Mathematica for Physicists and Engineers

Mathematica for Physicists and Engineers Hands-on textbook for learning how to use Mathematica to solve real-life problems in physics and engineering Mathematica for Physicists and Engineers provides the basic concepts of Mathematica for scientists and engineers, highlights Mathematica’s several built-in functions, demonstrates mathematical concepts that can be employed to solve problems in physics and engineering, and addresses problems in basic arithmetic to more advanced topics such as quantum mechanics. The text views mathematics and physics through the eye of computer programming, fulfilling the needs of students at master’s levels and researchers from a physics and engineering background and bridging the gap between the elementary books written on Mathematica and the reference books written for advanced users. Mathematica for Physicists and Engineers contains information on: Basics to Mathematica, its nomenclature and programming language, and possibilities for graphic output Vector calculus, solving real, complex and matrix equations and systems of equations, and solving quantum mechanical problems in infinite-dimensional linear vector spaces Differential and integral calculus in one and more dimensions and the powerful but elusive Dirac Delta function Fourier and Laplace transform, two integral transformations that are instrumental in many fields of physics and engineering for the solution of ordinary and partial differential equations Serving as a complete first course in Mathematica to solve problems in science and engineering, Mathematica for Physicists and Engineers is an essential learning resource for students in physics and engineering, master’s students in material sciences, geology, biological sciences theoretical chemists. Also lecturers in these and related subjects will benefit from the book.

Table of Contents:
Preface xiii Foreword xvii About the Authors xix 1 Preliminary Notions 1 1.1 Introduction 1 1.2 Versions of Mathematica 1 1.3 Getting Started 2 1.4 Simple Calculations 2 1.4.1 Arithmetic Operations 2 1.4.2 Approximate Numerical Results 3 1.4.3 Algebraic Calculations 3 1.4.4 Defining Variables 4 1.4.5 Using the Previous Results 5 1.4.6 Suppressing the Output 6 1.4.7 Sequences of Operations 6 1.5 Built-in Functions 7 1.6 Additional Features 9 1.6.1 Arbitrary-Precision Calculations 9 1.6.2 Value for Symbols 10 1.6.3 Defining Naming and Evaluating Functions 10 1.6.4 Composition of Functions 11 1.6.5 Conditional Assignment 12 1.6.6 Warnings and Messages 13 1.6.7 Interrupting Calculations 13 1.6.8 Using Symbols to Tag Objects 13 2 Basic Mathematical Operations 15 2.1 Introduction 15 2.2 Basic Algebraic Operations 15 2.3 Basic Trigonometric Operations 20 2.4 Basic Operations with Complex Numbers 21 3 Lists and Tables 25 3.1 Introduction 25 3.2 Lists 25 3.3 Arrays 26 3.4 Tables 26 3.5 Extracting the Elements from the Arrays/Tables 29 4 Two-Dimensional Graphics 31 4.1 Introduction 31 4.2 Plotting Functions of a Single Variable 31 4.3 Additional Commands 34 4.4 Plot Styles 44 4.5 Probability Distribution 58 4.5.1 Binomial Distribution 58 4.5.2 Poisson Distribution 58 4.5.3 Normal or Gaussian Distribution 59 4.6 Some More Useful Commands 61 5 Parametric, Polar, Contour, Density, and List Plots 65 5.1 Introduction 65 5.2 Parametric Plotting 65 5.3 Polar Plots 72 5.3.1 Polar Plots of Circles 72 5.3.2 Polar Plots of Ellipse, Parabola, and Hyperbola 72 5.4 Implicit Plot 80 5.5 Contour Plots 81 5.6 Density Plot 85 5.7 ListPlot and ListLinePlot 