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arXiv:1403.3272

Distributed Charge in a Coulomb Potential Well

Andrey Vasilyev

Published Mar 13, 2014• 0 citations
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Abstract

The behavior of a distributed charge in a Coulomb potential well is considered. It is shown that elements of a distributed charge can move along different trajectories, thus forming actually motion of a charge wave. Propagation over different trajectories is possible because elements of charge have on these trajectories the same energy. The overall velocity of motion of a charge wave is one half the velocity of individual elements in the wave, and the wave can move only along the $φ$ axis of a spherical coordinate system. The following features in the motion of a distributed charge are stressed: each point of the charge is a center from which elements of the charge propagate with the same velocity within the solid angle $2π$. The set of spherical functions needed to describe a distributed charge is specified, and it is shown that a charge cannot have spherical symmetry.

DOI
10.48550/arxiv.1403.3272

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The behavior of a distributed charge in a Coulomb potential well is considered. It is shown that elements of a distributed charge can move along different trajectories, thus forming actually motion of a charge wave. Propagation over different trajectories is possible because elements of charge have on these trajectories the same energy. The overall velocity of motion of a charge wave is one half the velocity of individual elements in the wave, and the wave can move only along the φφφ axis of a spherical coordinate system. The following features in the motion of a distributed charge are stressed: each point of the charge is a center from which elements of the charge propagate with the same velocity within the solid angle 2π2π2π. The set of spherical functions needed to describe a distributed charge is specified, and it is shown that a charge cannot have spherical symmetry.
The trajectory of motion of a scattering electron in the Coulomb potential from the wave function of the Schroedinger equation is presented in two ways, spherical polar coordinates and Temple coordinates, and is compared with each other and with the corresponding motion of classical mechanics. A good correspondence among dynamics by wave functions and the classical dynamics has been acknowledged by comparing computed examples.Detailed computing examples discriminate the optimal dynamics of the wave function that should be verified by an experiment.
We prove that radial wave functions of a charged quantum particle moving in a two-dimensional plane of the three-dimensional coordinate space and scattering by a Coulomb center at rest in the same plane are governed by the Coulomb equation with a half-integer index. We investigate the structure of these functions and consider three physically interesting limits: the non-Coulomb limit and high- and low-energy limits. We explicate the basic differences between two- and three-dimensional Coulomb scattering.
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.
A diffusion process for charge distributions in a phase space is examined. The corresponding charge moves in a force field and under an action of a random field. There are the diffusion motions for coordinates and for momenta. In our model, an inner state of the charge is defined by a complex vector. The vector rotates with a great constant angular velocity with respect to the proper time of the charge. A state of the diffusion process is a (complex-valued) wave function on the phase space. As in quantum mechanics, we assume that, for the wave functions, the superposition principle holds. The diffusion process averages out vectors of inner states from different points of the phase space. A differential equation for this diffusion process is founded and examined. We demonstrate that the motion (described by this process) decomposes into a fast motion and a slow motion. The fast motion reduces an arbitrary wave function to a function from a subspace whose elements are parameterized by complex-valued functions of coordinates. The slow motion occurs in this subspace and it is described by the Schr\''odinger equation. The parameters of the suggested model are estimated. The duration of the fast motion is of order 10−1110^{-11}10−11 s.
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.
In Part one of this Paper a hypothesis is forwarded of the electron charge in an atom existing in a distributed form. To check it by methods of electrodynamics and mechanics (without invoking the formalism of quantum mechanics and the concepts of the wave function and of the operators), the potential, kinetic, and total energies were calculated for three states of the hydrogen atom, which were found to agree closely with the available experimental data. The Part two of the Paper offers additional assumptions concerning various scenarios of motion of elements of the distributed electron charge which obey fully the laws of theoretical mechanics. The angular momentum of the ground-state hydrogen atom calculated in the frame of theoretical mechanics is shown to coincide with the spin which is ℏ/2\hbar/2ℏ/2.