Separable Boundary-Value Problems in Physics

Morten Willatzen and Lok C. Lew Yan Voon Separable Boundary-Value Problems in Physics

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Morten Willatzen and Lok C. Lew Yan Voon

Separable Boundary-Value Problems in Physics

WILEY-VCH Verlag GmbH & Co. KGaA

The Authors Prof. Morten Willatzen University of Southern Denmark Mads Clausen Institute Alsion 2 6400 Sønderborg Denmark Prof. Lok C. Lew Yan Voon Wright State University Dept. of Physics 3640 Colonel Glenn Hwy Dayton, OH 45435 USA

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Typesetting le-tex publishing services GmbH, Leipzig Printing and Binding Cover Design Adam-Design, Weinheim Printed in the Federal Republic of Germany Printed on acid-free paper ISBN Print 978-3-527-41020-0 ISBN oBook 978-3-527-63492-7 ISBN ePDF 978-3-527-63494-1 ISBN ePub 978-3-527-63493-4 ISBN Mobi 978-3-527-63495-8

V

Contents Preface XXI Part One

Preliminaries 1

1

Introduction 3

2 2.1 2.2 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.5.1 2.4 2.4.1 2.4.1.1 2.4.1.2 2.4.2 2.4.2.1 2.4.2.2 2.4.2.3 2.4.2.4 2.5 2.5.1 2.5.2 2.5.2.1 2.5.3 2.5.3.1 2.5.3.2 2.5.3.3 2.5.3.4

General Theory 7 Introduction 7 Canonical Partial Differential Equations 7 Differential Operators in Curvilinear Coordinates Metric 8 Gradient 9 Divergence 9 Circulation 9 Laplacian 9 Example 9 Separation of Variables 10 Two Dimensions 11 Rectangular Coordinate System 11 Other Coordinate Systems 12 Three Dimensions 15 Stäckel Matrix 16 Helmholtz Equation 17 Schrödinger Equation 19 Separable Coordinate Systems 19 Series Solutions 20 Singularities 20 Bôcher Equation 21 Example 21 Frobenius Method 22 One Regular Singular Point 24 Two Regular Singular Points 24 One Irregular Singular Point 24 Three Regular Singular Points 25

8

VI

Contents

2.6 2.6.1 2.6.2 2.7 2.7.1 2.7.2 2.7.3 2.7.4 2.7.5 2.7.6 2.7.7 2.8

Boundary-Value Problems 26 Boundary Conditions 26 Fourier Expansions 28 Physical Applications 30 Electrostatics 30 Photonics 30 Heat Conduction 31 Newtonian Gravitation 32 Hydrodynamics 33 Acoustics 33 Quantum Mechanics 35 Problems 36

Part Two Two-Dimensional Coordinate Systems 39 3 3.1 3.2 3.2.1 3.2.2 3.3 3.3.1 3.3.2 3.3.2.1 3.3.2.2 3.3.2.3 3.4 3.4.1 3.4.2 3.4.3 3.5 3.5.1 3.5.2 3.6

Rectangular Coordinates 41 Introduction 41 Coordinate System 41 Coordinates (x, y ) 41 Constant-Coordinate Curves 41 Differential Operators 42 Metric 42 Operators 43 Gradient 43 Divergence 43 Laplacian 43 Separable Equations 43 Laplace Equation 43 Helmholtz Equation 44 Schrödinger Equation 45 Applications 46 Electrostatics: Dirichlet Problem for a Conducting Strip 46 Quantum Mechanics: Dirichlet Problem for a Rectangular Box 47 Problems 49

4 4.1 4.2 4.2.1 4.2.2 4.3 4.3.1 4.3.2 4.3.2.1 4.3.2.2 4.3.2.3

Circular Coordinates 51 Introduction 51 Coordinate System 51 Coordinates 51 Constant-Coordinate Curves 51 Differential Operators 52 Metric 52 Operators 52 Gradient 52 Divergence 53 Laplacian 53

Contents

4.4 4.4.1 4.4.2 4.4.3 4.5 4.5.1 4.5.1.1 4.5.1.2 4.5.1.3 4.6

Separable Equations 53 Laplace Equation 53 Helmholtz Equation 54 Schrödinger Equation 55 Applications 56 Quantum Mechanics: Dirichlet and Neumann Problems for a Disk 56 Infinite-Barrier Solutions 56 Finite-Barrier Solutions 57 Infinite-Barrier Pie 58 Problems 59

5 5.1 5.2 5.2.1 5.2.2 5.3 5.3.1 5.3.2 5.3.2.1 5.3.2.2 5.3.2.3 5.4 5.4.1 5.4.2 5.4.3 5.5 5.5.1 5.5.1.1 5.6