85 5.8 LogPlot, LogLogPlot, ErrorListPlot 88 5.9 Least Square Fit 89 5.10 Plotting of Complex Numbers 92 6 Three-Dimensional Graphics 97 6.1 Introduction 97 6.2 Plotting Function of Two Variables 97 6.3 Parametric Plots 101 6.4 3D Plots in Cylindrical and Spherical Coordinates 102 6.5 ContourPlot3D 105 6.6 ListContourPlot3D 108 6.7 ListSurfacePlot3D 110 6.8 Surface of Revolution 112 6.9 Conicoids 114 7 Matrices 123 7.1 Introduction 123 7.2 Properties of Matrices 123 7.2.1 Matrix Multiplication 123 7.3 Types of Matrices 123 7.4 The Rank of the Matrix 124 7.5 Special Matrices 124 7.6 Creation of a Matrix and Matrix Operations 125 7.6.1 Extraction of the Submatrices or the Elements of the Matrices 126 7.7 Properties of the Special Matrices 133 7.8 Direct Sum of Matrices 137 7.9 Direct Product of Matrices 137 7.10 Examples from Group Theory 138 7.10.1 SO(3) Group 138 7.10.2 SU(n)Group 139 7.10.3 SU(2) Group 140 7.10.4 SU(3) Group 141 8 Solving Algebraic and Transcendental Equations 143 8.1 Introduction 143 8.2 Solving System of Linear Equations 143 8.2.1 Number of Equations Equal to Number of Unknowns 144 8.2.2 Number of Equations Less than the Number of Unknowns 146 8.2.3 Number of Equations More than Number of Unknowns 146 8.3 Nonlinear Algebraic Equations 147 8.4 Solving Complex Equations 149 8.5 Solving Transcendental Equations 153 9 Eigenvalues and Eigenvectors of a Matrix 161 9.1 Introduction 161 9.2 Eigenvalues and Eigenvectors 161 9.2.1 Distinct Eigenvalues Having Independent Eigenvectors 162 9.2.2 Multiple Eigenvalues Having Independent Eigenvectors 163 9.2.3 Multiple Eigenvalues Not Having Independent Eigenvectors 165 9.3 Cayley–Hamilton Theorem 166 9.4 Diagonalization of a Matrix 167 9.4.1 Gram–Schmidt Orthogonalization Method 167 9.4.2 Diagonalizability of a Matrix 169 9.4.3 Case of a Non-diagonalizable Matrix 170 9.5 Some More Properties of the Special Matrices 172 9.6 Power of a Matrix 173 9.6.1 Roots of a Matrix 174 9.6.2 Exponential of a Matrix 174 9.6.3 Logarithm of a Matrix 174 9.6.4 Matrix Power Series 174 9.7 Power of a Matrix by Diagonalization 174 9.8 Bilinear, Quadratic, and Hermitian Forms 177 9.9 Principal Axes Transformation 178 10 Differential Calculus 183 10.1 Introduction 183 10.2 Limits 183 10.2.1 Evaluation of the Limits Using L’Hospital’s Rule 184 10.2.2 Application of L’Hospital’s Rule for the “Indeterminate Form” ∞ 185 ∞ 10.2.3 Evaluation of the Limit Using Taylor’s Theorem of Mean 186 10.3 Differentiation 188 10.3.1 Computation of Partial Derivatives 191 10.3.2 Total Derivative 193 10.4 Derivatives of Functions in Parametric Forms 195 10.4.1 Chain Rule for a Function of Two Independent Variables 196 10.4.2 Chain Rule for a Function of Three Independent Variables 196 10.5 Rolle’s Theorem 198 10.6 Mean Value Theorem 198 10.7 Series 200 10.8 Maxima and Minima 209 10.8.1 First Derivative Test 210 10.8.2 Second Derivative Test 211 10.8.3 Maximum and Minimum Values of a Function in a Closed Interval 213 10.8.4 Maxima and Minima of Two Variables 218 10.9 Differential Equations 222 10.9.1 Simple Harmonic Oscillator 225 10.9.2 LCR Circuit – Discharging of a Condenser Through an LR Circuit 227 11 Integral Calculus 235 11.1 Introduction 235 11.1.1 Indefinite Integral 235 11.1.2 Definite Integral 235 11.1.3 Numerical Value of the Integral 235 11.1.4 Assumptions While Evaluating the Integral 236 11.1.5 Multiple Integrals 236 11.1.6 Triple Integral 236 11.2 Evaluation of Indefinite Integrals 236 11.3 Evaluation of Definite Integrals 238 11.3.1 Numerical Value of the