计算物理学 第2版 英文版【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

计算物理学 第2版 英文版PDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

Part Ⅰ Numerical Methods3

1 Error Analysis3

1.1 Machine Numbers and Rounding Errors3

1.2 Numerical Errors of Elementary Floating Point Operations6

1.2.1 Numerical Extinction7

1.2.2 Addition8

1.2.3 Multiplication9

1.3 Error Propagation9

1.4 Stability of Iterative Algorithms11

1.5 Example：Rotation12

1.6 Truncation Error13

1.7 Problems14

2 Interpolation15

2.1 Interpolating Functions15

2.2 Polynomial Interpolation16

2.2.1 Lagrange Polynomials17

2.2.2 Barycentric Lagrange Interpolation17

2.2.3 Newton's Divided Differences18

2.2.4 Neville Method20

2.2.5 Error of Polynomial Interpolation21

2.3 Spline Interpolation22

2.4 Rational Interpolation25

2.4.1 PadéApproximant25

2.4.2 Barycentric Rational Interpolation27

2.5 Multivariate Interpolation32

2.6 Problems33

3 Numerical Differentiation37

3.1 One-Sided Difference Quotient37

3.2 Central Difference Quotient38

3.3 Extrapolation Methods39

3.4 Higher Derivatives41

3.5 Partial Derivatives of Multivariate Functions42

3.6 Problems43

4 Numerical Integration45

4.1 Equidistant Sample Points46

4.1.1 Closed Newton-Cotes Formulae46

4.1.2 Open Newton-Cotes Formulae48

4.1.3 Composite Newton-Cotes Rules48

4.1.4 Extrapolation Method（Romberg Integration）49

4.2 Optimized Sample Points50

4.2.1 Clenshaw-Curtis Expressions50

4.2.2 Gaussian Integration52

4.3 Problems56

5 Systems of Inhomogeneous Linear Equations59

5.1 Gaussian Elimination Method60

5.1.1 Pivoting63

5.1.2 Direct LU Decomposition63

5.2 QR Decomposition64

5.2.1 QR Decomposition by Orthogonalization64

5.2.2 QR Decomposition by Householder Reflections66

5.3 Linear Equations with Tridiagonal Matrix69

5.4 Cyclic Tridiagonal Systems71

5.5 Iterative Solution of Inhomogeneous Linear Equations73

5.5.1 General Relaxation Method73

5.5.2 Jacobi Method73

5.5.3 Gauss-Seidel Method74

5.5.4 Damping and Successive Over-Relaxation75

5.6 Conjugate Gradients76

5.7 Matrix Inversion77

5.8 Problems78

6 Roots and Extremal Points83

6.1 Root Finding83

6.1.1 Bisection84

6.1.2 Regula Falsi（False Position）Method85

6.1.3 Newton-Raphson Method85

6.1.4 Secant Method86

6.1.5 Interpolation87

6.1.6 Inverse Interpolation88

6.1.7 Combined Methods91

6.1.8 Multidimensional Root Finding97

6.1.9 Quasi-Newton Methods98

6.2 Function Minimization99

6.2.1 The Ternary Search Method99

6.2.2 The Golden Section Search Method（Brent's Method）101

6.2.3 Minimization in Multidimensions106

6.2.4 Steepest Descent Method106

6.2.5 Conjugate Gradient Method107

6.2.6 Newton-Raphson Method107

6.2.7 Quasi-Newton Methods108

6.3 Problems110

7 Fourier Transformation113

7.1 Fourier Integral and Fourier Series113

7.2 Discrete Fourier Transformation114

7.2.1 Trigonometric Interpolation116

7.2.2 Real Valued Functions118

7.2.3 Approximate Continuous Fourier Transformation119

7.3 Fourier Transform Algorithms120

7.3.1 Goertzel's Algorithm120

7.3.2 Fast Fourier Transformation121

7.4 Problems125

8 Random Numbers and Monte Carlo Methods127

8.1 Some Basic Statistics127

8.1.1 Probability Density and Cumulative Probability Distribution127

8.1.2 Histogram128

8.1.3 Expectation Values and Moments129

8.1.4 Example：Fair Die130

8.1.5 Normal Distribution131

8.1.6 Multivariate Distributions132

8.1.7 Central Limit Theorem133

8.1.8 Example：Binomial Distribution133

8.1.9 Average of Repeated Measurements134

8.2 Random Numbers135

8.2.1 Linear Congruent Mapping135

8.2.2 Marsaglia-Zamann Method135

8.2.3 Random Numbers with Given Distribution136

8.2.4 Examples136

8.3 Monte Carlo Integration138

8.3.1 Numerical Calculation ofπ138

8.3.2 Calculation of an Integral139

8.3.3 More General Random Numbers140

8.4 Monte Carlo Method for Thermodynamic Averages141

8.4.1 Simple Sampling141

8.4.2 Importance Sampling142

