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

TL;DR

This paper introduces the LagrangeMesh Mathematica package, a tool that efficiently computes highly accurate spectra for one-dimensional quantum systems by implementing the Lagrange Mesh method, with user-friendly input and controllable precision.

Contribution

It presents a practical implementation of the Lagrange Mesh method in Mathematica, enabling quick and accurate spectral calculations for quantum systems.

Findings

High-accuracy spectra for quantum systems obtained

Package allows simple input of potential functions and intervals

Calculations are controllable in accuracy by the user

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.