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Physical Review A

Numerical simulations in stochastic mechanics

Marvin McClendon, HERSCHEL A. RABITZ

Published May 1, 1988• 38 citations
Publisher

Abstract

The stochastic differential equation of Nelson's stochastic mechanics is integrated numerically for several simple quantum systems. The calculations are performed with use of Helfand and Greenside's method and pseudorandom numbers. The resulting trajectories are analyzed both individually and collectively to yield insight into momentum, uncertainty principles, interference, tunneling, quantum chaos, and common models of diatomic molecules from the stochastic quantization point of view. In addition to confirming Shucker's momentum theorem, these simulations illustrate, within the context of stochastic mechanics, the position-momentum and time-energy uncertainty relations, the two-slit diffraction pattern, exponential decay of an unstable system, and the greater degree of anticorrelation in a valence-bond model as compared with a molecular-orbital model of ${\mathrm{H}}_{2}$. The attempt to find exponential divergence of initially nearby trajectories, potentially useful as a criterion for quantum chaos, in a periodically forced oscillator is inconclusive. A way of computing excited energies from the ground-state motion is presented. In all of these studies the use of particle trajectories allows a more insightful interpretation of physical phenomena than is possible within traditional wave mechanics.

DOI
10.1103/physreva.37.3479

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The stochastic differential equation of Nelson's stochastic mechanics is integrated numerically for several simple quantum systems. The calculations are performed with use of Helfand and Greenside's method and pseudorandom numbers. The resulting trajectories are analyzed both individually and collectively to yield insight into momentum, uncertainty principles, interference, tunneling, quantum chaos, and common models of diatomic molecules from the stochastic quantization point of view. In addition to confirming Shucker's momentum theorem, these simulations illustrate, within the context of stochastic mechanics, the position-momentum and time-energy uncertainty relations, the two-slit diffraction pattern, exponential decay of an unstable system, and the greater degree of anticorrelation in a valence-bond model as compared with a molecular-orbital model of H2{\mathrm{H}}_{2}H2​. The attempt to find exponential divergence of initially nearby trajectories, potentially useful as a criterion for quantum chaos, in a periodically forced oscillator is inconclusive. A way of computing excited energies from the ground-state motion is presented. In all of these studies the use of particle trajectories allows a more insightful interpretation of physical phenomena than is possible within traditional wave mechanics.
Nelson's stochastic mechanics formulates quantum dynamics as a real-time conservative diffusion process in which a particle undergoes Brownian-like motion with a fluctuation amplitude fixed by Planck's constant. While being mathematically equivalent to the Schrödinger formulation, this approach provides an alternative dynamical framework that enables the study of time-resolved quantities that are not straightforwardly defined within the standard operator-based approach. In the present work, Nelson's stochastic mechanics is employed to investigate tunneling-time statistics for bound states in double-well potentials. Using first-passage time theory within this framework, both the mean tunneling time, τˉ\barττˉ, and the full probability distribution, p(τ)p(τ)p(τ), are computed. The theoretical predictions are validated through extensive numerical simulations of stochastic trajectories for two representative potentials. For the square double-well potential, analytical expressions for τˉ\barττˉ are derived and are shown to be in excellent agreement with simulations. In the high-barrier limit, the results reveal a direct relation between the stochastic-mechanical and quantum-mechanical tunneling times, expressed as τQM=(π/2)τˉτ_{\mathrm{QM}} = (π/2)\barττQM​=(π/2)τˉ, where τQMτ_{\mathrm{QM}}τQM​ corresponds to half the oscillation period of the probability of finding the particle in either well. This relation is further confirmed for generic double-well systems through a WKB analysis. As a concrete application, the inversion dynamics of the ammonia molecule is analyzed, yielding an inversion frequency of approximately 24 GHz, in close agreement with experimental observations. These results highlight the potential of stochastic mechanics as a conceptually coherent and quantitatively consistent framework for analyzing tunneling phenomena in quantum systems.
