Computational Physics

by Giordano, Nicholas J.; Nakanishi, Hisao

• ISBN13: 9780131469907
• ISBN10: 0131469908
• Edition: 2nd
• Format: Hardcover
• Copyright: 2005-07-21
• Publisher: Pearson

Summary

Contains a wealth of topics to allow instructors flexibility in the choice of topics and depth of coverage: Examines projective motion with and without realistic air resistance. Discusses planetary motion and the three-body problem. Explores chaotic motion of the pendulum and waves on a string. Includes topics relating to fractal growth and stochastic systems. Offers examples on statistical physics and quantum mechanics. Contains ample explanations of the necessary algorithms students need to help them write original programs, and provides many example programs and calculations for reference.

Author Biography

Nicholas Giordano obtained his B.S. at Purdue University and his Ph.D. at Yale University. He has been on the faculty at Purdue since 1979, served as an Assistant Dean of Science from 2000-2003, and is currently the Hubert James Distinguished Professor of Physics. His research interests include electrical conduction, superconductivity, and magnetism in ultra-small metallic structures, along with musical acoustics and the physics of the piano. Ideas for this book grew out of the course on computational physics that he developed and taught in the early 1990s. Professor Giordano earned a Computational Science Education Award from the Department of Energy in 1997, and in 2004 was named Indiana Professor of the Year by the Carnegie Foundation for the Advancement of Teaching and the Council for the Advancement and Support of Education.

Hisao Nakanishi earned his B.S. from Brown University and his Ph.D. from Harvard University. His Ph.D. research concerned scaling and universality in a geometric phase transition called percolation and he has been interested in scale-invariance ever since. During his first postdoctoral work at Cornell he was introduced to the problem of surface critical phenomena such as wetting phase transitions, and later at the University of California, Santa Barbara, he started working on the statistics of diffusion and polymers in earnest. .In 1992 Professor Nakanishi was a part of the team that won a Gordon Bell Prize for the application of parallel computing to a problem in polymer statistics. More recently he has also put on another hat as a developer of a computer-based interactive exercise system which is used by a few thousand students at Purdue each year.

