PraiseforQuantumMechanics

“Whileit’sdefinitelyabookforpeoplewhohavesomemathbackground,itdoesn’trequiremuchanditunspoolsthebasicsslowly,thoughtfullyandwithexceptionalclarity.”

—AdamFrank,NPR’s13.7blog

“Ifyou’reeverbanishedtoadesertislandandallowedtotakejustonebook,hereitis.Givenenoughtime,withnodistractions,youcoulduseittoeventuallymasterquantummechanics....[E]venwithoutmasteringallthecalculationalcomplexities,acarefulreadattheveryleastoffersdeeperinsightintothelogicandmathematicalsubstanceofquantumphysicsthanyou’llgetfromanypopularaccount.”

—ScienceNews

“[QuantumMechanics]providesyouwiththe‘minimal’equipmentyouneedtounderstandwhatallthefuzzwithquantumoptics,quantumcomputing,andblackholeevaporationisabout....Ifyouwanttomakethestepfrompopularscienceliteraturetotextbooksandthegeneralscientific literature, thenthisbookseriesisamust-read.”

—BackReaction

“[V]erydetailedbutverywellwritten.”

—SanFranciscoBookReview

“Thewriting is freshand immediate,withplentyofdetailpackaged into the smoothnarrative. . . .[O]ntheirownterms,IfoundSusskindandFriedman’sexplanationscrispandsatisfying....Imaintainaclear recollection of the bewilderment with which I struggled through my own university quantum-mechanicscourses.Forstudentsinasimilarposition,tryingtodrawtogetherthefragmentsofformalisminto a clear conceptualwhole,Susskind andFriedman’spersuasiveoverview—and their insistenceonexplaining,withsharpmathematicaldetail,exactlywhatitisthatisstrangeaboutquantummechanics—maybejustwhatisneeded.”

—DavidSeery,Nature

“[QuantumMechanics]isevenbetterthanthefirstvolume,takingonamuchmoredifficultsubject....‘TheTheoreticalMinimum’phraseisareferencetoLandau,butit’sagoodcharacterizationofthisbookandthelecturesingeneral.Susskinddoesagoodjobofboilingthesesubjectsdowntotheircoreideasandexamples,andgivingacarefulexpositionoftheseinassimpletermsaspossible.Ifyou’vegottenataste for physics from popular books, this is a great place to start learningwhat the subject is reallyabout.”

—PeterWoit,NotEvenWrong

“[T]hebookwillworkwellasacompaniontextforuniversitystudentsstudyingquantummechanicsorthearmchairphysicistsfollowingSusskind’sYouTubelectures.”

—PublishersWeekly

“Either for those about to start a university physics course who want some preparation, or forsomeonewho finds popular science explanations too summary and is prepared to take on some quiteseriousmath...it’safascinatingadditiontothelibrary.”

—PopularScience(UK)

“Thisisquantummechanicsforreal.Thisisthegoodstuff,themostmysteriousaspectsofhowrealityworks,setoutwithcrystallineclarity.Ifyouwanttoknowhowphysicistsreallythinkabouttheworld,thisbookistheplacetostart.”

—SeanCarroll,physicist,CaliforniaInstituteofTechnology,andauthorofTheParticleattheEndoftheUniverse

Thisbook is thesecondvolumeof theTheoreticalMinimumseries.The firstvolume,The TheoreticalMinimum:WhatYouNeedtoKnowtoStartDoingPhysics,coveredclassicalmechanics,whichisthecoreofanyphysicseducation.WewillrefertoitfromtimetotimesimplyasVolumeI.Thissecondbookexplains quantummechanics and its relationship to classical mechanics. The books in this series runparallel to Leonard Susskind’s videos, available on the Web through Stanford University (seewww.theoreticalminimum.comfora listing).Whilecovering thesamegeneral topicsas thevideos, thebookscontainadditionaldetails,andtopicsthatdon’tappearinthevideos.

QUANTUMMECHANICS

AlsobyLeonardSusskind

TheTheoreticalMinimum

WhatYouNeedToKnowtoStartDoingPhysics(withGeorgeHrabovsky)

TheBlackHoleWar

TheCosmicLandscape

QUANTUMMECHANICS

TheTheoreticalMinimum

LEONARDSUSSKINDandARTFRIEDMAN

BASICBOOKSAMemberofthePerseusBooksGroup

NewYork

Copyright©2014byLeonardSusskindandArtFriedman

PublishedbyBasicBooks,AMemberofthePerseusBooksGroup

Allrightsreserved.Nopartofthisbookmaybereproducedinanymannerwhatsoeverwithoutwrittenpermissionexceptinthecaseofbriefquotationsembodiedincriticalarticlesandreviews.Forinformation,addressBasicBooks,250West57thStreet,15thFloor,NewYork,NY10107–1307.

BookspublishedbyBasicBooksareavailableatspecialdiscountsforbulkpurchasesintheUnitedStatesbycorporations,institutions,andotherorganizations.Formoreinformation,pleasecontacttheSpecialMarketsDepartmentatthePerseusBooksGroup,2300ChestnutStreet,Suite200,Philadelphia,PA19103,orcall(800)810–4145,ext.5000,ore-mailspecial.markets@perseusbooks.com.

DesignedbyArtFriedmanandLeonardSusskind

Hilbert’sPlacedrawingswerecreatedbyMargaretSloan.

ACIPcatalogrecordforthisbookisavailablefromtheLibraryofCongress.

ISBN(ebook):978-0-465-08061-8

ISBN(paperback):978-0-465-06290-4

10987654321

Forourparents,whomadeitallpossible:

IreneandBenjaminSusskindGeorgeandTrudyFriedman

Contents

Preface

Prologue

Introduction

1SystemsandExperiments

2QuantumStates

3PrinciplesofQuantumMechanics

4TimeandChange

5UncertaintyandTimeDependence

6CombiningSystems:Entanglement

7MoreonEntanglement

8ParticlesandWaves

9ParticleDynamics

10TheHarmonicOscillator

Appendix

Index

Preface

Albert Einstein, who was in many ways the father of quantum mechanics, had a notorious love-haterelationwiththesubject.HisdebateswithNielsBohr—BohrcompletelyacceptingofquantummechanicsandEinsteindeeplyskeptical—arefamous in thehistoryofscience. Itwasgenerallyacceptedbymostphysicists that Bohr won and Einstein lost. My own feeling, I think shared by a growing number ofphysicists,isthatthisattitudedoesnotdojusticetoEinstein’sviews.

BothBohr andEinsteinwere subtlemen.Einstein triedveryhard to show thatquantummechanicswas inconsistent; Bohr, however, was always able to counter his arguments. But in his final attackEinsteinpointed to something sodeep, socounterintuitive, so troubling, andyet soexciting, that at thebeginningofthetwenty-firstcenturyithasreturnedtofascinatetheoreticalphysicists.Bohr’sonlyanswertoEinstein’slastgreatdiscovery—thediscoveryofentanglement—wastoignoreit.

Thephenomenonofentanglementistheessentialfactofquantummechanics,thefactthatmakesitsodifferentfromclassicalphysics.Itbringsintoquestionourentireunderstandingaboutwhatisrealinthephysical world. Our ordinary intuition about physical systems is that if we know everything about asystem,thatis,everythingthatcaninprinciplebeknown,thenweknoweverythingaboutitsparts.Ifwehavecompleteknowledgeoftheconditionofanautomobile,thenweknoweverythingaboutitswheels,itsengine,itstransmission,rightdowntothescrewsthatholdtheupholsteryinplace.Itwouldnotmakesenseforamechanictosay,“IknoweverythingaboutyourcarbutunfortunatelyIcan’ttellyouanythingaboutanyofitsparts.”

Butthat’sexactlywhatEinsteinexplainedtoBohr—inquantummechanics,onecanknoweverythingaboutasystemandnothingaboutitsindividualparts—butBohrfailedtoappreciatethisfact.Imightaddthatgenerationsofquantumtextbooksblithelyignoredit.

Everyone knows that quantummechanics is strange, but I suspect very few people could tell youexactlyinwhatway.Thisbookisatechnicalcourseoflecturesonquantummechanics,butitisdifferentthanmostcoursesormosttextbooks.Thefocusisonthelogicalprinciplesandthegoalisnottohidetheutterstrangenessofquantumlogicbuttobringitoutintothelightofday.

I remind you that this book is one of several that closely follow my Internet course series, theTheoreticalMinimum.Mycoauthor,ArtFriedman,wasa student in these courses.ThebookbenefitedfromthefactthatArtwaslearningthesubjectandwasthereforeverysensitivetotheissuesthatmightbeconfusingto thebeginner.Duringthecourseofwriting,wehada lotoffun,andwe’ve tried toconveysomeofthatspiritwithabitofhumor.Ifyoudon’tgetit,ignoreit.

