230 CHAPTER14. THEADDITIONOFANGULARMOMENTUM
whichwenowquicklyverify. Suppose,e.g.,thatj 1 >j 2 ,sothatjmin=j 1 −j 2 .Then
N′ =
j (^1) ∑+j 2
j=j 1 −j 2
(2j+1)
=
j (^1) ∑+j 2
j=1
(2j+1)−
j 1 −∑j 2 − 1
j=1
(2j+1)
= 2
j (^1) ∑+j 2
j=1
j− 2
j 1 −∑j 2 − 1
j=1
j+(2j 2 +1) (14.54)
andusingthefactthat
∑L
l=1
l=
L(L+1)
2
(14.55)
wefind
N′ = (j 1 +j 2 )(j 1 +j 2 +1)−(j 1 −j 2 )(j 1 −j 2 −1)+(2j 2 +1)
= (2j 1 +1)(2j 2 +1)
= N (14.56)TheprocedureforfindingallthestatesisshowninFig.[14.3].Startingfromthe
highestweightstate,eq. (14.50),weapplyJ−totheleft-handsideandJ 1 −+J 2 −to
theright-handsidetogetΦj,j− 1. Proceedinginthisway,onefindsallthestateswith
jmax=j 1 +j 2. Next,thestatewithj=m=jmax− 1 musthavetheform
Φjmax− 1 ,jmax− 1 =aφ^1 j 1 j 1 φ^2 j 2 ,j 2 − 1 +bφ^1 j 1 ,j 1 − 1 φ^2 j 2 ,j 2 (14.57)becausethesearetheonlystateswithjz=jmax−1.Thetwoconstantsaandbare
determinedbyorthogonality
<Φjmax,jmax− 1 |Φjmax− 1 ,jmax− 1 >= 0 (14.58)andnormalization
<Φjmax− 1 ,jmax− 1 |Φjmax− 1 ,jmax− 1 >= 1 (14.59)
Havingdeterminedaandb, applyJ− tothelefthandside,andJ 1 −+J 2 −to the
righthandside, tofind Φjmax− 1 ,jmax− 2 ,and continueapplyingthe ladderoperator
successivelyto find allstates withjmax−1. Then determinethe threeconstants
a,b,cinthestate
Φjmax− 2 ,jmax− 2 =aφ^1 j 1 j 1 φ^2 j 2 ,j 2 − 2 +bφ^1 j 1 ,j 1 − 1 φ^2 j 2 ,j 2 − 1 +cφ^1 j 1 j 1 − 2 φ^2 j 2 ,j 2 (14.60)fromthetwoorthogonalityconditions
<Φjmax,jmax− 2 |Φjmax− 2 ,jmax− 2 > = 0
<Φjmax− 1 ,jmax− 2 |Φjmax− 2 ,jmax− 2 > = 0 (14.61)