Table of Contents

Preface	ix
About the Authors	xii
A First Numerical Problem	1	(17)
Radioactive Decay	1	(1)
A Numerical Approach	2	(1)
Design and Construction of a Working Program: Codes and Pseudocodes	3	(8)
Testing Your Program	11	(1)
Numerical Considerations	12	(2)
Programming Guidelines and Philosophy	14	(4)
Realistic Projectile Motion	18	(30)
Bicycle Racing: The Effect of Air Resistance	18	(7)
Projectile Motion: The Trajectory of a Cannon Shell	25	(6)
Baseball: Motion of a Batted Ball	31	(5)
Throwing a Baseball: The Effects of Spin	36	(8)
Golf	44	(4)
Oscillatory Motion and Chaos	48	(46)
Simple Harmonic Motion	48	(6)
Making the Pendulum More Interesting: Adding Dissipation, Non-linearity, and a Driving Force	54	(4)
Chaos in the Driven Nonlinear Pendulum	58	(8)
Routes to Chaos: Period Doubling	66	(4)
The Logistic Map: Why the Period Doubles	70	(5)
The Lorenz Model	75	(7)
The Billiard Problem	82	(6)
Behavior in the Frequency Domain: Chaos and Noise	88	(6)
The Solar System	94	(35)
Kepler's Laws	94	(7)
The Inverse-Square Law and the Stability of Planetary Orbits	101	(6)
Precession of the Perihelion of Mercury	107	(6)
The Three-Body Problem and the Effect of Jupiter on Earth	113	(5)
Resonances in the Solar System: Kirkwood Gaps and Planetary Rings	118	(5)
Chaotic Tumbling of Hyperion	123	(6)
Potentials and Fields	129	(27)
Electric Potentials and Fields: Laplace's Equation	129	(14)
Potentials and Fields Near Electric Charges	143	(5)
Magnetic Field Produced by a Current	148	(3)
Magnetic Field of a Solenoid: Inside and Out	151	(5)
Waves	156	(25)
Waves: The Ideal Case	156	(9)
Frequency Spectrum of Waves on a String	165	(4)
Motion of a (Somewhat) Realistic String	169	(5)
Waves on a String (Again): Spectral Methods	174	(7)
Random Systems	181	(54)
Why Perform Simulations of Random Processes?	181	(2)
Random Walks	183	(5)
Self-Avoiding Walks	188	(7)
Random Walks and Diffusion	195	(6)
Diffusion, Entropy, and the Arrow of Time	201	(5)
Cluster Growth Models	206	(6)
Fractal Dimensionalities of Curves	212	(6)
Percolation	218	(11)
Diffusion on Fractals	229	(6)
Statistical Mechanics, Phase Transitions, and the Ising Model	235	(35)
The Ising Model and Statistical Mechanics	235	(4)
Mean Field Theory	239	(5)
The Monte Carlo Method	244	(2)
The Ising Model and Second-Order Phase Transitions	246	(13)
First-Order Phase Transitions	259	(5)
Scaling	264	(6)
Molecular Dynamics	270	(33)
Introduction to the Method: Properties of a Dilute Gas	270	(15)
The Melting Transition	285	(9)
Equipartition and the Fermi-Pasta-Ulam Problem	294	(9)
Quantum Mechanics	303	(54)
Time-Independent Schrodinger Equation: Some Preliminaries	303	(4)
One Dimension: Shooting and Matching Methods	307	(16)
A Matrix Approach	323	(3)
A Variational Approach	326	(7)
Time-Dependent Schrodinger Equation: Direct Solutions	333	(12)
Time-Dependent Schrodinger Equation in Two Dimensions	345	(4)
Spectral Methods	349	(8)
Vibrations, Waves, and the Physics of Musical Instruments	357	(32)
Plucking a String: Simulating a Guitar	357	(5)
Striking a String: Pianos and Hammers	362	(5)
Exciting a Vibrating System with Friction: Violins and Bows	367	(5)
Vibrations of a Membrane: Normal Modes and Eigenvalue Problems	372	(10)
Generation of Sound	382	(7)
Interdisciplinary Topics	389	(67)
Protein Folding	389	(16)
Earthquakes and Self-Organized Criticality	405	(13)
Neural Networks and the Brain	418	(18)
Real Neurons and Action Potentials	436	(9)
Cellular Automata	445	(11)
APPENDICES
A Ordinary Differential Equations with Initial Values	456	(13)
A.1 First-Order, Ordinary Differential Equations	456	(4)
A.2 Second-Order, Ordinary Differential Equations	460	(4)
A.3 Centered Difference Methods	464	(3)
A.4 Summary	467	(2)
B Root Finding and Optimization	469	(10)
B.1 Root Finding	469	(3)
B.2 Direct Optimization	472	(1)
B.3 Stochastic Optimization	473	(6)
C The Fourier Transform	479	(14)
C.1 Theoretical Background	479	(2)
C.2 Discrete Fourier Transform	481	(2)
C.3 Fast Fourier Transform (FFT)	483	(3)
C.4 Examples: Sampling Interval and Number of Data Points	486	(2)
C.5 Examples: Aliasing	488	(2)
C.6 Power Spectrum	490	(3)
D Fitting Data to a Function	493	(7)
D.1 Introduction	493	(1)
D.2 Method of Least Squares: Linear Regression for Two Variables	494	(3)
D.3 Method of Least Squares: More General Cases	497	(3)
E Numerical Integration	500	(12)
E.1 Motivation	500	(1)
E.2 Newton-Cotes Methods: Using Discrete Panels to Approximate an Integral	500	(4)
E.3 Gaussian Quadrature: Beyond Classic Methods of Numerical Integration	504	(2)
E.4 Monte Carlo Integration	506	(6)
F Generation of Random Numbers	512	(8)
F.1 Linear Congruential Generators	512	(4)
F.2 Nonuniform Random Numbers	516	(4)
G Statistical Tests of Hypotheses	520	(7)
G.1 Central Limit Theorem and the Χ2 Distribution	521	(2)
G.2 Χ2 Test of a Hypothesis	523	(4)
H Solving Linear Systems	527	(14)
H.1 Solving A x = b, b, b ≠ 0	528	(7)
H.1.1 Gaussian Elimination	528	(2)
H.1.2 Gauss-Jordan elimination	530	(1)
H.1.3 LU decomposition	531	(2)
H.1.4 Relaxational method	533	(2)
H.2 Eigenvalues and Eigenfunctions	535	(6)
H.2.1 Approximate Solution of Eigensystems	537	(4)
Index	541