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208 CHAPTER13. ELECTRONSPIN

sothez-componentoftheforceis

Fz =

∂Bz

∂z

μz

=

eg

2 Mc

∂Bz

∂z

Sz (13.20)

Ameasurementof the deflection of the electron beam istherefore going to be a
measurement of thez-componentof electron spin. If s=^12 , thenthereareonly
twopossibilities: sz=±^12 ,andtheforceisorientedeitherupordownalongthez-
axis,withequalmagnitude.Eachelectronshouldbedeflectedalongoneofonlytwo
trajectories,asshowninFig.[13.1].Thisis,infactwhatisfoundexperimentally. The
Stern-Gerlachapparatusisthereforeawayofmeasuringthecomponentofelectron
spinalongagivenaxis,foranygivenelectron. Ifdeflectedupwards,theelectronis
inanSz=+^12 ̄h(or”spinup”)eigenstate.Iftheelectronisdeflecteddownwards,it
isinanSz=−^12 ̄h(or”spindown”)eigenstate.
Asalreadymentioned,anyspinningchargedobjecthasamagneticmoment,ac-
cordingtothelawsofclassicalelectromagnetism. Shouldwethenthinkoftheelectron
asbeingliterallyaspinningballofcharge? Therearetwomainproblemswithsucha
picture.First,ifweimaginethattheelectroniscomposedofsomesubstancewitha
constantcharge-to-massratioofe/M,then,generalizingonlyslightlythereasoning
thatappliestoacurrentloop,theelectronshouldhaveamagneticmoment

%μ=

e

2 Mc

S% (13.21)

insteadofwhatisactuallyfound,whichisabouttwicethatvalue

%μ=

e

2 Mc

gS% (13.22)

withgyromagneticratiog≈2. Thesecondreasonissimply thefactthat,inthis
picture,theelectronspinangularmomentumisjusttheorbitalangularmomentum
ofelectron”substance.”But,aswehavealreadyseen,orbitalangularmomentumcan
onlyhavevaluesl= 0 , 1 , 2 ,...;inparticular,l=^12 isruledout. Sothepictureofan
electronasaspinningvolumeof charged ”electronstuff”cannot easilyaccountfor
theoriginoftheelectronmagneticmoment.

13.1 Spin Wavefunctions

Thenextquestionishowtowritethewavefunctionofaneigenstateofspinangular
momentum.Iftheelectronwereaspinningballofcharge,thenitwouldbenaturalto
representthespinwavefunctionbysphericalharmonicsYssz(θ′,φ′),where∂tθ′isthe
classicalangularvelocityofthespinningball. Aswehaveseen,the”ballofcharge”