notes on quantum mechanics - k. schulten

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1、NotesonQuantumMechanicsK.SchultenDepartmentofPhysicsandBeckmanInstituteUniversityofIllinoisatUrbana–Champaign405N.MathewsStreet,Urbana,IL61801USA(April18,2000)PrefaceiPrefaceThefollowingnotesintroduceQuantumMechanicsatanadvancedleveladdressingstudentsof

2、Physics,Mathematics,ChemistryandElectricalEngineering.Theaimistoputmathematicalconceptsandtech-niqueslikethepathintegral,algebraictechniques,Liealgebrasandrepresentationtheoryatthereadersdisposal.Forthispurposeweattempttomotivatethevariousphysicalandmat

3、hematicalconceptsaswellasprovidedetailedderivationsandcompletesamplecalculations.Wehavemadeeveryeorttoincludeinthederivationsallassumptionsandallmathematicalstepsimplied,avoidingomissionofsupposedly`trivial'information.Muchoftheauthor'swritingeortwent

4、intoawebofcrossreferencesaccompanyingthemathe-maticalderivationssuchthattheintelligentanddiligentreadershouldbeabletofollowthetextwithrelativeease,inparticular,alsowhenmathematicallydicultmaterialispresented.Infact,theauthor'sdrivingforcehasbeenhisdesi

5、retopavethereader'swayintoterritoriesuncharteredpreviouslyinmostintroduc-torytextbooks,sincefewpractitionersfeelobligedtoeaseaccesstotheireld.AlsotheauthorembracedenthusiasticallythepotentialoftheTEXtypesettinglanguagetoenhancethepresentationofequation

6、sastomakethelogicalpatternbehindthemathematicsastransparentaspossible.Anysuggestiontoimprovethetextintherespectsmentionedaremostwelcome.Itisobvious,thateventhoughthesenotesattempttoservethereaderasmuchaswaspossiblefortheauthor,themaineorttofollowthetex

7、tandtomasterthematerialislefttothereader.ThenotesstartoutinSection1withabriefreviewofClassicalMechanicsintheLagrangeformulationandbuildonthistointroduceinSection2QuantumMechanicsinthecloselyrelatedpathintegralformulation.InSection3theSchr•odingerequatio

8、nisderivedandusedasanalternativedescriptionofcontinuousquantumsystems.Section4isdevotedtoadetailedpresentationoftheharmonicoscillator,introducingalgebraictechniquesandcomparingtheirusewithmoreconventionalmathematicalprocedures.In