Coulomb wave function

In mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument.

Coulomb wave equation

The Coulomb wave equation is

$\frac{d^2w}{d\rho^2}+\left(1-\frac{2\eta}{\rho}-\frac{L(L+1)}{\rho^2}\right)w=0$

where L is usually a non-negative integer. The solutions are called Coulomb wave functions. Putting x = 2iρ changes the Coulomb wave equation into the Whittaker equation, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments. Two special solutions called the regular and irregular Coulomb wave functions are denoted by F_L(η,ρ) and G_L(η,ρ), and defined in terms of the confluent hypergeometric function by

$F_L(\eta,\rho) = \frac{2^Le^{-\pi\eta/2}|\Gamma(L+1+i\eta)|}{\Gamma(2L+2)}\rho^{L+1}e^{-i\rho}M(L+1-i\eta,2L+2,2i\rho)$

References

• Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 14", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 538, ISBN 978-0486612720, MR 0167642 .
• Bateman, Harry (1953), Higher transcendental functions 1, McGraw-Hill .
• Jaeger, J. C.; Hulme, H. R. (1935), "The Internal Conversion of γ -Rays with the Production of Electrons and Positrons", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (The Royal Society) 148 (865): 708–728, ISSN 0080-4630, JSTOR 96298 
• Slater, Lucy Joan (1960), Confluent hypergeometric functions, Cambridge University Press, MR 0107026 .
• Thompson, I. J. (2010), "Coulomb Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248