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 (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 l=m and the usual case l≠m are spherical and circularly ring-shaped, respectively, by considering all variables r→=(r,θ,φ) 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 the two poles of the z-axis 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.
Highlights
IntroductionSince 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]
Since the ring-shaped noncentral potentials have potential applications in quantum chemistry and nuclear physics, it is not surprising that their relevant investigations have attracted much attention [1–20]
It is found that the graphics become compressed; that is, the space probability distributions elongate along with the x- and y-axis and the hole formed in the ring-shaped potentials expands towards the outside as the ring-shaped potential parameter b increases
Summary
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]. The main contributions mentioned above are concerned either with the radial part in the spherical shell (r, r + dr) or with the angular parts in volume angle dΩ [22, 23] This means that these studies are only related to one or two of three variables (r, θ, φ). To illustrate comprehensively the space probability distribution of the moving particle confined in the ring-shaped noncentral Coulomb potential, Advances in High Energy Physics 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.
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call