大二下专业课量子力学lecture10.pptx

IntroductoryQuantumMechanicsforMaterialsScientists

Spring,2013BinShanSchoolofMaterialsScienceandEngineeringHuazhongUniversityofScienceTechnology1

2Whatdidwelearnlasttime?Whatwewilllearntoday?OperatorsandMatricesAdjointoperatorHermitianoperatorHarmonicOscillatorOperatordefinitionBasicmathofoperators

3英文词汇表English中文Operator算符LinearOperator线性算符IdentityOperator单位算符Observable力学量Commutator对易Adjointoperator共轭算符Hermitianadjoint厄密共轭Compatibleoperator可对易的算符Uncertaintyprinciple不确定性原理

4Operators=Matrices?

5OperatorsandMatrixInquantummechanics,asystemisalwayspresumedtobeinsome“state”.Thestatemaybedescribedbysomesetofnumbersc1,c2,...(Thevectorforourwavefunctionwouldhaveaninfinitenumberofelements,oneforeachpointinspace,thoughwewillignorethemathematicalcomplicationsofthisinfinity,andthinkofthewavefunctionasafinitebutverylargevector.)

6Aconvenientwayofthinkingofanoperator(andonethatdefinesitasa“linear”operator)istothinkoftheoperatorasamatrixMthatmultipliesthevector.Inthisway,theresultdependspossiblyonallofthevaluesinthevectorthatdescribethestate.TransformsthecurrentvectortoanothervectorWhenthevectorisan“eigenvector”ofthematrix(orthestateisan“eigenstate”oftheoperator),wearesayingthatwehavetheremarkablesituationthatthisprocessresultsinavectorofexactlythesameformaswhenwestarted,multipliedonlybyaconstantthatisthe“eigenvalue”

7OperatorsInQuantumMechanics

8MathematicsofLinearOperatorsFunctionsofoperatorsForfunctionsofoperators,wefirstuseTaylorexpansion,andthensubstiuteexpansionvariablewiththecorrespondingoperatorForexample:

9IdentityOperatorsIdentityoperatorAveryimportantequationforIdentityoperatorCompletebasissets

10PositionOperatorEigenstates:

11MomentumOperatorEigenstates:

CommutatorCalculations

13Proof-Example

14OtherExamples

15Adjoin