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1、DensityOperatorinQuantumStatisticalMechanicsIgorKhavkineOctober11,20021EnsemblesOneofthefundamentalnotionsinstatisticalmechanicsisthatofanensemble.Wewillbrieflyreviewithere.Thissectioncanbeskippedifonefeelsfamiliarenoughwiththisconcept.Inclassicalmechanicswecanpreciselyspecifythe
2、stateofasystembyspecifyingthepositionqandmomentumpforeachparticle.Thespaceofallsuchconfigurationsiscalledthephasespace.Inquantummechanicswecanpreciselydescribethestateofasystembyspecifyingitsstatevector
3、ψi,oritswavefunctionψ(x)(notethatpreciselyspecifyingaquantumstateisnotthesame
4、aspreciselyspecifyingthevaluesofallitsphysicalobservables).Thespaceofallpossiblequantumstatesiscalledthesystem’sHilbertspace.Oncethestateofthesystemispreciselydefined,wecanusetheequationsofmotionprovidedbyeachtheorytopreciselypredictthestateofthesystematanyfuturetime.Forclassical
5、mechanics,theequationsofmotionaregivenbyNewton’slawsorbyHamilton’sequations.Forquantummechanics,theequationsofmotionaregivenbytheSchr¨odingerequation.Itisimportanttonotethatstatisticalmechanicsandensemblescanbeusedwithanytheoryofmechanics(beitclassical,quantum,orother)aslongasth
6、eprecisestatesandthedynamicsofthistheoryarewellunderstood.However,inmostcasesitisnotpossibletoknowtheprecisestate(ormicro-state)ofasystem.Wemayknow,forexample,somegrossoraveragepropertiesofthesystemthatmaycorrespondtomorethanonepossibleprecisestate.Sogivensomeimprecisespecificati
7、onofthestateofthesystem(oritsmacro-state),thesetofallprecisestatesthatcorrespondtoit(orthesetofallaccessiblestates)iscalledtheensemblerepresentingtheimprecisestateofthesystem.Theevolutionofeachprecisestateintheensemblecanthenbepredictedandsuitablemeansofmeasurementsoneachelement
8、oftheensemble(e.g.averageormostprobablevalue)canbetakentocorrespondtoexperimentalmeasurementsonthesystemrepresentedbytheensemble.Forexample,giventhatwewanttopredictthemeasurementofaphysicalobservableArepresentedbyagivenensemble,wecanusetheensembleaverageAtopredicttheoutcome:XA=P
9、sA(s),(1)swherethesumisoverallstatessthatareele
