Mathematics For Physicists

Description

This book covers the necessary aspects of mathematics for graduate students in physics and engineering. Advanced undergraduate students and researchers who intend to enter the field of theoretical physics can also pick up this book. The first eight chapters include variational method, Hilbert space and operators, ordinary linear differential equations, Bessel functions, Dirac delta function, the Green's function in mathematical physics, norm, integral equations. Beside these traditional contents, the last two chapters introduce some recent achievements of scientific research while presenting their mathematical background. Like the basis of number theory and its application in physics, material science and other scientific fields, the fundamental equations in spaces with arbitrary dimensions, not limited to Euclid space; Pseudo spherical coordinates. Plain terminologies were used to present the concept of metric, as well as new and interesting work on the Klein-Gorden equation and Maxwell equation. Introduction Preface Contents Chapter 1. Variational Method 1.1. Functional and Its Extremal Problems 1.1.1. The conception of functional 1.1.2. The extremes of functionals 1.2. The Variational of Functionals and the Simplest Euler Equation 1.2.1. The variational of functionals 1.2.2. The simplest Euler equation 1.3. The Cases of Multifunctions and Multivariates 1.3.1. Multifunctions 1.3.2. Multivariates 1.4. Functional Extremes under Certain Conditions 1.4.1. Isoperimetric problem 1.4.2. Geodesic problem 1.5. Natural Boundary Conditions 1.6. Variational Principle 1.6.1. Variational principle of classical mechanics 1.6.2. Variational principle of quantum mechanics 1.7. The Applications of the Variational Method in Physics 1.7.1. The applications in classical physics 1.7.2. The applications in quantum mechanics Exercises Chapter 2. Hilbert Space 2.1. Linear Space, Inner Product Space and Hilbert Space 2.1.1. Linear space 2.1.2. Inner product space 2.1.3. Hilbert space 2.2. Operators in Inner Product Spaces 2.2.1. Operators and adjoint operators 2.2.2. Self-adjoint operators 2.2.3. The alternative theorem for the solutions of linear algebraic equations 2.3. Complete Set of Orthonormal Functions 2.3.1. Three kinds of convergences 2.3.2. The completeness of a set of functions 2.3.3. N-dimensional space and Hilbert function space 2.3.4. Orthogonal polynomials 2.4. Polynomial Approximation 2.4.1. Weierstrass theorem 2.4.2. Polynomial approximation Exercises Chapter 3. Linear Ordinary Differential Equations of Second Order 3.1. General Theory 3.1.1. The existence and uniqueness of solutions 3.1.2. The structure of solutions of homogeneous equations 3.1.3. The solutions of inhomogeneous equations 3.2. Sturm-Liouville Eigenvalue Problem 3.2.1. The form of Sturm-Liouville equations 3.2.2. The boundary conditions of Sturm-Liouville equations 3.2.3. Sturm-Liouville eigenvalue problem 3.3. The Polynomial Solutions of Sturm-Liouville Equations 3.3.1. Possible forms of kernel and weight functions 3.3.2. The expressions in series and in derivatives of the polynomials 3.3.3. Generating functions 3.3.4. The completeness theorem of orthogonal polynomials as Sturm-Liouville solutions 3.3.5. Applications in numerical integrations 3.4. Equations and Functions that Relate to the Polynomial Solutions 3.4.1. Laguerre functions 3.4.2. Legendre functions 3.4.3. Chebyshev functions 3.4.4. Hermite functions 3.5. Complex Analysis Theory of the Ordinary Differential Equations of Second Order 3.5.1. Solutions of homogeneous equations 3.5.2. Ordinary differential equations of second order 3.6. Non-Self-Adjoint Ordinary Differential Equations of Second Order 3.6.1. Adjoint equations of ordinary differential equations 3.6.2. Sturm-Liouville operator 3.6.3. Complete