Energy spectrum of the hydrogen atom in a space with one compactified extra dimension, ${\mathbb{R}}^3\times S^1$

We investigate the consequences of one extra compactified dimension for the energy spectrum of the non-relativistic hydrogen atom with a potential defined by Gauss’ law, i.e. proportional to $1\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2$ in non-compactified 4d space. The calculations were performed numerically by diagonalizing the Hamiltonian in two different sets of basis vectors. The energy levels and electron probability density are plotted as a function of the compactification radius. The occurrence of several physical effects is discussed and interpreted.