内容

目次

Preface xix 1 Legendre Equation and Polynomials 1 1.1 Second–Order Differential Equations of Physics 1 1.2 Legendre Equation 2 1.2.1 Method of Separation of Variables 4 1.2.2 Series Solution of the Legendre Equation 4 1.2.3 Frobenius Method – Review 7 1.3 Legendre Polynomials 8 1.3.1 Rodriguez Formula 10 1.3.2 Generating Function 10 1.3.3 Recursion Relations 12 1.3.4 Special Values 12 1.3.5 Special Integrals 13 1.3.6 Orthogonality and Completeness 14 1.3.7 Asymptotic Forms 17 1.4 Associated Legendre Equation and Polynomials 18 1.4.1 Associated Legendre Polynomials Pm l (x) 20 1.4.2 Orthogonality 21 1.4.3 Recursion Relations 22 1.4.4 Integral Representations 24 1.4.5 Associated Legendre Polynomials for m < 0 26 1.5 Spherical Harmonics 27 1.5.1 AdditionTheorem of Spherical Harmonics 30 1.5.2 Real Spherical Harmonics 33 Bibliography 33 Problems 34 2 Laguerre Polynomials 39 2.1 Central Force Problems in Quantum Mechanics 39 2.2 Laguerre Equation and Polynomials 41 2.2.1 Generating Function 42 2.2.2 Rodriguez Formula 43 2.2.3 Orthogonality 44 2.2.4 Recursion Relations 45 2.2.5 Special Values 46 2.3 Associated Laguerre Equation and Polynomials 46 2.3.1 Generating Function 48 2.3.2 Rodriguez Formula and Orthogonality 49 2.3.3 Recursion Relations 49 Bibliography 49 Problems 50 3 Hermite Polynomials 53 3.1 Harmonic Oscillator in QuantumMechanics 53 3.2 Hermite Equation and Polynomials 54 3.2.1 Generating Function 56 3.2.2 Rodriguez Formula 56 3.2.3 Recursion Relations and Orthogonality 57 Bibliography 61 Problems 62 4 Gegenbauer and Chebyshev Polynomials 65 4.1 Wave Equation on a Hypersphere 65 4.2 Gegenbauer Equation and Polynomials 68 4.2.1 Orthogonality and the Generating Function 68 4.2.2 Another Representation of the Solution 69 4.2.3 The Second Solution 70 4.2.4 Connection with the Gegenbauer Polynomials 71 4.2.5 Evaluation of the Normalization Constant 72 4.3 Chebyshev Equation and Polynomials 72 4.3.1 Chebyshev Polynomials of the First Kind 72 4.3.2 Chebyshev and Gegenbauer Polynomials 73 4.3.3 Chebyshev Polynomials of the Second Kind 73 4.3.4 Orthogonality and Generating Function 74 4.3.5 Another Definition 75 Bibliography 76 Problems 76 5 Bessel Functions 81 5.1 Bessel’s Equation 83 5.2 Bessel Functions 83 5.2.1 Asymptotic Forms 84 5.3 Modified Bessel Functions 86 5.4 Spherical Bessel Functions 87 5.5 Properties of Bessel Functions 88 5.5.1 Generating Function 88 5.5.2 Integral Definitions 89 5.5.3 Recursion Relations of the Bessel Functions 89 5.5.4 Orthogonality and Roots of Bessel Functions 90 5.5.5 Boundary Conditions for the Bessel Functions 91 5.5.6 Wronskian of Pairs of Solutions 94 5.6 Transformations of Bessel Functions 95 5.6.1 Critical Length of a Rod 96 Bibliography 98 Problems 99 6 Hypergeometric Functions 103 6.1 Hypergeometric Series 103 6.2 Hypergeometric Representations of Special Functions 107 6.3 Confluent Hypergeometric Equation 108 6.4 Pochhammer Symbol and Hypergeometric Functions 109 6.5 Reduction of Parameters 113 Bibliography 115 Problems 115 7 Sturm–Liouville Theory 119 7.1 Self–Adjoint Differential Operators 119 7.2 Sturm–Liouville Systems 120 7.3 Hermitian Operators 121 7.4 Properties of Hermitian Operators 122 7.4.1 Real Eigenvalues 122 7.4.2 Orthogonality of Eigenfunctions 123 7.4.3 Completeness and the ExpansionTheorem 123 7.5 Generalized Fourier Series 125 7.6 Trigonometric Fourier Series 126 7.7 Hermitian Operators in Quantum Mechanics 127 Bibliography 129 Problems 130 8 Factorization Method 133 8.1 Another