Solving the one-dimensional time-independent Schrödinger equation with high accuracy: The LagrangeMesh Mathematica® package

Abstract

In order to find the spectrum associated with the one-dimensional Schrödinger equation, we discuss the Lagrange Mesh Method (LMM) and its numerical implementation. After presenting a general overview of the theory behind the LMM, we introduce the LagrangeMesh package: the numerical implementation of the LMM in Mathematica[Formula: see text]. Using few lines of code, the package enables a quick home-computer and highly accurate computation of the spectrum and provides a practical tool to study a large class of systems in quantum mechanics. The main properties of the package are (i) the input is the potential function and the interval on which it is defined; and (ii) the accuracy in calculations and final results is controllable by the user. Due to its high accuracy and simple usage, the package may be used as a research and educational tool. As illustration, a highly accurate spectrum of some relevant quantum systems is obtained by employing the commands that the package offers.