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Pauli Matrices

The Pauli matrices are


\begin{displaymath}
\sigma_1 = \left( \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{arra...
... \left( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array} \right),
\end{displaymath} (2.21)

where


\begin{displaymath}
\sigma_i\sigma_j = \delta_{ij} + i\epsilon_{ijk}\sigma_k .
\end{displaymath} (2.22)

where $\epsilon_{ijk}$ is the usual antisymmetric tensor


\begin{displaymath}
\epsilon_{ijk} = \left\{ \begin{array}{rl}
+1 & \textrm{for...
...xtrm{if two or more indices are the same} \end{array} \right.
\end{displaymath} (2.23)

The following identity is satisfied.


\begin{displaymath}
(\vec{a}\cdot\vec{\sigma})(\vec{b}\cdot\vec{\sigma}) =
\vec{a}\cdot\vec{b} I + i\vec{\sigma}\cdot(\vec{a}\times\vec{b}) ,
\end{displaymath} (2.24)

where $I$ is the $2\times 2$ identity matrix.



Douglas M. Gingrich (gingrich@ ualberta.ca)
2004-03-18