Form for the Sturm–Liouville Equation 133 8.2 Method of Factorization 135 8.3 Theory of Factorization and the Ladder Operators 136 8.4 Solutions via the Factorization Method 141 8.4.1 Case I (m > 0 and 𝜇(m) is an increasing function) 141 8.4.2 Case II (m > 0 and 𝜇(m) is a decreasing function) 142 8.5 Technique and the Categories of Factorization 143 8.5.1 Possible Forms for k(z,m) 143 8.5.1.1 Positive powers of m 143 8.5.1.2 Negative powers of m 146 8.6 Associated Legendre Equation (Type A) 148 8.6.1 Determining the Eigenvalues, 𝜆l 149 8.6.2 Construction of the Eigenfunctions 150 8.6.3 Ladder Operators for m 151 8.6.4 Interpretation of the L+ and L− Operators 153 8.6.5 Ladder Operators for l 155 8.6.6 Complete Set of Ladder Operators 159 8.7 Schrödinger Equation and Single–Electron Atom (Type F) 160 8.8 Gegenbauer Functions (Type A) 162 8.9 Symmetric Top (Type A) 163 8.10 Bessel Functions (Type C) 164 8.11 Harmonic Oscillator (Type D) 165 8.12 Differential Equation for the Rotation Matrix 166 8.12.1 Step–Up/Down Operators for m 166 8.12.2 Step–Up/Down Operators for m' 167 8.12.3 Normalized Functions with m = m' = l 168 8.12.4 Full Matrix for l = 2 168 8.12.5 Step–Up/Down Operators for l 170 Bibliography 171 Problems 171 9 Coordinates and Tensors 175 9.1 Cartesian Coordinates 175 9.1.1 Algebra of Vectors 176 9.1.2 Differentiation of Vectors 177 9.2 Orthogonal Transformations 178 9.2.1 Rotations About Cartesian Axes 182 9.2.2 Formal Properties of the Rotation Matrix 183 9.2.3 Euler Angles and Arbitrary Rotations 183 9.2.4 Active and Passive Interpretations of Rotations 185 9.2.5 Infinitesimal Transformations 186 9.2.6 Infinitesimal Transformations Commute 188 9.3 Cartesian Tensors 189 9.3.1 Operations with Cartesian Tensors 190 9.3.2 Tensor Densities or Pseudotensors 191 9.4 Cartesian Tensors and theTheory of Elasticity 192 9.4.1 Strain Tensor 192 9.4.2 Stress Tensor 193 9.4.3 Thermodynamics and Deformations 194 9.4.4 Connection between Shear and Strain 196 9.4.5 Hook’s Law 200 9.5 Generalized Coordinates and General Tensors 201 9.5.1 Contravariant and Covariant Components 202 9.5.2 Metric Tensor and the Line Element 203 9.5.3 Geometric Interpretation of Components 206 9.5.4 Interpretation of the Metric Tensor 207 9.6 Operations with General Tensors 214 9.6.1 Einstein Summation Convention 214 9.6.2 Contraction of Indices 214 9.6.3 Multiplication of Tensors 214 9.6.4 The Quotient Theorem 214 9.6.5 Equality of Tensors 215 9.6.6 Tensor Densities 215 9.6.7 Differentiation of Tensors 216 9.6.8 Some Covariant Derivatives 219 9.6.9 Riemann Curvature Tensor 220 9.7 Curvature 221 9.7.1 Parallel Transport 222 9.7.2 Round Trips via Parallel Transport 223 9.7.3 Algebraic Properties of the Curvature Tensor 225 9.7.4 Contractions of the Curvature Tensor 226 9.7.5 Curvature in n Dimensions 227 9.7.6 Geodesics 229 9.7.7 Invariance Versus Covariance 229 9.8 Spacetime and Four–Tensors 230 9.8.1 Minkowski Spacetime 230 9.8.2 Lorentz Transformations and Special Relativity 231 9.8.3 Time Dilation and Length Contraction 233 9.8.4 Addition of Velocities 233 9.8.5 Four–Tensors in Minkowski Spacetime 234 9.8.6 Four–Velocity 237 9.8.7 Four–Momentum and Conservation Laws 238 9.8.8 Mass of a Moving Particle 240 9.8.9 Wave Four–Vector 240 9.8.10 Derivative Operators in Spacetime 241 9.8.11 Relative Orientation of Axes in K and K Frames 241 9.9 Maxwell’s Equations in Minkowski Spacetime 243 9.9.1 Transformation of Electromagnetic Fields 246 9.9.2 Maxwell’s Equations in Terms