Quantum mechanical motion of classical particles

Institute for Radiophysics and Electronics,
National Academy of Science,
Kharkiv, Ukraine

We consider possible ways of quantum mechanical, or wave, properties of manifestation of moving point-like particles within the frame of classical physics. Different ways of derivations of Schrödinger Equation and physics behind these derivations are discussed in details. In particular, in [1] it has been shown that Schrödinger Equation may be derived from a generalized condition for stable trajectories of classical particle moving under potential force and fluctuation-dissipative environment. It has been shown that the energy of dissipative forces exactly equals to the Bohm's Quantum Potential (QP), while the above dissipative forces are the reason for converting Schrödinger Equation into its Langevin type equation with a stochastic term in the right hand side [2]. We have used stochastic Schrödinger Equation for quantum oscillator with dissipation to validate the above dissipative property of the QP. Quantum analogue (a quantum Langevin equation) of classical motion equation for a mean value of the momentum operator has been obtained on the basis of its exact solution. It has been shown that QP calculated with the help of wave function for the stochastic Schrödinger Equation for quantum oscillator with dissipation, produces the dissipative forces and fluctuations with zero mean value, which independently leads us to an analogues quantum Langevin equation. Deriving the Fischer information for the Quantum Potential and using Cramer-Rao inequality we directly arrive to the Heisenberg uncertainty principle for moving particles. We discuss a probable physical nature of the y-field source generating QP. We investigate physical reasons for the particle's wave properties which may be tied with its local complicated (but periodical) motion around a mass center and may be described in terms of "zitterbewegung" models [3]. In addition, we describe point-like particle within the frame of model for particle moving chaotically around a mass center, which may be a source for both wave and probabilistic properties simultaneously observed in Quantum Mechanics.

[1] V.D. Rusov, D.S. Vlasenko, and S.Ch. Mavrodiev, Ann. Phys. (2011), doi:10.1016/j.aop.2011.04.012.
[2] V.D.Rusov, K.A.Lukin and D.S. Vlasenko, Schrödinger Equation as Equation for Stable Motion of Classical Particles in Fluctuation-Dissipative Environment, Third Int. Conf. on Quantum Electrodynamics and Statistical Physics. Aug.2 –Sept.2, 2011, Kharkov, Ukraine. Book of Abstracts, p.127.
[3] G. Salesi. Non-relativistic classical mechanics for spinning particles arXiv:quant-ph/0412145v1. 2004.