QuantumPhysics.dvi

The state hasm=j−1, but also is annihilated byJ+, and so it is the highestmstate of the

representation with total angular momentumj 1 −^12. Thus, the tensor product decomposes

as follows,

D(j^1 )⊗D(1/2)=D(j^1 +1/2)⊕D(j^1 −^1 /2) j 1 ≥

1

2

(8.46)

8.9 Addition of two general angular momenta

We shall now carry out the addition of two general angular momenta, with irreducible

representations of spinsj 1 andj 2 , and associated Hilbert spacesH 1 andH 2. The tensor

product basis of the tensor productH=H 1 ⊗H 2 is given by

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

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

H |j 1 ,m 1 〉⊗|j 2 ,m 2 〉 ” (8.47)

Total angular momentumJis defined by

J=J 1 ⊗I 2 +I 1 ⊗J 2 (8.48)

At totalJ^3 =Jzeigenvaluem, we have

m=j 1 +j 2 |j 1 ,j 1 〉⊗|j 2 ,j 2 〉

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

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

··· ···

m=−j 1 −j 2 |j 1 ,−j 1 〉⊗|j 2 ,−j 2 〉 (8.49)

The unique state with the highest value ofm=j 1 +j 2 belongs to the irreducible represen-

tationD(j)withj=j 1 +j 2. Thus, we may identify

|j,+j〉 = |j 1 ,j 1 〉⊗|j 2 ,j 2 〉

|j,−j〉 = |j 1 ,−j 1 〉⊗|j 2 ,−j 2 〉 (8.50)

By applying the lowering operatorJ−=J 1 −⊗I 2 +I 1 ⊗J 2 −to this state, we find

J−|j,+j〉 = ̄h

√

2 j|j,j− 1 〉

=

(

J 1 −⊗I 2 +I 1 ⊗J 2 −

)

|j 1 ,j 1 〉⊗|j 2 ,j 2 〉

= ̄h

√

2 j 1 |j 1 ,j 1 − 1 〉⊗|j 2 ,j 2 〉+ ̄h

√

2 j 2 |j 1 ,j 1 〉⊗|j 2 ,j 2 − 1 〉 (8.51)