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

Authors:
Saeed Naif Turki Al Rashid at University of Anbar
Saeed Naif Turki Al Rashid
Mohammed AZ Habeeb
Mohammed AZ Habeeb
Mohammed AZ Habeeb
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Khalid Abdulwahab Ahmad at Al-Mustansiriya University
Khalid Abdulwahab Ahmad
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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 are 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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Supplementary resources (2)

Application of Scale Relativity to the Problem of a Particle in a Simple Harmonic Oscillator Potential
Data
July 2017
Saeed Al Rashid · Mohammed A Z Habeeb · Khalid Ahmed
Simple Harmonic Oscillator
Data
July 2017
Saeed Al Rashid · Mohammed A Z Habeeb · Khalid Ahmed
... In the framework of scale relativity, Herman [32] and later Al Rashid et al. [42] simulated QM particle in a box using the Langevin equations. Later, Al-Rashid et al. [43] simulated the quantum harmonic oscillator extending Herman's approach. The approach presented here can be used as an alternative to numerical solutions of the Schrödinger equation. ...
Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation
Article
Full-text available
  • Jun 2018
  • Entropy
The present work is concerned with the study of a generalized Langevin equation and its link to the physical theories of statistical mechanics and scale relativity. It is demonstrated that the form of the coefficients of the Langevin equation depends critically on the assumption of continuity of the reconstructed trajectory. This in turn demands for the fluctuations of the diffusion term to be discontinuous in time. This paper further investigates the connection between the scale-relativistic and stochastic mechanics approaches, respectively, with the study of the Burgers equation, which in this case appears as a stochastic geodesic equation for the drift. By further demanding time reversibility of the drift, the Langevin equation can also describe equivalent quantum-mechanical systems in a path-wise manner. The resulting statistical description obeys the Fokker–Planck equation of the probability density of the differential system, which can be readily estimated from simulations of the random paths. Based on the Fokker–Planck formalism, a new derivation of the transient probability densities is presented. Finally, stochastic simulations are compared to the theoretical results.
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... In the framework of scale relativity, Herman [23] simulated QM particle in a box using the Langevin equations. Later, Al-Rashid et al. [39] simulated the quantum harmonic oscillator extending Herman's approach. ...
Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation
Preprint
Full-text available
  • May 2018
The present work is concerned with the study of the generalized Langevin equation and its link to the physical theories of statistical mechanics and scale relativity. It is demonstrated that the form of the coefficients of Langevin equation depend critically on the assumption of continuity of the reconstructed trajectory. This in turn demands for the fluctuations of the diffusion term to be discontinuous in time. This paper further investigates the connection between the scale-relativistic and stochastic mechanics approaches, respectively, with the study of the Burgers equation, which in this case appears as a stochastic geodesic equation. By further demanding time reversibility of the drift the Langevin equation can also describe equivalent quantum-mechanical systems in a path-wise manner. The resulting statistical description obeys the Fokker-Plank equation of the probability density of the differential system, which can be readily estimated from Monte Carlo simulations of the random paths. Based on the Fokker-Plank formalism a new derivation of the transient probability densities is presented. Finally, stochastic simulations are compared to the theoretical results.
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NUMERICAL SIMULATIONS OF PARTICLE IN A DOUBLE OSCILLATORS
Article
Full-text available
  • Dec 2007
In the present work, we extend the Hermann and Al-Rashid works to the problem of particle in a double oscillators potential. In this problem, one can take a special case when oscillation quantum number (ν) is none negative integer. Computer programming is built to make numerical simulations to this problem. The probability density of finding particle in a double oscillators potential is calculated without using Schrödinger equation or any conventional quantum mechanics. This probability is compared with probability of conventional quantum mechanics.
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Some Applications of Scale Relativity Theory in Quantum Physics
Thesis
Full-text available
  • Jun 2006
The theory of scale relativity (ScR) is based on the extension of the principle of relativity of motion to include relativity of scale. Fractal space-time is the basis of this new theory as formulated by Nottale. Applications of this theory encompass diverse fields from microphysics, cosmology and complex systems. Previously, direct numerical simulations using this theory in quantum physics as performed by Hermann have resulted in the appearance of the correct quantum behavior without resorting to the Schrödinger equation. However these simulations were performed for only one limited case of a particle in an infinite one-dimensional square well. Hence, there is a need for more such applications using this theory to establish its validity in the quantum domain. In the present work, and along the lines of Hermann, ScR theory is applied to other standard one-dimensional quantum mechanical problems. These problems are: a particle in a finite one-dimensional square well, a particle in a simple harmonic oscillator (SHO) potential and a particle in a one-dimensional double-well potential. Some mathematical problems that arise when obtaining the solution to these problems were overcome by utilizing a novel mathematical connection between ScR theory and the well-known Riccati equation. Then, computer programmes were written using the standard MATLAB 7 code to numerically simulate the behavior of the quantum particle in the above potentials utilizing the solutions of the fractal equations of motion obtained from ScR theory. Several attempts were made to fix some of the parameters in the numerical simulations to obtain the best possible results in a practical computer CPU time within the limited local computer facilities. Comparison of the present results for the particle probability density in the three potentials with the corresponding results obtained from conventional quantum mechanics by solving the Schrödinger equation, shows very good 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 ScR theory can be used as a basis for describing the quantum behavior without reference to conventional quantum mechanics. Hence, it can also be concluded that the fractal nature of space-time, which is the basis of ScR theory, is the origin of the quantum behavior observed in these problems. More applications to potentials in more than one-dimension, including asymmetric potentials, would give greater confidence along these lines. Also, the novel mathematical connection between ScR theory and the Riccati equation, that was previously used in quantum mechanics without reference to ScR theory, needs further investigation in future work.
