Chapter 14
The Addition of Angular
Momentum
”Considertheliliesofthefield,howtheygrow.Theytoilnot,neitherdotheyspin...”
TheBookofMathew
InthepreviouschapterwefoundthattherewasacontributiontotheHydrogen
atomHamiltonianproportionaltotheoperator
L·S=
1
2
(J^2 −L^2 −S^2 ) (14.1)
InordertofindeigenstatesofL·S,weneedtofindeigenstatesofL^2 ,S^2 ,andthe
totalangularmomentumJ^2. Now,firstofall,theusualeigenstatesofL^2 ,Lz,S^2 ,Sz
thatwewrotedowninthelastLecture,namely
φ(r,θ,φ)χ+=Rnl(r)Ylm(θ,φ)[
1
0]
and φ(r,θ,φ)χ−=Rnl(r)Ylm(θ,φ)[
0
1](14.2)
are(mostly)noteigenstatesofL·S. ThisisbecauseL·S =LxSx+LySy+LzSz,
andwhilethewavefunctionsaboveareeigenstatesofLz andSz,theyare(mostly)
noteigenstatesofLx,LyandSx,Sy.Becausethesewavefunctionsarenoteigenstates
ofL·S,theyarenoteigenstatesofthetotalangularmomentumJ^2 ,either.
DefiningJx=Lx+Sxetc.,andusingthefactthattheorbitalangularmomentum
operatorscommutewiththespinangularmomentumoperators(theyactondifferent
degreesoffreedom),itseasytoseethatthecomponentsoftotalangularmomentum
satisfythenow-familiarcommutationrelations
[Jx,Jy] = i ̄hJz
[Jy,Jz] = i ̄hJx
[Jz,Jx] = i ̄hJy (14.3)