No abstract

This paper presents an accurate highly efficient method for solving the bound states in the one-dimensional Schrödinger equation with an arbitrary potential. We show that the bound state energies of a general potential well can be obtained from the scattering matrices of two associated scattering potentials. Such scattering matrices can be determined with high efficiency and accuracy, leading us to a new method to find the bound state energies. Moreover, it allows us to find the associated wavefunctions, their norm, and expected values. The method is validated by comparing solutions of the harmonic oscillator and the hydrogen atom with their analytical counterparts. The energies and eigenfunctions of Lennard-Jones potential are also computed and compared to others reported in the literature. This method is highly parallelizable and produces results that reach machine precision with low computational effort. A parallel implementation of this method written in FORTRAN is included in the Supplemental material to solve eigenstates of the Lennard-Jones potential.