QuantumPhysics.dvi

Finally, we need to make a change of basis to compare these representation matrices with

the ones found earlier, and we have

D(1)(Ja) =SLaS† (8.21)

where the unitary change of basis is given by

S=

1

√

2







1 −i 0

0 0 −

√

2

− 1 −i 0





 (8.22)

8.7 Addition of two spin 1/2 angular momenta

We begin with the simplest case of two spin 1/2 angular momenta,S 1 andS 2. The basic

assumption is that the two spin 1/2 degrees of freedom are completely independent from one

another. We may think of the spins of two electrons, whose statesare independent from one

another. The commutation relations are

[Sia,Sib] = i ̄h

∑^3

c=1

εabcSic i= 1, 2

[S 1 a,S 2 b] = 0 (8.23)

The Hilbert spaces of statesHifor each system admit the following basis vectors,

Hi |i,±〉 i= 1, 2 (8.24)

The dimension of each Hilbert space is 2, and the total number of states for the combined

system of two spins is their product, namely 4. The Hilbert space of the total systemHthus

has dimension 4. A natural basis of states forHis given by the tensor product of the basis

states for each spin 1/2 system,

H | 1 ,+〉⊗| 2 ,+〉 | 1 ,+〉⊗| 2 ,−〉

| 1 ,−〉⊗| 2 ,+〉 | 1 ,−〉⊗| 2 ,−〉 (8.25)

The total Hilbert spaceHis thus the tensor product of the factors,

H=H 1 ⊗H 2 (8.26)

The spin operators extend to this tensor product Hilbert space asfollows. The first spin

operatorS 1 really only acts on the first factor in the tensor product and it is theidentity