Advances in High Energy Physics

Volume 2017 (2017), Article ID 7937980, 19 pages

https://doi.org/10.1155/2017/7937980

## The Visualization of the Space Probability Distribution for a Moving Particle: In a Single Ring-Shaped Coulomb Potential

^{1}New Energy and Electronic Engineering, Yancheng Teachers University, Yancheng 224002, China^{2}Laboratorio de Información Cuántica, CIDETEC, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, 07700 Ciudad de México, Mexico

Correspondence should be addressed to Yuan You; moc.361@w_uoynauy, Chang-Yuan Chen; ten.361@yccctcy, and Shi-Hai Dong; moc.oohay@2hsgnod

Received 4 March 2017; Accepted 9 April 2017; Published 8 October 2017

Academic Editor: Saber Zarrinkamar

Copyright © 2017 Yuan You et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### 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-dimensional (contour) and three-dimensional (isosurface) visualizations for those specifically given quantum numbers (, , ) essentially related to those so-called* quasi*-quantum numbers (, , ) through changing the single ring-shaped Coulomb potential parameter *. *We find that the space probability distributions (isosurface) of a moving particle for the special case and the usual case are spherical and circularly ring-shaped, respectively, by considering all variables in spherical coordinates. We also study the features of the relative probability values of the space probability distributions. As an illustration, by studying the special case of the quantum numbers (, , ) = (6, 5, 1), we notice that the space probability distribution for a moving particle will move towards the two poles of the -axis as the relative probability value 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 increases.

#### 1. Introduction

Since the ring-shaped noncentral potentials have potential applications in quantum chemistry and nuclear physics (e.g., they might describe the molecular structure of benzene and interaction between the deformed nucleuses), it is not surprising that their relevant investigations have attracted much attention [1–20]. Based on a previous study, we have known that this type of ring-shaped noncentral potential can be solved in spherical coordinates and also the Hamiltonian system with the hidden symmetry makes the bound state energy levels possess an “accidental” degeneracy, which arises from the SU() invariance of the Schrödinger Hamiltonian [1]. Generally speaking, the most popular ring-shaped noncentral potentials are identified as the Coulomb or harmonic oscillator plus the ring-shaped part . In this work, we are concerned with only the single ring-shaped Coulomb potential in the limited space.

Many authors have obtained the radial and polar angular differential equations and also got their solutions in recent studies [7, 8, 13, 14, 21], but the whole space probability distributions of the moving particle in the single ring-shaped noncentral fields have never been reported due to the difficult computational skill that is required in the programming. The main contributions mentioned above are concerned either with the radial part in the spherical shell or with the angular parts in volume angle [22, 23]. This means that these studies are only related to one or two of three variables . To illustrate comprehensively the space probability distribution of the moving particle confined in the ring-shaped noncentral Coulomb potential, the aim of this work is to realize their two-dimensional (contour) and three-dimensional (isosurface) visualizations by considering all variables. Such studies have never been done to the best of our knowledge.

The rest of this work is organized as follows. In Section 2, we first present the solutions of the studied quantum system and give the concrete expressions of the angular wave functions in order to perform a comparison with the usual spherical harmonics and also show their distinct properties. In Section 3, we make use of the calculation formula of the space probability distribution to illustrate the visualizations of various cases for different choices of the parameters by overcoming the calculation skills in MATLAB program. In Section 4, we discuss the variation of the space probability distribution with the number of radial nodes and the variations with the relative probability value and those with the negative and positive ring-shaped Coulomb potential parameter . The expansion coefficients of the deformed spherical harmonics are calculated in Section 5. Some concluding remarks are given in Section 6.

#### 2. Exact Solutions to Single Ring-Shaped Coulomb Potential

The single ring-shaped Coulomb potential is given byand the Schrödinger equation is written asTake the wave function of the following form: where . Substitute (3) into (2) and get the respective radial and angular differential equations asTake a new variable transform ; hence, (4a) becomes Its solutions were given by the universal associated Legendre polynomials [22]:whereFor , notice that the differential equation (5) keeps invariant if is replaced by , and thus we know that is also the solution of (5) and its definition is given by [24, 25]

On the other hand, in terms of the spherical harmonics [26, 27],We can define deformed spherical harmonics with the following property: Obviously, is equal to when , and (8), (10), and (11) will reduce to the results in the central field. It is worth pointing out that all states are for a certain value* l*, but states are 2 (), or 1 (). This arises from the fact that the ring-shaped potential reduces the symmetry of the system and the degeneracy.

In Table 1, we list the analytical expressions of some given deformed spherical harmonics in the special cases of potential parameter and .