PraiseforQuantumMechanics“Whileit’sdefinitelyabookforpeoplewhohavesomemathbackground,

it doesn’t requiremuch and it unspools the basics slowly, thoughtfully andwithexceptionalclarity.”

—AdamFrank,NPR’s13.7blog

“If you’re ever banished to a desert island and allowed to take just onebook,hereitis.Givenenoughtime,withnodistractions,youcoulduseittoeventuallymaster quantummechanics… . [E]venwithoutmastering all thecalculational complexities, a careful read at the very least offers deeperinsight into the logic andmathematical substance of quantum physics thanyou’llgetfromanypopularaccount.”

—ScienceNews

“[QuantumMechanics] provides youwith the ‘minimal’ equipment youneed to understand what all the fuzz with quantum optics, quantumcomputing,andblackholeevaporationisabout….Ifyouwanttomakethestep from popular science literature to textbooks and the general scientificliterature,thenthisbookseriesisamust-read.”

—BackReaction

“[V]erydetailedbutverywellwritten.”

—SanFranciscoBookReview

“Thewritingisfreshandimmediate,withplentyofdetailpackagedintothe smooth narrative… . [O]n their own terms, I found Susskind andFriedman’s explanations crisp and satisfying… . I maintain a clearrecollection of the bewilderment with which I struggled through my ownuniversity quantum-mechanics courses. For students in a similar position,trying to draw together the fragments of formalism into a clear conceptualwhole, Susskind and Friedman’s persuasive overview—and their insistenceon explaining, with sharp mathematical detail, exactly what it is that isstrangeaboutquantummechanics—maybejustwhatisneeded.”

—DavidSeery,Nature

“[QuantumMechanics] is evenbetter than the first volume, takingon amuch more difficult subject… . ‘The Theoretical Minimum’ phrase is areference to Landau, but it’s a good characterization of this book and thelecturesingeneral.Susskinddoesagoodjobofboilingthesesubjectsdown

totheircoreideasandexamples,andgivingacarefulexpositionoftheseinassimple terms as possible. If you’ve gotten a taste for physics from popularbooks,thisisagreatplacetostartlearningwhatthesubjectisreallyabout.”

—PeterWoit,NotEvenWrong

“[T]he bookwillworkwell as a companion text for university studentsstudyingquantummechanicsorthearmchairphysicistsfollowingSusskind’sYouTubelectures.”

—PublishersWeekly

“Eitherforthoseabouttostartauniversityphysicscoursewhowantsomepreparation, or for someone who finds popular science explanations toosummary and is prepared to take on some quite serious math … it’s afascinatingadditiontothelibrary.”

—PopularScience(UK)

“This is quantum mechanics for real. This is the good stuff, the mostmysteriousaspectsofhowrealityworks,setoutwithcrystallineclarity.Ifyouwant to knowhowphysicists really think about theworld, this book is theplacetostart.”

—SeanCarroll, physicist,California InstituteofTechnology, andauthorofTheParticleattheEndoftheUniverse

ThisbookisthesecondvolumeoftheTheoreticalMinimumseries.Thefirstvolume,TheTheoreticalMinimum:WhatYouNeed toKnow toStartDoingPhysics, covered classical mechanics, which is the core of any physicseducation.We will refer to it from time to time simply as Volume I. Thissecond book explains quantum mechanics and its relationship to classicalmechanics.ThebooksinthisseriesrunparalleltoLeonardSusskind’svideos,available on the Web through Stanford University (seewww.theoreticalminimum.comforalisting).Whilecoveringthesamegeneraltopicsasthevideos,thebookscontainadditionaldetails,andtopicsthatdon’tappearinthevideos.

http://www.theoreticalminimum.com

QUANTUM

MECHANICS

AlsobyLeonardSusskind

TheTheoreticalMinimum

WhatYouNeedToKnowtoStartDoingPhysics(withGeorgeHrabovsky)

TheBlackHoleWar

TheCosmicLandscape

QUANTUM

MECHANICSTheTheoreticalMinimum

LEONARDSUSSKINDandARTFRIEDMAN

BASICBOOKS

AMemberofthePerseusBooksGroup

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

mailto:special.markets@perseusbooks.com

Forourparents,whomadeitallpossible:

IreneandBenjaminSusskindGeorgeandTrudyFriedman

Contents

Preface

Prologue

Introduction

1SystemsandExperiments

2QuantumStates

3PrinciplesofQuantumMechanics

4TimeandChange

5UncertaintyandTimeDependence

6CombiningSystems:Entanglement

7MoreonEntanglement

8ParticlesandWaves

9ParticleDynamics

10TheHarmonicOscillator

Appendix

Index

PrefaceAlbertEinstein,whowasinmanywaysthefatherofquantummechanics,hadanotoriouslove-haterelationwiththesubject.HisdebateswithNielsBohr—Bohr completely accepting of quantum mechanics and Einstein deeplyskeptical—arefamousinthehistoryofscience.Itwasgenerallyacceptedbymost physicists that Bohr won and Einstein lost. My own feeling, I thinkshared by a growing number of physicists, is that this attitude does not dojusticetoEinstein’sviews.

BothBohrandEinsteinweresubtlemen.Einsteintriedveryhardtoshowthatquantummechanicswasinconsistent;Bohr,however,wasalwaysabletocounterhisarguments.ButinhisfinalattackEinsteinpointedtosomethingsodeep, so counterintuitive, so troubling, and yet so exciting, that at thebeginning of the twenty-first century it has returned to fascinate theoreticalphysicists. Bohr’s only answer to Einstein’s last great discovery—thediscoveryofentanglement—wastoignoreit.

The phenomenon of entanglement is the essential fact of quantummechanics,thefactthatmakesitsodifferentfromclassicalphysics.Itbringsinto question our entire understanding about what is real in the physicalworld. Our ordinary intuition about physical systems is that if we knoweverythingaboutasystem,thatis,everythingthatcaninprinciplebeknown,thenweknoweverythingaboutitsparts.Ifwehavecompleteknowledgeoftheconditionofanautomobile,thenweknoweverythingaboutitswheels,itsengine,itstransmission,rightdowntothescrewsthatholdtheupholsteryinplace. It would notmake sense for amechanic to say, “I know everythingabout your car but unfortunately I can’t tell you anything about any of itsparts.”

But that’s exactly what Einstein explained to Bohr—in quantummechanics, one can know everything about a system and nothing about itsindividual parts—but Bohr failed to appreciate this fact. I might add thatgenerationsofquantumtextbooksblithelyignoredit.

Everyone knows that quantummechanics is strange, but I suspect veryfewpeoplecouldtellyouexactlyinwhatway.Thisbookisatechnicalcourseof lectures on quantummechanics, but it is different thanmost courses ormost textbooks.Thefocus ison the logicalprinciplesand thegoal isnot tohidetheutterstrangenessofquantumlogicbuttobringitoutintothelightofday.

I remind you that this book is one of several that closely follow myInternetcourseseries,theTheoreticalMinimum.Mycoauthor,ArtFriedman,wasastudentinthesecourses.ThebookbenefitedfromthefactthatArtwaslearningthesubjectandwasthereforeverysensitivetotheissuesthatmightbe confusing to thebeginner.During the courseofwriting,wehad a lot offun,andwe’vetriedtoconveysomeofthatspiritwithabitofhumor.Ifyoudon’tgetit,ignoreit.

LeonardSusskind

When I completed my master’s degree in computer science at Stanford, Icouldnot haveguessed that I’d return someyears later to attendLeonard’sphysicslectures.Myshort“career”inphysicsendedmanyyearsearlier,withthe completion ofmybachelor’s degree.Butmy interest in the subject hasremainedverymuchalive.

It appears that I have lots of company—the world seems filled withpeoplewhoaregenuinely,deeplyinterestedinphysicsbutwhoseliveshavetakenthemindifferentdirections.Thisbookisforallofus.

Quantum mechanics can be appreciated, to some degree, on a purelyqualitativelevel.Butmathematicsiswhatbringsitsbeautyintosharpfocus.We have tried to make this amazing body of work fully accessible tomathematically literate nonphysicists. I thinkwe’ve done a fairly good job,andIhopeyou’llagree.

Noonecompletesaproject like thiswithout lotsofhelp.ThepeopleatBrockman, Inc., have made the business end of things seem easy, and theproductionteamatPerseusBookshasbeentop-notch.Mysincerethanksgoto TJKelleher, RachelKing, and Tisse Takagi. It was our good fortune toworkwithatalentedcopyeditor,JohnSearcy.

I’m grateful to Leonard’s (other) continuing education students forroutinelyraisingthoughtful,provocativequestions,andformanystimulatingafter-class conversations. RobColwell, ToddCraig,Monty Frost, and JohnNash offered constructive comments on themanuscript. JeremyBranscomeand Russ Bryan reviewed the entire manuscript in detail, and identified anumberofproblems.

I thankmy family and friends for their kind support and enthusiasm. Iespeciallythankmydaughter,Hannah,formindingthestore.

Besides her love, encouragement, insight, and sense of humor, myamazingwife,MargaretSloan,contributedaboutathirdofthediagramsandbothHilbert’sPlaceillustrations.Thanks,Maggie.

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

ArtFriedman

PrologueArt looksoverhisbeerandsays,“Lenny, let’splayaroundof theEinstein-Bohrgame.”

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

“Fair enough.Here’smy first shot: God doesn’t play dice. 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,“OntheQuantumTheoryof Radiation,” argues that the emission of gamma rays is governed by astatisticallaw.

AProfessorandaFiddlerWalkintoaBarVolumeIwaspunctuatedbyshortconversationsbetweenLennyandGeorge,fictionalpersonaswhowerelooselybasedontwoJohnSteinbeckcharacters.ThesettingforthisvolumeoftheTheoreticalMinimumseriesisinspiredbythe stories of Damon Runyon. It’s a world filled with crooks, con artists,degenerates,smoothoperators,anddo-gooders.Plusafewordinaryfolks,justtrying toget through theday.TheactionunfoldsatapopularwateringholecalledHilbert’sPlace.

IntothissettingstrollLennyandArt,twogreenhornsfromCaliforniawhosomehowgotseparatedfromtheirtourbus.Wishthemluck.Theywillneedit.

WhattoBringYou don’t need to be a physicist to take this journey, but you should havesomebasicknowledgeofcalculusandlinearalgebra.Youshouldalsoknow

somethingaboutthematerialcoveredinVolumeI. It’sOKifyourmathisabit rusty. We’ll review and explain much of it as we go, especially thematerialonlinearalgebra.VolumeIreviewsthebasicideasofcalculus.

