Application of Gröbner basis in calculation of wave functions in nanostructures

Abstract

The knowledge of exact wave functions is required in calculating physical parameters such as optical dipole moments, scattering matrix elements, or in wave function engineering. In this report we describe how a system of algebraic equations that follows from the Schrödinger equation can be reduced to a Gröbner basis from which the exact wave function can be easily constructed. As an example, closed form solutions for a cylindrical electronic waveguide and double quantum well nanostructures are presented.