Quantum Mechanics for Mathematicians

takes

z=

z 1

z 2

→

αz+β

γz+δ

Such transformations are invertible if the determinant of the matrix is non-zero,
and one can show that these give conformal (angle-preserving) transformations
of the complex plane known as “M ̈obius transformations”. In chapter 40 we will
see that this group action appears in the theory of special relativity, where the
action on the sphere can be understood as transformations acting on the space
of light rays. When the matrix above is inSU(2) (γ=−β, δ=α, αα+ββ=
1), it can be shown that the corresponding transformation on the sphere is a
rotation of the sphere inR^3 , providing another way to understand the nature
ofSU(2) =Spin(3) as the double cover of the rotation groupSO(3).

7.5 The Bloch sphere

For another point of view on the relation between the two-state system with
H=C^2 and the geometry of the sphere (known to physicists as the “Bloch
sphere” description of states), the unit sphereS^2 ⊂R^3 can be mapped to
operators by
x→σ·x

For each pointx∈S^2 ,σ·xhas eigenvalues±1. Eigenvectors with eigenvalue
+1 are the solutions to the equation

σ·x|ψ〉=|ψ〉 (7.4)

and give a subspaceC⊂ H, giving another parametrization of the points in
CP(1). Note that one could equivalently consider the operators

Px=

1

2

( 1 −σ·x)

and look at the space of solutions to

Px|ψ〉= 0

It can easily be checked thatPxsatisfiesPx^2 =Pxand is a projection operator.
For a more physical interpretation of this in terms of the spin operators, one
can multiply 7.4 by^12 and characterize theC⊂ Hcorresponding tox∈S^2 as
the solutions to

S·x|ψ〉=

1

2

|ψ〉

Then the North pole of the sphere is a “spin-up” state, and the South pole is
a “spin-down” state. Along the equator one finds two points corresponding to
states with definite values forS 1 , as well as two for states that have definite
values forS 2.