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Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento

Comments on microscopic mechanics, generalizations of classical mechanics and Planck’s oscillators

M. Yussouff

Published Jul 1, 1983• 0 citations
Publisher
DOI
10.1007/bf02751817

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The possibility is discussed of inferring or simulating some aspects of quantum dynamics by adding classical statistical fluctuations to classical mechanics. We introduce a general principle of mechanical stability and derive a necessary condition for classical chaotic fluctuations to affect confined dynamical systems, on any scale, ranging from microscopic to macroscopic domains. As a consequence we obtain, both for microscopic and macroscopic aggregates, dimensional relations defining the minimum unit of action of individual constituents, yielding in all cases Planck action constant.
We apply the non-equilibrium statistical operator method within the mixed quantum–classical representation to the non-contact atomic force microscopy system in which a macroscopic tip is made to oscillate above a sample surface. In our treatment the tip is treated classically, whereas a microscopic system comprising atoms in the surface and in the nano-tip is considered quantum mechanically. Using a very general non-Markovian approach, we obtain both the Fokker–Planck equation for the tip and, independently, its equation of motion with friction terms and memory effects completely accounted for. In the classical limit (high temperatures) our equations turn exactly into the classical counterparts derived previously.
The generalized uncertainty principle (GUP) is a modification of Heisenberg's Principle predicted by several theories of quantum gravity. It consists of a modified commutator between the position and momentum. In this work, we compute potentially observable effects that GUP implies for the harmonic oscillator, coherent, and squeezed states in quantum mechanics. In particular, we rigorously analyze the GUP-perturbed harmonic oscillator Hamiltonian, defining new operators that act as ladder operators on the perturbed states. We use these operators to define the new coherent and squeezed states. We comment on potential applications.