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

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.

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EXACT SOLUTIONS OF NON-CENTRAL POTENTIALS

Modern Physics Letters B ◽

10.1142/s0217984910024134 ◽

2010 ◽

Vol 24 (16) ◽

pp. 1759-1767 ◽

Cited By ~ 1

Author(s):

XIAO-YAN GU ◽

MENG ZHANG ◽

JIAN-QIANG SUN

Keyword(s):

Exact Solutions ◽

Energy Spectra ◽

Quantization Rule ◽

Rule Approach ◽

Good Agreement

The extension of the quantization rule approach to non-central potentials is investigated. The energy spectra for the generalized Coulomb and oscillator systems are presented. The results are in good agreement with those obtained before.

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EXACT SOLUTIONS OF THE DIRAC EQUATION WITH A COULOMB PLUS SCALAR POTENTIAL IN 2 + 1 DIMENSIONS

International Journal of Modern Physics E ◽

10.1142/s0218301302001046 ◽

2002 ◽

Vol 11 (06) ◽

pp. 483-489 ◽

Cited By ~ 18

Author(s):

SHI-HAI DONG ◽

XIAO-YAN GU ◽

ZHONG-QI MA ◽

SHISHAN DONG

Keyword(s):

Dirac Equation ◽

Differential Equations ◽

Exact Solutions ◽

Coulomb Potential ◽

Scalar Potential ◽

Second Order ◽

First Order ◽

Second Order Differential Equations

The exact solutions of the (2+1)-dimensional Dirac equation with a Coulomb potential and a scalar one are analytically presented by studying the second-order differential equations obtained from a pair of coupled first-order ones. The eigenvalues are studied in some detail.

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Heun Polynomials and Exact Solutions for the Massless Dirac Particle in the C-Metric

Zeitschrift für Naturforschung A ◽

10.1515/zna-2017-0355 ◽

2018 ◽

Vol 73 (4) ◽

pp. 275-283

Author(s):

Priyasri Kar ◽

Ritesh K. Singh ◽

Ananda Dasgupta ◽

Prasanta K. Panigrahi

Keyword(s):

Dirac Equation ◽

Exact Solutions ◽

Algebraic Structure ◽

Differential Operators ◽

Equation Of Motion ◽

Dirac Particle ◽

Heun Equation ◽

Polar Parts

AbstractThe equation of motion of a massless Dirac particle in the C-metric leads to the general Heun equation (GHE) for the radial and the polar variables. The GHE, under certain parametric conditions, is cast in terms of a new set of su(1, 1) generators involving differential operators of degrees ±1/2 and 0. Additional Heun polynomials are obtained using this new algebraic structure and are used to construct some exact solutions for the radial and the polar parts of the Dirac equation.

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Symmetries of Relativistic Hydrogen Atom

Ukrainian Journal of Physics ◽

10.15407/ujpe64.12.1148 ◽

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.

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DIRAC ELECTRON IN A COULOMB FIELD IN (2+1) DIMENSIONS

Modern Physics Letters A ◽

10.1142/s0217732398000668 ◽

1998 ◽

Vol 13 (08) ◽

pp. 615-622 ◽

Cited By ~ 63

Author(s):

V. R. KHALILOV ◽

CHOON-LIN HO

Keyword(s):

Simple Model ◽

Dirac Equation ◽

Exact Solutions ◽

Coulomb Field ◽

Critical Charge ◽

Spatial Dimensions ◽

Dirac Electron

Exact solutions of Dirac equation in two spatial dimensions in the Coulomb field are obtained. Equation which determines the so-called critical charge of the Coulomb field is derived and solved for a simple model.

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

Nonlinear Phenomena in Complex Systems ◽

10.33581/1561-4085-2021-24-3-260-271 ◽

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.

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Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽

10.2298/fil1809347k ◽

2018 ◽

Vol 32 (9) ◽

pp. 3347-3354 ◽

Cited By ~ 1

Author(s):

Nematollah Kadkhoda ◽

Michal Feckan ◽

Yasser Khalili

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Exact Solutions ◽

Present Article ◽

Analytical Solutions ◽

Nonlinear Differential Equations ◽

Nonlinear Partial Differential Equations ◽

Rational Functions ◽

Expansion Method ◽

Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

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On the Exact Solutions of Two (3+1)-Dimensional Nonlinear Differential Equations

Journal of Function Spaces ◽

10.1155/2020/5740310 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8

Author(s):

Fanning Meng ◽

Yongyi Gu

Keyword(s):

Differential Equations ◽

Exact Solutions ◽

Shallow Water ◽

Traveling Wave ◽

Nonlinear Differential Equations ◽

Complex Method ◽

Meromorphic Solutions ◽

Complex Differential Equations ◽

Bkp Equation

In this article, exact solutions of two (3+1)-dimensional nonlinear differential equations are derived by using the complex method. We change the (3+1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equation and generalized shallow water (gSW) equation into the complex differential equations by applying traveling wave transform and show that meromorphic solutions of these complex differential equations belong to class W, and then, we get exact solutions of these two (3+1)-dimensional equations.

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Exact solutions of Dirac equation

Pramana ◽

10.1007/bf02872119 ◽

1979 ◽

Vol 12 (5) ◽

pp. 475-480 ◽

Cited By ~ 3

Author(s):

S V Kulkarni ◽

L K Sharma

Keyword(s):

Dirac Equation ◽

Exact Solutions

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Exact Solutions of a Spatially Dependent Mass Dirac Equation for Coulomb Field plus Tensor Interaction via Laplace Transformation Method

Advances in High Energy Physics ◽

10.1155/2012/873619 ◽

2012 ◽

Vol 2012 ◽

pp. 1-15 ◽

Cited By ~ 11

Author(s):

M. Eshghi ◽

M. Hamzavi ◽

S. M. Ikhdair

Keyword(s):

Dirac Equation ◽

Laplace Transformation ◽

Bound States ◽

Energy Eigenvalue ◽

Coulomb Field ◽

Transformation Method ◽

Arbitrary Spin ◽

Tensor Interaction ◽

Dependent Mass ◽

Laplace Transformation Method

The spatially dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials including a tensor interaction potential under the spin and pseudospin (p-spin) symmetric limits by using the Laplace transformation method (LTM). Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spin-orbit quantum number κ. Some numerical results are given too. The effect of the tensor interaction on the bound states is presented. It is shown that the tensor interaction removes the degeneracy between two states in the spin doublets. We also investigate the effects of the spatially-dependent mass on the bound states under the conditions of the spin symmetric limit and in the absence of tensor interaction (T=0).

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