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Journal of Physics A Mathematical and General

Stochastic mechanics of mixed states

Marc-Thierry Jaekel, D. Pignon

Published Jan 11, 1984• 7 citations
Publisher

Abstract

Nelson's stochastic interpretation of quantum mechanics is extended from the case of pure states to that of mixed states. It is shown that a pure probabilistic formalism, which applies the Newton-Nelson law to the initial position and velocity distributions, does not reproduce the time evolution predicted by quantum mechanics. In order to recover the latter, a new notion must be introduced, that of pure quantum states, over which the mixture has to be decomposed, and which then satisfy the Newton-Nelson law independently.

DOI
10.1088/0305-4470/17/1/015

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Nelson's stochastic interpretation of quantum mechanics is extended from the case of pure states to that of mixed states. It is shown that a pure probabilistic formalism, which applies the Newton-Nelson law to the initial position and velocity distributions, does not reproduce the time evolution predicted by quantum mechanics. In order to recover the latter, a new notion must be introduced, that of pure quantum states, over which the mixture has to be decomposed, and which then satisfy the Newton-Nelson law independently.
Nelson's assignment of a Markov process in configuration space to each Schr\"odinger Hamiltonian and each pure state is generalized to mixed states, arbitrary configuration spaces given by a maximal set of commuting observables, and general Hamiltonians given by integral kernels. Conditions for the quantum-mechanical description to be reconstructable from this stochastic description are discussed. It is shown that those stochastic expectation values which are stable with respect to the operation of coupling a dynamically independent but statistically correlated system to the given one are precisely those already expressible as quantum-mechanical expectation values.
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.
Bohm Mechanics and Nelson Stochastic Mechanics are confronted with Quantum Mechanics in presence of non-interacting subsystems. In both cases, it is shown that correlations at different times of compatible position observables on stationary states agree with Quantum Mechanics only in the case of product wave functions. By appropriate Bell-like inequalities it is shown that no classical theory, in particular no stochastic process, can reproduce the quantum mechanical correlations of position variables of non interacting systems at different times.