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A. D. Alhaidari

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

At least 55 records · Page 3Linked to original sources
arXiv2018

Bound states of a short-range potential with inverse cube singularity

We use the Tridiagonal Representation Approach (TRA) to obtain exact bound states solution (energy spectrum and wavefunction) of the Schrödinger equation for a three-parameter short-range potential with 1/r, 1/r^2 and 1/r^3 singularities at the origin. The solution is a finite series of square integrable functions with weighted coefficients that satisfy a three-term recursion relation. The solution of the recursion is the discrete version of a non-conventional orthogonal polynomial. We are currently preparing to use the results of this work to study the binding of an electron to a molecule with an effective electric quadrupole moment, which has the same 1/r^3 singularity.

quant-ph↗
arXiv2018

Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications

We introduce two ordinary second-order linear differential equations of the Laguerre- and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively. The expansion coefficients of the series satisfy three-term recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Most of these are well-known polynomials whereas few are not. We present physical applications of these differential equations in quantum mechanics.

math-ph↗
arXiv2017

Quantum mechanics with orthogonal polynomials

We present a formulation of quantum mechanics based on orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis in configuration space where the expansion coefficients are orthogonal polynomials in the energy. Information about the corresponding physical systems (both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable non-relativistic quantum mechanical problems becomes larger than in the conventional formulation (see, for example, Table 1 in the text).

quant-ph↗
arXiv2017

Solution of the spin and pseudo-spin symmetric Dirac equation in 1+1 space-time using the tridiagonal representation approach

The aim of this work is to find exact solutions of the Dirac equation in 1+1 space-time beyond the already known class. We consider exact spin (and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus (and minus) the vector potential. We also include pseudo-scalar potentials in the interaction. The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis, which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric. This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.

hep-th↗
arXiv2017

Extending the class of solvable potentials. IV Inverse square potential with a rich spectrum

This is the fourth article in a series where we succeed in enlarging the class of exactly solvable quantum systems. We do that by working in a complete set of square integrable basis that carries a tridiagonal matrix representation for the wave operator. Consequently, the matrix wave equation becomes a three-term recursion relation for the expansion coefficients of the wavefunction. Finding solutions of this recursion relation in terms of orthogonal polynomials is equivalent to solving the original problem. This method, called the Tridiagonal Representation Approach (TRA), gives a larger class of solvable potentials. Here, we obtain S-wave solutions for a new four-parameter 1/r2 singular but short-range radial potential with an elaborate configuration structure and rich spectral property. A particle scattered by this potential has to overcome a barrier then could be trapped within the potential valley in a resonance or bound state. Using complex scaling (complex rotation), we display the rich spectral property of the potential in the complex energy plane for non-zero angular momentum and show how this structure varies with the physical parameters.

quant-ph↗
arXiv2017

Energy Density Bands Engineering

We expand the quantum mechanical wavefunction in a complete set of square integrable orthonormal basis such that the matrix representation of the Hamiltonian operator is tridiagonal and symmetric. Consequently, the matrix wave equation becomes a symmetric three-term recursion relation for the expansion coefficients of the wavefunction whose solution is a set of orthogonal polynomials in the energy. The polynomials weight function is the energy density of the system constructed using the Green's function, which is written in terms of the Hamiltonian matrix elements. We study the distribution of zeros of these polynomials on the real energy line based exclusively on their three-term recursion relations. We show that the zeros are generally grouped into sets belonging to separated bands on the orthogonality interval. The number of these bands is equal to the periodicity (multiplicity) of the asymptotic values of the recursion coefficients and the location of their boundaries depend only on these asymptotic values. Bound states (if they exist) are located at discrete zeros found in the gaps between the density bands that are stable against variation in the order of the polynomial for very large orders. We give examples of systems with a single, double and triple energy density bands.

math-ph↗
arXiv2017

Energy spectrum of a generalized Scarf potential using the Asymptotic Iteration Method and the Tridiagonal Representation Approach

The well-known trigonometric Scarf potential is generalized by adding a sinusoidal term and then treated using the Asymptotic Iteration Method (AIM) and the Tridiagonal Representation Approach (TRA). The energy spectrum of the associated bound states are computed. For the AIM, we have improved convergence of the quantization condition that terminates the iterations asymptotically. This is accomplished by looking for the range of initial values of the space variable in the terminating condition that produces stable results (plateau of convergence). We have shown that with increasing iteration this plateau of convergence grows up rapidly to an optimal iteration number and then shrinks slowly to a point. The value of this point (or points) may depend on the physical parameters. The numerical results have been compared favorably with those resulting from the TRA.

quant-ph↗
arXiv2017

Solution of the nonrelativistic wave equation in the tridiagonal representation approach