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

©MargaretSloan

IntroductionClassical mechanics is intuitive; things move in predictable ways. Anexperienced ballplayer can take a quick look at a fly ball, and from itslocationanditsvelocity,knowwheretoruninordertobetherejustintimetocatch theball.Ofcourseasuddenunexpectedgustofwindmightfoolhim,butthat’sonlybecausehedidn’ttakeintoaccountallthevariables.Thereisanobviousreasonwhyclassicalmechanicsisintuitive:humans,andanimalsbefore them, havebeenusing itmany times every day for survival.But noone ever used quantum mechanics before the twentieth century. Quantummechanics describes things so small that they are completely beyond therangeof thehumansenses.So itstands toreason thatwedidnotevolveanintuition for thequantumworld.Theonlywaywecancomprehend it isbyrewiringour intuitionswithabstractmathematics.Fortunately, forsomeoddreason,wedidevolvethecapacityforsuchrewiring.

Ordinarily, we learn classical mechanics first, before even attemptingquantummechanics. But quantum physics is muchmore fundamental thanclassicalphysics.Asfarasweknow,quantummechanicsprovidesanexactdescriptionofeveryphysicalsystem,butsomethingsaremassiveenoughthatquantum mechanics can be reliably approximated by classical mechanics.That’sallthatclassicalmechanicsis:anapproximation.Fromalogicalpointof view, we should learn quantum mechanics first, but very few physicsteacherswouldrecommendthat.Eventhiscourseoflectures—theTheoreticalMinimum series—began with classical mechanics. Nevertheless, in thesequantumlectures,classicalmechanicswillplayalmostnoroleexceptneartheend, well after the basic principles of quantum mechanics have beenexplained. I think this is really the rightway todo it, not just logicallybutpedagogically aswell.Thatwaywedon’t fall into the trapof thinking thatquantummechanicsisbasicallyjustclassicalmechanicswithacoupleofnewgimmicks thrown in. By the way, quantummechanics is technically mucheasierthanclassicalmechanics.

Thesimplestclassicalsystem—thebasiclogicalunitforcomputerscience—is the two-state system. Sometimes it’s called a bit. It can representanythingthathasonlytwostates:acointhatcanshowheadsortails,aswitchthatisonoroff,oratinymagnetthatisconstrainedtopointeithernorthorsouth. As you might expect, especially if you studied the first lecture ofVolume I, the theory of classical two-state systems is extremely simple—boring,infact.Inthisvolume,we’regoingtobeginwiththequantumversion

of the two-state system, called a qubit, which is far more interesting. Tounderstandit,wewillneedawholenewwayofthinking—anewfoundationoflogic.

Lecture1

SystemsandExperimentsLennyandArtwanderintoHilbert’sPlace.

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

Lenny:Takeabreath.You’llgetusedtoit.

Art:Whichwayisup?

1.1 QuantumMechanicsIsDifferentWhat is so special about quantum mechanics? Why is it so hard tounderstand?Itwouldbeeasytoblamethe“hardmathematics,”andtheremaybe some truth in that idea. But that can’t be the whole story. Lots ofnonphysicistsareable tomasterclassicalmechanicsandfield theory,whichalsorequirehardmathematics.

Quantummechanicsdealswith thebehaviorofobjects sosmall thatwehumansareillequippedtovisualizethematall.Individualatomsareneartheupper end of this scale in terms of size. Electrons are frequently used asobjects of study.Our sensory organs are simply not built to perceive themotionofanelectron.Thebestwecando is to try tounderstandelectronsandtheirmotionasmathematicalabstractions.

“Sowhat?” says the skeptic. “Classical mechanics is filled to the brimwithmathematicalabstractions—pointmasses,rigidbodies,inertialreferenceframes,positions,momenta, fields,waves—the listgoesonandon.There’snothing new aboutmathematical abstractions.”This is actually a fair point,and indeed theclassicalandquantumworldshavesome important things incommon.Quantummechanics,however,isdifferentintwoways:

1. Different Abstractions. Quantum abstractions are fundamentallydifferentfromclassicalones.Forexample,we’llseethattheideaofastate in quantum mechanics is conceptually very different from itsclassical counterpart.States are representedbydifferentmathematicalobjectsandhaveadifferentlogicalstructure.

2. StatesandMeasurements. In theclassicalworld, therelationshipbetweenthestateofasystemandtheresultofameasurementonthat

system is very straightforward. In fact, it’s trivial. The labels thatdescribeastate(thepositionandmomentumofaparticle,forexample)arethesamelabelsthatcharacterizemeasurementsofthatstate.Toputitanotherway,onecanperformanexperimenttodeterminethestateofa system. In the quantum world, this is not true. States andmeasurements are two different things, and the relationship betweenthemissubtleandnonintuitive.

Theseideasarecrucial,andwe’llcomebacktothemagainandagain.

1.2 SpinsandQubitsTheconceptofspinisderivedfromparticlephysics.Particleshavepropertiesinadditiontotheirlocationinspace.Forexample,theymayormaynothaveelectriccharge,ormass.Anelectronisnotthesameasaquarkoraneutrino.But even a specific type of particle, such as an electron, is not completelyspecified by its location. Attached to the electron is an extra degree offreedomcalleditsspin.Naively,thespincanbepicturedasalittlearrowthatpoints insomedirection,but thatnaivepicture is tooclassical toaccuratelyrepresent the real situation. The spin of an electron is about as quantummechanicalasasystemcanbe,andanyattempttovisualizeitclassicallywillbadlymissthepoint.

Wecanandwillabstracttheideaofaspin,andforgetthatitisattachedtoanelectron.Thequantumspinisasystemthatcanbestudiedinitsownright.In fact, the quantum spin, isolated from the electron that carries it throughspace,isboththesimplestandthemostquantumofsystems.

The isolated quantum spin is an example of the general class of simplesystems we call qubits—quantum bits—that play the same role in thequantumworld as logical bits play in defining the state of your computer.Many systems—maybe even all systems—can be built up by combiningqubits.Thusinlearningaboutthem,wearelearningaboutagreatdealmore.

1.3 AnExperimentLet’smaketheseideasconcrete,usingthesimplestexamplewecanfind.Inthe first lecture of Volume I, we began by discussing a very simpledeterministic system:acoin that canshoweitherheads (H)or tails (T).Wecancall thisa two-statesystem,orabit,withthetwostatesbeingHandT.Moreformallyweinventa“degreeoffreedom”called thatcantakeontwovalues,namely+1and−1.ThestateHisreplacedby

andthestateTby

Classically, that’s all there is to the space of states.The system is either instate or and there is nothing in between. In quantummechanics,we’llthinkofthissystemasaqubit.

VolumeIalsodiscussedsimpleevolutionlawsthattellushowtoupdatethestatefrominstanttoinstant.Thesimplestlawisjustthatnothinghappens.Inthatcase,ifwegofromonediscreteinstant(n)tothenext(n+1),thelawofevolutionis

Let’sexposeahiddenassumptionthatwewerecarelessaboutinVolumeI.Anexperimentinvolvesmorethanjustasystemtostudy.Italsoinvolvesanapparatus A to make measurements and record the results of themeasurements. In the case of the two-state system, the apparatus interactswiththesystem(thespin)andrecordsthevalueof .Thinkoftheapparatusas a black box1 with a window that displays the 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 itsdirectionwillaffecttheoutcomesofourmeasurements.Webeginbypointingit along the z axis (Fig. 1.1). Initially, we have no knowledge of whether

or .Ourpurposeistodoanexperimenttofindoutthevalueof .

Beforetheapparatusinteractswiththespin,thewindowisblank(labeledwith a question mark in our diagrams). After it measures , the windowshowsa+1ora−1.Bylookingattheapparatus,wedeterminethevalueof .Thatwholeprocessconstitutesaverysimpleexperimentdesignedtomeasure.

Figure1.1:(A)Spinandcat-freeapparatusbeforeanymeasurementismade.(B)Spin and apparatus after onemeasurementhasbeenmade, resulting in

.Thespin isnowpreparedin the state. If thespin isnotdisturbed and the apparatus keeps the same orientation, all subsequentmeasurementswillgivethesameresult.Coordinateaxesshowourconventionforlabelingthedirectionsofspace.

Now that we’ve measured , let’s reset the apparatus to neutral and,withoutdisturbingthespin,measure again.AssumingthesimplelawofEq.1.1, we should get the same answer as we did the first time. The result

willbefollowedby .Likewisefor .Thesamewillbe true for any number of repetitions. This is good because it allows us toconfirm the result of an experiment.We can also say this in the followingway:ThefirstinteractionwiththeapparatusApreparesthesysteminoneofthetwostates.Subsequentexperimentsconfirmthatstate.Sofar,thereisnodifferencebetweenclassicalandquantumphysics.

Figure 1.2: The apparatus is flipped without disturbing the previouslymeasuredspin.Anewmeasurementresultsin .

Now let’s do something new. After preparing the spin by measuring itwithA,we turn the apparatusupsidedownand thenmeasure again (Fig.

1.2).Whatwefindisthatifweoriginallyprepared ,theupsidedownapparatusrecords .Similarly, ifweoriginallyprepared , theupsidedownapparatusrecords .Inotherwords,turningtheapparatusover interchanges and . From these results, we mightconclude that is a degree of freedom that is associated with a sense ofdirection in space.For example, if were an oriented vector of some sort,then it would be natural to expect that turning the apparatus over wouldreverse thereading.Asimpleexplanation is that theapparatusmeasures thecomponent 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 inwith50percentprobability.

Sofar,thereisstillnodifferencebetweenclassicalphysicsandquantumphysics.Thedifferenceonlybecomesapparentwhenwerotatetheapparatusthroughanarbitraryangle,say radians(90degrees).Theapparatusbeginsintheuprightposition(withtheup-arrowalongthezaxis).Aspinispreparedwith .Next,rotateAsothattheup-arrowpointsalongthexaxis(Fig.1.3),andthenmakeameasurementofwhatispresumablythexcomponentofthespin, .

Ifinfact reallyrepresentsthecomponentofavectoralongtheup-arrow,one would expect to get zero. Why? Initially, we confirmed that wasdirectedalongthezaxis,suggestingthatitscomponentalongxmustbezero.Butwe get a surprisewhenwemeasure : Instead of giving , theapparatusgiveseither or .Aisverystubborn—nomatterwhichwayitisoriented,itrefusestogiveanyanswerotherthan .Ifthespinreallyisavector,itisaverypeculiaroneindeed.

Nevertheless, we do find something interesting. Suppose we repeat theoperationmanytimes,eachtimefollowingthesameprocedure,thatis:

• BeginningwithAalongthezaxis,prepare .

• Rotatetheapparatussothatitisorientedalongthexaxis.

• Measure .

The repeated experiment spits out a randomseriesof plus-ones andminus-ones.Determinismhasbrokendown,butinaparticularway.Ifwedomanyrepetitions, we will find that the numbers of events andeventsarestatisticallyequal.Inotherwords, theaveragevalueof iszero.Insteadof theclassicalresult—namely, that thecomponentof along thexaxisiszero—wefindthattheaverageoftheserepeatedmeasurementsiszero.

Figure 1.4: The apparatus rotated by an arbitrary angle within theplane.Averagemeasurementresultis .

