第七章 谐振子.ppt

Chapter7TheHarmonicOscillator 谐振子 Manycomplicatedpotentialcanbeapproximatedinthevicinityoftheirequilibriumpointsbyaharmonicoscillator TheTaylorexpansionofV x atequilibriumpointx ais Hamitonnianfunctionofanoscillatorwithmassmandoscillatingfrequency 0canbewritten StationarySchrodingerequation Referencingthebookeditedby曾谨言 wesolvetheSchrodingerequation Introducetheno dimensionparameters 无量纲参数 Weget boundarycondition 1 2 Wegetanasymptoticsolution 试探解 Insert 2 to 1 get ThisisHermite 厄米 differentialequation Atthevicinityof 0 u isexpandedtheTaylorseries Only willsatisfiestheboundarycondition 4 Thereforethecondition 4 issatisfied wecangetthesolutionwhichisallowedinphysicsfield Accordingto 3 Energyeigenvalueofharmonicoscillator Energylevelisdiscrete Theenergygapisidentical Theenergylevelofgroundstate zeropointenergy isnotzero Thesolutionofequation 3 isHermitepolynomials 厄米多项式 Theeigenfuctionandenergyofharmonicoscillatorare Normalizedconstant SomemostsimpleHermitepolynomials H0 1 H1 2 H2 4 2 2 H3 8 3 12 ThebasicpropertiesofHermitepolynomials Thedefinition Twoimportantandusefulrelations n 0 n 1 n 2 Thefirstthreeeigenfunctionsofharmonicoscillator Thesymmetryproperty Whenniseven positiveparity n为偶数 偶宇称 Whennisodd negativeparity Ingeneral Groundstate Theenergyandwavefunctionofgroundstate n 0 Theprobabilityfindingaparticleatx 0ismaximum whichiscontrarytoclassicalparticle Foraclassicalharmonicoscillator whenx 0 itspotentialisminimumandkineticenergyismaximum hencetheintervalwhichitdelaysatx 0isshortest Inclassicalmechanics aparticlewithgroundstateenergyE0motionsintherange Accordingtoquantummechanics theprobabilityfindingaparticleoutsidetheclassicalallowedrangeis Zeropointenergyisadirectconsequenceoftheuncertaintyrelation Sincetheintegrand 被积函数 isanoddfunction Wecanwriteuncertaintyrelationagain Themeanenergy Theminimumenergyiszeropointenergy whichiscompatiblewithuncertaintyprinciple Thenormalizationeigenfunctionofharmonicoscillator Accordingtotheserelations weget ThedescriptionoftheHarmonicOscillatorbyCreationandAnnihilationoperators 产生算符和湮灭算符 Hence 1 2 Byadditionorsubtractionof 1 and 2 weget Wedefinetheoperators Hence iscalledtheloweringoperator 降幂算符 theraisingoperator 升幂算符 Thenumberoperator 数算符 Bysuccessivelyoperator on wecancalculatealltheeigenfunctions staringfromthegroundstate Forn 0 Theeigenfunctionofgroundstate Thenormalizedeigenfunction One dimensionHamiltonianharmonicoscillator Weintroduce Hence 3 RepresentationoftheOscillatorHamiltonianinTermsof and Accordingtothedefinitionsof and get WeobtainasimpleHamiltonianrepresentation Eigenvalue 基态 0所具有的零点能量为 2 而且我们知道谐振子的能量是等间隔的 n所具有的能量大于n 我们将该能量以能量量子 分成n份 谐振子场中的量子 称为声子 phonons 那么将 n称为n声子态 n phononstate 在Dirac s表象中表示为 表示声子数 零声子态 zero phononstate 称为真空 应用上面的表述 算符 和 作用于波函数可表示成 解释 如果 作用于波函数 则湮灭 annihilate 了一个声子 因而称为 湮灭算符 作用于函数 则产生一个声子 产生算符 4 Interpretationof and 由于 称为声子数算符 phononnumberoperator n为相应态的子数 声子表象的引入被称为二次量子化 而谐振子波场中的量子正是声子 如果与光子相类比的话 就更清楚了 Example1 UsingtherecursionofHermitepolynomials Provethefollowingexpressions Andaccordingtothese prove Solution n x istheeigenfunctionofharmonicoscillator andcanbewritten Hamitonnianofthecouplingharmonicoscillatorcanbewritten Example2 where x1 p1andx2 p2belongtodifferentfreedomdegree andset Problem theenergylevelofthiscouplingharmonicoscillator Solution ifthecouplingterm x1x2isnotexists thecouplingharmonicoscillatorbecomestwo dimensionoscillator andthenitsHamitanianisgivenby Usingseparatingvariable wecantransformtheabovequestionintothequestionoftwoindependentone dimensionharmonicoscillator thenitsenergylevelandeigenfunctionare where isenergyeigenfunctionofone dimensionoscillator Forthecouplingharmonicoscillator wecansimplifyittwoindependentharmonicoscillatorusingcoordinatetransformation soweset Wecaneasilyprovethefollowingexpressions ThereforeHamitanianbecomes Where Hence 1 UsingtherecursionofHermitepolynomials Provethefollowingexpressions Andaccordingtothese prove Exercise where 2 Aparticleisinthegroundstateofone dime