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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)

See also Dirac Matrices, Quaternion


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.




© 1996-9 Eric W. Weisstein
1999-05-26