The Pauli matrices

The Pauli matrices are a set of three complex, Hermitian matrices: \[\begin{equation} \require{physics} \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} % \qquad % \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} % \qquad % \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \thinspace . \end{equation}\]

They are used to define electron spin.

Using the \(x\)-, \(y\)- and \(z\)-matrices, we can define the raising and lowering matrices as follows: \[\begin{align} & \sigma_+ = \sigma_x + i \sigma_y = \begin{pmatrix} 0 & 2 \\ 0 & 0 \end{pmatrix} \\ % & \sigma_- = \sigma_x - i \sigma_y = \begin{pmatrix} 0 & 0 \\ 2 & 0 \end{pmatrix} \thinspace . \end{align}\]

The Pauli matrices obey the following commutator relations: \[\begin{equation} \comm{\sigma_i}{\sigma_j} = 2 i \varepsilon_{ijk} \sigma_k \end{equation}\] and the following anticommutator relations: \[\begin{equation} \comm{\sigma_i}{\sigma_j}_+ = 2 \delta_{ij} \vb{I}_2 \thinspace . \end{equation}\]

We should note that, even though they are matrices, we do not typeset them with a bold symbol \(\boldsymbol{\sigma}\): that notation is reserved for the vector of the Pauli matrices: \[\begin{equation} \boldsymbol{\sigma} = \begin{pmatrix} \sigma_x & \sigma_y & \sigma_z \end{pmatrix} \thinspace , \end{equation}\] such that the following inner product is defined: \[\begin{equation} \boldsymbol{\sigma} \vdot \vb{a} = \sigma_x a_x + \sigma_y a_y + \sigma_z a_z \end{equation}\] and the Dirac relation (also called the Dirac identity) is fulfilled: \[\begin{equation} (\boldsymbol{\sigma} \vdot \vb{a}) (\boldsymbol{\sigma} \vdot \vb{b}) = \qty(\vb{a} \vdot \vb{b}) \thinspace \vb{I}_2 + i \boldsymbol{\sigma} \vdot (\vb{a} \cross \vb{b}) \end{equation}\] for any two vector-like quantities \(\vb{a}\) and \(\vb{b}\).