Wave functions and energy spectra for the hydrogenic atom in $\mathbb {R}^3 \times \mathcal {M}$ R3×M

Abstract

We consider the hydrogenic atom in a space of the form \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3 \times \mathcal {M}$\end{document}R3×M, where \documentclass[12pt]{minimal}\begin{document}$\mathcal {M}$\end{document}M may be a generalized manifold obeying certain properties. We separate the solution to the governing time-independent Schrödinger equation into a component over \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3$\end{document}R3 and a component over \documentclass[12pt]{minimal}\begin{document}$\mathcal {M}$\end{document}M. Upon obtaining a solution to the relevant eigenvalue problems, we recover both the wave functions and energy spectrum for the hydrogenic atom over \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3 \times \mathcal {M}$\end{document}R3×M. We consider some specific examples of \documentclass[12pt]{minimal}\begin{document}$\mathcal {M}$\end{document}M, including the fairly simple D-dimensional torus \documentclass[12pt]{minimal}\begin{document}$T^D$\end{document}TD and the more complicated Kähler conifold \documentclass[12pt]{minimal}\begin{document}$\mathcal {K}$\end{document}K in order to illustrate the method. In the examples considered, we see that the corrections to the standard energy spectrum for the hydrogen atom due to the addition of higher dimensions scale as a constant times \documentclass[12pt]{minimal}\begin{document}$1/L^2$\end{document}1/L2, where L denotes the size of the additional dimensions. Thus, under the assumption of small compact extra dimensions, even the first energy corrections to the standard spectrum will be quite large.