LagrangeMesh-Package

Description

LagrangeMesh 1.0 is a Mathematica package devoted to solving numerically the one-dimensional time-independent Schrödinger equation with Dirichlet boundary conditions for an arbitrary one-dimensional domain. A complete description of the package can be found in

"Solving the One-Dimensional Time-Independent Schrödinger Equation with High Accuracy: The LagrangeMesh Mathematica Package" by J.C. del Valle, 2022 (https://arxiv.org/abs/2208.14340).

Prerequisites

Wolfram Mathematica 12,13,...

Notation

• $E$: eigenvalue/energy
• $m$: mass of the particle
• $x$: real variable
• $y(x)$: eigenfunction/wavefunction in variable x
• $V(x)$: confining potential

LagrangeMesh 1.0 is a Mathematica package devoted to solving numerically the one-dimensional time-independent Schrödinger equation with Dirichlet boundary conditions: $$-\frac{1}{2m} y''(x)+V(x)y(x)=Ey(x)\ ,\qquad y(0)=y(b)=0\ , \qquad a < x < b ,$$ for an arbitrary real interval $(a,b)$.

Installation

0. Download the file LagrangeMesh.wl

1. There two options:

(A) Full Installation

Open Mathematica, go to File -> Install. A window will prompt you for more information. Fill in as follows:

i. Type of Item to Install -> Package ii. Source -> Select file LagrangeMesh.wl iii. Install Name -> LagrangeMesh iv. Click OK.

Once the installation was successful, the file LagrangeMesh.wl can be deleted. To load the package in a given NoteBook, type Needs["LagrangeMesh`"] in the command line and evaluate the cell. Each time the kernel is restarted, the package has to be loaded as indicated.

(B) Loading the Package

This is a simple alternative that requires no full installation. Loading the package is achieved by evaluating <<"path/LagrangeMesh.wl" in the command line of the notebook we want to work with. The explicit form of path is found evaluating the command NotebookDirectory[]. This option only works if the package and the notebook (in which we will perform calculations) are contained in the same directory. Each time the kernel is restarted, package has to be loaded as indicated.

Commands

Once installed or loaded, the package provides the user with five commands:

• BuildMesh: Construction of mesh points and weights
• AvailableMeshQ: Checks available meshes and weights
• LagMeshEigenvalues: Computes eigenvalues
• LagMeshEigenfunctions: Computes eigenfunctions
• LagMeshEigensystem: Computes eigenvalues and eigenfunctions

Usage

Based on particular examples, we describe the usage of each command. Input and Output are shown in Mathematica's style. We assume that all evaluations are performed from the notebook called MyWorkNotebook.nb that is contained in a folder called MyDirectory. By default the mass is set m = 1.

(1) BuildMesh. It constructs the requested mesh points and weights for a given type of classical orthogonal polynomial. (Legendre,Laguerre,Hermite). Basic example:

    In[1]:= BuildMesh["Hermite",15,WorkingPrecision->20,Weights->True]
    Out[1]:= Hermite_15_WP_20.dat
    Hermite_15_WP_20_Weights.dat

As Output, shown in Out[1], the program prints on screen the name of two files that were generated and stored. Meshes and weights are automatically stored according to the following tree diagram:

DirectoryTree-eps-converted-to.pdf

All directories inside MyDirectory will be automatically created on the first use of the command BuildMesh.

(2) AvailableMeshQ. By specifying Type, Dimension, and WorkingPrecision, it delivers True on screen if the mesh was previously constructed. Otherwise, the Output is False. In addition, it can deliver an ordered table that shows all stored meshes. A basic example:

    In[2]:= AvailableMeshQ["Hermite",WorkingPrecision->20,Dimension->15]
    Out[2]:= True

(3) LagMeshEigenvalues. It computes the eigenvalues for a given potential defined on an interval. The number of mesh points must be specified as well as the requested lowest eigenvalues. If WorkingPrecision is not specified, calculations are performed with MachinePrecision. A basic example is presented below: the first three eigenvalues of the harmonic oscillator defined on the whole real line calculated with 15 mesh points and WorkinPrecision->20.

     In[3]:= LagMeshEigenvalues[1/2*x^2,{x,-Infinity,Infinity},3,15,WorkingPrecision->20]
    Out[3]:= {0.500000000000000000,1.50000000000000000,2.50000000000000000}

(4) LagMeshEigenfunctions. It computes the eigenfunctions for a given potential defined on an interval. The number of mesh points must be specified as well as the requested lowest eigenvalues. If WorkingPrecision is not specified, calculations are performed with MachinePrecision. A basic example is presented below: the first eigenfunction of the harmonic oscillator defined on the whole real line calculated with 15 mesh points and WorkinPrecision->20.

    In[4]:= LagMeshEigenfunctions[1/2*x^2,{x,-Infinity,Infinity},1,15,WorkingPrecision->20]
    Out[4]:= -0.75112554446494248283 E^(-(x^2/2))

(5) LagMeshEigensystem. It computes the eigenvalues and eigenfunctions simultaneously for a given potential defined on an interval. The number of mesh points must be specified as well as the requested lowest eigenvalues. If WorkingPrecision is not specified, calculations are performed with MachinePrecision. A basic example is presented below: the first eigenvalue and eigenfunction of the harmonic oscillator defined on the whole real line calculated with 15 mesh points and WorkinPrecision->20.

    In[5]:= LagMeshEigensystem[1/2*x^2,{x,-Infinity,Infinity},1,15,WorkingPrecision->20]
    Out[5]:= {0.500000000000000000, -0.75112554446494248283 E^(-(x^2/2))}

Authors

LagrangeMesh 1.0 (2022):

(C) J.C. del Valle, Version 1.0 (September,2022)
(Written in Mathematica 13 and tested in version 12 and 13)

Please report any malfunction of the package to jcdvaller@gmail.com