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Wave-particle duality in the damped harmonic oscillator PDF
Preview Wave-particle duality in the damped harmonic oscillator
7 0 0 Wave-particle duality 2 n in the a J 3 damped harmonic oscillator 1 1 v 2 8 0 1 0 7 0 / h p - t n a u q : v i X r a MasterthesisTheoreticalPhysics RadboudUniversityNijmegen StevenTeerenstra 2006 2 DEMATHEMATICUS In detijdsevolutievanQ Waar deVonNeumannalgebragegenereerdwordt Commutatief Gerepresenteerd Op deWienerruimte(Gaussischerepvande CCRoverL2(R)) Waar Qtvoldoetaan, Waar Btis eenBrownsebewegingmetvariatie2n, waar pure poe¨zie ontstaat Ditis de integraleversievandestochastischedifferentiaalvergelijking DusQt kange¨ıdentificeerdwordenmet EenOrnstein-Uhlenbeckproces QuantumMechanische Gedempte HarmonischeOscillator Hetbegrip verloren In woorden vergaan Mt isdan teidentificeren Meteen sterfteprocesop L2(Delta,P) De Guichardetruimte MetFock representatie Vangeduld En toewijding Loyaal inalle dimensies De mathematicus RogierTeerenstra,mei2006 Guide to the reader When writing this text, I had a two-fold aim and audience in mind which is roughly re- flectedinthedivisionofthetextintoamainpartandtheappendices. First of all, I wanted to give a gentle introduction to (elements of) quantum probability theoryusingquantizationofthedampedharmonicoscillatorasleitmotiv. Thisisthefocus in the main text. As audience I had (graduate) students in physics in mind, people like myselfwhenIstartedtheresearchofthisthesis,althoughalsootherswithsimilarlevelor interestarewelcometohavealook. Myaimwastoexplainsomeofthephysicalintuitionandmotivationbehindthe,sometimes overwhelming, math machinery of quantum probability theory in order to convey some of its intrinsic appeal. To this end, we start in the main text from a piece of quantum mechanicsthatistreatedinalmostallbasictextbooks:thequantizationofthe(undamped) harmonic oscillator from the Heisenberg and Schro¨dinger point of view. We show how bothtreatmentsarespecialinstancesofaquantumprobabilisticquantizationprocedure:the secondquantizationfunctor.Asacorrollary,quantumprobabilitytheoryisageneralization ofthequantummechanicsofHeisenberg,Dirac,Schro¨dingeretc.asweknowit. Wethen proceed to build up the quantum probability machinery needed to quantize the damped harmonicoscillator. Ihavetriedtostresstheideasandintuitioninthemaintextanddefer the details of the proofs and reasoning to the appendices, but I have not been entirely consistentinthat. Secondly,thisthesisalsoreflectstheresearchIhavedone. Actually,partsofthisresearch wereusedandextendedinlecturesthatmythesissupervisor,drHansMaassen,gaveina summerschoolinGrenoblein1998. Inparticular,Proposition4.1andProposition6.3of hislecturenotes[11]correspondtomytheorem2.5andlemma1.30. Thedetailsoftheir proofsaregiveninappendixBandA.18,respectively. Itgoeswithoutsayingthatmyownwritinghasbenefitedgreatlyfromthetheoreticalexpose´ thatdrMaassenprovidedinhislecturenotes[11]. StevenTeerenstra 5 6 Part I Harmonic Oscillator 7 Chapter 1 Quantum Mechanical Harmonic Oscillator 1.1 Dirac’s Quantization procedure Following Dirac’s quantization rules [5], we translate the (1-dimensional) classical har- monicoscillatorintoaquantummechanicalone: Observables Theclassicalconjugated(generalized)coordinatesforpositionandmomen- tum,qand p,arereplacedbyself-adjointoperatorsQandP(actingonsomeHilbert space H ). Any observable of the classical harmonic oscillator a= f(q,p) is re- placedbyitsquantumcounterpartA= f(Q,P)(whichishermitizedifnecessary). Inparticular,theHamiltonianofthequantummechanicaloscillatoris H= P2 +mw 2Q2. 