Solving Schrodinger Equation using Crank-Nicolson Method

Solving Schrodinger Equation using Crank-Nicolson Method 1
THE NUMERICAL SOLUTION FOR THE SCHRODINGER EQUATION USING CRANK-
NICOLSON METHOD
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Solving Schrodinger Equation using Crank-Nicolson Method 2
Table of Contents
1. CHAPTER 1: INTRODUCTION.....................................................................................................3
1.1. Research Background................................................................................................................3
1.2. Problem Statement....................................................................................................................5
1.3. Objectives of the Research........................................................................................................5
1.4. Significance of the Research......................................................................................................6
1.5. Research Questions....................................................................................................................7
1.6. Scope of the Study......................................................................................................................7
2. CHAPTER 2: LITERATURE REVIEW.........................................................................................8
2.1. Description of Schrodinger equation........................................................................................8
2.2. Description of finite difference methods..................................................................................9
2.3. Selected method.......................................................................................................................11
3. CHAPTER 3: METHODOLOGY..................................................................................................11
4. CHATER 4: EXPECTED RESULTS............................................................................................17
References................................................................................................................................................18

Solving Schrodinger Equation using Crank-Nicolson Method 3
1. CHAPTER 1: INTRODUCTION
Partial different equations are very common in mathematics, physics and chemistry fields. These
equations are used to model and describe a wide range of physical systems or chemical reactions,
which depict real world problems. One of the most prevalent and widely used partial differential
equations in the abovementioned fields is Schrodinger equation. Solving this equation gives a
wave function that makes it easier to examine the behavior or performance of a physical system
or chemical reaction. There are numerous methods that are used to solve Schrodinger equation.
This research proposal focuses on one of these methods – Crank-Nicolson method. Therefore,
the main purpose of this research proposal is to determine a numerical solution for the
Schrodinger equation using Crank-Nicolson method.
1.1. Research Background
This research proposal is about finding a numerical solution for the Schrodinger equation using
Crank-Nicolson method. Schrodinger equation is one of the essential mathematical tools used in
mathematics, physics and chemistry fields (Nakatsuji, 2012). The equation is a second order
differential equation or a partial differential equation. When it is solved, the solution gives a
wave function with information about the behavior of a particle in time and space. In other
words, Schrodinger equation is used to describe the behavior of physical systems (Okock &
Burns, 2015). Understanding the behavior of such systems is very important in different aspects
of life. This knowledge can be used to design systems that are safe and prone to events that are
likely to happen in the future and affect the system. For example, the knowledge can be used by
engineers to find the best designs and materials for constructing pipelines to be used in areas
with high temperature variations. In such a case, Schrodinger equation can be used to simulate
the behavior of the pipelines when exposed to different changes and establish appropriate

Solving Schrodinger Equation using Crank-Nicolson Method 4
measures that should be put into place to enhance the functionality, safety and durability of the
pipelines.
Several experimental studies conducted by different researchers suggested that atomic
particles exhibited wave-like properties. As a result of this, it was concluded that the behavior of
atomic particles could be explained using a wave equation, which is a Schrodinger equation. The
first person to write such an equation was Schrodinger. This wave equation became a subject of
discussion for many years and it was found that its eigenvalues were equal to the quantum
mechanical system’s energy levels. The eigenvalues were formulated from Fourier, which
expresses a mathematical function as the sum of infinite sequence of periodic functions
(Knyazev & Shcherbakova, 2017). After much discussion about the wave equation, it also
became accepted for use in probability distribution. The Schrodinger equation started being used
to determine acceptable energy levels of quantum mechanical systems and its wave function was
used to determine the probability of an atomic particle at a particular position and at a certain
time.
The biggest problem of Schrodinger equation was to find its correct solution (the wave
function) that had the right amplitudes such that when they were summed by superposition, they
gave the correct or anticipated solution. For many years, researchers struggled to develop
methods of solving Schrodinger equation (Popelier, 2011). In an attempt to simplify the problem,
the system’s wave function, which was the solution to the Schrodinger equation, was replaced by
an infinite series of wave functions for individual series. Schrodinger discovered that the
individual wave functions described the states of individual quantum systems and the amplitudes
of these wave functions provided very useful information about the state of the entire quantum