Physicist's Guide to Mathematica (eBook)

Patrick T. Tam (Autor)

eBook Download: EPUB

2011 | 2. Auflage
641 Seiten
Elsevier Science (Verlag)
978-0-08-092624-7 (ISBN)

Infos

For the engineering and scientific professional, A Physicist's Guide to Mathematica, 2/e provides an updated reference guide based on the 2007 new 6.0 release, providing an organized and integrated desk reference with step by step instructions for the most often used features of the software as it applies to research in physics.

For Professors teaching physics and other science courses using the Mathematica software, A Physicist's Guide to Mathematica, 2/e is the only fully compatible (new software release) Mathematica text that engages students by providing complete topic coverage, new applications, exercises and examples that enable the user to solve a wide range of physics problems.

. Does not require prior knowledge of Mathematica or computer programming
. Can be used as either a primary or supplemental text for upper-division physics majors and an Instructor's Solutions Manual is available
. Provides over 450 end-of-section exercises and end-of-chapter problems
. Serves as a reference suitable for chemists, physical scientists, and engineers
. Compatible with Mathematica Version 6, a recent major release
. Compact disk contains all of the Mathematica input and output in this book

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For the engineering and scientific professional, A Physicist's Guide to Mathematica, Second Edition provides an updated reference guide based on the 2007 new 6.0 release, providing an organized and integrated desk reference with step-by-step instructions for the most commonly used features of the software as it applies to research in physics. For professors teaching physics and other science courses using the Mathematica software, A Physicist's Guide to Mathematica, Second Edition is the only fully compatible (new software release) Mathematica text that engages students by providing complete topic coverage, new applications, exercises and examples that enable the user to solve a wide range of physics problems. - Does not require prior knowledge of Mathematica or computer programming- Can be used as either a primary or supplemental text for upper-division physics majors- Provides over 450 end-of-section exercises and end-of-chapter problems- Serves as a reference suitable for chemists, physical scientists, and engineers- Compatible with Mathematica Version 6, a recent major release

