4CHAPTER2.QUANTUMSTATESANDOBSERVABLESwecouldalsohavewrittenj"i;(2.3)an - PDF document

2CONTENTS 4CHAPTER2.QUANTUMSTATESANDOBSERVABLESwecouldalsohavewrittenj"i;(2.3)andj#i;(2.4)butletusstickwiththenumbersforourpurposes.Moreprecisely,theyarebasisvectorsinacomplexvectorspaceH,aswewillseeinasecond.Assuch,notonlyvectorsj0iandj1imakesense,butinfactanylinearsuperpositionj i=j0i+j1i2H'C2;(2.5)where;2Csuchthatjj2+jj2=1:(2.6)Forexamplej+i=1 p 2(j0i+j1i)(2.7)isalegitimatechoice.Orj i=sin(x)j0i+cos(x)j1i(2.8)forx2[0;2).Similarly,ji=1 p 2(j0i+ij1i)(2.9)andji=1 p 2(j0i�ij1i)(2.10)arevalidchoices.Theparametersandarehencecomplex,andEq.(2.6)reectsnormalizationofthevector,kjj ik2=h j i=jj2+jj2:(2.11)Superpositionswehavealreadyseenbefore,herewithcomplexcoefcients.Theyarecalledstatevectorsinquantummechanics.Here,theyariseinaquitesubtlefashion,however:AsituationoftheformasinEq.(2.7)doesnotcorre-spondtothesituationofthespinpointingeitherupordown.Itissomethingverydifferent.Itisasuperpositionofthespinpointingupanddown(omittingthehyphenationfromnowon).Wewillseewhatthismeansverysoon.Wehavesaidthatj0iandj1icorrespondtospinupanddown,respectively.Aswewillseeinmoredetaillater,comingbacktotheaboveboxes,j+i=1 p 2(j0i+j1i)(2.12) 6CHAPTER2.QUANTUMSTATESANDOBSERVABLESwith;2C,appropriatelynormalized,isagainalegitimatestatevector.2.1.2PaulioperatorsWehaveseen–evenifnotdiscussedinalldetail–thattheinternalstatescanbeassociatedwithvectors.WewillnowseethatphysicspropertiescanbeassociatedwithoperatorsA,sowithlinearmapsoftheformj i7!Aj i;(2.17)whereagainAj i2H.Beforewebecometooabstractatthispoint,letusconsideranexample:ThePauli-z-matrixzactsaszj0i=j0i;(2.18)zj1i=�j1i:(2.19)Wehencenotethatthevectorsj0iandj1iareeigenvectorsofz:Uptoacomplexnumber–here+1and�1,therespectiveeigenvalues–weobtainagainthesamevectorifweapplyztoit.TheabovementionedstatevectorsthatareonthenorthandsouthpolesoftheBlochspherehencecorrespondtoeigenvectorsofzwiththerespectiveeigenvalues.Similarly,wendthatthePauli-xmatrixxactsasxj+i=j+i;(2.20)xj�i=�j�i:(2.21)Finally,thePauli-y-matrixyhasthepropertythatyji=ji;(2.22)yji=�ji:(2.23)2.1.3ApreliminaryinterpretationoftheboxesWewillexplainthisinmoredetailbelow.Theshortandsomewhatellipticexplanationoftheabovesituationinvolvingtheboxesisasfollows:Proper-tiesareassociatedwithoperators,infactwithHermitianoperators,seebelow.SuchHermitianoperatorsarecalledobservables.PaulioperatorsareexamplesofHermitianoperators.AmeasurementalongthezaxiscorrespondstothePauli-z-matrix,andsimilarlyfortheotherPaulimatrices.Sotherstmea-surementcorrespondstoa“measurementoftheobservablePauli-z”.Afterthemeasurement,thesystemwillbeinaneigenvectoroftherespectiveobservable,theoutcomeofthemeasurementbeingtheeigenvalueoftheeigenvector.Forexample,wemeasurethespinalongthezaxis,hence“measuretheobservablePauli-z”.Ifwegetthevalue+1,wewillobtainthestatevectorj0i=zj0i(2.24) 8CHAPTER2.QUANTUMSTATESANDOBSERVABLES Statevectorsofnite-dimensionalquantumsystems:Thestatevectorofad-dimensionalquantumsystemcanbewrittenasavectorj