Numerically complemented analytic method for solving the time-independent one-dimensional Schrödinger equation

Abstract

A general method of solving the one-dimensional Schrödinger equation is developed. The first step is to construct an exactly solvable reference potential of several smoothly joined Morse-type components, which should be a good approximation to a given potential. The exact solutions for that reference Hamiltonian are then combined with a nonperturbative approach [R. G. Gordon, J. Chem. Phys. 51, 14 (1969)], which enables us to numerically solve the energy eigenvalue problem for the original potential to any desired accuracy. A full description of the analytical procedures is given and examples of both exact and numerical solutions, are presented.