Bound states for anisotropic potentials and masses

Abstract

A general method for finding the energies and wave functions for anisotropic masses and potentials has been studied. This method uses basis wave functions expanded in spherical harmonics. The coupled differential equations for the radial components of each basis function are integrated numerically. The energy eigenfunctions and their radial derivatives are then matched at a spherical boundary. Special treatment is needed to ensure linear independence of the basis functions at the boundary. The applicability of this method and the speed of convergence are tested on anisotropic harmonic oscillators. The method is then applied to the Coulomb potential with anisotropic masses. With a basis of five or fewer spherical harmonics, our method produces energies which converge to values lower than those previously reported. We have also obtained energy levels for the Coulomb potential with threefold mass anisotropy. This method should be applicable to other anisotropic problems with a single potential minimum. In particular, it should facilitate the employment of full-potential Green-function band theory.