Zero-Energy Limit of Coulomb Wavefunctions

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In the limit of zero energy, the Coulomb–Schrödinger equation in atomic units simplifies to



with the radial function satisfying


This can be solved exactly, giving a compact form for the radial function for angular momentum :


where is a Bessel function.

This Demonstration shows that the given limiting form is obtained, both for the discrete eigenfunctions in the limit and for the continuum eigenfunctions as . The eigenfunctions are shown in black, while the Bessel-function limit is drawn in red.

It is also amusing to see the behavior of the radial distribution function (RDF) for larger values of .


Contributed by: S. M. Blinder (June 2019)
Open content licensed under CC BY-NC-SA


The hydrogenic eigenfunctions, expressed in terms of confluent hypergeometric functions , are given in [1]. The normalization constants are trimmed so as to coincide with the Bessel function limit as for the discrete eigenfunctions and for the continuum. Thus we take


for the discrete and continuum eigenfunctions, respectively.

The limiting behavior as or can be deduced from the asymptotic limit [2]:

as .


[1] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, New York: Academic Press, 1957 pp. 21–25.

[2] "Confluent Hypergeometric Functions." NIST Digital Library of Mathematical Functions. (Jun 18, 2019)


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