_Brouwer=theory_of_many_particle_systems

《_Brouwer=theory_of_many_particle_systems》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、TheoryofMany-ParticleSystemsLecturenotesforP654,CornellUniversity,spring2005.cPietBrouwer,2005.Permissionisgrantedtoprintandcopythesenotes,ifkepttogetherwiththetitlepageandthiscopyrightnotice.Chapter1Preliminaries1.1ThermalaverageIntheliterature,twotypesoftheoriesareconst

2、ructedtodescribeasystemofmanyparticles:Theoriesofthemany-particlegroundstate,andtheoriesthataddressathermalaverageattemperatureT.Inthiscourse,we'lllimitourselvestonite-temperaturethermalequilibriaandtononequilibriumsituationsthatarisefromanite-temperatureequilibriumbysw

3、itch-ingonatime-dependentperturbationintheHamiltonian.Inallcasesofinteresttous,resultsatzerotemperaturecanbeobtainedasthezerotemperaturelimitofthenite-temperaturetheory.Below,webrie yrecallthebasicresultsofequilibriumstatisticalmechanics.ThethermalaverageofanobservableAi

4、sdenedashAi=trA^;^(1.1)where^isthedensitymatrixandA^istheoperatorcorrespondingtotheobservableA.Thedensitymatrix^describesthethermaldistributionoverthedierenteigenstatesofthesystem.Thesymboltrdenotesthetrace"ofanoperator,thesummationovertheexpectationvaluesoftheopera

5、toroveranorthonormalbasisset,XtrA=hnjAjni:(1.2)nThebasissetfjnigcanbethecollectionofmany-particleeigenstatesoftheHamiltonianH^,oranyotherorthonormalbasisset.Inthecanonicalensembleofstatisticalmechanics,thetraceistakenoverallstateswithNparticlesandonehas1H^=TH^=T^=e;Z

6、=tre;(1.3)Z12CHAPTER1.PRELIMINARIESwhereH^istheHamiltonian.UsingthebasisofN-particleeigenstatesjnioftheHamiltonianH^,witheigenvaluesEn,thethermalaveragehAicanthenbewrittenasPeEn=ThnjA^jnihAi=nP:(1.4)eEn=TnInthegrand-canonicalensemble,thetraceistakenoverallstates,irrespe

7、ctiveofparticlenumber,andonehas1(H^N^)=T(H^N^)=T=e;Z=tre:(1.5)ZHereisthechemicalpotentialandNistheparticlenumberoperator.UsuallywewillincludethetermN^intothedenitionoftheHamiltonianH^,sothatexpressionsforthethermalaverageinthecanonicalandgrandcanonicalensembles

8、areformallyidentical.1.2Schr•odinger,Heisenberg,andinteractionpictureThereexistformallydierentb