A Stochastic Mechanics Based on Bohm‧s Theory and its Connection with Quantum Mechanics
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We construct a stochastic mechanics by replacing Bohm‧s first-order ordinary differential equation of motion with a stochastic differential equation where the stochastic process is defined by the set of Bohmian momentum time histories from an ensemble of particles. We show that, if the stochastic process is a purely random process with n-th order joint probability density in the form of products of delta functions, then the stochastic mechanics is equivalent to quantum mechanics in the sense that the former yields the same position probability density as the latter. However, for a particular non-purely random process, we show that the stochastic mechanics is not equivalent to quantum mechanics. Whether the equivalence between the stochastic mechanics and quantum mechanics holds for all purely random processes but breaks down for all non-purely random processes remains an open question.
Key words:Bohmian mechanics stochastic differential equation stochastic process
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- 15.15. M. Jammer, The Philosophy of Quantum Mechanics (Wiley, New York, 1974).Google Scholar
- 18.18. E. Nelson, Stochastic Processes in Classical and Quantum Systems (Lecture Notes in Physics 262) (Springer, Berlin, 1986), pp. 438–469.Google Scholar
- 30.30. D. Bohm and B. J. Hiley, The Undivided Universe (Routledge, London, 1993).Google Scholar
- 32.32. P. R. Holland, The Quantum Theory of Motion (University Press, Cambridge, 1993).Google Scholar
- 33.33. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Corrected and Enlarged Edition (Academic, New York, 1992).Google Scholar
- 34.34. Y. O. Tan, Equivalence of Stochastic Mechanics with Bohmian Mechanics (Master Thesis, 2004).Google Scholar