Study of a hydrogenic atom near a rigid wall by using the finite element method

Abstract

A direct numerical solution of the Schrödinger equation for a hydrogenic atom near a rigid wall, and the corresponding graphics of some of the energies and wave functions are presented. The solution is obtained by using the finite-element method (FEM) with the appropriate boundary conditions. In particular, crossings of the von Neumann–Wigner type are observed and a classification is suggested of different eigenenergy sets, according to symmetries of the system. The limit with the nucleus being on the wall and the 3D bulk limit are also studied. A discussion of a possible role of the model for the treatment of atom–surface interactions is presented. This system can also be used as a model for 2D and 3D quantum devices, such as shallow state of impurity atoms or Wannier exciton near interfaces.