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A Physicist's Guide to Mathematica, 2nd Edition

Name: A Physicist's Guide to Mathematica, 2nd Edition
Author: Patrick T. Tam
ISBN: 9780080926247

by Patrick T. Tam

August 2011

Intermediate to advanced

641 pages

14h 48m

English

Academic Press

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Copyright
Dedication
Preface to the Second Edition
Preface to the First Edition
PurposeUsesOrganizationSuggestionsPrerequisitesComputer SystemsAcknowledgments
I. Mathematica with Physics
1. The First Encounter
1.1. The First Ten Minutes1.2. A Touch of Physics1.2.1. Numerical Calculations1.2.2. Symbolic Calculations1.2.3. Graphics1.3. Online Help1.4. Warning Messages1.5. Packages1.6. Notebook Interfaces1.6.1. Notebooks1.6.2. Entering Greek Letters1.6.3. Getting Help1.6.4. Preparing Input1.6.5. Starting and Aborting Calculations1.7. Problems
2. Interactive Use of Mathematica
2.1. Numerical Capabilities2.1.1. Arithmetic Operations2.1.2. Spaces and Parentheses2.1.3. Common Mathematical Constants2.1.4. Some Mathematical Functions2.1.5. Cases and Brackets2.1.6. Ways to Refer to Previous Results2.1.7. Standard Computations2.1.8. Exact versus Approximate Values2.1.9. Machine Precision versus Arbitrary Precision2.1.10. Special Functions2.1.11. Matrices2.1.12. Double Square Brackets2.1.13. Linear Least-Squares Fit2.1.14. Complex Numbers2.1.15. Random Numbers2.1.16. Numerical Solution of Polynomial Equations2.1.17. Numerical Integration2.1.18. Numerical Solution of Differential Equations2.1.19. Iterators2.1.20. Exercises2.2. Symbolic Capabilities2.2.1. Transforming Algebraic Expressions2.2.2. Transforming Trigonometric Expressions2.2.3. Transforming Expressions Involving Special Functions2.2.4. Using Assumptions2.2.5. Obtaining Parts of Algebraic Expressions2.2.6. Units, Conversion of Units, and Physical Constants2.2.7. Assignments and Transformation Rules2.2.8. Equation Solving2.2.9. Differentiation2.2.10. Integration2.2.11. Sums2.2.12. Power Series2.2.13. Limits2.2.14. Solving Differential Equations2.2.15. Immediate versus Delayed Assignments and Transformation Rules2.2.16. Defining Functions2.2.17. Relational and Logical Operators2.2.18. Fourier Transforms2.2.19. Evaluating Subexpressions2.2.20. Exercises2.3. Graphical Capabilities2.3.1. Two-Dimensional Graphics2.3.1.1. Basic Plots2.3.1.2. Options2.3.1.3. Multiple Plots2.3.1.4. FindRoot2.3.1.5. FindMinimum and FindMaximum2.3.1.6. Data Plots2.3.1.7. Parametric Plots2.3.1.8. Interactive Graphics Drawing2.3.2. Three-Dimensional Graphics2.3.2.1. Surface Plots2.3.2.2. Viewpoint2.3.3. Interactive Manipulation of Graphics2.3.4. Animation2.3.5. Exercise2.4. Lists2.4.1. Defining Lists2.4.2. Generating and Displaying Lists2.4.3. Counting List Elements2.4.4. Obtaining List and Sublist Elements2.4.5. Changing List and Sublist Elements2.4.6. Rearranging Lists2.4.7. Restructuring Lists2.4.8. Combining Lists2.4.9. Operating on Lists2.4.10. Using Lists in Computations2.4.11. Analyzing Data2.4.11.1. Basic Error Analysis2.4.11.2. Nonlinear Least-Squares Fit2.4.11.3. Interpolation2.4.12. Exercises2.5. Special Characters, Two-Dimensional Forms, and Format Types2.5.1. Special Characters2.5.1.1. Ways to Enter Special Characters2.5.1.2. Letters and Letterlike Forms2.5.1.3. OperatorsLogical OperatorsBracketing OperatorsOther Operators2.5.1.4. Structural Elements and Spacing Characters2.5.1.5. Similar-Looking Characters2.5.2. Two-Dimensional Forms2.5.2.1. Ways to Enter Two-Dimensional FormsPalettesControl KeysOrdinary CharactersCreate Table/Matrix2.5.2.2. Some Two-Dimensional Forms with Built-in Meaning2.5.2.3. Two-Dimensional Notation in Physics2.5.3. Input and Output Forms2.5.4. Exercises2.6. Problems