2m 2 Correspondenceprinciple The relation between two quantum observablesA= f(Q,P) andB=g(Q,P)oftheoscillatorsystemisexpressedbytheircommutator[A,B]:= AB BA. Ifa= f(q,p)andb=g(q,p)arethecorrespondingclassicalobservables − havingPoissonbracket ¶ ¶ ¶ ¶ a,b = f(q,p) g(q,p) g(q,p) f(q,p) { } ¶ q ¶ p −¶ q ¶ p thenthecorrespondenceprincipleholds: [A,B]=i~ a,b . (1.1) { }1 Asaspecialcase,Heisenberg’scommutationrelationemerges [Q,P]=i~ . (1.2) 1 Timeevolution If A=a(Q,P) is an observable that does not depend explicitly on time (i.e.doesnotdependontimeotherthanviathetimeevolutionofQenP),thenthe classicaldescriptionofthetimeevolution, da= a,h wherehistheclassicalHamil- dt { } tonian,istranslatedviathecorrespondenceprincipleintotheHeisenbergequationof motion dA 1 = [A,H], (1.3) dt i~ havingA(t)=eitH/~Ae itH/~ asformalsolution. − Inparticular, dQ =P/mand dP = Q. dt dt − States The states of the system are described by (normalized)vectors a in the Hilbert | i spaceH . 9 10 Chapter1.QuantumMechanicalHarmonicOscillator Measurements Any single measurement of an oscillator observable A yields a value a in the spectrum of A (as self-adjointoperator) i.e. there is a normalized eigenstate a such that A a =a a . Suppose for simplicity that the state of the system is | i | i | i completelydeterminedbyA, then these eigenstatesof A forma completesetin H i.e.everystate z canbedecomposedasa“superposition”ofeigenstatesofA: | i (cid:229) a az ifthespectrumofAisdiscrete, z = a| ih | i (1.4) | i ( da a az ifthespectrumofAiscontinuous. | ih | i R Quantumtheorystates thatwhen the system is in state z , the probabilityof mea- suringthevalueaforA,i.e.exactlyainthediscretecas|eiorwithintheresolutionD ofthemeasuringapparatusinthecontinuouscase,is az 2 ifthespectrumofAisdiscrete, |h | i| (1.5) a+D /2 az 2da D az 2 ifthespectrumofAiscontinuous. a D /2 |h | i| ≈ |h | i| R − DirectlyafterhavingmeasuredtheobservableA, thesystemisintheeigenstate a e e | i belongingtothevaluea(collapseofthewavefunction).Theprobabilityofmeasuring anothervaluea =adirectlyafter,isthenzero,sincea aa = Aaa = aA† a = ′ ′ ′ ′ 6 h | i h | i | aAa =a aa whichimplies(aseithera=0ora =0)that aa =0. ′ ′ ′ ′ ′ h | i h | i 6 6 h | i (cid:10) (cid:11) Expectedvalue Onaverage,measurementsofAonidenticalsystems,allinstate z ,will | i yieldanaveragevalueoftheobservableA,whichiscalledtheexpectedvalueofA andisdenotedby A z . h i Quantum theory states that A z = z Az / z z , which can be justified from the h i h | | i h | i probabilisticinterpretation(1.5)above.Forinstance,ifAhasacompletesetofeigen- vectorsandz isnormalized,then z Az =(cid:229) z a aAa az =(cid:229) z a ad az =(cid:229) a z a 2 h | | i a,ah | ih | | ih | i a,ah | i a,ah | i a |h | i| inthediscretecase,while e e e e e e z Az = dada z a aAa az = dada z a ad (a a) az h | | i h | ih | | ih | i h | i − h | i =Ra z a 2da R |h | i| R e e e e e e inthecontinuouscase. Inbothcases,weseethatthepossibleoutcomesofmeasure- mentsaareweightedbytheircorrespondingprobabilitiesinstatez . Maximalsystemofcommutingobservables: quantumnumbers Fortheonedimensional harmonicoscillator,everyeigenvalueqofthepositionoperatorQisnon-degenerate i.e.thereisessentiallyoneeigenstate q suchthatQ q =q q : allvectorsy H thatsatisfyQy =qy arescalarmultipl|esiof q . Asac|oinveni|enitconsequence,t∈here | i isanobservableAsuchthatmeasuringtheobservablemeansthatweknowthatthe systemcanonlybeinonestate,viz.the(oneandonly)eigenstate a thatcorresponds | i tothemeasuredeigenvaluea. Ingeneral,itisrareforaquantumsystemtohaveone observablethatcompletelydeterminesthesysteminthissense. Moreoftenthannot therearemoreessentiallydifferent(i.e.linearlyindependent)eigenvectorsforagiven eigenvalueai.e.theeigenspaceofaconsistingofallvectorsinH onwhichAacts asmultiplicationbyaisnotone-dimensional. Forexample,thex-positionofatwo- dimensionaloscillatordoesnotuniquelydetermineitsstate. Insuchcases,onelooks forobservables A thatcommutepairwise: [A,A ]=0. Forsuchobservables,an i i i j { } eigenvectorofoneA isatthesametimeaneigenvectorofanotherA [1,p.207],[17, i j p. 30],andthereforeasimultaneouseigenvector.Inparticular,suchobservablescan bemeasuredsimultaneously. Thesimultaneouseigenvectorscanbelabeledbytheir eigenvaluesa ,a ,...withrespecttothecommutingsetofobservablesA ,A ,...i.e. 1 2 1 2 A a ,a ,a ,... =a a ,a ,a ,... . Asetofcommutingobservables A iscalled i 1 2 3 i 1 2 3 i i | i | i { } maximalifthereisnoobservableBsuchthat[B,A]=0foralli(and,ofcourse,B i