# Pauli matrices

The Pauli matrices are a set of three Hermitian, unitary matrices used by Wolfgang Pauli in his theory of quantum-mechanical spin. They are given by:

 $\displaystyle\sigma_{1}$ $\displaystyle=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$ $\displaystyle\sigma_{2}$ $\displaystyle=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}$ $\displaystyle\sigma_{3}$ $\displaystyle=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$

They satisfy the following commutation and anticommutation identities:

 $\displaystyle\left[\sigma_{i},\sigma_{j}\right]$ $\displaystyle=2i\epsilon_{ijk}\sigma_{k}\text{where \epsilon_{ijk} is the % Levi-Civita symbol}$ $\displaystyle\{\sigma_{i},\sigma_{j}\}$ $\displaystyle=2\mathbf{I}\delta_{ij}\text{where \mathbf{I} is the identity % matrix and \delta_{ij} is the Kronecker delta}$

## 0.1 Delta notation

With the identity matrix I, the Pauli matrices form a group. When combined in this way, they are often given the symbols $\delta_{i}$, as follows:

 $\displaystyle\delta_{0}$ $\displaystyle=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$ $\displaystyle\delta_{1}$ $\displaystyle=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$ $\displaystyle\delta_{2}$ $\displaystyle=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}$ $\displaystyle\delta_{3}$ $\displaystyle=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$

This choice is useful when writing the Dirac matrices.

 Title Pauli matrices Canonical name PauliMatrices Date of creation 2013-03-22 17:57:01 Last modified on 2013-03-22 17:57:01 Owner invisiblerhino (19637) Last modified by invisiblerhino (19637) Numerical id 9 Author invisiblerhino (19637) Entry type Definition Classification msc 15A57 Synonym sigma matrices Related topic Spinor Related topic SchrodingersWaveEquation Related topic UnitaryGroup Related topic HermitianMatrix Related topic DiracMatrices Related topic DiracEquation