LeonardSusskind

WhenIcompletedmymaster’sdegreeincomputerscienceatStanford,IcouldnothaveguessedthatI’dreturn someyears later toattendLeonard’sphysics lectures.Myshort “career” inphysicsendedmany

yearsearlier,withthecompletionofmybachelor’sdegree.Butmyinterest in thesubjecthasremainedverymuchalive.

ItappearsthatIhavelotsofcompany—theworldseemsfilledwithpeoplewhoaregenuinely,deeplyinterestedinphysicsbutwhoseliveshavetakenthemindifferentdirections.Thisbookisforallofus.

Quantum mechanics can be appreciated, to some degree, on a purely qualitative level. Butmathematicsiswhatbringsitsbeautyintosharpfocus.Wehavetriedtomakethisamazingbodyofworkfullyaccessibletomathematicallyliteratenonphysicists.Ithinkwe’vedoneafairlygoodjob,andIhopeyou’llagree.

Noonecompletesaprojectlikethiswithoutlotsofhelp.ThepeopleatBrockman,Inc.,havemadethebusinessendofthingsseemeasy,andtheproductionteamatPerseusBookshasbeentop-notch.MysincerethanksgotoTJKelleher,RachelKing,andTisseTakagi.Itwasourgoodfortunetoworkwithatalentedcopyeditor,JohnSearcy.

I’m grateful to Leonard’s (other) continuing education students for routinely raising thoughtful,provocative questions, and for many stimulating after-class conversations. Rob Colwell, Todd Craig,MontyFrost, and JohnNash offered constructive comments on themanuscript. JeremyBranscome andRussBryanreviewedtheentiremanuscriptindetail,andidentifiedanumberofproblems.

I thankmyfamilyandfriendsfor theirkindsupportandenthusiasm.Iespecially thankmydaughter,Hannah,formindingthestore.

Besides her love, encouragement, insight, and sense of humor, my amazing wife,Margaret Sloan,contributedaboutathirdofthediagramsandbothHilbert’sPlaceillustrations.Thanks,Maggie.

Atthestartofthisproject,Leonard,sensingmyrealmotivation,remarkedthatoneofthebestwaystolearnphysicsistowriteaboutit.True,ofcourse,butIhadnoideahowtrue,andI’mgratefulthatIhadachancetofindout.Thanksamillion,Leonard.

ArtFriedman

Prologue

Artlooksoverhisbeerandsays,“Lenny,let’splayaroundoftheEinstein-Bohrgame.”

“OK,butI’mtiredoflosing.Thistime,youbeArtsteinandI’llbeL-Bore.Youstart.”

“Fairenough.Here’smyfirstshot:Goddoesn’tplaydice.Ha-ha,L-Bore,that’sonepointforme.”

“Notsofast,Artstein,notsofast.You,myfriend,werethefirstonetopointoutthatquantumtheoryisinherentlyprobabilistic.Hehhehheh,that’satwo-pointer!”

“Well,Itakeitback.”

“Youcan’t.”

“Ican.”

“Youcan’t.”

FewpeoplerealizethatEinstein,inhis1917paper,“OntheQuantumTheoryofRadiation,”arguesthattheemissionofgammaraysisgovernedbyastatisticallaw.

AProfessorandaFiddlerWalkintoaBarVolumeIwaspunctuatedbyshortconversationsbetweenLennyandGeorge,fictionalpersonaswhowerelooselybasedontwoJohnSteinbeckcharacters.ThesettingforthisvolumeoftheTheoreticalMinimumseries is inspired by the stories of Damon Runyon. It’s a world filled with crooks, con artists,degenerates,smoothoperators,anddo-gooders.Plusafewordinaryfolks, just tryingtoget throughtheday.TheactionunfoldsatapopularwateringholecalledHilbert’sPlace.

Into this setting strollLenny andArt, twogreenhorns fromCaliforniawho somehowgot separatedfromtheirtourbus.Wishthemluck.Theywillneedit.

WhattoBringYou don’t need to be a physicist to take this journey, but you should have some basic knowledge ofcalculusandlinearalgebra.YoushouldalsoknowsomethingaboutthematerialcoveredinVolumeI.It’sOKifyourmathisabitrusty.We’llreviewandexplainmuchofitaswego,especiallythematerialonlinearalgebra.VolumeIreviewsthebasicideasofcalculus.

Don’tletourlightheartedhumorfoolyouintothinkingthatwe’rewritingforairheads.We’renot.Ourgoalistomakeadifficultsubject“assimpleaspossible,butnosimpler,”andwehopetohavealittlefunalongtheway.SeeyouatHilbert’sPlace.

©MargaretSloan

Introduction

Classicalmechanicsisintuitive;thingsmoveinpredictableways.Anexperiencedballplayercantakeaquicklookataflyball,andfromitslocationanditsvelocity,knowwheretoruninordertobetherejustin time to catch the ball. Of course a sudden unexpected gust ofwindmight fool him, but that’s onlybecausehedidn’ttakeintoaccountallthevariables.Thereisanobviousreasonwhyclassicalmechanicsisintuitive:humans,andanimalsbeforethem,havebeenusingitmanytimeseverydayforsurvival.Butnooneeverusedquantummechanicsbeforethetwentiethcentury.Quantummechanicsdescribesthingssosmallthattheyarecompletelybeyondtherangeofthehumansenses.Soitstandstoreasonthatwedidnotevolve an intuition for the quantum world. The only way we can comprehend it is by rewiring ourintuitionswith abstractmathematics.Fortunately, for someodd reason,wedid evolve the capacity forsuchrewiring.

Ordinarily, we learn classical mechanics first, before even attempting quantum mechanics. Butquantumphysicsismuchmorefundamentalthanclassicalphysics.Asfarasweknow,quantummechanicsprovidesanexactdescriptionofeveryphysicalsystem,butsomethingsaremassiveenoughthatquantummechanicscanbereliablyapproximatedbyclassicalmechanics.That’sallthatclassicalmechanicsis:anapproximation. From a logical point of view,we should learn quantummechanics first, but very fewphysicsteacherswouldrecommendthat.Eventhiscourseoflectures—theTheoreticalMinimumseries—beganwithclassicalmechanics.Nevertheless, in thesequantumlectures,classicalmechanicswillplayalmost no role except near the end, well after the basic principles of quantum mechanics have beenexplained.I thinkthis isreallytherightwaytodoit,not just logicallybutpedagogicallyaswell.Thatwaywedon’tfall intothetrapofthinkingthatquantummechanicsisbasicallyjustclassicalmechanicswithacoupleofnewgimmicksthrownin.Bytheway,quantummechanicsistechnicallymucheasierthanclassicalmechanics.

Thesimplestclassicalsystem—thebasiclogicalunitforcomputerscience—isthetwo-statesystem.Sometimesit’scalledabit.Itcanrepresentanythingthathasonlytwostates:acointhatcanshowheadsortails,aswitchthatisonoroff,oratinymagnetthatisconstrainedtopointeithernorthorsouth.Asyoumight expect, especially if you studied the first lecture ofVolume I, the theory of classical two-statesystems is extremely simple—boring, in fact. In this volume, we’re going to begin with the quantumversionofthetwo-statesystem,calledaqubit,whichisfarmoreinteresting.Tounderstandit,wewillneedawholenewwayofthinking—anewfoundationoflogic.

Lecture1

SystemsandExperiments

LennyandArtwanderintoHilbert’sPlace.

Art:Whatisthis,theTwilightZone?Orsomekindoffunhouse?Ican’tgetmybearings.

Lenny:Takeabreath.You’llgetusedtoit.

Art:Whichwayisup?

1.1 QuantumMechanicsIsDifferent

Whatissospecialaboutquantummechanics?Whyisitsohardtounderstand?Itwouldbeeasytoblamethe“hardmathematics,”andtheremaybesometruthinthatidea.Butthatcan’tbethewholestory.Lotsofnonphysicists are able to master classical mechanics and field theory, which also require hardmathematics.

Quantummechanicsdealswith thebehaviorofobjectssosmall thatwehumansare illequippedtovisualizethematall.Individualatomsareneartheupperendofthisscaleintermsofsize.Electronsarefrequentlyusedasobjectsofstudy.Oursensoryorgansaresimplynotbuilttoperceivethemotionofanelectron. The best we can do is to try to understand electrons and their motion as mathematicalabstractions.

“Sowhat?”saystheskeptic.“Classicalmechanicsisfilledtothebrimwithmathematicalabstractions—pointmasses,rigidbodies,inertialreferenceframes,positions,momenta,fields,waves—thelistgoesonandon.There’snothingnewaboutmathematicalabstractions.”Thisisactuallyafairpoint,andindeedtheclassicalandquantumworldshavesomeimportantthingsincommon.Quantummechanics,however,isdifferentintwoways:

1. DifferentAbstractions.Quantumabstractionsarefundamentallydifferentfromclassicalones.Forexample,we’llseethattheideaofastateinquantummechanicsisconceptuallyverydifferentfromits classical counterpart. States are represented by different mathematical objects and have adifferentlogicalstructure.