Elliptic Coordinates 61 Introduction 61 Coordinate System 61 Coordinates (u, v ) 61 Constant-Coordinate Curves 62 Differential Operators 63 Metric 63 Operators 63 Gradient 63 Divergence 63 Laplacian 63 Separable Equations 64 Laplace Equation 64 Helmholtz Equation 64 Schrödinger Equation 65 Applications 66 Quantum Mechanics: Dirichlet Problem for an Ellipse 66 Finite-Barrier Solutions 66 Problems 68

6 6.1 6.2 6.2.1 6.2.2 6.3 6.3.1 6.3.2 6.3.2.1 6.3.2.2 6.3.2.3 6.4 6.4.1 6.4.2 6.4.3

Parabolic Coordinates 71 Introduction 71 Coordinate System 71 Coordinates (µ, ν) 71 Constant-Coordinate Curves 71 Differential Operators 72 Metric 72 Operators 72 Gradient 73 Divergence 73 Laplacian 73 Separable Equations 73 Laplace Equation 73 Helmholtz Equation 73 Schrödinger Equation 75

VII

VIII

Contents

6.5 6.5.1 6.6

Applications 75 Heat Conduction: Dirichlet Problem for the Laplace Equation 75 Problems 76

Part Three Three-Dimensional Coordinate Systems

79

7 7.1 7.2 7.2.1 7.2.2 7.3 7.3.1 7.3.2 7.3.2.1 7.3.2.2 7.3.2.3 7.3.2.4 7.3.3 7.4 7.4.1 7.4.2 7.4.3 7.5 7.5.1 7.6

Rectangular Coordinates 81 Introduction 81 Coordinate System 81 Coordinates (x, y, z) 81 Constant-Coordinate Surfaces 81 Differential Operators 82 Metric 82 Operators 82 Gradient 82 Divergence 82 Circulation 83 Laplacian 83 Stäckel Matrix 83 Separable Equations 83 Laplace Equation 83 Helmholtz Equation 85 Schrödinger Equation 86 Applications 87 Electrostatics: Dirichlet Problem for a Rectangular Box 87 Problems 89

8 8.1 8.2 8.2.1 8.2.2 8.3 8.3.1 8.3.2 8.3.2.1 8.3.2.2 8.3.2.3 8.3.2.4 8.3.3 8.4 8.4.1 8.4.2 8.4.3 8.5 8.5.1

Circular Cylinder Coordinates 91 Introduction 91 Coordinate System 91 Coordinates (r, φ, z) 91 Constant-Coordinate Surfaces 92 Differential Operators 92 Metric 92 Operators 93 Gradient 93 Divergence 93 Circulation 93 Laplacian 93 Stäckel Theory 93 Separable Equations 94 Laplace Equation 94 Helmholtz Equation 95 Schrödinger Equation 95 Applications 96 Heat Conduction: Dirichlet Problem for a Cylinder 96

Contents

8.5.2 8.5.2.1 8.6

Quantum Mechanics: Dirichlet Problem for a Cylinder 97 Infinite Barrier 97 Problems 97

9 9.1 9.2 9.2.1 9.2.2 9.3 9.3.1 9.3.2 9.3.2.1 9.3.2.2 9.3.2.3 9.3.2.4 9.3.3 9.4 9.4.1 9.4.2 9.4.3 9.5 9.5.1 9.5.2 9.6

Elliptic Cylinder Coordinates 99 Introduction 99 Coordinate System 99 Coordinates (u, v , z) 99 Constant-Coordinate Surfaces 100 Differential Operators 101 Metric 101 Operators 101 Gradient 101 Divergence 101 Circulation 102 Laplacian 102 Stäckel Matrix 102 Separable Equations 102 Laplace Equation 102 Helmholtz Equation 104 Schrödinger Equation 105 Applications 105 Hydrodynamics: Dirichlet Problem for an Elliptic Pipe 106 Quantum Mechanics: Dirichlet Problem for an Elliptic Cylinder 106 Problems 107

10 10.1 10.2 10.2.1 10.2.2 10.2.3 10.3 10.3.1 10.3.2 10.3.2.1 10.3.2.2 10.3.2.3 10.3.2.4 10.3.3 10.4 10.4.1 10.4.2 10.4.3 10.5 10.5.1

Parabolic Cylinder Coordinates 109 Introduction 109 Coordinate System 109 Coordinates (µ, ν, z) 109 Constant-Coordinate Surfaces 110 Other Geometrical Parameters 111 Differential Operators 112 Metric 112 Operators 112 Gradient 112 Divergence 112 Circulation 112 Laplacian 112 Stäckel Matrix 113 Separable Equations 113 Laplace Equation 113 Helmholtz Equation 114 Schrödinger Equation 115 Applications 115 Acoustics: Neumann Problem for a Cavity 116

