Coulomb Wave Function

The Coulomb wave function is a special case of the confluent hypergeometric function of the first kind. It gives the solution to the radial Schrödinger equation in the Coulomb potential (1/r) of a point nucleus


(Abramowitz and Stegun 1972; Zwillinger 1997, p. 122). The complete solution is


The Coulomb function of the first kind is




_1F_1(a;b;z) is the confluent hypergeometric function of the first kind, Gamma(z) is the gamma function. This function

The Coulomb function of the second kind is


where q_L, p_L, and a_k^L are defined in Abramowitz and Stegun (1972, p. 538).

The Coulomb wave functions of the first and second kind are implemented in the Wolfram Language as CoulombF[l, eta, r] and CoulombG[l, eta, r], respectively.

See also

Confluent Hypergeometric Function of the First Kind

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Abramowitz, M. and Antosiewicz, H. A. "Coulomb Wave Functions in the Transition Region." Phys. Rev. 96, 75-77, 1954.Abramowitz, M. and Rabinowitz, P. "Evaluation of Coulomb Wave Functions along the Transition Line." Phys. Rev. 96, 77-79, 1954.Abramowitz, M. and Stegun, I. A. (Eds.). "Coulomb Wave Functions." Ch. 14 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 537-544, 1972.Biedenharn, L. C.; Gluckstern, R. L.; Hull, M. H. Jr.; and Breit, G. "Coulomb Wave Functions for Large Charges and Small Velocities." Phys. Rev. 97, 542-554, 1955.Bloch, I.; Hull, M. H. Jr.; Broyles, A. A.; Bouricius, W. G.; Freeman, B. E.; and Breit, G. "Coulomb Functions for Reactions of Protons and Alpha-Particles with the Lighter Nuclei." Rev. Mod. Phys. 23, 147-182, 1951.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 631-633, 1953.National Bureau of Standards. Tables of Coulomb Wave Functions, Vol. 1, Applied Math Series 17. Washington, DC: U.S. Government Printing Office, 1952.Stegun, I. A. and Abramowitz, M. "Generation of Coulomb Wave Functions by Means of Recurrence Relations." Phys. Rev. 98, 1851-1852, 1955.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997.

Referenced on Wolfram|Alpha

Coulomb Wave Function

Cite this as:

Weisstein, Eric W. "Coulomb Wave Function." From MathWorld--A Wolfram Web Resource.

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