scholarly journals Investigations on the Relativistic Interactions in One-Electron Atoms with Modified Yukawa Potential for Spin 1/2 Particles

2017 ◽  
Vol 11 ◽  
pp. 29-44 ◽  
Author(s):  
Abdelmadjid Maireche
Keyword(s):  
Dirac Equation ◽  
Yukawa Potential ◽  
Relativistic Quantum ◽  
Star Product ◽  
Pseudospin Symmetry ◽  
Spin Symmetry ◽  
Real Space ◽  
Relativistic Quantum Mechanics ◽  
Noncommutative Space ◽  
Shift Method

Energy levels of one electron atoms have been re-examined by applying an alternative perturbative scheme in solving the modified Dirac equation (m.d.e.) for the modified Yukawa potential model with a arbitrary spin-orbit quantum number (see equation in the paper) by means Bopp’s shift method instead to solving (m.d.e.) with star product, in the framework of noncommutativity three dimensional real space (NC: 3D-RS). It is observed that the obtained corrections of energies are depended on the new discrete atomic quantum numbers (see equation in the paper) under spin-symmetry and pseudospin symmetry and two infinitesimal parameters (see equation in the paper) which induced by position-position noncommutativity. Furthermore, in limit of parameters (see equation in the paper), the new energy equations for modified Yukawa potential are consistent with the results of ordinary relativistic quantum mechanics for ordinary Yukawa potential. Keywords: Yukawa potential, noncommutative space, star product, Bopp’s shift method and Dirac equation.

scholarly journals A New Nonrelativistic Investigation for the Lowest Excitations States of Interactions in One-Electron Atoms, Muonic, Hadronic and Rydberg Atoms with Modified Inverse Power Potential

2016 ◽  
Vol 9 ◽  
pp. 33-46 ◽  
Author(s):  
Abdelmadjid Maireche
Keyword(s):  
Rydberg Atoms ◽  
Relativistic Quantum ◽  
Star Product ◽  
Real Space ◽  
Analytical Investigation ◽  
Inverse Power ◽  
Shift Method ◽  
Exact Solvability ◽  
Quantum Numbers ◽  
Standard Perturbation Theory

A new theoretical analytical investigation for the exact solvability of non-relativistic quantum spectrum systems at low energy for modified inverse power potential (m.i.p.) is discussed by means Boopp’s shift method instead to solving deformed Schrödinger equation with star product, in the framework of both noncommutativite two dimensional real space and phase (NC: 2D-RSP), the exact corrections for lowest excitations are found straightforwardly for interactions in one-electron atoms, muonic, hadronic and Rydberg atoms by means of the standard perturbation theory. Furthermore, the obtained corrections of energies are depended on the four infinitesimals parameters (θ,χ) and (θ,σ), which are induced by position-position and momentum-momentum noncommutativity, in addition to the discreet atomic quantum numbers (j=l±1/1,s=±1/2 andm) and we have also shown that, the old states are canceled and has been replaced by new degenerated 4(2l+1) sub-states.

scholarly journals Approximate κ-state solutions to the Dirac-Yukawa problem based on the spin and pseudospin symmetry

Open Physics ◽  
2012 ◽  
Vol 10 (2) ◽  
Author(s):  
Sameer Ikhdair
Keyword(s):  
Energy Spectrum ◽  
Dirac Equation ◽  
Yukawa Potential ◽  
Energy Levels ◽  
Pseudospin Symmetry ◽  
Spin Symmetry ◽  
Relativistic Limit ◽  
Special Cases ◽  
Wide Range ◽  
Screening Parameter

