Dirac Equation

The quantum electrodynamical law which applies to spin-1/2 particles and is the relativistic generalization of the Schrödinger equation. In 3+1 dimensions (three space dimensions and one time dimension), it is given by


(Bjorken and Drell 1964, p. 6), where h is h-bar, c is the speed of light, psi is the wavefunction, m is the mass of the particle, and alpha_i are the Dirac matrices (with alpha_4 being called beta by Bjorken and Drell 1964, p. 8; Berestetskii et al. 1982, p. 78).

The Dirac equation can also be written in the concise form


(Griffiths 1987, p. 216), where


gamma^mu are Dirac matrices in the "Dirac basis" (Griffiths 1987, p. 216), and Einstein summation has been used to sum over mu=0, 1, 2, 3.

In 1+1 dimensions, a generalization of the Dirac equation is given by the system of partial differential equations


(Alvarez et al. 1982; Zwillinger 1997, p. 137), where lambda=0 corresponds to the quantum electrodynamical equation.

See also

Klein-Gordon Equation, Schrödinger Equation

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Alvarez, A.; Pen-Yu, K.; and Vazquez, L. "The Numerical Study of a Nonlinear One-Dimensional Dirac Equation." Appl. Math. Comput. 18, 1-15, 1983.Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii, L. P. Quantum Electrodynamics, 2nd ed. Oxford, England: Pergamon Press, 1982.Bethe, H. A. and Salpeter, E. Quantum Mechanics of One- and Two-Electron Atoms. New York: Plenum, p. 37, 1977.Bjorken, J. D. and Drell, S. D. "The Dirac Equation." §1.3 in Relativistic Quantum Mechanics. New York: McGraw-Hill, pp. 6-9, 1964.Dirac, P. A. M. "The Quantum Theory of the Electron." Proc. Roy. Soc. London A117, 610-624, 1928.Dirac, P. A. M. "The Quantum Theory of the Electron, Part II." Proc. Roy. Soc. London A118, 351-361, 1928.Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, 1982.Griffiths, D. J. "The Dirac Equation" and "Solutions to the Dirac Equation." §7.1-7.2 in Introduction to Elementary Particles. New York: Wiley, pp. 213-222, 1987.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 137, 1997.

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Dirac Equation

Cite this as:

Weisstein, Eric W. "Dirac Equation." From MathWorld--A Wolfram Web Resource.

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