Abdulaziz D. Alhaidari

Solutions of the nonrelativistic wave equation with position-dependent effective mass

Given a spatially dependent mass distribution, we obtain potential functions for exactly solvable nonrelativistic problems. The energy spectrum of the bound states and their wave functions are written down explicitly. This is accomplished by mapping the wave equation for these systems into well-known exactly solvable Schrodinger equations with constant mass using point canonical transformation. The Oscillator, Coulomb, and Morse class of potentials are considered.

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

We obtain L2-series solutions of the three-dimensional Schrodinger 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 of the potential term, cos θ/r2 (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.

Charged particle in the field of an electric quadrupole in two dimensions

We obtain an 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, bound as well as 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.

Extending the class of solvable potentials. I. The infinite potential well with a sinusoidal bottom

This is the first in a series of papers where we succeed in enlarging the class of exactly solvable potentials in one and three dimensions by obtaining solutions for new relativistic and nonrelativistic problems. This is accomplished by constructing a matrix representation of the wave operator in a complete square integrable basis that makes it tridiagonal. Expanding the wave function in this basis makes the wave equation equivalent to a three-term recursion relation for the expansion coefficients. Consequently, finding solutions of the recursion relation is equivalent to solving the original problem. Doing so results in a larger class of solvable potentials. The usual diagonal representation constraint results in a reduction from the larger class to the conventional class of solvable potentials, giving the well-known energy spectra and the corresponding wave functions. Moreover, some of the new solvable problems show evidence of a Klauder-like phenomenon. In the present work, we give an exact solution for the infinite potential well with a bottom that has a sinusoidal shape.

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.

Confined Dirac fermions in a constant magnetic field

We obtain an exact solution of the Dirac equation in (2+1) dimensions in the presence of a constant magnetic field normal to the plane together with a two-dimensional Dirac-oscillator potential coupling. The solution space consists of positive- and negative-energy solutions, each of which splits into two disconnected subspaces depending on the sign of an azimuthal quantum number k=0,{+-}1,{+-}2,... and whether the cyclotron frequency is larger or smaller than the oscillator frequency. The spinor wave function is written in terms of the associated Laguerre polynomials. For negative k, the relativistic energy spectrum is infinitely degenerate due to the fact that it is independent of k. We compare our results with already published work and point out the relevance of these findings to a systematic formulation of the relativistic quantum Hall effect in a confining potential.

On the asymptotic solutions of the scattering problem

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.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.

Group-theoretical foundation of the J-matrix theory of scattering

Group-theoretical analysis shows that SO(2,1) is an underlying dynamical symmetry for all Hamiltonians that are compatible with the Jacobi matrix (J-matrix) formalism. The class of central potentials with this property is obtained including, but not limited to, the oscillator, Coulomb and Morse potentials. The L2 bases and J-matrix elements for these potentials are found. SO(2,1)-invariant transformation of the solutions of the recursion relation for one potential gives those for another potential in the class. Phase-shift and resonance calculations for a single-channel potential are carried out in the oscillator basis to illustrate the use of our results.

Scattering phase shift for relativistic exponential-type separable potentials

The J-matrix method of scattering is used to obtain analytic expressions for the phase shift of two classes of relativistic exponential-type separable potentials whose radial component is of the general form rν-1e-λr/2 or r 2νe-λ2r2/2, where λ is a range parameter and ν = 0, 1, or 2. The rank of these separable potentials is ν + 1. The nonrelativistic limit is obtained and shown to be identical to the nonrelativistic phase shift. An exact numerical evaluation for higher-order potentials (ν≥3) can also be obtained in a simple way as illustrated for the case ν = 3.

Study of resonances 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 approaches is presented in this work. It is an indirect algebraic method that studies resonance in a complex charge plane (Z-plane). The complex scaling (rotation) method is employed in the development of this formulation. A finite L2 basis is used in the numerical implementation of the method.