Some further considerations on deriving matrix elements of non-periodic potentials in the basis of the non-degenerate harmonic oscillator wavefunctions
a Laboratoire de Physique Quantique, Faculté des Sciences de Monastir, Route de Kairouan, 5000, Monastir, Tunisia
b Centre d'Etudes Fondamentales, Université de Perpignan, 52 Avenue de Villeneuve, 66860, Perpignan Cedex, France
Abstract:
Using the properties of the generating function of the Hermite polynomials, we have calculated the matrix elements for the Gaussian-type potential VG(x)=A exp{−B(x−C)2} and for the Morse-type potential VM(x)=De[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.