New Book - MATHEMATICA FOR PHYSICS

*To*: mathgroup at christensen.cybernetics.net*Subject*: [mg553] New Book - MATHEMATICA FOR PHYSICS*From*: steve at christensen.cybernetics.net (Steve Christensen)*Date*: Wed, 15 Mar 1995 16:02:57 -0600

Dear MathGroup: I would like to call your attention to the following new Mathematica book that was recently published, MATHEMATICA FOR PHYSICS. This book is intended for the advanced undergraduate and graduate physics student taking core courses in the physics curriculum. This text was developed with many years of student input. Although this book addresses physics problems, the introductory chapters are relevant for most Mathematica users. (In particular, many queries answered on the MathGroup are addressed in the first chapter.) The source code (without annotations) to selected problems is available on the MathSource server (item 0206-862 or http://mathsource.wri.com/ and search for Zimmerman) so you can "test drive" the problems. I have attached a description of the text, and the table of contents. Thank you, Fredrick I. Olness: olness at mail.physics.smu.edu ================================================ MATHEMATICA FOR PHYSICS ================================================ WHAT: A new book for doing Physics with Mathematica TITLE: MATHEMATICA FOR PHYSICS AUTHORS: Robert L. Zimmerman (U. of Oregon) Fredrick I. Olness (Southern Methodist University) with a foreword by Stephen Wolfram LENGTH: 436 pages ISBN 0-201-53796-6 For ordering information, call 1-800-822-6339 Addison-Wesley Publishing Company MathSource Number: 0206-862 Communication with the authors: Fredrick I. Olness: olness at mail.physics.smu.edu Robert L. Zimmerman: bob at zim.uoregon.edu TOPICS: General Physics Classical Mechanics Electrostatics Quantum Mechanics Oscillation Systems (Linear and Non-Linear) Special Relativity General Relativity Cosmology ================================================ GENERAL AUDIENCE ================================================ This book is intended for the advanced undergraduate and graduate physics student taking core courses in the physics curriculum. We expect this text to be a supplement to the standard course text. The student would use this book to get ideas on how to use Mathematica to solve the problems assigned by the instructor. Since we cover the canonical problems from the core courses, the student can practice with our solutions, and then modify our solutions to solve the particular problems assigned. This should help the student move up the Mathematica learning curve quickly. ================================================ About: MATHEMATICA FOR PHYSICS ================================================ Mathematica is a powerful mathematical software system for students, researchers, and anyone seeking an effective tool for mathematical analysis. Tools such as Mathematica have begun to revolutionize the way science is taught, and research performed. Now there is a book specifically for students and teachers of physics who wish to use Mathematica to visualize and display physics concepts and to generate numerical and graphical solutions to physics problems. Mathematica for Physics chooses the canonical problems from the physics curriculum, and solves these problems using Mathematica. This book takes the reader beyond the "textbook" solutions by challenging the student to cross check the results using the wide variety of Mathematica's analytical, numerical, and graphical tools. Throughout the book, the complexity of both the physics and Mathematica is systematically extended to broaden the tools the reader has at his or her disposal, and to broaden the range of problems that can be solved. As such, this text is an appropriate supplement for any of the core advanced undergraduate and graduate physics courses. Highlights include: Provides Mathematica solutions for the canonical problems in the physics curriculum. Covers essential problems in: Mechanics, Electrodynamics, Quantum Mechanics, Special and General Relativity, Cosmology, Elementary