QuantumPhysics.dvi

As a result, we have

|j,j− 1 〉=

1

√

2 j

(√

2 j 1 |j 1 ,j 1 − 1 〉⊗|j 2 ,j 2 〉+

√

2 j 2 |j 1 ,j 1 〉⊗|j 2 ,j 2 − 1 〉

)

(8.52)

The remaining linear combination at levelm=j−1 is given by

|j− 1 ,j− 1 〉=

1

√

2 j

(√

2 j 2 |j 1 ,j 1 − 1 〉⊗|j 2 ,j 2 〉−

√

2 j 1 |j 1 ,j 1 〉⊗|j 2 ,j 2 − 1 〉

)

(8.53)

and is the highestmvalue for the irreducible representationj−1. One may pursue this

process recursively, and find that

D(j^1 )⊗D(j^2 )=

j (^1) ⊕+j 2
j=k

D(j) (8.54)

It remains to determine the number k. One could do this directly. A convenient trick,

however, is to determinekby making sure that the dimensions work out correctly.

(2j 1 + 1)(2j 2 + 1) =

j (^1) ∑+j 2
j=k

(2j+ 1) (8.55)

Using

∑N

ℓ=0

(2j+ 1) = (N+ 1)^2 (8.56)

we readily find that

(2j 1 + 1)(2j 2 + 1) = (j 1 +j 2 + 1)^2 −k^2 ⇒ k^2 = (j 1 −j 2 )^2 (8.57)

so thatk=|j 1 −j 2 |. We recover easily the previous case wherej 2 =^12.

8.10 Systematics of Clebsch-Gordan coefficients

The above construction makes it clear that there are two ways of describing the states

of a system in which two angular momenta are added. The first is as the eigenstates of

J^21 ,J 13 ,J^22 ,J 23 , and we label these eigenstates as

|j 1 ,m 1 〉⊗|j 2 ,m 2 〉=|j 1 ,j 2 ;m 1 ,m 2 〉 (8.58)

But, alternatively, the same states may be label by the eigenvaluesof an equivalent set of

commuting observables,J^21 ,J^22 ,J^2 ,J^3 , and we label these states as

|j 1 ,j 2 ;j,m〉 (8.59)