i=d�1Xj=0jjji;(2.28)satisfyingnormalizationd�1Xj=0jjj2=1:(2.29) Indeed,thevectorsfj0i;:::;jd�1ig(2.30)areagainorthonormalbasisvectors,whileallotherstatevectorsaresuperpo-sitionsthereof.Theprinciplethatanyofthesevectorsareallowedstatevectorsissometimesalsoreferredtoasthesuperpositionprinciple.ThesevectorsformavectorspaceH,infactaHilbertspace(seeAppendix). Hilbertspaceofd-dimensionalquantumsystems:Thebasisvectorsfj0i;:::;jd�1igspantheHilbertspaceH'Cd. Veryimportantwillalsobethescalarproduct. Scalarproduct:Fortwostatevectorsj i=d�1Xj=0jjji;ji=d�1Xj=0jjji(2.31)wewritetheirstandardscalarproductash ji=hj i=d�1Xj=0jj:(2.32) Suchscalarproductsaresometimescalled“brackets”,whichiswhyvectors–inaninstanceofphysicshumor,judgeforyourself–arecalled“kets”.Dualvectors,fromthedualspaceH,arereferredtoas“bras”,h j=d�1Xj=0jhjj;(2.33) 10CHAPTER2.QUANTUMSTATESANDOBSERVABLESandadualvectorh j=d�1Xj=0jhjj(2.41)with[0:::d�1]:(2.42)2.2.2ObservablesObservables,so“observablequantities”,entitiesthatcaptureameasurementprescription,areHermitianoperators.Thisissuchanimportantstatementthatwegiveitabox: Observables:ObservablesinquantummechanicscorrespondtoHermi-tianoperators,A=Ay:(2.43) Theadjointcanbedenedviah j(Aji)=(h jA)ji(2.44)forallj i;ji.Suchobservablescanagainbewritteninmatrixform:Intermsofthebasisfj0i;:::;jd�1ig,thematrixformhastheentriesAj;k=hjjAjki:(2.45)Itisclearthatthismatrixformuniquelycharacterizestheoperator:Wecanmakeuseofthe“insertiontrick”A=1
A
1=Xj;khjjAjkijjihkj:(2.46)Inthefollowing,wewillalwaysidentifyanoperatorwithitsmatrixform.Thisisaverycommonidentication:WewillwriteAbothforthelinearoperatoritselfaswellasforthematrixthatreectsitgivenabasis.Thisissocommonandnaturalthatanyotherchoicewouldunnecessarilymakethenotationverycumbersome.Asanexercise,weformulatethematrixformofthePaulioperators.Therstoneisverymuchobvious:Wecanwritez=100�1:(2.47)Thevectorsj0iandj1iarealreadyeigenvectorsofz,soweshouldnotbesurprisedtoseethatthematrixformisdiagonal.Similarly,wendthematrix 12CHAPTER2.QUANTUMSTATESANDOBSERVABLES Compatibleobservables:TwoobservablesAandBarecompatible,if[A;B]=0:(2.54) 2.2.3EigenvectorsandeigenvaluesIndeed,thefactthatobservablesareHermitianmeansthattheireigenvaluesarereal. Eigenvaluedecompositionofobservables:Every(nite-dimensional)ob-servableAcanbewrittenasA=d�1Xj=0jjjihjj;(2.55)wheretheeigenvaluessatisfy0;:::;d�12RandtheeigenvectorsfjjigcanbechosentoformanorthonormalbasisofH. Thisdoesnotmean,ofcourse,thatallofthefjgarenecessarilydifferent.Iftwoormoreeigenvaluesareidentical,theyarecalleddegenerate.Therespec-tiveeigenvaluesthenspantheeigenspace,whichisnolongerone-dimensional.Eigenvaluesandeigenvectorsplayaveryimportantroleinquantummechan-ics:Theformeressentiallyas“measurementoutcomes”andthelatteras“post-measurementstatevectors”.Sinceeigenvaluedecompositionsaresoimpor-tant,westateatthispointanequivalentformoftheeigenvaluedecompositionofobservables: