  Traditionally, finite differences and finite element methods have been by
many regarded as the basic tools for obtaining numerical solutions in a variety
of quantum mechanical problems emerging in atomic, nuclear and particle
physics, astrophysics, quantum chemistry, etc. In recent years, however, an
alternative technique based on the semi-spectral methods has focused
considerable attention. The purpose of this work is first to provide the
necessary tools and subsequently examine the efficiency of this method in
quantum mechanical applications. Restricting our interest to time independent
two-body problems, we obtained the continuous and discrete spectrum solutions
of the underlying Schroedinger or Lippmann-Schwinger equations in both, the
coordinate and momentum space. In all of the numerically studied examples we
had no difficulty in achieving the machine accuracy and the semi-spectral
method showed exponential convergence combined with excellent numerical
stability.
