Preface xvii Acknowledgments xix Part I Basics 1 1 Introduction 3 1.1 Computational Physics and Science 3 1.2 This Book’s Subjects 4 1.3 Video Lecture Supplements 4 1.4 This Book’s Codes and Problems 5 1.5 Our Language: The Python Ecosystem 6 1.6 The Easy Way: Python Distributions 6 2 Software Basics 9 2.1 Making Computers Obey 9 2.2 Computer Number Representations 11 2.3 Python Mini Tutorial 18 2.4 Programming Warmup 25 2.5 Python’s Visualization Tools 30 2.6 Plotting Exercises 36 2.7 Code Listings 38 3 Errors and Uncertainties 44 3.1 Types of Errors 44 3.2 Experimental Error Investigation 49 3.3 Errors with Power Series 52 3.4 Errors in Bessel Functions 55 3.5 Code Listing 58 4 Monte Carlo Simulations 59 4.1 Random Numbers 59 4.2 Simulating a Random Walk 63 4.3 Spontaneous Decay 68 4.4 Testing and Generating Random Distributions 71 4.5 Code Listings 73 5 Differentiation and Integration 78 5.1 Differentiation Algorithms 78 5.2 Extrapolated Difference 80 5.3 Integration Algorithms 83 5.4 Gaussian Quadrature 89 5.5 Monte Carlo Integrations 91 5.6 Mean Value and N–D Integration 94 5.7 mc Variance Reduction 96 5.8 Importance Sampling and von Neumann Rejection 96 5.9 Code Listings 97 6 Trial-and-Error Searching and Data Fitting 100 6.1 Quantum Bound States I 100 6.2 Bisection Search 101 6.3 Newton–Raphson Search 102 6.4 Magnetization Search 105 6.5 Data Fitting 107 6.6 Fitting Exponential Decay 112 6.7 Least-Squares Fitting 113 6.8 Nonlinear Fit to a Resonance 118 6.9 Code Listings 120 7 Matrix Computing and N–D Searching 123 7.1 Masses on a String and N–D Searching 123 7.2 Matrix Generalities 126 7.3 Matrices in Python 129 7.4 Exercise: Tests Before Use 136 7.5 Solution to String Problem 139 7.6 Spin States and Hyperfine Structure 139 7.7 Speeding Up Matrix Computing ⊙ 141 7.8 Code Listing 144 8 Differential Equations and Nonlinear Oscillations 147 8.1 Nonlinear Oscillators 147 8.2 ODE Review 149 8.3 Dynamic Form of ODEs 150 8.4 ODE Algorithms 152 8.5 Solution for Nonlinear Oscillations 157 8.6 Extensions: Nonlinear Resonances, Beats, Friction 159 8.7 Code Listings 161 Part II Data Science 165 9 Fourier Analyses 167 9.1 Fourier Series 167 9.2 Fourier Transforms 170 9.3 Discrete Fourier Transforms 172 9.4 Noise Filtering 178 9.5 Fast Fourier Transform ⊙ 185 9.6 FFT Implementation 189 9.7 FFT Assessment 190 9.8 Code Listings 190 10 Wavelet and Principal Components Analysis 193 10.1 Part I: Wavelet Analysis 193 10.2 Wave Packets and Uncertainty Principle 195 10.3 Short-Time Fourier Transforms 197 10.4 Wavelet Transforms 198 10.5 Discrete Wavelet Transforms ⊙ 203 10.6 Part II: Principal Components Analysis 213 10.7 Code Listings 220 11 Neural Networks and Machine Learning 224 11.1 Part I: Biological and Artificial Neural Networks 225 11.2 A Simple Neural Network 226 11.3 A Graphical Deep Net 232 11.4 Part II: Machine Learning Software 234 11.5 TensorFlow and SkLearn Examples 235 11.6 ml Clustering 240 11.7 Keras: Python’s Deep Learning API 244 11.8 Image Processing with OpenCV 244 11.9 Explore ML Data Repositories 247 11.10 Code Listings 247 12 Quantum Computing (G. He, Coauthor) 254 12.1 Dirac Notation in Quantum Mechanics 254 12.2 From Bits to Qubits 255 12.3 Entangled and Separable States 257 12.4 Logic Gates 260 12.5 An Intro to QC Programming 264 12.6 Accessing the IBM Quantum Computer 270 12.7 Qiskit Plus IBM Quantum 272 12.8 The Quantum Fourier Transform 275 12.9 Oracle + Diffuser = Grover’s Search Algorithm 278 12.10 Shor’s Factoring ⊙ 281 12.11 Code Listings 284 Part III Applications 289 13 ODE Applications; Eigenvalues, Scattering, Trajectories 291 13.1 Quantum Eigenvalues for Arbitrary Potentials 291 13.2 Algorithm: ODE Solver + Search 293 13.3 Classical Chaotic Scattering 296 13.4 Projectile Motion with Drag 299 13.5 2- and 3-Body Planetary Orbits 301 13.6 Code Listings 303 14 Fractals and Statistical Growth Models 307 14.1 The Sierpiński Gasket 308 14.2 Growing Plants 310 14.3 Ballistic Deposition 312 14.4 Length of British Coastline 313 14.5 Correlated Growth 317 14.6 Diffusion-Limited Aggregation 318 14.7 Fractals in Bifurcations 320 14.8 Cellular Automata Fractals 320 14.9 Perlin Noise Adds Realism ⊙ 321 14.10 Code Listings 324 15 Nonlinear Population Dynamics 329 15.1 The Logistic Map, A Bug Population Model 329 15.2 Chaos 333 15.3 Bifurcation Diagrams 333 15.4 Measures of Chaos 336 15.5 Coupled Predator–Prey Models ⨀ 338 15.6 Code Listings 344 16 Nonlinear Dynamics of Continuous Systems 348 16.1 The Chaotic Pendulum 348 16.2 Phase Space 351 16.3 Chaotic Explorations 