Dirac Particle in the Coulomb Field on the Background of Hyperbolic Lobachevsky Model

2021 ◽  
Vol 24 (3) ◽  
pp. 260-271
Author(s):  
E. M. Ovsiyuk ◽  
A. D. Koral’kov ◽  
A. V. Chichurin ◽  
V.M. Red’kov
Keyword(s):  
Differential Equation ◽  
Gordon Equation ◽  
Order Differential Equation ◽  
Coulomb Field ◽  
Dirac Particle ◽  
Quantization Rule ◽  
Exact Analytical Solutions ◽  
Relativistic Hydrogen ◽  
Relativistic Hydrogen Atom

The known systems of radial equations describing the relativistic hydrogen atom on the base of the Dirac equation in Lobachevsky hyperbolic space is solved. The relevant 2-nd order differential equation has six regular singular points, its solutions of Frobenius type are constructed explicitly. To produce the quantization rule for energy values we have used the known condition for determination of the transcendental Frobenius solutions. This defines the energy spectrum which is physically interpretable and similar to the spectrum arising for the scalar Klein-Fock-Gordon equation in Lobachevsky space. In the present paper, exact analytical solutions referring to this spectrum are constructed. Convergence of the series involved is proved analytically and numerically. Squared integrability of the solutions is demonstrated numerically. It is shown that the spectrum coincides precisely with that previously found within the semi-classical approximation.

scholarly journals Dirac particle in the external coulomb field on the background of the Lobachevsky–Riemann space models

2021 ◽  
Vol 65 (2) ◽  
pp. 146-157
Author(s):  
E. M. Оvsiyuk ◽  
A. D. Koral’kov
Keyword(s):  
Hydrogen Atom ◽  
Dirac Equation ◽  
Differential Equations ◽  
Exact Solutions ◽  
Coulomb Field ◽  
Dirac Particle ◽  
Riemann Space ◽  
Energy Spectra ◽  
Spaces Of Constant Curvature ◽  
Quantization Rule

The known systems of the radial equations describing the hydrogen atom on the basis of the Dirac equation in the Lobachevsky–Riemann spaces of constant curvature are investigated. In the both geometrical models, the differential equations of second order with six regular singular points are found, and their exact solutions of Frobenius type are constructed. To produce the quantization rule for energy values we use the known condition which separates the transcendental Frobenius solutions. This provides us with the energy spectra that are physically interpretable and are similar to those for the Klein–Fock–Gordon particle in these space models. These spectra are similar to those that previously have appeared in studying the same systems of the equations with the use of the semi-classical approximation.

A Fast Method for the Determination of Emergent Intensities in Radiative Transfer Theory

1971 ◽  
Vol 2 (1) ◽  
pp. 42-43 ◽  
Author(s):  
C. J. Cannon
Keyword(s):  
Differential Equation ◽  
Radiative Transfer ◽  
Source Function ◽  
Order Differential Equation ◽  
Fast Method ◽  
First Order ◽  
Integration Schemes ◽  
Emergent Intensity ◽  
Difficult Part

One of the quantities usually required when solving the equation of radiative transfer is the intensity of radiation emerging from the surface of the medium under consideration. For multi-dimensional situations however, the methods presented to date have been numerical, and these first calculate the so-called source function Sv (r, Ω) as a function of position r, angle Ω and frequency v. This is generally the most difficult part of the exercise since an integro-difierential equation must be solved. The emergent intensity is then determined by solving a relatively simple first order differential equation by any of the well known numerical integration schemes. However, if the emergent intensity is required at a large number of angles, frequencies, and positions on the surface of the medium, and this is usually the case, the amount of computing needed may be considerable.

The Evaluation of Eigenvalues of a Differential Equation Arising in a Problem in Genetics

1962 ◽  
Vol 58 (4) ◽  
pp. 588-593 ◽  
Author(s):  
G. F. Miller ◽  
E. T. Goodwin
Keyword(s):  
Differential Equation ◽  
Asymptotic Behaviour ◽  
Order Differential Equation ◽  
Finite Size ◽  
Second Order ◽  
Random Drift ◽  
Smallest Eigenvalue ◽  
Second Order Differential Equation ◽  
Two Parameters

ABSTRACTThis paper concerns the determination of the smallest eigenvalue of a second order differential equation containing two parameters which arises in problems concerning genic selection under random drift in a population of finite size. A table of values is given, the method of computation is described, and the asymptotic behaviour for large values of one of the parameters is investigated.

scholarly journals Symmetries of Relativistic Hydrogen Atom

2019 ◽  
Vol 64 (12) ◽  
pp. 1148
Author(s):  
V. M. Simulik ◽  
I. O. Gordievich
Keyword(s):  
Hydrogen Atom ◽  
Dirac Equation ◽  
Spinor Field ◽  
Coulomb Field ◽  
Matrix Representations ◽  
Pure Matrix ◽  
Symmetry Operators ◽  
Non Local ◽  
Relativistic Hydrogen ◽  
Relativistic Hydrogen Atom

The Dirac equation in the external Coulomb field is proved to possess the symmetry determined by 31 operators, which form the 31-dimensional algebra. Two different fermionic realizations of the SO(1,3) algebra of the Lorentz group are found. Two different bosonic realizations of this algebra are found as well. All generators of the above-mentioned algebras commute with the operator of the Dirac equation in an external Coulomb field and, therefore, determine the algebras of invariance of such Dirac equation. Hence, the spin s = (1, 0) Bose symmetry of the Dirac equation for the free spinor field, proved recently in our papers, is extended here for the Dirac equation interacting with an external Coulomb field. A relativistic hydrogen atom is modeled by such Dirac equation. We are able to prove for the relativistic hydrogen atom both the fermionic and bosonic symmetries known from our papers in the case of a non-interacting spinor field. New symmetry operators are found on the basis of new gamma matrix representations of the Clifford and SO(8) algebras, which are known from our recent papers as well. Hidden symmetries were found both in the canonical Foldy–Wouthuysen and covariant Dirac representations. The found symmetry operators, which are pure matrix ones in the Foldy–Wouthuysen representation, become non-local in the Dirac model.