Circuits, Oscillating Systems. Uses the power of Mathematica to go beyond "textbook" solutions and bring the problems alive with animations, and other graphical tools. Emphasizes the graphical capability of Mathematica to develop the reader's intuition and visualization in problem solving. Introduces the reader to the aspects of Mathematica that are particularly useful for physics. ================================================ Brief Table of contents ================================================ Chpt 1: Getting Started Chpt 2: General Physics Chpt 3: Oscillating Systems Chpt 4: Lagrangians and Hamiltonians Chpt 5: Electrostatics Chpt 6: Quantum Mechanics Chpt 7: Relativity and Cosmology ================================================ BEST FEATURES OF THE BOOK ================================================ With Mathematica, the entire approach to problem solving can be drastically changed. We give some brief examples. DOUBLE PENDULUM: This is a topic that is generally treated as an "advanced" topic. With Mathematica, the solution is relatively straightforward. Once the solutions is obtained, the textbooks try to describe (in words) the general properties of the system, and the normal modes. (In particular, the property that the energy is transferred back and forth between the two segments of the pendulum.) With the animation capability of Mathematica, we do not need to lead the student to these conclusions, but we can point them in the general direction, and let them discover these results on their own by varying the amplitudes of the separate normal modes. E&M BOUNDARY VALUE PROBLEMS: For the beginning student, it is easy to become overwhelmed by boundary value problems. With the power of Mathematica, it is easy to show how straightforward these solutions are--especially with the help of the different coordinate systems built into Mathematica. When the student finishes the problem with pen and paper, they have only a set of formulas that may mean very little to the student. With Mathematica, we encourage the student to plot the final solution so that they can verify visually if the boundary conditions are satisfied. This techniques encourages the student to think about the solution, and not simply grind out the math. HYDROGEN ATOM: In the standard solution of the hydrogen atom, the student is completely lost in the mathematics. Mathematica is able to recognize that the solution of the radial equation is a Laguerre polynomial, assemble the constants to form the principal quantum number, and plot the solutions. The student then has the energy and the curiosity to numerically investigate the behavior of the wavefunctions, and consider the disastrous consequences of choosing a non-integral value for the principal quantum number. ================================================ BIO STATEMENT ================================================ Robert Zimmerman is a Professor of Physics and research associate in the Institute of Theoretical Science at the University of Oregon. He has written papers on Mathematical Physics, Elementary Particles, Astrophysics, Cosmology, and General Relativity. He has taught graduate courses in Mathematical Physics, Theoretical Mechanics, Electrodynamics, Quantum Mechanics, General Relativity and Cosmology. He received his Ph.D. from the University of Washington. Fredrick Olness is an Assistant Professor of Physics at Southern Methodist University in Dallas Texas. His research is in Theoretical High Energy Physics and he studies the Quantum Chromodynamic (QCD) theory of the strong interaction. He received his Ph.D. from the University of Wisconsin, received an SSC Fellowship in 1993, and is a member of the CTEQ collaboration--a novel collaboration of theorists and experimentalists. ================================================ TABLE OF CONTENTS ================================================ Foreword v Introduction vii Troubleshooting xi CHAPTER 1 Getting Started 1 1.1 Introduction 1 1.1.1 Computers as a Tool 1 1.1.2 Suggestions on Approaching the Exercises 