Separable Boundary-Value Problems in Physics

Innovative developments in science and technology require a thorough knowledge of applied mathematics, particularly in the field of differential equations and special functions. These are relevant in modeling and computing applications of electromagnetic theory and quantum theory, e.g. in photonics and nanotechnology. The problem of solving partial differential equations remains an important topic that is taught at both the undergraduate and graduate level. The proposed book has a very comprehensive coverage on partial differential equations in a variety of coordinate systems and geometry, and their solutions using the method of separation of variables. The treatment includes complete details on going from the basic theory (including separability conditions not presented in introductory texts) to full implementation for applications. A very good choice of examples is inspired by the authors? research on semiconductor nanostructures and metamaterials and include modern applications like quantum dots. The fluency of the text and the high quality of graphics make the topic easy accessible. The organization of the content by coordinate systems rather than by equation types is unique and offers an easy access. The authors consider recent research results which have led to a much increased pedagogical understanding of not just this topic but of many other related topics in mathematical physics, and which? like the explicit discussion on differential geometry shows - yet have not been treated in the older texts. To the benefit of the reader, a summary presents a convenient overview on all special functions covered. Homework problems are included as well as numerical algorithms for computing special functions. Thus this book can serve as a reference text for advanced undergraduate students, as a textbook for graduate level courses, and as a self-study book and reference manual for physicists, theoretically oriented engineers and traditional mathematicians. MA4300, PH2300 suitable for graduate level course; could serve as one of two main texts of a partial differential equations course Contents 6 Preface 22 Part One Preliminaries 23 1 Introduction 25 2 General Theory 29 2.1 Introduction 29 2.2 Canonical Partial Differential Equations 29 2.3 Differential Operators in Curvilinear Coordinates 30 2.4 Separation of Variables 32 2.5 Series Solutions 42 2.6 Boundary-Value Problems 48 2.7 Physical Applications 52 2.8 Problems 58 Part Two Two-Dimensional Coordinate Systems 61 3 Rectangular Coordinates 63 3.1 Introduction 63 3.2 Coordinate System 63 3.3 Differential Operators 64 3.4 Separable Equations 65 3.5 Applications 68 3.6 Problems 71 4 Circular Coordinates 73 4.1 Introduction 73 4.2 Coordinate System 73 4.3 Differential Operators 74 4.4 Separable Equations 75 4.5 Applications 78 4.6 Problems 81 5 Elliptic Coordinates 83 5.1 Introduction 83 5.2 Coordinate System 83 5.3 Differential Operators 85 5.4 Separable Equations 86 5.5 Applications 88 5.6 Problems 90 6 Parabolic Coordinates 93 6.1 Introduction 93 6.2 Coordinate System 93 6.3 Differential Operators 94 6.4 Separable Equations 95 6.5 Applications 97 6.6 Problems 98 Part Three Three-Dimensional Coordinate Systems 101 7 Rectangular Coordinates 103 7.1 Introduction 103 7.2 Coordinate System 103 7.3 Differential Operators 104 7.4 Separable Equations 105 7.5 Applications 109 7.6 Problems 111 8 Circular Cylinder Coordinates 113 8.1 Introduction 113 8.2 Coordinate System 113 8.3 Differential Operators 114 8.4 Separable Equations 116 8.5 Applications 118 8.6 Problems 119 9 Elliptic Cylinder Coordinates 121 9.1 Introduction 121 9.2 Coordinate System 121 9.3 Differential Operators 123 9.4 Separable Equations 124 9.5 Applications 127 9.6 Problems 129 10 Parabolic Cylinder Coordinates 131 10.1 Introduction 131 10.2 Coordinate System 131 10.3 Differential Operators 134 10.4 Separable Equations 135 10.5 Applications 137 10.6 Problems 146 11 Spherical Polar Coordinates 147 11.1 Introduction 147 11.2 Coordinate System 147 11.3 Differential Operators 148 11.4 Separable Equations 149 11.5 Applications 152 11.6 Problems 159 12 Prolate Spheroidal Coordinates 161 12.1 Introduction 161 12.2 Coordinate System 161 12.3 Differential Operators 