Application of Scale Relativity (ScR) Theory to the Problem of a Particle in a Finite One-Dimensional Square Well (FODSW) Potential

Al-Rashid, Saeed Naif Turki; Habeeb, M A Z; Ahmad, Khalid A.

doi:10.4236/jqis.2011.11002

Cited by 10 publications

(8 citation statements)

References 14 publications

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“…The validity of SR not restricted to the cases by Hermann [14] and Alrashid [18] [19] [20]. Besides, such applications are expected to reveal some novel concepts, such as the connection between SR and the Riccati equation [21] [22] [23] as revealed in the present work.…”

Section: Introductionmentioning

confidence: 83%

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Application of Scale Relativity to the Problem of a Particle in a Simple Harmonic Oscillator Potential

Al-Rashid¹,

Habeeb²,

Ahmed³

2017

JQIS

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In the present work, Scale Relativity (SR) is applied to a particle in a simple harmonic oscillator (SHO) potential. This is done by utilizing a novel mathematical connection between SR approach to quantum mechanics and the well-known Riccati equation. Then, computer programs were written using the standard MATLAB 7 code to numerically simulate the behavior of the quantum particle utilizing the solutions of the fractal equations of motion obtained from SR method. Comparison of the results with the conventional quantum mechanics probability density is shown to be in very precise agreement. This agreement was improved further for some cases by utilizing the idea of thermalization of the initial particle state and by optimizing the parameters used in the numerical simulations such as the time step and number of coordinate divisions. It is concluded from the present work that SR method can be used as a basis for description the quantum behavior without reference to conventional formulation of quantum mechanics. Hence, it can also be concluded that the fractal nature of space-time implied by SR, is at the origin of the quantum behavior observed in these problems. The novel mathematical connection between SR and the Riccati equation, which was previously used in quantum mechanics without reference to SR, needs further investigation in future work.

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Section: Introductionmentioning

confidence: 83%

“…Similarly, Al-Rashid [18] [19] [20], applied SR to the finite one-dimensional square well potential and special case in a double oscillator problems.…”

Section: Introductionmentioning

confidence: 99%

Application of Scale Relativity to the Problem of a Particle in a Simple Harmonic Oscillator Potential

Al-Rashid¹,

Habeeb²,

Ahmed³

2017

JQIS

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“…Dividing the real and imaginary parts in Equation ( U is the imaginary part of W ) one gets

({, \begin{array}{c} center & - D \cdot normalΔ italicU - ((), U \cdot \nabla) U = - \nabla u \\ center & \frac{\partial}{\partial t} U = 0 \end{array})

which (first equation) may be rewritten for a 1‐D path as a Riccati equation (Al‐Rashid et al ., ), c being a constant and y(x) an arbitrary function of x :

\frac{\partial}{\partial x} U (x) = - \frac{m}{ℏ} \cdot U^{2} (x) + \frac{2}{ℏ} \cdot (u (x) - c \cdot m)

\frac{\partial^{2}}{\partial x^{2}} y (x) = - \frac{2 m}{ℏ^{2}} \cdot (u (x) - c \cdot m) \cdot y (x) = 0

…”

Section: The Quantum Mechanical Picturementioning

confidence: 99%

Water Droplet Trajectories in a Sprinkler Jet Flow: The Quantum Hydrodynamic Framework

2014

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The design of a sprinkler irrigation system is always associated with a full understanding of the kinematics of the droplets during their aerial path. Resolving this problem involves both theoretical and experimental considerations. Among the theoretical studies the classical mechanical approach, based on Newton's law, offers a useful tool to describe the trajectories of water droplets from the sprinkler nozzle to the ground. The problem becomes even more complicated when not just a single droplet alone is assessed but a multi‐droplet system is accounted for. In addition to the inter‐parameter dependencies, an inter‐droplet reciprocal repulsion is also observed, mainly due to electrical interactions between the hydrogen and oxygen atoms of the different water molecules. In this context, the whole classic–quantum and single‐droplet versus multi‐droplet alternatives need to be discussed and pinpointed, and these are the main aims of the present paper which focuses on the theoretical part of the issue, thus highlighting new perspectives of a deeper comprehension in the spray flow‐related phenomena. On the whole, the new approach allows us to recast the classical fluid dynamic equations into so‐called quantum equations. Copyright © 2014 John Wiley & Sons, Ltd.

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“…As in Al‐Rashid et al . (), the first equation of system () for a 1‐D domain assumes the form of a Riccati equation:

\frac{d}{italicdx} Ux = \frac{- m}{scriptℏ} \cdot U^{2} x + \frac{2}{scriptℏ} \cdot ux - c \cdot m

where c is a case‐dependent constant of integration. Defining y ( x ) as an arbitrary function of x , the first equation in system () may be also written as (Al‐Rashid et al ., )

\frac{d^{2}}{d x^{2}} yx - \frac{2 \cdot m}{ℏ^{2}} \cdot ux - c \cdot m \cdot yx = 0

…”

Section: Single Particle Scale Relativity Theory Trajectorymentioning

confidence: 99%

Water Particle Kinematics Quantum Approach: A Challenge for Sprinkler Irrigation Systems

Wrachien

Lorenzini

Mambretti

2013

Irrigation and Drainage

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Designing a sprinkler irrigation plant is always associated with a full understanding of the kinematics of droplets during their aerial path. This requires a very complicated modelling of the problem, as many variables affect one another in contributing to the whole process. The literature offers different descriptive methods among which is the ballistic one, to which the authors have recently given a novel contribution, and which is also reported here. In addition to this, the present paper introduces two novel quantum approaches applied to describe droplet kinematics, based on the time-dependent Schrödinger equation and on the Scale Relativity Theory. Such an idea not only completes the classical description with a mean more tightly describing the microscopic phenomenon but also gives a broadly applicable tool to describe the actual kinematics of water droplets in sprinkler irrigation. RÉSUMÉLa conception d'une installation d'irrigation par aspersion est toujours associée à la pleine compréhension de la cinématique des gouttelettes d'eau au cours de leur parcours aérien. La modélisation du problème s'avère très compliquée, car autant de variables s'influencent mutuellement en contribuant à l'ensemble du processus. La littérature propose différentes méthodes descriptives, dont la méthode balistique, à laquelle les auteurs ont récemment donné une nouvelle contribution, qui est également rapportée ici. En plus de cela, le présent document présente deux nouvelles approches de mécanique quantique appliquée pour décrire la cinématique de gouttelettes, sur la base de l'équation de Schrödinger dépendante du temps et de la théorie de la relativité d'échelle. Une telle idée non seulement complète la description classique avec une plus étroite description du phénomène microscopique, mais aussi donne un outil largement applicable pour décrire la cinématique réelle de gouttelettes d'eau dans l'irrigation par aspersion. † La dynamique quantique appliquée aux particules d'eau: un défi pour les systèmes d'irrigation par aspersion. IRRIGATION AND DRAINAGE Irrig. and Drain. 62: 156-160 (2013)

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Application of Scale Relativity (ScR) Theory to the Problem of a Particle in a Finite One-Dimensional Square Well (FODSW) Potential

Cited by 10 publications

References 14 publications

Application of Scale Relativity to the Problem of a Particle in a Simple Harmonic Oscillator Potential

Application of Scale Relativity to the Problem of a Particle in a Simple Harmonic Oscillator Potential

Water Droplet Trajectories in a Sprinkler Jet Flow: The Quantum Hydrodynamic Framework

Water Particle Kinematics Quantum Approach: A Challenge for Sprinkler Irrigation Systems

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