set of non-self-adjoint ordinary differential equations of second order 3.7. The Conditions under Which Inhomogeneous Equations have Solutions Exercises Appendix 3A. Generalization of Sturm-Liouville Theorem to Dirac Equation Chapter 4. Bessel Functions 4.1. Bessel Equation 4.1.1. Bessel equation and its solutions 4.1.2. Bessel functions of the first and second kinds 4.2. Fundamental Properties of Bessel Functions 4.2.1. Recurrence relations of Bessel functions 4.2.2. Asymptotic formulas of Bessel functions 4.2.3. Zeros of Bessel functions 4.2.4. Wronskian 4.3. Bessel Functions of Integer Orders 4.3.1. Parity and the values at certain points 4.3.2. Generating function of Bessel functions of integer orders 4.4. Bessel Functions of Half-Integer Orders 4.5. Bessel Functions of the Third Kind and Spherical Bessel Functions 4.5.1. Bessel functions of the third kind 4.5.2. Spherical Bessel functions 4.6. Modified Bessel Functions 4.6.1. Modified Bessel functions of the first and second kinds 4.6.2. Modified Bessel functions of integer orders 4.7. Bessel Functions with Real Arguments 4.7.1. Eigenvalue problem of Bessel equation 4.7.2. Properties of eigenfunctions 4.7.3. Eigenvalue problem of spherical Bessel equation Exercises Chapter 5. The Dirac Delta Function 5.1. Definition and Properties of the Delta Function 5.1.1. Definition of the delta function 5.1.2. The delta function is a generalized function 5.1.3. The Fourier and Laplace transformations of the delta function 5.1.4. Derivative and integration of generalized functions 5.1.5. Complex argument in the delta function 5.2. The Delta Function as Weak Convergence Limits of Ordinary Functions 5.3. The Delta Function in Multidimensional Spaces 5.3.1. Cartesian coordinate system 5.3.2. The transform from Cartesian coordinates to curvilinear coordinates 5.4. Generalized Fourier Series Expansion of the Delta Function Exercises Chapter 6. Green's Function 6.1. Fundamental Theory of Green's Function 6.1.1. Definition of Green's function 6.1.2. Properties of Green's function 6.1.3. Methods of obtaining Green's function 6.1.4. Physical meaning of Green's function 6.2. The Basic Solution of Laplace Operator 6.2.1. Three-dimensional space 6.2.2. Two-dimensional space 6.2.3. One-dimensional space 6.3. Green's Function of a Damped Oscillator 6.3.1. Solution of homogeneous equation 6.3.2. Obtaining Green's function 6.3.3. Generalized solution of the equation 6.3.4. The case without damping 6.3.5. The influence of boundary conditions 6.4. Green's Function of Ordinary Differential Equations of Second Order 6.4.1. The symmetry of Green's function 6.4.2. Solutions of boundary value problem of ordinary differential equations of second order 6.4.3. Modified Green's function 6.4.4. Examples of solving boundary value problem of ordinary differential equations of second order 6.5. Green's Function in Multi-dimensional Spaces 6.5.1. Ordinary differential equations of second order and Green's function 6.5.2. Examples in two-dimensional space 6.6. Green's Function of Ordinary Differential Equation of First Order 6.6.1. Boundary value problem of inhomogeneous equations 6.6.2. Boundary value problem of homogeneous equations 6.6.3. Inhomogeneous equations and Green's function 6.6.4. General solutions of boundary value problem 6.7. Green's Function of Non-Self-Adjoint Equations 6.7.1. Adjoint Green's function 6.7.2. Solutions of inhomogeneous equations Exercises Chapter 7. Norm 7.1. Banach Space 7.1.1. Banach space 7.1.2. Hölder inequality 7.1.3. Minkowski inequality 7.2. Vector Norms 7.2.1. Vector norms 7.2.2. Equivalence between vector norms 7.3. Matrix Norms 7.3.1. Matrix norms 7.3.2. Spectral norm and spectral radius of matrices 7.4. Operator Norms 7.4.1. Operator norms 7.4.2. Adjoint operators 7.4.3. Projection operators Exercises