# The Visualization of the Space Probability Distribution for a Particle Moving in a Double Ring-Shaped Coulomb Potential

Authors: Yuan You; Fa-Lin Lu; Dong-Sheng Sun; Chang-Yuan Chen; Shi-Hai Dong
Journal: Advances in High Energy Physics
 (2018)

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## 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(n,l,m) are mainly relevant for thosequasi-quantum numbers (n′,l′,m′) via the double RSCP parameter c. The SPDs are of circular ring shape in spherical coordinates. The properties for the relative probability values (RPVs) P are also discussed. For example, when we consider the special case (n,l,m)=(6,5,0), the SPD moves towards two poles of z-axis when P increases. Finally, we discuss the different cases for the potential parameter b, which is taken as negative and positive values for c>0. Compared with the particular case b=0, the SPDs are shrunk for b=-0.5, while they are spread out for b=0.5.


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## Body

## 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 1/r2sin2⁡θ or the double ring-shaped part 1/r2sin2⁡θ+1/r2cos2⁡θ. 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 (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,θ,φ. 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 Section2. 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 P, and the RSCP parameter b (positive and negative) when c≠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(1)Vr,θ=-Ze2r+ħ22Mr2bsin2⁡θ+ccos2⁡θas plotted in Figures 1–3.Figure 1
V r , θ as the functions of variables θ and r.Figure 2
Potential functionVr,θ versus θ at r=0.1,1,10,100.Figure 3
Potential functionVr,θ versus r at θ=0.1°,30°,45°,89°.The Schrödinger equation with this potential is written as (ħ=M=e=1)(2)-12∇2-Zr+12r2bsin2⁡θ+ccos2⁡θΨr→=EΨr→.Take wave function as (3)Ψr→=12πurrHθe±imφ,m=0,1,2,….Substitute this into (2) and obtain the following differential equations:(4a)d2urdr2+2E+2Zr-λr2ur=0,(4b)1sin⁡θddθsin⁡θdHθdθ+λ-b+m2sin2⁡θ-ccos2⁡θHθ=0,where λ is a separation constant. Define x=cos⁡θ; (4b) is modified as (5)1-x2d2Hxdx2-2xdHxdx+l′l′+1-m′21-x2-cx2Hx=0.Its solutions are given by [30](6)Hl′m′x=Nl′m′Pl′m′γ1,x=Nl′m′1-x2m′/2xγ1∑ν=0k-1νΓk+γ1-ν+1Γ2l′-2ν+12l′ν!k-ν!Γ2k+2γ1-2ν+1Γl′-ν+1x2k-2ν,where (7)Nl′m′=2γ1k!2l′+1Γ2k+2γ1+1Γl′-k+12Γl′-k-γ1+1Γk+γ1+1Γ2l′-2k+1,m′=b+m2,l′=nθ+m′=2k+γ1+b+m2,λ=l′l′+1,m,k=0,1,2,…,(8)γ1=0or1,c=01+1+4c2,c>0.We are now in the position to consider (4a). Substituting λ=l′(l′+1) into (4a) and taking χ=τr,  s=2Z, and τ=Z-1/(2E), from (4a) we have(9)d2uχdχ2+sχ-14-l′l′+1χ2uχ=0,whose solutions are given by [18](10)un′l′r=1Γ2l′+2Za0Γn′+l′+1nr!n′21/22Zra0n′l′+1e-Zr/a0n′F-nr,2l′+2,2Zra0n′,where (11)n′=nr+l′+1=nr+2k+m′+3+1+4c2,c>0,nr,m,k=0,1,2,…nr+k+m′+1,c=0,nr,m,k=0,1,2,…and the Bohr radius a0=ħ2/Me2=1. The complete wave function has the form(12)Ψn′l′mr→=12πun′l′rrHl′m′cos⁡θe±imφ.


