Two-dimensional Schroedinger Equation Research Articles

Spontaneous generation of singularities in paraxial optical fields.

In nonrelativistic quantum mechanics, the spontaneous generation of singularities in smooth and finite wave functions is a well understood phenomenon also occurring for free particles. We use the familiar analogy between the two-dimensional Schrödinger equation and the optical paraxial wave equation to define a new class of square-integrable paraxial optical fields that develop a spatial singularity in the focal point of a weakly focusing thin lens. These fields are characterized by a single real parameter whose value determines the nature of the singularity. This novel field enhancement mechanism may stimulate fruitful research for diverse technological and scientific applications.

On a relation of pseudoanalytic function theory to the two-dimensional stationary Schrödinger equation and Taylor series in formal powers for its solutions

We consider the real stationary two-dimensional Schroedinger equation. With the aid of any its particular solution we construct a Vekua equation possessing the following special property. The real parts of its solutions are solutions of the original Schroedinger equation and the imaginary parts are solutions of an associated Schroedinger equation with a potential having the form of a potential obtained after the Darboux transformation. Using L. Bers theory of Taylor series for pseudoanalytic functions we obtain a locally complete system of solutions of the original Schroedinger equation which can be constructed explicitly for an ample class of Schroedinger equations. For example it is possible, when the potential is a function of one cartesian, spherical, parabolic or elliptic variable. We give some examples of application of the proposed procedure for obtaining a locally complete system of solutions of the Schroedinger equation. The procedure is algorithmically simple and can be implemented with the aid of a computer system of symbolic or numerical calculation.

Solving large systems of polynomial equations using REDUCE

Difficulties arising when numerically solving large systems of polynomial equations are discussed. Such systems determine the input data for the inverse task for the two-dimensional Schroedinger equation.

Propagation of quarks in the spatial direction in hot QCD.

The dynamics of light fermions propagating in a spatial direction at high temperatures can be described effectively by a two-dimensional Schroedinger equation with a heavy effective mass {ital m}{sub eff}={pi}{ital T}. Starting from QED, we discuss the transition from three- to two-dimensional positronium discussing the latter in detail including relativistic effects. In the case of QCD the problem is similar to that of heavy quarkonium. Our effective potential contains the usual Coulomb and confining parts as well as a perturbative spin-spin interaction. The resulting {ital {bar q}q} wave functions'' reproduce recent lattice data for the {rho} and {pi} channels. The physical meaning of such confinement'' is related to the nontrivial magnetic interaction of color currents in the quark-gluon plasma. Our results do not contradict the idea that the normal electric interaction of color charges is screened and produces no bound states in the usual sense.

Application of the finite-element method to the hydrogen atom in a box in an electric field

A better understanding of the problems of a pressurized atom and a crystal, both under the influence of a constant electric field, has been achieved through the numerical solution of the two-dimensional Schroedinger equation. The procedure consists of an adaptation of a finite-element package and gives fast and accurate results.

Finite element method for solving the two-dimensional schroedinger equation

A finite element package is presented that is able to treat two-dimensional Schroedinger equation problems over a finite region with an arbitrary potential and homogeneous boundary conditions. In order to present the applicability and accuracy of this approach, two cases for which the exact solution is known are solved: (i) a free particle in a spherical box, (ii) a hydrogen atom enclosed in a finite sphere. The results definitely indicate that the finite element method is both accurate and efficient and could serve as a useful tool in various single particle quantum mechanical problems.