We choose a complete set of square integrable functions as basis for the expansion of the wavefunction in configuration space such that the matrix representation of the nonrelativistic time-independent wave operator is tridiagonal and symmetric. Consequently, the matrix wave equation becomes a symmetric three-term recursion relation for the expansion coefficients of the wavefunction in this basis. The recursion relation is then solved exactly in terms of orthogonal polynomials in the energy. Some of these polynomials are not found in the mathematics literature. The asymptotics of these polynomials give the phase shift of the continuous energy scattering states and the spectrum for the discrete energy bound states. Depending on the space and boundary conditions, the basis functions are written in terms of either the Laguerre or Jacobi polynomials. The tridiagonal requirement limits the number of potential functions that yield exact solutions of the wave equation. Nonetheless, the class of exactly solvable problems in this approach is larger than the conventional class (see Table 12). We also give very accurate results for cases where the wave operator matrix is not tridiagonal but its elements could be evaluated either exactly or numerically with high precision.

quant-ph↗
arXiv2017

The infinite square well in a reformulation of Quantum mechanics without potential function

Using a recent reformulation of quantum mechanics where the potential function is not required, we are able to obtain the energy spectrum and wave function associated with the infinite square well analytically. Therefore, this work constitutes an example of how to establish the correspondence between this approach and the standard formulation of quantum mechanics.

math-ph↗
arXiv2016

Asymptotic Limit of Continuous Dual Hahn Polynomial

Using Darboux method, we obtain an asymptotic limit for Continuous Dual Hahn Polynomial.The basic concept is to construct a comparison function which is the singular part of the generating function; then expand the comparison function. With little simplification and modification,the asymptotic limit is achieved.

math-ph↗
arXiv2016

The Wilson-Racah Quantum System

Using a recent formulation of quantum mechanics without potential function, we present a four-parameter system associated with the Wilson and Racah polynomials. The continuum scattering states are written in terms of the Wilson polynomials whose asymptotics gives the scattering amplitude and phase shift. On the other hand, the finite number of discrete bound states are associated with the Racah polynomials. We are honored to dedicate this work to Prof. Hashim A. Yamani on the occasion of his 70th birthday.

quant-ph↗
arXiv2015

Energy spectrum for a short-range 1/r singular potential with a non-orbital barrier using the asymptotic iteration method

Using the asymptotic iteration method, we obtain the S-wave solution for a short-range three-parameter central potential with 1/r singularity and with a non-orbital barrier. To the best of our knowledge, this is the first attempt at calculating the energy spectrum for this potential, which was introduced by H. Bahlouli and A. D. Alhaidari and for which they obtained the "potential parameter spectrum". Our results are also independently verified using a direct method of diagonalizing the Hamiltonian matrix in the J-matrix basis.

quant-ph↗
arXiv2015

Solution of the Dirac equation in a curved space with static metric

Compatibility of symmetric quantization of the Dirac equation in a curved space with general covariance gives a special representation of the spin connections in which their dot product with the Dirac gamma matrices becomes equal to the "covariant divergence" of the latter. Requiring that the square of the equation gives the conventional Klein-Gordon equation in a curved space results in an operator algebra for the Dirac gamma matrices that involves the "covariant derivative" connections and the Riemann-Christoffel connections. In 1+1 space-time with static metric, we obtain exact solutions of this Dirac equation model for some examples. We also formulate the interacting theory of the model with various coupling modes and solve it in the same space for a given potential configuration.

quant-ph↗
arXiv2015

Compatibility of symmetric quantization with general covariance in the Dirac equation and spin connections

By requiring unambiguous symmetric quantization leading to the Dirac equation in a curved space, we obtain a special representation of the spin connections in terms of the Dirac gamma matrices and their space-time derivatives. We also require that squaring the equation give the Klein-Gordon equation in a curved space in its canonical from (without spinor components coupling and with no first order derivatives). These requirements result in matrix operator algebra for the Dirac gamma matrices that involves a universal curvature constant. We obtain exact solutions of the Dirac and Klein-Gordon equations in 1+1 space-time for a given static metric.

hep-th↗
arXiv2015

Quantum mechanics without potential function

In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schrödinger equation, which is solved for the wave function, bound states energy spectrum and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and there-parameter systems.

quant-ph↗
arXiv2015

Extending the class of solutions of the Dirac equation using tridiagonal matrix representations

The aim of this work is to find exact solutions of the one-dimensional Dirac equation that do not belong to the already known conventional class. We write the spinor wavefunction as a bounded infinite sum in a complete basis set, which is chosen such that the matrix representation of the Dirac wave operator becomes tridiagonal and symmetric. This makes the wave equation equivalent to a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. We solve the recursion relation and obtain the relativistic energy spectrum and corresponding state functions. Restriction to the diagonal representation results in the conventional class of solutions. As illustration, we consider the square well with half-sine potential bottom in the vector and scalar coupling under spin symmetry. We obtain the relativistic energy spectrum and show that the nonrelativistic limit agrees with results found elsewhere.

math-ph↗