Evaluation of Fluctuation Coefficients for Three Consecutive Term Recursive Basis Functions

Gözükırmızı, Coşar; Demіralp, Metin

doi:10.1063/1.2990899

Cited by 5 publications

(3 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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].…”

Section: Basic Motivationmentioning

confidence: 99%

See 1 more Smart Citation

Dynamical Causal Modeling from a Quantum Dynamical Perspective

Demiralp¹,

Demіralp²

2010

AIP Conference Proceedings

Self Cite

View full text Add to dashboard Cite

Recent research suggests that any set of first order linear vector ODEs can be converted to a set of specific vector ODEs adhering to what we have called "Quantum Harmonical Form (QHF)". QHF has been developed using a virtual quantum multi harmonic oscillator system where mass and force constants are considered to be time variant and the Hamiltonian is defined as a conic structure over positions and momenta to conserve the Hermiticity. As described in previous works, the conversion to QHF requires the matrix coefficient of the first set of ODEs to be a normal matrix. In this paper, this limitation is circumvented using a space extension approach expanding the potential applicability of this method. Overall, conversion to QHF allows the investigation of a set of ODEs using mathematical tools available to the investigation of the physical concepts underlying quantum harmonic oscillators. The utility of QHF in the context of dynamical systems and dynamical causal modeling in behavioral and cognitive neuroscience is briefly discussed.There is a large number of mathematical tools available for the investigation of the dynamics within quantum mechanical systems. 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.ODEs are considerably important for modeling various dynamical systems in engineering as well as physical, natural and social sciences. Specifically, Dynamical Causal Modelling (DCM) [21,22] in neuroscience is an excellent example. The governing equations are typically first order linear vector ODEs and are often used to investigate the effective connectivity between various brain areas in functional imaging. On the other hand, direct applicability or utility of quantum mechanics and quantum mechanical tools within neurophysiological research has been questioned [23]. Approaches entailing quantum treatment of exocytosis, links the neocortical activity with the existence of a large number of quantum probability amplitudes, where mental intention is thought to manifest itself neurally via an increase in the probability for exocytoses in a complete dendron [24]. Phenomenological discussion of quantum states within the brain a...

show abstract

Section: Basic Motivationmentioning

confidence: 99%

“…Our main goal is to connect (12) and (15). We can seek a linear mapping between the abovementioned evolution matrices as follows…”

Section: Conversion Of First Order General Linear Vector Odes To Quanmentioning

confidence: 99%

Dynamical Causal Modeling from a Quantum Dynamical Perspective

Demiralp¹,

Demіralp²

2010

AIP Conference Proceedings

Self Cite

View full text Add to dashboard Cite

show abstract

“…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].…”

mentioning

confidence: 99%

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

Demіralp¹

2010

AIP Conference Proceedings

Self Cite

View full text Add to dashboard Cite

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 ...

show abstract