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arXiv:physics/0110033

Electron trajectory in the hydrogen atom

Yoshio Nishiyama

Published Oct 10, 2001• 0 citations
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Abstract

A trajectory in the Schroedinger wave for an electron in an attractive Coulomb potential with the dynamical behavior is proposed and illustrated for a scattering and a bound state. The scattering cross section derived from the trajectories is almost exactly equal to that from the usual wave theory. The statistical nature of the result is examined. The period of one cycle in the bound state is exactly equal to that of the corresponding classical motion.

DOI
10.48550/arxiv.physics/0110033

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A trajectory in the Schroedinger wave for an electron in an attractive Coulomb potential with the dynamical behavior is proposed and illustrated for a scattering and a bound state. The scattering cross section derived from the trajectories is almost exactly equal to that from the usual wave theory. The statistical nature of the result is examined. The period of one cycle in the bound state is exactly equal to that of the corresponding classical motion.
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.
The trajectory of an electron in the hydrogen obeying the Dirac equation is proposed, and illustrated in the three-dimensional space for a bound and scattering state. This trajectory processes around an axis through the origin of the potential and the precession period is exactly equal to that derived from the Sommerfeld relativistic theory. The cross section in the scattering state of an electron is presented, and compared to that derived from the wave theory, showing a slight difference in the backward region, in general.
In this work we propose a new approach to the explanation of the nature of electron based on the corpuscular-wave monism using the further development of the optical-mechanical analogy to describe the physical reality. In this theory the motion of an electron is considered to occur along a trajectory the presence of which is a reflection of the existence of a particle, as well as it is assumed that any motion is defined by a wave V(x,t). It is assumed that there is an explicit relationship between the trajectory and wave equations of the electron, which are established on the basis of the local variational principle. In this approach, an electron wave propagating in free space takes along the electron trajectory. We used this theory to describe the electron motion in a hydrogen-like atom and found its stationary states. The energies of these states coincide with the known quantum mechanics solutions for the stationary energies of the hydrogen-like atom, however, in our approach the spatial trajectories of the electron have the form of the surfaces, which are formed in the region of nodes of the standing electron wave. These surfaces have the form of spheres for the spherical symmetrical electron states and the radii of these spheres coincide with the radii of the Bohr orbits of these states. Thus, in this approach the trajectory and wave measurements of the electron get a consistent spatial description that is inherent to the picture of the corpuscular-wave monism. We discuss the considerable correspondence of the proposed theory with the quantum mechanics results describing the stationary and non-stationary motions of the electron in the atom and their difference. We believe that measuring of the spatial configuration of the observed electron trajectory surfaces in an atom could be a deep examination of the standard quantum theory.
It is shown that a solution of the classical Kepler problem with a binomial potential explains the stationary orbits of the hydrogen atom, its discrete spectrum, and trajectories of scattered electrons with energies from several electron-volts to relativistic values. The Schrodinger equation with the binomial potential takes the form of a classical wave equation. Hence, the binomial potential is a more appropriate model for understanding atomic phenomena than the model based on the Coulomb potential.