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ReviewTheone-dimensionaltime-independentSchrodingerequationTheOne-dimensionalSquareWellPotentialTheOne-DimensionalHarmonicOscillatorTheFiniteSquareBarrierUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ReviewTheone-dimensionaltimeChapter4DiracNotationandOperatorTechniquesTheOperationRulesofOperatorTheMomentumOperatorsandTheAngularMomentumOperatorsTheEigenvaluesandEigenfunctionsofHermitionOperationTheRelationbetweenOperationandDynamicalQuantitiesDiracNotation§1§5§2§3§4backUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?Chapter4DiracNotationandOTheDefinitionofOperator
TheCharacterofOperatorTheOperationRulesofOperatorbackUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheDefinitionofOperatorThewhenanoperatoractsonawavefunctionitproducesanewwavefunction?u=voperator?actsonawavefunctionu,itproducesanewwavefunctionv，?isoperator.1）du/dx=v，d/dxisanoperator.2）xu=v，xisalsoanoperator.AnoperatormustactsonawavefunctionTheDefinitionofOperatorbackUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?whenanoperatoractsonawavlinearityOperator?(c1ψ1+c2ψ2)=c1?ψ1+c2?ψ2EqualOperatorIf?ψ=?ψ，So
?=?forarbitraryfunctionsψbackTheCharacterofOperatorPlusOperator(?+?)ψ=?ψ+?ψ=êψ?+?=êforarbitraryfunctionsψUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?linearityOperatorEqualOperaMultiplyOperator?(?
ψ)=(??)ψ=êψ??=êforarbitraryfunctionsψUsually??
≠
???and?donotcommuteis
ThecommutationrelationsbackInfactthemomentumandpositionoperatorsdonotcommuteisattheheartofquantummechanics.thecanonicalcommutationrelationUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?MultiplyOperator?(?ψ)=(Thesecommutationrelationsareattheheartofquantummechanics.backUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?Thesecommutationrelationsarcommutation
bracketcommutation
bracket[?,?]≡??-??1)[?,?]=-[?,?]2)[?,?+ê]=[?,?]+[?,ê]3)[?,?ê]=[?,?]ê+?[?,ê]4)[?,[?,ê]]+[?,[ê,?]]+[ê,[?,?]]=0ThelastequationisJacobiidenticalequation.backUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?commutationbracketcommutationWhen|x|→∞,ψ,→0ψ、φarearbitraryfunctionsTransposeoperatorsbackUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?When|x|→∞,ψ,→0ψ、φTranspoThehermitianconjugateoperatorsTransposeoperatorsAlsowriteThehermitianconjugateoperators?+:(?
?)+=?+
?+
(?
??...)+=...?+
?+
?+backUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ThehermitianconjugateoperatThehermitianoperatorsThehermitianoperators:character?+=?,?+=?(?+?)+=?++?+=(?+?)(??)+=?+?+=??≠
??only[?,?]=0,(??)+=??
backUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ThehermitianoperatorsTheherTheMomentumOperators
ThemomentumoperatorishermitianTheeigenvalueequationofmomentumTheuniverseNormalizableTheAngularMomentumOperators
TheAngularMomentumOperators
TheeigenvalueequationofAngularmomentumThecommutationrelationsofAngularmomentumbackTheMomentumOperatorsandTheAngularMomentumOperatorsUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheMomentumOperatorsbackTheTheMomentumOperatorsThemomentumoperatorishermitianTheeigenvalueequationofmomentumProof：When|x|→∞,ψ,→0UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheMomentumOperatorsThemomeI.SolutionsII.NormalizablecoefficientUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?I.SolutionsII.NormalizablxyzAA’oLTheuniverseNormalizableperiodicityboundaryconditionUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?xyzAA’oLTheuniverseNormalizac=L-3/2，WavefunctionsUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?c=L-3/2，WavefunctionsUniverTheAngularMomentumOperatorsTheAngularMomentumOperatorsThecartesiancoordinatesThesquareofAngularMomentumOperatorsTheclassicalangularmomentumcorrespondingquantumoperatorUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheAngularMomentumOperatorsxzThesphericalcoordinatesystem.ryThesphericalcoordinateUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?xzThesphericalcoordinatesyThecartesiancomponentsoftheangularmomentumvectorintermsofsphericalcoordinatesyieldingUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ThecartesiancomponentsUniverTheeigenvalueequationTheeigenvalueequation:LzNormalizablecoefficientUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheeigenvalueequationTheeigTheeigenvalueTheeigenfunctionsEqualzeroPeriodicityboundaryconditionUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheeigenvalueEqualzeroPeriodTheeigenvalueequation:
L2=（+1),where=0,1,2,...Solution:Ylm(,)UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?Theeigenvalueequation:L2=Theorthonormalcondition：Thedegeneration
oftheeigenvalue：m:0,±1,±2,±3,...,±(2+1)UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?Theorthonormalcondition：TheThecommutationrelationsofAngularmomentumProofUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ThecommutationrelationsofAHomeworksChapter4P127:6,8,
UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?HomeworksUniversityofElectr
AveragevalueofthehermitianoperatorsTheeigenvalueequationofthehermitianoperatorsTheorthogonalityoftheeigenfunctionsforhermitianoperatorsExampleTheEigenvaluesandEigenfunctionsofHermitionOperationbackUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?AveragevalueofthehermitiaThI：Averagevalueofthehermitianoperatorsmustbereal.Proof：AveragevalueofthehermitianoperatorsUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ThI：AveragevalueofthehermihermitianoperatorsMustberealTheeigenvalueequationofProof：TheeigenvalueequationofthehermitianoperatorshermitianoperatorsUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?hermitianoperatorsMustbereaThII：TheEigenvaluesofHermitionOperationarealltoberealProofAccordingtheoremI(I)
Allphysicallyobservablequantitiesareassociatedwithhermitianoperators.(II)
Whenthestateisaneigenstateofanoperator,measurementofitsassociatedobservablemustyieldtheeigenvaluebelongingtothateigenstate.UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ThII：TheEigenvaluesofHermiTHIII：Theeigenfunctionsareorthonormal,iftheeigenvaluesarenotdegenerate.Proof:Fm≠Fn，1.Theorthonormalconditionforquantumset：2.Theorthonormalconditionforcontinue：TheorthogonalityoftheeigenfunctionsforhermitianoperatorsUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?THIII：TheeigenfunctionsareTheexpectation
valuesofdynamicalquantitiesTheaverageofdynamicalquantities（1）Theeigenfunctionsofdynamicalquantitiesoperationformacompleteset.（2）Theexpectationvalueandprobabilityofthedynamicalquantities（3）
Theconditionfordynamicalquantitieshavecertainvalue
TheRelationbetweenOperationandDynamicalQuantitiesbackExampleUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?Theexpectationvaluesofdyna(1)Theeigenfunctionsofdynamicalquantitiesoperationformacompleteset.Example：Theeigenfunctionsofmomentumoperationformacompleteset.Theexpectation
valuesofdynamicalquantitiesUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?(1)Theeigenfunctionsofdyna（2）TheexpectationvalueandprobabilityofthedynamicalquantitiesUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?（2）TheexpectationvalueandTheconditionfordynamicalquantitieshavecertainvalueλ=λm
λm，|cm|2=1,|c1|2=|c2|2=...=0UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheconditionfordynamicalquTheaverageofdynamicalquantitiesUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheaverageofdynamicalquantExample1：find：（1）IsΨtheeigenstateofL2ornot？ （2）IsΨtheeigenstateofLz
ornot
？ （3）TheaverageofL2
； （4）TheexpectationvalueandprobabilityofL2andLz
Solution:UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?Example1：find：（1）IsΨthee（3）TheaverageofL2IUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?（3）TheaverageofL2IUniversiII（4）UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?II（4）UniversityofElectronicExample2：Whent=0，(x)=A[sin2kx+(1/2)coskx]
Findtheaverageofkineticenergyandmomentum.Solution:UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?Example2：Whent=0，(x)=A[UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?UniversityofElectronicScien1）theaverageofmomentum2）theaverageofkineticenergyUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?1）theaverageofmomentum2）tHomeworksChapter4P128:11,12,
UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?HomeworksUniversityofElectrThebasicsimplificationoftheDiracnotationistointroduceanabstractstatethatisaneigenvectorofthepositionoperator.Thatiswherethestate︱x〉iscalledaketanditsconjugateiscalledabra.WerequirethatDiracNotationUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ThebasicsimplificationofthWecannowdefineawavefunctiontobeUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?WecannowdefineawavefunctReviewTheone-dimensionaltime-independentSchrodingerequationTheOne-dimensionalSquareWellPotentialTheOne-DimensionalHarmonicOscillatorTheFiniteSquareBarrierUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ReviewTheone-dimensionaltimeChapter4DiracNotationandOperatorTechniquesTheOperationRulesofOperatorTheMomentumOperatorsandTheAngularMomentumOperatorsTheEigenvaluesandEigenfunctionsofHermitionOperationTheRelationbetweenOperationandDynamicalQuantitiesDiracNotation§1§5§2§3§4backUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?Chapter4DiracNotationandOTheDefinitionofOperator
TheCharacterofOperatorTheOperationRulesofOperatorbackUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheDefinitionofOperatorThewhenanoperatoractsonawavefunctionitproducesanewwavefunction?u=voperator?actsonawavefunctionu,itproducesanewwavefunctionv，?isoperator.1）du/dx=v，d/dxisanoperator.2）xu=v，xisalsoanoperator.AnoperatormustactsonawavefunctionTheDefinitionofOperatorbackUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?whenanoperatoractsonawavlinearityOperator?(c1ψ1+c2ψ2)=c1?ψ1+c2?ψ2EqualOperatorIf?ψ=?ψ，So
?=?forarbitraryfunctionsψbackTheCharacterofOperatorPlusOperator(?+?)ψ=?ψ+?ψ=êψ?+?=êforarbitraryfunctionsψUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?linearityOperatorEqualOperaMultiplyOperator?(?
ψ)=(??)ψ=êψ??=êforarbitraryfunctionsψUsually??
≠
???and?donotcommuteis
ThecommutationrelationsbackInfactthemomentumandpositionoperatorsdonotcommuteisattheheartofquantummechanics.thecanonicalcommutationrelationUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?MultiplyOperator?(?ψ)=(Thesecommutationrelationsareattheheartofquantummechanics.backUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?Thesecommutationrelationsarcommutation
bracketcommutation
bracket[?,?]≡??-??1)[?,?]=-[?,?]2)[?,?+ê]=[?,?]+[?,ê]3)[?,?ê]=[?,?]ê+?[?,ê]4)[?,[?,ê]]+[?,[ê,?]]+[ê,[?,?]]=0ThelastequationisJacobiidenticalequation.backUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?commutationbracketcommutationWhen|x|→∞,ψ,→0ψ、φarearbitraryfunctionsTransposeoperatorsbackUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?When|x|→∞,ψ,→0ψ、φTranspoThehermitianconjugateoperatorsTransposeoperatorsAlsowriteThehermitianconjugateoperators?+:(?
?)+=?+
?+
(?
??...)+=...?+
?+
?+backUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ThehermitianconjugateoperatThehermitianoperatorsThehermitianoperators:character?+=?,?+=?(?+?)+=?++?+=(?+?)(??)+=?+?+=??≠
??only[?,?]=0,(??)+=??
backUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ThehermitianoperatorsTheherTheMomentumOperators
ThemomentumoperatorishermitianTheeigenvalueequationofmomentumTheuniverseNormalizableTheAngularMomentumOperators
TheAngularMomentumOperators
TheeigenvalueequationofAngularmomentumThecommutationrelationsofAngularmomentumbackTheMomentumOperatorsandTheAngularMomentumOperatorsUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheMomentumOperatorsbackTheTheMomentumOperatorsThemomentumoperatorishermitianTheeigenvalueequationofmomentumProof：When|x|→∞,ψ,→0UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheMomentumOperatorsThemomeI.SolutionsII.NormalizablecoefficientUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?I.SolutionsII.NormalizablxyzAA’oLTheuniverseNormalizableperiodicityboundaryconditionUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?xyzAA’oLTheuniverseNormalizac=L-3/2，WavefunctionsUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?c=L-3/2，WavefunctionsUniverTheAngularMomentumOperatorsTheAngularMomentumOperatorsThecartesiancoordinatesThesquareofAngularMomentumOperatorsTheclassicalangularmomentumcorrespondingquantumoperatorUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheAngularMomentumOperatorsxzThesphericalcoordinatesystem.ryThesphericalcoordinateUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?xzThesphericalcoordinatesyThecartesiancomponentsoftheangularmomentumvectorintermsofsphericalcoordinatesyieldingUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ThecartesiancomponentsUniverTheeigenvalueequationTheeigenvalueequation:LzNormalizablecoefficientUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheeigenvalueequationTheeigTheeigenvalueTheeigenfunctionsEqualzeroPeriodicityboundaryconditionUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?TheeigenvalueEqualzeroPeriodTheeigenvalueequation:
L2=（+1),where=0,1,2,...Solution:Ylm(,)UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?Theeigenvalueequation:L2=Theorthonormalcondition：Thedegeneration
oftheeigenvalue：m:0,±1,±2,±3,...,±(2+1)UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?Theorthonormalcondition：TheThecommutationrelationsofAngularmomentumProofUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ThecommutationrelationsofAHomeworksChapter4P127:6,8,
UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?HomeworksUniversityofElectr
AveragevalueofthehermitianoperatorsTheeigenvalueequationofthehermitianoperatorsTheorthogonalityoftheeigenfunctionsforhermitianoperatorsExampleTheEigenvaluesandEigenfunctionsofHermitionOperationbackUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?AveragevalueofthehermitiaThI：Averagevalueofthehermitianoperatorsmustbereal.Proof：AveragevalueofthehermitianoperatorsUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ThI：AveragevalueofthehermihermitianoperatorsMustberealTheeigenvalueequationofProof：TheeigenvalueequationofthehermitianoperatorshermitianoperatorsUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?hermitianoperatorsMustbereaThII：TheEigenvaluesofHermitionOperationarealltoberealProofAccordingtheoremI(I)
Allphysicallyobservablequantitiesareassociatedwithhermitianoperators.(II)
Whenthestateisaneigenstateofanoperator,measurementofitsassociatedobservablemustyieldtheeigenvaluebelongingtothateigenstate.UniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?ThII：TheEigenvaluesofHermiTHIII：Theeigenfunctionsareorthonormal,iftheeigenvaluesarenotdegenerate.Proof:Fm≠Fn，1.Theorthonormalconditionforquantumset：2.Theorthonormalconditionforcontinue：TheorthogonalityoftheeigenfunctionsforhermitianoperatorsUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?THIII：TheeigenfunctionsareTheexpectation
valuesofdynamicalquantitiesTheaverageofdynamicalquantities（1）Theeigenfunctionsofdynamicalquantitiesoperationformacompleteset.（2）Theexpectationvalueandprobabilityofthedynamicalquantities（3）
Theconditionfordynamicalquantitieshavecertainvalue
TheRelationbetweenOperationandDynamicalQuantitiesbackExampleUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?Theexpectationvaluesofdyna(1)Theeigenfunctionsofdynamicalquantitiesoperationformacompleteset.Example：Theeigenfunctionsofmomentumoperationformacompleteset.Theexpectation
valuesofdynamicalquantitiesUniversityofElectronicScienceandTechnologyofChina2005-3-1Prof.ZhangXiaoxia?(1)Theeigenfunctionsofdyna（2）TheexpectationvalueandprobabilityofthedynamicalquantitiesUniversityofElectronicScienceandTechnologyofChina2005-3-1Pr
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