Computational Physics

Bookmark is added through MasterPDF. A complete introduction to the field of computational physics, with examples and exercises in the Python programming language. Computers play a central role in virtually every major physics discovery today, from astrophysics and particle physics to biophysics and condensed matter. This book explains the fundamentals of computational physics and describes in simple terms the techniques that every physicist should know, such as finite difference methods, numerical quadrature, and the fast Fourier transform. The book offers a complete introduction to the topic at the undergraduate level, and is also suitable for the advanced student or researcher who wants to learn the foundational elements of this important field. CONTENTS PREFACE 1 INTRODUCTION 2 PYTHON PROGRAMMING FOR PHYSICISTS 2.1 GETTING STARTED 2.2 BASIC PROGRAMMING 2.2.1 VARIABLES AND ASSIGNMENTS 2.2.2 VARIABLE TYPES 2.2.3 OUTPUT AND INPUT STATEMENTS 2.2.4 ARITHMETIC 2.2.5 FUNCTIONS, PACKAGES, AND MODULES 2.2.6 BUILT-IN FUNCTIONS 2.2.7 COMMENT STATEMENTS 2.3 CONTROLLING PROGRAMS WITH "IF" AND "WHILE" 2.3.1 THE IF STATEMENT 2.3.2 THE WHILE STATEMENT 2.3.3 BREAK AND CONTINUE 2.4 LISTS AND ARRAYS 2.4.l LISTS 2.4.2 ARRAYS 2.4.3 READING AN ARRAY FROM A FILE 2.4.4 ARITHMETIC WITH ARRAYS 2.4.5 SLICING 2.5 "FOR" LOOPS 2.6 USER-DEFINED FUNCTIONS 2.7 GOOD PROGRAMMING STYLE 3 GRAPHICS AND VISUALIZATION 3.1 GRAPHS 3.2 SCATTER PLOTS 3.3 DENSITY PLOTS 3.4 30 GRAPHICS 3.5 ANIMATION FURTHER EXERCISES 4 ACCURACY AND SPEED 4.1 VARIABLES AND RANGES 4.2 NUMERICAL ERROR 4.3 PROGRAM SPEED 5 INTEGRALS AND DERIVATIVES 5.1 FUNDAMENTAL METHODS FOR EVALUATING INTEGRALS 5.1.1 THE TRAPEZOIDAL RULE 5.1.2 SIMPSON'S RULE 5.2 ERRORS ON INTEGRALS 5.2.1 PRACTICAL ESTIMATION OF ERRORS 5.3 CHOOSING THE NUMBER OF STEPS 5.4 ROMBERG INTEGRATION 5.5 HIGHER-ORDER INTEGRATION METHODS 5.6 GAUSSIAN QUADRATURE 5.6.1 NONUNIFORM SAMPLE POINTS 5.6.2 SAMPLE POINTS FOR GAUSSIAN QUADRATURE 5.6.3 ERRORS ON GAUSSIAN QUADRATURE 5.7 CHOOSING AN INTEGRATION METHOD 5.8 INTEGRALS OVER INFINITE RANGES 5.9 MULTIPLE INTEGRALS 5.10 DERIVATIVES 5.10.1 FORWARD AND BACKWARD DIFFERENCES 5.10.2 ERRORS 5.10.3 CENTRAL DIFFERENCES 5.10.4 HIGHER-ORDER APPROXIMATIONS FOR DERIVATIVES 5.10.5 SECOND DERIVATIVES 5.10.6 PARTIAL DERIVATIVES 5.10.7 DERIVATIVES OF NOISY DATA 5.11 INTERPOLATION FURTHER EXERCISES 6 SOLUTION OF LINEAR AND NONLINEAR EQUATIONS 6.1 SIMULTANEOUS LINEAR EQUATIONS 6.1.1 GAUSSIAN ELIMINATION 6.1.2 BACKSUBSTITUTION 6.1.3 PIVOTING 6.1.4 LU DECOMPOSITION 6.1.5 CALCULATING THE INVERSE OF A MATRIX 6.1.6 TRIDIAGONAL AND BANDED MATRICES 6.2 EIGENVALUES AND EIGENVECTORS 6.3 NONLINEAR EQUATIONS 6.3.1 THE RELAXATION METHOD 6.3.2 RATE OF CONVERGENCE OF THE RELAXATION METHOD 6.3.3 RELAXATION METHOD FOR TWO OR MORE VARIABLES 6.3.4 BINARY SEARCH 6.3.5 NEWTON'S METHOD 6.3.6 THE SECANT METHOD 6.3.7 NEWTON'S METHOD FOR TWO OR MORE VARIABLES 6.4 MAXIMA AND MINIMA OF FUNCTIONS 6.4.1 GOLDEN RATIO SEARCH 6.4.2 THE GAUSS-NEWTON METHOD AND GRADIENT DESCENT 7 FOURIER TRANSFORMS 