons230–1aswaves37

QEDseequantumelectrodynamicsquantum26–7,73quantumcomputer164,218–9,248,263,279quantumelectrodynamics(QED)5,233–36,274,279experimentalverification237–42andhydrogen251

quantumeraser78–9,86–90,279quantumfieldtheory186,258,279quantuminterrogation136–141,192n.,279quantumleap73quantummechanicsseequantumphysicsquantumoptics2quantumphysics:definition3,4,279andfreeenergy249–53andhealth253–62asmagic245–9,251,263–4misuses245–65

quantumpotential186quantumstateseestateQuantumTantra255n.quantumteleportation194,195–220,273,280applications216–20

andclassicalphysics199,203–5andentanglement205–15,216experimentaldemonstration212–16andhomeopathy261–2limitations200–03

quantumuncertaintyseeuncertaintyprinciplequantumZenoeffect127–41,280,280andmeasurement131–40,135,168–9andquantuminterrogation136–141,192n.andZeno’sparadox129–30

quarks228,241n.qubits218,261

radiation,thermal25,281–2randomness74–6,86,102,116–18,194,256andEinstein75,107,170andmany-worldsinterpretation102,116–18,124–5

reality:andclassicalphysics71anddecoherence123–26limits49,57–60andmeasurement95–6andquantumphysics6,93

relativity5–6,30,174n.,228,280special29–30

Rohrer,Heinrich160Rosen,Nathan169,273,275

Schrödingerequation72–3,91,94,125,251,280andpotentialenergy153andwavefunctioncollapse104,105–6

Schrödinger,Erwin72n.Schrödinger’scat5,68–9,94,209,229,281Schwinger,Julian5,223n.,279ScientificAmerican86n.semiclassicalarguments281anduncertaintyprinciple46–49

‘shutupandcalculate’interpretation96–7,104Snow,Tiffany259solids,quantumtheory62–3sound:anddiffraction20–21,21,273andparticle–waveduality20–4andwavelength20–1,21

spin240

state69–70,197,281allowed61,63,64,72,73,91,208,271,272classical69–70entangled168–64andmeasurementseemeasurementandteleportation197–9,213–16,217–18,280anduncertaintyprinciple57seealsosuperpositionstates

STMseemicroscopes,scanningtunneling(STM)superpositionstates68–9,78,105,230,276,281anddecoherence123–4andmeasurement91,129,174–5,208–9andpolarizationoflight79–86,201–2andquantumelectrodynamics237–40andquantumeraser86–90andteleportation215,216

andwavefunction201

tauparticles241n.Tegmark,Max222teleportationseequantumteleportationtemperature,andenergy148Thompson,GeorgePaget36Thomson,J.J.36n.time:Plancktime232n..seealsoenergy–timeuncertaint

Tomonaga,Shin-Ichiro232n.279transactionalinterpretation97tunneling62,143–64,282andscanningtunnelingmicroscope160–4,162,163turningpoint152,153–4uncertainty155,157

uncertaintyprinciple5,43–66,166,282energy–timeuncertainty221–2,224–8,230–1,235–6,273,282andlimitsofreality49,57–60andpositionandmomentum43–9,51–3,57–60,61,155,224,282andsemiclassicalarguments46–9andzero-pointenergy64–6

universes,parallelseemany-worldsinterpretation

velocity:andkineticenergy148–149andparticles15,152anduncertaintyprinciple43–4,47,48,57–8,63

andwaves17virtualparticles6,65,221–2,229,282andantimatter221,230,271

experimentalverification237–43andquantumelectrodynamics233–36,238–43,274,279anduncertaintyprinciple229andzero-pointenergy221–2,224,230–1,251

wavepacket52–7wavefunction45,71–9,282–3branches100–1,108–15,117,124–5,129,136,256coherence114–17decoherence100–3,108–18,121andHeisenbergrelationships58–9infinite-sum57one-part78,87,89–90,105andpotentialenergy153andprobability49–57,72,73–4,76–7,85,87–8,91,113,124,154,282–3andSchrödingerequation72–3,91,280andtunneling158,159two-part78,87–89,89–90,133,138anduncertainty74

wavefunctioncollapse272,276andCopenhageninterpretation92–7,104,105,122,134–5,256andmany-worldsinterpretation105–8,122,124–5andmeasurement68,92–6,124,129

no-collapseinterpretations93n.andobserver68,93andSchrödingerequation104,105–6

wavelength12–13,17,79anddiffraction21–21,21,34,273andelectrons33,34,37–8,158andlight21,29,30–1,48andmomentum13,48,50–2,57–8,152–5andparticles/mass39andsound20–21,21

anduncertaintyprinciple53–7waves:amplitude16–17,79,80inclassicalphysics16–19andcoherence271–2diffraction20–22,23,273frequency17,25–8,79,225andinterferencepatterns18,22,23,275seealsolight;particle–waveduality;sound

Wigner,Eugene,“friend”experiment94–5

Wineland,D.J.132,134Wootters,William202,205

Young,Thomas22–4,34,36,40,86Zeilinger,Anton38–9,39,213–5,214ZenoofElea130zero-pointenergy148n.,245–6,283andfreeenergyschemes249–51anduncertainty60–66,230–1

andvirtualparticles221,224Zurek,Wojchiech202