Topics in mathematical physics

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1、TopicsinMathematicalPhysics:AJourneyThroughDifferentialEquationsYisongYangPolytechnicInstituteofNewYorkUniversityNonlinearequationsplayimportantrolesinfundamentalscienceandpresentgreatchallengesandopportunitiestoresearchmathematicians.Intheoreticalphysics,manyprofoundconcepts,p

2、redictions,andadvanceswerepioneeredthroughmathematicalinsightsgainedfromthestudyoftheequationsgoverningbasicphysicallaws.Whydosomematerialsdemonstratezeroelectricresistancewhencooled(superconductivity)?HowdothetwostrandsinanentangledDNAdoublehelixbecomeseparatedwhenheated(DNAd

3、enaturation)?Whydoestheuniversehaveafinitepast(bigbangcosmology)?Whyarealltheelectricchargesintegermultiplesofaminimalunit(chargequantization)?Whyisthatthebasicconstituentsofmatterknownasquarkscanneverbefoundinisolation(quarkconfinement)?Thesearesomeoftheexemplarysituationswhere

4、mathematicalinvestigationisessential.Thepurposeofthisseriesoflectures,tobegivenattheAppliedMathematicsGraduateSummerSchoolofBeijingUniversityinJuly2011,istoprovideavista-typeoverviewofabroadrangeofnonlinearequationsarisinginclassicalfieldtheory.Emphasiswillbegiventothemathemati

5、calstructureandphysicaldescriptionsofvariousbasicproblemsandtotheappreciationofthepoweroffunctionalanalysis,althoughcloseattentionwillalsobegiventothelinkswithotherareasofmathematics,includinggeometry,algebra,andtopology.Hereisatableofcontentsofthenotes.Contents1Hamiltoniansys

6、tems31.1Motionofamassiveparticle..............................31.2ThevortexmodelofKirchhoff............................51.3TheN-bodyproblem..................................71.4Hamiltonianfunctionandthermodynamics.....................81.5HamiltonianmodelingofDNAdenaturation....

7、.................102TheSchr¨odingerEquationandQuantumMechanics122.1Pathtoquantummechanics..............................122.2TheSchr¨odingerequation...............................142.3QuantumN-bodyproblem..............................202.4TheHartree–Fockmethod..................

8、.............222.5TheThomas–Fermiapproach....................