220 CHAPTER13. ELECTRONSPIN
Now
[J^2 ,L^2 ]=[J^2 ,S^2 ]= 0 (13.94)
sinceL^2 andS^2 commutewithallthecomponentsofJ%=L%+S%,andtherefore
[H,J^2 ]=[H,L^2 ]=[H,S^2 ]= 0 (13.95)RecallthatinsolvingfortheeigenstatesofH 0 ,wehadtofirstfindtheeigenstates
of L^2. For similarreasons, to findthe eigenstatesof H, weneed firstto find the
eigenstatesofJ^2 ,L^2 ,andS^2 .FindingtheeigenstatesoftotalangularmomentumJ^2 ,
giventheeigenstates
Ylmχ± (13.96)
oforbital(L^2 ,Lz)andspin(S^2 ,Sz)angularmomentumiscarriedoutbyasystematic
procedureknownastheAdditionofAngularMomentum,whichisournextorder
ofbusiness.