QuantumPhysics.dvi

Each group of states forms an orthogonal basis for the same Hilbert space. It is convenient

to normalize the states as follows,

〈j 1 ,j 2 ;m′ 1 ,m′ 2 |j 1 ,j 2 ;m 1 ,m 2 〉 = δm′ 1 ,m 1 δm′ 2 ,m 2

〈j 1 ,j 2 ;j′,m′|j 1 ,j 2 ;j,m〉 = δj′,jδm′,m (8.60)

We have already show that

|j 1 ,j 2 ;j,+j〉=|j 1 ,j 2 ; +j 1 ,+j 2 〉 j=j 1 +j 2

|j 1 ,j 2 ;j,−j〉=|j 1 ,j 2 ;−j 1 ,−j 2 〉 (8.61)

More generally, the above orthonormality conditions imply that the passage from one basis

to the other is by a unitary matrix,

|j 1 ,j 2 ;j,m〉=

∑

m 1

∑

m 2

|j 1 ,j 2 ;m 1 ,m 2 〉〈|j 1 ,j 2 ;m 1 ,m 2 |j 1 ,j 2 ;j,m〉 (8.62)

By choosing the phases of the states in both bases, all coefficientsmay in fact be chosen to

be real.

By applying the angular momentum generatorsJa=Ja 1 +J 2 ato both sides, we obtain

recursion relations between the matrix elements. ApplyingJ 3 , we get

m|j 1 ,j 2 ;j,m〉=

∑

m 1

∑

m 2

(m 1 +m 2 )|j 1 ,j 2 ;m 1 ,m 2 〉〈|j 1 ,j 2 ;m 1 ,m 2 |j 1 ,j 2 ;j,m〉 (8.63)

Taking the inner product with〈j 1 ,j 2 ;m′ 1 ,m′ 2 |, we obtain,

(m−m 1 −m 2 )〈|j 1 ,j 2 ;m 1 ,m 2 |j 1 ,j 2 ;j,m〉= 0 (8.64)

ApplyingJ±=J 1 ±+J 2 ±, we get

Nj,m±|j 1 ,j 2 ;j,m± 1 〉 =

∑

m 1

∑

m 2

Nj± 1 ,m 1 |j 1 ,j 2 ;m 1 ± 1 ,m 2 〉〈|j 1 ,j 2 ;m 1 ,m 2 |j 1 ,j 2 ;j,m〉

+

∑

m 1

∑

m 2

Nj± 2 ,m 2 |j 1 ,j 2 ;m 1 ,m 2 ± 1 〉〈|j 1 ,j 2 ;m 1 ,m 2 |j 1 ,j 2 ;j,m〉

where

Nj,m± =

√

j(j+ 1)−m(m±1) =

√

(j∓m)(j±m+ 1) (8.65)

Taking the inner product with〈j 1 ,j 2 ;m′ 1 ,m′ 2 |, we obtain,

Nj,m± 〈j 1 ,j 2 ;m 1 ,m 2 |j 1 ,j 2 ;j,m± 1 〉 = Nj± 1 ,m 1 ∓ 1 〈j 1 ,j 2 ;m 1 ∓ 1 ,m 2 |j 1 ,j 2 ;j,m〉

+Nj± 2 ,m 2 ∓ 1 〈j 1 ,j 2 ;m 1 ,m 2 ∓ 1 |j 1 ,j 2 ;j,m〉 (8.66)