IX

X

Contents

10.5.1.1 10.5.1.2 10.5.1.3 10.5.1.4 10.5.1.5 10.6

Case (a) 119 Case (b) 119 Case (c) 119 Relation between k and α 3 Results 120 Problems 124

11 11.1 11.2 11.2.1 11.2.2 11.3 11.3.1 11.3.2 11.3.2.1 11.3.2.2 11.3.2.3 11.3.2.4 11.3.3 11.4 11.4.1 11.4.2 11.4.3 11.5 11.5.1 11.5.1.1 11.5.1.2 11.5.1.3 11.5.1.4 11.5.1.5 11.5.1.6 11.5.1.7 11.5.1.8 11.6

Spherical Polar Coordinates 125 Introduction 125 Coordinate System 125 Coordinates (r, θ , φ) 125 Constant-Coordinate Surfaces 126 Differential Operators 126 Metric 126 Operators 126 Gradient 126 Divergence 127 Circulation 127 Laplacian 127 Stäckel Matrix 127 Separable Equations 127 Laplace Equation 127 Helmholtz Equation 128 Schrödinger Equation 129 Applications 130 Quantum Mechanics: Dirichlet Problem 130 Infinite-Barrier Spherical Dot 131 Finite-Barrier Spherical Dot 131 Quantum Ice Cream – Infinite Barrier 132 ν(
µ) D ν(µ) 133 E(
µ) D E(µ) 133 ν 
1/2 133 ν  jµj 133 Additional Constraints 134 Problems 137

12 12.1 12.2 12.2.1 12.2.2 12.3 12.3.1 12.3.2 12.3.2.1 12.3.2.2

Prolate Spheroidal Coordinates 139 Introduction 139 Coordinate System 139 Coordinates (α, β, φ and ξ , η, φ) 139 Constant-Coordinate Surfaces 140 Differential Operators 141 Metric 141 Operators 141 Gradient 141 Divergence 141

120

Contents

12.3.2.3 12.3.2.4 12.3.3 12.4 12.4.1 12.4.2 12.4.3 12.5 12.5.1 12.5.2 12.5.3 12.5.3.1 12.5.3.2 12.6

Circulation 142 Laplacian 142 Stäckel Matrix 142 Separable Equations 142 Laplace Equation 142 Helmholtz Equation 143 Schrödinger Equation 144 Applications 144 Dirichlet Problem for the Laplace Equation 145 Gravitation: Dirichlet–Neumann Problem 146 Quantum Mechanics: Dirichlet Problem 147 Infinite-Barrier Problem 147 Finite-Barrier Problem 150 Problems 154

13 13.1 13.2 13.2.1 13.2.2 13.3 13.3.1 13.3.2 13.3.2.1 13.3.2.2 13.3.2.3 13.3.2.4 13.3.3 13.4 13.4.1 13.4.2 13.4.3 13.5 13.5.1 13.5.2 13.6

Oblate Spheroidal Coordinates 155 Introduction 155 Coordinate System 155 Coordinates (α, β, ' and ξ , η, ') 155 Constant-Coordinate Surfaces 156 Differential Operators 157 Metric 157 Operators 157 Gradient 157 Divergence 158 Circulation 158 Laplacian 158 Stäckel Matrix 158 Separable Equations 159 Laplace Equation 159 Helmholtz Equation 159 Schrödinger Equation 160 Applications 161 Dirichlet Problem for the Laplace Equation 161 Asymptotic Solutions 162 Problems 163

14 14.1 14.2 14.2.1 14.2.2 14.2.3 14.3 14.3.1 14.3.2

Parabolic Rotational Coordinates 165 Introduction 165 Coordinate System 165 Coordinates (ξ , η, φ) 165 Constant-Coordinate Surfaces 166 Other Geometrical Parameters 166 Differential Operators 167 Metric 167 Operators 167

XI

XII

Contents

14.3.2.1 14.3.2.2 14.3.2.3 14.3.2.4 14.3.3 14.4 14.4.1 14.4.2 14.4.3 14.5 14.5.1

Gradient 167 Divergence 168 Circulation 168 Laplacian 168 Stäckel Matrix 168 Separable Equations 168 Laplace Equation 168 Helmholtz Equation 169 Schrödinger Equation 170 Applications 171 Heat Conduction: Boundary-Value Problem for the Laplace Equation 172 14.5.1.1 Dirichlet 172 14.5.2 Quantum Mechanics: Interior Dirichlet Problem 173 14.5.2.1 Numerical Results 177 14.6 Problems 179 15 15.1 15.2 15.2.1 15.2.2 15.3 15.3.1 15.3.2 15.3.2.1 15.3.2.2 15.3.2.3 15.3.2.4 15.3.3 15.4 15.4.1 15.4.2 15.4.3 15.5 15.5.1 15.6

