# Hydrogen Atom in Varying Dimensions

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This Demonstration considers the hypothetical problem of the hydrogen atom in -dimensional space (other than ). Specifically, we consider the radial function obeying the radial Schrödinger equation in atomic units

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Contributed by: S. M. Blinder (January 2021)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The general solution of the radial differential equation can be found using DSolve. Restrictions on the values of and follow from the requirement that the Laguerre functions reduce to polynomials, in order that remains finite as . The normalization constants can be determined using the integrals

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The problem has been considered in [1–3].

References

[1] F. Carusoa, J. Martins and V. Oguri, "On the Existence of Hydrogen Atoms in Higher Dimensional Euclidean Spaces," *Physics Letters A*, 377(9), 2013 pp. 694–698. doi:10.1016/j.physleta.2013.01.026.

[2] S. M. Al-Jaber, "Hydrogen Atom in N Dimensions," *International Journal of Theoretical Physics*, 37(4), 1998 pp. 1289–1298. doi:10.1023/A:1026679921970.

[3] X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong and W. Y. Ching, "Analytic Solution of a Two-Dimensional Hydrogen Atom. I. Nonrelativistic Theory," *Physical Review A,* 43(3), 1991 pp. 1186–1196. doi:10.1103/physreva.43.1186.

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