7.1 FOURIER SERIES 7.2 THE DISCRETE FOURIER TRANSFORM 7.2.1 POSITIONS OF THE SAMPLE POINTS 7.2.2 Two-DIMENSIONAL FOURIER TRANSFORMS 7.2.3 PHYSICAL INTERPRETATION OF THE FOURIER TRANSFORM 7.3 DISCRETE COSINE AND SINE TRANSFORMS 7.3.1 TECHNOLOGICAL APPLICATIONS OF COSINE TRANSFORMS 7.4 FAST FOURIER TRANSFORMS 7.4.1 FORMULAS FOR THE FFT 7.4.2 STANDARD FUNCTIONS FOR FAST FOURIER TRANSFORMS 7.4.3 FAST COSINE AND SINE TRANSFORMS FURTHER EXERCISES 8 ORDINARY DIFFERENTIAL EQUATIONS 8.1 FIRST-ORDER DIFFERENTIAL EQUATIONS WITH ONE VARIABLE 8.1.1 EULER'S METHOD 8.1.2 THE RUNGE-KUTTA METHOD 8.1.3 THE FOURTH-ORDER RUNGE-KUTTA METHOD 8.1.4 SOLUTIONS OVER INFINITE RANGES 8.2 DIFFERENTIAL EQUATIONS WITH MORE THAN ONE VARIABLE 8.3 SECOND-ORDER DIFFERENTIAL EQUATIONS 8.4 VARYING THE STEP SIZE 8.5 OTHER METHODS FOR DIFFERENTIAL EQUATIONS 8.5.1 THE LEAPFROG METHOD 8.5.2 TIME REVERSAL AND ENERGY CONSERVATION 8.5.3 THE VERLET METHOD 8.5.4 THE MODIFIED MIDPOINT METHOD 8.5.5 THE BULIRSCH-STOER METHOD 8.5.6 INTERVAL SIZE FOR THE BULIRSCH-STOER METHOD 8.6 BOUNDARY VALUE PROBLEMS 8.6.1 THE SHOOTING METHOD 8.6.2 THE RELAXATION METHOD 8.6.3 EIGENVALUE PROBLEMS FURTHER EXERCISES 9 PARTIAL DIFFERENTIAL EQUATIONS 9.1 BOUNDARY VALUE PROBLEMS AND THE RELAXATION METHOD 9.2 FASTER METHODS FOR BOUNDARY VALUE PROBLEMS 9.2.1 OVERRELAXATION 9.2.2 THE GAUSS-SEIDEL METHOD 9.3 INITIAL VALUE PROBLEMS 9.3.1 THE FTCS METHOD 9.3.2 NUMERICAL STABILITY 9.3.3 THE IMPLICIT AND CRANK-NICOLSON METHODS 9.3.4 SPECTRAL METHODS FURTHER EXERCISES 10 RANDOM PROCESSES AND MONTE CARLO METHODS 10.1 RANDOM NUMBERS 10.1.1 RANDOM NUMBER GENERATORS 10.1.2 RANDOM NUMBER SEEDS 10.1.2 RANDOM NUMBER SEEDS 10.1.3 RANDOM NUMBERS AND SECRET CODES 10.1.4 PROBABILITIES AND BIASED COINS 10.1.5 NONUNIFORM RANDOM NUMBERS 10.2 MONTE CARLO INTEGRATION 10.2.1 THE MEAN VALUE METHOD 10.2.2 INTEGRALS IN MANY DIMENSIONS 10.2.3 IMPORTANCE SAMPLING 10.3 MONTE CARLO SIMULATION 10.3.1 IMPORTANCE SAMPLING AND STATISTICAL MECHANICS 10.3.2 THE MARKOV CHAIN METHOD 10.4 SIMULATED ANNEALING FURTHER EXERCISES 11 USING WHAT YOU HAVE LEARNED APPENDIX A INSTALLING PYTHON APPENDIX B DIFFERENCES BETWEEN PYTHON VERSIONS APPENDIX C GAUSSIAN QUADRATURE APPENDIX D CONVERGENCE OF MARKOV CHAIN MONTE CARLO CALCULATIONS APPENDIX E USEFUL PROGRAMS E.1 GAUSSIAN QUADRATURE E.2 SOLUTION OF TRIDIAGONAL OR BANDED SYSTEMS OF EQUATIONS E.3 DISCRETE COSINE AND SINE TRANSFORMS E.4 COLOR SCHEMES INDEX INDEX This book explains the fundamentals of computational physics and describes the techniques that every physicist should know, such as finite difference methods, numerical quadrature, and the fast Fourier transform. The book offers a complete introduction to the topic at the undergraduate level, and is also suitable for the advanced student or researcher. The book begins with an introduction to Python, then moves on to a step-by-step description of the techniques of computational physics, with examples ranging from simple mechanics problems to complex calculations in quantum mechanics, electromagnetism, statistical mechanics, and more