Stochastic Parameters in Quantum Mechanical Systems

Marek Pechal

Johan G. B. Beumee, HERSCHEL A. RABITZ

An estimation method for unknown parameters in the initial conditions and the potential of a quantal system using the stochastic interpretation of quantum mechanics and some results in system theory are presented. According to this interpretation the possible trajectories of a particle through coordinate space may be represented by the realization of a stochastic process that satisfies a stochastic differential equation. The drift term in this equation is derived from the wave function and consequently contains all unknown parameters in the initial conditions and the potential. The main assumption of the paper is that a continuous sequence of position measurements on the trajectory of the particle can be identified with a realization of this stochastic process over the corresponding period of time. An application of the stochastic filtering theorems subsequently provides a minimum variance estimate of the unknown parameters in the drift conditional on this continuous sequence of measurements. As simple illustrations, this method is used to obtain estimates for the initial momentum of a free particle given measurements on its trajectory and to construct an estimator for the unknown parameters in a harmonic potential. It is shown that an optimal estimator exists if the stochastic processes are associated with a wave function from a potential of the Rellich type. In addition the a posteriori probability density of the parameters in the quantal system is calculated, assuming that all parameters involved prescribe a Rellich potential.

C. V. Heer

Stig Stenholm

Starting from the standard Lindblad form of a dissipative master equation, we identify a stochastic parameter in the quantum time evolution. This lacks dynamic evolution but its value is affected by the dissipative part of the master equation. An unravelling of the master equation in terms of quantum trajectories then shows this parameter performing the erratic variation characteristic of a stochastic variable. When the two evolutions separate, the stochastic Markov process can be extracted and solved independently and imposed on the parameter of the quantum evolution. This is the traditional form of treatment of stochastic variables in quantum mechanics. The relations to earlier works are established, and the approach is illustrated by simple model examples.

N. G. van Kampen

John J. Vastola, William R. Holmes

Stochastic mechanics---the study of classical stochastic systems governed by things like master equations and Fokker-Planck equations---exhibits striking mathematical parallels to quantum mechanics. In this article, we make those parallels more transparent by presenting a quantum mechanics-like formalism for deriving a path integral description of systems described by stochastic differential equations. Our formalism expediently recovers the usual path integrals (the Martin-Siggia-Rose-Janssen-De Dominicis and Onsager-Machlup forms) and is flexible enough to account for different variable domains (e.g. real line versus compact interval), stochastic interpretations, arbitrary numbers of variables, explicit time-dependence, dimensionful control parameters, and more. We discuss the implications of our formalism for stochastic biology.