Mathematica for Physicists and Engineers

K. B. Vijaya Kumar is a professor of physics at the N.M.A.M Institute of Technology, Nitte, India. His research is focused on theoretical and computational nuclear and particle physics. Antony P. Monteiro is working in the Department of Physics at St. Philomena College, Puttur, India. He has more than thirteen years of teaching experience and has authored several books in various fields of physics.

• Preface xiiiForeword xviiAbout the Authors xix1 Preliminary Notions 11.1 Introduction 11.2 Versions of Mathematica 11.3 Getting Started 21.4 Simple Calculations 21.4.1 Arithmetic Operations 21.4.2 Approximate Numerical Results 31.4.3 Algebraic Calculations 31.4.4 Defining Variables 41.4.5 Using the Previous Results 51.4.6 Suppressing the Output 61.4.7 Sequences of Operations 61.5 Built-in Functions 71.6 Additional Features 91.6.1 Arbitrary-Precision Calculations 91.6.2 Value for Symbols 101.6.3 Defining Naming and Evaluating Functions 101.6.4 Composition of Functions 111.6.5 Conditional Assignment 121.6.6 Warnings and Messages 131.6.7 Interrupting Calculations 131.6.8 Using Symbols to Tag Objects 132 Basic Mathematical Operations 152.1 Introduction 152.2 Basic Algebraic Operations 152.3 Basic Trigonometric Operations 202.4 Basic Operations with Complex Numbers 213 Lists and Tables 253.1 Introduction 253.2 Lists 253.3 Arrays 263.4 Tables 263.5 Extracting the Elements from the Arrays/Tables 294 Two-Dimensional Graphics 314.1 Introduction 314.2 Plotting Functions of a Single Variable 314.3 Additional Commands 344.4 Plot Styles 444.5 Probability Distribution 584.5.1 Binomial Distribution 584.5.2 Poisson Distribution 584.5.3 Normal or Gaussian Distribution 594.6 Some More Useful Commands 615 Parametric, Polar, Contour, Density, and List Plots 655.1 Introduction 655.2 Parametric Plotting 655.3 Polar Plots 725.3.1 Polar Plots of Circles 725.3.2 Polar Plots of Ellipse, Parabola, and Hyperbola 725.4 Implicit Plot 805.5 Contour Plots 815.6 Density Plot 855.7 ListPlot and ListLinePlot 855.8 LogPlot, LogLogPlot, ErrorListPlot 885.9 Least Square Fit 895.10 Plotting of Complex Numbers 926 Three-Dimensional Graphics 976.1 Introduction 976.2 Plotting Function of Two Variables 976.3 Parametric Plots 1016.4 3D Plots in Cylindrical and Spherical Coordinates 1026.5 ContourPlot3D 1056.6 ListContourPlot3D 1086.7 ListSurfacePlot3D 1106.8 Surface of Revolution 1126.9 Conicoids 1147 Matrices 1237.1 Introduction 1237.2 Properties of Matrices 1237.2.1 Matrix Multiplication 1237.3 Types of Matrices 1237.4 The Rank of the Matrix 1247.5 Special Matrices 1247.6 Creation of a Matrix and Matrix Operations 1257.6.1 Extraction of the Submatrices or the Elements of the Matrices 1267.7 Properties of the Special Matrices 1337.8 Direct Sum of Matrices 1377.9 Direct Product of Matrices 1377.10 Examples from Group Theory 1387.10.1 SO(3) Group 1387.10.2 SU(n)Group 1397.10.3 SU(2) Group 1407.10.4 SU(3) Group 1418 Solving Algebraic and Transcendental Equations 1438.1 Introduction 1438.2 Solving System of Linear Equations 1438.2.1 Number of Equations Equal to Number of Unknowns 1448.2.2 Number of Equations Less than the Number of Unknowns 1468.2.3 Number of Equations More than Number of Unknowns 1468.3 Nonlinear Algebraic Equations 1478.4 Solving Complex Equations 1498.5 Solving Transcendental Equations 1539 Eigenvalues and Eigenvectors of a Matrix 1619.1 Introduction 1619.2 Eigenvalues and Eigenvectors 1619.2.1 Distinct Eigenvalues Having Independent Eigenvectors 1629.2.2 Multiple Eigenvalues Having Independent Eigenvectors 1639.2.3 Multiple Eigenvalues Not Having Independent Eigenvectors 1659.3 Cayley–Hamilton Theorem 1669.4 Diagonalization of a Matrix 1679.4.1 Gram–Schmidt Orthogonalization Method 1679.4.2 Diagonalizability of a Matrix 1699.4.3 Case of a Non-diagonalizable Matrix 1709.5 Some More Properties of the Special Matrices 1729.6 Power of a Matrix 1739.6.1 Roots of a Matrix 1749.6.2 Exponential of a Matrix 1749.6.3 Logarithm of a Matrix 1749.6.4 Matrix Power Series 1749.7 Power of a Matrix by Diagonalization 1749.8 Bilinear, Quadratic, and Hermitian Forms 1779.9 Principal Axes Transformation 17810 Differential Calculus 18310.1 Introduction 18310.2 Limits 18310.2.1 Evaluation of the Limits Using L’Hospital’s Rule 18410.2.2 Application of L’Hospital’s Rule for the “Indeterminate Form” ∞ 185 ∞10.2.3 Evaluation of the Limit Using Taylor’s Theorem of Mean 18610.3 Differentiation 18810.3.1 Computation of