The Pauli matrices, also called the Pauli spin matrices, are complex matrices that arise in Pauli's treatment of spin in quantum mechanics. They are defined by

			(1)
			(2)
			(3)

(Condon and Morse 1929, p. 213; Gasiorowicz 1974, p. 232; Goldstein 1980, p. 156; Liboff 1980, p. 453; Arfken 1985, p. 211; Griffiths 1987, p. 115; Landau and Lifschitz 1991, p. 204; Landau 1996, p. 224).

The Pauli matrices are implemented in Mathematica as PauliMatrix[n], where , 2, or 3.

The Pauli spin matrices satisfy the identities

			(4)
			(5)
			(6)

where is the identity matrix, is the Kronecker delta, is the permutation symbol, the leading is the imaginary unit (not the index ), and Einstein summation is used in (6) to sum over the index (Arfken 1985, p. 211; Griffiths 1987, p. 139; Landau and Lifschitz 1991, pp. 204-205).

The Pauli matrices plus the identity matrix form a complete set, so any matrix can be expressed as

(7)

The associated matrices

			(8)
			(9)
			(10)

can also be defined.

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

Condon, E. U. and Morse, P. M. Quantum Mechanics. New York: McGraw-Hill, 1929.

Gasiorowicz, S. Quantum Physics. New York: Wiley, pp. 232-233, 1974.

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

Griffiths, D. J. Introduction to Elementary Particles. New York: Wiley, 1987.

Landau, L. D. and Lifschitz, E. M. Quantum Mechanics (Non-Relativistic Theory), 3rd ed. Oxford, England: Pergamon Press, 1991.

Landau, R. H. Quantum Mechanics II: A Second Course in Quantum Theory, 2nd ed. New York: Wiley, 1996.

Liboff, R. L. Introductory Quantum Mechanics. San Francisco, CA: Holden-Day, 1980.

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Weisstein, Eric W. "Pauli Matrices." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PauliMatrices.html