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 are mainly relevant for those quasi-quantum numbers via the double RSCP parameter . The SPDs are of circular ring shape in spherical coordinates. The properties for the relative probability values (RPVs) are also discussed. For example, when we consider the special case , the SPD moves towards two poles of -axis when increases. Finally, we discuss the different cases for the potential parameter , which is taken as negative and positive values for . Compared with the particular case , the SPDs are shrunk for , while they are spread out for .
1. Introduction
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 or the double ring-shaped part . 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 [15]. 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 or for the angular parts [27, 28]. The discussions mentioned above are only concerned with one or two of three variables . To show the SPD in all position spaces, we have studied the SPD of a single RSCP for two- and three-dimensional visualizations [29]. 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 [29] 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 , and the RSCP parameter (positive and negative) when 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 byas plotted in Figures 1–3.
The Schrödinger equation with this potential is written as ()Take wave function as Substitute this into (2) and obtain the following differential equations:where is a separation constant. Define ; (4b) is modified as Its solutions are given by [30]where
We are now in the position to consider (4a). Substituting into (4a) and taking , , and , from (4a) we havewhose solutions are given by [18]where and the Bohr radius . The complete wave function has the form
3. Two- and Three-Dimensional Visualizations of SPDs
As we know, the SPDs at the position are calculated by To show the SPD, let us transform (13) to popular Cartesian coordinates via the relations and . Thus, one is able to find the corresponding SPD .
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 discrete positions in the Cartesian space and studies density block, say , which is composed of all values for all positions. Here, is taken as 151. For states denoted by , we display their two- and three-dimensional visualizations for different states 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 and (see Table 1). The case corresponds to a single RSCP, which was discussed in our previous works [29, 30]. We take the unit in axis as the Bohr radial . It should be pointed out that we plot the figures only for the value of 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 , , .
Compared to the cases and , it is seen that the graphics are expanded. That is to say, the SPDs enlarge towards -axis and the hole is expanded outside when the potential parameter increases. We may understand it through considering (8). As we know, will increase relatively for a fixed . For , their isosurfaces are of circular ring shape.
We project the SPDs to a plane and find that they are symmetric to the -axis and -axis (see Table 2). In this work, the graphics are plotted only in the first quadrant by enlarging proportionally the probability 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 , , and 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 -axis and -axis.
4.2. Variation on RPV
To show the isosurface of the SPDs for various RPVs , the quantum numbers for two different cases and are taken as seen in Table 3. It is shown that the particle for smaller will be distributed to almost all spaces. However, the particle for larger will move to the poles in -axis.
4.3. Variations on Various Potential Parameters and
Considering given quantum numbers and , we know from (8) that will become bigger as increases and the parameter also becomes larger with increasing . As a result, this will result in increasing . In Table 4, the SPDs are plotted for state in the cases of and and and , respectively. Obviously, we see a big difference between them. When the potential parameter increases, the expansions of the SPDs along with -axis and -axis and the number of radial nodes are changed. However, when the parameter increases, the expansions of the SPDs are along with -axis.
The comparison is done for positive and negative and when (see Table 5). It is shown that the SPDs for the negative compared with the case are shrunk into the origin. However, the SPDs for are enlarged outside. We can understand it very well by studying the contributions of the potential parameter made on the Coulomb potential. The choice of the negative or positive 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.
5. Conclusions
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 by taking various values of the parameter . It is shown that the SPDs are of circular ring shape. On the other hand, the properties of the RPVs of the SPDs have also been discussed. As an example, we have studied the particular case, that is, , and found that the SPDs will move towards the poles of -axis when the RPVs increase.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
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.