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These matrices have the following properties, which can be easily verified by
inspection:

2
2
2
Property 1: σ = σ = σ = I (8.42)
1 2 3
where I is the 2 ⊗ 2 identity matrix.
Property 2: σ σ + σσ = σ σ + σσ = σσ + σσ = 0 (8.43)
1 2 2 1 1 3 3 1 2 3 3 2
Property 3: σσ = j σ ; σ σ = j σ ; σ σ = j σ (8.44)
1 2 3 2 3 1 3 1 2
r r
If we define the quantity σ⋅ v to mean:
r r
σ ⋅=v σ v + σ v + σ v (8.45)
1 1 22 33
r
that is, v = (v , v , v ), where the parameters v , v , v are represented as the
3
2
2
1
3
1
components of a vector, the following theorem is valid.
THEOREM
r r r r r r r r r
×
(σ ⋅ v )(σ ⋅ w ) = ( ⋅v w ) + jI σ ( ⋅ v w ) (8.46)
where the vectors’ dot and cross products have the standard definition.
PROOF The left side of this equation can be expanded as follows:

r r r r
(σ ⋅ v )(σ ⋅ w ) = (σ v + σ v + σ v )(σ w + σ w + σ w )
1 1 22 33 1 1 2 2 3 3
= (σ v w + σ vw + σ vw ) (σ σ+ v w + σ σ vw ) + (8.47)
2
2
2
1 1 1 2 2 2 3 3 3 1 2 1 2 2 1 2 1
+ (σ σ vw + σσ v w ) (σσ+ vw +σ σ vw )
1 3 1 3 3 1 3 1 2 3 32 3 3 2 3 2
Using property 1 of the Pauli’s matrices, the first parenthesis on the RHS of
Eq. (8.47) can be written as:

rr
(σ v w + σ vw + σ vw ) = (v w + vw + vw ) =I (v w⋅ )I (8.48)
2
2
2
1 1 1 2 2 2 3 3 3 1 1 2 2 3 3
Using properties 2 and 3 of the Pauli’s matrices, the second, third, and
fourth parentheses on the RHS of Eq. (8.47) can respectively be written as:

(σσ vw + σ σ v w ) = σ j (v w − v w ) (8.49)
1 2 1 2 2 1 2 1 3 1 2 2 1

© 2001 by CRC Press LLC