Diagonalizationofobservables:Every(nite-dimensional)observableAcanbewrittenasA=UDUy;(2.56)whereD=diag(0;:::;d�1),0;:::;d�12R,andUisunitary,sosatis-edUUy=UyU=1:(2.57) Unitaryoperatorsarethosethatpreservescalarproducts,andcorrespondtobasischangesoforthonormalbasis.Theabovestatementishenceamanifesta-tionoftheobservationthatintheireigenbasis,observablesarediagonal.The 14CHAPTER2.QUANTUMSTATESANDOBSERVABLESNow,theseoperatorsnolongerhavediscretespectra,sonodiscreteeigenval-ues,aswenowhavedim(H)=1.Wecanatthispointalreadygraspwhythisisthecase:eigenvaluescorrespondtomeasurementoutcomes.Incaseofposition,aparticlecantakeacontinuumofdifferentpositions,incaseofaone-dimensionalproblem,thisistheentirerealline.Andinfact,boththepositionandthemomentumoperatorhaveRastheirspectrum.Neitherthepositionnorthemomentumoperatorhaveeigenvectors(al-thoughmanybooksspellthemoutnevertheless,wewillcomebacktothat).Itstillmakessenseforx2Rtowritehxj i= (x);(2.61)evenifatthispointwetaketherighthandsidetobethedenitionforthelefthandsideoftheequalitysign(wedonotinsistthedualvectorshxjtobecontainedinanydualHilbertspace,butwestartworryingaboutthatdetailintheappendix).Thisrepresentationintermsofwavefunctionsisusuallycalledthepositionrepresentation. Wavefunctions:Singlespatialdegreesoffreedomaredescribedbywavefunctions :R!C:(2.62)TheysatisfyZ1�1j (x)j2=1:(2.63)j (x)j2canbeinterpretedastheprobabilitydensityforndingtheparticleatpositionx. Becauseweaskfor beingsquareintegrable,theHilbertspaceisH=L2(R)ofsquareintegrablefunctionsoverthereals.Soindeed,(x)=j (x)j2(2.64)forx2Rhasaprobabilityinterpretation.Yet,thewavefunctionis“alotmorethanamereprobabilitydistribution”,asitalsocontainsphaseinformation,asitisacomplex-valuedfunction.Inotherwords,thewavefunctiondoesnotreectthesituationthat“theparticleissomewhere,wemerelydonotknowwhereitis”.Thisstatement–evenifonecansometimesreadsuchstatementsintheliterature–iswrong.Whatistrue,however,isthattheprobabilityden-sityofndingparticlesinparticularplacescanbedeterminedfromthewavefunction.Again,statevectorsareelementsofHilbert(andhencevector)spaces:Thatistosaythatifj 1iisalegitimatestatevectorandj 2iaswell,thenanylinearcombinationofthem,appropriatelynormalized,isagainastatevector.Statedintermsofthepositionrepresentation,if 1:R!Cand 2:R!Carewave 16CHAPTER2.QUANTUMSTATESANDOBSERVABLESSimilarlytothepositionrepresentation,wecanintroducethemomentumrepresentation:Similarlyasabove,weconsiderhpj i=~ (p);(2.71)where :R!CistheFouriertransformof~ :R!C, (x)=1 p 2Z1�1~ (k)eikxdk:(2.72)2.2.6CombiningquantumsystemsHowdowedescribecompositequantumsystemsinquantumtheory?Clearly,theformalismmusthaveananswertothat.Wethinkofaparticlehavingseveraldegreesoffreedom.Orweaimatdescribingseveraldifferentparticlesatonce.Howdowecapturethissituation?Compositionofdegreesoffreedomisincorporatedbythetensorproductsinquantummechanics.Letusassumethatwehaveonedegreeoffreedomassociatedwithad1-dimensionalHilbertspaceH1=spanfj0i;:::;jd1�1ig:(2.73)Wethenconsideranother,seconddegreeoffreedom,comingalongwithad2-dimensionalHilbertspaceH2=spanfj0i;:::;jd2�1ig:(2.74)Thesespacescould,forexample,captureallsuperpositionsoftwospindegreesoffreedomoftwoparticlesdescribedbyquantummechanics.TheHilbertspaceofthejointsystemisthengivenbythetensorproductH=H1 