Solving the One-Dimensional Time-Independent Schr\"odinger Equation with High Accuracy: The LagrangeMesh Mathematica Package

Abstract

In order to find the spectrum associated with the one-dimensional Schr\"oodinger equation, we discuss the Lagrange Mesh method (LMM) and its numerical implementation for bound states. After presenting a general overview of the theory behind the LMM, we introduce the LagrangeMesh package: the numerical implementation of the LMM in Mathematica. Using few lines of code, the package enables a quick home-computer 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 basically the potential function and the interval on which is defined; and (ii) the accuracy in calculations and final results is controllable by the user. As illustration, a highly accurate spectrum of some relevant quantum systems is obtained by employing the commands that the package offers. In fact, the present work can be regarded as a user guide based on worked examples.