2. StatesandMeasurements.Intheclassicalworld,therelationshipbetweenthestateofasystemandtheresultofameasurementonthatsystemisverystraightforward.Infact, it’s trivial.Thelabelsthatdescribeastate(thepositionandmomentumofaparticle,forexample)arethesamelabelsthat

characterizemeasurementsof that state.Toput itanotherway,onecanperformanexperiment todeterminethestateofasystem.Inthequantumworld,thisisnottrue.Statesandmeasurementsaretwodifferentthings,andtherelationshipbetweenthemissubtleandnonintuitive.

Theseideasarecrucial,andwe’llcomebacktothemagainandagain.

1.2 SpinsandQubits

The concept of spin is derived from particle physics. Particles have properties in addition to theirlocationinspace.Forexample,theymayormaynothaveelectriccharge,ormass.Anelectronisnotthesameasaquarkoraneutrino.Butevenaspecifictypeofparticle,suchasanelectron,isnotcompletelyspecifiedbyitslocation.Attachedtotheelectronisanextradegreeoffreedomcalleditsspin.Naively,the spin can be pictured as a little arrow that points in some direction, but that naive picture is tooclassicaltoaccuratelyrepresenttherealsituation.Thespinofanelectronisaboutasquantummechanicalasasystemcanbe,andanyattempttovisualizeitclassicallywillbadlymissthepoint.

Wecanandwillabstracttheideaofaspin,andforgetthatitisattachedtoanelectron.Thequantumspinisasystemthatcanbestudiedinitsownright.Infact,thequantumspin,isolatedfromtheelectronthatcarriesitthroughspace,isboththesimplestandthemostquantumofsystems.

The isolated quantum spin is an example of the general class of simple systemswe call qubits—quantumbits—thatplay thesamerole in thequantumworldas logicalbitsplay indefining thestateofyourcomputer.Manysystems—maybeevenall systems—canbebuiltupbycombiningqubits.Thus inlearningaboutthem,wearelearningaboutagreatdealmore.

1.3 AnExperiment

Let’smaketheseideasconcrete,usingthesimplestexamplewecanfind.InthefirstlectureofVolumeI,webeganbydiscussingaverysimpledeterministicsystem:acointhatcanshoweitherheads(H)ortails(T).Wecancallthisatwo-statesystem,orabit,withthetwostatesbeingHandT.Moreformallyweinvent a “degreeof freedom”called that can takeon twovalues, namely+1 and−1.The stateH isreplacedby

andthestateTby

Classically,that’sallthereistothespaceofstates.Thesystemiseitherinstate or andthereisnothinginbetween.Inquantummechanics,we’llthinkofthissystemasaqubit.

VolumeI also discussed simple evolution laws that tell us how to update the state from instant toinstant.Thesimplestlawisjustthatnothinghappens.Inthatcase,ifwegofromonediscreteinstant(n)to

thenext(n+1),thelawofevolutionis

Let’sexposeahiddenassumptionthatwewerecarelessaboutinVolumeI.Anexperiment involvesmore than justasystemtostudy. Italso involvesanapparatusA tomakemeasurementsandrecord theresultsof themeasurements. In thecaseof the two-statesystem, theapparatus interactswith thesystem(thespin)andrecordsthevalueof .Thinkoftheapparatusasablackbox1withawindowthatdisplaysthe result of a measurement. There is also a “this end up” arrow on the apparatus. The up-arrow isimportant because it shows how the apparatus is oriented in space, and its direction will affect theoutcomesofourmeasurements.Webeginbypointingitalongthezaxis(Fig.1.1).Initially,wehavenoknowledgeofwhether or .Ourpurposeistodoanexperimenttofindoutthevalueof .

Beforetheapparatusinteractswiththespin,thewindowisblank(labeledwithaquestionmarkinourdiagrams). After it measures , the window shows a +1 or a −1. By looking at the apparatus, wedeterminethevalueof .Thatwholeprocessconstitutesaverysimpleexperimentdesignedtomeasure .

Figure1.1:(A)Spinandcat-freeapparatusbeforeanymeasurementismade.(B)Spinandapparatusafteronemeasurementhasbeenmade,resultingin .Thespinisnowpreparedinthe state.Ifthespinisnotdisturbedandtheapparatuskeepsthesameorientation,allsubsequentmeasurementswillgivethesameresult.Coordinateaxesshowourconventionforlabelingthedirectionsofspace.

Now that we’vemeasured , let’s reset the apparatus to neutral and, without disturbing the spin,measure again.AssumingthesimplelawofEq.1.1,weshouldgetthesameansweraswedidthefirsttime.Theresult willbefollowedby .Likewisefor .Thesamewillbetrueforanynumberofrepetitions.Thisisgoodbecauseitallowsustoconfirmtheresultofanexperiment.Wecanalsosaythisinthefollowingway:ThefirstinteractionwiththeapparatusAprepares thesysteminoneofthetwostates.Subsequentexperimentsconfirm thatstate.Sofar, there isnodifferencebetweenclassicalandquantumphysics.

Figure1.2:Theapparatusisflippedwithoutdisturbingthepreviouslymeasuredspin.Anewmeasurementresultsin .

Nowlet’sdosomethingnew.AfterpreparingthespinbymeasuringitwithA,weturntheapparatusupsidedownandthenmeasure again(Fig.1.2).Whatwefindisthatifweoriginallyprepared ,the upside down apparatus records . Similarly, ifwe originally prepared , the upsidedownapparatusrecords .Inotherwords,turningtheapparatusover interchanges and

.Fromtheseresults,wemightconcludethat isadegreeoffreedomthat isassociatedwithasenseofdirection in space.For example, if were anorientedvector of some sort, then itwouldbenaturaltoexpectthatturningtheapparatusoverwouldreversethereading.Asimpleexplanationisthatthe apparatusmeasures the component of the vector along an axis embedded in the apparatus. Is thisexplanationcorrectforallconfigurations?

Ifweareconvincedthatthespinisavector,wewouldnaturallydescribeitbythreecomponents: ,,and .Whentheapparatusisuprightalongthezaxis,itispositionedtomeasure .

Figure 1.3: The apparatus rotated by 90°. A new measurement results in with 50 percentprobability.

Sofar,thereisstillnodifferencebetweenclassicalphysicsandquantumphysics.Thedifferenceonlybecomesapparentwhenwerotatetheapparatusthroughanarbitraryangle,say radians(90degrees).Theapparatusbeginsintheuprightposition(withtheup-arrowalongthezaxis).Aspinispreparedwith

. Next, rotate A so that the up-arrow points along the x axis (Fig. 1.3), and then make ameasurementofwhatispresumablythexcomponentofthespin, .

Ifinfact reallyrepresentsthecomponentofavectoralongtheup-arrow,onewouldexpecttogetzero.Why? Initially,we confirmed that was directed along the z axis, suggesting that its component

alongxmustbezero.Butwegetasurprisewhenwemeasure :Insteadofgiving ,theapparatusgiveseither or .Aisverystubborn—nomatterwhichwayitisoriented,itrefusestogiveanyanswerotherthan .Ifthespinreallyisavector,itisaverypeculiaroneindeed.

Nevertheless,wedo find something interesting.Supposewe repeat theoperationmany times, eachtimefollowingthesameprocedure,thatis:

• BeginningwithAalongthezaxis,prepare .

• Rotatetheapparatussothatitisorientedalongthexaxis.

• Measure .

Therepeatedexperimentspitsoutarandomseriesofplus-onesandminus-ones.Determinismhasbrokendown,butinaparticularway.Ifwedomanyrepetitions,wewillfindthatthenumbersof eventsand eventsarestatisticallyequal.Inotherwords,theaveragevalueof iszero.Insteadoftheclassicalresult—namely,thatthecomponentof alongthexaxis iszero—wefind that theaverageoftheserepeatedmeasurementsiszero.

Figure 1.4: The apparatus rotated by an arbitrary anglewithin the plane.Averagemeasurementresultis .

Nowlet’sdothewholethingoveragain,butinsteadofrotatingAtolieonthexaxis,rotateittoanarbitrarydirectionalongtheunitvector2 .Classically,if wereavector,wewouldexpecttheresultoftheexperiment tobe thecomponentof along the axis. If liesatanangle withrespect toz, theclassicalanswerwouldbe =cos .Butasyoumightguess,each timewedo theexperimentweget

or .However,theresultisstatisticallybiasedsothattheaveragevalueis .Thesituation isofcoursemoregeneral.Wedidnothave tostartwithAorientedalongz.Pickany

direction andstartwiththeup-arrowpointingalong .Prepareaspinsothattheapparatusreads+1.Then,withoutdisturbing the spin, rotate theapparatus to thedirection , as shown inFig.1.4.A newexperimentonthesamespinwillgiverandomresults±1,butwithanaveragevalueequaltothecosineoftheanglebetween and .Inotherwords,theaveragewillbe .

