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Introduction to Quantum Statistical Mechanics

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Niveau: Supérieur, Doctorat, Bac+8
Introduction to Quantum Statistical Mechanics Alain Joye Institut Fourier, Universite de Grenoble 1, BP 74, 38402 Saint-Martin d'Heres Cedex, France This set of lectures is intended to provide a flavor of the physical ideas un- derlying some of the concepts of Quantum Statistical Mechanics that will be studied in this school devoted to Open Qantum Systems. Although it is quite possible to start with the mathematical definitions of notions such as ”bosons”, ”states”, ”Gibbs prescription” or ”entropy” for example and prove theorems about them, we believe it can be useful to have in mind some of the heuristics that lead to their precise definitions in order to develop some intuition about their properties. Given the width and depth of the topic, we shall only be able to give a very partial account of some of the key notions of Quantum Statistical Mechanics. Moreover, we do not intend to provide proofs of the statements we make about them, nor even to be very precise about the conditions under which these statements hold. The mathematics concerning these notions will come later. We only aim at giving plausibility arguments, borrowed from physical considerations or based on the analysis of simple cases, in order to give substance to the dry definitions. Our only hope is that the mathematically oriented reader will benefit somehow from this informal introduction, and that, at worse, he will not be too confused by the many admittedly hand waving arguments provided.

  • phase space

  • qi ?

  • physical fields

  • ???10 ?e ??

  • ?e

  • scalar potential

  • hamiltonian description

  • c2 ∂2


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IntroductiontoQuantumStatisticalMechanicsAlainJoyeInstitutFourier,Universite´deGrenoble1,BP74,38402Saint-Martind’He`resCedex,Francealain.joye@ujf-grenoble.frThissetoflecturesisintendedtoprovideaflavorofthephysicalideasun-derlyingsomeoftheconceptsofQuantumStatisticalMechanicsthatwillbestudiedinthisschooldevotedtoOpenQantumSystems.Althoughitisquitepossibletostartwiththemathematicaldefinitionsofnotionssuchas”bosons”,”states”,”Gibbsprescription”or”entropy”forexampleandprovetheoremsaboutthem,webelieveitcanbeusefultohaveinmindsomeoftheheuristicsthatleadtotheirprecisedefinitionsinordertodevelopsomeintuitionabouttheirproperties.Giventhewidthanddepthofthetopic,weshallonlybeabletogiveaverypartialaccountofsomeofthekeynotionsofQuantumStatisticalMechanics.Moreover,wedonotintendtoprovideproofsofthestatementswemakeaboutthem,noreventobeverypreciseabouttheconditionsunderwhichthesestatementshold.Themathematicsconcerningthesenotionswillcomelater.Weonlyaimatgivingplausibilityarguments,borrowedfromphysicalconsiderationsorbasedontheanalysisofsimplecases,inordertogivesubstancetothedrydefinitions.Ouronlyhopeisthatthemathematicallyorientedreaderwillbenefitsomehowfromthisinformalintroduction,andthat,atworse,hewillnotbetooconfusedbythemanyadmittedlyhandwavingargumentsprovided.Someofthemanygeneralreferencesregardinganaspectortheotheroftheselecturesareprovidedattheendofthesenotes.1QuantumMechanicsWeprovideinthissectionanintroductiontothequantumdescriptionofaphysicalsystem,startingfromtheHamiltoniandescriptionofClassicalMechanics.ThequantizationprocedureisillustratedforthestandardkineticpluspotentialHamiltonianbymeansoftheusualrecipe.Asetofpostulatesunderlyingthequantumdescriptionofsystemsisintroducedandmotivatedbymeansofthatspecialthoughimportantcase.Theseaspects,andmuchmore,aretreatedinparticularin[GJ]and[MR],forinstance.
2AlainJoye1.1ClassicalMechanicsLetusrecalltheHamiltonianversionofClassicalMechanicsinthefollowingtypicalsetting,neglectingthegeometricalcontentoftheformalism.ConsiderNparticlesinRdofcoordinatesqkRd,massesmkandmomentapkRd,k=1,∙∙∙,N,interactingbymeansofapotentialNdV:RR(1)q7→V(q).ThespaceRdNofthecoordinates(q1,q2,∙∙∙,qN),withqk,jR,j=1∙∙∙,dwhichweshallsometimesdenotecollectivelybyq(andsimilarlyforp),iscalledtheconfigurationspaceandthespaceΓ=RdN×RdN=R2dNofthevariables(q,p)iscalledthephasespace.Apoint(q,p)inphasespacecharacterizesthestateofthesystemandtheobservablesofthesystems,whicharethephysicalquantitiesonecanmeasureonthesystem,aregivenbyfunctionsdefinedonthephasespace.Forexample,thepotentialisanobservable.TheHamiltonianH:ΓRoftheabovesystemisdefinedbytheobservableN2pXH(p,q)=k+V(q1,q2,∙∙∙,qN),(2)k=12mkXwhereV(q1,q2,∙∙∙,qN)=Vij(|qiqj|),j<iwhichcoincideswiththesumofthekineticandpotentialenergies.Theequa-tionsofmotionreadforallk=1,∙∙∙,Nasq˙k=H(q,p),p˙k=H(q,p),with(q(0),p(0))=(q0,p0),(3)∂pk∂qkwhere∂qkdenotesthegradientwithrespecttoqk.Theequations(3)areequivalenttoNewton’sequations,withpk=mkq˙k,mkq¨k=V(q)with(q(0),q˙(0))=(q0,q˙0),qkforallk=1,∙∙∙,N.IncasetheHamiltonianistimeindependent,i.e.ifthepotentialVistimeindependent,thetotalenergyEofthesystemisconservedE=H(q(0),p(0))H(q(t),p(t)),t.(4)where(q(t),p(t))aresolutionsto(3)withinitialconditions(q(0),p(0)).Moregenerally,asystemissaidtobeHamiltonianifitsequationsofmotionsreadas(3)above.WeshallessentiallyonlydealwithsystemsgovernedbyHamiltoniansthataretime-independent.