Contents

1 Introduction

1.1 Computational modeling and visualization

1.2 The science and art of numerics

1.3 Fundamentals of programming and visualization

1.4 Exercises and Projects

1.A Floating point representation

1.B Python installation

1.C The Matplotlib plot function

1.D Basic NumPy array operations

2 Free fall and ordinary differential equations

2.1 Free fall with Euler's method

2.2 The Runge-Kutta methods

2.3 System of first order ODEs

2.4 The leapfrog method

2.5 Exercises and Projects

2.A Area preservation of the leapfrog method

2.B Program listings and descriptions

3 Realistic projectile motion with air resistance

3.1 Visualization of ideal projectile motion

3.2 Modeling air resistance

3.3 Linear air resistance

3.4 The Lambert W function

3.5 Quadratic air resistance and spin

3.6 Physics of ball sports

3.7 Shooting methods

3.8 Exercises and Projects

3.A Bisection and Newton's root finders

3.B Program listings and descriptions

4 Planetary motion and few-body problems

4.1 Motion of a planet

4.2 Properties of planetary motion

4.3 Precession of Mercury

4.4 Star wobbles and exoplanets

4.5 Planar three-body problems

4.6 The restricted three-body problem

4.7 Exercises and Projects

4.A Rotating frames and rate of change of vectors

4.B Rotation matrices

4.C Radial velocity transformation

4.D Program listings and descriptions

5 Nonlinear dynamics and chaos

5.1 A first model: the logistic map

5.2 Chaos

5.3 A nonlinear driven oscillator

5.4 The Lorenz flow

5.5 Power spectrum and Fourier transform

5.6 Fractals

5.7 Exercises and Projects

5.A Program listings and descriptions

6 Oscillations and waves

6.1 A damped harmonic oscillator

6.2 Vibrations of triatomic molecules

6.3 Displacement of a string under a load

6.4 Point source and finite element method

6.5 Waves on a string

6.6 Standing waves

6.7 Waves on a membrane

6.8 A falling tablecloth toward equilibrium

6.9 Exercises and Projects

6.A Program listings and descriptions

7 Electromagnetic fields

7.1 The game of electric field hockey

7.2 Electric potentials and fields

7.3 Laplace equation and finite element method

7.4 Boundary value problems with FEM

7.5 Meshfree methods for potentials and fields

7.6 Visualization of electromagnetic fields

7.7 Exercises and Projects

7.A Program listings and descriptions

8 Time-dependent quantum mechanics

8.1 Time-dependent Schrödinger equation

8.2 Direct simulation

8.3 Free fall, the quantum way

8.4 Two-state systems and Rabi flopping

8.5 Quantum waves in 2D

8.6 Exercises and Projects

8.A Numerical integration

8.B Program listings and descriptions

9 Time-independent quantum mechanics

9.1 Bound states by shooting methods

9.2 Periodic potentials and energy bands

9.3 Eigenenergies by FDM and FEM methods

9.4 Basis expansion method

9.5 Central field potentials

9.6 Quantum dot

9.7 Exercises and Projects

9.A Numerov's method

9.B The linear potential and Airy function

9.C Program listings and descriptions

10 Simple random problems

10.1 Random numbers and radioactive decay

10.2 Random walk

10.3 Brownian motion

10.4 Potential energy by Monte Carlo integration

10.5 Exercises and Projects

10.A Statistical theory of Brownian motion

10.B Nonuniform distributions

10.C Program listings and descriptions

11 Thermal systems

11.1 Thermodynamics of equilibrium

11.2 The Ising model

11.3 Thermal relaxation by simulated annealing

11.4 Molecular dynamics

11.5 Exercises and Projects

11.A Boltzmann factor and entropy

11.B Exact solutions of the 2D Ising model

11.C Program listings and descriptions

12 Classical and quantum scattering

12.1 Scattering and cross sections

12.2 Rainbow and glory scattering

12.3 Quantum scattering amplitude

12.4 Partial waves

12.5 Exercises and Projects

12.A Derivation of the deflection function

12.B Partial wave analysis

12.C Program listings and descriptions

List of programs

Bibliography

Index