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”