arXiv subjects

A. D. Alhaidari

Publications and source records attributed to A. D. Alhaidari.

At least 109 records · Page 6Linked to original sources

arXiv2005

Study of electromagnetic wave propagation in active medium and the equivalent Schrodinger equation with energy-dependent complex potential

We study the massless limit of the Klein-Gordon (K-G) equation in 1+1 dimensions with static complex potentials as an attempt to give an alternative, but equivalent, representation of plane electromagnetic (em) wave propagation in active medium. In the case of dispersionless em medium, the analogy dictates that the potential in the K-G equation is complex and energy-dependent. In the non-relativistic domain we study an analogous inertial system by considering wave packet propagation through a complex potential barrier and solve the time-independent Schrodinger equation with a potential that has the same energy dependence as that of the K-G equation. The behavior of these solutions is compared with those found elsewhere in the literature for the propagation of electromagnetic plane waves in a uniform active medium with complex dielectric constant. Our study concluded that the discrepancy between the time dependent and stationary results is due to the energy poles crossing the real axis in the complex energy plane. It was demonstrated unambiguously that there is a frequency (energy) and size dependent gain threshold above which the stationary results become unstable. This threshold corresponds to the value of the gain at which the pole crosses the real axis.

cond-mat.dis-nn↗

arXiv2005

Solution of the Dirac equation with non-minimal coupling to noncentral three-vector potential

We introduce non-minimal coupling to three-vector potential in the 3+1 dimensional Dirac equation. The potential is noncentral (angular-dependent) such that the Dirac equation separates completely in spherical coordinates. The relativistic energy spectrum and spinor wavefunctions are obtained for the case where the radial component of the vector potential is proportional to 1/r. The non-minimal coupling presented in this work is a generalization of that which was introduced by Moshinsky and Szczepaniak in the Dirac-Oscillator problem.

hep-th↗

arXiv2005

The three-dimensional Dirac-Oscillator in the presence of Aharonov-Bohm and magnetic monopole potentials

We study the Dirac equation in 3+1 dimensions with non-minimal coupling to isotropic radial three-vector potential and in the presence of static electromagnetic potential. The space component of the electromagnetic potential has angular (non-central) dependence such that the Dirac equation is completely separable in spherical coordinates. We obtain exact solutions for the case where the three-vector potential is linear in the radial coordinate (Dirac-Oscillator) and the time component of the electromagnetic potential vanishes. The relativistic energy spectrum and spinor eigenfunctions are obtained.

hep-th↗

arXiv2005

Solution of the Dirac equation by separation of variables in spherical coordinates for a class of three-parameter non-central electromagnetic potential

We consider the three-dimensional Dirac equation in spherical coordinates with coupling to static electromagnetic potential. The space components of the potential have angular (non-central) dependence such that the Dirac equation is separable in all coordinates. We obtain exact solutions for the case where the potential satisfies the Lorentz gauge fixing condition and its time component is the Coulomb potential. The relativistic energy spectrum and corresponding spinor wavefunctions are obtained. The Aharonov-Bohm and magnetic monopole potentials are included in these solutions.

hep-th↗

arXiv2004

Scattering and bound states for a class of non-central potentials

We obtain L2-series solutions of the nonrelativistic three-dimensional wave equation for a large class of non-central potentials that includes, as special cases, the Aharonov-Bohm, Hartmann, and magnetic monopole potentials. It also includes contributions from the potential term, cos(theta)/r^2 (in spherical coordinates). The solutions obtained are for all energies, the discrete (for bound states) as well as the continuous (for scattering states). The L2 bases of the solution space are chosen such that the matrix representation of the wave operator is tridiagonal. The expansion coefficients of the radial and angular components of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations resulting from the matrix wave equation.

quant-ph↗

arXiv2004

An extended class of L2-series solutions of the wave equation

We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support an infinite tridiagonal matrix representation of the wave operator. The class of solutions obtained as such includes the discrete (for bound states) as well as the continuous (for scattering states) spectrum of the Hamiltonian. The problem translates into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. These are written in terms of orthogonal polynomials, some of which are modified versions of known polynomials. The examples given, which are not exhaustive, include problems in one and three dimensions.