H2:(2.75)Itisspannedbytheorthonormalbasisvectorsfjji jki:j=0;:::;d1�1;k=0;:::;d2�1g:(2.76)Suchbasiselementsoftensorproductsaresometimesalsowrittenasfjj;ki:j=0;:::;d1�1;k=0;:::;d2�1g:(2.77)Thislooksmorecomplicatedthanitis:WhileanarbitrarysuperpositionofastatevectorfromH1canbewrittenasj 1i=d1Xj=0jjji(2.78)andanarbitrarysuperpositionofastatevectorfromH2isj 2i=d2Xj=0jjji;(2.79) 18CHAPTER2.QUANTUMSTATESANDOBSERVABLES2.3MeasurementAfterallthis,wearenallyinthepositiontocarefullyandpreciselydescribetheboxesthatwehaveencounteredabove.Fordoingso,weneedtopreciselyunderstandwhatmeasurementdoestoaquantumsystem.2.3.1MeasurementpostulateLetusstartbypreciselystatinghowmeasurementiscapturedinquantumthe-ory,andthendiscusswhatthismeansingreatdetail.Thesubsequenttypeofmeasurementiscalled“von-Neumannmeasurement”(toemphasizethatlateron,wewillhavealookatamoregeneralframeworktocapturemeasurement,whichishowevernotfundamentallymoregeneralthanthemeasurementpos-tulatethatwearegoingtoseenow). Measurementpostulate(forvon-Neumannmeasurements):WeconsiderameasurementoftheobservableAforaquantumsystempreparedinastatevectorj i2Cd.Letusassumethatnoneoftheeigenvalues0;:::;d�1ofA=d�1Xj=0jjjihjj(2.86)aredegenerate.Thentheprobabilitypjofobtainingtheoutcomejisgivenbypj=jh jjij2;(2.87)thestatevectorimmediatelyafterthemeasurementisgivenbyj ji=1 p pjhjj ijji:(2.88) Inotherwords,themeasurementvaluesaretheeigenvalues,andameasure-ment“collapses”thestatevectorintooneoftheeigenvectors.Thesituationofencounteringdegenerateeigenvaluesmerelyrequiresamildmodication:IncaseofadegenerateobservableA,witheigenvalues0;:::;d�1anddifferenteigenvalues0;:::;D�1,wecanwriteA=D�1Xk=0kk(2.89)wherek=Xj;j=kjjihjj(2.90) 20CHAPTER2.QUANTUMSTATESANDOBSERVABLES2.3.3AproperexplanationoftheboxesWearenowinthepositiontoproperlyexplainourinitialsituationinvolvingboxes–withthesingleexceptionoftheinitialpreparation,towhichwewillcomelater.TherstmeasurementofthePaulioperatorzcanhavetheout-comes0;1=1.Afterthemeasurement,thewavefunctionisprojectedontoj0iorj1i,dependentonthemeasurementoutcome.Thisiswhyanysubse-quentmeasurementofzwillprovidethesameoutcome:Onethestatevectoriscollapsedtoj0i,say,theprobabilityofobtainingtheoutcome1isagainp0=1:(2.99)Afancywayofsayingthisisthatzandzarecompatible,andwecanrepeatthemeasurementandarbitrarynumberoftimes,andwillgetthesamevalueoverandoveragain.Thisisdifferentincasewemeasurexinbetween.Say,wehaveobtainedtheoutcome0=1andthepost-measurementstatej0i.Thentheprobabilityofgettingthepost-measurement-statej+iisgivenbyjh+j0ij2=1 2;(2.100)andinthesamewayjh�j0ij2=1 