QuantumPhysics.dvi

As a result, we find that the state|j= 1,m= 0〉of the total system is given by

|j= 1,m= 0〉=

1

√

2

(

| 1 ,−〉⊗| 2 ,+〉+| 1 ,+〉⊗| 2 ,−〉

)

(8.33)

We may check the consistency of this process by applyingS−once more,

S−|j= 1,m= 0〉 = ̄h

√

2 |j= 1,m=− 1 〉

= (S 1 −⊗I 2 +I 1 ⊗S 2 −)

1

√

2

(

| 1 ,−〉⊗| 2 ,+〉+| 1 ,+〉⊗| 2 ,−〉

)

= ̄h

√

2 | 1 ,−〉⊗| 2 ,−〉 (8.34)

and this is in agreement with our earlier identification.

The remaining linear combination of the four states which is orthogonal to the three

j= 1 states may be normalized and is given by

1

√

2

(

| 1 ,−〉⊗| 2 ,+〉−| 1 ,+〉⊗| 2 ,−〉

)

(8.35)

This state is clearly annihilated by all three components ofS, and thus corresponds to the

unique|j= 0,m= 0〉state. Thus, we have proven by explicit calculation that

D(1/2)⊗D(1/2)=D(1)⊕D(0) (8.36)

8.8 Addition of a spin 1/2 with a general angular momentum

We shall now study the addition of a spin 1/2 with a general angular momentumJ 1 andS 2 ,

which commute with one another,

[J 1 a,S 2 b] = 0 a,b= 1, 2 , 3 (8.37)

The system 1, we shall restrict attention to the irreducible representation of spinj 1 , while

for system 2, it is the irreducible representation of spin 1/2. The associated Hilbert spaces

H 1 andH 2 are of dimensions 2j 1 + 1 and 2 respectively, and a canonical basis of states is

given by

H 1 |j 1 ,m 1 〉 m 1 =−j 1 ,−j 1 + 1,···,j 1 − 1 , j 1

H 2 | 2 ,±〉 (8.38)

A basis for the total Hilbert spaceH=H 1 ⊗H 2 is given by the tensor product of these basis

vectors,

H |j 1 ,m 1 〉⊗| 2 ,+〉 m 1 =−j 1 ,−j 1 + 1,···,j 1 − 1 , j 1

|j 1 ,m 1 〉⊗| 2 ,−〉 (8.39)