The Visualization of the Space Probability Distribution for a Particle Moving in a Double Ring-Shaped Coulomb Potential.
Since the ring-shaped noncentral potentials (RSNCPs) are used to describe the molecular structure of Benzene as well as the interaction between the deformed nucleuses, they have attracted much attention of many authors [1-14]. Generally, these RSNCPs are chosen as the sum of the Coulomb or harmonic oscillator and the single ring-shaped part 1/([r.sup.2][sin.sup.2][theta]) or the double ring-shaped part 1/([r.sup.2][sin.sup.2][theta]) + 1/([r.sup.2][cos.sup.2][theta]). In this work, what we are only interested in is the double RSCP, which may be used to describe the properties of ring-shaped organic molecule. The corresponding bound states were investigated by SUSY quantum mechanics and shape invariance . Recently, other complicated double RSCPs have also been proposed [16-25]. Many authors have obtained their solutions in [7,8,13,14,26]. Among them, the SPDs have been carried out, but their studies are treated either for the radial part in spherical shell (r, r + dr) or for the angular parts [27,28]. The discussions mentioned above are only concerned with one or two of three variables (r, [theta], [phi]). To show the SPD in all position spaces, we have studied the SPD of a single RSCP for two- and three-dimensional visualizations . In this work, our aim is to focus on the more comprehensive SPD for the particle moving in a double RSCP.
The plan of this paper is as follows. We present the exact solutions to the system in Section 2. In Section 3, we apply the SPD formula to show the visualizations using the similar technique in  getting over the difficulty appearing in the calculation skill when using MATLAB program. We discuss their variations on the number of radial nodes, the RPV P, and the RSCP parameter b (positive and negative) when c [not equal to] 0 in Section 4. We give our concluding remarks in Section 5.
2. Exact Solutions to a Double RSCP
In the spherical coordinates, the double RSCP is given by
V(r, [theta]) = -Z[e.sup.2]/r + [h.sup.2]/2M[r.sup.2](b/[sin.sup.2][theta] + c/[cos.sup.2][theta]) (1)
as plotted in Figures 1-3.
The Schrodinger equation with this potential is written as (h = M = e = 1)
[-1/2[[nabla].sup.2] - Z/r + 1/2[r.sup.2](b/[sin.sup.2][theta] + c/[cos.sup.2][theta])][PSI]([??]) = E[PSI]([??]). (2)
Take wave function as
[PSI]([??]) = 1/[square root of (2[pi])]u(r)/rH([theta])[e.sup.[+ or -]im[phi]], m = 0,1,2,... (3)
Substitute this into (2) and obtain the following differential equations:
[d.sup.2]u(r)/d[r.sup.2] + (2E + 2Z/r - [lambda]/[r.sup.2])u(r) = 0, (4a)
1/sin[theta] d/d[theta](sin[theta] dH([theta])/d[theta]) + ([lambda] - b + [m.sup.2]/[sin.sup.2][theta] - c/[cos.sup.2][theta])H([theta]) = 0, (4b)
where [lambda] is a separation constant. Define x = cos[theta]; (4b) is modified as
(1 - [x.sup.2])[d.sup.2]H(x)/d[x.sup.2] - 2x [dH(x)/dx] + (l'(l' + 1) - [(m').sup.2]/1 - [x.sup.2] - c/[x.sup.2])H(x) = 0. (5)
Its solutions are given by 
[mathematical expression not reproducible], (6)
[mathematical expression not reproducible], (7)
[mathematical expression not reproducible]. (8)
We are now in the position to consider (4a). Substituting [lambda] = l'(l' + 1) into (4a) and taking [chi] = [tau]r, s = 2Z, and [tau] = Z[square root of (-1/(2E))], from (4a) we have
[d.sup.2]u([chi])/d[chi square] + (s/[chi] - 1/4 - l'(l' + 1)/[chi square])u([chi]) = 0, (9)
whose solutions are given by 
[mathematical expression not reproducible], (10)
[mathematical expression not reproducible]. (11)
and the Bohr radius [a.sub.0] = [h.sup.2]/M[e.sup.2] = 1. The complete wave function has the form
[[PSI].sub.n'l'm]([??]) = [1/[square root of (2[pi])]][[u.sub.n'l'](r)/r] [H.sub.l'm'](cos [theta]) [e.sup.[+ or -]]im[phi]. (12)
3. Two- and Three-Dimensional Visualizations of SPDs
As we know, the SPDs at the position [??] = (r, [theta], [phi]) are calculated by
[rho] = [[absolute value of ([[PSI].sub.n'l'm']([??]))].sup.2] = [1/2[pi]] [[u.sup.2.sub.n'l'](r)/[r.sup.2]] [H.sup.2.sub.l'm'](cos [theta]). (13)
To show the SPD, let us transform (13) to popular Cartesian coordinates via the relations r = [square root of ([x.sup.2] + [y.sup.2] + [z.sup.2])] and cos [theta] = z/r. Thus, one is able to find the corresponding SPD [rho](x, y, z).
Taking a series of discrete positions, we may study the values of the respective SPD by numerical calculation. In order to make the graphic resolution better, one takes N discrete positions in the Cartesian space (x, y, z) and studies density block, say den(N,N,N), which is composed of all values [w.sub.n'l'm'] for all N x N x N positions. Here, N is taken as 151. For states denoted by (n', l', m'), we display their two- and three-dimensional visualizations for different states (n [less than or equal to] 6) using MATLAB program as shown in Tables 1 and 2.