## 3. Two- and Three-Dimensional Visualizations of SPDs

As we know, the SPDs at the positionr→=r,θ,φ are calculated by (13)ρ=Ψn′l′m′r→2=12πun′l′2rr2Hl′m′2cos⁡θ.To show the SPD, let us transform (13) to popular Cartesian coordinates via the relations r=x2+y2+z2 and  cos⁡θ=z/r. Thus, one is able to find the corresponding SPD ρx,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 takesN discrete positions in the Cartesian space (x,y,z) and studies density block, say den(N,N,N), which is composed of all values wn′l′m′ for all N×N×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≤6) using MATLAB program as shown in Tables 1 and 2.Table 1
The isosurface SPDs with a section plane.
n l M b = 0.5     c=0 b = 0.5    c=0.5 b = 0.5    c=5 2 1 0 3 1 0 3 2 1 4 1 0 4 2 1 4 3 0 4 3 2 5 1 0 5 2 1 5 3 0 5 3 2 5 4 1 5 4 3 6 1 0 6 2 1 6 3 0 6 3 2 6 4 1 6 4 3 6 5 0 6 5 2 6 5 4Table 2
The contour of the SPDs in the planeyoz.
n l m b = 0.5    c=0 b = 0.5    c=0.5 b = 0.5    c=5 2 1 0 3 1 0 3 2 1 4 1 0 4 2 1 4 3 0 4 3 2 5 1 0 5 2 1 5 3 0 5 3 2 5 4 1 5 4 3 6 1 0 6 2 1 6 3 0 6 3 2 6 4 1 6 4 3 6 5 0 6 5 2 6 5 4


## 4. Discussions on the SPD

### 4.1. Variation Caused by the Radial Nodes

We show the SPDs for various casesb=0.5 andc=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 a0. It should be pointed out that we plot the figures only for the value of nr=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 casesc=0 and b≠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≠m, their isosurfaces are of circular ring shape.We project the SPDs to a planeyoz 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 |Ψn′l′m′r→|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 RPVP

To show the isosurface of the SPDs for various RPVsP∈(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.Table 3
The SPDs for various RPVsP for (n,l,m)=(6,5,0)(b=0.5,c=0.5,10).
P(%) c = 0.5 c = 10 0.01 1 5 10 20 50 70 90 99


### 4.3. Variations on Various Potential Parametersb and c

Considering given quantum numbersm and nθ, we know from (8) that m′ will become bigger as bincreases and the parameter γ1 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.Table 4
The isosurface illustration of the state (5, 1, 0) with various values ofbandc.
b  (c=0.5) Isosurface illustration c  (b=0.5) Isosurface illustration 0  0 5  5 10  10 25  25 40  40 80  80The comparison is done for positive and negativeb<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.Table 5
SPDs for different cases of the value ofb for (4, 1, 0) when c=0.5.
b Isosurface illustration Contour illustration 0.5 0 −0.5


## 4.1. Variation Caused by the Radial Nodes

We show the SPDs for various casesb=0.5 andc=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 a0. It should be pointed out that we plot the figures only for the value of nr=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 casesc=0 and b≠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≠m, their isosurfaces are of circular ring shape.We project the SPDs to a planeyoz 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 |Ψn′l′m′r→|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 RPVP

To show the isosurface of the SPDs for various RPVsP∈(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.Table 3
The SPDs for various RPVsP for (n,l,m)=(6,5,0)(b=0.5,c=0.5,10).
P(%) c = 0.5 c = 10 0.01 1 5 10 20 50 70 90 99


## 4.3. Variations on Various Potential Parametersb and c

Considering given quantum numbersm and nθ, we know from (8) that m′ will become bigger as bincreases and the parameter γ1 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.Table 4
The isosurface illustration of the state (5, 1, 0) with various values ofbandc.
b  (c=0.5) Isosurface illustration c  (b=0.5) Isosurface illustration 0  0 5  5 10  10 25  25 40  40 80  80The comparison is done for positive and negativeb<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.Table 5
SPDs for different cases of the value ofb for (4, 1, 0) when c=0.5.
b Isosurface illustration Contour illustration 0.5 0 −0.5


## 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(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.



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