⚛️ quantum physics

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

This paper demonstrates that the Bohr model of the atom and the Schrödinger equation can be derived by modeling the electron's position as a Brownian motion controlled via stochastic optimal control, where minimizing a time-symmetric action yields drift fields that reproduce quantum energy levels, probability distributions, and exact angular momentum quantization.

Original authors: Vasil Yordanov

Published 2026-07-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Vasil Yordanov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the electron in a hydrogen atom not as a tiny planet orbiting a sun, and not as a fuzzy cloud of probability, but as a very jittery, very confused drunkard taking a walk. This is the core idea of a new paper by Vasil Yordanov, which tries to explain the strange rules of quantum mechanics by treating the electron's motion as a specific kind of random walk, known as Brownian motion.

The Drunkard's Two-Step Dance

In the old days, Niels Bohr imagined electrons hopping between neat, circular tracks like beads on a wire. Later, Erwin Schrödinger replaced those tracks with a wave equation that describes a "cloud" of where the electron might be. Yordanov's paper asks a different question: What if the electron is actually moving in real space, but its path is so jittery and random that it looks like a wave?

To make this work, the author treats the electron's movement like a Brownian motion—the same kind of zigzagging path a speck of dust takes when it's being bumped around by invisible water molecules. But here's the twist: because time can run forward or backward in this math, the electron has two "drifts" (or average directions it's leaning).

The Forward Drift: This is the direction the electron is being "steered" right now. Think of this as a control field, like a wind pushing the electron.
The Backward Drift: This is fixed by where the electron has been and where it is likely to be.

The paper argues that the "wind" (the forward drift) isn't just random; it's a smart feedback loop. It constantly adjusts based on the electron's current position and the shape of the probability cloud around it. By using a mathematical tool called stochastic optimal control (which is basically a fancy way of saying "finding the best path through a storm"), the author shows that if you set up this specific kind of random walk, the electron naturally settles into the exact patterns we see in quantum mechanics.

The Magic of the "Cloud"

The most exciting part of this paper is what happens when you actually run the numbers on a computer. The author didn't just write equations; they simulated the electron's journey.

When they simulated the electron in a hydrogen atom, the results were surprisingly precise:

The Map Matches the Territory: If you track where the simulated electron spends its time, the resulting map matches the famous "Born rule" (the standard quantum probability cloud) perfectly.
Energy Checks Out: The average energy the simulated electron has matches the exact energy levels predicted by quantum theory.
The Spin is Real (but the path isn't a simple circle): For electrons with a specific magnetic "spin" (quantum number $m = \pm 1$ ), the simulation reveals a fascinating duality. While the electron's position still wanders around to sample the entire probability cloud (it doesn't get stuck on a single ring), its drift (the "wind" pushing it) carries a constant, deterministic angular momentum of exactly $L_z = m\hbar$ . This means that even though the electron is jittering all over the place, every single step of its path is guided by a hidden "spin" that matches the quantized values Bohr guessed at a century ago. The circulation is a property of the guiding force, not a fixed circular track.

Here is the kicker: The paper shows that the angular momentum (the "spin" of the orbit) for these circulating electrons is exactly $L_z = m\hbar$ . This is the same quantization rule Bohr guessed at a century ago, but here it emerges naturally from the math of the random walk. The electron doesn't just look like it has quantized spin; in the simulation, the guiding drift carries exactly that amount of angular momentum along every single step of its path, even as the electron explores the full cloud.

What This Paper Says "No" To

It's important to know what this paper is not saying.

It's not a return to the old Bohr model: The author explicitly rejects the idea that electrons are just tiny balls on fixed, circular tracks like planets. The electron is still jittery and random; it just happens to have a "smart" random walk that creates the appearance of a wave.
It's not a complex wave function from the start: Unlike some other theories that start with complex numbers and imaginary speeds, this paper starts with a simple, real-valued random walk in physical space. The complex "wave" is something that gets reconstructed after the fact, not something built into the beginning.
It's not a "proof" that this is how nature works: The paper is careful to say this is a simulation and a mathematical derivation. It shows that this model can reproduce the results of quantum mechanics, but it doesn't claim to have proven that the universe actually works this way. It's a demonstration that the math holds up.

The Numbers Behind the Magic

The simulations used some very specific numbers to get the results right:

The time steps in the simulation were incredibly small, around $10^{-20}$ seconds (or $5 \times 10^{-21}$ seconds for some specific runs). This is more than a thousand times smaller than the natural time scale of a hydrogen atom (which is about $2.4 \times 10^{-17}$ seconds).
The electron's typical drift movement in one step was about $4 \times 10^{-4}$ times the Bohr radius ( $a_0$ ), and its random "jitter" was about $2 \times 10^{-2}$ times the Bohr radius.
For the circulating electrons ( $m = \pm 1$ ), the average speed was calculated to be roughly $6.44 \times 10^5$ m/s.
The rate at which the electron winds around the nucleus was found to be about $5.17$ radians per femtosecond (which is about 0.8 revolutions per femtosecond).

The Bottom Line

This paper is a clever detective story. It takes the weird, fuzzy rules of quantum mechanics and asks, "Could this just be a very specific kind of random walk?" By treating the electron's motion as a controlled, jittery dance, the author shows that the famous quantum rules—like energy levels and angular momentum—pop out naturally.