quant-ph↗

arXiv2004

Spinor superalgebra: Towards a theory for higher spin particles

We define a superalgebra S2(N/2) as a Z2 graded algebra of dimension 2N+3, where N is a positive, odd integer. The even component is a three-dimensional abelian subalgebra, while the odd component is made up of two N-dimensional, mutually conjugate algebras. For N = 1, two of the three even elements become identical, resulting in a four-dimensional superalgebra which is the graded extension of the SO(2,1) Lie algebra that has recently been introduced in the solution of the Dirac equation for spinn 1/2. Realization of the elements of S2(N/2) is given in terms of differential matrix operators acting on an N+1 dimensional space that could support a representation of the Lorentz space-time symmetry group for spin N/2. The N = 3 case results in a 4x4 matrix wave equation, which is linear and of first order in the space-time derivatives. We show that the "canonical" form of the Dirac Hamiltonian is an element of this superalgebra.

hep-th↗

arXiv2004

L2 series solutions of the Dirac equation for power-law potentials at rest mass energy

We obtain solutions of the three dimensional Dirac equation for radial power-law potentials at rest mass energy as an infinite series of square integrable functions. These are written in terms of the confluent hypergeometric function and chosen such that the matrix representation of the Dirac operator is tridiagonal. The "wave equation" results in a three-term recursion relation for the expansion coefficients of the spinor wavefunction which is solved in terms of orthogonal polynomials. These are modified versions of the Meixner-Pollaczek polynomials and of the continuous dual Hahn polynomials. The choice depends on the values of the angular momentum and the power of the potential.

math-ph↗

arXiv2004

Unified algebraic treatment of resonance

Energy resonance in scattering is usually investigated either directly in the complex energy plane (E-plane) or indirectly in the complex angular momentum plane (L-plane). Another formulation complementing these two approaches was introduced recently. It is an indirect algebraic method that studies resonances in a complex charge plane (Z-plane). This latter approach will be generalized to provide a unified algebraic treatment of resonances in the complex E-, L-, and Z-planes. The complex scaling (rotation) method will be used in the development of this approach. The resolvent operators (Green's functions) are formally defined in these three spaces. Bound states spectrum and resonance energies in the E-plane are mapped onto a discrete set of poles of the respective resolvent operator on the real line of the L- and Z-planes. These poles move along trajectories as the energy is varied. A finite square integrable basis is used in the numerical implementation of this approach. Stability of poles and trajectories against variation in all computational parameters is demonstrated. Resonance energies for a given potential are calculated and compared with those obtained by other studies.

physics.atom-ph↗

arXiv2004

Exact L2 series solution of the Dirac-Coulomb problem for all energies

We obtain exact solution of the Dirac equation with the Coulomb potential as an infinite series of square integrable functions. This solution is for all energies, the discrete as well as the continuous. The spinor basis elements are written in terms of the confluent hypergeometric functions and chosen such that the matrix representation of the Dirac-Coulomb operator is tridiagonal. The wave equation results in a three-term recursion relation for the expansion coefficients of the wavefunction which is solved in terms of the Meixner-Pollaczek polynomials.

hep-th↗

arXiv2004

L2 series solution of the relativistic Dirac-Morse problem for all energies

We obtain analytic solutions for the one-dimensional Dirac equation with the Morse potential as an infinite series of square integrable functions. These solutions are for all energies, the discrete as well as the continuous. The elements of the spinor basis are written in terms of the confluent hypergeometric functions. They are chosen such that the matrix representation of the Dirac-Morse operator for continuous spectrum (i.e., for scattering energies larger than the rest mass) is tridiagonal. Consequently, the wave equation results in a three-term recursion relation for the expansion coefficients of the wavefunction. The solution of this recursion relation is obtained in terms of the continuous dual Hahn orthogonal polynomials. On the other hand, for the discrete spectrum (i.e., for bound states with energies less than the rest mass) the spinor wave functions result in a diagonal matrix representation for the Dirac-Morse Hamiltonian.

