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Pauli Matrices

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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

sigma_1=sigma_x=P_1=[ 0 1; 1 0]
(1)
sigma_2=sigma_y=P_2=[ 0 -i; i 0]
(2)
sigma_3=sigma_z=P_3=[ 1 0; 0 -1]
(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 sigma are implemented in the Wolfram Language as PauliMatrix[n], where n=1, 2, or 3.

The Pauli spin matrices satisfy the identities

sigma_i^2=I
(4)
sigma_isigma_j+sigma_jsigma_i=2delta_(ij)I
(5)
sigma_isigma_j=Idelta_(ij)+iepsilon_(ijk)sigma_k,
(6)

where I is the 2×2 identity matrix, delta is the Kronecker delta, epsilon is the permutation symbol, the leading i is the imaginary unit (not the index i), and Einstein summation is used in (6) to sum over the index k (Arfken 1985, p. 211; Griffiths 1987, p. 139; Landau and Lifschitz 1991, pp. 204-205).

The Pauli matrices plus the 2×2 identity matrix I form a complete set, so any 2×2 matrix A can be expressed as

A=c_0I+c_1sigma_1+c_2sigma_2+c_3sigma_3.
(7)

The associated matrices

sigma_+=2[0 1; 0 0]
(8)
sigma_-=2[0 0; 1 0]
(9)
sigma^2=3[1 0; 0 1]
(10)

can also be defined.

See also

Dirac Matrices, Quaternion

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References

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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Pauli Matrices

Cite this as:

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

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