To receive full marks your code must include appropriate comments and documentation. Answers to questions should be clearly indicate

AS a remInder, only packages included below may be used when completing the assignment.

$[2]$:

#Import standard runctions

import numoy as nc import matolotlib.ovolot as plt

from matplotlib import rcParams #for adjusting plot parameters from tqdm$.$notebook import trange #Handy package to add a progress bar for loops $-$ you will need to install tqdm

From

I tadm.notebook import tadm

import time #For benchmarking

#Optional functions that are allowed. Uncomment to use.

iF import os

$*$ Import threading

# import multiprocess as mp #You may need to install this package. If you have problems you can try the alternative 'multiprocessing' packac

$$ from numba import iit #vou may need to install the numba package

The Schr$$dinger equation

when a particle of mass m is subjected to the influence of a potential $($r $,$t$),$ the schrodinger equation takes on the form

ih$-$wr$,$t$)=$

$-12+$ AW$($I$,$ t$)+$ V$($r$,$ t$)$ W$($I$,$ t$)$

Where A $=32/$Ax$?+$ d$2/$dy$2+22/$ dz$?$ is the Laplacian operator. If the potential energy is time$-$independent, then it is well known that the resulting eigenvalue

equation can be written as

$[-$ HA $+$ V$($r$)]($r$)=$ E$($r$)$

$(2),$

which is also commonly known as the time$-$independent Schr$$dinger equation $($TISE$).$

Consider the following second$-$order differential equation,

d$\text{'}$y $=-$g$($x$)$y$($x$)+$ s$($x$)=$ Z;

$(3)$

bounded by a box of width defined by $-$ Xmax $$ X $$ X max$$ To numerically solve the system we can divide the box into N small intervals of size ox$,$ with y; $=$ y$($x$).$ If we expand y$.$ as a Taylor series we obtain,

SS$+0$ U as x $->\backslash$deg c

b$./$to $1($x$)17$dx $=1$

c$.($x$)$ and $($x$)$ are continuous

Solve Eqn $(2)$ to find the eigenfunction bounded by X max $=10$a$,$ by using Equation $(6)$ for energy states Eo$,$ El$,$ E$2,$ E$3,$ E$4,$ Es$,$ using a shooting$-$method, where Eo

is the ground state, or lowest energy solution.

Your algorithm should use the following procedure

$1.$ Construct a grid with N divisions of equal width $($x$)$ indexed from i $=0,$ N$,$ with xo $=-$ Xmas andxN $=$ Xmas

$2.$ Begin with a trial value of E $=$ Eo $=$ EN$.$

$3.$ Use Eqn $(6)$ to advance the solution from Xo and n to a matching point X c

$4.$ Calculate &y $=$ Yo$,$c $-$ YN$,=(/)$o$,$c $-(/)$N c$,$ Where o$,$c refers to the solution started at xo evaluated at xc and N$,$c refers to the solution started at XN

and evaluated at Xc

$5.$ Vary E until Sv $=0$

$6.$ Ensure you have a continous eigenfunction and normalize the entire wavefunction

Your energy state solutions must be accurate to $10-8$

Make a table that presents the energy, in eV$,$ for each eigenstate and produce a plot that presents the eigenfunctions.