The visualization of the space probability distribution for a particle moving in a double ring-shaped Coulomb potential

TL;DR

This paper analytically solves and visualizes the space probability distribution of a particle in a double ring-shaped Coulomb potential, revealing how quantum numbers and potential parameters influence the distribution's shape.

Contribution

It provides the first analytical solutions and detailed visualizations of the space probability distribution for a double ring-shaped Coulomb potential, including effects of potential parameters.

Findings

SPDs are circular rings in spherical coordinates.

SPD shape shifts towards poles with increasing probability values.

Potential parameter b significantly affects SPD spread.

Abstract

The analytical solutions to a double ring-shaped Coulomb potential (RSCP) are presented. The visualizations of the space probability distribution (SPD) are illustrated for the two-(contour) and three-dimensional (isosurface) cases. The quantum numbers (n, l, m) are mainly relevant for those quasi quantum numbers (n' ,l' ,m' ) via the double RSCP parameter c. The SPDs are of circular ring shape in spherical coordinates. The properties for the relative probability values (RPVs) P are also discussed. For example, when we consider the special case (n, l, m)=(6, 5, 0), the SPD moves towards two poles of axis z when the P increases. Finally, we discuss the different cases for the potential parameter b which is taken as negative and positive values for c>0 . Compared with the particular case b=0 , the SPDs are shrunk for b=-0.5 while spread out for b=0.5.