Conical Coordinates 181 Introduction 181 Coordinate System 181 Coordinates (r, θ , λ) 181 Constant-Coordinate Surfaces 182 Differential Operators 183 Metric 183 Operators 183 Gradient 183 Divergence 183 Circulation 184 Laplacian 184 Stäckel Theory 184 Separable Equations 184 Laplace Equation 184 Helmholtz Equation 185 Schrödinger Equation 186 Applications 187 Electrostatics: Dirichlet and Neumann Problems on a Plane Angular Sector 187 Problems 189

16 16.1 16.2 16.2.1 16.2.2 16.3 16.3.1

Ellipsoidal Coordinates 191 Introduction 191 Coordinate System 192 Coordinates (ξ1 , ξ2 , ξ3 ) 193 Ellipsoid 194 Differential Operators 195 Metric 195

Contents

16.3.2 16.3.2.1 16.3.2.2 16.3.2.3 16.3.2.4 16.4 16.4.1 16.4.2 16.5 16.5.1 16.5.1.1 16.5.1.2 16.5.1.3 16.5.1.4 16.5.2 16.5.3 16.5.3.1 16.5.3.2 16.5.3.3 16.5.4 16.5.4.1 16.5.4.2 16.5.4.3 16.6

Operators 195 Gradient 195 Divergence 196 Circulation 196 Laplacian 196 Separable Equations 197 Laplace Equation 197 Helmholtz Equation 199 Applications 200 Interior Problem for the Laplace Equation 200 Ellipsoidal Harmonic of the First Species 201 Ellipsoidal Harmonic of the Second Species 202 Ellipsoidal Harmonic of the Third Species 202 Ellipsoidal Harmonic of the Fourth Species 203 Elliptic Functions 203 Dirichlet Problem for the Helmholtz Equation: ATZ Algorithm 204 First Solution to the Ellipsoidal Wave Equation 205 Second Solution to the Ellipsoidal Wave Equation 208 Ellipsoidal Domain 209 Quantum Mechanics: Interior Dirichlet Problem for an Ellipsoid 210 Characteristic Curves 211 Determination of γ Eigenvalues 212 Lamé Wave Functions 213 Problems 215

17 17.1 17.2 17.2.1 17.2.2 17.3 17.3.1 17.3.2 17.3.3 17.4 17.4.1 17.4.1.1 17.4.1.2 17.4.1.3 17.4.2 17.4.2.1 17.5 17.5.1 17.6

Paraboloidal Coordinates 217 Introduction 217 Coordinate System 217 Coordinates (µ, ν, λ) 217 Constant-Coordinate Surfaces 218 Differential Operators 219 Metric 219 Operators 219 Stäckel Matrix 220 Separable Equations 221 Laplace Equation 221 Separation of Variables 221 Series Solutions 221 Polynomial Solutions 223 Helmholtz Equation 227 Separation of Variables 227 Applications 227 Electrostatics: Dirichlet Problem for a Paraboloid 227 Problems 229

XIII

XIV

Contents

Part Four Advanced Formulations 231 18 18.1 18.2 18.2.1 18.2.2 18.2.3 18.2.4 18.3

Differential-Geometric Formulation 233 Introduction 233 Review of Differential Geometry 233 Curvilinear Coordinates 233 Gradient, Divergence, and Laplacian 236 Curl and Cross Products 238 Vector Calculus Expressions in General Coordinates 239 Problems 239

19

Quantum-Mechanical Particle Confined to the Neighborhood of Curves 241 Introduction 241 Laplacian in a Tubular Neighborhood of a Curve – Arc-Length Parameterization 241 Arc-Length Parameterization 241 Minimal Rotating Frame 243 Laplacian 245 Circular Cross Section 247 Application to the Schrödinger Equation 248 Solutions to the χ 2 and χ 3 Equations 249 Schrödinger Equation in a Tubular Neighborhood of a Curve – General Parameterization 250 Applications 251 Perturbation Theory Applied to the Curved-Structure Problem 259 Dirichlet Unperturbed Eigenstates 259 Evaluation of ∆λ n in the Case with Dirichlet Boundary Conditions 260 Eigenstate Perturbations 263 Neumann Unperturbed Eigenstates 263 Evaluation of ∆λ n in the Case with Neumann Boundary Conditions 264 Perturbation Theory in the General Parameterization Case 267 Comparison between Analytical Results and Perturbation Theory for Circular-Bent Rectangular Domains in Two Dimensions – Dirichlet Boundary Conditions 268 Rectangular Domain – No Bending 268 Rectangular Domain – With Bending 268 Comparison between Analytical Results and Perturbation Theory for Circular-Bent Rectangular Domains in Two Dimensions – Neumann Boundary Conditions 268 Rectangular Domain – No Bending 269 Rectangular Domain – With Bending 269 Problems 269