View
The Theory of Scale Relativity
Article
Full-text available
  • Aug 1992
Basing our discussion on the relative character of all scales in nature and on the explicit dependence of physical laws on scale in quantum physics, we apply the principle of relativity to scale transformations. This principle, in combination with its breaking above the Einstein-de Broglie wavelength and time, leads to the demonstration of the existence of a universal, absolute and impassable scale in nature, which is invariant under dilatation. This lower limit to all lengths is identified with the Planck scale, which now plays for scale the same role as is played by light velocity for motion. We get new scale transformations of a Lorentzian form and generalize the de Broglie and Heisenberg relations. As a consequence the high energy length and mass scales now decouple, energy and momentum tending to infinity when resolution tends to the Planck scale, which thus plays the role of the previous zero point. This theory solves the problem of divergence of charge and mass (self-energy) in electrodynamics, implies that the four fundamental couplings (including gravitation) converge at the Planck energy, improves the agreement of GUT predictions with experimental results, and allows one to get precise estimates of the values of the fundamental coupling constants.
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The Theory of Scale Relativity: Non-Dieren tiable Geometry and Fractal Space-Time1
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  • Aug 2004
The aim of the theory of scale relativity is to derive the physical behavior of a non-dieren tiable and fractal space-time and of its geodesics (with which particles are identied), under the constraint of the principle of the relativity of scales. We mainly study in this contribution the eects induced by internal fractal structures on the motion in standard space. We nd that the main consequence is the transformation of classical mechanics in a quantum mechanics. The various mathematical quantum tools (complex wave functions, spinors, bi-spinors) are built as manifestations of the non-dieren tiable geometry. Then the Schrodinger, Klein-Gordon and Dirac equations are successively derived as integrals of the geodesics equation, for more and more profound levels of de- scription. Finally we tentatively suggest a new development of the theory, in which quantum laws would hold also in the scale-space: in such an approach, one naturally denes a new conservative quantity, named 'complexergy', which measures the complexity of a system as regards its internal hierarchy of organization. We also give some examples of applications of these proposals in various sciences, and of their experimental and observational tests.
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Gauge Field theory in scale relativity
Article
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  • Dec 2013
The aim of the present article is to give physical meaning to the ingredients of standard gauge eld theory in the framework of the scale relativity theory. Owing to the principle of the relativity of scales, the scale-space is not absolute. Therefore, the scale variables are functions of the space-time coordinates, so that we expect a coupling between the displacement in space-time and the dilation/contraction of the scale variables, which are identied with gauge transformations. The gauge elds naturally appear as a new geometric contribution to the total variation of the scale variables. The gauge charges emerge as the generators of the scale transformation group applied to a generalized action (now identied with the scale relativistic in- variant) and are therefore the conservative quantities which nd their origin in the symmetries of the scale-space. We recover the expression for the covariant deriva- tive of non-Abelian gauge theory. Under the gauge transformations, the fermion multiplets and the boson eld transform in such a way that the Lagrangian, which is here derived instead of being set as a founding axiom, remains invariant. We have therefore obtained gauge theories as a consequence of scale symmetries issued from a geometric fractal space-time description, which we apply to peculiar examples of the electroweak and grand unied theories.
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First-Order Differential Equations
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First-order differential equations provide a rich example of differential equations of many forms, most of which we can solve easily in the formal sense, and many of which we can solve and actually get answers. From calculus, we need the rules of differentiation, both the formulas (for sums, products, quotients, chain rule, and so on) and the derivatives of the standard Mathematica functions (x n , trigonometric functions, logarithms, exponentials, hyperbolic functions, and the like), and techniques of integration. We will likely see an example of most of the kinds of integrals that you ever attempted. If this sounds like bad news, the good news is that Mathematica can do these integrations for you. You will serve as the mastermind, and Mathematica will do the labor. Your responsibility is to ensure the correctness of the work that you are having Mathematica do, but Mathematica will do these correctness and consistency checks for you. You control what is being done; Mathematica does the hard work. It is important that you manually do some examples of problems of each type. The reason for this is that if you have no idea how to do a problem yourself, then it is not too likely that you will know how to guide Mathematica through the solution process.
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Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies
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Numerical simulation of a quantum particle in a box
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It is shown how one can get numerical prediction of quantum mechanical particle behaviour without using the Schrödinger equation. The main steps of this development are the non-differentiability hypothesis, the equations of motion entailed by this hypothesis, and the numerical formulation of a simple one-dimensional problem: the particle in a box.
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