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European Journal of Physics

Sample paths in stochastic mechanics: some numerical exercises

D de Falco, E Pulignano

Published Jan 1, 1988• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0143-0807/9/1/006

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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 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.
Maintaining the position that the wave function ψψψ provides a complete description of state, the traditional formalism of quantum mechanics is augmented by introducing continuous trajectories for particles which are sample paths of a stochastic process determined (including the underlying probability space) by ψψψ. In the resulting formalism, problems relating to measurements and objective reality are solved as in Bohmian mechanics (without sharing its weak points). The pitfalls of Nelson's stochastic mechanics are also avoided.
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.
We have advocated in a previous paper (Godart M. arXiv: 1206.2917v2[quant-ph] ) a version of the stochastic theory of quantum mechanics. It is indirectly based on a method proposed by Nelson to associate a Markov process with any solution of the Schroedinger equation. The debate began very soon on the question to know if the new theory based on that stochastic procees was equivalent to the orthodox Copenhagen version. We conclude in this paper that the answer is in the negative in agreement with the opinion of several physicists. We show however that the elementary solutions of the Schroedinger equation can be recovered formally from the stochastic theory in a great number of particular cases. We confirm also that this equation is no longer valid when a magnetic field is at work and that it must then give way to some other equation.
Stochastic mechanics develops a formulation of quantum physics by assuming that a classical particle is also subjected to Brownian-like motion. Because this presumes that a particle's path is non-differentiable, making this transformation often involves implicitly replacing time-derivatives in classical mechanics by finite differences. In this paper, we will develop a stochastic mechanics by explicitly replacing time-derivatives in classical mechanics by finite differences, deriving the resulting dynamics and then introducing stochasticity. Like Nelson's original version of stochastic mechanics, our stochastic mechanics is also consistent with the Schrödinger Equation.