19.1 19.2 19.2.1 19.2.1.1 19.2.2 19.2.3 19.3 19.3.1 19.4 19.5 19.6 19.6.1 19.6.2 19.6.3 19.6.4 19.6.5 19.6.6 19.6.7

19.6.7.1 19.6.7.2 19.6.8

19.6.8.1 19.6.9 19.7

Contents

20 20.1 20.2 20.3 20.4 20.4.1 20.4.2 20.5

Quantum-Mechanical Particle Confined to Surfaces of Revolution 271 Introduction 271 Laplacian in Curved Coordinates 271 The Schrödinger Equation in Curved Coordinates 274 Applications 274 Truncated Cone 274 Elliptic Torus 277 Problems 281

21 21.1 21.2 21.3 21.4

Boundary Perturbation Theory 283 Nondegenerate States 283 Degenerate States 285 Applications 286 Problems 293

Appendix A Hypergeometric Functions 295 A.1 Introduction 295 A.2 Hypergeometric Equation 295 A.3 Hypergeometric Functions 296 A.3.1 First Solution 296 A.3.1.1 Examples 297 A.3.1.2 Properties 297 A.3.2 Second Solution 297 A.4 Confluent Hypergeometric Equation 298 A.5 Confluent Hypergeometric Functions 298 A.5.1 First Solution 298 A.5.1.1 Examples 298 A.5.2 Second Solution 299 A.5.3 Properties 299 A.6 Whittaker Functions 299 A.6.1 Whittaker Equation 299 A.6.2 Whittaker Functions 299 A.7 Associated Laguerre Functions 300 A.7.1 Associated Laguerre Equation 300 A.7.2 Associated Laguerre Function 300 A.7.3 Laguerre Equation 300 A.7.3.1 Alternative Representation 300 A.7.4 Generalized Laguerre Polynomials 300 A.8 Hermite Polynomial 301 A.8.1 Hermite Equation 301 A.8.2 Hermite Polynomials 301 A.8.3 Properties 302 A.9 Airy Functions 302 A.9.1 Airy Equation 303 A.9.2 Properties 303

XV

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Contents

Appendix B Baer Functions 305 B.1 Introduction 305 B.2 Baer Equation 305 B.3 Baer Functions 305 B.4 Baer Wave Equation 306 B.5 Baer Wave Functions 306 B.5.1 Orthogonality 306 Appendix C Bessel Functions 309 C.1 Introduction 309 C.2 Bessel Equations 309 C.3 Bessel Functions 310 C.3.1 ν Nonintegral 310 C.3.2 ν Integral 310 C.3.3 Properties 311 C.3.4 Hankel Functions 313 C.3.4.1 Properties 313 C.4 Modified Bessel Functions 314 C.4.1 Properties 314 C.5 Spherical Bessel Functions 315 C.5.1 Properties 316 C.6 Modified Spherical Bessel Functions 316 C.7 Bessel Wave Functions 316 C.7.1 Series Solution 317 C.7.2 Orthogonality 318 Appendix D Lamé Functions 321 D.1 Lamé Equations 321 D.2 Lamé Functions 322 D.2.1 First Kind 322 D.2.1.1 F(z) 322 p B(z) 324 D.2.1.2 F(z) D p z 2
a 2 p D.2.1.3 F(z) D z 2
a 2 z 2
b 2 B(z) D.2.2 Second Kind 326 D.3 Lamé Wave Equation 326 D.3.1 Moon–Spencer Form 327 D.3.2 Arscott’s Algebraic Form 327 Appendix E Legendre Functions 329 E.1 Introduction 329 E.2 Legendre Equation 329 E.3 Series Solutions 329 E.3.1 Recurrence Relation 329 E.3.2 Convergence 330 E.4 Legendre Polynomials 330 E.4.1 Normalization 330

325

Contents

E.4.2 E.4.2.1 E.4.2.2 E.4.2.3 E.4.2.4 E.4.3 E.4.4 E.4.5 E.5 E.5.1 E.5.2 E.6 E.6.1 E.6.2 E.6.2.1 E.6.3 E.6.4 E.6.5 E.6.6 E.6.7 E.7 E.7.1 E.7.2

