On quantal bound state solutions and potential energy surface fitting. A comparison of the finite element, Numerov-Cooley and finite difference methods

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This paper investigates several factors affecting the accuracy and efficiency of numerical determination of the bound state energy eigenvalues of the one dimensional Schrödinger equation. The efficiencies of the finite element method (FEM), the Numerov-Cooley method, and the finite difference method are compared. From this comparison, it is concluded that for potentials containing a single energy minimum, the Numerov-Cooley method is the most efficient, while for the most complex potentials the finite element method is superior due to its better numerical stability in the classically forbidden regions. The effects of various polynomial interpolation schemes on the calculated eigenvalues of potentials known only on a small number of points is examined. It is found that while higher order fits are superior to lower ones when the potential points are known accurately, they can introduce spurious information into the potential for inaccurately known points, and thus produce poor eigenvalues. Likewise, for accurately known potentials, a spline or Hermitian interpolation is better than a Lagrangian fit, but the Lagrangian functions are less susceptible to noise in a less well known case.