Quantum behavior of a classical particle subject to a random force

TL;DR

This paper explores how the Schrödinger equation can emerge from classical Newtonian mechanics with a random force, specifically in harmonic oscillators, linking stochastic classical dynamics to quantum behavior.

Contribution

It demonstrates that fluctuations in a classical harmonic oscillator under random forces can be described by the Schrödinger equation, extending to small potential perturbations and generalizing to other potentials.

Findings

Fluctuations match Schrödinger equation for harmonic oscillators.

Result applies to small potential perturbations with preserved periodicity.

Noise spectrum can be tuned for all oscillator frequencies.

Abstract

We give a partial answer to the question whether the Schrodinger equation can be derived from the Newtonian mechanics of a particle in a potential subject to a random force. We show that the fluctuations around the classical motion of a one dimensional harmonic oscillator subject to a random force can be described by the Schrodinger equation for a period of time depending on the frequency and the energy of the oscillator. We achieve this by deriving the postulates of Nelson's stochastic formulation of quantum mechanics for a random force depending on a small parameter. We show that the same result applies to small potential perturbations around the harmonic oscillator as long as the total potential preserves the periodicity of motion with a small shift in frequency. We also show that the noise spectrum can be chosen to obtain the result for all oscillator frequencies for fixed mass. We discuss heuristics to generalize the result for a particle in one dimension in a potential where the motion can be described using action-angle variables.