Nowlet’sdothewholethingoveragain,butinsteadofrotatingAtolieonthe x axis, rotate it to an arbitrary direction along the unit vector2 .Classically,if wereavector,wewouldexpecttheresultoftheexperimenttobethecomponentof alongthe axis.If liesatanangle withrespecttoz, theclassicalanswerwouldbe =cos .Butasyoumightguess,eachtimewedotheexperimentweget or .However,theresultisstatisticallybiasedsothattheaveragevalueis .

Thesituationisofcoursemoregeneral.WedidnothavetostartwithAorientedalongz.Pickanydirection andstartwith theup-arrowpointingalong . Prepare a spin so that the apparatus reads +1. Then, withoutdisturbing thespin, rotate theapparatus to thedirection , as shown inFig.1.4.Anewexperimentonthesamespinwillgiverandomresults±1,butwith

anaveragevalueequaltothecosineoftheanglebetween and .Inotherwords,theaveragewillbe .

ThequantummechanicalnotationforthestatisticalaverageofaquantityQ is Dirac’s bracket notation . We may summarize the results of ourexperimental investigationasfollows:IfwebeginwithAorientedalongand confirm that , then subsequent measurement with A orientedalong givesthestatisticalresult

What we are learning is that quantum mechanical systems are notdeterministic—theresultsofexperimentscanbestatisticallyrandom—but ifwe repeat an experiment many times, average quantities can follow theexpectationsofclassicalphysics,atleastuptoapoint.

1.4 ExperimentsAreNeverGentleEvery experiment involves an outside system—an apparatus—that mustinteract with the system in order to record a result. In that sense, everyexperimentisinvasive.Thisistrueinbothclassicalandquantumphysics,butonlyquantumphysicsmakesabigdealoutofit.Whyisthatso?Classically,anidealmeasuringapparatushasavanishinglysmalleffectonthesystemitismeasuring.Classicalexperimentscanbearbitrarilygentleandstillaccuratelyand reproducibly record the results of the experiment. For example, thedirectionofanarrowcanbedeterminedbyreflectinglightoffthearrowandfocusingittoformanimage.Whileitistruethatthelightmusthaveasmallenoughwavelengthtoformanimage,thereisnothinginclassicalphysicsthatprevents the image from being made with arbitrarily weak light. In otherwords,thelightcanhaveanarbitrarilysmallenergycontent.

In quantum mechanics, the situation is fundamentally different. Anyinteraction that is strong enough to measure some aspect of a system isnecessarily strong enough todisrupt someother aspect of the same system.Thus, you can learn nothing about a quantum system without changingsomethingelse.

This shouldbeevident in theexamples involvingA and . Supposewebeginwith alongthezaxis.Ifwemeasure againwithAorientedalongz,wewill confirm the previous value.We can do this over and overwithoutchangingtheresult.Butconsiderthispossibility:Betweensuccessive

measurements along the z axis, we turn A through 90 degrees, make anintermediatemeasurement, and turn it back to its original direction.Will asubsequentmeasurementalongthezaxisconfirmtheoriginalmeasurement?Theanswerisno.Theintermediatemeasurementalongthexaxiswillleavethespininacompletelyrandomconfigurationasfarasthenextmeasurementisconcerned.Thereisnowaytomaketheintermediatedeterminationofthespinwithoutcompletelydisruptingthefinalmeasurement.Onemightsaythatmeasuringonecomponentofthespindestroystheinformationaboutanothercomponent.Infact,onesimplycannotsimultaneouslyknowthecomponentsof the spinalong twodifferentaxes,not ina reproducibleway inanycase.There is something fundamentally different about the state of a quantumsystemandthestateofaclassicalsystem.

1.5 PropositionsThespaceofstatesofaclassicalsystemisamathematicalset.Ifthesystemisa coin, the space of states is a set of two elements, H and T. Using setnotation,wewouldwrite{H,T}.Ifthesystemisasix-sideddie,thespaceofstates has six elements labeled {1, 2, 3, 4, 5, 6}.The logic of set theory iscalledBooleanlogic.Booleanlogicisjustaformalizedversionofthefamiliarclassicallogicofpropositions.

A fundamental idea inBoolean logic is thenotionof a truth-value.Thetruth-value of a proposition is either true or false. Nothing in between isallowed. The related set theory concept is a subset. Roughly speaking, aproposition is true forall theelements in its corresponding subset and falseforall theelementsnot in thissubset.Forexample, if theset represents thepossiblestatesofadie,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}.

There are rules for combining propositions into more complexpropositions, the most important being or, and, and not. We just saw anexampleofnot,whichgetsappliedtoasinglesubsetorproposition.And isstraightforward, andapplies toapairofpropositions.3 It says they arebothtrue.Appliedtotwosubsets,andgivestheelementscommontoboth,thatis,the intersection of the two subsets. In the die example, the intersection ofsubsetsAandB is thesubsetofelements thatarebothoddand less than4.Fig.1.5usesaVenndiagramtoshowhowthisworks.

Theorruleissimilartoand,buthasoneadditionalsubtlety.Ineverydayspeech, thewordor is generallyused in the exclusive sense—theexclusiveversion is true if oneor the other of twopropositions is true, but not both.However, Boolean logic uses the inclusive version of or, which is true ifeitherorbothofthepropositionsaretrue.Thus,accordingtotheinclusiveor,theproposition

AlbertEinsteindiscoveredrelativityorIsaacNewtonwasEnglish

istrue.Sois

AlbertEinsteindiscoveredrelativityorIsaacNewtonwasRussian.

Theinclusiveorisonlywrongifbothpropositionsarefalse.Forexample,

AlbertEinsteindiscoveredAmerica4orIsaacNewtonwasRussian.

Theinclusiveorhasasettheoreticinterpretationastheunionoftwosets:itdenotes thesubsetcontaininganythingineitherorbothof thecomponentsubsets.Inthedieexample,(AorB)denotesthesubset{1,2,3,5}.

Figure 1.5: An Example of the Classical model of State Space. Subset Arepresents theproposition“thedieshowsanodd-numberedface.”SubsetB:

“Thedieshowsanumber<4.”DarkshadingshowstheintersectionofAandB,whichrepresents theproposition(AandB).WhitenumbersareelementsoftheunionofAwithB,representingtheproposition(AorB).

1.6 TestingClassicalPropositionsLet’sreturntothesimplequantumsystemconsistingofasinglespin,andthevarious propositions whose truth we could test using the apparatus A.Considerthefollowingtwopropositions:

A:Thezcomponentofthespinis+1.

B:Thexcomponentofthespinis+1.

Each of these is meaningful and can be tested by orienting A along theappropriateaxis.Thenegationofeach isalsomeaningful.Forexample, thenegationofthefirstpropositionis

notA:Thezcomponentofthespinis−1.

Butnowconsiderthecompositepropositions

(AorB):Thezcomponentofthespinis+1orthexcomponentofthespinis+1.

(AandB):Thezcomponentofthespinis+1andthexcomponentofthespinis+1.

Consider howwewould test the proposition (AorB). If spins behavedclassically(andofcoursetheydon’t),wewouldproceedasfollows:5

• Gently measure and record the value. If it is +1, we arefinished: theproposition (AorB) is true. If is−1, continue to thenextstep.

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

Thereisanalternativeprocedure,whichistointerchangetheorderofthetwomeasurements. To emphasize this reversal of ordering, we’ll call the newprocedure(BorA):

• Gentlymeasure andrecordthevalue.Ifitis+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 twoordersofoperationgive the sameanswer.Thereasonforthisisthatmeasurementscanbearbitrarilygentle—sogentlethatthey do not affect the results of subsequent measurements. Therefore, theproposition(AorB)hasthesamemeaningastheproposition(BorA).

1.7 TestingQuantumPropositionsNowwecometothequantumworldthatIdescribedearlier.Letusimagineasituation in which someone (or something) unknown to us has secretlyprepared a spin in the state.Our job is to use the apparatusA todeterminewhethertheproposition(AorB)istrueorfalse.Wewilltryusingtheproceduresoutlinedabove.

Webeginbymeasuring .Since theunknownagenthas set thingsup,wewilldiscoverthat . It isunnecessarytogoon:(AorB) is true.Nevertheless, we could test just to see what happens. The answer isunpredictable.Werandomlyfind that or .Butneitheroftheseoutcomesaffectsthetruthofproposition(AorB).

Butnowlet’sreversetheorderofmeasurement.Asbefore,we’llcallthereversedprocedure(BorA),andthistimewe’llmeasure first.Becausetheunknownagentsetthespinto+1alongthezaxis,themeasurementof israndom.If it turnsout that ,weare finished: (BorA) is true.Butsupposewefindtheoppositeresult, .Thespinisorientedalongthe−xdirection.Let’spauseherebriefly, tomakesureweunderstandwhat justhappened.As a result of our firstmeasurement, the spin is no longer in itsoriginal state . It is in a new state, which is either or

.Pleasetakeamomenttoletthisideasinkin.Wecannotoverstateitsimportance.

Nowwe’re ready to test thesecondhalfofproposition (BorA).Rotatethe apparatus A to the z axis and measure . According to quantummechanics, the result will be randomly ±1. This means that there is a 25percentprobabilitythattheexperimentproduces and . Inotherwords,withaprobabilityof ,wefindthat(BorA)isfalse;thisoccursdespitethefactthatthehiddenagenthadoriginallymadesurethat .

Evidently,inthisexample,theinclusiveorisnotsymmetric.Thetruthof(AorB)maydependontheorderinwhichweconfirmthetwopropositions.Thisisnotasmallthing;itmeansnotonlythatthelawsofquantumphysicsaredifferentfromtheirclassicalcounterparts,butthattheveryfoundationsoflogicaredifferentinquantumphysicsaswell.

Whatabout (AandB)?Supposeour firstmeasurement yieldsandthesecond, .Thisisofcourseapossibleoutcome.Wewouldbeinclinedtosaythat(AandB)istrue.Butinscience,especiallyinphysics,thetruth of a proposition implies that the proposition can be verified bysubsequent observation. In classical physics, the gentleness of observationsimpliesthatsubsequentexperimentsareunaffectedandwillconfirmanearlierexperiment.AcointhatturnsupHeadswillnotbeflippedtoTailsbytheactof observing it—at least not classically.Quantummechanically, the secondmeasurement ruinsthepossibilityofverifyingthefirst.Oncehas been prepared along the x axis, anothermesurement of will give arandomanswer.Thus(AandB) isnotconfirmable: thesecondpieceof theexperimentinterfereswiththepossibilityofconfirmingthefirstpiece.

Ifyouknowabitaboutquantummechanics,youprobablyrecognizethatwe are talking about the uncertainty principle. The uncertainty principledoesn’tapplyonlytopositionandmomentum(orvelocity);itappliestomanypairs of measurable quantities. In the case of the spin, it applies topropositionsinvolvingtwodifferentcomponentsof .Inthecaseofpositionandmomentum,thetwopropositionswemightconsiderare:

Acertainparticlehaspositionx.

