Some further considerations on deriving matrix elements of non-periodic potentials in the basis of the non-degenerate harmonic oscillator wavefunctions

Abstract

Using the properties of the generating function of the Hermite polynomials, we have calculated the matrix elements for the Gaussian-type potential V G (x)=A exp{−B(x−C) 2} and for the Morse-type potential V M( x)= D e[1−exp(− ax)] 2 in the basis of the non-degenerate harmonic oscillator wavefunctions. The final forms of these matrix elements are very simple to use and hence suitable for the numerical resolution of the Schrödinger equation for multiple-well potentials or anharmonic oscillators.