Partial Derivatives 19110.3.2 Total Derivative 19310.4 Derivatives of Functions in Parametric Forms 19510.4.1 Chain Rule for a Function of Two Independent Variables 19610.4.2 Chain Rule for a Function of Three Independent Variables 19610.5 Rolle’s Theorem 19810.6 Mean Value Theorem 19810.7 Series 20010.8 Maxima and Minima 20910.8.1 First Derivative Test 21010.8.2 Second Derivative Test 21110.8.3 Maximum and Minimum Values of a Function in a Closed Interval 21310.8.4 Maxima and Minima of Two Variables 21810.9 Differential Equations 22210.9.1 Simple Harmonic Oscillator 22510.9.2 LCR Circuit – Discharging of a Condenser Through an LR Circuit 22711 Integral Calculus 23511.1 Introduction 23511.1.1 Indefinite Integral 23511.1.2 Definite Integral 23511.1.3 Numerical Value of the Integral 23511.1.4 Assumptions While Evaluating the Integral 23611.1.5 Multiple Integrals 23611.1.6 Triple Integral 23611.2 Evaluation of Indefinite Integrals 23611.3 Evaluation of Definite Integrals 23811.3.1 Numerical Value of the Integral 23811.3.2 Options for Integration 23911.4 Two and Three-Dimensional Integrals 24011.5 Evaluation of the Integral in Polar Coordinates 24211.6 Evaluation of Special Integrals 24211.7 Orthogonal Polynomials 24811.8 Area Between Curves 25211.9 Application of Green’s Theorem in a Plane 25611.10 Area of Surfaces of Revolution 25712 Dirac Delta Function 26312.1 Introduction 26312.2 The Limiting Form of the Dirac Delta Function 26312.3 Integral Representation of the Dirac Delta Function 26512.4 Some Important Properties of the Dirac Delta Function 26712.5 The Three-Dimensional Dirac Delta Function 27013 Fourier Transforms 27313.1 Introduction 27313.2 Fourier Transforms 27313.3 Scaling Property 28013.4 Shifting Property 28013.5 Fourier Sine and Cosine Transforms 28113.6 Fourier Transform of the Derivative 28213.7 Inverse Fourier Transform 28213.8 Convolution 28313.9 Convolution Theorem for Fourier Transforms 29113.10 Parseval’s Theorem 29314 Laplace Transforms 29514.1 Introduction 29514.2 Some Simple Examples 29614.3 Properties of the Laplace Transforms 29714.3.1 Linearity 29714.3.2 Shifting Property 29714.3.3 Scaling Property 29714.4 Laplace Transform of the Derivative 29814.5 Laplace Transform of Certain Special Functions 29914.6 The Laplace Transform of Error and Complementary Error Functions 30014.7 The Evaluation of a Certain Class of Definite Integrals Using Laplace Transforms 30014.8 The Inverse Laplace Transform 30214.8.1 Inverse Laplace Transform of Standard Functions 30314.8.2 Shifting Properties 30314.8.3 Inverse Laplace Transforms of Derivatives 30514.9 Solving the Differential Equation by Laplace Transform 30614.10 Convolution Theorem 30714.11 Graphical Treatment of the Convolution 30815 Vectors 31515.1 Introduction 31515.2 Properties 31515.3 Vector Differentiation 31915.4 Directional Derivative 32015.5 Unit Vector Normal to the Surface 32015.6 Gradient, Divergence, and Curl in the Cartesian Coordinate System 32015.6.1 Gradient 32015.6.2 Divergence 32115.6.3 Curl 32115.6.4 Laplacian Operator (∇ 2) 32115.6.5 Examples 32215.7 Expressing the Gradient, Divergence, and Curl in Other Coordinate Systems 32615.7.1 Spherical Coordinate System 32615.7.2 Cylindrical Coordinate System 33015.8 Vector Plots 33716 Linear Vector Spaces and Quantum Mechanics 34316.1 Introduction 34316.2 Linear Independence, Basis, and Dimension 34316.3 Dimension of the Vector Space 34316.4 Basis of the Vector Space 34316.5 Completeness 34416.6 Scalar Product in a Linear Vector Space 34416.7 Norm of the Vector 34416.8 Orthonormal Basis 34416.9 Linear Independence of Functions 34816.10 Hilbert Space 34916.11 Completeness in Functional Space 35016.12 The Dirac Ket and Bra Notation 35116.12.1 The Scalar Product of Kets and Bras 35116.12.2 Schwartz Inequality 35216.12.3 The Orthonormal States 35216.12.4 Basis 35216.12.5 Probability Density 35216.13 The Hermitian and Skew-Hermitian Operators in Dirac Ket and Bra Notation 35216.14 Expectation Values 35316.15 Matrix Representation of the Linear Operator 35917 Application of Mathematica to Quantum Mechanics 36117.1 Introduction 36117.2 A Particle in a One-Dimensional Box 36117.3 A Particle in a Two-Dimensional Box 36517.4 The Hydrogen Atom Problem 36817.4.1 The Orthonormal Property of the Hydrogen Atom Wave Functions 37117.5 The One-Dimensional Linear Harmonic Oscillator Atom Problem 37317.6 Three-Dimensional Harmonic Oscillator 37717.7 Miscellaneous Problems 382References 385Index 387