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Quantum Wave Function Based on String Theory for Frictional Medium to Obtain Collision Probability, Energy Operator and Schrodinger Equation
Issue:
Volume 5, Issue 3, September 2019
Pages:
52-57
Received:
24 March 2019
Accepted:
21 May 2019
Published:
12 August 2019
Abstract: Schrodinger equation suffer from being not sensitive to the mechanical properties as well as the electric and magnetic properties of matter. This set back can be cured by starting the derivation using the function sensitive to these parameters. Treating particle as vibrating strings a useful expression for the velocity was found using the equation of motion. Then the relation of current density with a velocity and electric field intensity was utilized to obtain the electric field intensity in a frictional medium. Using the analogy of the electric field and quantum wave function, the wave function was obtained and found to give the conventional expression for the collision probability with relaxation time twice the classical one. Another approach was tackled by obtaining a useful expression of the total energy of strings for resistive collisional medium. This expression utilizes the wave function of quantum particle in a frictional medium to obtain collision probability formula. Fortunately this latter approach gives a relaxation time equal to the classical one. The same wave function is used to find Hamiltonian operator for the both steady state and perturbed state by friction. Fortunately both Hamiltonians satisfy hermiticty condition. The hermiticty condition for the perturbed states however needs splitting the Hamiltonian into unpertured and perturbed part.. The perturbed term satisfies uncertainty principle. The energy expression for the resistive medium resembles that of Einstein and RLC circuits. Schrodinger equation for the frictional medium was also found, where it reduces to the ordinary one when friction disappear.
Abstract: Schrodinger equation suffer from being not sensitive to the mechanical properties as well as the electric and magnetic properties of matter. This set back can be cured by starting the derivation using the function sensitive to these parameters. Treating particle as vibrating strings a useful expression for the velocity was found using the equation ...
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Study on a Spinorial Representation of Linear Canonical Transformation
Raoelina Andriambololona,
Ravo Tokiniaina Ranaivoson,
Hanitriarivo Rakotoson
Issue:
Volume 5, Issue 3, September 2019
Pages:
58-65
Received:
15 June 2019
Accepted:
30 July 2019
Published:
26 August 2019
Abstract: This paper is mainly focused on the description of an approach for establishing a spinorial representation of linear canonical transformations. It can be considered as a continuation of our previous works concerning linear canonical transformations and phase space representation of quantum theory. The said method is based on the development of an adequate parameterization of linear canonical transformations which permits to represent them with special pseudo-orthogonal transformations. Obtaining this pseudo-orthogonal representation makes it possible to establish the spinorial representation exploiting the well-known relation existing between special pseudo-orthogonal and spin groups. The cases of one dimension and general multidimensional theories are both studied. The design of the pseudo-orthogonal transformation associated to a linear canonical transformation is achieved by introducing adequate operators which are linear combinations of reduced momentum and coordinate operators. It is shown that a linear canonical transformation is equivalent to a special pseudo-orthogonal transformation defined in the set formed by these adequate operators. The spinorial representation is then deduced by defining a composite operator which is linear combinations of the tensorial products of the generators of the Clifford algebra with the adequate operators defining the special pseudo-orthogonal representation. It is established that unlike the case of a spinorial representation associated with an ordinary commutative vector space, the main invariant corresponding to the transformation is not the square of the composite operator but a higher degree polynomial function of it.
Abstract: This paper is mainly focused on the description of an approach for establishing a spinorial representation of linear canonical transformations. It can be considered as a continuation of our previous works concerning linear canonical transformations and phase space representation of quantum theory. The said method is based on the development of an a...
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Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition
Konan Firmin N'gohisse,
Diabate Nabongo,
Lassane Traoré
Issue:
Volume 5, Issue 3, September 2019
Pages:
66-71
Received:
17 July 2019
Accepted:
12 August 2019
Published:
26 August 2019
Abstract: Theoretical study of the phenomenon of blow-up solutions for semilinear Schrödinger equations has been the subject of investigations of many authors. It is said that the maximal time interval of existence of the solution blows up in a finite time when this time is finite, and the solution develops a singularity in a finite time. In fact, semilinear Schrhödinger equation models a lot of physical phenomenon such as nonlinear optics, energy transfer in molecular systems, quantum mechanics, seismology, plasma physics. In the past, certain authors have used numerical methods to study the phenomenon of blow-up for semilinear Schrödinger equations. They have considered the same problem and one proves that the energy of the system is conserved, and the method used to show blow-up solutions are based on the energy's method. This paper proposes a method based on a modification of the method of Kaplan using eigenvalues and eigenfunctions to show that the semidiscrete solution blows up in a finite time under some assumptions. The semidiscrete blow-up time is also estimate. Similar results are obtain replacing the reaction term by another form to generalise the result. Finally, this paper propose two schemes for some numerical experiments and a graphics is given to illustrate the analysis.
