# 1.1. Spin-1/2 and the Bloch sphere¶

In this section, we elaborate a bit on the states of a particle whose Hilbert space is two-dimensional; this is the simplest non-trivial example of a Hilbert space. The standard example of such a system is a spin-1/2 particle, but many other realisations of systems with a two-dimensional Hilbert space are possible. Systems with such a Hilbert space are denoted as 'two-level systems' (TLS).

Any Hermitian operator in this space can be represented as a hermitian matrix. This means that such a matrix has in principle 4 degrees of freedom (the diagonal elements must be real, and the off-diagonal elements must be each-other's complex conjugate). This means that any hermitian operator can be written as a linear combination of four basis operators. These are taken to be the unit matrix and the three Pauli matrices:

The Pauli matrices satisfy the properties:
where the indices and stand for , or . The braces
are generally used for the *anti-commutator* in these notes.
From this it follows in particular that . If we
apply a space rotation, the spin also changes. This change is expressed
by the rotation operator
,
which represents a rotation over an angle
about an axis with direction
. Although this looks like a complicated expression
to work out, the property enables us to turn
it into a simple expression. For a rotation over an angle about
the -axis, we have
where the equality can be verified from the Taylor expansions of the
exponential function and of the sine and cosine.

A vector in the spin-1/2 Hilbert space is represented by a vector This vector is characterised by three real numbers (two complex numbers contain four real numbers, minus one because of the normalisation condition). Furthermore, when we multiply this vector by a phase factor , the state does not change. Therefore by choosing, say, to be real, there are only two numbers left. These can be represented by a point on a 3D sphere (which itself is a two-dimensional manifold) -- see figure below. The point is defined by the two polar angles and .

The point on the sphere is simply given by the expectation value of the
three Pauli matrices:
and similar for and . The point with these coordinates is the
*polarisation*.

Up to an overall phase factor, the relation between the components
and and the polar angles and is given by
as can be verified (see
problem 1.9). The sphere of polarisation points is called
*Bloch sphere*.