Abstract We revisit the Bohr model of the atom 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 (FP) 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 of the control problem 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–FP system reduces to the Schrödinger equation once the standard single-valuedness condition is imposed on the reconstructed phase. 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 of the electron’s stochastic motion around the nucleus show the coordinate distributions converging to the Born marginals and the time-averaged energies reproducing the quantum expectation values. For the 2 p eigenstates with magnetic quantum number m = ± 1 , 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 ℏ . The angular-momentum quantization postulated in the Bohr model thus reappears as a property of the simulated stochastic motion.
Abstract We revisit the Bohr model of the atom 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 (FP) 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 of the control problem 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–FP system reduces to the Schrödinger equation once the standard single-valuedness condition is imposed on the reconstructed phase. 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 of the electron’s stochastic motion around the nucleus show the coordinate distributions converging to the Born marginals and the time-averaged energies reproducing the quantum expectation values. For the 2 p eigenstates with magnetic quantum number m = ± 1 , 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 ℏ . The angular-momentum quantization postulated in the Bohr model thus reappears as a property of the simulated stochastic motion.
Abstract This study develops a strictly classical time-delayed model of the hydrogen atom, with retarded Coulomb forces and finite-difference inertia over a state-dependent delay, to reproduce key quantum features without quantum postulates. The delay law is anchored in the Abraham, Lorentz and Dirac self-time and fixed by matching the Rydberg constant, thereby determining a neutral delay scale and an effective energy frequency proportionality without inserting Planck’s constant into the equations of motion. With this single calibration, a neutral mean-power condition and a phase-locking rule between orbital motion and delay generate a discrete ladder of radii and energies reproducing the Rydberg spectrum without imposing angular-momentum quantisation. Neutral radial and azimuthal phases, constructed from the retarded trajectory, define two indices that act as classical analogues of principal and orbital quantum numbers and satisfy simple sum rules for elliptic orbits, mirroring Sommerfeld’s extension of the Bohr model. Interpreting the neutral delay as a minimal time window for inferring position and velocity yields a classical position–momentum error product of order Planck’s constant. In weakly driven settings, non-zero mean neutral power turns the delayed inertial channel into an energy reservoir, allowing deterministic level-to-level transitions between long-lived neutral orbits and preventing the catastrophic radiative collapse of the Rutherford atom.
The quantum molecular trajectory is the deterministic trajectory, arising from the Bohm theory, that describes the instantaneous positions of the nuclei of molecules by assuring the agreement with the predictions of quantum mechanics. Therefore, it provides the suitable framework for representing the geometry and the motions of molecules without neglecting their quantum nature. However, the quantum molecular trajectory is extremely demanding from the computational point of view, and this strongly limits its applications. To overcome such a drawback, we derive a stochastic representation of the quantum molecular trajectory, through projection operator techniques, for the degrees of freedom of an open quantum system. The resulting Fokker-Planck operator is parametrically dependent upon the reduced density matrix of the open system. Because of the pilot role played by the reduced density matrix, this stochastic approach is able to represent accurately the main features of the open system motions both at equilibrium and out of equilibrium with the environment. To verify this procedure, the predictions of the stochastic and deterministic representation are compared for a model system of six interacting harmonic oscillators, where one oscillator is taken as the open quantum system of interest. The undeniable advantage of the stochastic approach is that of providing a simplified and self-contained representation of the dynamics of the open system coordinates. Furthermore, it can be employed to study the out of equilibrium dynamics and the relaxation of quantum molecular motions during photoinduced processes, like photoinduced conformational changes and proton transfers.
Abstract The Dirac equation, central to relativistic quantum mechanics, governs spin- 1 2 particles and their antiparticles, with each spinor component satisfying a Klein–Gordon-type equation—the quantum counterpart of the relativistic mass-shell condition. In our prior work (Yordanov 2024 Sci. Rep. 14 6507), we derived the Dirac equation using stochastic optimal control (SOC) theory by linearizing both the Lagrangian’s kinetic term and the Hamilton–Jacobi–Bellman (HJB) equation, but that earlier linearized derivation did not preserve the relativistic mass-shell condition. Here, we formulate the SOC problem starting from a specified covariant Lagrangian for a single relativistic charged particle in an external electromagnetic field, in which the standard relativistic square-root kinetic term and the minimal electromagnetic coupling are retained in their original form. In addition, the Lagrangian is supplemented by a covariant spin-field coupling term describing the interaction of the particle’s intrinsic spin with the electromagnetic field. Because this spin-field term is matrix-valued, it is incorporated through scalarization so that the HJB problem remains scalar. To retain a scalar HJB formulation, the present construction is restricted to electromagnetic backgrounds for which σ μ ν F μ ν admits a spacetime-independent eigenspinor; the Dirac equation is then obtained in the corresponding fixed spin sector. This yields a relativistically consistent SOC derivation in which the ℏ → 0 limit of the HJB equation recovers the proper-time relativistic Hamilton–Jacobi equation, whose stationary form yields the classical mass-shell condition, while the stationary HJB equation yields the quantum-corrected mass-shell relation. We illustrate the theory with stochastic simulations of the Dirac–Landau problem (electron in a uniform magnetic field): at the analytic optimal drift the average stochastic action attains a local minimum, and the average action components are consistent with the Dirac–Landau values within statistical uncertainty.
This chapter illustrates Niels Bohr's ideas in some detail. In the formulation of his model, Bohr assumed that the electron in the hydrogen atom moves under the influence of the Coulomb attraction between it and the positively charged nucleus, as assumed in classical mechanics. The Bohr model was very successful in explaining the discrete atomic spectra of one-electron atoms. The Bohr model is, of course, a theoretical model of an atom. Using Bohr's quantization postulate, find the linear velocity of the electron's sphere. To find the radius of the orbit, Bohr postulated that the angular momentum of the electron is quantized, i.e., it can only take values that are integer multiples of ħ. The chapter begins with considering the classical equation of motion for the electron in a circular orbit, which is based on Newton's laws of motion and Coulomb's law of electric force.
Following the de Broglie's idea of pilot wave, this paper treats quantum mechanics as a problem of stochastic optimal guidance law design. The guidance scenario considered in the quantum world is that an electron is the flight vehicle to be guided and its accompanying pilot wave is the guidance law to be designed so as to guide the electron to a random target driven by Wiener process, while minimizing a cost-to-go function. After solving the stochastic optimal guidance problem by differential dynamic programming, we point out that the optimal pilot wave guiding the particle's motion is just the wavefunction ψ(t,x), a solution to Schrödinger equation. Quantum trajectories under the action of the optimal guidance law are solved and their statistical distribution is shown to coincide with the prediction of the probability density function ψ*ψ.
The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto c-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.