2 1.2 Arithmetic and Algebra 2 1.2.1 Arithmetic and Notation 2 1.2.2 Algebra 3 1.2.3 Mapping Expressions 3 1.2.4 Rules 4 1.2.5 Conjugation 5 1.2.6 User-defined Complex Conjugate Rule 5 1.2.7 Algebraic Equations 5 1.2.8 Threading Expressions 7 1.3 Functions and Procedures 7 1.3.1 User-defined Functions 7 1.3.2 Discontinuous Functions 8 1.3.3 Nonanalytic Functions 8 1.3.4 Rules 9 1.3.5 Procedures 10 1.4 Miscellaneous 10 1.4.1 Packages 10 1.4.2 Contexts 11 1.4.3 Protecting Commands 12 1.5 Calculus 12 1.5.1 Integration 12 1.5.2 Analytic Solutions of Differential Equations 13 1.5.3 Changing Variables and Pure Functions 13 1.5.4 Numerical Solutions of Differential Equations 14 1.6 Graphics 15 1.6.1 Animated Plots 15 1.6.2 Vector Field Plots 16 1.6.3 Shadowing 16 1.6.4 Three-dimensional Graphics 18 1.6.5 Space Curve 18 1.7 Exercises 18 CHAPTER 2 General Physics 23 2.1 Introduction 23 2.2 General Physics 23 2.2.1 Newtonian Motion 23 2.2.2 Electricity, Magnetism, and Circuits 24 2.3 Commands 24 2.3.1 Packages 24 2.3.2 User-defined Procedures 25 Procedure to find a least-squares fit to a set of data 25 Example: Slope of a Straight Line 25 Example: An Exponential 25 Procedure to find the electric potential for point charges 26 Example: Dipole 26 2.3.3 Protect User-defined Procedures 26 2.4 Problems 26 2.4.1 Projectile Motion in a Constant Gravitational Field 26 Problem 1: Escape Velocity 26 Problem 2: Projectile in a Uniform Gravitational Field 27 Problem 3: Projectile with Air Resistance 31 Problem 4: Rocket with Varying Mass 36 2.4.2 Projectile Motion in Rotating Reference Frames 42 Problem 1: Coriolis and Centrifugal Forces 42 Problem 2: Foucault Pendulum 47 2.4.3 Electricity and Magnetism 52 Problem 1: Charged Disk 52 Problem 2: Uniformly Charged Sphere 54 Problem 3: Electric Dipole 60 Problem 4: Magnetic Vector Potential for a Linear Current 63 2.4.4 Circuits 66 Problem 1: Series RC Circuit 66 Problem 2: Series RL Loop 68 Problem 3: RLC Loop 71 2.4.5 Modern Physics 75 Problem 1: The Bohr Atom 75 Problem 2: Relativistic Collision 77 2.5 Unsolved Problems 79 CHAPTER 3 Oscillating Systems 83 3.1 Introduction 83 3.1.1 Oscillations 83 Potentials 83 Phase Planes 83 Small Oscillations and Normal Modes 84 3.2 Commands 84 3.2.1 Packages 84 3.2.2 User-defined Procedures 85 Series expansion for second-order equation 85 Example 1: Second order solution 85 Example 2: How the diffSeriesOne routine works 86 Phase plot for one-dimensional system 87 Example: Phase plots for harmonic motion 87 Time behavior of phase plot for a one-dimensional system 88 Example: Time-evolved phase plots for harmonic motion 88 doublePlot : Phase plots and time evolution for a one-dimensional system 89 Example: Phase plots and time evolution for harmonic motion 89 Fourier spectrum of a one-dimensional oscillating system 89 Example: Fast Fourier transform 90 Eigenvalues and eigenvectors for small oscillating systems 90 Example: Two coupled particles 91 Animation for linear motion 93 Example: Linear harmonic oscillator 93 3.2.3 Protect User-defined Procedures 94 3.3 Problems 94 3.3.1 Linear Oscillations 94 Problem 1: Analysis of Linear Oscillator 94 Problem 2: Solution of Linear Oscillator 97 Problem 3: Damped Linear Oscillator 99 Problem 4: Damped Harmonic Oscillator and Driving Forces 104 3.3.2 Nonlinear Oscillations 109 Problem 1: Duffing's Oscillator Equation 109 Problem 2: Forced Duffing Oscillator for Double-well Potential 114 Problem 3: van der Pole Oscillator and Limiting Cycles 118 Problem 4: Motion of a Damped, Forced Nonlinear Pendulum 121 3.3.3 Small Oscillations 124 Problem 1: Two Coupled Harmonic Oscillators 124 Problem 2: Three Coupled Harmonic Oscillators 130 Problem 3: Double Pendulum 135 3.4 Unsolved Problems 139 CHAPTER 4 Lagrangians and Hamiltonians 143 4.1 Introduction 143 4.1.1 Lagrange's Equations 143 Generalized Coordinates and Constraints 143 Lagrangian 144 Nonholonomic Constraints and Lagrangian Multipliers 144 4.1.2 Hamilton's and Hamilton-Jacobi Equations 144 Hamilton's Equations 144 Hamilton-Jacobi Technique 