Representations 331 Hypergeometric Function 331 Rodrigue’s Formula 331 Generating Function 331 Schaefli Integral Representation 331 Special Values 332 Orthogonality 332 Expansions 332 Legendre Function 333 Hypergeometric Representation 333 Properties 333 Associated Legendre Functions 333 Associated Legendre Equation 333 Associated Legendre Functions 334 Properties 335 Associated Legendre Polynomials 335 Generating Function 335 Recurrence Relations 335 Parity 336 Orthogonality 336 Spherical Harmonics 336 Definition 336 Orthogonality 337

Appendix F Mathieu Functions 339 F.1 Introduction 339 F.2 Mathieu Equation 339 F.3 Mathieu Function 340 F.3.1 Properties 341 F.3.2 Orthogonality 341 F.3.3 Periodic Solution for Small q 343 F.4 Characteristic Equation 343 F.4.1 Recurrence Relations 344 F.4.1.1 (Even, π) Solutions 344 F.4.1.2 (Even, 2π) Solutions 345 F.4.1.3 (Odd, π) Solutions 345 F.4.1.4 (Odd, 2π) Solutions 346 F.4.1.5 (Even, π) Solutions 346 F.4.1.6 (Even, 2π) Solutions 347 F.4.1.7 (Odd, π) Solutions 347 F.4.1.8 (Odd, 2π) Solutions 347 F.4.2 Continued Fraction Solution 347 F.4.2.1 (Even, π) Solutions 348 F.4.2.2 (Even, 2π) Solutions 348 F.4.2.3 (Odd, π) Solutions 348

XVII

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Contents

F.4.2.4 F.4.2.5 F.4.2.6 F.4.2.7 F.4.2.8 F.5 F.6

(Odd, 2π) Solutions 348 (Even, π) Solutions 348 (Even, 2π) Solutions 349 (Odd, π) Solutions 349 (Odd, 2π) Solutions 349 Mathieu Functions of Fractional Order 349 Nonperiodic Second Solutions 350

Appendix G Spheroidal Wave Functions 351 G.1 Introduction 351 G.2 Spheroidal Wave Equation 351 G.3 Spheroidal Wave Functions 352 G.3.1 Prolate Angular Functions 352 G.3.1.1 Recurrence Relation 352 G.3.1.2 Eigenvalue Problem 354 G.3.1.3 Continued Fractions 355 Appendix H Weber Functions 357 H.1 Weber Equation 357 H.2 Weber Functions 358 H.2.1 Properties 359 Appendix I Elliptic Integrals and Functions 361 I.1 Elliptic Integrals 361 I.1.1 Elliptic Integral of the First Kind 361 I.1.2 Elliptic Integral of the Second Kind 362 I.1.3 Elliptic Integral of the Third Kind 362 I.1.4 Complete Elliptic Integrals 363 I.1.4.1 Limiting Values 363 I.2 Jacobian Elliptic Functions 363 I.2.1 Notation 364 I.2.2 Degeneracy 365 I.2.2.1 k ! 0 365 I.2.2.2 k ! 1 365 I.2.3 Relations 365 I.2.4 Derivatives 365 I.2.5 Parity 365 I.2.6 Addition Theorems 365 I.2.6.1 snz 365 I.2.6.2 cnz 365 I.2.6.3 dnz 366 I.2.7 K 0 366 I.2.8 Special Values 366 I.2.9 Period 366

Contents

I.2.10 Behavior near the Origin and i K 0 I.2.10.1 Near the Origin 367 I.2.10.2 Near i K 0 367 References 369 Index

375

366

XIX

XXI

Preface We became interested in the research that has has led to this book in 2000, when the two of us met at the ferry terminal in Tsim Tsai Tsui, Hong Kong, and discussed the problem of separability of partial differential equations. This was followed by a research visit by L.C.L.Y.V. to the Mads Clausen Institute at Syddansk Universitet in 2003, a visit funded by the Balslev Foundation. It is only fitting that L.C.L.Y.V. was invited back to the Mathematical Modeling Group of the Mads Clausen Institute on the beautiful new campus of Syddansk Universitet at Alsion to finish work on the book. Our interaction during that time has led to numerous publications, including a few on the topic of this book and another book on the electronic properties of semiconductors. Whereas our earlier work followed the exposition of Morse and Feshbach and that of Moon and Spencer closely, we have since incorporated a more general differential-geometric approach. Both approaches are featured in this book. As mathematical physicists, it was a pleasure to put together a book that blends together knowledge in mathematics and physics going back 100 years. The research and book writing has received generous financial support over the years. The work of M.W. has been supported by Syddansk Universitet and Sønderborg Kommune. The work of L.C.L.Y.V. has been funded by the National Science Foundation (USA), the Balslev Foundation, and Sønderborg Kommune. L.C.L.Y.V. would also like to thank the College of Science and Mathematics at Wright State University for release from duties to write this book and the hospitality of the Mads Clausen Institute at Syddansk Universitet, where most of the writing took place. Two individuals have contributed to some parts of this work. First, Prof. Jens Gravesen was an indispensable collaborator in our work on the differentialgeometric formulation and this is obvious from his coauthorship of many of our joint papers in this area. Second, we would like to thank Lars Duggen for his help in making some of the figures in the book. Of course, none of this would have been possible without the encouragement and support of our families. Finally, we would like to thank our editors at WileyVCH for their wonderful job, not only with the nice product, but also with their professionalism in keeping us on track. October 2010

