[Pekka.]_Advanced_quantum_mechanics(BookFi.org)

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1、1IntroductionStatevectorsSGSternGerlachexperimentinrealityÙzBasedonthemeasurementsonecanevaluatethez-componentsSzoftheangularmomentumoftheatomsSBtodetectorandfindoutthatovenN•fortheupperdistributionSz=¯h/2.collimator•forthelowerdistributionSz=−¯h/2.IntheSternGe

2、rlachexperimentInquantummechanicswesaythattheatomsareinthe•silveratomsareheatedinanoven,fromwhichtheyangularmomentumstates¯h/2and−¯h/2.escapethroughanarrowslit,Thestatevectorisamathematicaltoolusedtorepresentthestates.Atomsreachingthedetectorcanbedescribed,•th

3、eatomspassthroughacollimatorandenteraninhomogenousmagneticfield,weassumethefieldtoforexample,bytheket-vectors

4、Sz;↑iand

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

6、andhSz;↓

7、.Statevectorsareassumed

8、•tobeacompletedescriptionofthedescribedsystem,•adetectormeasurestheintensityoftheelectronsemergingfromthemagneticfieldasafunctionofz.•toformalinear(Hilbert)space,sotheassociatedmathematicsisthetheoryof(infinitedimensional)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,measurementtheatomsareinadefiniteangu

14、lar•thepotentialenergyofamagneticmomentinthemomentumstate,i.e.inthemeasurementthestatemagneticfieldBis−µ·B,sotheforceactinginthec↑

15、Sz;↑i+c↓

16、Sz;↓iz-directiononthesilveratomsis∂Bzcoll