scholarly journals Approximate Solutions to the Klein-Fock-Gordon Equation for the Sum of Coulomb and Ring-Shaped-Like Potentials

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Sh. M. Nagiyev ◽  
A. I. Ahmadov ◽  
V. A. Tarverdiyeva
Keyword(s):  
Gordon Equation ◽  
Dynamical Symmetry ◽  
Approximate Solutions ◽  
Wave Functions ◽  
Energy Spectra ◽  
Mechanical Problem ◽  
Radial Wave ◽  
Equal Scalar ◽  
Analytical Expressions ◽  
Radial Wave Equation

We consider the quantum mechanical problem of the motion of a spinless charged relativistic particle with mass M, described by the Klein-Fock-Gordon equation with equal scalar Sr→ and vector Vr→ Coulomb plus ring-shaped potentials. It is shown that the system under consideration has both a discrete at E<Mc2 and a continuous at E>Mc2 energy spectra. We find the analytical expressions for the corresponding complete wave functions. A dynamical symmetry group SU1,1 for the radial wave equation of motion is constructed. The algebra of generators of this group makes it possible to find energy spectra in a purely algebraic way. It is also shown that relativistic expressions for wave functions, energy spectra, and group generators in the limit c⟶∞ go over into the corresponding expressions for the nonrelativistic problem.

Exact solution of the relativistic finite-difference equation for the Coulomb plus a ring-shaped-like potential

2019 ◽  
Vol 34 (17) ◽  
pp. 1950089 ◽  
Author(s):  
Sh. M. Nagiyev ◽  
A. I. Ahmadov
Keyword(s):  
Energy Spectrum ◽  
Finite Difference ◽  
Jacobi Polynomials ◽  
Relativistic Quantum ◽  
Three Dimensional ◽  
Dynamical Symmetry ◽  
Wave Functions ◽  
Relativistic Quantum Mechanics ◽  
Radial Part ◽  
Radial Wave

In this paper, a three-dimensional problem of the motion of a charged relativistic particle in a noncentral Coulomb plus ring-shaped potential is studied. Our investigation is based on a finite-difference version of relativistic quantum mechanics. The energy eigenvalues and the corresponding wave functions are obtained analytically. It is shown that radial part and the angular part of the wave functions are expressed through the Meixner–Pollaczek polynomials and Jacobi polynomials, respectively. All relativistic expressions, for example, radial wave functions and energy spectrum, have the correct nonrelativistic limit. We also build a dynamical symmetry group for the radial part of the equation of motion, which allows us to find the energy spectrum purely algebraically.

scholarly journals Results of calculations of atomic wave functions IV—Results for F - , Al +3 , and Rb +

1935 ◽  
Vol 151 (872) ◽  
pp. 96-105 ◽  
Author(s):  
Douglas Rayner Hartree
Keyword(s):  
Energy Parameter ◽  
Wave Functions ◽  
Numerical Accuracy ◽  
Radial Wave ◽  
Atomic Wave ◽  
Consistent Field ◽  
Self Consistent Field ◽  
The One ◽  
High Degree ◽  
Radial Wave Equation

In three previous papers, results of calculations of atomic wave functions, carried out by the method of the self-consistent field to a fairly high degree of numerical accuracy (for work of this kind), have been given for a number of atoms. The present paper gives further results of this kind for F - , Al +3 , and Rb + . Similar calculations are in progress for Ag + , and the results of the preliminary stages of the calculation have been given by Miss Black. The results here given are presented in the same form as in previous papers, namely:― (1) Unnormalized radial wave functions P, and the values of the normalization integral ∫ 0 ∞ P 2 dr , and of the energy parameter ε in the one-electron radial wave equation, for each function P; also values of P/ r l + 1 for small r .

A REALIZATION OF DYNAMIC GROUP FOR AN ELECTRON IN A UNIFORM MAGNETIC FIELD

2002 ◽  
Vol 11 (04) ◽  
pp. 265-271 ◽  
Author(s):  
SHISHAN DONG ◽  
SHI-HAI DONG
Keyword(s):  
Magnetic Field ◽  
Relativistic Electron ◽  
Uniform Magnetic Field ◽  
Wave Functions ◽  
Matrix Elements ◽  
Eigenvalues And Eigenfunctions ◽  
Radial Wave ◽  
Creation And Annihilation Operators ◽  
The Matrix ◽  
Analytical Expressions

