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Advances in High Energy Physics

The Visualization of the Space Probability Distribution for a Moving Particle: In a Single Ring-Shaped Coulomb Potential

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

Published Jan 1, 2017• 3 citations
Publisher

Abstract

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.

DOI
10.1155/2017/7937980

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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 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 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.
Our work utilizes the quantum model of the hydrogen atom which is based on the Schrödinger equation with Coulomb potential. Specifically, we concentrate on the angular components of the wave eigenfunctions derived from this model. We consider the quantum states with n ≤ 4. In order to visualize the orbital shapes of these states, we built in the spherical coordinates system their 3D geometric representations. Furthermore, we use the corresponding spherical harmonics, to calculate the θ nodal values that describe the configurations of these orbital states.
As a manifold is embedded into a higher dimensional Euclidean space, quantum mechanics gives various embedding quantities. In the present study, two embedding quantities for a two-dimensional curved surface in the three-dimensional flat space, the geometric momentum and the geometric potential, are derived in a unified manner. For a particle moving on a two-dimensional sphere or a free rotation of a spherical top, the projections of both the geometric momentum p\mathbf{p}p and the angular momentum L\mathbf{L}L onto a certain Cartesian axis form a complete set of commuting observables as [pi,Li]=0[p_{i},L_{i}]=0[pi​,Li​]=0 (i=1,2,3i=1,2,3i=1,2,3), thus constituting a dynamical (pi,Li)(p_{i},L_{i})(pi​,Li​) representation for the states on the two-dimensional spherical surface. The geometric momentum distribution of the states represented by spherical harmonics is successfully obtained, and this distribution for a homonuclear diatomic molecule seems within the resolution power of present momentum spectrometers and can be measured to probe the embedding effect.
We have studied the path integral solution of a system of particle moving in certain class of non-central potential without using 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 2- dimensional simple harmonic oscillators. The energy spectrum for this system is calculated analytically. Hartmann ring-shaped potential and compound Coulomb plus Aharanov- Bohm potential have also been studied as special cases.