Mathematica for Physicists and Engineers

Matrices 7.7 Properties of the Special Matrices 7.8 The Direct Sum of Matrices 7.9 The Direct Product of Matrices 7.10 Examples from Group Theory 7.10.1 SO(3) Group 7.10.2 SU(n) Group 7.10.3 SU(2) Group 7.10.4 SU(3) Group CHAPTER 8 - Solving Algebraic and Transcendental Equations 8.1 Solving Systems of Linear Equations 8.1.1 Number of Equations Equal to the Number of Unknowns 8.1.2 Number of Equations Less Than the Number of Unknowns 8.1.3 Number of Equations More Than the Number of Unknowns 8.2 Non- Linear Algebraic Equations 8.3 Solving Transcendental Equations CHAPTER 9 - Eigenvalues and Eigenvectors of a Matrix: Matrix Diagonalization 9.1 Introduction 9.2 Eigenvalues and Vectors of a Matrix 9.2.1 Distinct Eigenvalues having Independent Eigenvectors 9.2.2 Multiple Eigenvalues having Independent Eigenvectors 9.2.3 Multiple Eigenvalues not having Independent Eigenvectors 9.3 The Cayley-Hamilton Theorem 9.4 Diagonalization of a Matrix 9.4.1 Gram-Schmidt Orthogonalization Method 9.4.2 Diagonalizability of a Matrix 9.4.3 Case of a Non-diagonalizable Matrix 9.5 Some More Properties of Special Matrices 9.6 Power of a Matrix 9.6.1 Roots of a Matrix 9.6.2 Exponential of a Matrix 9.6.3 Logarithm of a Matrix 9.7 Power of a Matrix by Diagonalization 9.8 Bilinear, Quadratic and Hermitian Forms 9.9 Principal Axes Transformation CHAPTER 10 - Differential Calculus 10.1 Introduction 10.2 Limits 10.2.1 Evaluation of the Limits Using L' Hospital?s Rule 10.2.2 Application of L' Hospital?s Rule for "Indeterminate Form 10.2.3 Evaluation of the Limit Using Taylor's Theorem of Mean 10.3 Differentiation 10.3.1 Computation of Partial Derivatives 10.3.2 Total Derivative 10.4 Derivatives of Functions in Parametric Forms 10.4.1 Chain Rule for a Function of Two Independent Variables 10.4.2 Chain Rule for a Function of Three Independent Variables 10.5 Rolle's Theorem 10.6 Mean Value Theorem 10.7 Series 10.8 Maxima and Minima 10.8.1 First Derivative Test 10.8.2 Second Derivative Test 10.8.3 Maximum and Minimum Values of a Function in a Closed Interval 10.8.4 Maxima and Minima of Two Variables 10.9 Differential Equations 10.9.1 Simple Harmonic Oscillator 10.9.2 LCR Circuit- Discharging of a Condenser through LR Circuit CHAPTER 11 - Integral Calculus 11.1 Introduction 11.1.1 Indefinite Integral 11.1.2 Definite Integral 11.1.3 Numerical Value of the Integral 11.1.4 Assumptions in Evaluating the Integral 11.1.5 Multiple Integrals 11.1.6 Triple Integral 11.2 Evaluation on Indefinite Integrals 11.3 Evaluation of Definite Integrals 11.3.1 Numerical Value of the Integral 11.3.2 Options for Integration 11.4 Two and Three-Dimensional Integrals 11.5 Evaluation of the Integral in Polar Coordinates 11.6 Evaluation of Special Integrals 11.7 Orthogonal Polynomials 11.8 The Area between Curves 11.9 The Application of Green's Theorem in a Plane CHAPTER 12 - Dirac Delta Function 12.1 Introduction 12.2 The Limiting Form of Dirac Delta Function 12.3 Integral Representation of the Dirac Delta Function 12.4 Some Important Properties of the Dirac Delta Function 12.5 The Three-Dimensional Dirac Delta Function CHAPTER 13 - Fourier Transform 13.1 Introduction 13.2 Fourier Transforms 13.3 Scaling Property 13.4 Shifting Property 13.5 Fourier Sine and Cosine Transforms 13.6 Fourier Transform of the Derivative 13.7 Inverse Fourier Transform 13.8 Convolution 13.9 Convolution Theorem for Fourier Transforms 13.10 Parseval's Theorem CHAPTER 14 - Laplace Transforms 14.1 Introduction 14.2 Some Simple Examples 14.3 Properties of the Laplace Transform 14.3.1 Linearity 14.3.2 Shifting Property 14.3.3 Scaling 14.4 Laplace Transform of Derivative 14.5 Laplace Transform of Certain Special Functions 14.6 Laplace Transform of Error and Complementary Error Functions 14.7 The Evaluation of Certain Class of Definite Integrals Using Laplace Transform 14.8 The Inverse Laplace Transform 14.8.1 Inverse Laplace Transform of Standard Functions 14.8.2 Shifting Property 14.8.3 Inverse Laplace Transforms of Derivatives 14.9 Solving the Differential Equation by Laplace Transform 14.10 Convolution Theorem 14.11 Graphical Treatment of the Convolution CHAPTER 15 - Vectors 15.1 Definition and Properties 15.2 Vector Differentiation 15.3 Directional Derivative 15.4 Unit Vector Normal to the Surface 15.5 Gradient, Divergence and Curl in Cartesian Coordinate system 15.5.1 Gradient 15.5.2 Divergence 15.5.3 Curl 15.5.4 Laplacian Operator 15.5.5 Examples 15.6 Expressing the Gradient, Divergence and Curl in Other Coordinate Systems 15.6.1 Spherical Coordinate System 15.6.2 Cylindrical Coordinate System 15.7 Vector Plots CHAPTER 16 - Linear Vector Spaces and Quantum Mechanics 16.1 Introduction 16.2 Linear Independence, Basis and Dimension 16.3 Dimension of the Vector Space 16.4 Basis of the Vector Space 16.5 Completeness 16.6 Scalar Product in a Linear Vector Space 16.7 Norm of the Vector 16.8 Orthonormal Basis 16.9 Linear Independence of Functions 16.10 Hilbert Space 16.11 Completeness in Functional Space 16.12 Dirac Ket and Bra Notation 16.12.1 The Scalar Properties of the Kets and Bras 16.12.2 Schwartz Inequality 16.12.3 The Orthonormal States 16.12.4 Basis 16.12.5 Probability Density 16.13 The Hermitian and Skew-Hermitian Operators in Dirac Ket and Bra Notation 16.14 Expectation Values 16.15 Matrix Representation of the Linear Operator CHAPTER 17 - Applications of Mathematica to Quantum Mechanics 17.1 Introduction 17.2 A Particle in a One-Dimensional Box 17.3 A Particle in a Two-Dimensional Box 17.4 The Hydrogen Atom Problem 17.4.1 The Orthonormal Property of the Hydrogen Atom Wave Functions 17.5 One-Dimensional Linear Harmonic Oscillator Atom Problem 17.6 Three-Dimensional Harmonic Oscillator Problem 17.7 Miscellaneous Problems References Index