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



using the form of the -dimensional Laplacian (Laplace–Beltrami operator) in hyperspherical coordinates. Here represents a generalized angular momentum quantum number, (reducing to for and to for ). The normalized solutions of the radial equation work out to

, ,

where is an associated Laguerre polynomial. The corresponding energy eigenvalues are given by


independent of . The allowed values of are .

Choosing "radial functions" shows plots of and the corresponding radial distribution functions (RDF) for selected values of , and (with . Choosing "3D plot of RDF" shows a three-dimensional plot of the RDF in which the dimension can be varied continuously (including through nonphysical noninteger values). This shows how the wavefunctions of the hydrogen atom spread out into space of increasing dimensionality.


Contributed by: S. M. Blinder (January 2021)
Open content licensed under CC BY-NC-SA



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


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


[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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