ABSTRACT
J m mΦ = + Φ= and this, by consideration of Eq.(34.5), gives sj mmm += (34.7) Since the maximum values of m and are l and s, the maximum will be so
that the condition sm jm ,sl +
jm j ≤ yields
( ) slmmj s +=+= maxmax (34.8) and the corresponding eigenfunction is
( ) ( ), ,l s l s ls n llR r Y θ ϕ χ+ +Ψ = Φ = For there are two mutually orthogonal states written as ,1−+= slm j smmΦ
1,,1 , −− ΦΦ slsl and these can be combined to give two linearly independent
jjmΨ states corresponding to the possible choices slj += , 11 −=−+= jslm j : , 1l s l s+ + −Ψ
1−+= slj , jslm j =−+= 1 : 1, 1l s l s+ − + −Ψ For there will be three mutually orthogonal 2−+= slm j smmΦ states denoted by
slj −=min (34.9) so that the possible values of j are sljsl +≤≤− (34.10) The relation between the two representations of the total angular momentum eigenstates is usually given by linking the two sets of eigenfunctions
jjmΨ and smmΦ in the form
m m CΨ = Φ∑∑ (34.11)
jm lsmmC are called the Clebsch-Gordan coefficients.