3. Programming in Mathematica
3.1. Expressions3.1.1. Atoms3.1.2. Internal Representation3.1.3. Manipulation3.1.3.1. Obtaining Parts of Expressions3.1.3.2. Changing Parts of Expressions3.1.3.3. Rearranging Expressions3.1.3.4. Restructuring Expressions3.1.3.5. Operating on Expressions3.1.3.6. Manipulating Equations3.1.4. Exercises3.2. Patterns3.2.1. Blanks3.2.2. Naming Patterns3.2.3. Restricting Patterns3.2.3.1. Types3.2.3.2. Tests3.2.3.3. Conditions3.2.4. Structural Equivalence3.2.5. Attributes3.2.6. Defaults3.2.7. Alternative or Repeated Patterns3.2.8. Multiple Blanks3.2.9. Exercises3.3. Functions3.3.1. Pure Functions3.3.2. Selecting a Definition3.3.3. Recursive Functions and Dynamic Programming3.3.4. Functional Iterations3.3.5. Protection3.3.6. Upvalues and Downvalues3.3.7. Exercises3.4. Procedures3.4.1. Local Symbols3.4.2. Conditionals3.4.3. Loops3.4.3.1. Changing Values of Variables3.4.3.2. Do, While, and For3.4.4. Named Optional Arguments3.4.5. An Example: Motion of a Particle in One Dimension3.4.6. Exercises3.5. Graphics3.5.1. Graphics Objects3.5.2. Two-Dimensional Graphics3.5.2.1. Two-Dimensional Graphics Primitives3.5.2.2. Two-Dimensional Graphics Directives3.5.2.3. Two-Dimensional Graphics Options3.5.2.4. Wave Motion3.5.3. Three-Dimensional Graphics3.5.3.1. Three-Dimensional Graphics Primitives3.5.3.2. Three-Dimensional Graphics Directives3.5.3.3. Three-Dimensional Graphics Options3.5.3.4. Crystal Structure3.5.4. Exercises3.6. Programming Styles3.6.1. Procedural Programming3.6.2. Functional Programming3.6.3. Rule-Based Programming3.6.4. Exercises3.7. Packages3.7.1. Contexts3.7.2. Context Manipulation3.7.3. A Sample Package3.7.3.1. The Problem3.7.3.2. The Package3.7.3.3. Analysis of the Package3.7.4. Template for Packages3.7.5. Exercises
II. Physics with Mathematica
4. Mechanics
4.1. Falling Bodies4.1.1. The Problem4.1.2. Physics of the Problem4.1.3. Solution with Mathematica4.2. Projectile Motion4.2.1. The Problem4.2.2. Physics of the Problem4.2.3. Solution with Mathematica4.3. The Pendulum4.3.1. The Problem4.3.2. Physics of the Problem4.3.2.1. The Plane Pendulum4.3.2.2. The Damped Pendulum4.3.2.3. The Damped, Driven Pendulum4.3.3. Solution with Mathematica4.3.3.1. The Plane Pendulum4.3.3.2. The Damped Pendulum4.3.3.3. The Damped, Driven Pendulum4.3.3.4. ChaoticPendulum’: A Mathematica Package4.4. The Spherical Pendulum4.4.1. The Problem4.4.2. Physics of the Problem4.4.3. Solution with Mathematica4.4.3.1. θ0 = 120, 0 = 0, ϕ0 = 45, 0 = 04.4.3.2. θ0 = 135, 0 = 0, ϕ0 = 90, 0 = 21/44.4.3.3. θ0 = 135, 0 = 2.5, ϕ0 = 90, 0 = 1.5 ×21/44.4.3.4. θ0 = 120, 0 = 0.75, ϕ0 = 90, 0 = 2.0 × 21/44.5. Problems
5. Electricity and Magnetism
5.1. Electric Field Lines and Equipotentials5.1.1. The Problem5.1.2. Physics of the Problem5.1.2.1. Electric Field Lines5.1.2.2. Equipotentials5.1.2.3. Electric Field Lines and Equipotentials for Two Point Charges5.1.3. Solution with Mathematica5.1.3.1. q1 = q2 = + q5.1.3.2. q1 = +2q and q2 = –q5.2. Laplace’s Equation5.2.1. The Problem5.2.2. Physics of the Problem5.2.2.1. Analytical Solution5.2.2.2. Numerical Solution5.2.3. Solution with Mathematica5.2.3.1. Analytical Solution5.2.3.2. Numerical Solution5.3. Charged Particle in Crossed Electric and Magnetic Fields5.3.1. The Problem5.3.2. Physics of the Problem5.3.3. Solution with Mathematica5.4. Problems

