14.1. THEGENERALMETHOD 231
andthenormalizationcondition
<Φjmax− 2 ,jmax− 2 |Φjmax− 2 ,jmax− 2 >= 1 (14.62)Ladderoperatorsareappliedagain,andthewholeprocedureiscontinueduntilall
ofthestateswithtotalangularmomentumj=|j 1 −j 2 |havebeenfound. Atthat
point,stop.
Animportantapplicationof thisprocedureistheaddition ofthespinangular
momentumoftwospin^12 particles;acasethatturnsupagainandagaininatomic,
nuclear,andparticlephysics.Inthiscase,therearefoureigenstatesofS^21 ,S 1 z,S 22 ,S 2 z,
denoted
χ^1 ±χ^2 ± (14.63)
WewanttofindallpossibleeigenstatesofS^2 ,Sz,S^21 ,S^22. Inthiscase
smax=1
2
+
1
2
= 1 smin=|1
2
−
1
2
|= 0 (14.64)
Thehighestweightstateis
Φ 11 =χ^1 +χ^2 + (14.65)
Applyingtheladderoperators
S−Φ 11 = (S 1 −+S 2 −)χ^1 +χ^2 +
√
2 ̄hΦ 10 = (S 1 −χ^1 +)χ^2 ++χ^1 +(S 2 −χ^2 +)
= ̄h[χ^1 −χ^2 ++χ^1 +χ+ 2 ] (14.66)tofind
Φ 10 =1
√
2
[χ^1 +χ^2 −+χ^1 −χ^2 +] (14.67)Againapplytheladderoperators
S−Φ 10 =1
√
2
(S 1 −+S 2 −)[χ^1 +χ^2 −+χ^1 −χ^2 +]̄h√
2 Φ 1 − 1 =
1
√
2
[(S 1 −χ^1 +)χ^2 −+(S 1 −χ^1 −)χ^2 ++χ^1 +(S 2 −χ^2 −)+χ^1 −(S 2 −χ^2 +)]=
̄h
√
2[χ^1 −χ^2 −+ 0 + 0 +χ^1 −χ^2 −] (14.68)whichgives,asitshould,thelowestweightstate
Φ 1 − 1 =χ^1 −χ^2 − (14.69)Therearethreestateswithj=1,whichareknownasthetripletspinstates.
Theoneremainingstateatj= 0 isknown,forobviousreasons,asthesingletspin
state. Thesingletstate
Φ 00 =aχ^1 +χ^2 −+bχ^1 −χ^2 + (14.70)