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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
