arXiv subjects

A. D. Alhaidari

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

At least 91 records · Page 5Linked to original sources

arXiv2009

Dirac particle in a square well and in a box

We obtain an exact solution of the 1D Dirac equation for a square well potential of depth greater then twice the particle's mass. The energy spectrum formula in the Klein zone is surprisingly very simple and independent of the depth of the well. This implies that the same solution is also valid for the potential box (infinitely deep well). In the nonrelativistic limit, the well-known energy spectrum of a particle in a box is obtained. We also provide in tabular form the elements of the complete solution space of the problem for all energies.

quant-ph↗

arXiv2008

The rotating Morse potential model for diatomic molecules in the J-matrix representation: II. The S-matrix approach

This is the second article in which we study the rotating Morse potential model for diatomic molecules using the tridiagonal J-matrix approach. Here, we improve further the accuracy of computing the bound states and resonance energies for this potential model from the poles of the S-matrix for arbitrary angular momentum. The calculation is performed using an infinite square integrable basis that supports a tridiagonal matrix representation for the reference Hamiltonian, which is included in the computations analytically without truncation. Our method has been applied to both the regular and inverted Morse potential with favorable results in comparison with available numerical data. We have also shown that the present method adds few significant digits to the accuracy obtained from the finite dimensional approach (e.g. the complex rotation method). Moreover, it allows us to handle easily both analytic and non-analytic potentials as well as 1/r singular potentials.

physics.chem-ph↗

arXiv2008

Sine function with a cosine attitude

We give a revealing expose that addresses an important issue in scattering theory of how to construct two asymptotically sinusoidal solutions of the wave equation with a phase shift using the same basis having the same boundary conditions at the origin. Analytic series representations of these solutions are obtained. In 1D, one of the solutions is an even function that behaves asymptotically as sin(x), whereas the other is an odd function, which is asymptotically cos(x). The latter vanishes at the origin whereas the derivative of the former becomes zero there. Eliminating the lowest N terms of the series makes these functions vanishingly small in an interval around the origin whose size increases with N. We employ the tools of the J-matrix method of scattering in the construction of these solutions in one and three dimensions.

quant-ph↗

arXiv2008

Infinite potential well with a sinusoidal bottom

We construct a tridiagonal matrix representation of the wave operator that maps the wave equation into a three-term recursion relation for the expansion coefficients of the wavefunction. Finding a solution of the recursion relation is equivalent to solving the original problem. Consequently, a larger class of solvable potentials is obtained. The usual diagonal representation constraint results in a reduction to the conventional class of solvable potentials. To exhibit the power of this approach, we give an exact solution for the infinite potential well with sinusoidal bottom.

quant-ph↗

arXiv2007

Solution of the wave equation in a tridiagonal representation space

We use variable transformation from the real line to finite or semi-infinite spaces where we expand the regular solution of the 1D time-independent Schrodinger equation in terms of square integrable bases. We also require that the basis support an infinite tridiagonal matrix representation of the wave operator. By this requirement, we deduce a class of solvable potentials along with their corresponding bound states and stationary wavefunctions expressed as infinite series in terms of these bases. This approach allows for simultaneous treatment of the discrete (bound states) as well as the continuous (scattering states) spectrum on the same footing. 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, illustrate the power of this approach in dealing with 1D quantum problems.

quant-ph↗

arXiv2007

Charged particle in the field an electric quadrupole in two dimensions

We obtain analytic solution of the time-independent Schrodinger equation in two dimensions for a charged particle moving in the field of an electric quadrupole. The solution is written as a series in terms of special functions that support a tridiagonal matrix representation for the angular and radial components of the wave operator. This solution is for all energies, the discrete (for bound states) as well as the continuous (for scattering states). The expansion coefficients of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations. The charged particle could become bound to the quadrupole only if its moment exceeds a certain critical value.

physics.atom-ph↗

arXiv2007

Taming the Yukawa potential singularity: improved evaluation of bound states and resonance energies

Using the tools of the J-matrix method, we absorb the 1/r singularity of the Yukawa potential in the reference Hamiltonian, which is handled analytically. The remaining part, which is bound and regular everywhere, is treated by an efficient numerical scheme in a suitable basis using Gauss quadrature approximation. Analysis of resonance energies and bound states spectrum is performed using the complex scaling method, where we show their trajectories in the complex energy plane and demonstrate the remarkable fact that bound states cross over into resonance states by varying the potential parameters.

physics.atom-ph↗

arXiv2007

Analytic solution of the Schrodinger equation for an electron in the field of a molecule with an electric dipole moment