354 16.4 Other Chaotic Systems 358 16.5 Code Listings 364 17 Thermodynamics Simulations and Feynman Path Integrals 365 17.1 An Ising Magnetic Chain 365 17.2 Metropolis Algorithm 368 17.3 Fast Equilibration via Wang–Landau Sampling ⊙ 372 17.4 Path Integral Quantum Mechanics ⊙ 374 17.5 Lattice Path Integration 377 17.6 Implementation 381 17.7 Code Listings 385 18 Molecular Dynamics Simulations 391 18.1 MD Versus Thermodynamics 394 18.2 Initial, Boundary, and Large r Conditions 394 18.3 Verlet Algorithms 396 18.4 MD for 16 Particles 400 18.5 Code Listing 402 19 General Relativity 408 19.1 Einstein’s Field Equations 408 19.2 Gravitational Deflection of Light 412 19.3 Planetary Orbits in GR Gravity 414 19.4 Visualizing Wormholes 418 19.5 Problems 420 19.6 Code Listings 420 20 Integral Equations 425 20.1 Nonlocal Potential Binding 425 20.2 Momentum-Space Schrödinger Equation 425 20.3 Scattering in Momentum Space ⊙ 429 20.4 Code Listings 434 Part IV PDE Applications 437 21 PDE Review, Electrostatics and Relaxation 439 21.1 Review 439 21.2 Laplace’s Equation 441 21.3 Finite-Difference Algorithm 444 21.4 Alternate Capacitor Problems 447 21.5 Electric Field Visualization 449 21.6 Code Listings 450 22 Heat Flow and Leapfrogging 452 22.1 The Parabolic Heat Equation 452 22.2 Time Stepping (Leapfrog) Algorithm 454 22.3 Newton’s Radiative Cooling 457 22.4 The Crank–Nicolson Algorithm 458 22.5 Code Listings 462 23 String and Membrane Waves 464 23.1 A Vibrating String’s Hyperbolic Wave Equation 464 23.2 Time-Stepping Algorithm 466 23.3 von Neumann Stability Analysis 468 23.4 Beyond The Simple Wave Equation 469 23.5 Vibrating Membrane (2D Waves) 474 23.6 Analytical Solution 475 23.7 Numerical Solution 476 23.8 Code Listings 478 24 Quantum Wave Packets and EM Waves 480 24.1 Time-Dependent Schrödinger Equation 480 24.2 Split-Time Algorithm 482 24.3 Special Schrödinger Algorithm 484 24.4 Quantum Chaos 485 24.5 E&M Waves: Finite Difference Time Domain 488 24.6 Maxwell’s Equations 488 24.7 Split-Time FDTD 489 24.8 More E&M Problems 492 24.9 Code Listings 496 25 Shock and Soliton Waves 501 25.1 The Continuity and Advection Equations 502 25.2 Shock Waves via Burgers’ Equation 503 25.3 Including Dispersion 505 25.4 KdeV Solitons 506 25.5 Pendulum Chain Solitons 510 25.6 Continuum Limit, the Sine-Gordon Equation 512 25.7 Code Listings 516 26 Fluid Hydrodynamics 518 26.1 Navier–Stokes Equation 518 26.2 Flow Through Parallel Plates 520 26.3 Navier–Stokes Difference Equation 522 26.4 Vorticity Form of Navier–Stokes Equation 523 26.5 Assessment and Exploration 527 26.6 Code Lisitings 529 27 Finite Element Electrostatics ⊙ 531 27.1 The Potential of Two Metal Plates 531 27.2 Finite Element Method 532 27.3 1D FEM Problems 536 27.4 2D FEM Exercises 537 27.5 Code Listings 539 Appendix Codes and Animations 543 References 546 Index 555

The classic in the field for more than 25 years, now with increased emphasis on data science and new chapters on quantum computing, machine learning (AI), and general relativity Computational physics combines physics, applied mathematics, and computer science in a cutting-edge multidisciplinary approach to solving realistic physical problems. It has become integral to modern physics research because of its capacity to bridge the gap between mathematical theory and real-world system behavior. Computational Physics provides the reader with the essential knowledge to understand computational tools and mathematical methods well enough to be successful. Its philosophy is rooted in “learning by doing”, assisted by many sample programs in the popular Python programming language. The first third of the book lays the fundamentals of scientific computing, including programming basics, stable algorithms for differentiation and integration, and matrix computing. The latter two-thirds of the textbook cover more advanced topics such linear and nonlinear differential equations, chaos and fractals, Fourier analysis, nonlinear dynamics, and finite difference and finite elements methods. A particular focus in on the applications of these methods for solving realistic physical problems. Readers of the fourth edition of Computational Physics will also find: An exceptionally broad range of topics, from simple matrix manipulations to intricate computations in nonlinear dynamicsA whole suite of supplementary material: Python programs, Jupyter notebooks and videos Computational Physics is ideal for students in physics, engineering, materials science, and any subjects drawing on applied physics.