163 12.4 Separable Equations 164 12.5 Applications 166 12.6 Problems 176 13 Oblate Spheroidal Coordinates 177 13.1 Introduction 177 13.2 Coordinate System 177 13.3 Differential Operators 179 13.4 Separable Equations 181 13.5 Applications 183 13.6 Problems 185 14 Parabolic Rotational Coordinates 187 14.1 Introduction 187 14.2 Coordinate System 187 14.3 Differential Operators 189 14.4 Separable Equations 190 14.5 Applications 193 14.6 Problems 201 15 Conical Coordinates 203 15.1 Introduction 203 15.2 Coordinate System 203 15.3 Differential Operators 205 15.4 Separable Equations 206 15.5 Applications 209 15.6 Problems 211 16 Ellipsoidal Coordinates 213 16.1 Introduction 213 16.2 Coordinate System 214 16.3 Differential Operators 217 16.4 Separable Equations 219 16.5 Applications 222 16.6 Problems 237 17 Paraboloidal Coordinates 239 17.1 Introduction 239 17.2 Coordinate System 239 17.3 Differential Operators 241 17.4 Separable Equations 243 17.5 Applications 249 17.6 Problems 251 Part Four Advanced Formulations 253 18 Differential-Geometric Formulation 255 18.1 Introduction 255 18.2 Review of Differential Geometry 255 18.3 Problems 261 19 Quantum-Mechanical Particle Con.ned to the Neighborhood of Curves 263 19.1 Introduction 263 19.2 Laplacian in a Tubular Neighborhood of a Curve – Arc-Length Parameterization 263 19.3 Application to the Schrödinger Equation 270 19.4 Schrödinger Equation in a Tubular Neighborhood of a Curve – General Parameterization 272 19.5 Applications 273 19.6 Perturbation Theory Applied to the Curved-Structure Problem 281 19.7 Problems 291 20 Quantum-Mechanical Particle Con.ned to Surfaces of Revolution 293 20.1 Introduction 293 20.2 Laplacian in Curved Coordinates 293 20.3 The Schrödinger Equation in Curved Coordinates 296 20.4 Applications 296 20.5 Problems 303 21 Boundary Perturbation Theory 305 21.1 Nondegenerate States 305 21.2 Degenerate States 307 21.3 Applications 308 21.4 Problems 315 Appendix A Hypergeometric Functions 317 Appendix B Baer Functions 327 Appendix C Bessel Functions 331 Appendix D Lamé Functions 343 Appendix E Legendre Functions 351 Appendix F Mathieu Functions 361 Appendix G Spheroidal Wave Functions 373 Appendix H Weber Functions 379 Appendix I Elliptic Integrals and Functions 383 References 391 Index 397 3527410201,9783527410200 Wiley-VCH,2011 Contents......Page 6 Preface......Page 22 Part One Preliminaries......Page 23 1 Introduction......Page 25 2.2 Canonical Partial Differential Equations......Page 29 2.3 Differential Operators in Curvilinear Coordinates......Page 30 2.4 Separation of Variables......Page 32 2.5 Series Solutions......Page 42 2.6 Boundary-Value Problems......Page 48 2.7 Physical Applications......Page 52 2.8 Problems......Page 58 Part Two Two-Dimensional Coordinate Systems......Page 61 3.2 Coordinate System......Page 63 3.3 Differential Operators......Page 64 3.4 Separable Equations......Page 65 3.5 Applications......Page 68 3.6 Problems......Page 71 4.2 Coordinate System......Page 73 4.3 Differential Operators......Page 74 4.4 Separable Equations......Page 75 4.5 Applications......Page 78 4.6 Problems......Page 81 5.2 Coordinate System......Page 83 5.3 Differential Operators......Page 85 5.4 Separable Equations......Page 86 5.5 Applications......Page 88 5.6 Problems......Page 90 6.2 Coordinate System......Page 93 6.3 Differential Operators......Page 94 6.4 Separable Equations......Page 95 6.5 Applications......Page 97 6.6 Problems......Page 98 Part Three Three-Dimensional Coordinate Systems......Page 101 7.2 Coordinate System......Page 103 7.3 Differential Operators......Page 104 7.4 Separable Equations......Page 105 7.5 Applications......Page 109 7.6 Problems......Page 111 8.2 Coordinate System......Page 113 8.3 Differential Operators......Page 114 8.4 Separable Equations......Page 116 8.5 Applications......Page 118 8.6 Problems......Page 119 9.2 Coordinate System......Page 121 9.3 Differential Operators......Page 123 9.4 Separable Equations......Page 124 9.5 Applications......Page 127 9.6 Problems......Page 129 10.2 Coordinate System......Page 131 10.3 Differential Operators......Page 134 10.4 Separable Equations......Page 135 10.5 Applications......Page 137 10.6 Problems......Page 146 11.2 Coordinate System......Page 147 11.3 Differential