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Journal of Applied Sciences

Fractional Quantum Problem Based on Caputo-Fabrizio Derivative

Bouzenna Fatma El-Genbazi, Meftah Mohammed T

Published Jan 15, 2022• 0 citations
Publisher

Abstract

Background and Objective: In the context of the novel definition of the fractional derivative, we solve the Fractional Schrödinger equation based on the non-singular Caputo-Fabrizio derivative definition.The quantum particle is submitted to a step potential.We focus our study to find the reflection and transmission coefficients.Materials and Methods: To reach this goal, we first work out the fractional differential equation defined in terms of the Caputo-Fabrizio derivative.Results: Hence, according to the obtained wave functions of fractional Schrödinger equations, the reflection and transmission coefficient are calculated.Conclusion: Interestingly, our findings collapse to the standard results when the fractional order of derivative become integer numbers.

DOI
10.3923/jas.2022.42.47

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Background and Objective: In the context of the novel definition of the fractional derivative, we solve the Fractional Schrödinger equation based on the non-singular Caputo-Fabrizio derivative definition.The quantum particle is submitted to a step potential.We focus our study to find the reflection and transmission coefficients.Materials and Methods: To reach this goal, we first work out the fractional differential equation defined in terms of the Caputo-Fabrizio derivative.Results: Hence, according to the obtained wave functions of fractional Schrödinger equations, the reflection and transmission coefficient are calculated.Conclusion: Interestingly, our findings collapse to the standard results when the fractional order of derivative become integer numbers.
This study investigates the time-fractional Schrödinger equation (TFSE) for a free particle using the Caputo fractional derivative. The model extends the classical Schrödinger equation by incorporating memory effects through a non-integer order time derivative. By applying the Fourier transform method, an explicit solution is derived in terms of the Mittag–Leffler function, which generalizes the classical exponential evolution. The analysis reveals that the resulting dynamics are non-unitary, leading to a time-dependent total probability that deviates from the conservation property of standard quantum mechanics. Numerical illustrations demonstrate the influence of the fractional order ν on wave packet evolution and probability behavior. The results highlight the role of fractional calculus in modeling non-Markovian quantum systems and provide insight into the transition from fractional to classical dynamics as v→1. .
Fractional calculus is becoming increasingly important nowadays in studying and understanding fundamental physical phenomena, both simple and complex, through the formulation of generalized models. In the present work, a treatment of non-integer order is developed for the study of the wave function that describes the accessible states of a one-dimensional particle confined in a finite well with finite and constant walls, using a conformable derivative in describing the temporal evolution of the states of the system and Caputo-Fabrizio fractional derivative in describing its spatial evolution. By plotting the complete fractional solution ψ(x,t) for given values ​​of the Kernel indices α, β, γ, we recover the flat-wave front oscillations predicted by the integer-order calculation, whose superposition represents the particle state with uniform probability in all space. However, for another range of values ​​of the indices of the integral transformation, damping effects are observed, which do not seem to have a quantum mechanical physical meaning in themselves, but they do have a special physical-mathematical interest since such effects are related to the nature of the Caputo-Fabrizio operator, which as a non-local operator saves memory, that is, non-local spatial effects entail long-range interactions in a physical region, so the generalized wave function obtained does not depend exclusively on a point in space but on a region of space. When calculating the probability density, the fractional wave function shows a definite parity when γ is unitary, just as expected for a symmetric potential well. The generalized model obtained by incorporating a greater number of free parameters can represent an appropriate alternative formulation in the description of the probabilistic behavior of simple and complex quantum mechanical systems. Palabras clave: Ecuación de Schrödinger fraccional; Derivada fraccionaria de Caputo, derivada fraccionaria de Caputo-Fabrizio, derivada general conformable de Khalil.
The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a "free particle" obeying fractional dynamics, obtained by replacing the integer order derivatives with respect to time by those of fractional order.The equations of motion contain quantities which have "fractional" dimensions, chosen such that the "energy" has the correct dimension [𝑀𝐿 2 /𝑇 2 ].The action 𝑆 is defined as a fractional time integral of the Lagrangian, and a "fractional Planck constant" is introduced.The TFSE corresponds to a "subdiffusion" equation with an imaginary fractional diffusion constant and reproduces the regular Schrodinger equation in the limit of integer order.The present work corrects a number of errors in Naber's work.The correct continuity equation for the probability density is derived and a Green function solution for the case of a "free particle" is obtained.The total probability for a "free" particle is shown to go to zero in the limit of infinite time, in contrast with Naber's result of a total probability greater than unity.A generalization to the case of a particle moving in a potential is also given.
The ‘diffraction in space’ and the ‘diffraction in time’ phenomena are considered in regard to a continuously open, and a closed shutter that is opened at an instant in time, respectively. The purpose of this is to provide a background to the principal theme of this article, which is to extend the ‘quantum shutter problem’ for the case when the wave function is determined by the fundamental solution to a partial differential equation with a fractional derivative of space or of time. This involves the development of Green’s function solutions for the space- and time-fractional Schrödinger equation and the time-fractional Klein–Gordon equation (for the semi-relativistic case). In each case, the focus is on the development of primarily one-dimensional solutions, subject to an initial condition which controls the dynamical behaviour of the wave function. Coupled with variations in the fractional order of the fractional derivatives, illustrative example results are provided that are based on presenting space-time maps of the wave function; specifically, the probability density of the wave function. In this context, the paper provides a case study of fractional quantum mechanics and control using fractional calculus.
In this paper, we illustrate an application of the Laplace transformation for finding the quantum mechanical Reflection and Transmission coefficients for a particle through a one-dimensional vertical step potential. Quantum mechanics is one of the branches of physics in which the physical problems are solved by algebraic and analytic methods. By applying the Laplace transformation, we can find the quantum mechanical Reflection and Transmission coefficients for a particle through a one-dimensional vertical step potential. Generally, the Laplace transformation has been applied in different areas of science and engineering and makes it easier to solve the problems inengineering applications. It is a mathematical tool which has been put to use for solving the differential equations without finding their general solutions. It has applications in nearly all science and engineering disciplines like analysis of electrical circuits, heat and mass transfer, fluid dynamics, nuclear physics, process controls, quantum mechanical problems,etc.