2:(2.101)Sotheprobabilityofgettingeachoutcomeisprecisely1=2.Then,ifweobtainj+iinthemeasurementoftheobservablex,thentheprobabilityofgettingj0iinasubsequentzmeasurementisagain1=2,forthesamereasons–onlywiththerolesofthetwovectorsreversed.Havingunderstoodthemeasurementpostulate,theabovesituationisveryclear.Thisnotionofameasurementofx“disturbing”ameasurementofzisamanifestationoftheobservablesxandznotcommuting:Theyareincompat-ible.Compatibleobservablescanberepeatedlymeasuredwithoutalteringtheoutcome,whilethisisnotsoforincompatibleobservables.Theorderofmea-surementalsomatters,anditdoessimplynotmakesensetoaskwhetherthespinis“trulypointingupordown”incaseonehasjustperformedameasure-mentalongthexaxis.Suchpropertiesaresimplynotdenedwithinquantumtheory.2.3.4ThreereadingsoftheHeisenberguncertaintyrelationThisnotionofoutcomesofmeasurementsbeing“uncertain”canbemademorepreciseintermsoftheso-calleduncertaintyrelation.Therearefewstatementsinquantumtheory,yet,thataresooftenmisunderstoodastheuncertaintyre-lation.Recently,alawyercametomeandasked,well,istheuncertaintyinadjudicationnotjustamanifestationoftheHeisenbergprinciple,that“noth-ingispreciselydened”?Well,actually,no. 22CHAPTER2.QUANTUMSTATESANDOBSERVABLES Heisenberg'suncertaintyrelation:TwoobservablesAandBofaquantumsystempreparedinj isatisfytheuncertaintyrelation
A

B1 2jh j[A;B]j ij:(2.111) Thatistosay,theproductofthetwouncertaintiescannottakeanarbitrarilysmallvalue–unlessthetwoobservablesAandBcommute.Otherwise,theproductofthetwouncertaintiesofAandBcannotbearbitrarilysmall.Thisisaremarkableobservation.Whenappliedtopositionandmomentum,thisprinciplereads
X

P~ 2;(2.112)whichmeansthatwavefunctionscannotbearbitrarilynarrowbothinpositionandmomentum.Thismeansthatifwepreparemanyquantumsystemsandrstestimate
Amanytimes,suchthatweknowitlaterwithhighstatisticalsignicance,andthenestimate
B:Thentheproductofthetwouncertaintieswillalwaysbelargerthantheabovegivenvalue.Quantummechanicssimplydoesnotallowforanysmalleruncertainties.Soinaway,onecannot“knowthevaluesoftwonon-commutingobservablesatonce”.Precisely,whatismeant,however,isthatoneindependentlypreparesthesystemsandmeasureseitherAorB,andthenanalysesthedata.ThisistherstreadingoftheHeisenberguncertaintyrelation.Thisisalsothederivationthatmostbooksonquantummechanicsoffer.Interestingly,theexplanationgivenisquiteoftenincompatiblewiththeabovederivation.ThisisthenotionofameasurementofoneobservableAmakestheoutcomeofanothernon-commutingobservableBlesscertain.Thisisthesec-ondreadingoftheuncertaintyrelation.WehencemeasureonthesamesystemrstA,thenB.Ifwe“knowthevalueofAprecisely,thenthemeasurementofBwillbealotdisturbed”.ThisisthereadingofthefamousHeisenbergmi-croscope:HeisenbergdiscussedhowameasurementofXdisturbslaterlatermeasurementsofthenon-commutingobservableP.Thisisalsotrue,butthisisnotquitewhatwehavederivedabove(whichmadeuseoftheindependentandidenticaldistribution(i.i.d.)oftheinitialpreparation).OnecanalsoderiveuncertaintiesfortheHeisenberguncertaintyprincipleinthissecondreading,buttheprefactorontherighthandsidewillbeslightlydifferent.Athirdreadingwhichweonlybrieymentionistheonewheretriestojointlymeasuretwoobservablesinasinglegeneralizedmeasurement.Wehavetodelaythisdiscussionasweareatthispointnotquiteclearaboutwhatageneralizedmeasurementis.Justfortherecords:HereoneaimsatobtainingasmuchinformationaspossibleaboutAandBinasinglerunofameasurement,