Thatsameparticlehasmomentump.

Fromthese,wecanformthetwocompositepropositions

Theparticlehaspositionxandtheparticlehasmomentump.

Theparticlehaspositionxortheparticlehasmomentump.

Awkwardastheyare,bothofthesepropositionshavemeaningintheEnglishlanguage,andinclassicalphysicsaswell.However,inquantumphysics,thefirst of thesepropositions is completelymeaningless (not evenwrong), andthesecondonemeanssomethingquitedifferentfromwhatyoumightthink.Itall comes down to a deep logical difference between the classical andquantumconceptsofthestateofasystem.Explainingthequantumconceptofstate will require some abstract mathematics, so let’s pause for a briefinterlude on complex numbers and vector spaces. The need for complexquantities will become clear later on, when we study the mathematicalrepresentationofspinstates.

1.8 MathematicalInterlude:ComplexNumbersEveryonewhohasgotten this far in theTheoreticalMinimumseriesknows

aboutcomplexnumbers.Nevertheless,Iwillspendafewlinesremindingyouoftheessentials.Fig.1.6showssomeoftheirbasicelements.

A complex number z is the sum of a real number and an imaginarynumber.Wecanwriteitas

Figure 1.6: Two Common Ways to Represent Complex Numbers. In theCartesian representation, x and y are the horizontal (real) and vertical(imaginary)components.Inthepolarrepresentation,r istheradius,and isthe anglemadewith the x axis. In each case, it takes two real numbers torepresentasinglecomplexnumber.

where x and y are real and . Complex numbers can be added,multiplied, and divided by the standard rules of arithmetic. They can bevisualizedaspointsonthecomplexplanewithcoordinatesx,y.Theycanalsoberepresentedinpolarcoordinates:

Adding complex numbers is easy in component form: just add thecomponents.Similarly,multiplyingthemiseasy in theirpolarform:Simplymultiplytheradiiandaddtheangles:

Every complex number z has a complex conjugate that is obtained bysimplyreversingthesignoftheimaginarypart.

If

then

Multiplyingacomplexnumberanditsconjugatealwaysgivesapositiverealresult:

Itisofcoursetruethateverycomplexconjugateisitselfacomplexnumber,but it’s often helpful to think of z and as belonging to separate “dual”numbersystems.Dualheremeans that foreveryz there is aunique andviceversa.

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

1.9 MathematicalInterlude:VectorSpaces

1.9.1 AxiomsForaclassicalsystem,thespaceofstatesisaset(thesetofpossiblestates),

and the logic of classical physics isBoolean. That seems obvious and it isdifficulttoimagineanyotherpossibility.Nevertheless,therealworldoperatesalong entirely different lines, at least whenever quantum mechanics isimportant.Thespaceofstatesofaquantumsystemisnotamathematicalset;6it is a vector space. Relations between the elements of a vector space aredifferent from those between the elements of a set, and the logic ofpropositionsisdifferentaswell.

BeforeItellyouaboutvectorspaces,Ineedtoclarifythetermvector.Asyouknow,weusethistermtoindicateanobjectinordinaryspacethathasamagnitude and a direction. Such vectors have three components,corresponding to the three dimensions of space. I want you to completelyforgetaboutthatconceptofavector.Fromnowon,wheneverIwanttotalkaboutathingwithmagnitudeanddirectioninordinaryspace,Iwillexplicitlycallita3-vector.Amathematicalvectorspaceisanabstractconstructionthatmayormaynot have anything to dowith ordinary space. Itmayhave anynumber of dimensions from 1 to ∞ and it may have components that areintegers,realnumbers,orevenmoregeneralthings.

ThevectorspacesweusetodefinequantummechanicalstatesarecalledHilbertspaces.Wewon’tgivethemathematicaldefinitionhere,butyoumayaswell add this term to your vocabulary.When you come across the termHilbertspaceinquantummechanics,itreferstothespaceofstates.AHilbertspacemayhaveeitherafiniteoraninfinitenumberofdimensions.

Inquantummechanics,avectorspaceiscomposedofelements calledket-vectorsorjustkets.Herearetheaxiomswewillusetodefinethevectorspaceofstatesofaquantumsystem(zandwarecomplexnumbers):

1. Thesumofanytwoket-vectorsisalsoaket-vector:

2. Vectoradditioniscommutative:

3. Vectoradditionisassociative:

4. Thereisauniquevector0suchthatwhenyouaddittoanyket,itgivesthesameketback:

5. Givenanyket ,thereisauniqueket suchthat

6. Givenanyket andanycomplexnumberz,youcanmultiplythemtogetanewket.Also,multiplicationbyascalarislinear:

7. Thedistributivepropertyholds:

Axioms6and7takentogetherareoftencalledlinearity.

Ordinary 3-vectors would satisfy these axioms except for one thing:Axiom6allowsavectortobemultipliedbyanycomplexnumber.Ordinary3-vectorscanbemultipliedbyrealnumbers(positive,negative,orzero)butmultiplicationbycomplexnumbersisnotdefined.Onecanthinkof3-vectorsasformingarealvectorspace,andketsasformingacomplexvectorspace.Our definition of ket-vectors is fairly abstract. As we will see, there arevariousconcretewaystorepresentket-vectorsaswell.

1.9.2 FunctionsandColumnVectorsLet’slookatsomeconcreteexamplesofcomplexvectorspaces.Firstofall,considerthesetofcontinuouscomplex-valuedfunctionsofavariablex.CallthefunctionsA(x).Youcanaddanytwosuchfunctionsandmultiplythemby

complex numbers. You can check that they satisfy all seven axioms. Thisexample shouldmake it obvious thatwe are talking about somethingmuchmoregeneralthanthree-dimensionalarrows.

Two-dimensional columnvectorsprovide another concrete example.Weconstructthembystackingupapairofcomplexnumbers, and ,intheform

andidentifyingthis“stack”withtheket-vector .Thecomplexnumbersarethecomponentsof .Youcanaddtwocolumnvectorsbyaddingtheircomponents:

Moreover,youcanmultiplythecolumnvectorbyacomplexnumberzjustbymultiplyingthecomponents,

Columnvectorspacesofanynumberofdimensionscanbeconstructed.Forexample,hereisafive-dimensionalcolumnvector:

Normally,wedonotmixvectorsofdifferentdimensionality.

1.9.3 BrasandKetsAswehaveseen, thecomplexnumbershaveadualversion: in the formofcomplexconjugatenumbers.Inthesameway,acomplexvectorspacehasadualversionthatisessentiallythecomplexconjugatevectorspace.Foreveryket-vector ,thereisa“bra”vectorinthedualspace,denotedby .Whythestrangetermsbraandket?Shortly,wewilldefineinnerproductsofbrasand kets, using expressions like to formbra-kets or brackets. Innerproductsareextremelyimportantinthemathematicalmachineryofquantummechanics,andforcharacterizingvectorspacesingeneral.

Bravectorssatisfythesameaxiomsastheket-vectors,buttherearetwothingstokeepinmindaboutthecorrespondencebetweenketsandbras:

1. Suppose is thebracorresponding to theket , and isthebracorrespondingtotheket .Thenthebracorrespondingto

is

2. If z is a complex number, then it is not true that the bracorresponding to is . You have to remember to complex-conjugate.Thus,thebracorrespondingto

is

In the concrete examplewhere kets are represented by column vectors, thedualbrasarerepresentedbyrowvectors,withtheentriesbeingdrawnfromthe complex conjugate numbers. Thus, if the ket is represented by thecolumn

thenthecorrespondingbra isrepresentedbytherow

1.9.4 InnerProductsYouarenodoubtfamiliarwiththedotproductdefinedforordinary3-vectors.The analogous operation for bras and kets is the inner product. The innerproductisalwaystheproductofabraandaketanditiswrittenthisway:

The result of this operation is a complex number. The axioms for innerproductsarenottoohardtoguess:

1. Theyarelinear:

2. Interchangingbrasandketscorrespondstocomplexconjugation:

Exercise1.1:

a) Usingtheaxiomsforinnerproducts,prove

b) Prove isarealnumber.

Intheconcreterepresentationofbrasandketsbyrowandcolumnvectors,theinnerproductisdefinedintermsofcomponents:

Theruleforinnerproductsisessentiallythesameasfordotproducts:addtheproductsofcorrespondingcomponentsofthevectorswhoseinnerproductisbeingcalculated.

Exercise1.2: Show that the innerproduct definedbyEq.1.2 satisfiesalltheaxiomsofinnerproducts.

Usingtheinnerproduct,wecandefinesomeconceptsthatarefamiliarfromordinary3-vectors:

• NormalizedVector:A vector is said to be normalized if its innerproductwithitselfis1.Normalizedvectorssatisfy,

For ordinary 3-vectors, the term normalized vector is usuallyreplacedbyunitvector,thatis,avectorofunitlength.

• OrthogonalVectors:Twovectorsaresaidtobeorthogonaliftheirinnerproductiszero. and areorthogonalif

Thisistheanalogofsayingthattwo3-vectorsareorthogonaliftheirdotproductiszero.

1.9.5 OrthonormalBasesWhenworkingwithordinary3-vectors, it isextremelyuseful to introduceaset of three mutually orthogonal unit vectors and use them as a basis toconstructanyvector.Asimpleexamplewouldbetheunit3-vectorsthatpointalongthex,y,andzaxes.Theyareusuallycalled , ,and .Eachisofunitlength and orthogonal to the others. If you tried to find a fourth vectororthogonal to these three, there wouldn’t be any—not in three dimensionsanyway.However, if thereweremore dimensions of space, therewould bemorebasisvectors.Thedimensionofaspacecanbedefinedasthemaximumnumberofmutuallyorthogonalvectorsinthatspace.

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

Thesameprincipleistrueforcomplexvectorspaces.Onecanbeginwithanynormalizedvectorandthenlookforasecondone,orthogonaltothefirst.Ifyou findone, then the space isat least two-dimensional.Then look forathird, fourth, andsoon.Eventually,youmay runoutofnewdirectionsandtherewillnotbeanymoreorthogonalcandidates.Themaximumnumberofmutually orthogonal vectors is the dimension of the space. For columnvectors,thedimensionissimplythenumberofentriesinthecolumn.

Let’sconsideranN-dimensionalspaceandaparticularorthonormalbasis

ofket-vectorslabeled .7Thelabelirunsfrom1toN.Consideravector,writtenasasumofbasisvectors:

The are complex numbers called the components of the vector, and tocalculatethemwetaketheinnerproductofbothsideswithabasisbra :

Next,weusethefactthatthebasisvectorsareorthonormal.Thisimpliesthat if i is not equal to j, and if i = j. In other words,.ThismakesthesuminEq.1.4collapsetooneterm:

Thus,weseethatthecomponentsofavectorarejustitsinnerproductswiththebasisvectors.WecanrewriteEq.1.3intheelegantform

1“Blackbox”meanswehavenoknowledgeofwhat’sinsidetheapparatusorhowitworks.Butrestassured,itdoesnotcontainacat.

