The laws of physics in d spatial dimensions are interesting and often lead to insights concerning the laws of physics in three spatial dimensions. A hydrogenic bound system in d-dimensions is investigated where d is any real positive number, not necessarily an integer, and where a d-dimensional radial Schrödinger equation is defined by the analytic continuation of the Schrödinger equation used in integer dimensions, with a d-dimensional potential defined by Gauss’ law. The fundamental constants and physical relationships in d dimensions are arrived at by assuming the electric flux obeys a Gausslike law in d dimensions. This fixes the potential and makes it possible to write down the d-dimensional Schrödinger equation. This equation is then solved numerically for the ground-state energy eigenvalues, for several different dimensions, using a fourth-order Runge–Kutta predictor algorithm. The solutions to this equation are compared to the analogous Bohr model energies in d dimensions and to the results obtained by others at well-known integer dimensions. There is good qualitative agreement between the two theories in all dimensions. In addition, the wave functions for the Schrödinger equation are numerically determined and normalized in order to locate the most probable orbital radius. These values are compared to the d-dimensional Bohr radii.
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Andrew et al. (1990) studied this question.