The eigenvalues and eigenfunctions of the Schrödinger equation with a non-relativistic electron in a uniform magnetic field are presented. A realization of the creation and annihilation operators for the radial wave-functions is carried out. It is shown that these operators satisfy the commutation relations of an SU(1,1) group. Closed analytical expressions are evaluated for the matrix elements of different functions ρ2 and [Formula: see text].

scholarly journals Relativistic solution in D-dimensions to a spin-zero particle for equal scalar and vector ring-shaped Kratzer potential

Open Physics ◽  
2008 ◽  
Vol 6 (1) ◽  
Author(s):  
Sameer Ikhdair ◽  
Ramazan Sever
Keyword(s):  
Bound States ◽  
Gordon Equation ◽  
Wave Functions ◽  
Three Dimensions ◽  
Vector Potentials ◽  
Klein Gordon ◽  
Energy Bound ◽  
Uvarov Method ◽  
Equal Scalar ◽  
Exact Energy

AbstractThe Klein-Gordon equation in D-dimensions for a recently proposed ring-shaped Kratzer potential is solved analytically by means of the conventional Nikiforov-Uvarov method. The exact energy bound states and the corresponding wave functions of the Klein-Gordon are obtained in the presence of the non-central equal scalar and vector potentials. The results obtained in this work are more general and can be reduced to the standard forms in three dimensions given by other works.

Analytical solution of the Duffin - Kemmer - Petiau equation for the sum of the Manning - Rosen and the Yukawa class potential

2021 ◽  
pp. 140-150
Author(s):  
S.M. Aslanova ◽  
Keyword(s):  
Hypergeometric Functions ◽  
Energy Levels ◽  
Energy Eigenvalue ◽  
Bound State ◽  
Wave Functions ◽  
State Solution ◽  
Bound State Solution ◽  
Radial Wave ◽  
Quantum Numbers ◽  
Analytical Expressions

This paper presents an analytical bound-state solution to the Duffin - Kemmer - Petiau equation for the new putative combined Manning - Rosen and Yukawa class potentials. Using the developed scheme to approximate and overcome the difficulties arising in the centrifugal part of the potential, the bound-state solution of the modified Duffin - Kemmer - Petiau equation is found. Analytical expressions of energy eigenvalue and the corresponding radial wave functions are obtained for an arbitrary value of the orbital quantum number l . Also, eigenfunctions are expressed in terms of hypergeometric functions. It is shown that energy levels and eigenfunctions are quite sensitive to the choice of radial and orbital quantum numbers.

Diatomic molecule energies of the modified Rosen−Morse potential energy model

2014 ◽  
Vol 92 (4) ◽  
pp. 341-345 ◽  
Author(s):  
Hong-Ming Tang ◽  
Guang-Chuan Liang ◽  
Lie-Hui Zhang ◽  
Feng Zhao ◽  
Chun-Sheng Jia
Keyword(s):  
Morse Potential ◽  
Potential Model ◽  
Vibrational Energy ◽  
Energy Levels ◽  
Wave Functions ◽  
Energy Spectra ◽  
Radial Wave ◽  
Empirical Potential ◽  
Vibrational Energy Levels ◽  
Π State

We solve the Schrödinger equation with the modified Rosen−Morse empirical potential model to obtain rotation-vibrational energy spectra and unnormalized radial wave functions. The vibrational energy levels calculated with the modified Rosen−Morse potential model for the 61Πu state of the 7Li2 molecule and the X3Π state of the SiC radical are in better agreement with the Rydberg−Klein−Rees data than the predictions of the Morse potential model.

scholarly journals Bound state solution of the Klein–Fock–Gordon equation with the Hulthén plus a ring-shaped-like potential within SUSY quantum mechanics

2018 ◽  
Vol 33 (33) ◽  
pp. 1850203 ◽  
Author(s):  
A. I. Ahmadov ◽  
Sh. M. Nagiyev ◽  
M. V. Qocayeva ◽  
K. Uzun ◽  
V. A. Tarverdiyeva
Keyword(s):  
Quantum Mechanics ◽  
Gordon Equation ◽  
Jacobi Polynomials ◽  
Scalar Potential ◽  
Bound State ◽  
Wave Functions ◽  
State Solution ◽  
Bound State Solution ◽  
Radial Wave ◽  
Energy Eigenvalues