ThequantummechanicalnotationforthestatisticalaverageofaquantityQisDirac’sbracketnotation

.Wemay summarize the results of our experimental investigation as follows: Ifwe beginwithAorientedalong andconfirmthat ,thensubsequentmeasurementwithAorientedalong givesthestatisticalresult

What we are learning is that quantum mechanical systems are not deterministic—the results ofexperimentscanbestatisticallyrandom—butifwerepeatanexperimentmanytimes,averagequantitiescanfollowtheexpectationsofclassicalphysics,atleastuptoapoint.

1.4 ExperimentsAreNeverGentle

Everyexperimentinvolvesanoutsidesystem—anapparatus—thatmustinteractwiththesysteminordertorecordaresult.Inthatsense,everyexperimentisinvasive.Thisistrueinbothclassicalandquantumphysics, but only quantum physics makes a big deal out of it. Why is that so? Classically, an idealmeasuringapparatushasavanishinglysmalleffectonthesystemit ismeasuring.Classicalexperimentscanbearbitrarilygentleandstillaccuratelyandreproducibly record the resultsof theexperiment.Forexample,thedirectionofanarrowcanbedeterminedbyreflectinglightoffthearrowandfocusingittoformanimage.Whileitistruethatthelightmusthaveasmallenoughwavelengthtoformanimage,thereis nothing in classical physics that prevents the image frombeingmadewith arbitrarilyweak light. Inotherwords,thelightcanhaveanarbitrarilysmallenergycontent.

Inquantummechanics,thesituationisfundamentallydifferent.Anyinteractionthatisstrongenoughtomeasuresomeaspectofasystemisnecessarilystrongenoughtodisruptsomeotheraspectof thesamesystem.Thus,youcanlearnnothingaboutaquantumsystemwithoutchangingsomethingelse.

ThisshouldbeevidentintheexamplesinvolvingAand .Supposewebeginwith alongthezaxis.Ifwemeasure againwithAorientedalongz,wewillconfirmthepreviousvalue.Wecandothisover and over without changing the result. But consider this possibility: Between successivemeasurementsalongthezaxis,weturnAthrough90degrees,makeanintermediatemeasurement,andturnit back to its original direction.Will a subsequentmeasurement along the z axis confirm the originalmeasurement?Theanswerisno.Theintermediatemeasurementalongthexaxiswillleavethespininacompletelyrandomconfigurationasfarasthenextmeasurementisconcerned.Thereisnowaytomaketheintermediatedeterminationofthespinwithoutcompletelydisruptingthefinalmeasurement.Onemightsaythatmeasuringonecomponentofthespindestroystheinformationaboutanothercomponent.Infact,one simply cannot simultaneously know the components of the spin along two different axes, not in areproducibleway in anycase.There is something fundamentallydifferent about the stateof aquantumsystemandthestateofaclassicalsystem.

1.5 Propositions

Thespaceofstatesofaclassicalsystemisamathematicalset.Ifthesystemisacoin,thespaceofstatesisasetoftwoelements,HandT.Usingsetnotation,wewouldwrite{H,T}.Ifthesystemisasix-sided

die, the space of states has six elements labeled {1, 2, 3, 4, 5, 6}. The logic of set theory is calledBooleanlogic.Booleanlogicisjustaformalizedversionofthefamiliarclassicallogicofpropositions.

AfundamentalideainBooleanlogicisthenotionofatruth-value.Thetruth-valueofapropositioniseithertrueorfalse.Nothinginbetweenisallowed.Therelatedsettheoryconceptisasubset.Roughlyspeaking, a proposition is true for all the elements in its corresponding subset and false for all theelementsnotinthissubset.Forexample,ifthesetrepresentsthepossiblestatesofadie,onecanconsidertheproposition

A:Thedieshowsanodd-numberedface.

Thecorrespondingsubsetcontainsthethreeelements{1,3,5}.

Anotherpropositionstates

B:Thedieshowsanumberlessthan4.

Thecorrespondingsubsetcontainsthestates{1,2,3}.

Everypropositionhasitsopposite(alsocalleditsnegation).Forexample,

notA:Thediedoesnotshowanodd-numberedface.

Thesubsetforthisnegatedpropositionis{2,4,6}.

Therearerulesforcombiningpropositionsintomorecomplexpropositions,themostimportantbeingor,and,andnot.We justsawanexampleofnot,whichgetsapplied toasinglesubsetorproposition.Andisstraightforward,andappliestoapairofpropositions.3Itsaystheyarebothtrue.Appliedtotwosubsets,andgives theelementscommon toboth, that is, the intersection of the twosubsets. In thedieexample,theintersectionofsubsetsAandB isthesubsetofelementsthatarebothoddandlessthan4.Fig.1.5usesaVenndiagramtoshowhowthisworks.

Theor rule is similar toand, but has one additional subtlety. In everyday speech, thewordor isgenerallyusedintheexclusivesense—theexclusiveversionistrueifoneortheotheroftwopropositionsistrue,butnotboth.However,Booleanlogicusestheinclusiveversionofor,whichistrueifeitherorbothofthepropositionsaretrue.Thus,accordingtotheinclusiveor,theproposition

AlbertEinsteindiscoveredrelativityorIsaacNewtonwasEnglish

istrue.Sois

AlbertEinsteindiscoveredrelativityorIsaacNewtonwasRussian.

Theinclusiveorisonlywrongifbothpropositionsarefalse.Forexample,

AlbertEinsteindiscoveredAmerica4orIsaacNewtonwasRussian.

The inclusive or has a set theoretic interpretation as the union of two sets: it denotes the subset

containinganythingineitherorbothofthecomponentsubsets.Inthedieexample,(AorB)denotes thesubset{1,2,3,5}.

Figure1.5:AnExampleoftheClassicalmodelofStateSpace.SubsetArepresentstheproposition“thedie showsanodd-numbered face.”SubsetB: “Thedie showsanumber<4.”Dark shading shows theintersectionofAandB,whichrepresentstheproposition(AandB).Whitenumbersareelementsof theunionofAwithB,representingtheproposition(AorB).

1.6 TestingClassicalPropositions

Let’sreturntothesimplequantumsystemconsistingofasinglespin,andthevariouspropositionswhosetruthwecouldtestusingtheapparatusA.Considerthefollowingtwopropositions:

A:Thezcomponentofthespinis+1.

B:Thexcomponentofthespinis+1.

EachoftheseismeaningfulandcanbetestedbyorientingAalongtheappropriateaxis.Thenegationofeachisalsomeaningful.Forexample,thenegationofthefirstpropositionis

notA:Thezcomponentofthespinis−1.

Butnowconsiderthecompositepropositions

(AorB):Thezcomponentofthespinis+1orthexcomponentofthespinis+1.

(AandB):Thezcomponentofthespinis+1andthexcomponentofthespinis+1.

Considerhowwewouldtesttheproposition(AorB).Ifspinsbehavedclassically(andofcoursetheydon’t),wewouldproceedasfollows:5

• Gentlymeasure and record thevalue. If it is+1,weare finished: theproposition (AorB) istrue.If is−1,continuetothenextstep.

• Gentlymeasure .Ifitis+1,thentheproposition(AorB)istrue.Ifnot,thismeansthatneither znor xwasequalto+1,and(AorB)isfalse.

There is an alternative procedure, which is to interchange the order of the two measurements. Toemphasizethisreversalofordering,we’llcallthenewprocedure(BorA):

• Gentlymeasure andrecord thevalue. If it is+1wearefinished:Theproposition(BorA) istrue.If is−1continuetothenextstep.

• Gentlymeasure .Ifitis+1,then(BorA)istrue.Ifnot,itmeansthatneither xnor wasequalto+1,and(BorA)isfalse.

In classical physics, the two orders of operation give the same answer. The reason for this is thatmeasurements can be arbitrarily gentle—so gentle that they do not affect the results of subsequentmeasurements.Therefore,theproposition(AorB)hasthesamemeaningastheproposition(BorA).

1.7 TestingQuantumPropositions

NowwecometothequantumworldthatIdescribedearlier.Letusimagineasituationinwhichsomeone(orsomething)unknowntoushassecretlypreparedaspin in the state.Our job is touse theapparatusA to determine whether the proposition (A or B) is true or false. We will try using theproceduresoutlinedabove.

Webeginbymeasuring .Sincetheunknownagenthassetthingsup,wewilldiscoverthat.Itisunnecessarytogoon:(AorB)istrue.Nevertheless,wecouldtest justtoseewhathappens.Theanswer is unpredictable.We randomly find that or .But neither of these outcomesaffectsthetruthofproposition(AorB).

