Schrodinger's Equation is Universal, Dark Matter and Double Diffusion

TL;DR

This paper demonstrates that classical elastic collisions can be modeled to produce equations akin to Schrödinger's, linking classical mechanics, quantum mechanics, and potential insights into dark matter and relativity.

Contribution

It introduces a novel classical collision framework that reproduces Schrödinger's equation and explores implications for dark matter and relativistic corrections.

Findings

Classical collisions can be represented with symmetric and anti-symmetric terms similar to Nelson's measure.

Schrödinger's equation emerges from stochastic modeling of particle velocities.

The model suggests potential links between classical mechanics, quantum phenomena, and dark matter detection.

Abstract

This paper considers a main particle and an incident particle classical mechanics elastic collision preserving energy and momentum while ignoring the angular momentum, spin or other particle characteristics. The main result of the paper shows that the colliding two particle classical Hamiltonian energy can be represented in four weighted individual particle in symmetric and anti-symmetric (osmotic) terms similar to the quadratic Nelson measure used in the derivation of the Schrodinger wave function. Following Nelson, if the second particle behavior can be captured in a potential and the ingoing and outgoing velocities of the main particle are modelled using stochastic differential equations the motion of the main particle satisfies the Schrodinger's equation. The diffusion variance of this equation is replaced by a related ratio of masses and the assumed variance. The first example attempts to reconcile this result with quantum mechanics by considering the Schrodinger equation in the presence of only one type of incident particle. The main particle energy levels become multiples of the incident particle and the energy expression for the entire system agrees with quantum mechanics but there are differences with the stochastic equation. The Schrodinger equation can also be used to represent corrections for Newton's equation and suggests a user profile to be used in the search for Dark Matter. An alternative solution to the collision model also shows relativistic properties as the interactions suggest corrections to the Minkowski equation in Einstein's Special Relativity. It is also possible to use the classical Schrodinger's equation both on the main and incident particle simultaneously leading to a correlated set of wave equations with different diffusion parameters.