Existence and Uniqueness Results for Nonlinear Differential Equation Arising in Non-Newtonian Fluid Flow

2000 ◽  
Author(s):  
K. Vajravelu ◽  
J. R. Cannon ◽  
D. Rollins
Keyword(s):  
Differential Equation ◽  
Fluid Flow ◽  
Existence And Uniqueness ◽  
Order Differential Equation ◽  
Perturbation Technique ◽  
Integral Form ◽  
Rotating Cylinder ◽  
Exact Analytical Solutions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results

Abstract Solution for a nonlinear second order differential equation, arising in a viscoelastic fluid flow at a rotating cylinder, is obtained. Furthermore, using the Shauder theory and the perturbation technique existence, uniqueness and analyticity results are established. Moreover, the exact analytical solutions (in integral form) are compared with the corresponding numerical ones.

A Model of the Solar Convection Zone

1972 ◽  
Vol 2 (2) ◽  
pp. 92-92 ◽  
Author(s):  
B. E. Waters ◽  
R. Van Der Borght
Keyword(s):  
Differential Equation ◽  
Convection Zone ◽  
Order Differential Equation ◽  
Wave Number ◽  
Length Scales ◽  
Model Calculations ◽  
Solar Convection Zone ◽  
Solar Convection ◽  
Second Order Differential Equation

In a previous paper it was shown how one could improve upon the Böhm-Vitense model of the solar convection zone by the inclusion of four different length scales and by the determination of these length scales with the use of the quasi-Vitense model as developed by Unno. In this way the vertical wave number kz, associated with a characteristic eddy, can be determined by the integration of a second order differential equation. The integrations have to be started at a suitable depth and all model calculations depend critically on the assumed structure of the top layer.

scholarly journals Pauli approximation for a vector particle with anomalous magnetic moment in an external Coulomb field

2020 ◽  
Vol 56 (4) ◽  
pp. 419-435
Author(s):  
Ya. A. Voynova ◽  
N. G. Krylova ◽  
E. M. Оvsiyuk
Keyword(s):  
Differential Equation ◽  
Magnetic Moment ◽  
Anomalous Magnetic Moment ◽  
Order Differential Equation ◽  
Coulomb Field ◽  
Energy Spectra ◽  
Vector Particle ◽  
Nonrelativistic Approximation ◽  
Interaction Terms ◽  
Additional Interaction

Herein, a spin 1 particle with anomalous magnetic moment in an external Coulomb field is studied. We start with the relativistic tensor system of the Proca type in Cartesian coordinates. In these equations the Γ parameter is present related to an additional characteristic of the particle. In the case of an external magnetic field, it is interpreted as an anomalous magnetic moment. In the presence of an external electric field, additional interaction terms are presented as well; moreover, the terms of the first and second orders in parameter Γ appear. The case of an external Coulomb field is considered in detail. In the nonrelativistic approximation a Pauli type equation is obtained. In the nonrelativistic equation the separation of the variables with the use of spherical vectors is realized. One separate 2-nd order differential equation is found, in which additional interaction terms are missing. Besides, we derive systems of two coupled 2-nd order equations wherein linear and quadratic in parameter Γ interaction terms are presented. Previously, another approach was developed for analyzing the vector particle with anomalous magnetic moment. It was based on the use of tetrad formalism and separation of the variables in the Duffin – Kemmer equation with the help of the Wigner function. The nonrelativistic approximation was performed directly in the system of radial equations. Besides, previously formal Frobenius type solutions for an arising 4-th order differential equation were constructed; however, physically interpretable energy spectra were not found. We have proved that the radial equations derived by different methods are the same up to a simple liner transformation over two radial functions. In this paper, we have obtained a simpler 4-th order equation, the construction of Frobenius solutions becomes technically easier, but physical energy spectra are not found either.

Spinless Particle with Darwin–Cox Structure in an External Coulomb Field

2020 ◽  
Vol 23 (4) ◽  
pp. 357-373
Author(s):  
A. D. Koral’kov ◽  
E. M. Ovsiyuk ◽  
V. V. Kisel ◽  
A. V. Chichurin ◽  
Ya. A. Voynova ◽  
...  
Keyword(s):  
Angular Momentum ◽  
Power Series ◽  
Bound States ◽  
Gordon Equation ◽  
Coulomb Field ◽  
Spinless Particle ◽  
Quantization Rule ◽  
Spherical Region ◽  
Recurrent Relations ◽  
Order Algebraic

Generalized Klein–Fock–Gordon equation for a spinless particle with the Darwin–Cox structure, which takes into account distribution of the electric charge of a particle inside a finite spherical region is studied in presence of an external Coulomb field. There have been constructed exact Frobenius type solutions of the derived equations, convergence of the relevant power series with 8-term recurrent relations has been studied. As an analytical quantization rule is taken the so-called transcendency conditions. It provides us with a 4-th order algebraic equation with respect to energy values, which has four sets of roots. One set of roots, 0 < En;k < 1, depending on the angular momentum n = 0; 1; 2; : : : and the main quantum number n = 0; 1; 2; : : : may be interpreted as corresponding to some bound states of the particle in a Coulomb field. In the same manner, a generalized nonrelativistic Schr¨odinger equation for such a particle is studied, the final results are similar.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

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