Solving Schr\"odinger Equation with Scattering Matrices. Bound states of Lennard-Jones Potential

TL;DR

This paper introduces a highly efficient, parallelizable method for solving the Schrödinger equation by using scattering matrices, accurately determining bound states, wavefunctions, and expectations for arbitrary potentials, including Lennard-Jones.

Contribution

The paper presents a novel approach that links bound state energies to scattering matrices, enabling precise and efficient solutions for the Schrödinger equation with arbitrary potentials.

Findings

Accurately computed bound states for harmonic oscillator and hydrogen atom.

Validated method by comparing Lennard-Jones potential results with literature.

Achieved machine precision results with low computational effort.

Abstract

This paper presents an accurate highly efficient method for solving the bound states in the one-dimensional Schr\"odinger 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.