Printer Friendly

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

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/([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 [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, [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 [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 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 [30]

[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 [18]

[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.

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.

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.


[1] C. Quesne, "A new ring-shaped potential and its dynamical invariance algebra," Journal of Physics A: Mathematical and General, vol. 21, no. 14, pp. 3093-3101, 1988.

[2] Y. C. Chen and S. D. Sun, "Exact solutions of a ring-shaped oscillator," Acta Photonica Sinica, vol. 30, no. 104, 2001.

[3] C. Y. Chen, "General formulas and recurrence formulas for radial matrix elements of ring shaped oscillator," Acta Physica Sinica, vol. 30, no. 539, 2001.

[4] S.-H. Dong, G.-H. Sun, and M. Lozada-Cassou, "An algebraic approach to the ring-shaped non-spherical oscillator," Physics Letters A, vol. 328, no. 4-5, pp. 299-305, 2004.

[5] J.-Y. Guo, J.-C. Han, and R.-D. Wang, "Pseudospin symmetry and the relativistic ring-shaped non-spherical harmonic oscillator," Physics Letters A, vol. 353, no. 5, pp. 378-382, 2006.

[6] A. S. Zhedanov, "Hidden symmetry algebra and overlap coefficients for two ring-shaped potentials," Journal of Physics A: Mathematical and General, vol. 26, no. 18, pp. 4633-4641, 1993.

[7] H. Hartmann and R. Schuck, "Spin-orbit coupling for the motion of a particle in a ring-shaped potential," International Journal of Quantum Chemistry, vol. 18, no. 1, pp. 125-141, 1980.

[8] H. Hartmann, "Die Bewegung eines Korpers in einem ringformigen Potentialfeld," Theoretical Chemistry Accounts, vol. 24, no. 2-3, pp. 201-206, 1972.

[9] C. C. Gerry, "Dynamical group for a ring potential," Physics Letters A, vol. 118, no. 9, pp. 445-447, 1986.

[10] G. G. Blado, "Supersymmetric treatment of a particle subjected to a ring-shaped potential," International Journal of Quantum Chemistry, vol. 58, no. 5, pp. 431-439, 1996.

[11] L. Chetouani, L. Guechi, and T. F. Hammann, "Algebraic treatment of a general noncentral potential," Journal of Mathematical Physics, vol. 33, no. 10, pp. 3410-3418, 1992.

[12] B. P. Mandal, "Path integral solution of noncentral potential," International Journal of Modern Physics A, vol. 15, no. 8, pp. 1225-1234, 2000.

[13] C.-Y. Chen, D.-S. Sun, and C.-L. Liu, "The general calculation formulas and the recurrence relations of radial matrix elements for Hartmann potential," Physics Letters A, vol. 317, no. 1-2, pp. 80-86, 2003.

[14] C.-Y. Chen, C.-L. Liu, and D.-S. Sun, "The normalized wave-functions of the Hartmann potential and explicit expressions for their radial average values," Physics Letters A, vol. 305, no. 6, pp. 341-348, 2002.

[15] L. F. Lu, C. G. Zhuang, and Y. C. Chen, "Exact solutions of the Schrodinger equation with double ring-shaped potential," Journal of Physics B: Atomic and Molecular Physics, vol. 23, p. 493, 2006.

[16] A. Hautot, "Exact motion in noncentral electric fields," Journal of Mathematical Physics, vol. 14, no. 10, pp. 1320-1327, 1973.

[17] A. de Souza Dutra and M. Hott, "Dirac equation exact solutions for generalized asymmetrical Hartmann potentials," Physics Letters A, vol. 356, no. 3, pp. 215-219, 2006.

[18] C. Berkdemir, "A novel angle-dependent potential and its exact solution," Journal of Mathematical Chemistry, vol. 46, no. 1, pp. 139-154, 2009.

[19] F. L. Lu and C. Y. Chen, "Bound states of the Schrodinger equation for the Pooschl-Teller double-ring-shaped Coulomb potential," Chinese Physics B, vol. 19, no. 10, Article ID 100309, 2010.

[20] G.-H. Sun and S.-H. Dong, "Exact solutions of dirac equation for a new spherically asymmetrical singular oscillator," Modern Physics Letters A, vol. 25, no. 33, pp. 2849-2857, 2010.

[21] F. T. Chen and M. C. Zhang, "Exact solutions of the Schrodinger equation for a ring-shaped non-central potential," Acta Physica Sinica, vol. 59, no. 6819, 2010.

[22] H. Hassanabadi, Z. Molaee, and S. Zrrinkamar, "Relativistic vector bosons under poschl-teller double-ring-shaped coulomb potential," Modern Physics Letters A, vol. 27, no. 39, Article ID 1250228, 2012.

[23] S. Bskkeshizadeh and V. Vahidi, "Exact solution of the Dirac equation for the Coulomb potential plus NAD potential by using the Nikiforov-Uvarov method," Adv. Studies Theor. Phys, vol. 6, p. 733, 2012.

[24] A. g. Arda and R. Sever, "Non-central potentials, exact solutions and Laplace transform approach," Journal of Mathematical Chemistry, vol. 50, no. 6, pp. 1484-1494, 2012.

[25] E. Maghsoodi, H. Hassanabadi, and S. Zarrinkamar, "Exact solutions of the Dirac equation with Poschl--Teller doublering-shaped Coulomb potential via the Nikiforov--Uvarov method," Chinese Physics B, vol. 22, no. 3, Article ID 030302, 2013.

[26] C.-Y. Chen, F.-L. Lu, D.-S. Sun, Y. You, and S.-H. Dong, "Exact solutions to a class of differential equation and some new mathematical properties for the universal associated-Legendre polynomials," Applied Mathematics Letters, vol. 40, pp. 90-96, 2015.

[27] R. A. Sari, A. Suparmi, and C. Cari, "Solution of Dirac equation for Eckart potential and trigonometric Manning Rosen potential using asymptotic iteration method," Chinese Physics B, vol. 25, no. 1, Article ID 010301, 2015.

[28] D.-S. Sun, Y. You, F.-L. Lu, C.-Y. Chen, and S.-H. Dong, "The quantum characteristics of a class of complicated double ring-shaped non-central potential," Physica Scripta, vol. 89, no. 4, Article ID 045002, 2014.

[29] Y. You, F. Lu, D. Sun, C. Chen, and S. Dong, "The Visualization of the Space Probability Distribution for a Moving Particle: In a Single Ring-Shaped Coulomb Potential," Advances in High Energy Physics, vol. 2017, pp. 1-19, 2017.

[30] C.-Y. Chen, F.-L. Lu, D.-S. Sun, and S.-H. Dong, "Analytic solutions of the double ring-shaped Coulomb potential in quantum mechanics," Chinese Physics B, vol. 22, no. 10, Article ID 100302, 2013.

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;, Chang-Yuan Chen;, and Shi-Hai Dong;

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.
COPYRIGHT 2018 Hindawi Limited
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2018 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
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
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2018
Previous Article:S-Wave Heavy Quarkonium Spectra: Mass, Decays, and Transitions.
Next Article:The Third Five-Parametric Hypergeometric Quantum-Mechanical Potential.

Terms of use | Privacy policy | Copyright © 2019 Farlex, Inc. | Feedback | For webmasters