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arXiv:1803.09313 · Advances in High Energy Physics

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

Yuan You, Fa‐Lin Lu, Dong‐Sheng Sun, Chang-Yuan Chen, Shi‐Hai Dong

Published Jan 1, 2018• 7 citations
arXivHTMLPDF

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 those quasi-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 .

DOI
10.1155/2018/8307486

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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 those quasi-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 .
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.
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-(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 and the usual case are spherical and circular ring-shaped, respectively by considering all variables 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 two poles of axis z 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.
The identification on probabilities of nonrelativistic particles moved in doubled-ring shaped Coulomb oscillator (DRSCO) potential has been analyzed by the definition on Shannon entropy information to get the quantify of the quantum entanglement and its correlations. We proposed the new modified potential called DRSCO potential that have been solved in recent research paper and the solution of ground state wave function in axial part was used to get its entropy information. The position-space and momentum-space information entropy were investigated by numerical results by using GNU Octave 4.2.2 and have been shown graphicaly.
Schrodinger equation for a doubled ring-shaped coulomb oscillator potential is investigated using supersymmetric quantum mechanics approach. The three dimensional Schrodinger equation for a doubled ring-shaped coulomb oscillator potential in cylindrical coordinates is separated into three parts that contains one dimensional Schrodinger type equation which are solved using supersymmetric operator and the idea of shape invariant potential. The energy spectrum and the total wave functions are obtained.
The path integral for the potential is evaluated in three different coordinate systems, i. e. in cartesian coordinates, in polar coordinates and in parabolic coordinates. The equivalence between the different approaches is shown and in each case the energyspectrum and the wave functions are explicitly calculated. Furthermore we discuss special cases, including the ring-potential, the Coulomb potential with an Aharonov-Bohm solenoid, and the genuine Coulomb problem. We also point out the separability of the path integral formulation of the ring-potential in spheroidal coordinates, and of the Coulomb potential in spheroidal and spheroconical coordinates, respectively.
We have studied the path integral solution of a system of particle moving in a certain class of noncentral potential without using the Kustannheimo–Stiefel transformation. The Hamiltonian of the system has been converted to a separable Hamiltonian of Liouville type in parabolic coordinates and has further reduced to a Hamiltonian corresponding to two two-dimensional simple harmonic oscillators. The energy spectrum for this system is calculated analytically. The Hartmann ring-shaped potential and compound Coulomb plus the Aharanov–Bohm potential have also been studied as special cases.