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The Schrödinger Equation and the Scale Principle

Rodolfo A. Frino

Published Jul 1, 2014• 1 citation
OpenAlex

Abstract

Earlier this year (May) I wrote a paper entitled Scale Factors and the Scale Principle. In this paper I formulated a new law which describes a number of fundamental quantum mechanical laws. Since then I found that other quantum mechanical laws including the Bohr postulate and the De Broglie wavelength formula also obey the scale principle. Later I proved that this new law also describes the formula for the Schwarzschild radius, the equation for Einstein’s relativistic energy and Newton’s law of universal gravitation. Now I discovered that the Schrodinger’s equation can also be explained in terms of the present formulation.

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Earlier this year I wrote a paper entitled Scale Factors and the Scale Principle. In that paper I formulated a new law which describes nature at both quantum and cosmic scales. This article shows that both the Bohr postulate and the De Broglie condition are special cases of the abovementioned formulation.

•Jul 1, 2014•6 citations
Earlier this year (May) I wrote a paper entitled Scale Factors and the Scale Principle. In this paper I formulated a new law which describes a number of fundamental quantum mechanical laws. Since then I found that other quantum mechanical laws including the Bohr postulate and the De Broglie wavelength formula also obey the scale principle. Later I proved that this new law also describes the formula for the Schwarzschild radius, the equation for Einstein’s relativistic energy and Newton’s law of universal gravitation. Now I discovered that the Schrodinger’s equation can also be explained in terms of the present formulation.
This paper is about a new paradigm shift in physics. According to Einstein, Newton’s theory of gravity is a first approximation of general relativity, which is Einstein’s theory of gravity, in the limit, which is when gravitational fields are weak and the velocities of masses (relative to a coordinate system) are small compared to the velocity of light. Since Schrödinger was aware of de Broglie’s relation when he formulated his quantum wave mechanics, it is acceptable to enforce the application of de Broglie’s relation in Schrödinger’s quantum wave mechanics. This enforcement leads to a new paradigm shift in physics.
In quantum mechanics, there are two very famous formulas. One is the energy formula of the bose particle, called Planck’s law. The other is the wavelength formula, which is called the de Broy wavelength. According to Einstein’s mass-energy equation, we have studied Planck’s law and De Bloy’s wavelength, and generalized it to the De Bloy’s wavelength formula from low speed to light speed. Then, on this basis, the smallest particle is defined as mass quantum. The new wavelength formula is obtained from the mass quantum and converted into the frequency formula. The generalized Planck’s law is obtained.
Despite its successes, quantum mechanics (QM) has remained mysterious to all who have encountered it. Starting with Bohr and progressing into the present, the departure from intuitive, physical reality has widened. The connection between QM and reality is more than just a “philosophical” issue. It reveals that QM is not a correct or complete theory of the physical world and that inescapable internal inconsistencies and incongruities arise when attempts are made to treat it as physical as opposed to a purely mathematical “tool.” Some of these issues are discussed in a review by F. Laloe [Am. J. Phys. 69, 655 (2001)]. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of e moving in the Coulombic field of the proton and the wave equation as modified by Schrodinger, a classical approach is explored that yields a remarkably accurate model and provides insight into physics on the atomic level. The proverbial view, deeply seated in the wave-particle duality notion, that there is no large-scale physical counterpart to the nature of the electron may not be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantummechanical problems. Specifically, a theory of classical quantum mechanics (CQM) is derived from first principles that successfully applies physical laws on all scales. Rather than using the postulated Schrodinger boundary condition “Ψ → 0 as r → ∞,” which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell’s equations, the classical wave equation is solved with the constraint that the bound (n = 1)state electron cannot radiate energy. By further application of Maxwell’s equations to electromagnetic and gravitational fields at particle production, the Schwarzschild metric is derived from the classical wave equation, which modifies general relativity to include conservation of spacetime in addition to momentum and matter/energy. The result gives a natural relationship among Maxwell’s equations, special relativity, and general relativity. CQM holds over a scale of spacetime of 85 orders of magnitude — it correctly predicts the nature of the universe from the scale of the quarks to that of the cosmos. A review is given by G. Landvogt [Internat. J. Hydrogen Energy 28, 1155 (2003)].
Bohr’s model of the hydrogen atom was successful in that it gave us a radically new way to look at atoms. However, it has serious shortcomings. It could not be used to explain the spectra of He or any multielectron atom. It could not predict the intensities of the H absorption and emission lines. With de Broglie’s hypothesis that matter was also wavelike,1 there arose a question at the 1925 Solvey conference: What is the wave equation? De Broglie could not answer this; however, over the next year Erwin Schro¨dinger, working in Vienna, published a series of papers in which he deduced the general form of the equation that bears his name and applied it successfully to the hydrogen atom.2,3 What emerged was a new set of postulates, much like Newton’s, that laid the foundations of quantum theory.