III$-$ Solving the Schr$$dinger equation in $1$D numerically

Introduction

$$ We will be using the FDM $($Finite Difference Method$)$ technique to discretize the Schr$$dinger equation since it is by far the easiest approach to work with.

$$ Let us consider the quantum well problem where we need to solve the $1$D eigenvalue Schr$$dinger equation:

$$ It is not possible to find an analytical solution to this equation if the potential U$($z$)$ does not take a simple form $($like a constant potential, linear, $1/$r$,$ or parabolic$).$

$$ We propose to compute the solution $\backslash$phi $($could be any eigenvectors$)$ on a numerical grid composed of N equidistant points. The continuum solution is reformulated as follows:

where we denote $\backslash$phi i$\backslash$phi $($zi$).$ The discretized space is also known as the computational domain.

$1$D FDM discretization process using a numerical grid. In the discretized space $\backslash$phi $($z$)$ is only known at the nodes z $=$ z$1,$z$2,...,$zN$.$

$$ The FDM approximation consists of replacing the Laplacian operator $($second derivative in the differential equation$)$ by a discrete one which only depends on the values of the functions at the neighboring nodes. We assume that our numerical grid is uniform and we denote $$a$$ the grid step i$.$e$.$

a $=$ L$/($N$1)=|$zi$+1$zi $|=|$zi$$zi$-1|$

Using a Taylor development at z $=$ zi and using the first neighboring nodes, we obtain:

$$ By adding those two expressions, we obtain an approximation for the discretized Laplacian operator $($second derivative$)$:

The discretized FDM Schr$$dinger equation looks like then:

which represents a system of N linear equations!

This system can then be formulated into the following system matrix $($with t$=2/(2$ma$2))$

which looks like an eigenvalue system Ax$=\backslash$lambda x as seen in linear algebra! $($A is tridiagonal of size NxN and each eigenvector xn is of size Nx$1$ associated with the eigenvalue $\backslash$lambda n$).$

We note the following:

$$In practice, the boundary conditions $\backslash$phi $(0)=\backslash$phi $($L$)=0$ can be enforced by artificially assuming a very high potential at the boundaries i$.$e$.$ U$1=$UN $=$ U$\backslash$infty $=100$eV $($so no electron can go in there$).$

$$ Once constructed, the eigenvalue system matrix can be solved using numerical routines such as eig in Matlab or equivalent eigh in Numpy$/$Scipy $($Python$),$ or with the use of more efficient numerical libraries in C$/$C$++$ or Fortran $($such as Intel$-$MKL$).$ Here is a Python example on how to easily build a $10$x$10$ tridiagonal $($made up$)$ matrix and solve all its eigenvalues and eigenvectors:

import numpy as np

N$=10$

diagA$=$np$.$full$($N$,4)$ # constant vector

offdiagA$=$np$.$full$($N$-1,-1)$ # constant vector

A$=$np$.$diag$($offdiagA$,-1)+$ np$.$diag$($diagA$,0)+$ np$.$diag$($offdiagA$,1)$ # tridiagonal matrix A NxN

e$,$X$=$np$.$linalg.eigh$($A$)$ # return N eigenvalue vector e and the NxN eigenvector matrix X

print$($e$)$ # N eigenvalues

print$($X$[$:$,0])$ # $1$st eigenvector

$$ The potential U which is assumed to be known at all discretization nodes, can be added to the diagonal term of the matrix. In contrast to an analytical treatment of the problem, U can take any forms! and the eigenvalue$/$eigenvector solutions $\{\backslash$epsi n$,\backslash$phi n$\}$ will then be obtained numerically. As an example, let us assume U $=0($quantum well$)$ and plot the solution vectors $|\backslash$phi $1|2,|\backslash$phi $2|2$ and $|\backslash$phi $3|2($all the component vectors are squared$)$ using L$=3$nm and N$=300$ points of discretization, we get:

Questions

Use your preferred programming language to implement this problem $($I suggest python$)$ and solve numerically the eigenvalue problem for the situations described below:

$*$Any plots provided must have your name in the Title,

$*$You must include a copy of your codes at the end of this document.

$1.$ Using L $=3$nm$,$ N $=300,$ U$($z$)=0($zero potential$),$ and U$\backslash$infty $=100$eV for boundary conditions

$($given here in eV$....$but must be in Joule in the matrix!$).$

a$-$ Provide the first $3$ calculated eigenvalues $($in eV$)$ side by side with their analytical $($true$)$ value $($quantum well formula$).$ Remark: If they do not look the same $(+/-$ numerical approximation errors$),$ there is a problem in your code and you still have to work on it before going to the next questions.

b$-$ Plot the resulting curves $|\backslash$phi n$|2$ for n $=1,2,3$ in $[0,$ L$]($on the same graph$).$ Remark: results should be similar to the ones provided above.