preprints.org > physical sciences > theoretical physics > doi: 10.20944/preprints202403.0206.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 1 March 2024 / Approved: 4 March 2024 / Online: 5 March 2024 (13:23:14 CET)

This paper presents a quantum formulation for classical abstract dynamical systems (ADS), defined by coupled sets of first order differential equations. They are referred as “abstract” because their dynamical variables can be of different interrelated nature, not necessarily corresponding to physics, such as populations, socioeconomic variables, behavioural variables, etc. A first order classical Hamiltonian in canonical moments can be derived for ADS by using Dirac’s dynamics for singular Hamiltonian systems. And also a corresponding first order Schrödinger equation (which involves the existence of a system Planck constant particular of each system) can be derived from this Hamiltonian. However, Bohm and Hiley’s reinterpretation of quantum mechanics produces no further information about the mathematical formulation of ADS. However, a second order classical Hamiltonian in canonical moments can be also derived for ADS, as well as a corresponding second order Schrödinger equation. In this case, Bohm and Hiley’s reinterpretation of quantum mechanics provides a quantum Hamiltonian that does provide the quantum formulation for ADS, which provides new quantum variables interrelated dynamically with the classical variables. An application case is presented: the one-dimensional autonomous system given by the logistic dynamics, where the differences between the classical and the quantum trajectories are highlighted.

dynamical systems, general systems, classical dynamics, Schrödinger equation, quantum formulation, Bohm and Hiley’s interpretation, quantum dynamics, logistic function.

Physical Sciences, Theoretical Physics

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