Noncentral potentials and spherical harmonics using supersymmetry and shape invariance

Abstract

It is shown that the operator methods of supersymmetric quantum mechanics and the concept of shape invariance can profitably be used to derive properties of spherical harmonics in a simple way. The same operator techniques can also be applied to several problems with non-central vector and scalar potentials. As examples, we analyze the bound state spectra of an electron in a Coulomb plus an Aharonov-Bohm field and/or in the magnetic field of a Dirac monopole.

Relativistic solution in D-dimensions to a spin-zero particle for equal scalar and vector ring-shaped Kratzer potential

“…The path integral for particles moving in non-central potentials is evaluated to derive the energy spectrum of this system analytically [43]. In addition, the idea of SUSY and shape invariance is also used to obtain exact solutions of such non-central but separable potentials [44,45]. Very recently, the conventional Nikiforov-Uvarov (NU) method [46] has been used to give a clear recipe for how to obtain an explicit exact boundstates solutions for the energy eigenvalues and their corresponding wave functions in terms of orthogonal polynomials for a class of non-central potentials [47].…”

Romanovski polynomials in selected physics problems

“…Romanovski polynomials are not so widespread as the others in applications. But in recent years several problems have been solved in terms of this family of polynomials (Schrödinger equation with the hyperbolic Scarf and the trigonometric Rosen-Morse potentials [6,7], Klein-Gordon equation with equal vector and scalar potentials [8], certain classes of non-central potential problems as well [9]) and so they deserve a closer look and to be placed on equal footing with the classical orthogonal polynomials. In this context, our goal is threefold.…”

Exactly solvable problems of quantum mechanics and their spectrum generating algebras: A review

“…[18], where it was shown to be generated by the superpotential W = l tanh z. (N.B., this is not the same as the trigonometric Pöschl-Teller I potential we studied in the introduction.)…”

“…Thus we have an example where we started from a very well known algebra, and showed that it is connected to a solvable one-dimensional Schrödinger equation through the transformation in Eq. (18). Other transformations will produce other potentials.…”