Inhalt

Front Cover 1
A Physicist’s Guide to Mathematica 4
Copyright Page 5
Table of Contents 8
Preface to the Second Edition 14
Preface to the First Edition 16
Part I: Mathematica with Physics 22
Chapter 1. The First Encounter 24
1.1 The First Ten Minutes 24
1.2 A Touch of Physics 27
1.2.1 Numerical Calculations 27
1.2.2 Symbolic Calculations 27
1.2.3 Graphics 27
1.3 Online Help 28
1.4 Warning Messages 30
1.5 Packages 31
1.6 Notebook Interfaces 33
1.6.1 Notebooks 33
1.6.2 Entering Greek Letters 33
1.6.3 Getting Help 34
1.6.4 Preparing Input 35
1.6.5 Starting and Aborting Calculations 36
1.7 Problems 36
Chapter 2. Interactive Use of Mathematica 40
2.1 Numerical Capabilities 40
2.1.1 Arithmetic Operations 40
2.1.2 Spaces and Parentheses 41
2.1.3 Common Mathematical Constants 41
2.1.4 Some Mathematical Functions 42
2.1.5 Cases and Brackets 43
2.1.6 Ways to Refer to Previous Results 43
2.1.7 Standard Computations 44
2.1.8 Exact versus Approximate Values 45
2.1.9 Machine Precision versus Arbitrary Precision 46
2.1.10 Special Functions 48
2.1.11 Matrices 48
2.1.12 Double Square Brackets 50
2.1.13 Linear Least-Squares Fit 51
2.1.14 Complex Numbers 53
2.1.15 Random Numbers 53
2.1.16 Numerical Solution of Polynomial Equations 54
2.1.17 Numerical Integration 55
2.1.18 Numerical Solution of Differential Equations 60
2.1.19 Iterators 64
2.1.20 Exercises 65
2.2 Symbolic Capabilities 79
2.2.1 Transforming Algebraic Expressions 79
2.2.2 Transforming Trigonometric Expressions 82
2.2.3 Transforming Expressions Involving Special Functions 85
2.2.4 Using Assumptions 85
2.2.5 Obtaining Parts of Algebraic Expressions 88
2.2.6 Units, Conversion of Units, and Physical Constants 90
2.2.7 Assignments and Transformation Rules 93
2.2.8 Equation Solving 97
2.2.9 Differentiation 101
2.2.10 Integration 107
2.2.11 Sums 111
2.2.12 Power Series 115
2.2.13 Limits 117
2.2.14 Solving Differential Equations 118
2.2.15 Immediate versus Delayed Assignments and Transformation Rules 120
2.2.16 Defining Functions 121
2.2.17 Relational and Logical Operators 126
2.2.18 Fourier Transforms 129
2.2.19 Evaluating Subexpressions 133
2.2.20 Exercises 135
2.3 Graphical Capabilities 165
2.3.1 Two-Dimensional Graphics 165
2.3.2 Three-Dimensional Graphics 195
2.3.3 Interactive Manipulation of Graphics 200
2.3.4 Animation 203
2.3.5 Exercise 210
2.4 Lists 247
2.4.1 Defining Lists 247
2.4.2 Generating and Displaying Lists 248
2.4.3 Counting List Elements 250
2.4.4 Obtaining List and Sublist Elements 253
2.4.5 Changing List and Sublist Elements 257
2.4.6 Rearranging Lists 258
2.4.7 Restructuring Lists 259
2.4.8 Combining Lists 262
2.4.9 Operating on Lists 264
2.4.10 Using Lists in Computations 265
2.4.11 Analyzing Data 276
2.4.12 Exercises 289
2.5 Special Characters, Two-Dimensional Forms, and Format Types 308
2.5.1 Special Characters 309
2.5.2 Two-Dimensional Forms 317
2.5.3 Input and Output Forms 327
2.5.4 Exercises 330
2.6 Problems 335
Chapter 3. Programming in Mathematica 350
3.1 Expressions 350
3.1.1 Atoms 350
3.1.2 Internal Representation 352
3.1.3 Manipulation 355
3.1.4 Exercises 373
3.2 Patterns 381
3.2.1 Blanks 382
3.2.2 Naming Patterns 383
3.2.3 Restricting Patterns 384
3.2.4 Structural Equivalence 391
3.2.5 Attributes 392
3.2.6 Defaults 394
3.2.7 Alternative or Repeated Patterns 397
3.2.8 Multiple Blanks 398
3.2.9 Exercises 399
3.3 Functions 407
3.3.1 Pure Functions 407
3.3.2 Selecting a Definition 413
3.3.3 Recursive Functions and Dynamic Programming 415
3.3.4 Functional Iterations 419
3.3.5 Protection 423
3.3.6 Upvalues and Downvalues 425
3.3.7 Exercises 429
3.4 Procedures 435
3.4.1 Local Symbols 436
3.4.2 Conditionals 438
3.4.3 Loops 444
3.4.4 Named Optional Arguments 449
3.4.5 An Example: Motion of a Particle in One Dimension 456
3.4.6 Exercises 467
3.5 Graphics 473
3.5.1 Graphics Objects 473
3.5.2 Two-Dimensional Graphics 476
3.5.3 Three-Dimensional Graphics 498
3.5.4 Exercises 534
3.6 Programming Styles 537
3.6.1 Procedural Programming 540
3.6.2 Functional Programming 545
3.6.3 Rule-Based Programming 548
3.6.4 Exercises 555
3.7 Packages 558
3.7.1 Contexts 558
3.7.2 Context Manipulation 562
3.7.3 A Sample Package 564
3.7.4 Template for Packages 571
3.7.5 Exercises 572
Part II: Physics with Mathematica 574
Chapter 4. Mechanics 576
4.1 Falling Bodies 576
4.1.1 The Problem 576
4.1.2 Physics of the Problem 577
4.1.3 Solution with Mathematica 577
4.2 Projectile Motion 579
4.2.1 The Problem 579
4.2.2 Physics of the Problem 579
4.2.3 Solution with Mathematica 580
4.3 The Pendulum 582
4.3.1 The Problem 582
4.3.2 Physics of the Problem 583
4.3.3 Solution with Mathematica 584
4.4 The Spherical Pendulum 591
4.4.1 The Problem 591
4.4.2 Physics of the Problem 591
4.4.3 Solution with Mathematica 594
4.5 Problems 601
Chapter 5. Electricity and Magnetism 604
5.1 Electric Field Lines and Equipotentials 604
5.1.1 The Problem 604
5.1.2 Physics of the Problem 604
5.1.3 Solution with Mathematica 607
5.2 Laplace’s Equation 612
5.2.1 The Problem 612
5.2.2 Physics of the Problem 612
5.2.3 Solution with Mathematica 615
5.3 Charged Particle in Crossed Electric and Magnetic Fields 623
5.3.1 The Problem 623
5.3.2 Physics of the Problem 623
5.3.3 Solution with Mathematica 624
5.4 Problems 628
Chapter 6. Quantum Physics 632
6.1 Blackbody Radiation 632
6.1.1 The Problem 632
6.1.2 Physics of the Problem 632
6.1.3 Solution with Mathematica 633
6.2 Wave Packets 637
6.2.1 The Problem 637
6.2.2 Physics of the Problem 637
6.2.3 Solution with Mathematica 638
6.3 Particle in a One-Dimensional Box 643
6.3.1 The Problem 643
6.3.2 Physics of the Problem 643
6.3.3 Solution with Mathematica 645
6.4 The Square Well Potential 647
6.4.1 The Problem 647
6.4.2 Physics of the Problem 647
6.4.3 Solution with Mathematica 650
6.5 Angular Momentum 660
6.5.1 The Problem 660
6.5.2 Physics of the Problem 660
6.5.3 Solution with Mathematica 665
6.6 The Kronig–Penney Model 668
6.6.1 The Problem 668
6.6.2 Physics of the Problem 668
6.6.3 Solution with Mathematica 669
6.7 Problems 671
Appendix 674
A The Last Ten Minutes 674
B Operator Input Forms 676
C Solutions to Exercises 680
D Solutions to Problems 724
References 730
Index 734

