Derivation of the Schrödinger Equation from Newtonian Mechanics

TLDR

Newton's law F=ma describes external field influence, analogous to the Ornstein‑Uhlenbeck theory of macroscopic Brownian motion with friction. The study tests whether each particle of mass m undergoes frictionless Brownian motion with diffusion coefficient ħ/(2m). The hypothesis naturally yields the Schrödinger equation, yet interprets it classically with continuous trajectories and an incomplete wave function, and indicates that, despite this, the two theories may be equivalent within a limited measurement framework.

Abstract

We examine the hypothesis that every particle of mass $m$ is subject to a Brownian motion with diffusion coefficient $\frac{\ensuremath{\hbar}}{2m}$ and no friction. The influence of an external field is expressed by means of Newton's law $\mathbf{F}=m\mathbf{a}$, as in the Ornstein-Uhlenbeck theory of macroscopic Brownian motion with friction. The hypothesis leads in a natural way to the Schr\"odinger equation, but the physical interpretation is entirely classical. Particles have continuous trajectories and the wave function is not a complete description of the state. Despite this opposition to quantum mechanics, an examination of the measurement process suggests that, within a limited framework, the two theories are equivalent.