Coulomb Wavefunctions

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

Coulomb wavefunctions are solutions of the Schrödinger equation for scattering of charged particles by a positively charged nucleus, described by a Coulomb potential. They are important for applications in the quantum theory of scattering, particularly for nuclear physics. Coulomb wavefunctions can be expressed in terms of confluent hypergeometric functions or, alternatively, Whittaker functions. Two special solutions, called the regular and irregular Coulomb wavefunctions, according to their analytic behavior at the nucleus, are denoted by and , respectively.

Contributed by: Enrique Zeleny (August 2014)
Open content licensed under CC BY-NC-SA




[1] L. J. Slater, "Coulomb Wave Functions," Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (M. Abramowitz and I. A. Stegun, eds.), 9th ed., New York: Dover, 1972 pp. 537–544.

[2] I. J. Thompson. "Coulomb Functions." Digital Library of Mathematical Functions. (Apr 25, 2014)

[3] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions, New York: Cambridge University Press, 2010.

[4] QuantumDot. "Irregular Confluent Hypergeometric Functions (Spherical Coulomb Wavefunctions)." Mathematica Stack Exchange. (Mar 22, 2013)

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.