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Tables of Contents for Computational Methods in Physics, Chemistry and Mathematical Biology
Chapter/Section Title
Page #
Page Count
Preface
xi
Acknowledgements
xiii
About the author
xv
About the book
xvii
Introduction
xix
Numerical Solutions to Schrodinger's Equation
1
44
Particles and Classical Mechanics
1
5
Waves and Quantum Mechanics
6
5
Analytical Solutions to Schrodinger's Equation
11
5
The Shooting Method
16
3
Initial Conditions
19
1
Computational Implementation
20
2
Application to Parabolic Potentials
22
3
Improved Energy Eigenvalues
25
3
Further Convergence Tests: Pushing Accuracy to its Limits
28
3
Deduction of the Wave Functions
31
2
Three-Dimensional Potentials
33
1
Application to the Hydrogen Atom
34
11
Approximate Methods
45
26
Small Changes
45
1
Non-Degenerate Perturbation Theory
46
2
Computing the First-Order Correction
48
4
Computing the Second-Order Correction
52
3
Two-Dimensional Perturbing Potentials
55
3
The Variational Method
58
2
Application to the Hydrogen Atom
60
4
Application to Biased Quantum Wells
64
7
Matrix Methods
71
20
Basis sets
71
3
Expansion
74
2
Formation of the Matrix Equation
76
1
Solution of the Matrix Equation
77
1
Computational Implementation for Quantum Wells
78
4
Convergence Tests: Application to Parabolic Potentials
82
2
Solution of Schrodinger's Equation in Several Dimensions
84
2
Degenerate Perturbation Theory
86
5
Deterministic Simulations
91
24
Complex Time Dependent Systems
91
1
Classical Mechanics and Space Rockets
92
4
Simulation of Launch Trajectory
96
2
Verification of Computational Parameters
98
2
Fun with Simulations
100
1
Introduction to Diffusion
101
3
Theory
104
1
Boundary Conditions
105
1
Numerical Implementation
105
2
Convergence Tests
107
1
Constant Diffusion Coefficients and Universality
108
1
Non-linear Diffusion
109
6
Stochastic Simulations
115
16
Stochastic or Monte Carlo Simulations
115
1
Random Numbers
116
1
How Random is Random?
117
1
Monte Carlo Simulations of Electron Scattering
118
3
Tests and Limits
121
2
Simulations of Three-Level Systems
123
2
Changing Probabilities and Pauli Exclusion
125
6
Percolation Theory
131
18
Complex Many-Body Interacting Systems
131
1
Disease Propagation
132
1
Computational Implementation
133
4
The Dependence of Infection upon Immunisation
137
3
The Limit of Large Populations and Critical Densities
140
2
Mixed Magnetic Phases
142
7
Evolutionary Methods
149
20
Genetic Algorithms
149
1
The Fitness Function
150
1
Random (Monte Carlo) Generation of Genomes
150
6
Reproduction and Natural Selection
156
4
Implementation of a Genetic Algorithm
160
4
Mutation and Conclusion
164
5
Molecular Dynamics
169
20
Modelling Molecules and Crystals
169
1
Atom-Atom Interactions and Interaction Potentials
170
3
Simulations of Atomic Ensembles
173
2
Computational Implementation
175
4
Initial Simulations and Convergence Tests
179
1
Larger Ensembles and Reproducing Nature
180
2
Non-spherical Interaction Potentials
182
1
Organic Molecules
183
6
Appendix A: Fortran implementation of the shooting method
189
2
Appendix B: Δ2 in spherical polar coordinates
191
2
Appendix C: A comment on the computer sourcecodes
193
2
Appendix D: Note for tutors
195
2
References
197
2
Index
199