S 1 |ψ〉=−^12 |ψ〉
S 2 |ψ〉=−^12 |ψ〉
S 3 |ψ〉=−^12 |ψ〉S 1 |ψ〉= +^12 |ψ〉S 2 |ψ〉= +^12 |ψ〉S 3 |ψ〉= +^12 |ψ〉Figure 7.4: The Bloch sphere.For later applications of the spin representation, we would like to make for
eachxa choice of solution to equation 7.4, getting a map
u+:x∈S^2 →|ψ〉=u+(x)∈H=C^2such that
(σ·x)u+(x) =u+(x) (7.5)
This equation determinesu+only up to multiplication by anx-dependent scalar.
A standard choice is
u+(x) =1
√
2(1 +x 3 )(
1 +x 3
x 1 +ix 2)
=
(
cosθ 2
eiφsinθ 2)
(7.6)
whereθ,φare standard spherical coordinates (which will be discussed in section
8.3). This particular choice has two noteworthy characteristics:
- One can check that it satisfies
u+(Rx) = Ωu+(x)whereR= Φ(Ω) is the rotation corresponding to anSU(2) elementΩ =
(
cosθ 2 −e−iφsinθ 2
eiφsinθ 2 cosθ 2