QMGreensite_merged

222 CHAPTER14. THEADDITIONOFANGULARMOMENTUM

SothismeansthatJz=Lz+SzmustcommutewithJ^2. Whatwewanttodoisto
constructeigenstatesofJ^2 ,Jz,L^2 ,S^2 ,denoted|jjzls>,aslinearcombinationsofthe
eigenstatesofL^2 ,Lz,S^2 ,Szdenotedby|lmssz>,whosecorrespondingeigenfunctions
areshownin(14.2). Inotherwords,we wanttofindthesetofClebsch-Gordon
coefficientsCjlmssjzlszsuchthat

|jjzls>=

∑

sz,m=jz−sz

Cjlmssjzlsz|lmssz> (14.4)

or,inwavefunctionnotation

Φjjz=

∑

sz,m=jz−sz

CjlmssjzlszYlmχsz (14.5)

(InthecaseoftheHydrogenatom,s=^12 .Butthetechniquewearegoingtousewill
workforanyvalueofs.)
Thetrickistonoticethat oneofthe|lmssz >states,knownas the”highest
weightstate”,isalsoaneigenstateofJ^2. Supposeweaskwhichstateistheeigen-
statewiththehighest eigenvalueof Jz. NowJz=Lz+Sz,andanyeigenstateof
Lz,SzisalsoaneigenstateofJz,althoughitisusuallynotaneigenstateofJ^2. The
eigenstatewith thehighest eigenvalue of Jz has to bethe statewith thehighest
eigenvalueofLzandSz,andthereisonlyonesuchstate

”highestweightstate”=Yllχ+ (14.6)

whichhasjz=l+^12 .Nowifthehighestjzisl+^12 ,thenthismustalsobethehighest
possiblevalueofj thatcanbeconstructedfromtheYlmχsz. Butthereisonlyone
statewithjz = l+s,so thismustalsobetheeigenstateoftheoperatorJ^2 with
j=l+s.Soweconcludethat

Φjj=Yllχ+ (j=l+s) (14.7)

(Byexactlythesamereasoning,thereisalsoalowestweightstate

Φj,−j=Yl,−lχ− (j=l+s) (14.8)

withjz=−(l+s).)

Exercise: Using

J^2 = L^2 +S^2 + 2 L·S

Jz = Lz+Sz (14.9)

andexpressingLx,Ly,Sx,Syintermsof ladderoperators, show explicitlythat the
highestweightstateisaneigenstateofJ^2 ,i.e.

J^2 Yllχ+ = jmax(jmax+1) ̄h^2 Yllχ+ where jmax=l+

1

2

JzYllχ+ = jmaxh ̄Yllχ+ (14.10)