Relaxational approach to solving the Schrödinger equation

A relaxational approach to solving the Schrödinger equation is developed that utilizes split boundary conditions. The method solves the wave equation for the eigenvalues and eigenfunctions simultaneously. Advantages over numerical integration offered with the technique include simplification of the selection of boundary conditions and initial eigensystem estimates. The power of this technique is demonstrated by numerical solution of the radial Schrödinger equation for several important or problematic potentials, including the Saxon-Wood potential, the laser-dressed Coulombic potential, the regulated Coulombic potential, the Morse potential, and the Lennard-Jones potential. In order to demonstrate this method’s applicability to even highly nonlinear systems, the ground-state energy levels of the helium and lithium atomic systems are solved utilizing a self-consistent-field approximation. The application of the method to two- and three-dimensional systems is then briefly discussed.