Textauszug

Preface to the First Edition

Traditionally, the upper-division theoretical physics courses teach the formalisms of the theories, the analytical technique of problem-solving, and the physical interpretation of the mathematical solutions. Problems of historical significance, pedagogical value, or if possible, recent research interest are chosen as examples. The analytical methods consist mainly of working with models, making approximations, and considering special or limiting cases. The student must master the analytical skills, because they can be used to solve many problems in physics and, even in cases where solutions cannot be found, can be used to extract a great deal of information about the problems. As the computer has become readily available, these courses should also emphasize computational skills, since they are necessary for solving many important, real, or “fun” problems in physics. The student ought to use the computer to complement and reinforce the analytical skills with the computational skills in problem-solving and, whenever possible, use the computer to visualize the results and observe the effects of varying the parameters of the problem in order to develop a greater intuitive understanding of the underlying physics.

The pendulum in classical mechanics serves as an example to elucidate these ideas. The plane pendulum is used as a model. It consists of a particle under the action of gravity and constrained to move in a vertical circle by a massless rigid rod. For small angular deviations, the equation of motion can be linearized and solved easily. For finite angular oscillations, the motion is nonlinear. Yet it can still be studied analytically in terms of the energy integral and the phase diagram. The period of motion is expressed in terms of an elliptic integral. The integral can be expanded in a power series, and for small angular oscillations the expansion converges rapidly. However, numerical methods and computer programming are necessary for determining the motion of a damped, driven pendulum. The student can use the computer to explore and simulate the motion of the pendulum with different sets of values for the parameters in order to gain a deeper intuitive understanding of the chaotic dynamics of the pendulum.

Normally, physics juniors and seniors have taken a course in a low-level language such as FORTRAN or Pascal and possibly also a course in numerical analysis. Nevertheless, attempts to introduce numerical methods and computer programming into the upper-division theoretical physics courses have been largely unsuccessful. Mastering the symbols and syntactic rules of these low-level languages is straightforward; but programming with them requires too many lines of complicated and convoluted code in order to solve interesting problems. Consequently, rather than enhancing the student’s problem-solving skills and physical intuition, it merely adds a frustrating and ultimately nonproductive burden to the student already struggling in a crowded curriculum.

