Derivation of the Schr\"odinger equation from classical stochastic dynamics

TL;DR

This paper derives the Schrödinger equation from classical stochastic dynamics by linking the wave function's real and imaginary parts to classical variables, showing how quantum behavior emerges from classical stochastic processes.

Contribution

It presents a novel derivation of the Schrödinger equation starting from classical stochastic equations, connecting classical and quantum descriptions.

Findings

Derivation of the Schrödinger equation from classical stochastic equations.

Establishes a relationship between the stochastic wave function and quantum wave function.

Shows how the quantum Liouville equation arises from classical stochastic dynamics.

Abstract

From classical stochastic equations of motion we derive the quantum Schr\"odinger equation. The derivation is carried out by assuming that the real and imaginary parts of the wave function $\phi$ are proportional to the coordinates and momenta associated to the degrees of freedom of an underlying classical system. The wave function $\phi$ is assumed to be a complex time dependent random variable that obeys a stochastic equation of motion that preserves the norm of $\phi$. The quantum Liouville equation is obtained by considering that the stochastic part of the equation of motion changes the phase of $\phi$ but not its absolute value. The Schr\"odinger equation follows from the Liouville equation. The wave function $\psi$ obeying the Schr\"odinger equation is related to the stochastic wave function by $|\psi|^2=\langle|\phi|^2\rangle$.