ho_fs_ident.nb:10/14/04::22:47:34 1 HarmonicOscillator,a,a†,FockSpace,IdenticleParticles, Bose/Fermi ThissetoflecturesintroducesthealgebraictreatmentoftheHarmonicOscillatorandappliestheresulttoastring,a prototypicalsystemwithalargenumberofdegreesoffreedom.ThatsystemisusedtointroduceFockspace,discuss systemsofidenticleparticlesandintroduceBose/Fermiannihilationandcreationoperators. Á HarmonicOscillator ü ClassicSHO TheclassicalHamiltonianforthesimpleharmonicoscillatoris 1 2 k 2 1 2 mw2 2 H = ÅÅÅÅÅÅÅÅÅ2 m p + ÅÅÅÅ2 x = ÅÅÅÅÅÅÅÅÅ2 m p + ÅÅÅÅÅÅÅÅÅÅÅÅ2 x Thisleadstosimpleharmonicmotionwithfrequencyw = k m . r s ü QM:wavemechanics ∑ Makethereplacementp = -i ÅÅÅÅ∑xÅÅÅ ,andsolveforthewavefunctions.Forexample,Merzbacherchapter5. ü QM:operatorapproach Introduceraisingandloweringoperators( aanda† )andsolvesimplealgebraiceigenvalueproblem.Note:insome contexts(fieldtheory)a, a† arealsoknownasannihilationandcreationoperators. ü Setupofproblem,introductionofa, a†, and N ü ForconveniencesimplifyH Define:p' = ÅÅÅÅÅÅÅÅpÅÅÅÅÅÅÅ andx' = mw x,inwhichcaseH canberewrittenas mw r r w 2 2 H = ÅÅ2ÅÅÅ p' + x' + / ü Defineaanda† . Further,definetheoperator a = ÅÅÅÅ1ÅÅÅÅÅÅ x' + ip' 2 r + / andsincexandparehermitian,theadjointis a† = ÅÅÅÅ1ÅÅÅÅÅÅ x' - ip' . 2 r + / ho_fs_ident.nb:10/14/04::22:47:34 2 Alsonotethatx'andp'canberewrittenas x' = ÅÅÅÅ1ÅÅÅÅÅÅ a† + a andp' = ÅÅÅÅÅÅÅÅi ÅÅ a† - a . 2 2 r + / r + / Thecommutatorofaanda† is † 1 a, a = ÅÅ2ÅÅ x' + ip', x' - ip' = i p', x' = i p, x = 1 # ' # ' # ' # ' and † 1 1 2 2 1 2 2 1 a a = ÅÅ2ÅÅ x' - ip' x' + ip' = ÅÅ2ÅÅ x' + p' + i x', p' = ÅÅ2ÅÅ x' + p' - ÅÅ2ÅÅ † 1 1 2 2 1 2 2 1 a a = ÅÅÅÅ2 +x' + ip'/ +x' - ip'/ = ÅÅÅÅ2 +x' + p' - i#x', p''/ = ÅÅÅÅ2 + x' + p' / + ÅÅÅÅ2 + / + / + # '/ + / ü DefineN andrewriteH ItisconvenienttorecastH w 2 2 H = ÅÅ2ÅÅÅ p' + x' † 1 += w a a /+ ÅÅÅÅ2 1 = w+N + ÅÅÅÅ2 / + / wherethe"number"operatorisN = a† a.Itshouldbeobviousthat H, N = 0,andsoH andN canbesimultaneously diagonalized.Determiningthespectrumofenergyeigenstatescanbereclas# ' sifiedasdeterminingthespectrumofN . ü Thespectrumofstates Define n asthenormalizedeigenstatesofN ,andletitbeunderstoodthatthestatesarelabeledbytheeigenvalue,i.e. ? N n = n n ? ? ü Shownispositivedefinite Considerthequantity n N n = n n n = n. ; « « ? ; « ? Itisconvenienttodefine b = a n .SinceN = a† awealsohave ? ? n N n = n a† a n = b b ¥ 0. ; « « ? ; « « ? ; « ? Itfollowsthatnisrealandpositive-definite. ü Showthataisaloweringoperator. a, N = a a† a - a† a a # ' = +a a† / a -+ a† /a a = +a, a†/ a + / = #a ' ho_fs_ident.nb:10/14/04::22:47:34 3 Again,let b = a n ,sothat ? ? N b = N a n , ? = N?, a + a N n = +#-a +'a n n / ? = +n - 1 a n/ ? = +n - 1/ b ? + / ? i.e. b isananeigenstateofN ,witheigenvalue n - 1 ,or ? + / a n = cn n - 1 ? ? wherecn issomeasyetundeterminedcoefficient. Wecanevaluatecn byconsidering † 2 n = n N n = n a a n = cn , ; « « ? ; « « ? if whichgivescn = e n .Conventionallyf = 0,whichgives r a n = n n - 1 . r ? ? ü anda† isaraisingoperator † † † † Similarly N, a = a anda n = bn+1 n + 1 ,wherebn = n aswell.a actsasaraisingoperator- r # ' ? ? a† n = n + 1 n + 1 . r ? ? Itisoftenmoreconvenientinthisform.a† n - 1 = n n ,wherewecaneasilysee † † r N n = a a n = a n n - 1 = n n . ? ? r ? ? ? ? ü spectrum Sofarwehaveshowedhowtoconstructasetofstates n withnvaluesseparatedbyintegers.Thereare,however, manysuchsets,butonlyoneisaviablesetofstatesfortheSHO.Recallthatweha ? vetheconstraintthatn ¥ 0. Supposenisintheinterval 0, 1 .Thenoperatingwithawouldgiveastatewithn < 0,whichisnotallowed.Theonly possiblityisthatoperatingwith+ a/gives0,butthatwouldviolatetherelationforcn -unlessn = 0.Itseemstheonly possiblityisforntobeintegral.Inthiscasewecansatisfytheboundaryconditionbya 0 = 0 0 = 0.Thespectrum ofstatesisthengivenby n , n = 0, 1, 2. .. ? ? ? Thisseemsveryreasonable.AstheHamiltonianispositivedefinite,theexpectationvalueisrequiredtobepositive. 1 Evenwiththeextracontributionof ÅÅ2ÅÅ itisnotunreasonablethatNisalsopositive. Onecanbuildnormalizedstatesbyiterativeuseoftheraisingoperator. † n n = ÅÅÅÅÅÅÅÅÅÅÅÅa 0 + n/! ? r ? ho_fs_ident.nb:10/14/04::22:47:34 4 ü Matrixformforoperators H, N, a, a† .Itisstraightforwardtoexpresstheoperatorsinmatrixform(seeMerzbacher).H andN arediagonal.a anda† areoffdiagonal.e.g. ∫ 0 1 0 ∫ 1 0 1 0 ÅÅÅÅ 0 ∫ r r 2 L \ L 1 0 2 0 \ 3 M 0 0 2 0 ] M ] H = wL 0 ÅÅÅÅ 0 \,a = M ],x = M r r ],... M 2 ] M r ] M ] M ] M 0 0 0 3 ] M 0 2 0 3 ] M ª 0 ∏ ] M ] M r r ] M ] M ª r∏ ] M ª ∏ ] M ] M 0 0 ] M 0 3 ] N ^ MMM ] M r ] N ^ N ^ ora = S n n - 1 n . n r ? ; ü Usefulrelation Ausefulfactfordoingsomemanipulations(forexampleproblem2.18)is † ∑ † a, f a = ÅÅÅÅ∑aÅņÅÅÅ f a # + /' + / ∑ Thisissimilarto k, x = -i ÅÅÅÅ∑xÅÅÅ # ' Á FockSpace-"2ndquantization" Theharmonicoscillatorprovidesastartingpointfordiscussinganumberofmoreadvancedtopics,including multiparticlestates,identicleparticlesandfieldtheory. Asanintroduction,considertheproblemofquantizingaclassicalstring(e.g.aguitarstring).Thisisdonebytreating theoscillatingmodesofthestringasasetofharmonicoscillators.EachH.O.canbequantized,sothatthequantum stateofthestringisgivenbyspecifyingthequantumstateofeachoscillator.Thisissometimereferredtoas"second quantization".Presumablythetermismeanttosuggestthatthefirstquantizationisdeterminingtheeigenmodesofthe system,andthesecondquantizationisdeterminingtheexcitationlevelofeachmode. Analternativelanguagefordiscussingthequantizedstringistolabeltheexcitationsoftheindividualmodesas particles.Thislanguagenaturallycarriesovertoanynumberofclassicalsystemsthatexhibitoscillatorybehavior, includingtransversephononsonastring,longitudinalphononsthroughamedium,andphotonsasquantizedoscillations oftheelectromagneticradiation.TheseparticlesareallexamplesofBosons.Bosesystemsmayexhibitalargedegreeof excitation.Intheparticlelanguagethisisequivalenttodiscussingasystemwithalargenumberofparticles.the particlesareidenticle,althoughtheymaybefoundindifferentstates.Accordingly,itisnaturaltodiscussthequantum theoryofidenticalparticlesatthistime.TheconceptofannihilationandcreationoperatorsforBosons,canbeextended todescribeFermisystemsaswell. ü Quantizesimplestring Theenergyforaclassicalstringisgivenbythesumofthekineticandpotentialenergies. E = T + U ho_fs_ident.nb:10/14/04::22:47:34 5 Tobespecific,considerboundaryconditionswherethestringisstretchedbetweentwofixedendpointsatx = 0, L , thestringhasmassdensityr,andtensionk.Letthepositionofstringbey x, t .Then + / + / r L ° 2 T = ÅÅÅÅÅ2 0 y ¼ k L 2 U = ÅÅÅÅ2 0 y' ¼ ü Expansionineigenfunctions Next, expandyintermsofnormalizedeigenfunctionsforthestring. 