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

<p><i><b>Rubin H. Landau, PhD, </b> is Professor Emeritus in the Department of Physics at Oregon State University, Corvallis, Oregon, USA. In his long and distinguished research career he has been instrumental in the development of computational physics as a defined subject, and founded both the Computational Physics Degree Program and the Northwest Alliance for Computational Science and Engineering.</i> <p><i><b>Manuel J. Páez, PhD, </b> is a Professor in the Department of Physics at the University of Antioquia in Medellin, Colombia. He teaches courses in both physics and programming, and he and Professor Landau have collaborated on pathbreaking computational physics investigations.</i> <p><i><b>Cristian C. Bordeianu, PhD, </b>taught Physics and Computer Science at the Military College “Stefan cel Mare,” Campulung Moldovenesc, Romania.</i>

<p> <b>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</b> <p>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. <p><i>Computational Physics </i>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. <p>Readers of the fourth edition of <i>Computational Physics </i>will also find: <ul><li>An exceptionally broad range of topics, from simple matrix manipulations to intricate computations in nonlinear dynamics</li><li>A whole suite of supplementary material: Python programs, Jupyter notebooks and videos</li></ul> <p><i>Computational Physics </i>is ideal for students in physics, engineering, materials science, and any subjects drawing on applied physics.

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