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

Abstract

We revisit the Bohr model through Brownian motion of the electron and the principles of stochastic optimal control. The electron is assumed to have a definite but random position, represented by a single real-valued stochastic process in physical space whose probability density obeys the Fokker-Planck equation. Because Brownian paths are not differentiable, the process carries two mean drifts, one for each direction of time. We treat the forward drift as the control field, while the backward drift is fixed by the density of the same process. The running cost combines the two drifts into a time-symmetric kinetic term, and through the backward drift it inherits a dependence on the density, so the value becomes a functional on density space. Bellman's dynamic-programming principle requires the control to minimize the expected action from every intermediate time and density onward. The drift therefore emerges as a feedback law on position and density, rather than from the global stationarity of a stochastic action. The resulting law-dependent HJB-Fokker-Planck system reduces to the Schrödinger equation. For stationary hydrogen states the theory yields explicit drift fields in spherical coordinates and reproduces the standard radial and angular kinetic-energy averages of the quantum operator formalism. Direct trajectory-level simulations show the coordinate distributions converging to the Born marginals and the time-averaged energies reproducing the quantum expectation values. For the 2p eigenstates with magnetic quantum number $m=\pm1$, a phase-driven azimuthal drift makes the simulated trajectories circulate at the analytically predicted rate, and the angular momentum accumulated from the raw trajectory increments converges to exactly $L_z=m\hbar$. The angular-momentum quantization postulated in the Bohr model thus reappears as a property of the simulated stochastic motion.