Pauli spin matrices

The Pauli spin matrices (named after physicist Wolfgang Ernst Pauli) are a set of unitary Hermitian matrices which form an orthogonal basis (along with the identity matrix) for the real Hilbert space of 2 × 2 Hermitian matrices and for the complex Hilbert spaces of all 2 × 2 matrices. They are usually denoted:

$\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{y}={\begin{pmatrix}0&-{\mathit {i}}\\{\mathit {i}}&0\end{pmatrix}},\quad \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$

Algebraic properties

$\sigma _{x}^{2}=\sigma _{y}^{2}=\sigma _{z}^{2}=I$

For i = 1, 2, 3:

${\mbox{det}}(\sigma _{i})=-1\,$

${\mbox{Tr}}(\sigma _{i})=0\,$

${\mbox{eigenvalues}}=\pm 1\,$

Commutation relations

$\sigma _{1}\sigma _{2}=i\sigma _{3}\,\!$

$\sigma _{3}\sigma _{1}=i\sigma _{2}\,\!$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_2\sigma_3 = i\sigma_1\,\!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i\sigma_j = -\sigma_j\sigma_i\mbox{ for }i\ne j\,\!}

The Pauli matrices obey the following commutation and anticommutation relations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} [\sigma_i, \sigma_j] &=& 2 i\,\varepsilon_{i j k}\,\sigma_k \\[1ex] \{\sigma_i, \sigma_j\} &=& 2 \delta_{i j} \cdot I \end{matrix}}

where $\varepsilon _{ijk}$ is the Levi-Civita symbol, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_{ij}} is the Kronecker delta, and I is the identity matrix.

The above two relations can be summarized as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i \sigma_j = \delta_{ij} \cdot I + i \varepsilon_{ijk} \sigma_k. \,}