Butnowlet’sreversetheorderofmeasurement.Asbefore,we’llcallthereversedprocedure(BorA),andthistimewe’llmeasure first.Becausetheunknownagentsetthespinto+1alongthezaxis,themeasurementof israndom.Ifitturnsoutthat ,wearefinished:(BorA)istrue.Butsupposewe find the opposite result, . The spin is oriented along the −x direction. Let’s pause herebriefly,tomakesureweunderstandwhatjusthappened.Asaresultofourfirstmeasurement,thespinisnolongerinitsoriginalstate .Itisinanewstate,whichiseither or .Pleasetakeamomenttoletthisideasinkin.Wecannotoverstateitsimportance.

Nowwe’rereadytotestthesecondhalfofproposition(BorA).RotatetheapparatusAtothezaxisandmeasure .Accordingtoquantummechanics,theresultwillberandomly±1.Thismeansthatthereisa25percentprobabilitythattheexperimentproduces and .Inotherwords,withaprobability of ,we find that (BorA) is false; this occurs despite the fact that the hidden agent hadoriginallymadesurethat .

Evidently,inthisexample,theinclusiveorisnotsymmetric.Thetruthof(AorB)maydependontheorderinwhichweconfirmthetwopropositions.Thisisnotasmallthing;itmeansnotonlythatthelawsofquantumphysicsaredifferentfromtheirclassicalcounterparts,but that theveryfoundationsof logicaredifferentinquantumphysicsaswell.

Whatabout (AandB)?Supposeour firstmeasurementyields and the second, .

Thisisofcourseapossibleoutcome.Wewouldbeinclinedtosaythat(AandB)istrue.Butinscience,especiallyinphysics,thetruthofapropositionimpliesthatthepropositioncanbeverifiedbysubsequentobservation.Inclassicalphysics,thegentlenessofobservationsimpliesthatsubsequentexperimentsareunaffectedandwillconfirmanearlierexperiment.AcointhatturnsupHeadswillnotbeflippedtoTailsby the act of observing it—at least not classically. Quantum mechanically, the second measurement

ruins the possibility of verifying the first. Once has been prepared along the x axis,anothermesurementof will give a random answer. Thus (AandB) is not confirmable: the secondpieceoftheexperimentinterfereswiththepossibilityofconfirmingthefirstpiece.

If you know a bit about quantummechanics, you probably recognize thatwe are talking about theuncertainty principle. The uncertainty principle doesn’t apply only to position and momentum (orvelocity); it applies to many pairs of measurable quantities. In the case of the spin, it applies topropositions involving two different components of . In the case of position andmomentum, the twopropositionswemightconsiderare:

Acertainparticlehaspositionx.

Thatsameparticlehasmomentump.

Fromthese,wecanformthetwocompositepropositions

Theparticlehaspositionxandtheparticlehasmomentump.

Theparticlehaspositionxortheparticlehasmomentump.

Awkwardastheyare,bothofthesepropositionshavemeaningintheEnglishlanguage,andinclassicalphysicsaswell.However,inquantumphysics,thefirstofthesepropositionsiscompletelymeaningless(notevenwrong),andthesecondonemeanssomethingquitedifferentfromwhatyoumight think.Itallcomesdown toadeep logicaldifferencebetween theclassical andquantumconceptsof the stateof asystem.Explainingthequantumconceptofstatewillrequiresomeabstractmathematics,solet’spauseforabrief interludeoncomplexnumbersandvectorspaces.Theneedforcomplexquantitieswillbecomeclearlateron,whenwestudythemathematicalrepresentationofspinstates.

1.8 MathematicalInterlude:ComplexNumbers

Everyone who has gotten this far in the Theoretical Minimum series knows about complex numbers.Nevertheless,Iwillspendafewlinesremindingyouoftheessentials.Fig.1.6showssomeoftheirbasicelements.

Acomplexnumberzisthesumofarealnumberandanimaginarynumber.Wecanwriteitas

Figure1.6:TwoCommonWaystoRepresentComplexNumbers.IntheCartesianrepresentation,xandyarethehorizontal(real)andvertical(imaginary)components.Inthepolarrepresentation,ristheradius,and is the anglemadewith thex axis. In each case, it takes two real numbers to represent a singlecomplexnumber.

wherex andy are real and .Complexnumbers canbe added,multiplied, anddividedby thestandardrulesofarithmetic.Theycanbevisualizedaspointsonthecomplexplanewithcoordinatesx,y.Theycanalsoberepresentedinpolarcoordinates:

Addingcomplexnumbersiseasyincomponentform:justaddthecomponents.Similarly,multiplyingthemiseasyintheirpolarform:Simplymultiplytheradiiandaddtheangles:

Everycomplexnumberzhasacomplexconjugate thatisobtainedbysimplyreversingthesignoftheimaginarypart.If

then

Multiplyingacomplexnumberanditsconjugatealwaysgivesapositiverealresult:

Itisofcoursetruethateverycomplexconjugateisitselfacomplexnumber,butit’softenhelpfultothinkofzand asbelongingtoseparate“dual”numbersystems.Dualheremeansthatforeveryzthereisaunique andviceversa.

ThereisaspecialclassofcomplexnumbersthatI’llcall“phase-factors.”Aphase-factorissimplyacomplexnumberwhoser-componentis1.Ifzisaphase-factor,thenthefollowinghold:

1.9 MathematicalInterlude:VectorSpaces

1.9.1 AxiomsForaclassicalsystem,thespaceofstatesisaset(thesetofpossiblestates),andthelogicofclassicalphysicsisBoolean.Thatseemsobviousanditisdifficulttoimagineanyotherpossibility.Nevertheless,therealworldoperatesalongentirelydifferentlines,atleastwheneverquantummechanicsisimportant.The space of states of a quantum system is not a mathematical set;6 it is a vector space. Relationsbetween theelementsofavectorspacearedifferent fromthosebetween theelementsofaset,and thelogicofpropositionsisdifferentaswell.

BeforeItellyouaboutvectorspaces,Ineedtoclarifythetermvector.Asyouknow,weusethistermto indicate an object in ordinary space that has amagnitude and a direction. Such vectors have threecomponents,correspondingtothethreedimensionsofspace.Iwantyoutocompletelyforgetabout thatconceptofavector.Fromnowon,wheneverIwanttotalkaboutathingwithmagnitudeanddirectioninordinaryspace,Iwillexplicitlycallita3-vector.Amathematicalvectorspaceisanabstractconstructionthatmayormaynothaveanythingtodowithordinaryspace.Itmayhaveanynumberofdimensionsfrom1to∞anditmayhavecomponentsthatareintegers,realnumbers,orevenmoregeneralthings.

ThevectorspacesweusetodefinequantummechanicalstatesarecalledHilbertspaces.Wewon’tgivethemathematicaldefinitionhere,butyoumayaswelladdthis termtoyourvocabulary.WhenyoucomeacrossthetermHilbertspaceinquantummechanics,itreferstothespaceofstates.AHilbertspacemayhaveeitherafiniteoraninfinitenumberofdimensions.

Inquantummechanics,avectorspace iscomposedofelements calledket-vectorsor justkets.Hereare theaxiomswewillusetodefinethevectorspaceofstatesofaquantumsystem(zandwarecomplexnumbers):

1. Thesumofanytwoket-vectorsisalsoaket-vector:

2. Vectoradditioniscommutative:

3. Vectoradditionisassociative:

4. Thereisauniquevector0suchthatwhenyouaddittoanyket,itgivesthesameketback:

5. Givenanyket ,thereisauniqueket suchthat

6. Given any ket and any complex number z, you can multiply them to get a new ket. Also,multiplicationbyascalarislinear:

7. Thedistributivepropertyholds:

Axioms6and7takentogetherareoftencalledlinearity.

Ordinary3-vectorswouldsatisfytheseaxiomsexceptforonething:Axiom6allowsavectortobemultiplied by any complex number. Ordinary 3-vectors can be multiplied by real numbers (positive,negative,orzero)butmultiplicationbycomplexnumbers isnotdefined.Onecan thinkof3-vectorsasformingarealvectorspace,andketsasformingacomplexvectorspace.Ourdefinitionofket-vectorsisfairlyabstract.Aswewillsee,therearevariousconcretewaystorepresentket-vectorsaswell.

1.9.2 FunctionsandColumnVectors

Let’s look at some concrete examples of complex vector spaces. First of all, consider the set ofcontinuouscomplex-valuedfunctionsofavariablex.CallthefunctionsA(x).Youcanaddanytwosuchfunctionsandmultiplythembycomplexnumbers.Youcancheckthattheysatisfyallsevenaxioms.Thisexample should make it obvious that we are talking about something much more general than three-dimensionalarrows.

