Hydrogen Atom in a Fuzzy Spherical Cavity

Abstract

The fuzzy onion model formed by connecting a set of concentric fuzzy spheres of increasing radius is motivated by studies of quantum space but can also be used to study standard physics. The main feature of the model is that functions in three-dimensional space -- like scalar fields or wavefunctions -- are expressed in terms of Hermitian matrices of a certain structure. Relevant equations are then matrix equations, and some problems, such as searching for the energy spectrum for fixed quantum numbers $(l,m)$, can be expressed as an eigenvalue problem. We show how this simple approach can reproduce the results of other studies analyzing the hydrogen atom in a spherical cavity. We also test the effect of the short-distance quantum structure of the space on these solutions -- not looking for the phenomenological consequences, as the scale of quantum space is many orders below the order of the Bohr radius, but to understand the effect of quantum space in general. We observe a set of solutions without a classical counterpart which have been suggested also in a former theoretical study.