Morten Willatzen Lok C. Yan Voon

Part One

Preliminaries

3

1 Introduction This is a textbook about how to solve boundary-value problems in physics using the method of separation of variables which goes beyond the few simple coordinate systems presented in most textbook discussions. Our goal is to present an applicationoriented approach to the study of the general theory of the method of separation of variables, whereby the variety of separable orthogonal coordinate systems is included to illustrate various aspects of the theory (e.g., lesser known coordinate systems, the coupling of separation constants, and solving for the boundary-value problem particularly for many-parameter surfaces) and also to discuss the variety of special functions that can result (e.g., from transcendental to Lamé functions). We will add, right upfront, that this is not a text about special functions, though sufficient results about the latter are included to make the text as self-contained as possible. In numerous areas of science and engineering, one has to solve a partial differential equation (PDE) for some fairly regular shape. Examples include Newtonian gravity for an ellipsoidal meteorite [1], the temperature distribution over a paraboloidal aircraft cone [2], the electric field in the vicinity of the brain modeled as an ellipsoid [3], and the electronic structure of spherical quantum dots [4]. A very powerful method is the method of separation of variables, whereby the PDE is separated into ordinary differential equations (ODEs). The latter then need to be solved, often in the form of power series, leading to special functions such as the Legendre functions and the Baer functions, and, finally, boundary conditions are applied. Even when the shape deviates from the ideal regular shape, a preliminary investigation using the regular shape is often useful both as a validation technique for some other, more numerical approach and as a first step in a, for example, perturbative approach to the exact solution. Indeed, according to Morse and Feshbach [5], the method of separation of variables is only one of two generally practical methods of solution, the other being the integral solution. Furthermore, practically all mathematical physics texts discuss the method heuristically applied to one or more of the following coordinate systems: rectangular, circular cylindrical, and spherical polar. Nevertheless, the restriction to a few coordinate systems hides a number of features of the method as well as, of course, its range of applicability. Discussion of more advanced features of the method has been reserved to a few texts [5–9]. Thus, the separability of the Helmholtz equation in 11 orthogonal coordinate systems is not generally known in spite of the utility of many of these coordinate systems for Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

4

1 Introduction

applications. Even the formal definition of “separation of variables” is rarely given. It has been argued that such a definition is needed before general results can be demonstrated [10, 11]. In this book, the problem of separating the Laplacian in various orthogonal coordinate systems in Euclidean 3-space is presented and the resulting ODEs for a number of PDEs of physical interest are given. Explicit solutions in terms of special functions are then described. Various physical problems are discussed in detail, including in acoustics, in heat conduction, in electrostatics, and in quantum mechanics, as the corresponding PDEs represent three general forms to which many other differential equations reduce (Laplace, Helmholtz, and Schrödinger). Furthermore, they represent two classes of differential equations (elliptic and hyperbolic) and different types of boundary conditions. A unique feature of our book is the part devoted to the differential geometric formulation of PDEs and their solutions for various kinds of confined geometries and boundary conditions. Such a treatment, though not entirely new, has recently been extended by a few authors, including us, and has mostly only appeared in the research literature. There are obviously many applications of the method of separation of variables, particularly for the common rectangular, circular cylindrical, and spherical polar coordinate systems. The general theory has also been worked out and discussed in the mathematical physics literature. Our treatment follows closely the books by Morse and Feshbach [5] and Moon and Spencer [6] in covering more than just the standard coordinate systems. The former gives an exposition of the method as applied to the Laplace, Helmholtz, and Schrödinger equations, whereas the latter lists the coordinate systems, resulting ODEs, and series solutions in a very compact and formal form, leading occasionally to less practical solutions (see, e.g., the “corrections” in [12]). We extend their treatments by giving many examples of boundaryvalue problems and include some more recent results mostly in the field of nanotechnology. Our book is not a comprehensive review of all the special-function literature, nor is the formal mathematical theory presented. The former is done in the many books on special functions, whereas the latter is presented in a nice book by Miller [8]. It is also worthwhile pointing out that the method of separation of variables has been applied to other PDEs such as the Dirac equation and the Klein–Gordon equation. One of the foci of the book is to emphasize that there are three distinct separability problems: that of the differential equations, that of the separation constants, and that of the boundary conditions. The separability of the differential equations is addressed by presenting the results in 11 coordinate systems (even though there can be separability in additional coordinate systems for special cases such as the Laplace equation). The consequence of a varying degree of separability of the separation constants is made clear in connection with the boundary-value problem; this is an aspect that is missing in Moon and Spencer’s treatment. Finally, the separability of the boundary conditions relates to the choice of the coordinate system. Last but not least, we present a variety of computational algorithms for the more difficult boundary-value problems that should be of practical help to readers for a complete solution to such problems. In this respect, we show the limited practical value of the series