The simulations suggest that if you were to watch a single electron in this model, you wouldn't see a fuzzy cloud. You'd see a particle zipping around, sometimes getting trapped in one lobe of an atom for a while before jumping to another, but always guided by a drift that carries the exact right amount of spin and energy. It's a fresh, playful way to look at an old problem, showing that the "weirdness" of the quantum world might just be the result of a very sophisticated kind of randomness.

Technical Summary: Revisiting the Bohr Model of the Atom through Brownian Motion of the Electron

Problem Statement
The paper addresses the historical tension between classical orbital mechanics and quantum atomic spectra, specifically within the hydrogen atom. While the Bohr model imposed quantized orbits on classical dynamics and Schrödinger's wave mechanics replaced these with a probabilistic wave equation, the authors ask whether the stationary quantum properties of hydrogen can be represented by a single real stochastic motion of the electron in physical space. The goal is to derive the Schrödinger equation and quantum expectation values not from a wave function postulate, but from a density-space stochastic optimal control framework applied to a real diffusion process.

Methodology
The authors extend Nelson's stochastic mechanics into a density-space stochastic optimal control theory. The core methodology involves the following steps:

Stochastic Formulation: The electron is modeled as a real-valued Markov diffusion in physical space with a diffusion coefficient $D = \hbar/(2m)$ . The motion is characterized by two drift fields: a forward drift $b_+$ (treated as the control field) and a backward drift $b_-$ (fixed by the probability density $P$ via time-reversal consistency).
Law-Dependent Control: Unlike standard control problems where the cost depends only on position, the running cost here is law-dependent. The authors employ a time-symmetric Lagrangian $L = \frac{m}{2}b_+ \cdot b_- - V$ , which, when expressed in terms of the forward drift and density, creates a cost functional dependent on the law of the process.
Density-Space Bellman Principle: Optimality is imposed recursively via Bellman's dynamic programming principle on the space of probability densities. The value function is a functional $V(t, P)$ rather than a simple function of position. This leads to a coupled system of a local Hamilton–Jacobi–Bellman (HJB) equation and the Fokker–Planck equation.
Reconstruction of Quantum Mechanics: By defining a shifted local Bellman field and transforming variables to a phase $S$ and density $P$ , the authors show that the HJB–Fokker–Planck system reduces to the Madelung equations. Imposing the standard single-valuedness condition on the reconstructed phase recovers the Schrödinger equation.
Hydrogen Atom Simulation: For stationary hydrogen states, the authors derive closed-form expressions for the optimal drift fields in spherical coordinates based on the known wave functions $\psi_{n\ell m}$ . They then perform direct trajectory-level simulations of the electron's stochastic motion using the Euler–Maruyama scheme, without solving the HJB system numerically.

Key Contributions
The paper makes three primary contributions:

Theoretical Formulation: It formulates a density-space Bellman problem generated by forward–backward kinematics, deriving a local law-dependent HJB–Fokker–Planck system. It demonstrates that the Schrödinger equation emerges from this system once the single-valuedness condition is applied to the reconstructed phase.
Analytical Drift Fields: For stationary hydrogen states, the authors obtain explicit closed-form expressions for the optimal drift fields in spherical coordinates. They analytically prove that the stochastic radial and angular kinetic-energy averages coincide exactly with the expectation values calculated via standard quantum mechanical operators.
Trajectory-Level Simulations: The paper presents the first trajectory-level simulations of hydrogen eigenstates with non-zero angular momentum ( $m \neq 0$ ). It quantitatively recovers electron energies and angular momentum directly from the simulated trajectories.

Results

Convergence to Born Rule: The coordinate distributions (radial, polar, and azimuthal) of the simulated single-electron trajectories converge to the Born marginals predicted by the wave function. The paper notes that convergence times depend on the presence of nodal surfaces; for states with nodes (e.g., $2s$ or $2p$ ), the ergodic mixing time is longer due to rare transitions between metastable regions (lobes or shells).
Energy Recovery: The time-averaged kinetic, potential, and total energies calculated from the simulated trajectories converge to the analytical quantum expectation values.
Angular Momentum Quantization: For the $2p$ $2 p$ eigenstates with magnetic quantum number $m = \pm 1$ $m = \pm 1$ , the simulations reveal a phase-driven azimuthal drift.
- The trajectories circulate at a rate analytically predicted by the theory.
- Crucially, the angular momentum accumulated from the raw trajectory increments converges to exactly $L_z = m\hbar$ .
- This demonstrates that the Bohr–Sommerfeld quantization postulate reappears as a deterministic property of the stochastic motion: the drift field carries exactly $L_z = m\hbar$ along every trajectory, even though the trajectory itself samples the full Born density.

Significance and Claims
The paper claims to provide a direct link between stochastic optimal control and quantum mechanics, showing that the Schrödinger equation is the dynamic programming equation for a specific time-symmetric control problem on density space.

The authors emphasize that their approach recovers the Bohr model's angular momentum quantization not as an external postulate, but as an emergent property of the simulated stochastic motion. Specifically, for $m \neq 0$ states, the "circulation" is invisible in the static Born density (which is identical for $m=+1$ and $m=-1$ ) but is fully resolved at the level of the single stochastic trajectory. The work suggests that the quantum phase acts through the optimal drift to enforce deterministic angular momentum values along the stochastic path, offering a new perspective on the transition from classical orbital concepts to quantum probability.

The paper remains modest regarding the scope, focusing strictly on the hydrogen atom as a test case to validate the theoretical framework and the recovery of standard quantum results (energies, distributions, and angular momentum) from a purely stochastic, trajectory-based perspective.