vii
TABLE OF CONTENTS
CHAPTER
1
TITLE
PAGE
COVER
i
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xi
LIST OF SYMBOLS
xiii
LIST OF APPENDICES
xvi
INTRODUCTION
1
1.1
An Introduction to Quantum Mechanics
1
1.2
Schrödinger’s Equation as a Gateway to
1
Quantum Mechanical Studies
1.3
Problem Background
3
1.4
Statement of Problem
5
1.5
Simulation Using C Programming Language
5
1.6
Objectives of Study
6
1.7
Scope of Study
6
1.8
Significance of Study
6
1.9
Thesis Organization
7
vii
2
LITERATURE REVIEW
8
2.1
Discovery of Schrödinger Equation
8
2.2
Numerical Methods for Solving Schrödinger
9
Equation
2.3
Meshless Element Free Galerkin (MEFG)
11
Method
3
2.4
Fourier grid Hamiltonian (FGH) Method
12
2.5
Summary
18
MESHLESS ELEMENT FREE GALERKIN
19
METHOD
3.1
Introduction
19
3.2
Basis Set
19
3.2.1
Plane Wave Basis Set
20
3.2.2
Localized Basis Set – Meshless
22
Basis Set
3.3
Meshless Element Free Galerkin Method
23
(MEFG)
3.4
Moving Least Square (MLS) Approximation
24
3.5
Weight Function
29
3.6
Problem Formulation
32
3.7
Generalized Eigenvalue Problems
36
3.7.1
37
3.8
4
Solving a Standard Eigensystem
Summary
39
RESULT ANALYSIS AND DISCUSSION
40
4.1
Introduction
40
4.2
Problem Formulation for H 2 Molecule
40
4.3
Numerical Implementation
41
4.3.1
Convergence Testing
42
4.3.2
Problem Solving for H 2 Molecule
43
4.4
Results and Analysis
46
viii
4.5
5
Summary
52
CONCLUSION AND RECOMMENDATION
53
5.1
Conclusion for the Case Study Result
53
5.2
Overall Conclusion
53
5.3
Recommendation
54
REFERENCES
55
APPENDICES
59
x
LIST OF TABLES
TABLE NO.
4.1
TITLE
PAGE
Comparison of eigenvalues calculated using
the MEFG method, FGH method and exact
analytic formula for a Morse potential
representing H 2
47
xi
LIST OF FIGURES
FIGURE NO.
TITLE
3.1
Wavefronts of plane wave in 3D space
3.2
Overlapping domain of influence and local
node numbering at point x
3.3
21
23
The approximation function 𝑢ℎ (𝑥) and the
nodal parameters 𝑢𝑖 in the MLS
approximation
3.4
PAGE
25
MLS shape function in 1D space for the node
at x = 0 obtained using five nodes evenly
distributed in the support domain of [-1, 1].
Quadratic spline weight function is used. (a)
MLS shape function; (b) derivative of the
shape function
3.5
28
Weight functions and resulting MLS
approximations. (a) Constant weight function
and corresponding approximation for 𝑢ℎ (𝑥).
(b) Constant weight function with compact
support and corresponding approximation.