of Potentials 246 9.9.3 Covariance of Newton’s Dynamic Theory 247 Bibliography 248 Problems 249 10 Continuous Groups and Representations 257 10.1 Definition of a Group 258 10.1.1 Nomenclature 258 10.2 Infinitesimal Ring or Lie Algebra 259 10.2.1 Properties of rG 260 10.3 Lie Algebra of the Rotation Group R(3) 260 10.3.1 Another Approach to rR(3) 262 10.4 Group Invariants 264 10.4.1 Lorentz Transformations 266 10.5 Unitary Group in Two Dimensions U(2) 267 10.5.1 Special Unitary Group SU(2) 269 10.5.2 Lie Algebra of SU(2) 270 10.5.3 Another Approach to rSU(2) 272 10.6 Lorentz Group and Its Lie Algebra 274 10.7 Group Representations 279 10.7.1 Schur’s Lemma 279 10.7.2 Group Character 280 10.7.3 Unitary Representation 280 10.8 Representations of R(3) 281 10.8.1 Spherical Harmonics and Representations of R(3) 281 10.8.2 Angular Momentum in Quantum Mechanics 281 10.8.3 Rotation of the Physical System 282 10.8.4 Rotation Operator in Terms of the Euler Angles 282 10.8.5 Rotation Operator in the Original Coordinates 283 10.8.6 Eigenvalue Equations for Lz, L±, and L2 287 10.8.7 Fourier Expansion in Spherical Harmonics 287 10.8.8 Matrix Elements of Lx, Ly, and Lz 289 10.8.9 Rotation Matrices of the Spherical Harmonics 290 10.8.10 Evaluation of the dlm'm(𝛽) Matrices 292 10.8.11 Inverse of the dlm'm(𝛽) Matrices 292 10.8.12 Differential Equation for dlm'm(𝛽) 293 10.8.13 AdditionTheorem for Spherical Harmonics 296 10.8.14 Determination of Il in the AdditionTheorem 298 10.8.15 Connection of Dlmm' (𝛽) with Spherical Harmonics 300 10.9 Irreducible Representations of SU(2) 302 10.10 Relation of SU(2) and R(3) 303 10.11 Group Spaces 306 10.11.1 Real Vector Space 306 10.11.2 Inner Product Space 307 10.11.3 Four–Vector Space 307 10.11.4 Complex Vector Space 308 10.11.5 Function Space and Hilbert Space 308 10.11.6 Completeness 309 10.12 Hilbert Space and QuantumMechanics 310 10.13 Continuous Groups and Symmetries 311 10.13.1 Point Groups and Their Generators 311 10.13.2 Transformation of Generators and Normal Forms 312 10.13.3 The Case of Multiple Parameters 314 10.13.4 Action of Generators on Functions 315 10.13.5 Extension or Prolongation of Generators 316 10.13.6 Symmetries of Differential Equations 318 Bibliography 321 Problems 322 11 Complex Variables and Functions 327 11.1 Complex Algebra 327 11.2 Complex Functions 329 11.3 Complex Derivatives and Cauchy–Riemann Conditions 330 11.3.1 Analytic Functions 330 11.3.2 Harmonic Functions 332 11.4 Mappings 334 11.4.1 Conformal Mappings 348 11.4.2 Electrostatics and Conformal Mappings 349 11.4.3 Fluid Mechanics and Conformal Mappings 352 11.4.4 Schwarz–Christoffel Transformations 358 Bibliography 368 Problems 368 12 Complex Integrals and Series 373 12.1 Complex Integral Theorems 373 12.1.1 Cauchy–GoursatTheorem 373 12.1.2 Cauchy IntegralTheorem 374 12.1.3 CauchyTheorem 376 12.2 Taylor Series 378 12.3 Laurent Series 379 12.4 Classification of Singular Points 385 12.5 ResidueTheorem 386 12.6 Analytic Continuation 389 12.7 Complex Techniques in Taking Some Definite Integrals 392 12.8 Gamma and Beta Functions 399 12.8.1 Gamma Function 399 12.8.2 Beta Function 401 12.8.3 Useful Relations of the Gamma Functions 403 12.8.4 Incomplete Gamma and Beta Functions 403 12.8.5 Analytic Continuation of the Gamma Function 404 12.9 Cauchy Principal Value Integral 406 12.10 Integral Representations of Special Functions 410 12.10.1 Legendre Polynomials 410 12.10.2 Laguerre Polynomials 411 12.10.3 Bessel Functions 413 Bibliography 416 