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Theory of Potential Energy Minima for Electron Orbits in Neighboring Atoms

David Bruce Hack

Published Jan 1, 1961• 0 citations
OpenAlex

Abstract

A few decades ago the electron was thought by some to be smeared out over space so that it could not be regarded as a discrete particle.\nThe distribution curve given by solutions of Schroedinger's wave equation is now widely interpreted as a probability of position, rather than as a true "picture" of a single electron. To even use the words "probability of position" implies that an individual electron must have a unique position in space and time. If then an electron has position in space and time the next logical step for curious theorists would seem to be to ask themselves what new models can be built up from that idea.

Related papers

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This article explores a possible physical interpretation of the wavefunction. It zooms in on the physical model it implies for an electron in free space (further building on Schroedinger's ideas in regard to the apparent Zitterbewegung of an electron), but then also discusses the wavefunction for particles (including non-charged particles) in general. While it basically concludes that the mainstream interpretation of quantum physics (the Copenhagen interpretation) is and remains the most parsimonious explanation, it also argues that one or two extra assumptions – the wavefunction as a two-dimensional self-sustaining electromagnetic or gravitational oscillation in space – would make more frivolous explanations (many worlds, pilot wave, etcetera) redundant.

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The Three-dimensional Schro¨dinger Equations

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In the previous chapter we saw that for a particle in a one-dimensional potential well, physically acceptable solutions to the one-dimensional Schro¨dinger equation are possible only for particular discrete values of the total energy. Moreover, in the case of an electron in a well of atomic dimensions, the spacings between these energy levels are in qualitative agreement with the separations experimentally observed in atoms; we also interpreted the square of the wave function as a probability distribution for the position of the particle. These ideas led us to phenomena, such as quantum-mechanical tunnelling, that have been observed experimentally. The real world, however, is three-dimensional and, although one-dimensional examples often provide useful insights and analogies, we shall have to extend our theory into three dimensions before we can make quantitative predictions of most experimental results. In the present chapter, therefore, we shall set up the three-dimensional Schro¨dinger equation and obtain its solutions in a number of cases, culminating in a discussion of the hydrogen atom where we shall find that theory and experiment agree to a remarkable degree of accuracy.

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Cambridge University Press eBooks•May 11, 2023•0 citations

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Abstract Introduces the Schrödinger wave equation as the key equation governing the behaviour of electrons, and presents simple solutions (known as wave functions) as electron waves, and packets of electron waves, which can represent localised, particle-like electrons. Considers the behaviour of electron waves on meeting a potential barrier and trapped in potential wells with finite and infinite walls, and introduces the idea of energy levels. Discusses the accepted interpretation of the electron wave function, namely that its modulus square leads to the probability of finding an electron in a given region of space. Considers further the uncertainty relationship and associated philosophical implications.

•Aug 26, 2024•0 citations

Wrong Potential Energy Term in Schrödinger’s Equation for Hydrogen Atom

Gurcharn Singh Sandhu

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.

Journal of Modern Physics•Jan 1, 2018•3 citations
A few decades ago the electron was thought by some to be smeared out over space so that it could not be regarded as a discrete particle.\nThe distribution curve given by solutions of Schroedinger's wave equation is now widely interpreted as a probability of position, rather than as a true "picture" of a single electron. To even use the words "probability of position" implies that an individual electron must have a unique position in space and time. If then an electron has position in space and time the next logical step for curious theorists would seem to be to ask themselves what new models can be built up from that idea.
This article explores a possible physical interpretation of the wavefunction. It zooms in on the physical model it implies for an electron in free space (further building on Schroedinger's ideas in regard to the apparent Zitterbewegung of an electron), but then also discusses the wavefunction for particles (including non-charged particles) in general. While it basically concludes that the mainstream interpretation of quantum physics (the Copenhagen interpretation) is and remains the most parsimonious explanation, it also argues that one or two extra assumptions – the wavefunction as a two-dimensional self-sustaining electromagnetic or gravitational oscillation in space – would make more frivolous explanations (many worlds, pilot wave, etcetera) redundant.
In the previous chapter we saw that for a particle in a one-dimensional potential well, physically acceptable solutions to the one-dimensional Schro¨dinger equation are possible only for particular discrete values of the total energy. Moreover, in the case of an electron in a well of atomic dimensions, the spacings between these energy levels are in qualitative agreement with the separations experimentally observed in atoms; we also interpreted the square of the wave function as a probability distribution for the position of the particle. These ideas led us to phenomena, such as quantum-mechanical tunnelling, that have been observed experimentally. The real world, however, is three-dimensional and, although one-dimensional examples often provide useful insights and analogies, we shall have to extend our theory into three dimensions before we can make quantitative predictions of most experimental results. In the present chapter, therefore, we shall set up the three-dimensional Schro¨dinger equation and obtain its solutions in a number of cases, culminating in a discussion of the hydrogen atom where we shall find that theory and experiment agree to a remarkable degree of accuracy.
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.
Abstract Introduces the Schrödinger wave equation as the key equation governing the behaviour of electrons, and presents simple solutions (known as wave functions) as electron waves, and packets of electron waves, which can represent localised, particle-like electrons. Considers the behaviour of electron waves on meeting a potential barrier and trapped in potential wells with finite and infinite walls, and introduces the idea of energy levels. Discusses the accepted interpretation of the electron wave function, namely that its modulus square leads to the probability of finding an electron in a given region of space. Considers further the uncertainty relationship and associated philosophical implications.
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.