8.4.3 Metropolis Algorithm142

8.5 Problems144

9 Eigenvalue Problems147

9.1 Direct Solution148

9.2 Jacobi Method148

9.3 Tridiagonal Matrices150

9.3.1 Characteristic Polynomial of a Tridiagonal Matrix151

9.3.2 Special Tridiagonal Matrices151

9.3.3 The QL Algorithm156

9.4 Reduction to a Tridiagonal Matrix157

9.5 Large Matrices159

9.6 Problems160

10 Data Fitting161

10.1 Least Square Fit162

10.1.1 Linear Least Square Fit163

10.1.2 Linear Least Square Fit with Orthogonalization165

10.2 Singular Value Decomposition167

10.2.1 Full Singular Value Decomposition168

10.2.2 Reduced Singular Value Decomposition168

10.2.3 Low Rank Matrix Approximation170

10.2.4 Linear Least Square Fit with Singular Value Decomposition172

10.3 Problems175

11 Discretization of Differential Equations177

11.1 Classification of Differential Equations178

11.1.1 Linear Second Order PDE178

11.1.2 Conservation Laws179

11.2 Finite Differences180

11.2.1 Finite Differences in Time181

11.2.2 Stability Analysis182

11.2.3 Method of Lines183

11.2.4 Eigenvector Expansion183

11.3 Finite Volumes185

11.3.1 Discretization of fluxes188

11.4 Weighted Residual Based Methods190

11.4.1 Point Collocation Method191

11.4.2 Sub-domain Method191

11.4.3 Least Squares Method192

11.4.4 Galerkin Method192

11.5 Spectral and Pseudo-spectral Methods193

11.5.1 Fourier Pseudo-spectral Methods193

11.5.2 Example：Polynomial Approximation194

11.6 Finite Elements196

11.6.1 One-Dimensional Elements196

11.6.2 Two-and Three-Dimensional Elements197

11.6.3 One-Dimensional Galerkin FEM201

11.7 Boundary Element Method204

12 Equations of Motion207

12.1 The State Vector208

12.2 Time Evolution of the State Vector209

12.3 Explicit Forward Euler Method210

12.4 Implicit Backward Euler Method212

12.5 Improved Euler Methods213

12.6 Taylor Series Methods215

12.6.1 Nordsieck Predictor-Corrector Method215

12.6.2 Gear Predictor-Corrector Methods217

12.7 Runge-Kutta Methods217

12.7.1 Second Order Runge-Kutta Method218

12.7.2 Third Order Runge-Kutta Method218

12.7.3 Fourth Order Runge-Kutta Method219

12.8 Quality Control and Adaptive Step Size Control220

12.9 Extrapolation Methods221

12.10 Linear Multistep Methods222

12.10.1 Adams-Bashforth Methods222

12.10.2 Adams-Moulton Methods223

12.10.3 Backward Differentiation（Gear）Methods223

12.10.4 Predictor-Corrector Methods224

12.11 Verlet Methods225

12.11.1 Liouville Equation225

12.11.2 Split-Operator Approximation226

12.11.3 Position Verlet Method227

12.11.4 Velocity Verlet Method227

12.11.5 St？rmer-Verlet Method228

12.11.6 Error Accumulation for the St？rmer-Verlet Method229

12.11.7 Beeman's Method230

12.11.8 The Leapfrog Method231

12.12 Problems232

Part Ⅱ Simulation of Classical and Quantum Systems239

13 Rotational Motion239

13.1 Transformation to a Body Fixed Coordinate System239

13.2 Properties of the Rotation Matrix240

13.3 Properties of W,Connection with the Vector of Angular Velocity242

13.4 Transformation Properties of the Angular Velocity244

13.5 Momentum and Angular Momentum246

13.6 Equations of Motion of a Rigid Body246

13.7 Moments of Inertia247

13.8 Equations of Motion for a Rotor248

13.9 Explicit Methods248

13.10 Loss of Orthogonality250

13.11 Implicit Method251

13.12 Kinetic Energy of a Rotor255

13.13 Parametrization by Euler Angles255

13.14 Cayley-Klein Parameters,Quaternions,Euler Parameters256

13.15 Solving the Equations of Motion with Quaternions259

13.16 Problems260

14 Molecular Mechanics263

14.1 Atomic Coordinates264

14.2 Force Fields266

14.2.1 Intramolecular Forces267

14.2.2 Intermolecular Interactions269

14.3 Gradients270

14.4 Normal Mode Analysis274

14.4.1 Harmonic Approximation274

14.5 Problems276

15 Thermodynamic Systems279

15.1 Simulation of a Lennard-Jones Fluid279

15.1.1 Integration of the Equations of Motion280

15.1.2 Boundary Conditions and Average Pressure281