Problems 416 13 Fractional Calculus 423 13.1 Unified Expression of Derivatives and Integrals 425 13.1.1 Notation and Definitions 425 13.1.2 The nth Derivative of a Function 426 13.1.3 Successive Integrals 427 13.1.4 Unification of Derivative and Integral Operators 429 13.2 Differintegrals 429 13.2.1 Grünwald’s Definition of Differintegrals 429 13.2.2 Riemann–Liouville Definition of Differintegrals 431 13.3 Other Definitions of Differintegrals 434 13.3.1 Cauchy Integral Formula 434 13.3.2 Riemann Formula 439 13.3.3 Differintegrals via Laplace Transforms 440 13.4 Properties of Differintegrals 442 13.4.1 Linearity 443 13.4.2 Homogeneity 443 13.4.3 Scale Transformations 443 13.4.4 Differintegral of a Series 443 13.4.5 Composition of Differintegrals 444 13.4.5.1 Composition Rule for General q and Q 447 13.4.6 Leibniz Rule 450 13.4.7 Right– and Left–Handed Differintegrals 450 13.4.8 Dependence on the Lower Limit 452 13.5 Differintegrals of Some Functions 453 13.5.1 Differintegral of a Constant 453 13.5.2 Differintegral of [x − a] 454 13.5.3 Differintegral of [x − a]p (p > −1) 455 13.5.4 Differintegral of [1 − x]p 456 13.5.5 Differintegral of exp(±x) 456 13.5.6 Differintegral of ln(x) 457 13.5.7 Some Semiderivatives and Semi–Integrals 459 13.6 Mathematical Techniques with Differintegrals 459 13.6.1 Laplace Transform of Differintegrals 459 13.6.2 Extraordinary Differential Equations 463 13.6.3 Mittag–Leffler Functions 463 13.6.4 Semidifferential Equations 464 13.6.5 Evaluating Definite Integrals by Differintegrals 466 13.6.6 Evaluation of Sums of Series by Differintegrals 468 13.6.7 Special Functions Expressed as Differintegrals 469 13.7 Caputo Derivative 469 13.7.1 Caputo and the Riemann–Liouville Derivative 470 13.7.2 Mittag–Leffler Function and the Caputo Derivative 473 13.7.3 Right– and Left–Handed Caputo Derivatives 474 13.7.4 A Useful Relation of the Caputo Derivative 475 13.8 Riesz Fractional Integral and Derivative 477 13.8.1 Riesz Fractional Integral 477 13.8.2 Riesz Fractional Derivative 480 13.8.3 Fractional Laplacian 482 13.9 Applications of Differintegrals in Science and Engineering 482 13.9.1 Fractional Relaxation 482 13.9.2 Continuous Time RandomWalk (CTRW) 483 13.9.3 Time Fractional Diffusion Equation 486 13.9.4 Fractional Fokker–Planck Equations 487 Bibliography 489 Problems 490 14 Infinite Series 495 14.1 Convergence of Infinite Series 495 14.2 Absolute Convergence 496 14.3 Convergence Tests 496 14.3.1 Comparison Test 497 14.3.2 Ratio Test 497 14.3.3 Cauchy Root Test 497 14.3.4 Integral Test 497 14.3.5 Raabe Test 499 14.3.6 CauchyTheorem 499 14.3.7 Gauss Test and Legendre Series 500 14.3.8 Alternating Series 503 14.4 Algebra of Series 503 14.4.1 Rearrangement of Series 504 14.5 Useful Inequalities About Series 505 14.6 Series of Functions 506 14.6.1 Uniform Convergence 506 14.6.2 Weierstrass M–Test 507 14.6.3 Abel Test 507 14.6.4 Properties of Uniformly Convergent Series 508 14.7 Taylor Series 508 14.7.1 Maclaurin Theorem 509 14.7.2 BinomialTheorem 509 14.7.3 Taylor Series with Multiple Variables 510 14.8 Power Series 511 14.8.1 Convergence of Power Series 512 14.8.2 Continuity 512 14.8.3 Differentiation and Integration of Power Series 512 14.8.4 Uniqueness Theorem 513 14.8.5 Inversion of Power Series 513 14.9 Summation of Infinite Series 514 14.9.1 Bernoulli Polynomials and their Properties 514 14.9.2 Euler–Maclaurin Sum Formula 516 14.9.3 Using ResidueTheorem to Sum Infinite Series 519 14.9.4 Evaluating Sums of Series by Differintegrals 