2Thestandardnotation foraunitvector (oneofunit length) is toplacea“hat”above thesymbolrepresentingthevector.

3Andmaybedefinedformultiplepropositions,butwe’llonlyconsidertwo.Thesamegoesforor.4OK,perhapsEinsteindiddiscoverAmerica.Buthewasnotthefirst.5Recall that the classical meaning of is different from the quantum mechanical meaning.

Classically, isastraightforward3-vector; xand zrepresentitsspatialcomponents.

6Tobealittlemoreprecise,wewillnotfocusontheset-theoreticpropertiesofstatespaces,eventhoughtheymayofcourseberegardedassets.

7Mathematically,basisvectorsarenotrequiredtobeorthonormal.However,inquantummechanicstheygenerallyare.Inthisbook,wheneverwesaybasis,wemeananorthonormalbasis.

Lecture2

QuantumStatesArt:Oddlyenough,thatbeermademyheadstopspinning.Whatstatearewein?

Lenny:IwishIknew.Doesitmatter?

Art:Itmight.Idon’tthinkwe’reinCaliforniaanymore.

2.1 StatesandVectorsIn classical physics, knowing the state of a system implies knowingeverything that is necessary to predict the future of that system. As we’veseen in the last lecture, quantum systems are not completely predictable.Evidently,quantumstateshaveadifferentmeaningthanclassicalstates.Veryroughly,knowingaquantumstatemeansknowingasmuchascanbeknownabouthowthesystemwasprepared.Inthelastchapter,wetalkedaboutusinganapparatustopreparethestateofaspin.Infact,weimplicitlyassumedthattherewasnomorefinedetail tospecifyor thatcouldbespecifiedabout thestateofthespin.

Theobviousquestion toask iswhether theunpredictability isdue toanincompleteness inwhatwecallaquantumstate.Therearevariousopinionsaboutthismatter.Hereisasampling:

• Yes, the usual notion of quantum state is incomplete. There are“hidden variables” that, if only we could access them, would allowcompletepredictability.Therearetwoversionsofthisview.InversionA, the hidden variables are hard tomeasure but in principle they areexperimentally available to us. In versionB, becausewearemadeofquantummechanicalmatterandthereforesubject totherestrictionsofquantum mechanics, the hidden variables are, in principle, notdetectable.

• No, the hiddenvariables concept does not leadus in a profitabledirection.Quantummechanics isunavoidablyunpredictable.Quantummechanicsisascompleteacalculusofprobabilitiesasispossible.Thejobofaphysicististolearnandusethiscalculus.

Idon’tknowwhat theultimateanswer to thisquestionwillbe,orevenif itwill prove to be a useful question.But for our purposes, it’s not important

what any particular physicist believes about the ultimate meaning of thequantumstate.Forpracticalreasons,wewilladoptthesecondview.

In practice, what thismeans for the quantum spin of Lecture 1 is that,whentheapparatusAactsandtellsusthat or ,thereisnomoretoknow,orthatcanbeknown.Likewise, ifwerotateA andmeasure

or , there is no more to know. Likewise for or anyothercomponentofthespin.

2.2 RepresentingSpinStatesNowit’s time to tryourhandat representingspinstatesusingstate-vectors.Ourgoalistobuildarepresentationthatcaptureseverythingweknowaboutthe behavior of spins.At this point, the processwill bemore intuitive thanformal.Wewilltrytofitthingstogetherthebestwecan,basedonwhatwe’vealreadylearned.Pleasereadthissectioncarefully.Believeme,itwillpayoff.

Let’sbeginbylabelingthepossiblespinstatesalongthethreecoordinateaxes. IfA is oriented along the z axis, the two possible states that can beprepared correspond to . Let’s call them up and down and denotethembyket-vectors and .Thus,whentheapparatusisorientedalongthezaxisandregisters+1,thestate hasbeenprepared.

On the other hand, if the apparatus is oriented along the x axis andregisters−1,thestate hasbeenprepared.We’llcallitleft.IfAisalongtheyaxis,itcanpreparethestates and (inandout).Yougettheidea.

The idea that there are no hidden variables has a very simplemathematicalrepresentation:thespaceofstatesforasinglespinhasonlytwodimensions.Thispointdeservesemphasis:

Allpossiblespinstatescanberepresentedinatwo-dimensionalvectorspace.

We could, somewhat arbitrarily,1 choose and as the two basisvectorsandwriteanystateasalinearsuperpositionofthesetwo.We’lladoptthatchoicefornow.Let’susethesymbol foragenericstate.Wecanwritethisasanequation,

where and are the componentsof along the basis directions

and .Mathematically,wecanidentifythecomponentsof as

Theseequationsareextremelyabstract,anditisnotatallobviouswhattheirphysical significance is. I am going to tell you right nowwhat theymean:Firstofall, canrepresentanystateofthespin,preparedinanymanner.Thecomponents and arecomplexnumbers;bythemselves,theyhavenoexperimentalmeaning,but theirmagnitudesdo.Inparticular, and

havethefollowingmeaning:

• Giventhatthespinhasbeenpreparedinthestate ,andthattheapparatusisorientedalongz,thequantity is theprobabilitythatthe spin would be measured as . In other words, it is theprobabilityofthespinbeingupifmeasuredalongthezaxis.

• Likewise, is the probability that would be down ifmeasured.

The values, or equivalently and , are called probabilityamplitudes.Theyarethemselvesnotprobabilities.Tocomputeaprobability,their magnitudes must be squared. In other words, the probabilities formeasurementsofupanddownaregivenby

NoticethatIhavesaidnothingaboutwhat isbeforeitismeasured.Beforethemeasurement,allwehaveisthevector ,whichrepresentsthepotentialpossibilitiesbutnottheactualvaluesofourmeasurements.

Twootherpointsare important:First,note that and aremutuallyorthogonal.Inotherwords,

The physical meaning of this is that, if the spin is prepared up, then theprobability to detect it down is zero, and vice versa. This point is soimportant,I’llsayitagain:Twoorthogonalstatesarephysicallydistinctandmutuallyexclusive.Ifthespinisinoneofthesestates,itcannotbe(haszeroprobabilitytobe)intheotherone.Thisideaappliestoallquantumsystems,notjustspin.

But don’t mistake the orthogonality of state-vectors for orthogonaldirections in space. In fact, the directions up anddown are not orthogonaldirectionsinspace,eventhoughtheirassociatedstate-vectorsareorthogonalinstatespace.

The second important point is that for the total probability to come outequaltounity,wemusthave

Thisisequivalenttosayingthatthevector isnormalizedtoaunitvector:

This is a very general principle of quantum mechanics that extends to allquantumsystems:thestateofasystemisrepresentedbyaunit(normalized)vector in avector spaceof states.Moreover, the squaredmagnitudesof thecomponents of the state-vector, along particular basis vectors, representprobabilitiesforvariousexperimentaloutcomes.

2.3 AlongthexAxisWesaidbeforethatwecanrepresentanyspinstateasalinearcombinationofthebasisvectors and .Let’strydoingthisnowforthevectors and,whichrepresentspinspreparedalongthexaxis.We’llstartwith .As

you recall fromLecture1, ifA initially prepares , and is then rotated tomeasure , therewillbeequalprobabilities forupanddown.Thus,and mustbothbeequalto .Asimplevectorthatsatisfiesthisruleis

Thereissomeambiguityinthischoice,butaswewillseelater,itisnothingmorethantheambiguityinourchoiceofexactdirectionsforthexandyaxes.

Next,let’slookatthevector .Hereiswhatweknow:whenthespinhasbeenpreparedintheleftconfiguration,theprobabilitiesfor areagainequalto .Thatisnotenoughtodeterminethevalues and ,butthereisanother condition thatwe can infer. Earlier, I told you that and areorthogonal for the simple reason that, if the spin is up, it’s definitely notdown.Butthereisnothingspecialaboutupanddownthatisnotalsotrueofrightandleft.Inparticular,ifthespinisright,ithaszeroprobabilityofbeingleft.Thus,byanalogywithEq.2.3,

Thisprettymuchfixes intheform

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

Again, thereissomeambiguityinthechoiceof .Thisiscalledthephase

ambiguity.Supposewemultiply byanycomplexnumberz.Thatwillhavenoeffectonwhetheritisorthogonalto , thoughingeneraltheresultwillnolongerbenormalized(haveunitlength).Butifwechoose (wherecanbeanyrealnumber), thentherewillbenoeffectonthenormalizationbecause hasunitmagnitude. Inotherwords, will remainequal to1.Sinceanumberof the form iscalledaphase-factor, theambiguity is called the phase ambiguity. Later, we will find out that nomeasurablequantityissensitivetotheoverallphase-factor,andthereforewecanignoreitwhenspecifyingstates.

2.4 AlongtheyAxisFinally, thisbringsus to and , thevectors representing spinsorientedalongtheyaxis.Let’slookattheconditionstheyneedtosatisfy.First,

Thisconditionstatesthatinandoutarerepresentedbyorthogonalvectorsinthesamewaythatupanddownare.Physically,thismeansthatifthespinisin,itisdefinitelynotout.

There are additional restrictions on the vectors and . Using therelationshipsexpressed inEqs.2.1and2.2, and the statistical results of ourexperiments,wecanwritethefollowing:

Inthefirsttwoequations, takestheroleof fromEqs.2.1and2.2. Inthe second two, takes that role.Theseconditions state that if the spin isorientedalongy,andisthenmeasuredalongz,itisequallylikelytobeupor

down.Weshouldalsoexpectthatifthespinweremeasuredalongthexaxis,it would be equally likely to be right or left. This leads to additionalconditions:

Theseconditionsaresufficient todeterminetheformof thevectors and,apartfromthephaseambiguity.Hereistheresult:

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

It’s interesting that two of the components in Eqs. 2.10 are imaginary. Ofcourse,we’vesaidallalongthatthespaceofstatesisacomplexvectorspace,butuntilnowwehavenothad tousecomplexnumbers inourcalculations.Are thecomplexnumbers inEqs.2.10aconvenienceoranecessity?Givenour framework for spin states, there is noway around them. It’s somewhattedious todemonstrate this,but thestepsarestraightforward.Thefollowingexercisegivesyoua roadmap.Theneed forcomplexnumbers is ageneralfeatureofquantummechanics,andwe’llseemoreexamplesaswego.

Exercise 2.3: For the moment, forget that Eqs. 2.10 give us workingdefinitions for and in terms of and , and assume that thecomponents ,β,γ,andδareunknown:

a) UseEqs.2.8toshowthat

b) UsetheaboveresultandEqs.2.9toshowthat

c) Showthat and musteachbepureimaginary.

If is pure imaginary, then and β cannot both be real. The samereasoningappliesto .

