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arXiv:1305.6553

Wave Mechanics: Behavior of a Distributed Electron Charge in an Atom

Andrey Vasilyev

Published Apr 4, 2013• 0 citations
arXivHTMLPDF

Abstract

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 $\hbar/2$.

DOI
10.48550/arxiv.1305.6553

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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.
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.
An alternative method of deriving the values of the observables of atomic systems is presented. Rather than using operators and eigenvalues the local variables method uses the continuity equation together with current densities derived from wave functions that are solutions of the Dirac or Pauli equation. The method is applied to atomic hydrogen using the usual language of quantum mechanics rather than that of geometric algebra with which the method is often associated. The picture of the atom that emerges is one in which the electron density as a whole is rotating about a central axis. The results challenge some assumptions of conventional quantum mechanics. Electron spin is shown to be a property of the dynamical motion of the electron and not an intrinsic property of the electron, the ground state of hydrogen is shown to have an orbital angular momentum of ℏ, and excited states are shown to have angular momenta that are different from the eigenvalues of the usual quantum mechanical operators. The uncertainty relations are found not to be applicable to the orthogonal components of the angular momentum. No double electron spin gyromagnetic ratio is required to account for the observed magnetic moments, and the behavior of the atom in a magnetic field is described entirely in kinetic terms.
Aspects of an electrodynamical interpretation of the wave function suggested by Schrodinger are described. According to this conception, electron charge is continuously distributed throughout the interior of the atomic system. A proof is given that classical electrodynamics holds within an atom. The Schrodinger atom is shown to be the only model in which electrons do not lose their energy through emission when they move around the nucleus. The value of the self-field of the distributed electron charge is estimated. Practical applications of this concept are noted, including a new trend in quantum electrodynamics. Experimental and theoretical confirmation of the model of an atom with continuously distributed charge is presented.
In the 1920s, the old quantum theory was gradually being superseded by the new quantum theory. The cornerstone of the old theory was Bohr’s model of the hydrogen atom, which predicted that an electron cannot orbit the proton in the hydrogen atom in any arbitrary fashion. Orbits are “quantized,” meaning that only certain sizes, shapes, and magnetic properties are allowed. The principal quantum number n determined the allowed radii of the orbits, the orbital quantum number l determined the allowed shapes, and the magnetic quantum number m determined the magnetic behavior. Additionally, there is a fourth quantum number s which denotes the fact that the electron has an additional angular momentum, loosely associated with self rotation about its own axis, and that is quantized in units of (1/2)~. The old quantum theory was useful to infer the existence of discrete energy levels in atoms, calculate energy spacings between these levels, and therefore allowed one to interpret atomic spectra. The new quantum theory appeared to be more revolutionary and more powerful. It was triggered by Heisenberg’s discovery of matrix mechanics and Schro¨dinger’s discovery of wave mechanics. These two formalisms would not only predict the quantization of energy and provide a prescription to determine the energy difference between the levels (and thus explain the multiplicity of atomic spectra), but also allow one to calculate easily probabilities of transitions between different quantized energy states. At first, matrix mechanics and wave mechanics looked entirely different in their mathematical appearance and physical meaning. However, Schro¨dinger and Eckart [1] independently showed that the two theories are mathematically equivalent. Toward the end of 1926, Dirac unified the two theories using the concept of state vector and thus established the transformation theory of quantum mechanics. This ultimately had a profound implication for the quantum mechanical (mathematical) recipe to treat spin, as we will show in this chapter. The transformation theory is the mathematical recipe to handle modern quantum mechanics. In Heisenberg’s matrix mechanics, a physical quantity is expressed by a matrix, whereas in Schro¨dinger’s wave mechanics, a physical quantity is expressed by a linear operator. In the unified transformation theory, physical quantities are represented by abstract linear operators called Dirac’s q-numbers, which are linear operators in an infinite-dimensional linear space. Depending upon which types of orthogonal coordinate systems are used in this linear space, either matrix mechanics or wave mechanics emerges.
This chapter develops the wave mechanics formalism. The emphasis here is on symmetries and conservation laws: parity, linear and angular momentum, and the electromagnetic interaction. The only specific physical application is the completion of the study of an isolated hydrogen atom, with some discussion of the motion of a particle in a magnetic field. The chapter also outlines the general assumptions of quantum wave mechanics, which may be summarized as follows: the state of a physical system is represented by a wave function and each measurable attribute of the system is represented by a linear self-adjoint operator in the space of functions. To apply these general assumptions to a given physical system, one must give a specific prescription for the observables and their algebra, and one must adopt a definite form for the Hamiltonians as a function of the observables.