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Eleventh International Conference on Geometry, Integrability and Quantization June 5–10, 2009, Varna, Bulgaria Ivaïlo M. Mladenov, Gaetano Vilasi and Akira Yoshioka, Editors Avangard Prima, Sofia 2010, pp 181–197 Geometry, Integrability and XI Quantization ON THE UNCERTAINTY RELATIONS IN STOCHASTIC MECHANICS∗ DIMITAR A. TRIFONOV, BLAGOVEST A. NIKOLOV† and IVAÏLO M. MLADENOV‡ Institute of Nuclear Research, Bulgarian Academy of Sciences Tzarigradsko chaussee 72, 1184 Sofia, Bulgaria † Department of Physics, Constantin Preslavski University 9712 Shumen, Bulgaria ‡ Institute of Biophysics, Bulgarian Academy of Sciences Acad. G. Bonchev Str., Bl. 21, 1113 Sofia, Bulgaria Abstract. It is shown that the Bohm equations for the phase S and squared modulus ρ of the quantum mechanical wave function can be derived from the classical ensemble equations admiting an aditional momentum ps of the form proportional to the osmotic velocity in the Nelson stochastic mechanics and using the variational principle with appropriate change of variables. The possibility to treat gradS and ps as two parts of the momentum of quantum ensemble particles is considered from the view point of uncertainty relations of Robertson - Schrödinger type on the examples of the stochastic image of quantum mechanical canonical coherent and squeezed states. 1. Introduction The uncertainty (indeterminacy) principle in quantum physics, which quantitatively is expressed in the form of uncertainty relations (URs) [13, 14, 24, 25] is commonly regarded as the most radical departure from the classical physics. However in the recent decades publications have appeared [5, 11, 12, 21, 23] in which inequalities are introduced in Nelson stochastic mechanics (SM) [19] and discussed as Heisenberg-type URs. The equations of motion in this mechanics coincide with the David Bohm equations [1] (the continuity equation and the ∗ Reprinted from JGSP 16 (2009) 57–75. 181