Effects of confinement on two-dimensional hydrogenic atoms

In this work we study quantum confinement effects on two-dimensional hidrog enic atoms (2DHA) as a simple example that, at the same time, is higly rele vant to current research of quantum phenomena in mesoscopic systems. Our mo del considers that the electron is confined to move to the interior of a ci rcular region centered on the nucleus: confined two-dimensional hydrogenic atom (C2DHA). In this case it is not possible to obtain explicit analytic s olutions in a closed form, reason for which we use a first order perturbati on theory, as well as a variational approach. For the later method, the tes t functions are proposed based on the wave functions of a free 2DHA, but al so including cut-off terms that make them satisfy at the same lime the boun dary conditions and the cilyndrical symmetry of the system. Our results ten d to the expected ones when the radius of the confining region tends either to zero or to infinity: free particle in a box or 2DHA, respectively. The first order perturbation results give very simple expressions where the com petition between the kinetic energy and the coulombian attraction can be re adily appreciated. Comparing both approaches we see that for small radii we obtain very similar results, but that the variational approach presents nu merical difficulties, for the first excited state with cero angular momentu m. This behaviour presents itself for radii of the confinig circular box th at are comparable with the mean expected value of the radial coordinate in the ground state of a free 2DHA, a 2D Bohr radius. Besides, it is observed that confinement effects are more severe in two than in three-dimensional s ystems.