AbstractUsing an approximation scheme to deal with the centrifugal (pseudo-centrifugal) term, we solve the Dirac equation with the screened Coulomb (Yukawa) potential for any arbitrary spin-orbit quantum number κ. Based on the spin and pseudospin symmetry, analytic bound state energy spectrum formulas and their corresponding upper- and lower-spinor components of two Dirac particles are obtained using a shortcut of the Nikiforov-Uvarov method. We find a wide range of permissible values for the spin symmetry constant C s from the valence energy spectrum of particle and also for pseudospin symmetry constant C ps from the hole energy spectrum of antiparticle. Further, we show that the present potential interaction becomes less (more) attractive for a long (short) range screening parameter α. To remove the degeneracies in energy levels we consider the spin and pseudospin solution of Dirac equation for Yukawa potential plus a centrifugal-like term. A few special cases such as the exact spin (pseudospin) symmetry Dirac-Yukawa, the Yukawa plus centrifugal-like potentials, the limit when α becomes zero (Coulomb potential field) and the non-relativistic limit of our solution are studied. The nonrelativistic solutions are compared with those obtained by other methods.

scholarly journals Spin and pseudospin solutions to Dirac equation and its thermodynamic properties using hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Ituen B. Okon ◽  
E. Omugbe ◽  
Akaninyene D. Antia ◽  
C. A. Onate ◽  
Louis E. Akpabio ◽  
...  
Keyword(s):  
Dirac Equation ◽  
Thermodynamic Properties ◽  
Energy Equation ◽  
Pseudospin Symmetry ◽  
Spin Symmetry ◽  
Bound State ◽  
Compact Form ◽  
Relativistic Energy ◽  
Quadratic Potential ◽  
Special Cases

AbstractIn this research article, the modified approximation to the centrifugal barrier term is applied to solve an approximate bound state solutions of Dirac equation for spin and pseudospin symmetries with hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential using parametric Nikiforov–Uvarov method. The energy eigen equation and the unnormalised wave function were presented in closed and compact form. The nonrelativistic energy equation was obtain by applying nonrelativistic limit to the relativistic spin energy eigen equation. Numerical bound state energies were obtained for both the spin symmetry, pseudospin symmetry and the non relativistic energy. The screen parameter in the potential affects the solutions of the spin symmetry and non-relativistic energy in the same manner but in a revised form for the pseudospin symmetry energy equation. In order to ascertain the accuracy of the work, the numerical results obtained was compared to research work of existing literature and the results were found to be in excellent agreement to the existing literature. The partition function and other thermodynamic properties were obtained using the compact form of the nonrelativistic energy equation. The proposed potential model reduces to Hulthen and exponential inversely quadratic potential as special cases. All numerical computations were carried out using Maple 10.0 version and Matlab 9.0 version softwares respectively.

scholarly journals Quantum backflow in solutions to the Dirac equation of the spin-1 2 free particle

2018 ◽  
Vol 33 (32) ◽  
pp. 1850186 ◽  
Author(s):  
Hong-Yi Su ◽  
Jing-Ling Chen
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Relativistic Particle ◽  
Relativistic Quantum ◽  
Free Particle ◽  
Negative Energy ◽  
Dirac Particle ◽  
Relativistic Quantum Mechanics ◽  
Probability Current ◽  
Negative Probability

It was known that a free, non-relativistic particle in a superposition of positive momenta can, in certain cases, bear a negative probability current — hence termed quantum backflow. Here, it is shown that more variations can be brought about for a free Dirac particle, particularly when negative-energy solutions are taken into account. Since any Dirac particle can be understood as an antiparticle that acts oppositely (and vice versa), quantum backflow is found to arise in the superposition (i) of a well-defined momentum but different signs of energies, or more remarkably (ii) of different signs of both momenta and energies. Neither of these cases has a counterpart in non-relativistic quantum mechanics. A generalization by using the field-theoretic formalism is also presented and discussed.

scholarly journals An extended Dirac equation in noncommutative spacetime

2016 ◽  
Vol 31 (15) ◽  
pp. 1650089 ◽  
Author(s):  
R. Vilela Mendes
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Relativistic Quantum ◽  
Large Mass ◽  
Exterior Algebra ◽  
Relativistic Quantum Mechanics ◽  
Spacetime Geometry ◽  
Noncommutative Spacetime ◽  
Extended Dirac Equation

Stabilizing, by deformation, the algebra of relativistic quantum mechanics a noncommutative spacetime geometry is obtained. The exterior algebra of this geometry leads to an extended massless Dirac equation which has both a massless and a large mass solution. The nature of the solutions is discussed as well as the effects of coupling the two solutions.

