PHA S 0 0 2 9 F I N A L A S S I G NMENT 2 0 2 2
TH E I N F I N I T E “ U N S QUA R E ” P OT E NT I A L
L A S T R E V I S I O N MAR CH 3 , 2 0 2 2
M . S Z UM I L O @U C L . AC . U K
Contents
1 Introduction 2
2 Boundary value problems: eigenvalue solutions 2
3 Infinite square well 3
3.1 Task for session 8: Finding the ground state energy 3
3.2 Finding the ground state wavefunction 5
3.3 Finding the higher energy states 5
4 The infinite “unsquare” well 6
5 General assignment advice 7
5.1 Assessment 7
5.2 Referencing and bibliography 7
5.3 Submission 8
5.4 Page limit 9
phas0029 final assignment 2022 2
1 Introduction
In the course so far we’ve used a wide variety of techniques to give you a flavour
of the sort of problems that can be tackled using computational approaches.
In session 8, we’re going to look at one more new aspect of this: including
boundary values in the solving of differential equations. You will then apply this to
the problem for the PHAS0029 final assignment: solving the quantum-
mechanical
problem of a particle in an infinite potential well. You will first solve for an infinite
square potential well, and use the familiar analytical solutions for this system to
check that your results are correct. You will then be able to extend this to
calculate for systems where there is no simple analytical solution.
Before commencing any of the tasks below, make sure you have gone through and
understood the contents of the Jupyter Notebook script for session 8, which will
guide you through the process of finding roots of a function using the secant
method, and hence solving differential equations with boundary value conditions,
both of which you will need to solve this problem. It is also recommended that
you read this entire script through before commencing the Session 8 task outlined
in section 3.1.
2 Boundary value problems: eigenvalue solutions
In this task we are going to be applying our new knowledge of boundary value
problems to the one-dimensional time-independent Schr dinger equation,