15.1.3 Initial Conditions and Average Temperature281

15.1.4 Analysis of the Results282

15.2 Monte Carlo Simulation287

15.2.1 One-Dimensional Ising Model287

15.2.2 Two-Dimensional Ising Model289

15.3 Problems290

16 Random Walk and Brownian Motion293

16.1 Markovian Discrete Time Models293

16.2 Random Walk in One Dimension294

16.2.1 Random Walk with Constant Step Size295

16.3 The Freely Jointed Chain296

16.3.1 Basic Statistic Properties297

16.3.2 Gyration Tensor299

16.3.3 Hookean Spring Model300

16.4 Langevin Dynamics301

16.5 Problems303

17 Electrostatics305

17.1 Poisson Equation305

17.1.1 Homogeneous Dielectric Medium306

17.1.2 Numerical Methods for the Poisson Equation307

17.1.3 Charged Sphere309

17.1.4 Variable ε311

17.1.5 Discontinuous ε313

17.1.6 Solvation Energy of a Charged Sphere314

17.1.7 The Shifted Grid Method314

17.2 Poisson-Boltzmann Equation315

17.2.1 Linearization of the Poisson-Boltzmann Equation317

17.2.2 Discretization of the Linearized Poisson-Boltzmann Equation318

17.3 Boundary Element Method for the Poisson Equation318

17.3.1 Integral Equations for the Potential318

17.3.2 Calculation of the Boundary Potential321

17.4 Boundary Element Method for the Linearized Poisson-Boltzmann Equation324

17.5 Electrostatic Interaction Energy（Onsager Model）325

17.5.1 Example：Point Charge in a Spherical Cavity326

17.6 Problems327

18 Waves329

18.1 Classical Waves329

18.2 Spatial Discretization in One Dimension332

18.3 Solution by an Eigenvector Expansion334

18.4 Discretization of Space and Time337

18.5 Numerical Integration with a Two-Step Method338

18.6 Reduction to a First Order Differential Equation340

18.7 Two-Variable Method343

18.7.1 Leapfrog Scheme343

18.7.2 Lax-Wendroff Scheme345

18.7.3 Crank-Nicolson Scheme347

18.8 Problems349

19 Diffusion351

19.1 Particle Flux and Concentration Changes351

19.2 Diffusion in One Dimension353

19.2.1 Explicit Euler（Forward Time Centered Space）Scheme353

19.2.2 Implicit Euler（Backward Time Centered Space）Scheme355

19.2.3 Crank-Nicolson Method357

19.2.4 Error Order Analysis358

19.2.5 Finite Element Discretization360

19.3 Split-Operator Method for Multidimensions360

19.4 Problems362

20 Nonlinear Systems363

20.1 Iterated Functions364

20.1.1 Fixed Points and Stability364

20.1.2 The Lyapunov Exponent366

20.1.3 The Logistic Map367

20.1.4 Fixed Points of the Logistic Map367

20.1.5 Bifurcation Diagram369

20.2 Population Dynamics370

20.2.1 Equilibria and Stability370

20.2.2 The Continuous Logistic Model371

20.3 Lotka-Volterra Model372

20.3.1 Stability Analysis372

20.4 Functional Response373

20.4.1 Holling-Tanner Model375

20.5 Reaction-Diffusion Systems378

20.5.1 General Properties of Reaction-Diffusion Systems378

20.5.2 Chemical Reactions378

20.5.3 Diffusive Population Dynamics379

20.5.4 Stability Analysis379

20.5.5 Lotka-Volterra Model with Diffusion380

20.6 Problems382

21 Simple Quantum Systems385

21.1 Pure and Mixed Quantum States386

21.1.1 Wavefunctions387

21.1.2 Density Matrix for an Ensemble of Systems387

21.1.3 Time Evolution of the Density Matrix388

21.2 Wave Packet Morion in One Dimension389

21.2.1 Discretization of the Kinetic Energy390

21.2.2 Time Evolution392

21.2.3 Example：Free Wave Packet Morion402

21.3 Few-State Systems403

21.3.1 Two-State System405

21.3.2 Two-State System with Time Dependent Perturbation408

21.3.3 Superexchange Model410

21.3.4 Ladder Model for Exponential Decay412

21.3.5 Landau-Zener Model414

21.4 The Dissipative Two-State System416

21.4.1 Equations of Morion for a Two-State System416

21.4.2 The Vector Model417

21.4.3 The Spin-1/2 System418

21.4.4 Relaxation Processes—The Bloch Equations420

21.4.5 The Driven Two-State System421

21.4.6 Elementary Qubit Manipulation428

21.5 Problems430

Appendix Ⅰ Performing the Computer Experiments433

Appendix Ⅱ Methods and Algorithms435

References441

Index449