The Dirac Equation

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Relativistic Quantum ◽  
Type Equation ◽  
Nonrelativistic Limit ◽  
Relativistic Quantum Mechanics ◽  
Electronic Systems ◽  
Nonrelativistic Case ◽  
Relativistic Description ◽  
Basic Features

The purpose of this chapter is to introduce the Dirac equation, which will provide us with a basis for developing the relativistic quantum mechanics of electronic systems. Thus far we have reviewed some basic features of the classical relativistic theory, which is the foundation of relativistic quantum theory. As in the nonrelativistic case, quantum mechanical equations may be obtained from the classical relativistic particle equations by use of the correspondence principle, where we replace classical variables by operators. Of particular interest are the substitutions In terms of the momentum four-vector introduced earlier, this yields In going from a classical relativistic description to relativistic quantum mechanics, we require that the equations obtained are invariant under Lorentz transformations. Other basic requirements, such as gauge invariance, must also apply to the equations of relativistic quantum mechanics. We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrödinger-type equation containing spin.

A time operator in the simulations of the Dirac equation

2019 ◽  
Vol 34 (22) ◽  
pp. 1950114 ◽  
Author(s):  
M. Bauer
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Relativistic Quantum ◽  
Einstein Condensate ◽  
Time Operator ◽  
Relativistic Quantum Mechanics ◽  
Electron Motion ◽  
Klein Paradox ◽  
Bose Einstein Condensate ◽  
Bose Einstein

Simulations of the Dirac equation have allowed to mimic measurably the predicted unusual characteristics of the electron motion, e.g. Zitterbewegung and Klein paradox that are beyond current technical capabilities. In this paper, it is shown that a Bose–Einstein condensate experiment carried out corroborates these results, but in addition exhibits a particular feature of an observable represented by a Dirac self-adjoint time operator introduced in relativistic quantum mechanics.

A new theoretical study of the deformed unequal scalar and vector Hellmann plus modified Kratzer potentials within the deformed Klein–Gordon equation in RNCQM symmetries

2021 ◽  
Author(s):  
Abdelmadjid Maireche
Keyword(s):  
Quantum Mechanics ◽  
Yukawa Potential ◽  
Relativistic Quantum ◽  
Scalar Potential ◽  
Relativistic Quantum Mechanics ◽  
Energy Eigenvalues ◽  
Quantum Numbers ◽  
Noncommutative Quantum Mechanics ◽  
Hellmann Potential ◽  
Klein Gordon

In this paper, within the framework of relativistic quantum mechanics and using the improved approximation scheme to the centrifugal term for any [Formula: see text]states via Bopp’s shift method and standard perturbation theory, we have obtained the modified energy eigenvalues of a newly proposed modified unequal vector and scalar Hellmann plus modified Kratzer potentials (DUVSHMK-Ps) for some diatomic N2, I2, CO, NO, O2 and HCl molecules. This study includes corrections of the first-order in noncommutativity parameters [Formula: see text]. This potential is a superposition of the attractive Coulomb Yukawa potential plus the Kratzer potential and new central terms appear as a result of the effects of noncommutativity properties of space–space. The obtained energy eigenvalues appear as a function of noncommutativity parameters, the strength parameters [Formula: see text] and [Formula: see text] of the (scalar vector) Hellmann potential, the screening range parameter [Formula: see text], the dissociation energy of the vector, and scalar potential [Formula: see text], the equilibrium inter-nuclear distance [Formula: see text] in addition to the atomic quantum numbers [Formula: see text]. Furthermore, we obtained the corresponding modified energy of DUVSHMK-Ps in the symmetries of non-relativistic noncommutative quantum mechanics (NRNCQM). In both relativistic and non-relativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM.

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