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.

A study is made of the ’’conservation of the total cross section’’ and the ’’equivalence of the total cross section’’ rules for scattering from H2. It is shown that these rules are a better approximation than the random phase approximation would indicate. Cross section formulas are given for scattering atoms from mj state selected molecules and it is shown that total cross sections for state selected molecules depend on the anisotropic part of the interaction potential, while the spin-averaged total cross section often depends only on the spherically symmetric part of the interaction potential. The total spin-averaged cross section is thus independent of the initial rotation state of the molecule and depends only on the relative collision energy. It is further demonstrated that isotopic substitution, which shifts the center of mass changing the symmetric part of the interaction potential, has too small an effect on the total cross section to be useful as a means of determining the anisotropy of the potential.