We relax the usual diagonal constraint on the matrix representation of the eigenvalue wave equation by allowing it to be tridiagonal. This results in a larger solution space that incorporates an exact analytic solution for the non-central electric dipole potential cos(theta)/r^2, which was known not to belong to the class of exactly solvable potentials. As a result, we were able to obtain an exact analytic solution of the three-dimensional time-independent Schrodinger equation for a charged particle in the field of a point electric dipole that could carry a nonzero net charge. This problem models the interaction of an electron with a molecule (neutral or ionized) that has a permanent electric dipole moment. The solution is written as a series of square integrable functions that support a tridiagonal matrix representation for the angular and radial components of the wave operator. Moreover, this solution is for all energies, the discrete (for bound states) as well as the continuous (for scattering states). The expansion coefficients of the radial and angular components of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations. For the Coulomb-free case, where the molecule is neutral, we calculate critical values for its dipole moment below which no electron capture is allowed. These critical values are obtained not only for the ground state, where it agrees with already known results, but also for excited states as well.

physics.chem-ph↗

arXiv2007

The rotating Morse potential model for diatomic molecules in the tridiagonal J-matrix representation: I. Bound states

This is the first in a series of articles in which we study the rotating Morse potential model for diatomic molecules in the tridiagonal J-matrix representation. Here, we compute the bound states energy spectrum by diagonalizing the finite dimensional Hamiltonian matrix of H2, LiH, HCl and CO molecules for arbitrary angular momentum. The calculation was performed using the J-matrix basis that supports a tridiagonal matrix representation for the reference Hamiltonian. Our results for these diatomic molecules have been compared with available numerical data satisfactorily. The proposed method is handy, very efficient, and it enhances accuracy by combining analytic power with a convergent and stable numerical technique.

physics.chem-ph↗

arXiv2007

Relativistic scattering with spatially-dependent effective mass in the Dirac equation

We formulate an algebraic relativistic method of scattering for systems with spatially dependent mass based on the J-matrix method. The reference Hamiltonian is the three-dimensional Dirac Hamiltonian but with a mass that is position-dependent and having a constant asymptotic limit. Additionally, this effective mass distribution is locally represented in a finite dimensional function subspace. The spinor couples to spherically symmetric vector and pseudo scalar potentials that are short-range such that they are accurately represented by their matrix elements in the same finite dimensional subspace. We calculate the relativistic phase shift as a function of energy for a given configuration and study the effect of spatial variation of the mass on the energy resonance structure.

physics.atom-ph↗

arXiv2007

Relativistic extension of the complex scaling method

We construct a tridiagonal matrix representation for the three dimensions Dirac-Coulomb Hamiltonian that provides for a simple and straightforward relativistic extension of the complex scaling method. Besides the Coulomb interaction, additional vector, scalar, and pseudo-scalar coupling to short-range potentials could also be included in the same representation. Using that, we are able to obtain highly accurate values for the relativistic bound states and resonance energies. A simple program code is developed to perform the calculation for a given charge, angular momentum and potential configuration. The resonance structure in the complex relativistic energy plane is also shown graphically. Illustrative examples are given and we verify that in the nonrelativistic limit one obtains known results. As an additional advantage of this tridiagonal representation, we use it to obtain a highly accurate evaluation of the relativistic bound states energies for the Woods-Saxon potential (as a model of nuclear interaction) with the nucleus treated as solid sphere of uniform charge distribution.

quant-ph↗

arXiv2006

Density of States Extracted from Modified Recursion Relations

We evaluate the density of states (DOS) associated with tridiagonal symmetric Hamiltonian matrices and study the effect of perturbation on one of its entries. Analysis is carried out by studying the resulting three-term recursion relation and the corresponding orthogonal polynomials of the first and second kind. We found closed form expressions for the new DOS in terms of the original one when perturbation affects a single diagonal or off-diagonal site or a combination of both. The projected DOS is also calculated numerically and its relation to the average DOS is explored both analytically and numerically.

cond-mat.mtrl-sci↗

arXiv2006

Supersymmetric Jaynes-Cummings model and its exact solutions

The super-algebraic structure of a generalized version of the Jaynes-Cummings model is investigated. We find that a Z2 graded extension of the so(2,1) Lie algebra is the underlying symmetry of this model. It is isomorphic to the four-dimensional super-algebra u(1/1) with two odd and two even elements. Differential matrix operators are taken as realization of the elements of the superalgebra to which the model Hamiltonian belongs. Several examples with various choices of superpotentials are presented. The energy spectrum and corresponding wavefunctions are obtained analytically.

