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

TL;DR

This paper presents exact solutions and visualizations of the space probability distribution for a particle in a ring-shaped Coulomb potential, revealing how distributions change with quantum numbers and potential parameters.

Contribution

It provides the first detailed visualization of space probability distributions in a ring-shaped Coulomb potential and analyzes their dependence on quantum numbers and potential parameters.

Findings

Distributions are spherical or ring-shaped depending on quantum numbers.

Probability distribution shifts towards poles as probability increases.

Principal component of spherical harmonics decreases with increasing potential parameter.

Abstract

We first present the exact solutions of the single ring-shaped Coulomb potential and then realize the visualizations of the space probability distribution for a moving particle within the framework of this potential. We illustrate the two-(contour) and three-dimensional (isosurface) visualizations for those specifically given quantum numbers (n, l, m) essentially related to those so-called quasi quantum numbers (n',l',m') through changing the single ring-shaped Coulomb potential parameter b. We find that the space probability distributions (isosurface) of a moving particle for the special case and the usual case are spherical and circular ring-shaped, respectively by considering all variables in spherical coordinates. We also study the features of the relative probability values P of the space probability distributions. As an illustration, by studying the special case of the quantum numbers (n, l, m)=(6, 5, 1) we notice that the space probability distribution for a moving particle will move towards two poles of axis z as the relative probability value P increases. Moreover, we discuss the series expansion of the deformed spherical harmonics through the orthogonal and complete spherical harmonics and find that the principal component decreases gradually and other components will increase as the potential parameter b increases.