145 4.1.3 Commands 146 Packages 146 User-defined Rules 147 HyperbolicToComplex and ComplexToHyperbolic 147 Example: HyperbolicToComplex and ComplexToHyperbolic 147 User-defined Procedures 147 Finding Lagrange's equations 147 Example: Lagrange's equation for a particle in a potential V[x] 148 Finding the canonical momentum, Hamiltonian, and equations of motion 148 Example 1: Hamilton's equations of motion in one dimension 149 Example 2: Hamilton's equations of motion in two dimensions 149 Example 3: How Hamilton works 150 Finding the canonical momentum, Hamilton's principal function, and Hamilton-Jacobi equations 151 Example 1: One-dimensional particle in a potential V[x] 151 Example 2: Two-dimensional particle in a potential V[x] 152 Example 3: How HamiltonJacobi works 152 Series expansion solution for second-order equation 153 Series expansion solution for two first-order equations 153 Example 1: Expansion of harmonic oscillator 154 Example 2: How firstDiffSeries works 154 First-order perturbation solution 155 Example 1: Perturbed harmonic oscillator 156 Example 2: Details of firstOrderPert 156 4.1.4 Protect User-defined Procedures 158 4.2 Problems 158 4.2.1 Lagrangian Problems 158 Problem 1: Atwood Machine 158 Problem 2: Bead Sliding on a Rotating Wire 161 Problem 3: Bead on a Rotating Hoop 165 Problem 4: Hoop Rolling on an Incline 171 Problem 5: Sphere Rolling on a Fixed Sphere 174 Problem 6: Mass Falling Through a Hole in a Table 178 4.2.2 Orbiting Bodies 183 Problem 1: Equivalent One-body Problem 183 Problem 2: Kepler Problem 188 Problem 3: Precessing Ellipse and Generalized Kepler Problem 192 Problem 4: Numerical Solution for Orbits with Central Forces 195 Problem 5: Quadripole Potential and Perturbative Solutions 199 4.2.3 Hamilton and Hamilton-Jacobi Problems 206 Problem 1: Harmonic Oscillator and Hamilton's Equations 206 Problem 2: Hamilton's Equations in Cylindrical and Spherical Coordinates 210 Problem 3: Spherical Pendulum and Hamilton's Equations 213 Problem 4: Harmonic Oscillator and Hamilton-Jacobi Equations 218 Problem 5: Kepler's Problem and Hamilton-Jacobi Equations 221 4.3 Unsolved Problems 225 CHAPTER 5 Electrostatics 229 5.1 Introduction 229 5.1.1 Electric Field and Potential 229 Electric field 229 Electrostatic potential 229 5.1.2 Laplace's Equation 230 Cartesian coordinates 230 Cylindrical coordinates 230 Spherical coordinates 230 5.1.3 Commands 231 Packages 231 User-defined procedures 231 Operator: TrigToY 231 Example 232 Operator: TrigToP 232 Example 233 Monopole 233 Example 233 PotentialExpansion 234 Example: Asymptotic potential of two-point charges 234 VEPlot 234 Example: Equipotential surface and electric field of two-point charges 235 MultipoleSH 236 Example: Potential of non-axially symmetric charge density 236 MultipoleP 237 Example: Potential of an axially symmetric charge distribution 238 5.1.4 Protect User-defined Procedures 238 5.2 Problems 238 5.2.1 Point Charges, Multipoles, and Image Problems 238 Problem 1: Superposition of point charges 238 Problem 2: Point charges and grounded plane 242 Problem 3: Point charges and grounded sphere 245 Problem 4: Line charge and grounded plane 249 Problem 5: Multipole expansion of a charge distribution 252 5.2.2 Cartesian and Cylindrical Coordinates 258 Problem 1 : Separation of variables in Cartesian and cylindrical coordinates 258 Problem 2 : Potential and a rectangular groove 261 Problem 3: Rectangular conduit 265 Problem 4: Potential inside a rectangular box with five sides at zero potential 269 Problem 5: Conducting cylinder with a potential on the surface 274 5.2.3 Legendre Polynomials and Spherical Harmonics 278 Problem 1: A charged ring 278 Problem 2: Grounded sphere in an electric field 285 Problem 3: Sphere with an axially symmetric charge distribution 288 Problem 4: Sphere with a given axially symmetric potential 292 Problem 5: Sphere with upper hemisphere V and lower hemisphere - .75pt V 295 5.3 Unsolved Problems 299 CHAPTER 6 Quantum Mechanics 303 6.1 Introduction 303 6.1.1 Foundations of Quantum Mechanics 