180 CHAPTER11. ANGULARMOMENTUM
Confirmationof theseresultscomesfromNature. Allelementaryparticles, for
example, have an intrinsic spinangular momentum. The magnitudeof this spin
angularmomentumdependsonthetypeofelementaryparticle,butitisalwaysone
ofthevalues(11.52). Theπ-mesonhasspin0,correspondingtol=0. Electrons,
protons,neutrons,andquarkshave”spin^12 ”,i.e.theirtotalspinangularmomentum
hasamagnitude
|S|=√
3
4̄h (11.54)correspondingtol=^12 .Theρandωmesonshave”spin1”,i.e.
|S|=√
2 ̄h (11.55)corresponding to l = 1. The∆-hyperon has”spin^32 ”(l =^32 ), andso on. The
componentofspinangularmomentumalongaparticularaxiscanalsobemeasured
experimentallyforthoseparticleswhichhaveamagneticmoment.Thecomponentof
magneticmomentalongagivenaxisisproportionaltothespinangularmomentum
along that axis. Experimentally, the values of the componentSz along,e.g., the
z-axis, obeythe relation(11.53). Thereare,however, some importantdifferences
betweenintrinsicspinangularmomentum(whosemagnitudecanneverchangefora
giventypeofparticle)andorbitalangularmomentum,whichcanchangebydiscrete
amounts. Wewillreservefurtherdiscussionofspinangularmomentumforlaterin
thecourse.
Apartfromthefactthatboththetotalmagnitude|L|andcomponentLzcome
indiscreteamounts(11.51),angularmomentuminquantummechanicsdiffersfrom
angularmomentuminclassicalmechanicsinseveralways.Ofthese,themoststriking
isthefactthattheangularmomentuminthez-direction(orinanyotherdirection)is
alwayssmallerthanthemagnitudeoftotal(non-zero)angularmomentum. Lz=l ̄h
isthelargestpossiblevalueforLz,and
l ̄h<√
l(l+1) ̄h (11.56)whichmeansthattheangularmomentumcanneverbecompletelyalignedinapar-
ticulardirection. Infact,iftheangularmomentumdidpointinadefinitedirection,
thenallcomponentsLx, Ly, Lzwouldbedefinite.Butthesecomponentscannotbe
simultaneouslydefinite,becausethecorrespondingoperatorsdon’tcommute. These
meansthatthereisnophysical stateinwhichthe angularmomentumpointsina
particulardirection.
Inclassicalphysics,wevisualizeangularmomentum%Lasavector. Inquantum
physics,itisbettertovisualizetheangularmomentumassociatedwithagiveneigen-
stateφlm asacone. Inaneigenstatethemagnitudeofangularmomentum|L|and
thez-componentLzarefixed,
|L|=√
l(l+1) ̄h Lz=mh ̄ (11.57)