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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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