(c) Continuous weight function and
corresponding approximation
3.6
Influence domain and the cubic spline weight
function
4.1
30
31
Schematic flow of MEFG method for onedimensional time independent Schrödinger
equation
42
xi
4.2
Eigenvectors at 𝜐 = 0 determined by (a)
MEFG method, (b) FGH method, and (c)
analytical solution
4.3
48
Eigenvectors at 𝜐 = 5 determined by (a)
MEFG method, (b) FGH method, and (c)
analytical solution
4.4
49
Eigenvectors at 𝜐 = 15 determined by (a)
MEFG method, (b) FGH method, and (c)
analytical solution
51
xiii
LIST OF SYMBOLS
ℎ
-
Plank constant
ℏ
-
𝑚
-
Planck constant divide by 2𝜋
Ψ(𝐫, 𝑡)
-
Wavefunction dependent on time
𝐫
-
𝑡
-
Vector space in 𝐢, 𝐣 and 𝐤 direction
𝑉
-
Potential field or potential energy
𝜓
-
Wavefunction or eigenfunction
𝑇(𝑡)
-
𝑖, 𝑗, 𝑙
-
Function involve of time 𝑡
𝐸
-
Energy of the particle or
Mass of atom/molecule
Time
Integer index
eigenenergy/eigenvalue
�
ℋ
-
Hamiltonian operator
i
-
Complex number
𝑦, 𝑝𝑗 (𝑥), 𝑎𝑗 (𝑥̅ )
-
𝑥
-
Arbitrary function of 𝑥
𝐴1 , 𝐴2 , 𝑘𝑎 , 𝑁, 𝑐1
-
Arbitrary constants
𝑓
-
frequency
𝑐
-
Speed of wave
𝜆𝑒
-
Wave length
Δ𝑥
-
Spacing between two grid/nodal points
𝑥𝑖
-
Real space coordinate at node i
A
-
Arbitrary observable
𝐴̂
-
Arbitrary operator
-
Total number of grid/nodal points
L
-
Length of domain
𝑛
Real space coordinate
xiv
𝑥�
-
Coordinate operator
𝑘
-
Reciprocal space coordinate
-
Kinetic energy operator
𝜆𝑚𝑎𝑥
-
Maximum wave length
𝐼̂𝑥
-
Identity operator
𝛿𝑖𝑗
-
Knonecker delta function
𝑓𝑙
-
Discrete Fourier Transform
𝛿(𝑥 − 𝑥 ′ )
-
Dirac Delta function
�𝑖𝑗
𝐻
-
Hamiltonian operator in discrete form
𝐻𝑖𝑗0
-
Renormalized Hamiltonian matrix
𝑤
-
Circular frequency
𝑘𝑤𝑎𝑣𝑒
-
Wave number
𝐴𝑛𝑜𝑟𝑚𝑎𝑙
-
Normalizing constant
𝑢(𝑥)
-
Approximated wave function
𝑢ℎ (𝑥)
-
MLS approximants
Ω
-
Domain
𝐏(𝑥)
-
Complete polynomial of order
q
-
Number of terms in the basis
𝐚(𝑥)
-
Unknown coordinate-dependent coefficients
𝑢𝑖
-
Nodal parameters
𝐽
-
𝑤(𝑥 − 𝑥𝑖 )
-
Weighted discrete 𝐿2 norm
𝐀(𝑥), 𝐁(𝑥), 𝐊, 𝐕, 𝑽̇(𝑥),
-
Arbitrary matrix and vectors
𝐮
-
Φ𝑖 (𝑥)
𝑇�
𝐌, 𝐇, 𝐒, 𝐓, 𝐱 , 𝐑, 𝐂, 𝐲, 𝐘
Weight function
𝚽(𝑥)
-
Vector of a set of 𝑢𝐼
-
Shape functions
𝑑𝑖
-
Distance between a nodal point at its
𝑑𝑚𝑖
-
Domain of influence size of the Ith node
-
Scaling parameter
𝑐𝑖
-
Maximum distance to the nearest neighbor
𝑑𝑚𝑎𝑥
Vector of a set of shape functions
neighborhood
xv
𝑟
-
Normalized radius
𝐋
-
Lower triangular matrix
𝐈
-
Identity matrix
𝜆
-
Eigenvalue
𝑅𝑚𝑖𝑛
-
Lower bound of a domain
𝑅𝑚𝑎𝑥
-
Upper bound of a domain
𝐷𝑒
-
Well depth
𝛽, 𝛾
-
Constant
𝑥𝑒
-
Equilibrium internuclear distance (bond
𝜈𝑒
-
vibrational constant
𝜇
-
Reduced mass
𝑚1
-
Mass for first atom
𝑚2
-
Mass for second atom
𝐇𝟏 , 𝐌𝟏 , 𝐊 𝟏 , 𝐕𝟏 , 𝐮𝟏
-
Reduced matrices and vector
𝜐
-
Quantum number
length)
xvi
LIST OF APPENDICES
APPENDIX
TITLE
PAGE
A1
Atomic units
59
A2
Atomic Rydberg units
61
B
Morse potential
62