Coulomb Wave Function

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

$${d^2W\over d\rho^2} + \left[{1-{2\eta\over\rho}-{L(L+1)\over\rho^2}}\right]W = 0.$$ (1)

The complete solution is

$$W=C_1F_L(\eta,\rho)+C_2G_L(\eta,\rho).$$ (2)

The Coulomb function of the first kind is

$$F_L(\eta,\rho) = C_L(\eta)\rho^{L+1}e^{-i\rho}{}_1F_1(L+1-i\eta;2L+2;2i\rho),$$ (3)

where

$$C_L(\eta)\equiv {2^Le^{-\pi\eta/2}\vert\Gamma(L+1+i\eta)\vert\over\Gamma(2L+2)},$$ (4)

${}_1F_1(a;b;z)$ is the Confluent Hypergeometric Function, $\Gamma(z)$ is the Gamma Function, and the Coulomb function of the second kind is

$G_L(\eta,\rho) = {2\eta\over {C_0}^2(\eta)} F_L(\eta,\rho)\left[{\ln(2\rho)+{q_L(\eta)\over p_L(\eta)}}\right]$
$+{1\over (2L+1)C_L(\eta)}\rho^{-L} \sum_{K=-L}^\infty a_k^L(\eta)\rho^{K+L},\quad$	(5)

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

References

Abramowitz, M. and Stegun, C. 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.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 631-633, 1953.