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Journal of Surface Investigation X-ray Synchrotron and Neutron Techniques

Binomial potential of the electron-proton interaction as an alternative to the Coulomb law

V. K. Gudym, E. V. Andreeva

Published Apr 1, 2007• 2 citations
Publisher

Abstract

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.

DOI
10.1134/s1027451007020218

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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.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
This chapter illustrates Niels Bohr's ideas in some detail. In the formulation of his model, Bohr assumed that the electron in the hydrogen atom moves under the influence of the Coulomb attraction between it and the positively charged nucleus, as assumed in classical mechanics. The Bohr model was very successful in explaining the discrete atomic spectra of one-electron atoms. The Bohr model is, of course, a theoretical model of an atom. Using Bohr's quantization postulate, find the linear velocity of the electron's sphere. To find the radius of the orbit, Bohr postulated that the angular momentum of the electron is quantized, i.e., it can only take values that are integer multiples of ħ. The chapter begins with considering the classical equation of motion for the electron in a circular orbit, which is based on Newton's laws of motion and Coulomb's law of electric force.
Even after nine decades of successful run of the Quantum Mechanics (QM), different viewpoints on foundational problems of Quantum Physics are still being actively debated. That is because mathematical logic of QM often defies the physical intuition which constitutes the main spirit of Physics. De Broglie’s hypothesis of matter waves implied that the dynamic characteristics of a micro particle in motion, can be ascribed to the wave characteristics of the wavelet accompanying the particle. The Schrödinger equation models the matter-wave interactions through wavefunction ψ and effectively serves as the foundation of QM. Even though mathematical structure of the Schrödinger equation is sound and elegant, here we show a conceptual mistake in the development of this equation wherein the physical situation has not been correctly modeled in the equation. The Coulomb potential energy of the proton electron pair in Hydrogen atom is essentially the negative interaction energy between their superposed electrostatic fields which is inversely proportional to their instantaneous separation distance. Assuming the proton to be relatively fixed at the origin of an appropriate coordinate system, the potential energy of the orbiting electron will be a function of instantaneous position coordinates of the electron. This has not been properly modeled in the Schrödinger equation. The resulting errors in the solution have been quantitatively demonstrated in this paper. We have stressed the necessity of incorporating a specific correction in the potential energy term of the Schrödinger equation, after which it may facilitate the adoption of Bohmian QM.
The hydrogen atom with the Coulomb interaction is one of the exactly solvable non-relativistic quantum models. Unlike many other exactly solvable models it describes a real physical object providing the formulas for energy levels and stationary state wave functions of a real hydrogen atom. In this paper we modify the model replacing the Coulomb interaction by the interaction of the proton and the electron with the classical electromagnetic field serving as an intermediary transmitting the electromagnetic interaction of these two charged quantum particles.
This paper tries to prove that the existence and properties of the electron’s spin can be established not only as a consequence of relativistic quantum mechanics, but also as a consequence of non-relativistic quantum mechanics of the Hydrogen atom or Hydrogenic atoms. It is, in fact, an extended model of the Hydrogen atom or Hydrogenic atoms. This new model starts from the assumption that to the foundation of atomic structure there is not only a potential of electrical nature, but also a potential of magnetic nature, which, together with the Coulomb potential, determine the movement of electrons around the atomic nucleus. As a consequence of this new model is obtained a non-relativistic theory of Hydrogen atom which takes into account the spin of the electron and leads us to a non-relativistic theory of the atom in agreement with the Stern-Gerlach experiment results. As well the model can be used in order to explain the spectrum of Alkali-atoms.