This study offers a comprehensive exploration of quantum mechanics by integrating theoretical analysis with simulation-based experimentation to investigate fundamental principles such as energy quantization, operator expectations, and wavefunction behavior. Utilizing a mixed-methods approach, the research employed numerical solutions to the time-independent Schrödinger equation and qualitative analysis of foundational quantum postulates to model and interpret the behavior of discrete quantum states. Nine detailed datasets were generated, each containing over 20 quantum states, capturing energy levels, expectation values of position and momentum, and probability density maxima. The analysis revealed consistent non-linear spacing in energy levels, symmetrical behavior in position and momentum distributions, and the emergence of wavefunction localization in high-probability states. Twelve complex visualizations—including line, bar, pie, scatter, and hybrid plots—further illustrated the dynamic relationships among observables. These results confirm core theoretical constructs such as the Heisenberg uncertainty principle and the correspondence between potential confinement and probability density peaks. Violin plots and scatter matrices identified subtle structural symmetries and degeneracies in state distributions, while hybrid line-bar visualizations highlighted state-dependent transitions in expectation values. Additionally, the study demonstrated the scalability and accuracy of simulation frameworks in modeling quantum systems and emphasized their pedagogical value in visualizing abstract quantum behaviors. Collectively, the results substantiate quantum mechanical theory while offering data-driven insights into operator correlations and system evolution. This research not only reinforces classical interpretations but also paves the way for further exploration in quantum computing, photonic systems, and educational simulations.
Nelson's stochastic mechanics associates a stochastic process to each solution of the Schrodinger equation. For a few simple examples (gaussian wavepackets and superpositions of gaussian wavepackets), the authors exhibit numerically generated sample paths of the associated Nelson processes. In doing so, they have to solve a few stochastic differential equations: the simplicity of the algorithm needed for this purpose is emphasised and is of some pedagogical interest by itself. As to stochastic mechanics, they address themselves the question of how to read, from the sample paths thus generated, information about various observables of the quantum system, obtaining complete answers about position and momentum and partial, but stimulating, evidence about parity and orbital angular momentum.
The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. The Schroedinger equation and the so-called Nelson equations are used to determine the diffusion tensor and the drift vectors characteristic of such a process. It is then possible to write down the forward and backward Kolmogorov equations that are used to determine the conditional probability density, as well as the Fokker-Planck equation that is used to determine the normal probability density. This method is applied to several simple cases and the results obtained are compared with those of the orthodox theory. Among the most important differences let us mention that any system evolving freely from any original state returns spontaneously to its ground state, that the definitions of the particles velocities and momenta are impossible because the sample functions of a diffusion Markov process nowhere possess a derivative with respect to time and that the so-called collapse of the wave function is a mere mirage explained by the updating choice between new and older initial conditions in the resolution of partial differential equations. We finally attempt to extend the proposed theory to the domain of the relativistic quantum mechanics. It is promising but is clearly unfinished because it has not been possible up to now to solve the Nelson and Kolmogorov equations, except in the very simple case of a free particle.
Stochastic mechanics is based on the hypothesis that all matter is subject to universal modified Brownian motion. In this report, we calculated probability density distributions using concepts of stochastic mechanics independent of bootstrapping with a known wave function. We calculate a velocity field directly from the potential and total energy and then use the resultant velocity field to do a stochastic Langevin integration over histories to create the probability density distribution for particle position. Using the harmonic oscillator as a minimally sufficient system for our exposition, we explored the effects of spatial and time discretization on solution noise. We explore the effect of energy defect off of the ground state energy on the velocity field, which dictates how a particle interacts with the background of stochastic fluctuations in position, and describes how repulsive drift (negative defect) and constructive oscillation (positive defect) end a stable state as its basin of stability in the velocity field shrinks with increasing energy defect. The results suggest a pathway for future development of stochastic mechanics as a numerical strategy to describe the structure of physical quantum systems for applications in chemistry, materials and information sciences.
Quantum mechanics is one of the most successful theories of modern physics. Nelson’s stochastic mechanics description offers an alternative approach to non-relativistic quantum mechanics based on Newtonian mechanics. In this thesis, we derive the stochastic mechanical equivalent to the quantum-mechanical Rayleigh-Ritz principle. This principle is then used to build a genetic algorithm that calculates the osmotic velocity together with the ground state energy by first minimizing the derived energy functional and then solving a Riccati equation. This new and efficient algorithm is then used to solve the ground state of two tweezer potentials used in levitodynamics. These are then compared to each other and with the harmonic oscillator by using methods from time series analysis. Additionally, we use the Itô-formula to derive two coupled stochastic differential equations, which yields a phase space description of quantum mechanics without violating the Heisenberg uncertainty principle.