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:

[more]

,

in atomic units with curvature parameter . In this model, the continuum is replaced by a closely spaced discrete spectrum, resembling the eigenvalues of the particle-in-a-box. The system is described by hyperspherical coordinates in four-dimensional Euclidean space, such that the Cartesian coordinates are given by

,

,

,

,

with

, , .

The Coulomb potential is given by

,

while the Laplacian can be written

,

where is the usual three-dimensional angular momentum operator. The Schrödinger equation can be solved exactly. The radial equation reduces to

.

The (unnormalized) solutions are given by

,

,

where is a Jacobi polynomial.

The left graphic shows plots of the radial function for selected values of , , and . The right graphic shows the corresponding hydrogenic functions in Euclidean space, which represent the limit as the curvature approaches 0, or .

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Contributed by: S. M. Blinder (May 2018)
Open content licensed under CC BY-NC-SA


Snapshots


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: Mathematical 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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