2.5 CountingParametersIt’salways important toknowhowmanyindependentparameters it takes tocharacterize a system.For example, the generalized coordinatesweused inVolume I (referred to as ) each represented an independent degree offreedom. That approach freed us from the difficult job of writing explicitequationstodescribephysicalconstraints.Alongsimilarlines,ournexttaskistocountthenumberofphysicallydistinctstatesthereareforaspin.Iwilldoitintwoways,toshowthatyougetthesameanswereitherway.

Thefirstwayissimple.Pointtheapparatusalonganyunit3-vector2 andprepareaspinwith alongthataxis.If ,youcanthinkofthespinasbeingorientedalongthe axis.Thus,theremustbeastateforeveryorientation of the unit 3-vector . How many parameters does it take tospecifysuchanorientation?Theanswerisofcoursetwo.Ittakestwoanglestodefineadirectioninthree-dimensionalspace.3

Now, let’s consider the same question from another perspective. Thegeneralspinstateisdefinedbytwocomplexnumbers, and .Thatseems

toadduptofourrealparameters,witheachcomplexparametercountingastworealones.But recall that thevectorhas tobenormalizedas inEq.2.4.Thenormalization conditiongivesusone equation involving real variables,andcutsthenumberofparametersdowntothree.

AsIsaidearlier,wewilleventuallysee that thephysicalpropertiesofastate-vectordonotdependontheoverallphase-factor.Thismeansthatoneofthe threeremainingparameters is redundant, leavingonly two—thesameasthenumberofparametersweneedtospecifyadirectioninthree-dimensionalspace.Thus,thereisenoughfreedomintheexpression

todescribeallthepossibleorientationsofaspin,eventhoughthereareonlytwopossibleoutcomesofanexperimentalonganyaxis.

2.6 RepresentingSpinStatesasColumnVectorsSo far,we have been able to learn a lot by using the abstract forms of ourstate-vectors, that is, and and so forth. These abstractions help usfocus on mathematical relationships without worrying about unnecessarydetails.However,soonwewillneedtoperformdetailedcalculationsonspinstates, and for that we’ll need to write our state-vectors in column form.Becauseof“phase indifference,” thecolumnrepresentationsarenotunique,andwe’lltrytochoosethesimplestandmostconvenientoneswecanfind.

Asusual,we’llstartwith and .Weneedthemtohaveunitlength,and to be mutually orthogonal. A pair of columns that satisfies theserequirementsis

With thesecolumnvectors inhand, itwillbeeasy tocreatecolumnvectorsfor and usingEqs.2.5 and 2.6, and for and using Eqs. 2.10.We’lldothatinthenextlecture,wheretheseresultsareneeded.

2.7 PuttingItAllTogetherWehavecoveredalotofgroundinthislecture.Beforemovingon,let’stakestockofwhatwe’vedone.Ourgoalwas tosynthesizewhatweknowaboutspinsandvectorspaces.Wefiguredouthowtousevectorstorepresentspinstates,andintheprocesswegotaglimpseofthekindofinformationastate-vectorcontains(anddoesnotcontain!).Hereisabriefoutlineofwhatwedid:

• Based on our knowledge of spin measurements, we chose threepairsofmutuallyorthogonalbasisvectors.Pair-wise,wenamedthem

and , and ,and and .Becausethebasisvectorsand representphysicallydistinctstates,wewereabletoassert thatthey are mutually orthogonal. In other words, . The sameholdsfor and ,andalsofor and .

• Wefoundthatittakestwoindependentparameterstospecifyaspinstate,andthenwearbitrarilychoseoneoftheorthogonalpairs, and

,asourbasisvectors for representingall spinstates—even thoughthetwocomplexnumbersinastate-vectorrequirefourrealnumberstospecifythem.Howdidwegetawaywiththis?Wewerecleverenoughto notice that these four numbers are not all independent.4 Thenormalizationconstraint(totalprobabilitymustequal1)eliminatesoneindependentparameter,and“phaseindifference”(thephysicsofastate-vectorisunaffectedbyitsoverallphase-factor)eliminatesasecond.

• Havingchosen and asourmainbasis vectors,we figuredout how to represent the other two pairs of basis vectors as linearcombinations of and , using additional orthogonality andprobability-basedconstraints.

• Finally,weestablishedawaytorepresentourmainbasisvectorsascolumns. This representation is not unique. In the next lecture, we’lluseour and columnvectorstoderivecolumnvectorsforthetwootherbases.

While achieving these concrete results,we got a chance to see some state-vector mathematics in action and learn something about how thesemathematicalobjectscorrespondtophysicalspins.Althoughwewillfocuson

spin, the same concepts and techniques apply to other quantum systems aswell.Pleasetakealittletimetoassimilatethematerialwe’vecoveredsofarbeforemovingontothenextlecture.AsIsaidatthebeginning,itwillreallypayoff.

1Thechoiceisnottotallyarbitrary.Thebasisvectorsmustbeorthogonaltoeachother.2Keepinmindthat3-vectorsarenotbrasorkets.3Recallthatsphericalcoordinatesusetwoanglestorepresenttheorientationofapointinrelationto

theorigin.Latitudeandlongitudeprovideanotherexample.4Pleaseindulgeinaself-satisfiedgrin.

Lecture3

PrinciplesofQuantumMechanicsArt: I’m not like you, Lenny. My brain just wasn’t built for quantummechanics.

Lenny:Nah,minewasn’t either. Just can’t really visualize the stuff.But I’lltellyou,Ionceknewaguywhothoughtjustlikeanelectron.

Art:Whathappenedtohim?

Lenny:Art,allI’mgonnatellyouisthatitsurewasn’tpretty.

Art:Hmm,Iguessthatgenedidn’tfly.

No,wewere not built to sense quantumphenomena; not the samewaywewere built to sense classical things like force and temperature. But we arevery adaptable creatures and we’ve been able to substitute abstractmathematics for the missing senses that might have allowed us to directlyvisualize quantummechanics.And eventuallywe do develop new kinds ofintuition.

This lecture introduces theprinciplesofquantummechanics. Inorder todescribethoseprinciples,we’llneedsomenewmathematicaltools.Let’sgetstarted.

3.1 MathematicalInterlude:LinearOperators

3.1.1 MachinesandMatricesStates in quantummechanics are mathematically described as vectors in avector space. Physical observables—the things that you can measure—aredescribedbylinearoperators.We’lltakethatasanaxiom,andwe’llfindoutlater (inSection3.1.5) that operators corresponding to physical observablesmustbeHermitianaswell as linear.Thecorrespondencebetweenoperatorsandobservablesissubtle,andunderstandingitwilltakesomeeffort.

Observablesarethethingsyoumeasure.Forexample,wecanmakedirectmeasurements of the coordinates of a particle; the energy, momentum, orangular momentum of a system; or the electric field at a point in space.Observables are also associatedwith a vector space, but they are not state-vectors. They are the things youmeasure— would be an example—and

they are represented by linear operators. John Wheeler liked to call suchmathematical objectsmachines.He imagined amachinewith two ports: aninputportandanoutputport.Intheinputportyouinsertavector,suchas. The gears turn and themachine delivers a result in the output port. Thisresultisanothervector,say .

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

Not every machine is a linear operator. Linearity implies a few simpleproperties. To begin with, a linear operator must give a unique output foreveryvectorinthespace.Wecanimagineamachinethatgivesanoutputforsome vectors, but just grinds up others and gives nothing. This machinewouldnot be a linear operator.Somethingmust comeout for anythingyouputin.

ThenextpropertystatesthatwhenalinearoperatorMactsonamultipleofan inputvector, itgives the samemultipleof theoutputvector.Thus, if

,andzisanycomplexnumber,then

Theonlyotherruleisthat,whenMactsonasumofvectors,theresultsaresimplyaddedtogether:

Togiveaconcreterepresentationoflinearoperators,wereturntotherowandcolumnvectorrepresentationofbra-andket-vectorsthatweusedinLecture1. The row-column notation depends on our choice of basis vectors. If thevectorspaceisN-dimensional,wechooseasetofNorthonormal(orthogonalandnormalized)ket-vectors.Let’slabelthem ,andtheirdualbra-vectors

.

Wearenowgoingtotaketheequation

and write it in component form. As we did in Eq. 1.3, we’ll represent anarbitraryket asasumoverbasisvectors:

Here,we’reusingjasanindexratherthanisoyouwon’tbetemptedtothinkthat we’re talking about the in spin state. Now, we’ll represent in thesamewayandplugbothofthesesubstitutionsinto .Thatgives

Thelaststepistotaketheinnerproductofbothsideswithaparticularbasisvector ,resultingin

Tomake senseof this result, remember that is zero if j andk are notequal, and 1 if they are equal. That means that the sum on the right sidecollapsestoasingleterm, .

Ontheleftside,weseeasetofquantities .Wecanabbreviate with the symbol . Notice that each is just a complex

number.Toseewhy,thinkofMoperatingon togivesomenewket-vector.Theinnerproductof withthisnewket-vectormustbeacomplexnumber.Thequantities arecalledthematrixelementsofMandareoftenarrangedintoasquareN×Nmatrix.Forexample,ifN=3,wecanwritethesymbolicequation

This equation involves a slight abuse of notation that would give a puristindigestion.Theleftsideisanabstractlinearoperatorandtherightsideisaconcreterepresentationofitinaparticularbasis.Equatingthemissloppybutitshouldnotcauseconfusion.

Nowlet’srevisitEq.3.1andreplace with .Weget

Wecanwritethisinmatrixformaswell.Eq.3.3becomes

You’re probably familiar with the rule for matrixmultiplication, but I willremindyoujust incase.Tocomputethefirstentryontheright,β1,takethefirstrowofthematrixand“dot”itintothe column:

Forthesecondentry,dotthesecondrowofthematrixwiththe column:

And so on. If you are not familiar with matrix multiplication, run to your

computerandlookituprightaway.It’sacrucialpartofourtoolkit,andIwillassumeyouknowitfromnowon.

Therearebothadvantagesanddisadvantagestorepresentingvectorsandlinear operators concretely with columns, rows, and matrices (knowncollectively as components). The advantages are obvious. Componentsprovideacompletelyexplicitsetofarithmeticrulesforworkingthemachine.The disadvantage is that they depend on a specific choice of basis vectors.Theunderlyingrelationshipsbetweenvectorsandoperatorsisindependentoftheparticularbasiswechoose,andtheconcreterepresentationobscuresthatfact.

