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
![(d^2W)/(drho^2)+[1-(2eta)/rho-(L(L+1))/(rho^2)]W=0](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FCoulombWaveFunction%2FNumberedEquation1.svg){kind=small} |
(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
![G_L(eta,rho)=(2eta)/(C_0^2(eta))F_L(eta,rho)[ln(2rho)+(q_L(eta))/(p_L(eta))]
+1/((2L+1)C_L(eta))rho^(-L)sum_(K=-L)^inftya_k^L(eta)rho^(K+L),](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FCoulombWaveFunction%2FNumberedEquation5.svg) |
(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.
See also
Confluent Hypergeometric Function of the First Kind
Explore with Wolfram|Alpha
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
Cite this as:
Weisstein, Eric W. "Coulomb Wave Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CoulombWaveFunction.html