Quantum Dynamics of Multi Harmonic Oscillators Described by Time Variant Conic Hamiltonian and their Use in Contemporary Sciences

Abstract

This work focuses on the dynamics of a system of quantum multi harmonic oscillators whose Hamiltonian is conic in positions and momenta with time variant coefficients. While it is simple, this system is useful for modeling the dynamics of a number of systems in contemporary sciences where the equations governing spatial or temporal changes are described by sets of ODEs. The dynamical causal models used readily in neuroscience can be indirectly described by these systems. In this work, we want to show that it is possible to describe these systems using quantum wave function type entities and expectations if the dynamic of the system is related to a set of ODEs. WHY QUANTUM MECHANICAL TOOLS?Dynamics of quantum mechanical systems are governed by linear PDEs unless a control problem is considered [1,2,3,4,5]. Such equations are handled by the Hilbert space, and therefore, linear vector space tools. This is one and perhaps the most important reason why quantum mechanics is desired to be put on the stage when it is possible. It is also feasible to establish a bridge between quantum mechanics and sets of nonlinear ODEs. To this end, the time courses of the expectation values of certain operators can be investigated [6,7]. In this approach, if certain expectations can be expressed in terms of simpler entities, wave functions can be by-passed. This approximation is generally based on the fluctuation free matrix representation [8,9,10,11,12] and integration. The use of the fluctuation concept facilitates function approximations [13,14,15] and linear or multilinear array decompositions [16,17,18,19]. The fluctuation terms which are defined over certain operators are responsible for the creation of the difference between quantum and classical mechanics and the classical mechanics limit is the case where all fluctuations die and uncertainties vanish. The expectation dynamical equations of motion appear to be a set of ODEs with appropriate initial conditions. They are generally nonlinear. This urges us to use certain virtual quantum systems to describe the set of ODEs. This is the basic motivation of this work. QUANTUM MULTIHARMONIC OSCILLATOR SYSTEM WITH TIME VARIANT PARAMETERSHarmonic oscillators, in general, are described by a conic (second degree polynomial) potential function and polynomialcoefficients (especially the second degree monomial coefficient). These coefficients are considered time invariant in the modelling of atomic and molecular systems where the bond structure of the system is time invariant unless an external time varying influence is exerted on the system. Beyond this limitation, the kinetic energy terms are assumed to be composed of only second degree polynomials in momentum with constant coefficients. These correspond to the mass structure of the oscillators since mass values remain unchanged during the motion of the oscillator. This is true as long as an external mass transferring influence is not applied to the system. All these suggest that the Hamiltonian of a harmonic oscillator has, in general, a ...

Dynamical Causal Modeling from a Quantum Dynamical Perspective

“…The expectations of the momentum and position operators satisfy the following set of ODEs as it is described in our other paper [1] to be presented in this conferencė…”

“…These recalls suffice for our purposes. More details can be found in the aforementioned reference [1].…”

“…Recently our research group has been successful in relating ODEs to dynamics of quantum mechanical systems [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] through the equations of motion for the expectations (expected values) of certain operators corresponding to observable entities such as position and momentum of quantum harmonic oscillators. Recent work has shown that, a subset of all linear vector ODEs with varying matrix coefficients and inhomogeneities can be converted to a set of two linear vector ODEs with certain matrix coefficients using a multi harmonic oscillator system with a time dependent Hamiltonian which has a conic structure in positions and momenta [1]. This framework requires the matrix coefficients of the original ODE to be normal, in other words, diagonalizable which is a fairly severe restriction excluding all types of matrices which cannot be diagonalized but can be reduced to Jordan canonical form from our analysis.…”