228 CHAPTER14. THEADDITIONOFANGULARMOMENTUM
splittingisverysmall;wecan seethatthecorrectiontoatomicenergiesisonthe
orderofthesquareofthefinestructureconstant,α^2 ,whichislessthanonepartin
10 ,000.
Atthispoint,youmayfeelalittleletdown.Didwegotoallthetroubleoflearning
howtoaddangularmomenta,justtocompute atiny,arcanesplittingofhydrogen
atomspectrallines? Theanswer,ofcourse,isno.Thetechniqueforaddingangular
momentumisoneofthemostimportanttoolsthereisinatomic,molecular,nuclear,
particle,andcondensedmatterphysics,andthereasonissimple: Thingsaremade
ofotherthings,andthoseotherthingsspin.Atoms,forexample,consistofelectrons
andanucleus. Liketheelectron,atomicnucleialsospin.Sothepossiblespinangular
momentaofagivenatomisdeterminedbyaddingthetotalangularmomentumof
the electron(J = L+S)to the spinangular momentum of thenucleus. Nuclei
aremadeofprotonsandneutrons,eachofwhichhavespin^12. Theenergylevelsof
nucleidependonthetotalspinofthenuclei,andthisrequiresaddingthespinsofthe
protonsandneutrons,togetherwiththeirorbitalangularmomentuminthenucleus.
Likewise,protonsandneutrons(andmanyothershort-livedparticles)arecomposed
ofthreequarks. Eachquarkhasspin^12. Todeterminewhichcombinationofthree
quarkswouldgivespin^12 protonsandneutrons,andwhichwouldleadtohigherspin
objectslikehyperons,itsnecessarytobeabletoaddthespinangularmomentaof
thequarks.Inshort,itsimportanttolearnhowtoaddangularmomentum,because
allphysicalobjectsarecomposedofspinningparticles. Viewedattheatomiclevel
everythingspins,includingtheliliesofthefield.
14.1 The General Method
Consideracompositesystemconsistingoftwosubsystems,andsupposeeachsubsys-
temhasacertainangularmomentum. Itcouldbethatthesystemisanatom,one
subsystemisalltheelectrons,andtheothersubsystemisthenucleus. Orperhapsthe
systemisadeuteriumnucleus,consistingofoneproton,oroneneutron. Ormaybe
thesystemisapimeson,consistingofonequark, andoneantiquark. Depending
ontheparticularcase,theangularmomentumof eachsubsystemmightbetheor-
bitalangularmomentumofthesubsystem,thespinofthesubsystem,ormaybethe
total(spinplusorbital)angularmomentumofthesubsystem. Thepointis,adding
angularmomentumisanalgebraictechnique, anditreallydoesn’tmatterwhether
weareaddingspintospin,orbitaltoorbital,orbitaltospin,etc. Wewilljustde-
notetheangularmomentumofonesystembyJ 1 ,andtheangularmomentumofthe
othersystembyJ 2 ,withtheunderstandingthat thesecan beanytypeofangular
momentum.
ItisassumedthatweknowtheeigenstatesofJ 12 ,J 1 z,J 22 ,J 2 z,whichwedenote{ψj^11 m 1 ψ^2 j 2 m 2 } (14.45)