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Progress in physics

Sampling the Hydrogen Atom

Norman Graves

Published Feb 1, 2012• 0 citations
OpenAlex

Abstract

A model is proposed for the hydrogen atom in which the electron is an objectively real particle orbiting at very near to light speed. The model is based on the postulate that certain velocity terms associated with orbiting bodies can be considered as being affected by relativity. This leads to a model for the atom in which the stable electron orbits are associated with orbital velocities where Gamma is n/α, leading to the idea that it is Gamma that is quantized and not angular momentum as in the Bohr and other models. The model provides a mechanism which leads to quantization of energy levels within the atom and also provides a simple mechanical explanation for the Fine Structure Constant. The mechanism is closely associated with the Sampling theorem and the related phenomenon of aliasing developed in the mid-20th century by engineers at Bell labs.

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A Mechanistic Model for the Hydrogen Atom

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•Mar 3, 2017•0 citations
A model is proposed for the hydrogen atom in which the electron is an objectively real particle orbiting at very near to light speed. The model is based on the postulate that certain velocity terms associated with orbiting bodies can be considered as being affected by relativity. This leads to a model for the atom in which the stable electron orbits are associated with orbital velocities where Gamma is n/α, leading to the idea that it is Gamma that is quantized and not angular momentum as in the Bohr and other models. The model provides a mechanism which leads to quantization of energy levels within the atom and also provides a simple mechanical explanation for the Fine Structure Constant. The mechanism is closely associated with the Sampling theorem and the related phenomenon of aliasing developed in the mid-20th century by engineers at Bell labs.
A model is proposed for the hydrogen atom in which the orbiting electron is seen as an objectively real particle. The model is based on the postulate that certain velocity terms can be treated as being affected by relativity. The model provides a mechanism which drives the quantization process and so leads to the discrete energy levels of the atom. The Rydberg formula, being empirically derived, represents the yardstick by which any model for the hydrogen atom must be judged. Rather than develop a model and test it against the Rydberg formula, the approach taken here is to use the Rydberg formula itself as the basis for such a model. The model effectively unifies quantum mechanics with classical mechanics as well as providing a simple mechanical explanation of the Somerfield Fine Structure Constant.
Consideration of the Rydberg Series leads to a model for the hydrogen atom in which the variable of quantisation is the Lorentz factor, Gamma, and not angular momentum as postulated by Niels Bohr and integral to current theories. This is consistent with an atom in which the electron is seen as an objectively real particle having deterministic position and velocity. The electron is seen to orbit at a constant radius irrespective of the energy state of the atom and so there is no change in potential energy between energy states. All of the energy changes are therefore kinetic in nature. The model also provides a simple physical explanation of the hitherto mysterious Fine Structure Constant.
The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter. According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.
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.