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1、Heisenberg’suncertaintyrelation:ViolationandreformulationMasanaoOzawaGraduateSchoolofInformationScience,NagoyaUniversity,Chikusa-ku,Nagoya,464-8601,JapanE-mail:ozawa@is.nagoya-u.ac.jpAbstract.TheuncertaintyrelationformulatedbyHeisenbergin1927describesatrad
2、e-offbetweentheerrorofameasurementofoneobservableandthedisturbancecausedonanothercomplementaryobservablesothattheirproductshouldbenolessthanalimitsetbyPlanck’sconstant.In1980,Braginsky,Vorontsov,andThorneclaimedthatthisrelationleadstoasensitivitylimitforgra
3、vitationalwavedetectors.However,in1988amodelofpositionmeasurementwasconstructedthatbreaksboththislimitandHeisenberg’srelation.Here,wediscusstheproblemsastohowwereformulateHeisenberg’srelationtobeuniversallyvalidandhowweexperimentallyquantifytheerrorandthed
4、isturbancetorefutetheoldrelationandtoconfirmthenewrelation.1.Heisenberg’sEDRThediscoveryofquantummechanicsintroducednon-commutativityinalgebraiccalculusofobservables;thecanonicalcommutationrelation(CCR)[Q,P]=i¯h(1)isrequiredtoholdbetweenacoordinateQofaparti
5、cleanditsmomentumP,wherethecommutator[Q,P]isdefinedby[Q,P]=QP−PQ.In1927,Heisenbergproposedanoperationalmeaningofthenon-commutativity:“themorepreciselythepositionisdetermined,thelesspreciselythemomentumisknown,andconversely”[1,p.64].@arXiv:1402.5601v1[quant-
6、ph]23Feb2014Bythefamousγraymicroscopethoughtexperimenthederivedtherelation¯hε(Q)η(P)≥,(2)2whereε(Q)isthe“meanerror”ofapositionmeasurementandη(P)isthetherebycaused“discontinuouschange”inthemomentumP:Letε(Q)betheprecisionwithwhichthevalueQisknown(ε(Q)is,say,
7、themeanerrorofQ),thereforeherethewavelengthofthelight.Letη(P)betheprecisionwithwhichthevaluePisdeterminable;thatis,here,thediscontinuouschangeofPintheComptoneffect[1,p.64].Here,“meanerror”isnaturallyunderstoodtobe“root-mean-square(rms)error”asintroducedbyGa
8、uss[2],and“discontinuouschange”isoftencalled“meandisturbance.”HeisenbergclaimedthatEq.(2)isa“straightforwardmathematicalconsequence”ofEq.(1)[1,p.65]andgaveitsmathematicaljustification[1,p.69].2.Heisenberg’sder