Operators......Page 148 11.4 Separable Equations......Page 149 11.5 Applications......Page 152 11.6 Problems......Page 159 12.2 Coordinate System......Page 161 12.3 Differential Operators......Page 163 12.4 Separable Equations......Page 164 12.5 Applications......Page 166 12.6 Problems......Page 176 13.2 Coordinate System......Page 177 13.3 Differential Operators......Page 179 13.4 Separable Equations......Page 181 13.5 Applications......Page 183 13.6 Problems......Page 185 14.2 Coordinate System......Page 187 14.3 Differential Operators......Page 189 14.4 Separable Equations......Page 190 14.5 Applications......Page 193 14.6 Problems......Page 201 15.2 Coordinate System......Page 203 15.3 Differential Operators......Page 205 15.4 Separable Equations......Page 206 15.5 Applications......Page 209 15.6 Problems......Page 211 16.1 Introduction......Page 213 16.2 Coordinate System......Page 214 16.3 Differential Operators......Page 217 16.4 Separable Equations......Page 219 16.5 Applications......Page 222 16.6 Problems......Page 237 17.2 Coordinate System......Page 239 17.3 Differential Operators......Page 241 17.4 Separable Equations......Page 243 17.5 Applications......Page 249 17.6 Problems......Page 251 Part Four Advanced Formulations......Page 253 18.2 Review of Differential Geometry......Page 255 18.3 Problems......Page 261 19.2 Laplacian in a Tubular Neighborhood of a Curve – Arc-Length Parameterization......Page 263 19.3 Application to the Schrödinger Equation......Page 270 19.4 Schrödinger Equation in a Tubular Neighborhood of a Curve – General Parameterization......Page 272 19.5 Applications......Page 273 19.6 Perturbation Theory Applied to the Curved-Structure Problem......Page 281 19.7 Problems......Page 291 20.2 Laplacian in Curved Coordinates......Page 293 20.4 Applications......Page 296 20.5 Problems......Page 303 21.1 Nondegenerate States......Page 305 21.2 Degenerate States......Page 307 21.3 Applications......Page 308 21.4 Problems......Page 315 Appendix A Hypergeometric Functions......Page 317 Appendix B Baer Functions......Page 327 Appendix C Bessel Functions......Page 331 Appendix D Lamé Functions......Page 343 Appendix E Legendre Functions......Page 351 Appendix F Mathieu Functions......Page 361 Appendix G Spheroidal Wave Functions......Page 373 Appendix H Weber Functions......Page 379 Appendix I Elliptic Integrals and Functions......Page 383 References......Page 391 Index......Page 397

Innovative developments in science and technology require a thorough knowledge of applied mathematics, particularly in the field of differential equations and special functions. These are relevant in modeling and computing applications of electromagnetic theory and quantum theory, e.g. in photonics and nanotechnology. The problem of solving partial differential equations remains an important topic that is taught at both the undergraduate and graduate level.

Separable Boundary-Value Problems in Physics is an accessible and comprehensive treatment of partial differential equations in mathematical physics in a variety of coordinate systems and geometry and their solutions, including a differential geometric formulation, using the method of separation of variables. With problems and modern examples from the fields of nano-technology and other areas of physics.

The fluency of the text and the high quality of graphics make the topic easy accessible. The organization of the content by coordinate systems rather than by equation types is unique and offers an easy access.

The authors consider recent research results which have led to a much increased pedagogical understanding of not just this topic but of many other related topics in mathematical physics, and which like the explicit discussion on differential geometry shows - yet have not been treated in the older texts. To the benefit of the reader, a summary presents a convenient overview on all special functions covered. Homework problems are included as well as numerical algorithms for computing special functions. Thus this book can serve as a reference text for advanced undergraduate students, as a textbook for graduate level courses, and as a self-study book and reference manual for physicists, theoretically oriented engineers and traditional mathematicians.