Solving Schrodinger Equation with Scattering Matrices. Bound States of Lennard-Jones Potential

This work presents an accurate and highly efficient method for solving the bound states in the one-dimensional Schrodinger equation considering arbitrary potentials. We show that the bound state energies of a general potential well can be found from the scattering matrices of two associated scattering potentials. Such matrices can be determined with high efficiency and remarkable accuracy, providing a new method to obtain bound state energies, as well as the associated wavefunctions, their norm, and expected values. The method is validated by comparing its solutions for the harmonic oscillator and the hydrogen atom with their analytical counterparts. Likewise, by comparing it with the Numerov shooting method, we show advantages of the proposed method in terms of computational time and accuracy. The energies and eigenfunctions of the Lennard-Jones potential are also computed and compared to others reported in the literature, with excellent agreement. This method is easily parallelizable and its results reach machine precision with low computational effort. A parallel implementation of this method to solve eigenstates of the Lennard-Jones potential is included in the Supplemental material.