In this paper, the bound state solution of the modified Klein–Fock–Gordon equation is obtained for the Hulthén plus ring-shaped-like potential by using the developed scheme to overcome the centrifugal part. The energy eigenvalues and corresponding radial and azimuthal wave functions are defined for any [Formula: see text] angular momentum case on the conditions that scalar potential is whether equal and nonequal to vector potential, the bound state solutions of the Klein–Fock–Gordon equation of the Hulthén plus ring-shaped-like potential are obtained by Nikiforov–Uvarov (NU) and supersymmetric quantum mechanics (SUSY QM) methods. The equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is revealed owing to both methods. The energy levels and the corresponding normalized eigenfunctions are represented in terms of the Jacobi polynomials for arbitrary [Formula: see text] states. A closed form of the normalization constant of the wave functions is also found. It is shown that the energy eigenvalues and eigenfunctions are sensitive to [Formula: see text] radial and [Formula: see text] orbital quantum numbers.

scholarly journals Relativistic Symmetries of Hulthén Potential Incorporated with Generalized Tensor Interactions

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
A. N. Ikot ◽  
H. Hassanabadi ◽  
E. Maghsoodi ◽  
S. Zarrinkamar
Keyword(s):  
State Energy ◽  
Bound State ◽  
Tensor Interaction ◽  
Wave Functions ◽  
Energy Spectra ◽  
Bound State Energy ◽  
Hulthén Potential ◽  
Radial Wave ◽  
Hulthen Potential ◽  
Uvarov Method

Spin and pseudospin symmetries of the Dirac equation for a Hulthén potential with a novel tensor interaction, that is, a combination of the Coulomb and Yukawa potentials, are investigated using the Nikiforov-Uvarov method. The bound-state energy spectra and the radial wave functions are approximately obtained in the case of spin and pseudospin symmetries. The tensor interactions and the degeneracy-removing role are presented in details.

scholarly journals Analytical Bound State Solutions of the Klein-Fock-Gordon Equation for the Sum of Hulthén and Yukawa Potential within SUSY Quantum Mechanics

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
A. I. Ahmadov ◽  
S. M. Aslanova ◽  
M. Sh. Orujova ◽  
S. V. Badalov
Keyword(s):  
Quantum Mechanics ◽  
Gordon Equation ◽  
Jacobi Polynomials ◽  
Yukawa Potential ◽  
Scalar Potential ◽  
Bound State ◽  
Wave Functions ◽  
Relativistic Energy ◽  
Radial Wave ◽  
Energy Eigenvalues

The relativistic wave equations determine the dynamics of quantum fields in the context of quantum field theory. One of the conventional tools for dealing with the relativistic bound state problem is the Klein-Fock-Gordon equation. In this work, using a developed scheme, we present how to surmount the centrifugal part and solve the modified Klein-Fock-Gordon equation for the linear combination of Hulthén and Yukawa potentials. In particular, we show that the relativistic energy eigenvalues and corresponding radial wave functions are obtained from supersymmetric quantum mechanics by applying the shape invariance concept. Here, both scalar potential conditions, which are whether equal and nonequal to vector potential, are considered in the calculation. The energy levels and corresponding normalized eigenfunctions are represented as a recursion relation regarding the Jacobi polynomials for arbitrary l states. Beyond that, a closed form of the normalization constant of the wave functions is found. Furthermore, we state that the energy eigenvalues are quite sensitive with potential parameters for the quantum states. The nonrelativistic and relativistic results obtained within SUSY QM overlap entirely with the results obtained by ordinary quantum mechanics, and it displays that the mathematical implementation of SUSY quantum mechanics is quite perfect.

scholarly journals Mass Spectrum and Decay Constants of Heavy Quarkonia

2021 ◽  
Vol 58 (2) ◽  
pp. 73-83
Author(s):  
Tasawer Shahzad Ahmad ◽  
Talab Hussain ◽  
M. Atif Sultan
Keyword(s):  
Wave Equation ◽  
Mass Spectrum ◽  
Hyperfine Splitting ◽  
Potential Model ◽  
Wave Functions ◽  
Heavy Quarkonia ◽  
Radial Wave ◽  
Decay Constants ◽  
Relativistic Potential ◽  
Radial Wave Equation

In this paper, a non-relativistic potential model is used to find the solution of radial Schrodinger wave equation by using Crank Nicolson discretization for heavy quarkonia ( ̅, ̅). After solving the Schrodinger radial wave equation, the mass spectrum and hyperfine splitting of heavy quarkonia are calculated with and without relativistic corrections. The root means square radii and decay constants for S and P states of c ̅ and ̅ mesons by using the realistic and simple harmonic oscillator wave functions. The calculated results of mass, hyperfine splitting, root means square radii and decay constants agreed with experimental and theoretically calculated results in the literature.

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