522 14.10 Asymptotic Series 523 14.11 Method of Steepest Descent 525 14.12 Saddle–Point Integrals 528 14.13 Padé Approximants 535 14.14 Divergent Series in Physics 539 14.14.1 Casimir Effect and Renormalization 540 14.14.2 Casimir Effect and MEMS 542 14.15 Infinite Products 542 14.15.1 Sine, Cosine, and the Gamma Functions 544 Bibliography 546 Problems 546 15 Integral Transforms 553 15.1 Some Commonly Encountered Integral Transforms 553 15.2 Derivation of the Fourier Integral 555 15.2.1 Fourier Series 555 15.2.2 Dirac–Delta Function 557 15.3 Fourier and Inverse Fourier Transforms 557 15.3.1 Fourier–Sine and Fourier–Cosine Transforms 558 15.4 Conventions and Properties of the Fourier Transforms 560 15.4.1 Shifting 561 15.4.2 Scaling 561 15.4.3 Transform of an Integral 561 15.4.4 Modulation 561 15.4.5 Fourier Transform of a Derivative 563 15.4.6 Convolution Theorem 564 15.4.7 Existence of Fourier Transforms 565 15.4.8 Fourier Transforms inThree Dimensions 565 15.4.9 ParsevalTheorems 566 15.5 Discrete Fourier Transform 572 15.6 Fast Fourier Transform 576 15.7 Radon Transform 578 15.8 Laplace Transforms 581 15.9 Inverse Laplace Transforms 581 15.9.1 Bromwich Integral 582 15.9.2 Elementary Laplace Transforms 583 15.9.3 Theorems About Laplace Transforms 584 15.9.4 Method of Partial Fractions 591 15.10 Laplace Transform of a Derivative 593 15.10.1 Laplace Transforms in n Dimensions 600 15.11 Relation Between Laplace and Fourier Transforms 601 15.12 Mellin Transforms 601 Bibliography 602 Problems 602 16 Variational Analysis 607 16.1 Presence of One Dependent and One Independent Variable 608 16.1.1 Euler Equation 608 16.1.2 Another Form of the Euler Equation 610 16.1.3 Applications of the Euler Equation 610 16.2 Presence of More than One Dependent Variable 617 16.3 Presence of More than One Independent Variable 617 16.4 Presence of Multiple Dependent and Independent Variables 619 16.5 Presence of Higher–Order Derivatives 619 16.6 Isoperimetric Problems and the Presence of Constraints 622 16.7 Applications to Classical Mechanics 626 16.7.1 Hamilton’s Principle 626 16.8 Eigenvalue Problems and Variational Analysis 628 16.9 Rayleigh–RitzMethod 632 16.10 Optimum Control Theory 637 16.11 BasicTheory: Dynamics versus Controlled Dynamics 638 16.11.1 Connection with Variational Analysis 641 16.11.2 Controllability of a System 642 Bibliography 646 Problems 647 17 Integral Equations 653 17.1 Classification of Integral Equations 654 17.2 Integral and Differential Equations 654 17.2.1 Converting Differential Equations into Integral Equations 656 17.2.2 Converting Integral Equations into Differential Equations 658 17.3 Solution of Integral Equations 659 17.3.1 Method of Successive Iterations: Neumann Series 659 17.3.2 Error Calculation in Neumann Series 660 17.3.3 Solution for the Case of Separable Kernels 661 17.3.4 Solution by Integral Transforms 663 17.3.4.1 Fourier Transform Method 663 17.3.4.2 Laplace Transform Method 664 17.4 Hilbert–Schmidt Theory 665 17.4.1 Eigenvalues for Hermitian Operators 665 17.4.2 Orthogonality of Eigenfunctions 666 17.4.3 Completeness of the Eigenfunction Set 666 17.5 Neumann Series and the Sturm–Liouville Problem 668 17.6 Eigenvalue Problem for the Non–Hermitian Kernels 672 Bibliography 672 Problems 672 18 Green’s Functions 675 18.1 Time–Independent Green’s Functions in One Dimension 675 18.1.1 Abel’s Formula 677 18.1.2 Constructing the Green’s Function 677 18.1.3 Differential Equation for the Green’s Function 679 18.1.4 Single–Point