6. Quantum Physics
6.1. Blackbody Radiation6.1.1. The Problem6.1.2. Physics of the Problem6.1.3. Solution with Mathematica6.1.3.1. u(λ, T) at Several Temperatures6.1.3.2. Wien’s Displacement Law6.1.3.3. λmax for Solar Radiation6.2. Wave Packets6.2.1. The Problem6.2.2. Physics of the Problem6.2.3. Solution with Mathematica6.3. Particle in a One-Dimensional Box6.3.1. The Problem6.3.2. Physics of the Problem6.3.3. Solution with Mathematica6.3.3.1. Function Definitions6.3.3.2. Animation6.4. The Square Well Potential6.4.1. The Problem6.4.2. Physics of the Problem6.4.2.1. Analytical Solution6.4.2.2. Numerical Solution6.4.3. Solution with Mathematica6.4.3.1. Analytical Solution6.4.3.2. Numerical Solution6.5. Angular Momentum6.5.1. The Problem6.5.2. Physics of the Problem6.5.2.1. Angular Momentum in Quantum Mechanics6.5.2.2. Orbital Angular Momentum6.5.2.3. The Eigenvalue Problem6.5.3. Solution with Mathematica6.6. The Kronig–Penney Model6.6.1. The Problem6.6.2. Physics of the Problem6.6.3. Solution with Mathematica6.7. Problems
A. The Last Ten Minutes
B. Operator Input Forms
C. Solutions to Exercises
Section 2.1.20Section 2.2.20Section 2.3.5Section 2.4.12Section 2.5.4Section 3.1.4Section 3.2.9Section 3.3.7Section 3.4.6Section 3.5.4Section 3.6.4Section 3.7.5
D. Solutions to Problems
Section 1.7Section 2.6Section 4.5Section 5.4Section 6.7
References

Content preview from A Physicist's Guide to Mathematica, 2nd Edition

Chapter 4. Mechanics

Falling Bodies

The Problem

A small body falls downward with an initial velocity v₀ from a height h near the surface of the earth, as in Figure 4.1.1. For low velocities (less than approximately 24 m/s), the effect of air resistance may be approximated by a frictional force proportional to the velocity. Find the displacement and velocity of the body, and determine the terminal velocity. Plot the speed as a function of time for several initial velocities.

A small body falling downward with an initial velocity v0 from a height h.

Figure 4.1.1. A small body falling downward with an initial velocity v₀ from a height h.

Physics of the Problem

Let the x-axis be directed upward, as in Figure 4.1.1. The net force ...

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