Effects on the Non-Relativistic Dynamics of a Charged Particle Interacting with a Chern-Simons Potential

TL;DR

This paper numerically investigates a two-dimensional hydrogen atom with a Chern-Simons potential, analyzing its wave functions, eigenvalues, and dependence on photon mass, revealing weak binding similar to Rydberg atoms.

Contribution

It provides the first numerical solutions for a 2D hydrogen atom with a Chern-Simons potential, exploring how topological photon mass affects its energy spectrum and structure.

Findings

Eigenvalues are in the range -3.5e-3 eV to -9.0e-2 eV.

The atom exhibits very weak binding, similar to Rydberg states.

Energy and radius depend on the photon mass scale.

Abstract

The hydrogen atom in two dimensions, described by a Schr\"odinger equation with a Chern-Simons potential, is numerically solved. Both its wave functions and eigenvalues were determined for small values of the principal quantum number $n$. The only possible states correspond to $l=0$. How the result depends on the topological mass of the photon is also discussed. In the case $n=1$, the energy of the fundamental state corresponding to different choice for the photon mass scale are found to be comprehended in the interval $-3,5 \times 10^{-3} eV \leq E \leq -9,0 \times 10^{-2} eV$, corresponding to a mean radius of the electron in the range $ (5.637 \pm 0.005) \times 10^{-8}$~cm $\leq <r> \leq (48.87 \pm 0.03) \times 10^{-8}$~cm. In any case, the planar atom is found to be very weekly bounded showing some features similar to the Rydberg atoms in three dimensions with a Coulombian interaction.