Two-dimensionalcolumnvectorsprovideanotherconcreteexample.Weconstruct thembystackingupapairofcomplexnumbers, and ,intheform

andidentifyingthis“stack”withtheket-vector .Thecomplexnumbers arethecomponentsof .Youcanaddtwocolumnvectorsbyaddingtheircomponents:

Moreover,youcanmultiplythecolumnvectorbyacomplexnumberzjustbymultiplyingthecomponents,

Column vector spaces of any number of dimensions can be constructed. For example, here is a five-dimensionalcolumnvector:

Normally,wedonotmixvectorsofdifferentdimensionality.

1.9.3 BrasandKetsAswehaveseen,thecomplexnumbershaveadualversion:intheformofcomplexconjugatenumbers.In

thesameway,acomplexvectorspacehasadualversionthatisessentiallythecomplexconjugatevectorspace. For every ket-vector , there is a “bra” vector in the dual space, denoted by .Why thestrangetermsbraandket?Shortly,wewilldefineinnerproductsofbrasandkets,usingexpressionslike

to form bra-kets or brackets. Inner products are extremely important in the mathematicalmachineryofquantummechanics,andforcharacterizingvectorspacesingeneral.

Bravectorssatisfythesameaxiomsastheket-vectors,buttherearetwothingstokeepinmindaboutthecorrespondencebetweenketsandbras:

1. Suppose isthebracorrespondingtotheket ,and is thebracorrespondingto theket.Thenthebracorrespondingto

is

2. Ifzisacomplexnumber,thenitisnottruethatthebracorrespondingto is .Youhavetoremembertocomplex-conjugate.Thus,thebracorrespondingto

is

Intheconcreteexamplewhereketsarerepresentedbycolumnvectors,thedualbrasarerepresentedbyrowvectors,with theentriesbeingdrawnfromthecomplexconjugatenumbers.Thus, if theket isrepresentedbythecolumn

thenthecorrespondingbra isrepresentedbytherow

1.9.4 InnerProductsYouarenodoubtfamiliarwiththedotproductdefinedforordinary3-vectors.Theanalogousoperationforbrasandketsistheinnerproduct.Theinnerproductisalwaystheproductofabraandaketanditiswrittenthisway:

Theresultofthisoperationisacomplexnumber.Theaxiomsforinnerproductsarenottoohardtoguess:

1. Theyarelinear:

2. Interchangingbrasandketscorrespondstocomplexconjugation:

Exercise1.1:a) Usingtheaxiomsforinnerproducts,prove

b) Prove isarealnumber.

Intheconcreterepresentationofbrasandketsbyrowandcolumnvectors,theinnerproductisdefinedintermsofcomponents:

Theruleforinnerproductsisessentiallythesameasfordotproducts:addtheproductsofcorrespondingcomponentsofthevectorswhoseinnerproductisbeingcalculated.

Exercise 1.2: Show that the inner product defined by Eq. 1.2 satisfies all the axioms of innerproducts.

Usingtheinnerproduct,wecandefinesomeconceptsthatarefamiliarfromordinary3-vectors:

• Normalized Vector: A vector is said to be normalized if its inner product with itself is 1.Normalizedvectorssatisfy,

Forordinary3-vectors, the termnormalizedvector is usually replacedbyunitvector, that is, avectorofunitlength.

• OrthogonalVectors:Twovectorsaresaidtobeorthogonaliftheirinnerproductiszero. andareorthogonalif

Thisistheanalogofsayingthattwo3-vectorsareorthogonaliftheirdotproductiszero.

1.9.5 OrthonormalBasesWhen working with ordinary 3-vectors, it is extremely useful to introduce a set of three mutuallyorthogonalunitvectorsandusethemasabasistoconstructanyvector.Asimpleexamplewouldbetheunit3-vectors thatpointalongthex,y,andzaxes.Theyareusuallycalled , ,and .Each isofunitlength andorthogonal to the others. If you tried to find a fourth vector orthogonal to these three, therewouldn’t be any—not in threedimensions anyway.However, if thereweremoredimensionsof space,therewouldbemorebasisvectors.Thedimensionofaspacecanbedefinedasthemaximumnumberofmutuallyorthogonalvectorsinthatspace.

Obviously,thereisnothingspecialabouttheparticularaxesx,y,andz.Aslongasthebasisvectorsareofunitlengthandaremutuallyorthogonal,theycompriseanorthonormalbasis.

Thesameprincipleistrueforcomplexvectorspaces.Onecanbeginwithanynormalizedvectorandthen look for a second one, orthogonal to the first. If you find one, then the space is at least two-dimensional.Thenlookforathird,fourth,andsoon.Eventually,youmayrunoutofnewdirectionsandtherewillnotbeanymoreorthogonalcandidates.Themaximumnumberofmutuallyorthogonalvectorsisthe dimension of the space. For column vectors, the dimension is simply the number of entries in thecolumn.

Let’sconsideranN-dimensionalspaceandaparticularorthonormalbasisofket-vectorslabeled .7

Thelabelirunsfrom1toN.Consideravector ,writtenasasumofbasisvectors:

The arecomplexnumberscalledthecomponentsofthevector,andtocalculatethemwetaketheinnerproductofbothsideswithabasisbra :

Next,weusethefactthatthebasisvectorsareorthonormal.Thisimpliesthat ifiisnotequaltoj,and ifi=j.Inotherwords, .ThismakesthesuminEq.1.4collapsetooneterm:

Thus,weseethat thecomponentsofavectorare just its innerproductswith thebasisvectors.WecanrewriteEq.1.3intheelegantform

1“Blackbox”meanswehavenoknowledgeofwhat’sinsidetheapparatusorhowitworks.Butrestassured,itdoesnotcontainacat.2Thestandardnotationforaunitvector(oneofunitlength)istoplacea“hat”abovethesymbolrepresentingthevector.3Andmaybedefinedformultiplepropositions,butwe’llonlyconsidertwo.Thesamegoesforor.4OK,perhapsEinsteindiddiscoverAmerica.Buthewasnotthefirst.5Recallthattheclassicalmeaningof isdifferentfromthequantummechanicalmeaning.Classically, isastraightforward3-vector; x

and zrepresentitsspatialcomponents.6Tobealittlemoreprecise,wewillnotfocusontheset-theoreticpropertiesofstatespaces,eventhoughtheymayofcourseberegarded

assets.7Mathematically, basis vectors are not required to be orthonormal. However, in quantum mechanics they generally are. In this book,

wheneverwesaybasis,wemeananorthonormalbasis.

Lecture2

QuantumStates

Art:Oddlyenough,thatbeermademyheadstopspinning.Whatstatearewein?

Lenny:IwishIknew.Doesitmatter?

Art:Itmight.Idon’tthinkwe’reinCaliforniaanymore.

2.1 StatesandVectors

In classical physics, knowing the state of a system implies knowing everything that is necessary topredict thefutureof thatsystem.Aswe’veseenin thelast lecture,quantumsystemsarenotcompletelypredictable. Evidently, quantum states have a different meaning than classical states. Very roughly,knowingaquantumstatemeansknowingasmuchascanbeknownabouthowthesystemwasprepared.Inthelastchapter,wetalkedaboutusinganapparatustopreparethestateofaspin.Infact,weimplicitlyassumedthattherewasnomorefinedetailtospecifyorthatcouldbespecifiedaboutthestateofthespin.

Theobviousquestiontoaskiswhethertheunpredictabilityisduetoanincompletenessinwhatwecallaquantumstate.Therearevariousopinionsaboutthismatter.Hereisasampling:

• Yes,theusualnotionofquantumstateisincomplete.Thereare“hiddenvariables”that,ifonlywecouldaccess them,wouldallowcompletepredictability.Thereare twoversionsof thisview. Inversion A, the hidden variables are hard to measure but in principle they are experimentallyavailable tous. InversionB, becausewe aremadeof quantummechanicalmatter and thereforesubject to the restrictions of quantum mechanics, the hidden variables are, in principle, notdetectable.

• No,thehiddenvariablesconceptdoesnotleadusinaprofitabledirection.Quantummechanicsisunavoidably unpredictable. Quantummechanics is as complete a calculus of probabilities as ispossible.Thejobofaphysicististolearnandusethiscalculus.

Idon’tknowwhat theultimateanswer to thisquestionwillbe,or even if itwillprove tobeausefulquestion.Butforourpurposes,it’snotimportantwhatanyparticularphysicistbelievesabouttheultimatemeaningofthequantumstate.Forpracticalreasons,wewilladoptthesecondview.

Inpractice,whatthismeansforthequantumspinofLecture1isthat,whentheapparatusAactsandtellsusthat or ,thereisnomoretoknow,orthatcanbeknown.Likewise,ifwerotate

Aandmeasure or ,thereisnomoretoknow.Likewisefor oranyothercomponentofthespin.