Integral 238 11.3.2 Options for Integration 239 11.4 Two and Three-Dimensional Integrals 240 11.5 Evaluation of the Integral in Polar Coordinates 242 11.6 Evaluation of Special Integrals 242 11.7 Orthogonal Polynomials 248 11.8 Area Between Curves 252 11.9 Application of Green’s Theorem in a Plane 256 11.10 Area of Surfaces of Revolution 257 12 Dirac Delta Function 263 12.1 Introduction 263 12.2 The Limiting Form of the Dirac Delta Function 263 12.3 Integral Representation of the Dirac Delta Function 265 12.4 Some Important Properties of the Dirac Delta Function 267 12.5 The Three-Dimensional Dirac Delta Function 270 13 Fourier Transforms 273 13.1 Introduction 273 13.2 Fourier Transforms 273 13.3 Scaling Property 280 13.4 Shifting Property 280 13.5 Fourier Sine and Cosine Transforms 281 13.6 Fourier Transform of the Derivative 282 13.7 Inverse Fourier Transform 282 13.8 Convolution 283 13.9 Convolution Theorem for Fourier Transforms 291 13.10 Parseval’s Theorem 293 14 Laplace Transforms 295 14.1 Introduction 295 14.2 Some Simple Examples 296 14.3 Properties of the Laplace Transforms 297 14.3.1 Linearity 297 14.3.2 Shifting Property 297 14.3.3 Scaling Property 297 14.4 Laplace Transform of the Derivative 298 14.5 Laplace Transform of Certain Special Functions 299 14.6 The Laplace Transform of Error and Complementary Error Functions 300 14.7 The Evaluation of a Certain Class of Definite Integrals Using Laplace Transforms 300 14.8 The Inverse Laplace Transform 302 14.8.1 Inverse Laplace Transform of Standard Functions 303 14.8.2 Shifting Properties 303 14.8.3 Inverse Laplace Transforms of Derivatives 305 14.9 Solving the Differential Equation by Laplace Transform 306 14.10 Convolution Theorem 307 14.11 Graphical Treatment of the Convolution 308 15 Vectors 315 15.1 Introduction 315 15.2 Properties 315 15.3 Vector Differentiation 319 15.4 Directional Derivative 320 15.5 Unit Vector Normal to the Surface 320 15.6 Gradient, Divergence, and Curl in the Cartesian Coordinate System 320 15.6.1 Gradient 320 15.6.2 Divergence 321 15.6.3 Curl 321 15.6.4 Laplacian Operator (∇ 2) 321 15.6.5 Examples 322 15.7 Expressing the Gradient, Divergence, and Curl in Other Coordinate Systems 326 15.7.1 Spherical Coordinate System 326 15.7.2 Cylindrical Coordinate System 330 15.8 Vector Plots 337 16 Linear Vector Spaces and Quantum Mechanics 343 16.1 Introduction 343 16.2 Linear Independence, Basis, and Dimension 343 16.3 Dimension of the Vector Space 343 16.4 Basis of the Vector Space 343 16.5 Completeness 344 16.6 Scalar Product in a Linear Vector Space 344 16.7 Norm of the Vector 344 16.8 Orthonormal Basis 344 16.9 Linear Independence of Functions 348 16.10 Hilbert Space 349 16.11 Completeness in Functional Space 350 16.12 The Dirac Ket and Bra Notation 351 16.12.1 The Scalar Product of Kets and Bras 351 16.12.2 Schwartz Inequality 352 16.12.3 The Orthonormal States 352 16.12.4 Basis 352 16.12.5 Probability Density 352 16.13 The Hermitian and Skew-Hermitian Operators in Dirac Ket and Bra Notation 352 16.14 Expectation Values 353 16.15 Matrix Representation of the Linear Operator 359 17 Application of Mathematica to Quantum Mechanics 361 17.1 Introduction 361 17.2 A Particle in a One-Dimensional Box 361 17.3 A Particle in a Two-Dimensional Box 365 17.4 The Hydrogen Atom Problem 368 17.4.1 The Orthonormal Property of the Hydrogen Atom Wave Functions 371 17.5 The One-Dimensional Linear Harmonic Oscillator Atom Problem 373 17.6 Three-Dimensional Harmonic Oscillator 377 17.7 Miscellaneous Problems 382 References 385 Index 387