physics.atom-ph↗

arXiv2006

Representation reduction and solution space contraction in quasi-exactly solvable systems

In quasi-exactly solvable problems partial analytic solution (energy spectrum and associated wavefunctions) are obtained if some potential parameters are assigned specific values. We introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters. To obtain a larger solution space one varies the energy over a discrete set (the spectrum). A unified treatment that includes the standard as well as the new class of quasi-exactly solvable problems is presented and few examples (some of which are new) are given. The solution space is spanned by discrete square integrable basis functions in which the matrix representation of the Hamiltonian is tridiagonal. Imposing quasi-exact solvability constraints result in a complete reduction of the representation into the direct sum of a finite and infinite component. The finite is real and exactly solvable, whereas the infinite is complex and associated with zero norm states. Consequently, the whole physical space contracts to a finite dimensional subspace with normalizable states.

quant-ph↗

arXiv2006

Scattering theory with a natural regularization: Rediscovering the J-matrix method

In three dimensional scattering, the energy continuum wavefunction is obtained by utilizing two independent solutions of the reference wave equation. One of them is typically singular (usually, near the origin of configuration space). Both are asymptotically regular and sinusoidal with a phase difference (shift) that contains information about the scattering potential. Therefore, both solutions are essential for scattering calculations. Various regularization techniques were developed to handle the singular solution leading to different well-established scattering methods. To simplify the calculation the regularized solutions are usually constructed in a space that diagonalizes the reference Hamiltonian. In this work, we start by proposing solutions that are already regular. We write them as infinite series of square integrable basis functions that are compatible with the domain of the reference Hamiltonian. However, we relax the diagonal constraint on the representation by requiring that the basis supports an infinite tridiagonal matrix representation of the wave operator. The hope is that by relaxing this constraint on the solution space a larger freedom is achieved in regularization such that a natural choice emerges as a result. We find that one of the resulting two independent wavefunctions is, in fact, the regular solution of the reference problem. The other is uniquely regularized in the sense that it solves the reference wave equation only outside a dense region covering the singularity in configuration space. However, asymptotically it is identical to the irregular solution. We show that this natural and special regularization is equivalent to that already used in the J-matrix method of scattering.

quant-ph↗

arXiv2005

New method for evaluating integrals involving orthogonal polynomials: Laguerre polynomial and Bessel function example

Using the theory of orthogonal polynomials, their associated recursion relations and differential formulas we develop a method for evaluating new integrals. The method is illustrated by obtaining a closed-form expression for the value of an integral that involves the Bessel function and associated Laguerre polynomial. The result, which is given by Eq. (19) in the text, is written in terms of the Gegenbauer (ultra-spherical) polynomial.

math-ph↗

arXiv2005

Density function associated with nonlinear bifurcating map

In the class of nonlinear one-parameter real maps we study those with bifurcation that exhibits period doubling cascade. The fixed points of such a map form a finite discrete real set with dimension (2^n)m, where m is the (odd) number of "primary branches" of the map in the non-chaotic region and n is a non-negative integer. We associate with this map a nonlinear dynamical system whose Hamiltonian matrix is real, tridiagonal and symmetric. The density of states of the system is calculated and shown to have a number of separated bands equals to (2^n-1)m for n not equal 0, in which case the density has m bands. The location of the bands depends only on the map parameter and the odd/even ordering of the fixed points in the set. Polynomials orthogonal with respect to this density (weight) function are constructed. The logistic map is taken as an illustrative example.

math-ph↗

arXiv2005

Dirac and Klein-Gordon equations with equal scalar and vector potentials

We study the three-dimensional Dirac and Klein-Gordon equations with scalar and vector potentials of equal magnitudes as an attempt to give a proper physical interpretation of this class of problems which has recently been accumulating interest. We consider a large class of these problems in which the potentials are noncentral (angular-dependent) such that the equations separate completely in spherical coordinates. The relativistic energy spectra are obtained and shown to differ from those of well-known problems that have the same nonrelativistic limit. Consequently, such problems should not be misinterpreted as the relativistic extension of the given potentials despite the fact that the nonrelativistic limit is the same. The Coulomb, Oscillator and Hartmann potentials are considered. This shows that although the nonrelativistic limit is well-defined and unique, the relativistic extension is not. Additionally, we investigate the Klein-Gordon equation with uneven mix of potentials leading to the correct relativistic extension. We consider the case of spherically symmetric exponential-type potentials resulting in the s-wave Klein-Gordon-Morse problem.

hep-th↗