math-ph↗

arXiv2004

The Dirac-Oscillator Green's function

We obtain the two-point Green's function for the relativistic Dirac-Oscillator problem. This is accomplished by setting up the relativistic problem in such a way that makes comparison with the nonrelativistic problem highly transparent and results in a map of the latter into the former. The relativistic bound states energy spectrum is obtained by locating the energy poles of this Green's function in a simple and straightforward manner.

hep-th↗

arXiv2004

Solution of the relativistic Dirac-Hulthen problem

The one-particle three-dimensional Dirac equation with spherical symmetry is solved for the Hulthen potential. The s-wave relativistic energy spectrum and two-component spinor wavefunctions are obtained analytically. Conforming to the standard feature of the relativistic problem, the solution space splits into two distinct subspaces depending on the sign of a fundamental parameter in the problem. Unique and interesting properties of the energy spectrum are pointed out and illustrated graphically for several values of the physical parameters. The square integrable two-component wavefunctions are written in terms of the Jacobi polynomials. The nonrelativistic limit reproduces the well-known nonrelativistic energy spectrum and results in Schrodinger equation with a "generalized" three-parameter Hulthen potential, which is the sum of the original Hulthen potential and its square.

hep-th↗

arXiv2004

Studying resonance in the complex charge plane

Potential resonances are usually investigated either directly in the complex energy plane or indirectly in the complex angular momentum plane. Another formulation complementing these two is presented in this work. It is an indirect method which studies resonances in a complex charge plane (Z-plane). The complex scaling (rotation) method is employed in the development of this formulation. Bound states spectrum and resonance energies are mapped onto a discrete set of poles of a resolvent operator on the real line of the Z-plane. These poles move along trajectories as we vary the energy. A finite L2 basis is used in the numerical implementation of the method. Stability of poles and trajectories against variations in all computational parameters is demonstrated. Resonances for a given potential example are calculated and compared with those obtained elsewhere. In this presentation, modest effort will be devoted to mathematical rigor.

math-ph↗

arXiv2003

Solution of the Dirac equation with position-dependent mass in the Coulomb field

We obtain exact solution of the Dirac equation for a charged particle with position-dependent mass in the Coulomb field. The effective mass of the spinor has a relativistic component which is proportional to the square of the Compton wavelength and varies as 1/r. It is suggested that this model could be used as a tool in the renormalization of ultraviolet divergences in field theory. The discrete energy spectrum and spinor wave-function are obtained explicitly.

math-ph↗

arXiv2003

Enlarging the class of exactly solvable nonrelativistic problems

We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support a tridiagonal matrix representation of the wave operator. Doing so results in exactly solvable problems with a class of potentials which is larger than, and/or generalization of, what is already known. In addition, we found new representations for the solution space of some well known potentials. The problem translates into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. Some of these solutions are new kind of orthogonal polynomials. The examples given, which are not exhaustive, are for problems in one and three dimensions. The analytic solutions obtained by this approach include the discrete as well as the continuous spectrum of the Hamiltonian.

math-ph↗

arXiv2003

Retraction of "On a theory of half-integral spin particles with gauge invariant coupling to the electromagnetic potential: Towards higher spin QED"

The representation of the superalgebra so(2,1) that resulted in the relativistic wave equation (2.1) and Eq. (2.4) is fully reducible. In fact, its even part that lead to Eq. (3.1) is the direct sum of two spin 1/2 representations of the Lorentz group and does not represent spin 3/2 particle as we claimed. Consequently, we retract our claim and withdraw the manuscript.

hep-th↗

arXiv2003

Retraction of "Half-integer spin representations of the graded extension of so(2,1) Lie algebra: Spin 3/2 particle in the Dirac-Oscillator potential"

The representation of the superalgebra SO(2,1) which is given by Eq. (3.1) and which resulted in the relativistic wave equation (4.1) is fully reducible. In fact, its even part is the direct sum of two spin 1/2 representations of the Lorentz group and does not represent spin 3/2 particle as we claimed. Consequently, we retract our claim and withdraw the manuscript.

hep-th↗