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# 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 () of a point nucleus

 (1)

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

 (2)

The Coulomb function of the first kind is

 (3)

where

 (4)

is the confluent hypergeometric function of the first kind, is the gamma function. This function

The Coulomb function of the second kind is

 (5)

where , , and 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.

Confluent Hypergeometric Function of the First Kind

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## References

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. https://mathworld.wolfram.com/CoulombWaveFunction.html