Abstract: Theoretical study of the phenomenon of blow-up solutions for semilinear Schrödinger equations has been the subject of investigations of many authors. It is said that the maximal time interval of existence of the solution blows up in a finite time when this time is finite, and the solution develops a singularity in a finite time. In fact, semilinear...
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Numerical Study of Creeping Flow Through Sinusoidally Periodic Tube
Asif Mahmud,
Suhana Perveen,
Md. Nazmul Hasan,
Md. Samsuzzoha,
Nazmul Islam
Issue:
Volume 5, Issue 3, September 2019
Pages:
72-81
Received:
29 April 2019
Accepted:
4 September 2019
Published:
19 September 2019
Abstract: There has been renewed interest in the flow behaviour within tubes with periodically varying cross-section with the recognition that they can be used as particle separation devices. In this paper, we present a numerical study of the effect of tube geometry on creeping flow of viscous incompressible fluid through sinusoidally constricted periodic tube which is axisymmetric but longitudinally asymmetric. The boundary element method is used to solve for the flow in the tube by specifying the pressure drop across the ends of the tube. The boundary element equations have been formulated for an infinite periodic tube by writing the velocity in terms of the integrals over the tube boundary and is used to calculate the force on the tube boundary, to obtain the detailed velocity distribution within the tube and to determine the effect of amplitude and wavelength of corrugation on the structure of the flow. We have found that the highest axial velocity is at throat region and lowest axial velocity is at expansion region. Also, we have discovered that the maximum radial velocity occurs at diverging cross-section and minimum radial velocity occurs at converging cross-section. The tangential force on the tube wall is examined for different amplitudes and wavelengths of corrugation and observed that the tangential force is greater in the constricted region than in the expansion region. The physical quantities (such as velocity and force) increase with increasing amplitude and decrease with increasing wavelength. Finally, we have compared our results with the work of Hemmat and Borhan [3] and have found good agreement with them.
Abstract: There has been renewed interest in the flow behaviour within tubes with periodically varying cross-section with the recognition that they can be used as particle separation devices. In this paper, we present a numerical study of the effect of tube geometry on creeping flow of viscous incompressible fluid through sinusoidally constricted periodic tu...
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A Proof on the Conjecture of Twin Primes
Issue:
Volume 5, Issue 3, September 2019
Pages:
82-84
Received:
15 July 2019
Accepted:
27 July 2019
Published:
20 September 2019
Abstract: Although the mathematicians all over the world offered hard explorations of more than one hundred years, the proof of using pure mathematical theories on the conjecture of twin primes has not born in the world. This paper is trying to apply computer program to prove that corresponding to infinite primes p, there are infinite p+2 primes. As a mathematical proof, the paper uses the concept of mapping to connect the computer program and the pure mathematical theory. With the requirement of a mathematical proof, in accord with the restriction of the integer of which the computer allows to take, an assumption is suggested, and on the basis of it, using the program of C language the paper presents, or regarding the C program as the mapping from infinite p primes to infinite p+2 primes, the paper proves that corresponding to infinite primes p, there are infinite p+2 primes; namely, the conjecture of twin primes is true.
Abstract: Although the mathematicians all over the world offered hard explorations of more than one hundred years, the proof of using pure mathematical theories on the conjecture of twin primes has not born in the world. This paper is trying to apply computer program to prove that corresponding to infinite primes p, there are infinite p+2 primes. As a mathem...
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Forced Nutation for Rigid Earth Model with Different Theories
Mohamed Soliman,
Hadia Hassan Selim,
Inal Adham Hassan
Issue:
Volume 5, Issue 3, September 2019
Pages:
85-96
Received:
13 August 2019
Accepted:
10 September 2019
Published:
23 September 2019
Abstract: Where Earth is not strictly rigid body but can responds to any effects that tend to its rotation and shape, we will explain, in the present paper, the goal which is to define the forced nutation for a rigid Earth model using two different theories. We will formulate a first order Hamiltonian of a deformable Earth for its rotational motion around the Sun through the contribution of triaxial symmetry of the Earth. The formulation of the theory will be formed twice times. Firstly, deduce the tidal affect’s forces by Luni - Solar attraction coupling with the Earth’s geopotential force. Secondly, through the formulation, we will neglect the coupling between the different effects (the geopotential Earth force effect and the Luni - Solar attraction force), so, we will find the transform of the Hamiltonian for each force separately. The analytical solution for the formulated Hamiltonian will be derived for the two cases by using perturbation technique of Lie - Hori series. Once can get the analytical solution by getting the generation function, we will derive the nutation series analytically and numerically for each case and conclude over the results.
Abstract: Where Earth is not strictly rigid body but can responds to any effects that tend to its rotation and shape, we will explain, in the present paper, the goal which is to define the forced nutation for a rigid Earth model using two different theories. We will formulate a first order Hamiltonian of a deformable Earth for its rotational motion around th...
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