MeasurementandtheInterpretationofQMVIII.pdf

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1、MeasurementandtheInterpretationofQuantumMechanicsMinHuang,PhDCHE@TongjiUniv.•Everydescriptionofanexperimentonamicroscopicsystem(evensinglemoleculespectroscopy)isessentiallystatistical.•Typicallyoneperformsanexperimentonasampleconsistingofsimilarmicrosc

2、opicsystems.•suchanexperimentasequivalenttoperformingasequenceofmeasurementsoneach(nowsupposedlyidentical)microscopicsysteminisolation.•Thisgeneratesadefiniteresultforeachindividualexperiment,andthestatisticsofthedistributionofresultsisdescribedbyquant

3、ummechanics.•quantummechanicsdescribesthestatisticaloutcomeofanexperimentperformedonanensembleofidenticalmicrosystems•Andthewavefunctioninquantummechanicsdescribesthepropertiesoftheensemble.•itwill,ofcause,involvesomeinteractionbetweendifferentmicrosys

4、temsorbetweenmicrosystemsandtheenvironment,andsoforth.However,thisisbeyondthescopeofthislecturenote.•Thisistheso-calledensembleinterpretationofquantummechanics.MeasurementsinQuantumMechanics•AssumethatifwemakerepeatedmeasurementsofthequantityAonagivene

5、nsemble,wehavepossibleoutcomesa1,a2,...,.•TheprobabilityP(ai)tofindthevalueaiforthegivenensembleisNi/Ntot,whereNiisthenumberoftimesweobtainaioutofatotalnumberofmeasurementsNtot.•Inthelimitofaverylargesetofmeasurementstheseprobabilitieswillconvergetodef

6、initenumbers0≤P(ai)≤1.•themostlikelyoutcomeofjustonemeasurement，NiApaiaiaiiiNtot•whereAisarandomvariable,aiisthepossiblediscretevaluesofA,Ntotisthenumberofmeasurements(Ntot»1)andNiisthenumberoftimestheoutcomeaiisfound.Probabilityisnormalized,ther

7、efore,1paa1iiiNtoti•Whentheensembleishuge,onthecontinuumlimit,dpAdA•whichistheprobabilitythattheresultofonemeasurementliesbetweenAandA+dA.(A)istheprobabilitydensity,whichisnormalized,AdA1•Themomentsofthedistributionis,AnAnAdA

8、•firstmoment=expectationvalueofA,AAAdA•secondmomentdeterminesthestandarddeviationsofthedistribution,•If,thenAAA222sAAA•scontrolsthespreadofthemeasurementsdata.•InQuantumMechanics,strictrulesmustbegivenfordiscussingme