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

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 $\le <r>\le (48.87\pm 0.03)\times {10}^{-8}$~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.