Electronic energy levels and wave-functions evolution from 3-D to 2-D of a hydrogen atom confined by two parallel planes

Abstract

The compression of an atom produced by two planes induces a change in its electronic structure that evolves from a free atom in 3-D to a 2-D atom. This behavior is of importance in low-dimensional materials and high compression produced by an anvil cell. In this work, we study the evolution of the energy levels and electronic wave-functions of a hydrogen atom placed between two impenetrable planes as a function of the inter-plane separation through a numerical approach. As the inter-plane separation is reduced, the electron motion is restricted along the direction normal to the planes, similar to a particle in a box, while leaving the electron to move unrestricted along the planes, thus, breaking the spherical geometry of the H atom caused by the planes’ compression. The energy levels evolve from 3-D, described by nlm quantum numbers to a 2-D described by n′ml′, where l′ is the quantum number for a particle in a box along the z direction and n′ is the principal quantum number of the 2-D atom radial direction. We evaluate the energy levels from 3-D to 2-D and the radial average distance ⟨ρ⟩ in cylindrical coordinates, as a function of the inter-plane separation D along the z-direction. We find that as the inter-plane separation is reduced, the angular momentum quantum number l merges to the z-component of the angular momentum and it produces two branches, a symmetric for l-even and one anti-symmetric for l-odd, connected to a particle in a box quantum number l′ along the z-axis with implications in the atom photo-luminescence, resulting from the symmetry of the system. Furthermore, states with l-odd merge with states with l-even, as they have the same energy and average distance when D → 0. We provide an Aufbau principle for it. Our results agree to the analytical solutions at the 3-D and 2-D limiting cases.