3.1.2 EigenvaluesandEigenvectorsIngeneral,whenalinearoperatoractsonavector,itwillchangethedirectionofthevector.Thismeansthatwhatcomesoutofthemachinewillnotjustbetheinputvectormultipliedbyanumber.Butforaparticularlinearoperator,therewillbecertainvectorswhosedirectionsare thesamewhen theycomeout as they were when they went in. These special vectors are calledeigenvectors.ThedefinitionofaneigenvectorofMisavector suchthat

Thedoubleuseof isadmittedlyalittleconfusing.Firstofall, (asopposedto )isanumber—generallyacomplexone,butstillanumber.Ontheotherhand, is a ket-vector. Furthermore, it is a ket with a very specialrelationship toM.When is fed into themachineM, all that happens isthatitgetsmultipliedbythenumber .I’llgiveyouanexample.IfMisthe2×2matrix

thenit’seasytoseethatthevector

just getsmultipliedby3whenM acts on it. Try it out.M also happens tohaveanothereigenvector:

WhenM acts on this eigenvector, it multiplies the vector by a differentnumber,namely−1.Ontheotherhand,ifMactsonthevector

thevectorisnotsimplymultipliedbyanumber.Maltersthedirectionofthevectoraswellasitsmagnitude.

Justas thevectors thatgetmultipliedbynumberswhenMactsonthemare called eigenvectors ofM, the constants that multiply them are calledeigenvalues. In general, the eigenvalues are complex numbers. Here is anexamplethatyoucanworkoutforyourself.Takethematrix

andshowthatthevector

isaneigenvectorwitheigenvalue−i.

Linearoperatorscanalsoactonbra-vectors.Thenotationformultiplying

byMis

I will keep the discussion short by telling you the rule for this type ofmultiplication. It is most simple in component form. Remember that bra-vectorsarerepresentedincomponentformasrowvectors.Forexample, thebra mightberepresentedby

Theruleisagainjustmatrixmultiplication.Withaslightabuseofnotation,

3.1.3 HermitianConjugationYoumight think that if then , but if youdoyouare wrong. The problem is complex conjugation. Even when Z is just acomplexnumber, if , it isnotgenerally true that .You have to complex-conjugate Z when going from kets to bras:

. Of course, ifZ happens to be a real number, then complexconjugation has no effect—every real number is equal to its own complexconjugate.

Whatwe need is a concept of complex conjugation for operators. Let’slookattheequation incomponentnotation,

andformitscomplexconjugate,

Wewould like towrite this equation inmatrix form, using bras instead ofkets.Indoingthis,wehavetorememberthatbra-vectorsarerepresentedbyrows, not columns. For the result to work out correctly, we also need torearrangethecomplexconjugateelementsofthematrixM.ThenotationforthisrearrangementisM†,asexplainedbelow.Ournewequationis

Lookcarefullyat thedifferencebetween thematrix in thisequationand thematrix in Eq. 3.6. You will see two differences. The most obvious is thecomplexconjugationofeachelement,butyoucanalsoseeadifferenceintheelementindices.Forexample,whereyouseem23inEq.3.6,yousee inEq.3.7.Inotherwords,therowsandcolumnshavebeeninterchanged.

Whenwechangeanequationfromtheketformtothebraform,wemustmodifythematrixintwosteps:

1. Interchangetherowsandthecolumns.

2. Complex-conjugateeachmatrixelement.

Inmatrixnotation,interchangingrowsandcolumnsiscalledtransposingandisindicatedbyasuperscriptT.Thus,thetransposeofthematrixMis

Noticethattransposingamatrixflipsitaboutthemaindiagonal(thediagonal

fromtheupperlefttothelowerright).

The complex conjugate of a transposed matrix is called its Hermitianconjugate,denotedbyadagger.Youcouldthinkofthedaggerasahybridofthestar-notationusedincomplexconjugationandtheTusedintransposition.Insymbols,

Tosummarize:ifMactsontheket togive , thenitfollowsthatactsonthebra togive .Insymbols:

If

then

3.1.4 HermitianOperatorsRealnumbersplayaspecialroleinphysics.Theresultsofanymeasurementsare realnumbers.Sometimes,wemeasure twoquantities,put them togetherwithan i (forminga complexnumber), and call this number the result of ameasurement. But it’s actually just a way of combining two realmeasurements. If we want to be pedantic, we might say that observablequantitiesareequaltotheirowncomplexconjugates.That’sofcoursejustafancywayof saying they are real.We are going to findout very soon thatquantummechanical observables are represented by linear operators.Whatkindoflinearoperators?Thekindthataretheclosestthingtoarealoperator.Observables in quantummechanics are represented by linear operators thatare equal to their own Hermitian conjugates. They are called Hermitianoperators after the French mathematician Charles Hermite. Hermitianoperatorssatisfytheproperty

Intermsofmatrixelements,thiscanbestatedas

Inotherwords, if you flip aHermitianmatrix about themaindiagonal andthentakeitscomplexconjugate,theresultisthesameastheoriginalmatrix.Hermitianoperators(andmatrices)havesomespecialproperties.Thefirstisthattheireigenvaluesareallreal.Let’sproveit.

Suppose and represent an eigenvalue and the correspondingeigenvectoroftheHermitianoperatorL.Insymbols,

Then,bythedefinitionofHermitianconjugation,

However,sinceLisHermitian,itisequalto .Thus,wecanrewritethetwoequationsas

and

NowmultiplyEq.3.8by andEq.3.9by .Theybecome

and

Obviously,forbothequationstobetrue, mustequal .Inotherwords,(andthereforeanyeigenvalueofaHermitianoperator)mustbereal.

3.1.5 HermitianOperatorsandOrthonormalBases

We come now to the basic mathematical theorem—I will call it thefundamental theorem—that serves as a foundation of quantum mechanics.The basic idea is that observable quantities in quantum mechanics arerepresented byHermitian operators. It’s a very simple theorem, but it’s anextremelyimportantone.Wecanstateitmorepreciselyasfollows:

TheFundamentalTheorem

• TheeigenvectorsofaHermitianoperatorareacompleteset.Thismeansthatanyvectortheoperatorcangeneratecanbeexpandedasasumofitseigenvectors.

• If and aretwounequaleigenvaluesofaHermitianoperator,thenthecorrespondingeigenvectorsareorthogonal.

• Even if the two eigenvalues are equal, the correspondingeigenvectorscanbechosentobeorthogonal.Thissituation,wheretwodifferenteigenvectorshavethesameeigenvalue,hasaname:it’scalleddegeneracy. Degeneracy comes into play when two operators havesimultaneouseigenvectors,asdiscussedlateroninSection5.1.

Onecansummarizethefundamentaltheoremasfollows:Theeigenvectorsofa Hermitian operator form an orthonormal basis. Let’s prove it, beginningwiththesecondbulletitem.

Accordingtothedefinitionofeigenvectorsandeigenvalues,wecanwrite

Now,usingthefactthatLisHermitian(itsownHermitianconjugate),wecanflipthefirstequationintoabraequation.

Thus,

By now, the trick should be obvious, but I will spell it out. Take the firstequationandformitsinnerproductwith .Then,takethesecondequationandformitsinnerproductwith .Theresultis

Bysubtracting,weget

Therefore,if and aredifferent,theinnerproduct mustbezero.Inotherwords,thetwoeigenvectorsmustbeorthogonal.

Next,let’sprovethatevenif ,thetwoeigenvectorscanbechosentobeorthogonal.Suppose

Inotherwords,therearetwodistincteigenvectorswiththesameeigenvalue.Itshouldbeclearthatanylinearcombinationofthetwoeigenvectorsisalsoan eigenvector with the same eigenvalue. With this much freedom, it isalwayspossibletofindtwoorthogonallinearcombinations.

Let’s see how. Consider an arbitrary linear combination of these twoeigenvectors:

OperatingonbothsideswithL,weget

andfinally

This equationdemonstrates that any linear combinationof and isalsoaneigenvectorofL,withthesameeigenvalue.Byassumption,thesetwovectorsarelinearlyindependent—otherwise,theywouldnotrepresentdistinctstates.WewillalsosupposethattheyspanthesubspaceofeigenvectorsofLthathaveeigenvalue .Thereisastraightforwardprocess,calledtheGram-Schmidtprocedure, forfindinganorthonormalbasisforasubspace,givenasetof independentvectors thatspansthesubspace.InplainEnglish,wecanfindtwoorthonormaleigenvectorsbywritingthemasalinearcombinationof

and . We outline the Gram-Schmidt procedure below, in Section3.1.6.

Thefinalpartofthetheoremstatesthattheeigenvectorsarecomplete.Inother words, if the space is N-dimensional, there will be N orthonormaleigenvectors.TheproofiseasyandIwillleaveittoyou.

Exercise3.1: Prove the following: If avector space isN-dimensional,an orthonormal basis of N vectors can be constructed from theeigenvectorsofaHermitianoperator.

3.1.6 TheGram-SchmidtProcedureSometimeswe encounter a set of linearly independent eigenvectors thatdonot form an orthonormal set. This typically happens when a system hasdegenerate states—distinct states that have the same eigenvalue. In thatsituation, we can always use the linearly independent vectors we have, tocreateanorthonormalsetthatspansthesamespace.ThemethodistheGram-SchmidtprocedureIalludedtoearlier.Fig.3.1illustrateshowitworksforthesimplecaseoftwolinearlyindependentvectors.Westartwiththetwovectors

and ,andfromtheseweconstructtwoorthonormalvectors, and .

Figure 3.1: The Gram-Schmidt Procedure. Given two linearly independentvectors, and ,thatarenotnecessarilyorthogonal,wecanconstructtwoorthonormal vectors, and . is an intermediate result used in theconstructionprocess.Wecanextend thisprocedure to largersetsof linearlyindependentvectors.

The first step is todivide by itsown length, ,whichgivesusaunitvectorparallelto .We’llcallthatunitvector ,and becomesthefirstvectorinourorthonormalset.Next,weproject ontothedirectionof by forming the expression . Now,we subtract

from .We’llcalltheresultofthissubtraction .YoucanseeinFig.3.1that isorthogonalto .Lastly,wedivide byitsownlengthtoformthesecondmemberofourorthonormalset, .Itshouldbeclearthatwecanextend this procedure to larger sets of linearly independent vectors inmoredimensions.For instance, ifwehada third linearly independentvector,say

,pointingoutofthepage,wewouldsubtractitsprojectionsontoeachoftheunitvectors and ,andthendividetheresultbyitsownlength.1

3.2 ThePrinciplesWearenowfullyprepared tostate theprinciplesofquantummechanics, sowithoutfurtherado,let’sdoit.

Theprinciplesallinvolvetheideaofanobservable,andtheypresupposetheexistenceofanunderlyingcomplexvectorspacewhosevectorsrepresentsystem states. In this lecture, we present the four principles that do notinvolvetheevolutionofstate-vectorswithtime.InLecture4,wewilladdafifthprinciplethataddressesthetimedevelopmentofsystemstates.

Anobservablecouldalsobecalledameasurable.It’sathingthatyoucan

measure with a suitable apparatus. Earlier, we spoke about measuring thecomponents of a spin, , , and . These are examples of observables.We’llcomebacktothem,butfirstlet’slookattheprinciples:

• Principle 1: The observable ormeasurable quantities of quantummechanicsarerepresentedbylinearoperatorsL.