Mathematica, a system developed recently for doing mathematics by computer, promises to empower the student to solve a wide range of problems including those that are important, real, or “fun,” and to provide an environment for the student to develop intuition and a deeper understanding of physics. In addition to numerical calculations, Mathematica performs symbolic as well as graphical calculations and animates two- and three-dimensional graphics. The numerical capabilities broaden the problem-solving skills of the student; the symbolic capabilities relieve the student from the tedium and errors of “busy” or long-winded derivations; the graphical capabilities and the capabilities for “instant replay” with various parameter values for the problem enable the student to deepen his or her intuitive understanding of physics. These astounding interactive capabilities are sufficiently powerful for handling most problems and are surprisingly easy to learn and use. For complex and demanding problems, Mathematica also features a high-level programming language that can make use of more than a thousand built-in functions and that embraces many programming styles such as functional, rule-based, and procedural programming. Furthermore, to provide an integrated technical computing environment, the Macintosh and Windows versions for Mathematica support documents called “notebooks.” A notebook is a “live textbook.” It is a file containing ordinary text, Mathematica input and output, and graphics. Mathematica, together with the user-friendly Macintosh and Windows interfaces, is likely to revolutionize not only how but also what we teach in the upper-division theoretical physics courses.

Purpose

The primary purpose of this book is to teach upper-division and graduate physics students as well as professional physicists how to master Mathematica, using examples and approaches that are motivating to them. This book does not replace Stephen Wolfram’s Mathematica: A System for Doing Mathematics by Computer [Wol91] for Mathematica version 2 or The Mathematica Book [Wol96] for version 3. The encyclopedic nature of these excellent references is formidable, indeed overwhelming, for novices. My guidebook prepares the reader for easy access to Wolfram’s indispensable references. My book also shows that Mathematica can be a powerful and wonderful tool for learning, teaching, and doing physics.

Uses

This book can serve as the text for an upper-division course on Mathematica for physics majors. Augmented with chemistry examples, it can also be the text for a course on Mathematica for chemistry majors. (For the last several years, a colleague in the chemistry department and I have team-taught a Mathematica course for both chemistry and physics majors.) Part I, “Mathematica with Physics,” provides sufficient material for a two-unit, one-semester course. A three-unit, one-semester course can cover Part I, sample Part II, “Physics with Mathematica,” require a polished Mathematica notebook from each student reporting a project, and include supplementary material on introductory numerical analysis discussed in many texts (see [KM90], [DeV94], [Gar94], and [Pat94]). Exposure to numerical analysis allows the student to appreciate the limitations (i.e., the accuracy and stability) of numerical algorithms and understand the differences between numerical and symbolic functions, for example, between NSolve and Solve, NIntegrate and Integrate, as well as NDSolve and DSolve. Experience suggests that a three-hour-per-week laboratory is essential to the success of both the two-and three-unit courses. For the degree requirement, either course is an appropriate addition to, if not replacement for, the existing course in a low-level language such as C, Pascal, or FORTRAN.

If a course on Mathematica is not an option, a workshop merits consideration. A twoday workshop can cover Chapter 1, “The First Encounter,” and Chapter 2, “Interactive Use of Mathematica,” and a one-week workshop can also include Chapter 3, “Programming in Mathematical Of course, further digestion of the material may be necessary after one of these accelerated workshops.

For students who are Mathematica neophytes, this book can also be a supplemental text for upper-division theoretical physics courses on mechanics, electricity and magnetism, and quantum physics. For Mathematica to enrich rather than encroach upon the curriculum, it must be introduced and integrated into these courses gradually and patiently throughout the junior and senior years, beginning with the interactive capabilities. While the interactive capabilities of Mathematica are quite impressive, in order to realize its full power the student must grasp its structure and master it as a programming language. Be forewarned that learning these advanced features as part of the regular courses, while possible, is difficult. A dedicated Mathematica course is usually a more gentle, efficient, and effective way to learn this computer algebra system.

Finally, the book can be used as a self-paced tutorial for advanced physics students and professional physicists who would like to learn Mathematica on their own. While the sections in Part I should be studied consecutively, those in Part II, each focusing on a particular physics problem, are independent of each other and can be read in any order. The reader may find the solutions to exercises and problems in Appendices D and E helpful.

Organization

Part I gives a practical, physics-oriented, and self-contained introduction to Mathematica. Chapter 1 shows the beginner how to get started with Mathematica and discusses the notebook front end. Chapter 2 introduces the numerical, symbolic, and graphical capabilities of Mathematica. Although these features of Mathematica are dazzling, Mathematica’s real power rests on its programming capabilities. While Chapter 2 considers many elements of Mathematica’s programming language, Chapter 3 treats in depth five key programming elements: expressions,...

Produkt-Details

• Erscheint lt. Verlag: 9.8.2011
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Computerprogramme / Computeralgebra
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• ISBN-10: 0-08-092624-X / 008092624X
• ISBN-13: 978-0-08-092624-7 / 9780080926247