2 n p x y x, t = S yn t ÅÅÅÅÅ Sin ÅÅÅÅÅÅÅÅÅÅÅ n L L + / + / + / Then,thekineticandpotentialenergytermscanbereexpressedasasumovermodes. L L ° 2 ° 2 n p x ° 2 m p x 0 y = S yn ÅÅÅÅÅL Sin ÅÅÅÅÅÅÅÅÅÅÅL S ym ÅÅÅÅÅL Sin ÅÅÅÅÅÅÅÅÅÅÅÅÅL 0 n m - + /1 - + /1 ¼ à ° ° 2 L n p x m p x = S yn ym ÅÅÅÅÅ Sin ÅÅÅÅÅÅÅÅÅÅÅ Sin ÅÅÅÅÅÅÅÅÅÅÅÅÅ n,m L 0 L L ° ° ¼ + / + / = S yn ym dnm n,m ° 2 = S yn n L L y'2 = S y ÅÅÅÅÅ2 ÅÅÅÅÅÅÅÅn p Cos ÅÅÅÅÅÅÅÅÅÅÅn p x S y ÅÅÅÅÅ2 ÅÅÅÅÅÅÅÅÅm p Cos ÅÅÅÅÅÅÅÅÅÅÅÅÅm p x 0 n L L L m L L L 0 n m - + / + /1 - + / + /1 ¼ à 2 n p m p L n p x m p x = S yn ym ÅÅÅÅÅ ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ Cos ÅÅÅÅÅÅÅÅÅÅÅ Cos ÅÅÅÅÅÅÅÅÅÅÅÅÅ n,m L L L 0 L L n p+ m/ p+ / ¼ + / + / = S yn ym ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ dnm n,m L L n p 2 + 2/ + / = S ÅÅÅÅÅÅÅÅ yn n L + 2 / 2 = S kn yn n ü string=sumoverharmonicoscillators TheseresultscanbeusedtoreexpresstheHamiltonianasasumoverharmonicoscillators. 1 ° 2 2 2 H = ÅÅÅÅ S r yn + k kn yn 2 n wn 2 2 † 1 = S ÅÅÅÅÅÅÅÅ pn + qn = S wn an an + ÅÅÅÅ n 2 n 2 + / + / where ho_fs_ident.nb:10/14/04::22:47:34 6 ° pn = yn r wn r qn = yn t r wn 1r an = ÅÅÅÅÅÅÅÅÅÅ qn + i pn 2 r † 1 + / a = ÅÅÅÅÅÅÅÅÅÅ qn - i pn n 2 r + / wn = r k kn = c kn = w0 n rc p w0 = ÅÅÅÅÅÅÅÅL s ü Fockspace:basisstatesforthestring Fromtheabovediscussion,acontinuousstringcanbedescribedbyalarge(infinite)numberofQMharmonic oscillators,oneharmonicoscillatorforeacheigenfunctiondescribingthestring'smotion.Denotetheoscillatorsbythe subscripti .ThebasisstatesforeachoscillatorcanbechosentobetheeigenstatesofthenumberoperatorNi ,andthe basisstatesforthestringmaybetakentobeaproductoftheindividualoscillatorbasisstates.Astringbasisstatecan thereforebedescribedbyaninfinitedimensionalvectorspecifyingthestateofeachoftheharmonicoscillators. n = n1 ≈ n2 ≈ n3 ≈ ... = n1, n2, n3, ... ¹¶? ? ? ? ? andeachoftheni maybeanon-negativeinteger.Anarbitrarystateofthestring a wouldbegivenbyasuperposition ¶ ¶ ¶ ? a? = S S S ... cn n? n1=0 n2=0 n3=0 ¹¶ ¹¶ wherethecn arecomplexnumbers. ¹¶ Thelowestenergystateofthestringisonewherealltheni are0.Notethatthelowestenergystatehasinfiniteenergy, wi fromsummingupallthezeropointenergiesforeachoftheindividualoscillators,E0 = S ÅÅÅÅÅÅÅ = ¶.Itisconventionalto i 2 ignorethisinfinity,notingthatabsoluteenergyscalesarenotobservable,onlyrelativeenergyscales.Withthis subtraction,theenergyofastringstateis E = n H n = n Si Ni wi n = Si ni wi ;¹¶ « « ¹¶? ;¹¶ « « ¹¶? Thebasisstatesofthestringcanbebuiltupfromthegroundstatebyoperatingwithappropriateraisingoperators.For example, † 1, 0, 0, ... = a1 0, 0, 0. .. ? † ? Theoperatora1 raisestheexcitationlevelofthelowestmodeby1. Itisimportanttodistinguishtheeffectofraisingtheexcitationamplitudeofagivenmode,fromtheeffectofshifting excitationstoahighermode.Inthecontextofaguitarstring,shiftingtoahighermodecorrespondstoahigher frequencyorahigher"note".Increasingtheexcitationofagivenmode,correspondstomakingthatnote"louder".For example,supposethestringisinthen1 = 1state, 1, 0, 0, ... .Theenergyisw0 .Wecanincreasetheenergyto2 w0 nd