Chapter 8. Integral Equations 8.1. Fundamental Theory of Integral Equations 8.1.1. Definition and classification of integral equations 8.1.2. Relations between integral equations and differential equations 8.1.3. Theory of homogeneous integral equations 8.2. Iteration Technique for Linear Integral Equations 8.2.1. The second kind of Fredholm integral equations 8.2.2. The second kind of Volterra integral equations 8.3. Iteration Technique of Inhomogeneous Integral Equations 8.3.1. Iteration procedure 8.3.2. Lipschitz condition 8.3.3. Use of contraction 8.3.4. Anharmonic vibration of a spring 8.4. Fredholm Linear Equations with Degenerated Kernels 8.4.1. Separable kernels 8.4.2. Kernels with a finite rank 8.4.3. Expansion of kernel in terms of eigenfunctions 8.5. Integral Equations of Convolution Type 8.5.1. Fredholm integral equations of convolution type 8.5.2. Volterra integral equations of convolution type 8.6. Integral Equations with Polynomials 8.6.1. Fredholm integral equations with polynomials 8.6.2. Generating function method Exercises Chapter 9. Application of Number Theory in Inverse Problems in Physics 9.1. Chen-Möbius Transformation 9.1.1. Introduction 9.1.2. Möbius transformation 9.1.3. Chen-Möbius transformation 9.2. Inverse Problem in Phonon Density of States in Crystals 9.2.1. Inversion formula 9.2.2. Low-temperature approximation 9.2.3. High-temperature approximation 9.3. Inverse Problem in the Interaction Potential between Atoms 9.3.1. One-dimensional case 9.3.2. Two-dimensional case 9.3.3. Three-dimensional case 9.4. Additive Möbius Inversion and Its Applications 9.4.1. Additive Möbius inversion of functions and its applications 9.4.2. Additive Möbius inversion of series and its applications 9.5. Inverse Problem in Crystal Surface Relaxation and Interfacial Potentials 9.5.1. Pair potentials between an isolated atom and atoms in a semi-infinite crystal 9.5.2. Relaxation of atoms at a crystal surface 9.5.3. Inverse problem of interfacial potentials 9.6. Construction of Biorthogonal Complete Function Sets Exercises Appendix 9A. Some Values of Riemann ζ Function Appendix 9B. Calculation of Reciprocal Coefficients Chapter 10. Fundamental Equations in Spaces with Arbitrary Dimensions 10.1. Euclid Spaces with Arbitrary Dimensions 10.1.1. Cartesian coordinate system and spherical coordinates 10.1.2. Gradient, divergence and Laplace operator 10.2. Green's Functions of the Laplace Equation and Helmholtz Equation 10.2.1. Green's function of the Laplace equation 10.2.2. Green's function of the Helmholtz equation 10.3. Radial Equations under Central Potentials 10.3.1. Radial equation under a central potential in multidimensional spaces 10.3.2. Helmholtz equation 10.3.3. Infinitely deep spherical potential 10.3.4. Finitely deep spherical potential 10.3.5. Coulomb potential 10.3.6. Harmonic potential 10.3.7. Molecular potential with both negative powers 10.3.8. Molecular potential with positive and negative powers 10.3.9. Attractive potential with exponential decay 10.3.10. Conditions that the radial equation has analytical solutions 10.4. Solutions of Angular Equations 10.4.1. Four-dimensional space 10.4.2. Five-dimensional space 10.4.3. N-dimensional space 10.5. Pseudo Spherical Coordinates 10.5.1. Pseudo coordinates in four-dimensional space 10.5.2. Solutions of Laplace equation 10.5.3. Five- and six-dimensional spaces 10.6. Non-Euclidean Space 10.6.1. Metric tensor 10.6.2. Five-dimensional Minkowski space and four-dimensional de Sitter space 10.6.3. Maxwell equations in de Sitter spacetime Exercises Appendix 10A. Hypergeometric Equation and Hypergeometric Functions References 15 Answers of Selected Exercises Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Author Index BCDEFGHJKLMN PRSTVWY Subject Index L₂,ε,l₂,n-,p-; ab c def ghi jklm nopqr st uvwz