4. Discussions on the SPD
4.1. Variation Caused by the Radial Nodes. We show the SPDs for various cases b = 0.5 and c = 0, 0.5, 5 (see Table 1). The case c = 0 corresponds to a single RSCP, which was discussed in our previous works [29,30]. We take the unit in axis as the Bohr radial [a.sub.0]. It should be pointed out that we plot the figures only for the value of [n.sub.r] = n - l - 1 equal to integer. To display the inside structure of the graphics, we create a section plane but need not consider SPDs numerical values in the regions x < 0, y < 0, z > 0.
Compared to the cases c = 0 and b [not equal to] 0, it is seen that the graphics are expanded. That is to say, the SPDs enlarge towards z-axis and the hole is expanded outside when the potential parameter c increases. We may understand it through considering (8). As we know, m' will increase relatively for a fixed m. For l [not equal to] m, their isosurfaces are of circular ring shape.
We project the SPDs to a plane yoz and find that they are symmetric to the y-axis and z-axis (see Table 2). In this work, the graphics are plotted only in the first quadrant by enlarging proportionally the probability [[absolute value of ([[PSI].sub.n'l'm']([??])].sup.2] and by making the maximum value as 100, in which the interval is taken as 10. A corresponding balance among the density distributions exists in the directions of axes x, y, and z because the sum of density distributions has to be equal to one when considering the normalization condition. It is clear that each figure becomes expanded along with y-axis and z-axis.
4.2. Variation on RPV P. To show the isosurface of the SPDs for various RPVs P [member of] (0,100)%, the quantum numbers (n, l, m) = (6, 5, 0) for two different cases b = 0.5 and c = 0. 5, 10 are taken as seen in Table 3. It is shown that the particle for smaller P will be distributed to almost all spaces. However, the particle for larger P will move to the poles in z-axis.
4.3. Variations on Various Potential Parameters b and c. Considering given quantum numbers m and [n.sub.[theta]], we know from (8) that m' will become bigger as b increases and the parameter [[gamma].sub.1] also becomes larger with increasing c. As a result, this will result in increasing l'. In Table 4, the SPDs are plotted for state (n, l, m) = (5, 1, 0) in the cases of c = 0.5 and b = 0, 5, 10, 25, 40, 80 and b = 0.5 and c = 0, 5, 10, 25, 40, 80, respectively. Obviously, we see a big difference between them. When the potential parameter b increases, the expansions of the SPDs along with x-axis and y-axis and the number of radial nodes are changed. However, when the parameter c increases, the expansions of the SPDs are along with z-axis.
The comparison is done for positive and negative b <0 and b > 0 when c = 0.5 (see Table 5). It is shown that the SPDs for the negative b = -0.5 compared with the case b = 0 are shrunk into the origin. However, the SPDs for b = 0.5 are enlarged outside. We can understand it very well by studying the contributions of the potential parameter b made on the Coulomb potential. The choice of the negative or positive b determines the attractive Coulomb potential that is bigger or smaller relatively. Thus, the attractive force that acts on the particle will be larger or smaller. Finally, this will result in the SPDs that are shrunk or expanded.
The analytical solutions to the double RSCP have been obtained and then the visualization of the SPDs for this potential is performed. The contour and isosurface visualizations have been illustrated for quantum numbers (n', l', m') by taking various values of the parameter c. It is shown that the SPDs are of circular ring shape. On the other hand, the properties of the RPVs P of the SPDs have also been discussed. As an example, we have studied the particular case, that is, (n, l, m) = (6, 5, 0), and found that the SPDs will move towards the poles of z-axis when the RPVs P increase.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
This work is supported by the National Natural Science Foundation of China under Grant no. 11275165, partially by 20180677-SIP-IPN, and by CONACYT, Mexico, under Grant no. 288856-CB-2016. Professor Yuan You acknowledges Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-Aged Teachers and Presidents for support.
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Yuan You (iD), (1) Fa-Lin Lu, (1) Dong-Sheng Sun, (1) Chang-Yuan Chen (iD), (1) and Shi-Hai Dong (iD) (2)
(1) New Energy and Electronic Engineering, Yancheng Teachers University, Yancheng 224002, China
(2) Laboratorio de Informacian Cuantica, CIDETEC, Instituto Politacnico Nacional, Unidad Profesional Adolfo Lopez Mateos, 07700 Ciudad de Mexico, Mexico
Correspondence should be addressed to Yuan You; email@example.com, Chang-Yuan Chen; firstname.lastname@example.org, and Shi-Hai Dong; email@example.com
Received 1 October 2017; Accepted 20 February 2018; Published 24 April 2018
Academic Editor: Saber Zarrinkamar
Caption: Figure 1: V(r, [theta]) as the functions of variables [theta] and r.
Caption: Figure 2: Potential function V(r, [theta]) versus [theta] at r = 0.1,1,10,100.
Caption: Figure 3: Potential function V(r, [theta]) versus r at [theta] = 0.1[degrees], 30[degrees], 45[degrees], 89[degrees].
Caption: Table 1: The isosurface SPDs with a section plane.
Caption: Table 2: The contour of the SPDs in the plane yoz.
Caption: Table 3: The SPDs for various RPVs P for (n, l, m) = (6, 5, 0)(b = 0.5, c = 0.5, 10).
Caption: Table 4: The isosurface illustration of the state (5, 1, 0) with various values of b and c.
Caption: Table 5: SPDs for different cases of the value of of b for (4, 1, 0) when c = 0.5.
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|Title Annotation:||Research Article|
|Author:||You, Yuan; Lu, Fa-Lin; Sun, Dong-Sheng; Chen, Chang-Yuan; Dong, Shi-Hai|
|Publication:||Advances in High Energy Physics|
|Date:||Jan 1, 2018|
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