Revisiting the Two-Dimensional Hydrogen Atom: Azimuthal Wavefunctions for Illustrating s, p, d, and f Orbitals

Abstract

The two-dimensional (2D) hydrogen atom is a fundamental atomic model that is important for various technologies based on 2D materials. Here, the atomic model is revisited to enhance understanding of the hydrogen wavefunctions. Unlike in previous studies, we propose an alternative expression of azimuthal wavefunctions, which are the eigenstates of the square of angular momentum and exhibit rotational symmetry. Remarkably, our expression leads to the rotation and oscillation along the azimuthal direction of the probability densities, which do not appear in the conventional wavefunctions. These behaviors are validated by the numerical results obtained through the 2D finite difference approach. Variation in oscillator strengths due to the rotation of wavefunctions is observed in our proposed 2D hydrogen wavefunctions, whereas those due to the conventional wavefunctions remain constant. More importantly, the proposed wavefunctions’ advantage is illustrating the orbital shapes of the planar hydrogen states, whose orientation is labeled here using Cartesian representation for the first time. This study can be applied to visualize the orbital characteristics of the states in quantum confinement with a radial potential.