Pauli Matrices

Matrices which arise in Pauli's treatment of spin in quantum mechanics. They are defined by

$\displaystyle \sigma_1$	$\textstyle =$	$\displaystyle \sigma_x\equiv{\hbox{\sf P}}_1\equiv\left[\begin{array}{cc}0 & 1\\ 1 & 0\end{array}\right]$	(1)
$\displaystyle \sigma_2$	$\textstyle =$	$\displaystyle \sigma_y\equiv{\hbox{\sf P}}_2\equiv\left[\begin{array}{cc}0 & i\\ -i & 0\end{array}\right]$	(2)
$\displaystyle \sigma_3$	$\textstyle =$	$\displaystyle \sigma_z\equiv{\hbox{\sf P}}_3\equiv\left[\begin{array}{cc}1 & 0\\ 0 & -1\end{array}\right].$	(3)

The Pauli matrices plus the $2\times 2$ Identity Matrix I form a complete set, so any $2\times 2$ matrix A can be expressed as

$\begin{displaymath} {\hbox{\sf A}}=c_0{\hbox{\sf I}}+c_1\sigma_1+c_2\sigma_2+c_3\sigma_3. \end{displaymath}$

(4)

The associated matrices

$\displaystyle \sigma_+$	$\textstyle \equiv$	$\displaystyle 2\left[\begin{array}{cc}0 & 1\\ 0 & 0\end{array}\right]$	(5)
$\displaystyle \sigma_-$	$\textstyle \equiv$	$\displaystyle 2\left[\begin{array}{cc}0 & 0\\ 1 & 0\end{array}\right]$	(6)
$\displaystyle \sigma^2$	$\textstyle \equiv$	$\displaystyle 3\left[\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right]$	(7)

can also be defined. The Pauli spin matrices satisfy the identities

$\begin{displaymath} \sigma_i\sigma_j ={\hbox{\sf I}}\delta_{ij}+\epsilon_{ijk}i\sigma_k \end{displaymath}$

(8)

$\begin{displaymath} \sigma_i\sigma_j+\sigma_j\sigma_i = 2\sigma_{ij} \end{displaymath}$

(9)

$\begin{displaymath} \sigma_xp_x+\sigma_yp_y+\sigma_zp_z =\sqrt{{p_x}^2 +{p_y}^2 +{p_z}^2}. \end{displaymath}$

(10)

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 211-212, 1985.

Goldstein, H. ``The Cayley-Klein Parameters and Related Quantities.'' Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 156, 1980.