2.2 RepresentingSpinStates

Now it’s time to try our hand at representing spin states using state-vectors. Our goal is to build arepresentation thatcaptureseverythingweknowabout thebehaviorof spins.At thispoint, theprocesswillbemoreintuitivethanformal.Wewilltrytofitthingstogetherthebestwecan,basedonwhatwe’vealreadylearned.Pleasereadthissectioncarefully.Believeme,itwillpayoff.

Let’sbeginbylabelingthepossiblespinstatesalongthethreecoordinateaxes.IfAisorientedalongthezaxis,thetwopossiblestatesthatcanbepreparedcorrespondto .Let’scall themupanddownanddenotethembyket-vectors and .Thus,whentheapparatusisorientedalongthezaxisandregisters+1,thestate hasbeenprepared.

Ontheotherhand,iftheapparatusisorientedalongthexaxisandregisters−1,thestate hasbeenprepared.We’llcallitleft.IfAisalongtheyaxis,itcanpreparethestates and (inandout).Yougettheidea.

Theideathattherearenohiddenvariableshasaverysimplemathematicalrepresentation:thespaceofstatesforasinglespinhasonlytwodimensions.Thispointdeservesemphasis:

Allpossiblespinstatescanberepresentedinatwo-dimensionalvectorspace.

Wecould,somewhatarbitrarily,1choose and asthetwobasisvectorsandwriteanystateasalinearsuperpositionofthesetwo.We’lladoptthatchoicefornow.Let’susethesymbol foragenericstate.Wecanwritethisasanequation,

where and arethecomponentsof alongthebasisdirections and .Mathematically,wecanidentifythecomponentsof as

Theseequationsareextremelyabstract,anditisnotatallobviouswhattheirphysicalsignificanceis.Iam going to tell you right nowwhat theymean: First of all, can represent any state of the spin,preparedinanymanner.Thecomponents and arecomplexnumbers;bythemselves,theyhavenoexperimental meaning, but their magnitudes do. In particular, and have the followingmeaning:

• Giventhatthespinhasbeenpreparedinthestate ,andthattheapparatusisorientedalongz,thequantity istheprobabilitythatthespinwouldbemeasuredas .Inotherwords,itistheprobabilityofthespinbeingupifmeasuredalongthezaxis.

• Likewise, istheprobabilitythat wouldbedownifmeasured.

The values,orequivalently and ,arecalledprobabilityamplitudes.Theyarethemselvesnot probabilities. To compute a probability, their magnitudes must be squared. In other words, theprobabilitiesformeasurementsofupanddownaregivenby

NoticethatIhavesaidnothingaboutwhat isbeforeit ismeasured.Beforethemeasurement,allwehave is the vector , which represents the potential possibilities but not the actual values of ourmeasurements.

Twootherpointsareimportant:First,notethat and aremutuallyorthogonal.Inotherwords,

Thephysicalmeaningofthisisthat,ifthespinispreparedup, thentheprobabilitytodetectitdown iszero,andviceversa.Thispoint is so important, I’ll say itagain:Twoorthogonalstatesarephysicallydistinctandmutuallyexclusive.Ifthespinisinoneofthesestates,itcannotbe(haszeroprobabilitytobe)intheotherone.Thisideaappliestoallquantumsystems,notjustspin.

But don’tmistake the orthogonality of state-vectors for orthogonal directions in space. In fact, thedirectionsupanddownarenotorthogonaldirectionsinspace,eventhoughtheirassociatedstate-vectorsareorthogonalinstatespace.

Thesecondimportantpointisthatforthetotalprobabilitytocomeoutequaltounity,wemusthave

Thisisequivalenttosayingthatthevector isnormalizedtoaunitvector:

Thisisaverygeneralprincipleofquantummechanicsthatextendstoallquantumsystems:thestateofasystem is representedbyaunit (normalized)vector inavector spaceof states.Moreover, the squaredmagnitudesofthecomponentsofthestate-vector,alongparticularbasisvectors,representprobabilitiesforvariousexperimentaloutcomes.

2.3 AlongthexAxis

Wesaidbeforethatwecanrepresentanyspinstateasalinearcombinationofthebasisvectors and.Let’strydoingthisnowforthevectors and ,whichrepresentspinspreparedalongthexaxis.

We’ll start with . As you recall from Lecture 1, ifA initially prepares , and is then rotated tomeasure ,therewillbeequalprobabilitiesforupanddown.Thus, and mustbothbeequalto .Asimplevectorthatsatisfiesthisruleis

Thereissomeambiguityinthischoice,butaswewillseelater,itisnothingmorethantheambiguityinourchoiceofexactdirectionsforthexandyaxes.

Next,let’slookatthevector .Hereiswhatweknow:whenthespinhasbeenpreparedintheleftconfiguration,theprobabilitiesfor areagainequalto .That isnotenoughtodeterminethevalues

and ,butthereisanotherconditionthatwecaninfer.Earlier,Itoldyouthat and areorthogonalforthesimplereasonthat,ifthespinisup,it’sdefinitelynotdown.Butthereisnothingspecialaboutupanddown that is not also trueof right and left. Inparticular, if the spin is right, it has zeroprobabilityofbeingleft.Thus,byanalogywithEq.2.3,

Thisprettymuchfixes intheform

Exercise2.1: Provethatthevector inEq.2.5isorthogonaltovector inEq.2.6.

Again, there is some ambiguity in the choice of . This is called the phase ambiguity. Supposewemultiply byanycomplexnumberz.Thatwillhavenoeffectonwhetheritisorthogonalto ,thoughingeneraltheresultwillnolongerbenormalized(haveunitlength).Butifwechoose (wherecanbeanyrealnumber),thentherewillbenoeffectonthenormalizationbecause hasunitmagnitude.Inotherwords, will remainequal to1.Sinceanumberof theform iscalledaphase-factor, the ambiguity is called the phase ambiguity. Later, we will find out that no measurablequantityissensitivetotheoverallphase-factor,andthereforewecanignoreitwhenspecifyingstates.

2.4 AlongtheyAxis

Finally,thisbringsusto and ,thevectorsrepresentingspinsorientedalongtheyaxis.Let’slookattheconditionstheyneedtosatisfy.First,

Thisconditionstatesthatinandoutarerepresentedbyorthogonalvectorsinthesamewaythatupanddownare.Physically,thismeansthatifthespinisin,itisdefinitelynotout.

Thereareadditionalrestrictionsonthevectors and .UsingtherelationshipsexpressedinEqs.2.1and2.2,andthestatisticalresultsofourexperiments,wecanwritethefollowing:

Inthefirsttwoequations, takestheroleof fromEqs.2.1and2.2.Inthesecondtwo, takesthatrole.Theseconditionsstatethatifthespinisorientedalongy,andisthenmeasuredalongz,itisequallylikelytobeupordown.Weshouldalsoexpectthatifthespinweremeasuredalongthexaxis,itwouldbeequallylikelytoberightorleft.Thisleadstoadditionalconditions:

These conditions are sufficient to determine the form of the vectors and , apart from the phaseambiguity.Hereistheresult:

Exercise2.2: Provethat and satisfyalloftheconditionsinEqs.2.7,2.8,and2.9.Aretheyuniqueinthatrespect?

It’sinterestingthattwoofthecomponentsinEqs.2.10areimaginary.Ofcourse,we’vesaidallalongthatthespaceofstatesisacomplexvectorspace,butuntilnowwehavenothadtousecomplexnumbersinour calculations. Are the complex numbers in Eqs. 2.10 a convenience or a necessity? Given ourframeworkforspinstates,thereisnowayaroundthem.It’ssomewhattedioustodemonstratethis,butthestepsarestraightforward.Thefollowingexercisegivesyouaroadmap.Theneedforcomplexnumbersisageneralfeatureofquantummechanics,andwe’llseemoreexamplesaswego.

Exercise2.3: Forthemoment,forget thatEqs.2.10giveusworkingdefinitionsfor and intermsof and ,andassumethatthecomponents ,β,γ,andδareunknown:

a) UseEqs.2.8toshowthat

b) UsetheaboveresultandEqs.2.9toshowthat

c) Showthat and musteachbepureimaginary.

If ispureimaginary,then andβcannotbothbereal.Thesamereasoningappliesto .

2.5 CountingParameters

It’salways important toknowhowmany independentparameters it takes tocharacterizea system.Forexample, the generalized coordinates we used in Volume I (referred to as ) each represented anindependentdegreeoffreedom.Thatapproachfreedusfromthedifficultjobofwritingexplicitequationstodescribephysicalconstraints.Alongsimilar lines,ournext task is tocount thenumberofphysicallydistinctstatesthereareforaspin.Iwilldoitintwoways,toshowthatyougetthesameanswereitherway.

