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 $2\times 2$ 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: $$\begin{aligned} \unicode{120793} &= \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right) \\ \sigma_x &= \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) \\ \sigma_y &= \left( \begin{array}{cc} 0 & -{\rm i}\\ {\rm i}& 0 \end{array}\right) \\ \sigma_z &= \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right). \end{aligned}$$

The Pauli matrices satisfy the properties: $$\left\{ \sigma_j, \sigma_k \right\} = \sigma_j \sigma_k + \sigma_k \sigma_j = 2\delta_{jk}$$ where the indices $j$ and $k$ stand for $x$, $y$ or $z$. The braces $\{,\}$ are generally used for the anti-commutator in these notes. From this it follows in particular that $\sigma_j^2 = \unicode{120793}$. If we apply a space rotation, the spin also changes. This change is expressed by the rotation operator $\exp\left({\rm i}\boldsymbol{\alpha} \cdot \boldsymbol{\sigma}/2\right)$, which represents a rotation over an angle $\left| \boldsymbol{\alpha} \right|$ about an axis with direction $\boldsymbol{\alpha}$. Although this looks like a complicated expression to work out, the property $\sigma_j^2 = \unicode{120793}$ enables us to turn it into a simple expression. For a rotation over an angle $\alpha$ about the $z$-axis, we have $$\exp\left( {\rm i}\alpha \sigma_z/2\right) = \cos(\alpha/2) \unicode{120793} + {\rm i}\sin(\alpha/2) \sigma_z,$$ 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 $$\left| \psi \right\rangle = \left( \begin{array}{c} a \\ b \end{array}\right) \text{ with } \left|a\right|^2 + \left|b\right|^2 = 1.$$ 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 $\exp({\rm i}\gamma)$, the state does not change. Therefore by choosing, say, $a$ 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 $\vartheta$ and $\varphi$.

The Bloch sphere

The point on the sphere is simply given by the expectation value of the three Pauli matrices: $$\left\langle\psi \left| \sigma_x \right| \psi \right\rangle= a^* b + a b^*$$ and similar for $y$ and $z$. The point with these coordinates is the polarisation.

Up to an overall phase factor, the relation between the components $a$ and $b$ and the polar angles $\vartheta$ and $\varphi$ is given by $$\begin{aligned} a &= \exp(-{\rm i}\varphi/2) \cos(\vartheta/2) ; \\ b &= \exp({\rm i}\varphi/2) \sin(\vartheta/2), \end{aligned}$$ as can be verified (see problem 1.9). The sphere of polarisation points is called Bloch sphere.