[Pekka.]_Advanced_quantum_mechanics(BookFi.org)

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1、1IntroductionStatevectorsSGSternGerlachexperimentinrealityùzBasedonthemeasurementsonecanevaluatethez-componentsSzoftheangularmomentumoftheatomsSBtodetectorand?ndoutthatovenN?fortheupperdistributionSz=ˉh/2.collimator?forthelowerdistributionSz=?ˉh/2.IntheSternGe

2、rlachexperimentInquantummechanicswesaythattheatomsareinthe?silveratomsareheatedinanoven,fromwhichtheyangularmomentumstatesˉh/2and?ˉh/2.escapethroughanarrowslit,Thestatevectorisamathematicaltoolusedtorepresentthestates.Atomsreachingthedetectorcanbedescribed,?th

3、eatomspassthroughacollimatorandenteraninhomogenousmagnetic?eld,weassumethe?eldtoforexample,bytheket-vectors

4、Sz;↑iand

5、Sz;↓i.Associatedwiththeket-vectorstherearedualbra-vectorsbeuniforminthexy-planeandtovaryinthez-direction,hSz;↑

6、andhSz;↓

7、.Statevectorsareassumed

8、?tobeacompletedescriptionofthedescribedsystem,?adetectormeasurestheintensityoftheelectronsemergingfromthemagnetic?eldasafunctionofz.?toformalinear(Hilbert)space,sotheassociatedmathematicsisthetheoryof(in?nitedimensional)Weknowthatlinearspaces.?46ofthe47electro

9、nsofasilveratomformaWhentheatomsleavetheoventhereisnoreasontosphericallysymmetricshellandtheangularexpecttheangularmomentumofeachatomtobemomentumoftheelectronoutsidetheshelliszero,orientedalongthez-axis.Sincethestatevectorsformasothemagneticmomentduetotheorbit

10、almotionoflinearspacealsothesuperpositiontheelectronsiszero,c↑

11、Sz;↑i+c↓

12、Sz;↓i?themagneticmomentofanelectroniscS,whereSisthespinofanelectron,isastatevectorwhichobviouslydescribesanatomwithangularmomentumalongbothpositiveandnegative?thespinsofelectronscancelpair

13、wise,z-axis.ThemagnetintheSternGerlachexperimentcanbe?thusthemagneticmomentμofansilveratomisthoughtasanapparatusmeasuringthez-componentofalmostsolelyduetothespinofasingleelectron,i.e.theangularmomentum.Wesawthataftertheμ=cS,measurementtheatomsareinade?niteangu

14、lar?thepotentialenergyofamagneticmomentinthemomentumstate,i.e.inthemeasurementthestatemagnetic?eldBis?μ·B,sotheforceactinginthec↑

15、Sz;↑i+c↓

16、Sz;↓iz-directiononthesilveratomsis?Bzcoll