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Mathematical Equivalence of the Real Wave Quantum Mechanics Algorithm with Schrodinger's Equation

Ewins Ron

Published Apr 14, 2026• 0 citations
Publisher

Abstract

Provides a mathematical justification for the previously noted (see 'The Distributed Electron') congruence between the RWQM algorithm and solutions to Schrodinger's equation in one dimenion. Presents a differential equation for 1-d RWQM and compares it with Schrodinger's equation presented in similarly dimensionless form. Applies both equations to the 1-d Simple Harmonic Oscillator and cofirms identical results.

DOI
10.5281/zenodo.19570045

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Provides a mathematical justification for the previously noted (see 'The Distributed Electron') congruence between the RWQM algorithm and solutions to Schrodinger's equation in one dimenion. Presents a differential equation for 1-d RWQM and compares it with Schrodinger's equation presented in similarly dimensionless form. Applies both equations to the 1-d Simple Harmonic Oscillator and cofirms identical results.
Real Wave Quantum Mechanics (RWQM) seeks to demonstrate that standard Quantum Mechanics (QM) can be derived from first principles based entirely on the interactions of real waves in an underlying universal medium. This paper extends previous work on 1 and 2 dimensional applications which used discrete linear/grid structures of entities known as Field Rotators (FRs). It is shown that FRs can be extended continuously in the 3rd dimension to give threadlike forms named FReds. Although continuous, it is shown that the 3rd dimension can be discretized to simplify calculations. Results are similar to those from Schr\"odinger's equation and can be viewed interactively from within the document or directly from the Zenodo site by downloading the appropriate html file. The meaning and role of phase in standard QM is made clear and intuitive by the RWQM approach.
This chapter examines some explicitly one-dimensional systems, including the finite square well, the harmonic oscillator, and the application of harmonic oscillator theory to molecular vibrations. It analyzes the computer solutions of the Erwin Schrodinger equation. Although there are no real one-dimensional square wells in nature, many real systems have potentials that resemble the square well. Near the center of force the different wave functions must differ from one another or else they would not satisfy the Schrodinger equation for different values of energy. In contemporary research almost all the manipulation of the Schrodinger equation is done not analytically but rather by computer using numerical methods. A major advantage of the dimensionless form of the Schrodinger equation is that a single solution for one kind of system is trivially easy to extend to all similar systems. The dimensionless form of the Schrödinger equation can be solved numerically if it is approximated by a so-called difference equation.
An alternative method for obtaining analytical solutions of the time-independent Schrödinger equation (TISE) is developed and applied to 1-dimensional model quantum systems of interest in atomic and molecular physics. The method is based on proposing a wave function that is a product of two functions, one of them an integrating factor. The differential equation resulting from operating the TISE on the product wave function is compared with a differential equation with a known solution. This comparison produces a system of differential equations for the integrating factor. This system of equations is easily solved for model quantum systems such as the harmonic oscillator, the hydrogen atom and the Morse oscillator. The method presented simplifies some approaches shown in textbooks, based on asymptotic analyses of the TISE, and power series methods with recurrence relations. In addition, the method presented here facilitates the understanding of the relationship between the ordinary differential equations of the mathematical physics and the TISE of physical models as the harmonic oscillator, the Hydrogen atom, and the Morse oscillator.