# 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

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Contributed by: S. M. Blinder (June 2019)

Open content licensed under CC BY-NC-SA

## Details

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

and

for the discrete and continuum eigenfunctions, respectively.

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

as .

References

[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) dlmf.nist.gov/13.8.E9.

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