The first way is simple. Point the apparatus along any unit 3-vector2 and prepare a spin withalongthataxis.If ,youcanthinkofthespinasbeingorientedalongthe axis.Thus,

theremustbeastateforeveryorientationof theunit3-vector .Howmanyparametersdoes it taketospecify such an orientation?The answer is of course two. It takes two angles to define a direction inthree-dimensionalspace.3

Now,let’sconsiderthesamequestionfromanotherperspective.Thegeneralspinstateisdefinedbytwo complex numbers, and . That seems to add up to four real parameters,with each complexparametercountingas tworealones.Butrecall that thevectorhas tobenormalizedas inEq.2.4.Thenormalizationconditiongivesusoneequationinvolvingrealvariables,andcutsthenumberofparametersdowntothree.

AsIsaidearlier,wewilleventuallyseethatthephysicalpropertiesofastate-vectordonotdependontheoverallphase-factor.Thismeansthatoneofthethreeremainingparametersisredundant,leavingonly two—the same as the number of parameterswe need to specify a direction in three-dimensionalspace.Thus,thereisenoughfreedomintheexpression

todescribeallthepossibleorientationsofaspin,eventhoughthereareonlytwopossibleoutcomesofanexperimentalonganyaxis.

2.6 RepresentingSpinStatesasColumnVectors

Sofar,wehavebeenabletolearnalotbyusingtheabstractformsofourstate-vectors,thatis, andandsoforth.Theseabstractionshelpusfocusonmathematicalrelationshipswithoutworryingabout

unnecessarydetails.However,soonwewillneedtoperformdetailedcalculationsonspinstates,andforthatwe’llneed towriteourstate-vectors incolumnform.Becauseof“phase indifference,” thecolumnrepresentationsarenotunique,andwe’lltrytochoosethesimplestandmostconvenientoneswecanfind.

As usual, we’ll start with and . We need them to have unit length, and to be mutuallyorthogonal.Apairofcolumnsthatsatisfiestheserequirementsis

Withthesecolumnvectorsinhand,itwillbeeasytocreatecolumnvectorsfor and usingEqs.2.5and2.6, and for and usingEqs.2.10.We’ll do that in the next lecture,where these results areneeded.

2.7 PuttingItAllTogether

Wehavecoveredalotofgroundinthislecture.Beforemovingon,let’stakestockofwhatwe’vedone.Our goalwas to synthesizewhatwe know about spins and vector spaces.We figured out how to usevectors torepresentspinstates,andin theprocesswegotaglimpseof thekindof informationastate-vectorcontains(anddoesnotcontain!).Hereisabriefoutlineofwhatwedid:

• Basedonourknowledgeofspinmeasurements,wechosethreepairsofmutuallyorthogonalbasisvectors. Pair-wise,we named them and , and , and and . Because the basisvectors and represent physically distinct states, we were able to assert that they aremutuallyorthogonal.Inotherwords, .Thesameholdsfor and ,andalsofor and

.

• Wefoundthat it takestwoindependentparameterstospecifyaspinstate,andthenwearbitrarilychoseoneoftheorthogonalpairs, and ,asourbasisvectorsforrepresentingallspinstates—eventhoughthetwocomplexnumbersinastate-vectorrequirefourrealnumberstospecifythem.Howdidwegetawaywiththis?Wewerecleverenoughtonoticethatthesefournumbersarenotall independent.4 The normalization constraint (total probability must equal 1) eliminates oneindependentparameter,and“phaseindifference”(thephysicsofastate-vectorisunaffectedbyitsoverallphase-factor)eliminatesasecond.

• Havingchosen and asourmainbasisvectors,wefiguredouthowtorepresenttheothertwo

pairs of basis vectors as linear combinations of and , using additional orthogonality andprobability-basedconstraints.

• Finally,weestablishedawaytorepresentourmainbasisvectorsascolumns.Thisrepresentationisnotunique.Inthenextlecture,we’lluseour and columnvectorstoderivecolumnvectorsforthetwootherbases.

Whileachievingtheseconcreteresults,wegotachancetoseesomestate-vectormathematics inactionand learn something about how thesemathematical objects correspond to physical spins.Althoughwewillfocusonspin,thesameconceptsandtechniquesapplytootherquantumsystemsaswell.Pleasetakealittletimetoassimilatethematerialwe’vecoveredsofarbeforemovingontothenextlecture.AsIsaidatthebeginning,itwillreallypayoff.

1Thechoiceisnottotallyarbitrary.Thebasisvectorsmustbeorthogonaltoeachother.2Keepinmindthat3-vectorsarenotbrasorkets.3Recall that spherical coordinates use two angles to represent the orientation of a point in relation to the origin.Latitude and longitude

provideanotherexample.4Pleaseindulgeinaself-satisfiedgrin.

Lecture3

PrinciplesofQuantumMechanics

Art:I’mnotlikeyou,Lenny.Mybrainjustwasn’tbuiltforquantummechanics.

Lenny:Nah,minewasn’teither.Justcan’treallyvisualizethestuff.ButI’lltellyou,Ionceknewaguywhothoughtjustlikeanelectron.

Art:Whathappenedtohim?

Lenny:Art,allI’mgonnatellyouisthatitsurewasn’tpretty.

Art:Hmm,Iguessthatgenedidn’tfly.

No,wewerenotbuilttosensequantumphenomena;notthesamewaywewerebuilttosenseclassicalthingslikeforceandtemperature.Butweareveryadaptablecreaturesandwe’vebeenabletosubstituteabstract mathematics for the missing senses that might have allowed us to directly visualize quantummechanics.Andeventuallywedodevelopnewkindsofintuition.

This lecture introduces theprinciplesof quantummechanics. Inorder todescribe thoseprinciples,we’llneedsomenewmathematicaltools.Let’sgetstarted.

3.1 MathematicalInterlude:LinearOperators

3.1.1 MachinesandMatricesStates in quantum mechanics are mathematically described as vectors in a vector space. Physicalobservables—thethings thatyoucanmeasure—aredescribedbylinearoperators.We’ll takethatasanaxiom,andwe’ll findout later (inSection3.1.5) thatoperatorscorresponding tophysicalobservablesmustbeHermitianaswellas linear.Thecorrespondencebetweenoperatorsandobservables issubtle,andunderstandingitwilltakesomeeffort.

Observables are the things you measure. For example, we can make direct measurements of thecoordinatesofaparticle;theenergy,momentum,orangularmomentumofasystem;ortheelectricfieldatapointinspace.Observablesarealsoassociatedwithavectorspace,buttheyarenotstate-vectors.Theyare the thingsyoumeasure— wouldbe an example—and they are representedby linear operators.JohnWheelerlikedtocallsuchmathematicalobjectsmachines.Heimaginedamachinewithtwoports:

aninputportandanoutputport.Intheinputportyouinsertavector,suchas .Thegearsturnandthemachinedeliversaresultintheoutputport.Thisresultisanothervector,say .

Let’sdenotetheoperatorbytheboldfaceletterM(for“machine”).HereistheequationtoexpressthefactthatMactsonthevector togive :

Noteverymachineisalinearoperator.Linearityimpliesafewsimpleproperties.Tobeginwith,alinearoperatormustgiveauniqueoutputforeveryvectorinthespace.Wecanimagineamachinethatgivesanoutputforsomevectors,butjustgrindsupothersandgivesnothing.Thismachinewouldnotbealinearoperator.Somethingmustcomeoutforanythingyouputin.

ThenextpropertystatesthatwhenalinearoperatorMactsonamultipleofaninputvector,itgivesthesamemultipleoftheoutputvector.Thus,if ,andzisanycomplexnumber,then

Theonlyotherruleisthat,whenMactsonasumofvectors,theresultsaresimplyaddedtogether:

To give a concrete representation of linear operators, we return to the row and column vectorrepresentationofbra-andket-vectorsthatweusedinLecture1.Therow-columnnotationdependsonourchoice of basis vectors. If the vector space is N-dimensional, we choose a set of N orthonormal(orthogonalandnormalized)ket-vectors.Let’slabelthem ,andtheirdualbra-vectors .

Wearenowgoingtotaketheequation

andwriteitincomponentform.AswedidinEq.1.3,we’llrepresentanarbitraryket asasumoverbasisvectors:

Here,we’reusingjasanindexratherthanisoyouwon’tbetemptedtothinkthatwe’retalkingaboutthein spin state. Now, we’ll represent in the same way and plug both of these substitutions into

.Thatgives

Thelaststepistotaketheinnerproductofbothsideswithaparticularbasisvector ,resultingin

Tomakesenseofthisresult,rememberthat iszeroifjandkarenotequal,and1iftheyareequal.Thatmeansthatthesumontherightsidecollapsestoasingleterm, .

Ontheleftside,weseeasetofquantities .Wecanabbreviate withthesymbol.Notice thateach is just a complexnumber.To seewhy, thinkofM operatingon togive

somenewket-vector.The