Boundary Conditions 679 18.1.5 Green’s Function for the Operator d2¨Mdx2 680 18.1.6 Inhomogeneous Boundary Conditions 682 18.1.7 Green’s Functions and Eigenvalue Problems 684 18.1.8 Green’s Functions and the Dirac–Delta Function 686 18.1.9 Helmholtz Equation with Discrete Spectrum 687 18.1.10 Helmholtz Equation in the Continuum Limit 688 18.1.11 Another Approach for the Green’s function 697 18.2 Time–Independent Green’s Functions inThree Dimensions 701 18.2.1 Helmholtz Equation in Three Dimensions 701 18.2.2 Green’s Functions inThree Dimensions 702 18.2.3 Green’s Function for the Laplace Operator 704 18.2.4 Green’s Functions for the Helmholtz Equation 705 18.2.5 General Boundary Conditions and Electrostatics 710 18.2.6 Helmholtz Equation in Spherical Coordinates 712 18.2.7 Diffraction from a Circular Aperture 716 18.3 Time–Independent PerturbationTheory 721 18.3.1 Nondegenerate PerturbationTheory 721 18.3.2 Slightly Anharmonic Oscillator in One Dimension 726 18.3.3 Degenerate PerturbationTheory 728 18.4 First–Order Time–Dependent Green’s Functions 729 18.4.1 Propagators 732 18.4.2 Compounding Propagators 732 18.4.3 Diffusion Equation with Discrete Spectrum 733 18.4.4 Diffusion Equation in the Continuum Limit 734 18.4.5 Presence of Sources or Interactions 736 18.4.6 Schrödinger Equation for Free Particles 737 18.4.7 Schrödinger Equation with Interactions 738 18.5 Second–Order Time–Dependent Green’s Functions 738 18.5.1 Propagators for the ScalarWave Equation 741 18.5.2 Advanced and Retarded Green’s Functions 743 18.5.3 ScalarWave Equation 745 Bibliography 747 Problems 748 19 Green’s Functions and Path Integrals 755 19.1 Brownian Motion and the Diffusion Problem 755 19.1.1 Wiener Path Integral and Brownian Motion 757 19.1.2 Perturbative Solution of the Bloch Equation 760 19.1.3 Derivation of the Feynman–Kac Formula 763 19.1.4 Interpretation of V(x) in the Bloch Equation 765 19.2 Methods of Calculating Path Integrals 767 19.2.1 Method of Time Slices 769 19.2.2 Path Integrals with the ESKC Relation 770 19.2.3 Path Integrals by the Method of Finite Elements 771 19.2.4 Path Integrals by the “Semiclassical” Method 772 19.3 Path Integral Formulation of Quantum Mechanics 776 19.3.1 Schrödinger Equation For a Free Particle 776 19.3.2 Schrödinger Equation with a Potential 778 19.3.3 Feynman Phase Space Path Integral 780 19.3.4 The Case of Quadratic Dependence on Momentum 781 19.4 Path Integrals Over Lévy Paths and Anomalous Diffusion 783 19.5 Fox’s H–Functions 788 19.5.1 Properties of the H–Functions 789 19.5.2 Useful Relations of the H–Functions 791 19.5.3 Examples of H–Functions 792 19.5.4 Computable Form of the H–Function 796 19.6 Applications of H–Functions 797 19.6.1 Riemann–Liouville Definition of Differintegral 798 19.6.2 Caputo Fractional Derivative 798 19.6.3 Fractional Relaxation 799 19.6.4 Time Fractional Diffusion via R–L Derivative 800 19.6.5 Time Fractional Diffusion via Caputo Derivative 801 19.6.6 Derivation of the Lévy Distribution 803 19.6.7 Lévy Distributions in Nature 806 19.6.8 Time and Space Fractional Schrödinger Equation 806 19.6.8.1 Free Particle Solution 808 19.7 Space Fractional Schrödinger Equation 809 19.7.1 Feynman Path Integrals Over Lévy Paths 810 19.8 Time Fractional Schrödinger Equation 812 19.8.1 Separable Solutions 812 19.8.2 Time Dependence 813 19.8.3 Mittag–Leffler Function and the Caputo Derivative 814 19.8.4 Euler Equation for the Mittag–Leffler Function 814 Bibliography 817 Problems 818 Further Reading 825 Index 827