I realize that this is the kind of hopelessly abstract statement thatmakes people give up on quantum mechanics and take up surfinginstead.Don’tworry—itsmeaningwillbecomeclearbytheendofthelecture.

We’llsoonseethatLmustalsobeHermitian.Someauthors regardthisasapostulate,orbasicprinciple.Wehavechoseninsteadtoderiveitfromtheotherprinciples.Theendresultisthesameeitherway:theoperatorsthatrepresentobservablesareHermitian.

• Principle 2: The possible results of a measurement are theeigenvalues of the operator that represents the observable.We’ll calltheseeigenvalues .Thestateforwhichtheresultofameasurementisunambiguously isthecorrespondingeigenvector .Don’tunpackyoursurfboardjustyet.

Here’sanotherwaytosay it: if thesystemis in theeigenstate ,theresultofameasurementisguaranteedtobe i.

• Principle3:Unambiguouslydistinguishable states are representedbyorthogonalvectors.

• Principle 4: If is the state-vector of a system, and theobservableLismeasured,theprobabilitytoobservevalue is

I’llremindyouthatthe aretheeigenvaluesofL,and arethecorrespondingeigenvectors.

These brief statements are hardly self-explanatory, and we’ll need to fleshthem out. For the moment, let’s accept the first item, namely that everyobservable is identifiedwith a linear operator.We can alreadybegin to seethatanoperatorisawayofpackagingupstatesalongwiththeireigenvalues,whichare thepossible resultsofmeasuring thosestates.These ideasshould

becomeclearaswemoveforward.

Let’s recall some important points from our earlier discussion of spins.First of all, the result of ameasurement is generally statistically uncertain.However, foranygivenobservable, thereareparticular states forwhich theresultisabsolutelycertain.Forexample,ifthespin-measuringapparatusAisoriented along the z axis, the state always leads to the value .Likewise,thestate nevergivesanythingbut .Principle1givesusanewwaytolookatthesefacts.Itimpliesthateachobservable( , ,and ) is identifiedwith a specific linear operator in the two-dimensionalspaceofstatesdescribingthespin.

Whenanobservableismeasured,theresultisalwaysarealnumberdrawnfrom a set of possible results. For example, if the energy of an atom ismeasured,theresultwillbeoneoftheestablishedenergylevelsoftheatom.Forthefamiliarcaseofthespin,thepossiblevaluesofanyofthecomponentsare ±1. The apparatus never gives any other result. Principle 2 defines therelation between the operator representing an observable and the possiblenumericaloutputsofthemeasurement.Namely,theresultofameasurementis always one of the eigenvalues of the corresponding operator. Thus, eachcomponentofthespinoperatormusthavetwoeigenvaluesequalto±1.2

Principle 3 is the most interesting. At least I find it so. It speaks ofunambiguouslydistinctstates, akey idea thatwehavealreadyencountered.Twostatesarephysicallydistinctifthereisameasurementthatcantellthemapartwithout ambiguity. For example, and can be distinguished bymeasuring .Ifyouarehandedaspinandtoldthatitiseitherinthestateor thestate , to findoutwhichof the twostates is the rightone,allyouhavetodoisalignAwiththezaxisandmeasure .Thereisnopossibilityofa mistake. The same is true for and . You can distinguish them bymeasuring .

Butsupposeinsteadthatyouaretoldthespinisinoneofthetwostates, or (up or right). There is nothing you can measure that will

unambiguouslytellyouthespin’struestate.Measuring won’tdoit.Ifyouget , it is possible that the initial statewas since there is a 50percentprobabilityofgettingthisanswerinthestate .Forthatreason,and aresaidtobephysicallydistinguishable,but and arenot.Onemightsaythattheinnerproductoftwostatesisameasureoftheinabilitytodistinguish themwith certainty. Sometimes this inner product is called theoverlap. Principle 3 requires physically distinct states to be represented by

orthogonalstate-vectors,thatis,vectorswithnooverlap.Thus,forspinstates,but

Finally, Principle 4 quantifies these ideas in a rule that expresses theprobabilities for various outcomes of an experiment. If we assume that asystemhasbeenpreparedinstate ,andsubsequentlytheobservableL ismeasured,thentheoutcomewillbeoneoftheeigenvalues oftheoperatorL.But,ingeneral,thereisnowaytotellforcertainwhichofthesevalueswillbe observed. There is only a probability—let us call it —that theoutcomewillbe .Principle4tellsushowtocalculatethatprobability,andit is expressed in termsof theoverlapof and .More precisely, theprobabilityisthesquareofthemagnitudeoftheoverlap:

or,equivalently,

Youmightbewonderingwhytheprobabilityisnottheoverlapitself.Whythesquareoftheoverlap?Keepinmindthattheinnerproductoftwovectorsisnot always positive, or even real. Probabilities, on the other hand, are bothpositiveandreal.Soitwouldnotmakesensetoidentify with .But thesquareof themagnitude, , isalwayspositiveandrealandthuscanbeidentifiedwiththeprobabilityofagivenoutcome.

Animportantconsequenceoftheprinciplesisasfollows:

TheoperatorsthatrepresentobservablesareHermitian.

Thereasonforthisistwofold.First,sincetheresultofanexperimentmustbearealnumber,theeigenvaluesofanoperatorLmustalsobereal.Secondly,the eigenvectors that represent unambiguously distinguishable results musthavedifferenteigenvalues,andmustalsobeorthogonal.TheseconditionsaresufficienttoprovethatLmustbeHermitian.

3.3 AnExample:SpinOperatorsItmaybehardtobelieve,butsinglespins—assimpleastheyare—stillhavealotmoretoteachusaboutquantummechanics,andweplantomilkthemfor

allthey’reworth.Ourgoalinthissectionistowritedownthespinoperatorsinconcreteform,as2×2matrices.Then,we’llgettoseehowtheyworkinspecificsituations.We’llbuildupourspinoperatorsandstate-vectorsshortly.But beforewe dive into the details, I’d like to say a littlemore about howoperators are related to physicalmeasurements.The relationship is a subtleone,andwe’llsaymoreaboutitaswego.

As you know, physicists recognize various types of physical quantities,such as scalars and vectors. It should come as no surprise, then, that anoperator associated with themeasurement of a vector (such as spin) has avectorcharacterofitsown.

Inourtravelssofar,wehaveseenmorethanonekindofvector.The3-vector is the most straightforward and serves as a prototype. It’s amathematical representation of an arrow in three-dimensional space, and isoften represented by three real numbers, written out as a column matrix.Becausetheircomponentsarereal-valued,3-vectorsarenotquiterichenoughto represent quantum states. For that, we need bras and kets, which havecomplex-valuedcomponents.

What sort of vector is the spin operator ? It is definitely not a state-vector (a bra or a ket). It’s not exactly a 3-vector either, but it does have astrongfamilyresemblancebecauseit’sassociatedwithadirectioninspace.Infact,wewillfrequentlyuse asthoughitwereasimple3-vector.However,we’lltrytokeepthingsstraightbycalling a3-vectoroperator.

Butwhatdoesthatactuallymean?Inphysicalterms,itmeansthis:Justasa spin-measuring apparatus can only answer questions about a spin’sorientation in a specific direction, a spin operator can only provideinformation about the spin component in a specific direction.Tophysicallymeasurespininadifferentdirection,weneedtorotatetheapparatustopointinthenewdirection.Thesameideaappliestothespinoperator—ifwewantitto tell us about the spin component in a new direction, it too must be“rotated,” but this kind of rotation is accomplished mathematically. Thebottom line is that there is a spin operator for each direction in which theapparatuscanbeoriented.

3.4 ConstructingSpinOperatorsNow,let’sworkoutthedetailsofspinoperators.Thefirstgoalistoconstructoperators to represent the components of spin, , , and . Thenwe’llbuild on those results to construct an operator that represents a spincomponentinanydirection.Asusual,webeginwith .Weknowthat has

definite, unambiguous values for the states and , and that thecorrespondingmeasurementvaluesare and .Hereiswhatthefirstthreeprinciplestellus:

• Principle 1: Each component of is represented by a linearoperator.

• Principle 2: The eigenvectors of are and . Thecorrespondingeigenvaluesare+1and−1.Wecanexpressthiswiththeabstractequations

• Principle3:States and are orthogonal to each other.Thiscanbeexpressedas

Recallingourcolumnrepresentationsof and fromEqs.2.11and2.12,wecanwriteEqs.3.12inmatrixformas

and

Thereisonlyonematrixthatsatisfiestheseequations.Ileaveitasanexercisetoprove

or,moreconcisely,

Exercise3.2: ProvethatEq.3.16istheuniquesolutiontoEqs.3.14and3.15.

This is our very first example of a quantummechanical operator. Let’ssummarizewhatwentintoit.First,someexperimentaldata:therearecertainstates that we called and , in which the measurement of givesunambiguous results ±1. Next, the principles told us that and areorthogonalandareeigenvectorsofa linearoperator .Finally,we learnedfrom the principles that the corresponding eigenvalues are the observed (ormeasured)values,again±1.That’sallittakestoderiveEq.3.17.

Canwedo the same for theother twocomponentsof spin, and ?Yes,wecan.3Theeigenvectorsof are and ,witheigenvalues+1and−1respectively.Inequationform,

Recallthat and arelinearsuperpositionsof and :

Substitutingtheappropriatecolumnvectorsfor and ,weget

TomakeEqs.3.18concrete,wecanwritetheminmatrixform:

and

Ifyouwritetheseequationsoutinlonghandform,theyturnintofoureasilysolvedequationsforthematrixelements , , ,and .

Hereisthesolution:

or

Finally,wecandothesamefor .Theeigenvectorsof aretheinandoutstates and :

Incomponentform,theseequationsbecome

andaneasycalculationgives

Tosummarize,thethreeoperators , ,and arerepresentedbythethreematrices

Thesethreematricesareveryfamousandcarrythenameoftheirdiscoverer.TheyarethePaulimatrices.4

3.5 ACommonMisconceptionThis is a convenient time to warn you about a potential hazard. Thecorrespondence between operators and measurements is fundamental inquantummechanics.Itisalsoveryeasytomisunderstand.Here’swhatistrueaboutoperatorsinquantummechanics:

1. Operators are the things we use to calculate eigenvalues andeigenvectors.

2. Operators act on state-vectors (which are abstract mathematicalobjects),notonactualphysicalsystems.

3. Whenanoperatoractsonastate-vector, itproducesanewstate-vector.

Having said what is true about operators, I want to warn you about acommonmisconception. It is often thought thatmeasuring anobservable isthe same as operating with the corresponding operator on the state. For

example, suppose we are interested in measuring an observable L. Themeasurementissomekindofoperationthattheapparatusdoestothesystem,but that operation is in no way the same as acting on the state with theoperator L. For example, if the state of the system before we do themeasurement is , it is not correct to say that the measurement of Lchangesthestateto .