Revisiting the Bohr Model of the Atom through Brownian Motion of the Electron

TL;DR

This paper proposes a refined stochastic model of the hydrogen atom where the electron's motion is described by Brownian motion, reproducing quantum predictions without wave function collapse.

Contribution

It introduces a stochastic equation of motion for the electron based on the Fokker-Planck equation, aligning classical stochastic processes with quantum mechanics.

Findings

Reproduces average kinetic energies consistent with quantum theory

Confirms stability of electron trajectories through numerical simulations

Recovers probability distributions matching the Born rule

Abstract

In this work, we refine the Bohr model of the hydrogen atom by describing the motion of the electron through a single real-valued stochastic process, effectively realizing Brownian motion under the Born rule. Our approach derives the electron's stochastic equation of motion from the Fokker-Planck equation while ensuring the particle always maintains a definite - albeit random - position. This feature obviates the need for wave function collapse as invoked in the Copenhagen interpretation. Instead, the wave function serves as a tool to compute the electron's drift velocity. Building on this, we develop modified stochastic equations in spherical coordinates, tailored to the spherical symmetry of the hydrogen atom. We show that these equations reproduce the correct average radial and angular kinetic energies, matching operator-based quantum mechanical predictions. Numerical simulations confirm the stability of electron trajectories and, as expected, recover the probability distribution prescribed by the Born rule. At very short timescales, however, single-electron probability distributions may deviate from wave function-based predictions due to insufficient ensemble averaging. Taken together, these findings offer an alternative perspective on atomic structure and highlight the potential of using wave function-derived drift velocities within a single stochastic process to capture quantum dynamics in the hydrogen atom.