303 Historical beginnings 303 Time-independent quantum mechanics 304 6.1.2 Commands 304 Packages 304 User-defined procedures and rules 304 Change of variables procedure 304 Example: Change of variable 305 Series expansion solution for second-order equation 305 HyperbolicToComplex and ComplexToHyperbolic 306 Complex conjugate rule 306 Example: Complex exponential 306 User-defined solutions of differential equations 307 Solution: Hermite polynomials 307 Example: Hermite solution 307 Solution: Legendre polynomials 307 Example: Legendre solution 307 User-defined one-dimensional wave properties 308 One-dimensional wave function with constant potential 308 6.1.3 User-defined Three-dimensional Quantum Equations 308 Operator: schrodinger 308 Example: Spherical potential and spherical wave function 308 Operator: hamiltonian 309 Example: Harmonic oscillator 309 Operator: flux 309 Example: Plane wave flux 309 6.1.4 Protect User-defined Operators 310 6.2 Problems 310 6.2.1 One-dimensional Schrodinger's Equation 310 Problem 1: Particle bound in an infinite potential well 310 Problem 2: Particle bound in a finite potential well 314 Problem 3: Particle hitting a finite step potential 322 Problem 4: Particle propagating towards a rectangular potential 328 Problem 5: The one-dimensional harmonic oscillator 336 6.2.2 Three-dimensional Schrodinger's Equation 341 Problem 1: Three-dimensional harmonic oscillator in Cartesian coordinates 341 Problem 2: Schrodinger's equation for spherically symmetric potentials 344 Problem 3: Particle in an infinite, spherical box 349 Problem 4: Particle with negative energy in a finite, spherical box 353 Problem 5: The hydrogen atom in spherical coordinates 359 Problem 6: Separation in cylindrical and paraboloidal coordinates 364 6.3 Unsolved Problems 369 CHAPTER 7 Relativity and Cosmology 371 7.1 Introduction 371 7.1.1 Special Relativity 371 The two basic postulates of special relativity 372 Lorentz transformations 372 Covariant equations and tensors 373 Cartesian coordinates and ``flat'' spacetime 373 7.1.2 General Relativity and Cosmology 374 Spacetime metric 374 Field equations 374 Free-falling test particles and light trajectories 374 Robertson-Walker cosmology 375 7.1.3 Commands 375 Packages 375 User-defined metric, boost, and velocity parameters 375 Metric 375 Rule for relativistic velocity parameters 376 Boost along the x-axis 376 User-defined geometric procedures 376 Christoffel symbols 376 Example: Christoffel symbols for a pseudo-Euclidean metric 377 Curvature tensor 377 Example: Curvature tensor for pseudo-Euclidean metric 378 Ricci tensor 378 Example: Ricci tensor for Go del metric 378 Killing's equations 379 Example: Killing vector equations in pseudo-Euclidean space 379 Einstein tensor 380 Example: Einstein tensor for wave metric 380 Geodesic equations 380 Example: Geodesics for a pseudo-Euclidean metric 381 User-defined metrics and Christoffel symbols 381 Schwarzschild metric 381 Kerr metric 382 Protect user-defined operators 384 7.2 Problems 384 7.2.1 Special Relativity Problems 384 Problem 1: Decay of a particle 384 Problem 2: Two-particle collision 385 Problem 3: Compton scattering 386 Problem 4: Moving mirror and generalized Snell's law 389 Problem 5: One-dimensional motion of a relativistic particle with constant acceleration 392 Problem 6: Two-dimensional motion of a relativistic particle in a uniform electric field 395 7.2.2 General Relativity and Cosmology 398 Problem 1: Schwarzschild solution in null coordinates 398 Problem 2: The horizons and surfaces of infinite redshift 400 Problem 3: Killing vectors and constants of motion 402 Problem 4: Potential analysis for timelike geodesics 405 Problem 5: Time it takes to fall into a black hole 408 Problem 6: Circular geodesics for the Schwarzschild metric 414 Problem 7: Field equations for Robertson-Walker cosmology 416 Problem 8: Zero-pressure cosmological models 421 Problem 9: The expansion and age of the standard model 424 7.3 Unsolved Problems 430 Index 433

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