geometrical domain of spin 12 probability mass functionnew

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1、NonamemanuscriptNo.(willbeinsertedbytheeditor)KarthikBharath·SwarnamalaSirsi·A.R.UshaDeviGeometricalDomainofSpin-1/2ProbabilityMassFunctionAbstractThequantumanalogueoftheclassicalcharacteristicfunctionforaspin1/2assemblyisconsideredandtheprobabilitymassfunctionoftherandomvectorassociatedwiththeass

2、emblyisderived.ItisseenthatthepositiveregionsofWignerandMargenau-Hillquasidistributionsforthethreecomponentsofspin,correspondtoatrivariateprobabilitymassfunction.WeidentifythedomainofthesepositiveregionsasanOctahedroninscribedintheBlochspherewithitsverticesonthesurfaceofthesphere.Itisinthisdomaint

3、hataquantumcharacteristicfunctioncharacterizingthequasidistribution,admitsaprobabilitymassfunctioninIR3.ItisalsoobservedthattheclassicalvariatesX1,X2,X3correspondingtothe3spinoperatorsσ1,σ2,σ3inthedomain,areindependentifftheBlochvectorliesonanyoneoftheaxes.KeywordsSpin-1/2statisticalassembly;Quasip

4、robabilitydistributionPACS03.65.-w1INTRODUCTIONThestudyofquasiprobabilitydistributionsasquantumanaloguestophasespacedistributionsinquantummechanicalsystemshasbeenwelldocumented[1].Thesedistributionshaveservedasaveryhelpfultoolinanalyzingthestatisticaldistributionsofspinningsubmicroscopicparticlesl

5、ikeelectrons,protonsetcintovariousquantummechanicalstatesandalsoinprovidinginsightsintotheconnectionsarXiv:0711.3617v1[quant-ph]22Nov2007betweenclassicalandquantummechanics.TheWignerfunctiongivesthedescriptionofaquantumstateintermsofageneralizedfunctionwhichactsaquantumanaloguetoprobabilitydistrib

6、utionsinparticlephasespace[2].Itwasobservedthatthisfunctionassumednegativevaluesinsomedomainsofthephasespaceforcertainquantumstates.TheWignerfunctionishenceaquasiprobabilitydis-tributionforthepositionandmomentumofamechanicalsystemdeterminingthequantumstateˆρ.TheSudarshan-Glauber’sPdistribution[3][

7、4]andtheHusimi-Kano’sQdistribution[5][6],adoptingadensitymatrixrepresentationofthequantumstate,haveattemptedtoobtaindistributionswhichcorrespondtopositive,normalizedclassicaldistributionfunctionsinphasespace.Butt