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
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.
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.
) of a point nucleus
![(d^2W)/(drho^2)+[1-(2eta)/rho-(L(L+1))/(rho^2)]W=0](/images/equations/CoulombWaveFunction/NumberedEquation1.svg)
is the confluent hypergeometric
function of the first kind, and 
is the gamma function.
![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),](/images/equations/CoulombWaveFunction/NumberedEquation5.svg)
, 
, and 
are defined in Abramowitz and Stegun (1972, p. 538).
