# Hydrogen Atom in Curved Space

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In 1940, Schrödinger considered the hypothetical problem of a hydrogen atom in a space of positive curvature [1, 2]. The actual curvature of space is far too feeble to have any detectable physical effect on an atomic scale, but this might become relevant in the neighborhood of a black hole or neutron star. Schrödinger was able to derive an exact solution with a remarkably simple energy spectrum:

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Contributed by: S. M. Blinder (May 2018)

Open content licensed under CC BY-NC-SA

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## Details

The canonical partition function for the hydrogen atom had long presented a paradox since the summation over just the bound states

is divergent. However, by using the curved-space energies

,

a finite result can be derived [3], with the radius of curvature of the four-dimensional hypersphere serving as a "metaphor" for the volume in three-dimensional space, with the correspondence

.

References

[1] E. Schrödinger, "A Method of Determining Quantum-Mechanical Eigenvalues and Eigenfunctions," *Proceedings of the Royal Irish Academy. Section A: Mathematica*l and Physicial Sciences, 46, 1940 pp. 9–16. www.jstor.org/stable/20490744.

[2] N. Bessis and G. Bessis, "Electronic Wavefunctions in a Space of Constant Curvature," *Journal of Physics A: Mathematical General*, 12(11), 1979 pp. 1991–1997.

[3] S. M. Blinder, "Canonical Partition Function for the Hydrogen Atom in Curved Space," *Journal of Mathematical Chemistry*, 19(1), 1996 pp. 43–46. doi:10.1007/BF01165129.

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