1 Introduction

solutions in the book by Moon and Spencer and the usefulness but also restricted applicability of the algorithms given by Zhang and Jin [13]. This aspect is also not covered in the book by Morse and Feshbach. The book is divided into four parts. The first part deals with the general theory of the method of separation of variables and also has a brief summary of the areas of physical applications discussed in the book. Part Two presents the technique in two dimensions. The solutions of the resulting ODEs are discussed in some detail, particularly when a special function appears for the first time. Part Three considers the three-dimensional coordinate systems, which include the simple threedimensional extension of the two-dimensional systems of Part Two (rectangular and cylindrical systems) and of systems with rotational symmetry, and also the lesser known conical, ellipsoidal, and paraboloidal systems. Part Four provides an alternative formulation of the method of separation of variables in terms of differential geometry. Illustrations are provided for problems with nanowire structures and a recent perturbative theory is discussed in detail. Finally, a few key results on special functions are included in the appendices. Functions that appear directly as solutions to the separated ODEs are described in separate appendices (except for Appendix I on elliptic functions) and other useful functions which show up occasionally are collected in Appendix A on the hypergeometric function. In summary, it is intended that this book not only contains the standard introductory topics to the study of separation of variables but will also provide a bridge to the more advanced research literature and monographs on the subject. The fundamental material presented and a few of the coordinate systems can serve as a textbook for a one-semester course on PDEs either at the senior undergraduate level or at the graduate level. It is also expected to complement the many books that have already been published on boundary-value problems and special functions (e.g., [5–9, 14–19]), particularly in the treatment of the Helmholtz problem. The chapters not covered in a course would be appropriate for self-study and even serve as sources of ideas for both undergraduate- and graduate-level research projects.

5

7

2 General Theory 2.1 Introduction

It is widely believed that the first systematic study of the conditions required for a partial differential equation (PDE) to be separable was carried out by Stäckel [20] for the nonlinear Hamilton–Jacobi equation. This procedure was applied by Robertson to the time-independent Schrödinger equation [21], leading to the so-called Stäckel–Robertson separability conditions. Eisenhart subsequently showed that the Schrödinger equation is separable in exactly 11 curvilinear orthogonal coordinate systems, all derived from confocal quadrics [22–24]. In this chapter, we will summarize the types of PDEs to be discussed together with some possible physical applications of the said equations. We will also present key results on curvilinear differential operators and the general separability conditions in Euclidean 2- and 3-spaces, as well as the Frobenius method for series solutions.

2.2 Canonical Partial Differential Equations

We will look at the mathematical solutions to three types of canonical PDEs: r 2 ψ D 0 (Laplace equation) ,

(2.1)

r ψ C k ψ D 0 (Helmholtz equation) ,

(2.2)

r ψ C k (r)ψ D 0 (Schrödinger equation) ,

(2.3)

2 2

2 2

where r 2 is the Laplacian operator, ψ is a scalar field (we will only rarely mention other types of field such as vector fields), and k 2 is either a constant or a function of the spatial coordinates. The Laplace equation arises in potential-field problems such as electrostatics and Newtonian gravitation. The Helmholtz equation arises as the time-independent part of the wave equation, r 2 Ψ (r, t)


1 @2 Ψ (r, t) D0, c2 @t 2

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

8

2 General Theory

and the diffusion equation, r 2 Ψ (r, t)


1 @Ψ (r, t) D0. K @t

The Schrödinger equation is similar to the Helmholtz equation except for the generalization of the wave number k to be position dependent. As given, it is the timeindependent version of the time-dependent Schrödinger equation,


„2 2 @Ψ (r, t) r Ψ (r, t) C V(r)Ψ (r, t) D i„ . 2m @t

This, of course, does not include all of the physical theories. For example, firstorder differential equations such as the Dirac equation and higher-order equations such as for the mechanics of beam bending will not be discussed to keep this book manageable and focused.

2.3 Differential Operators in Curvilinear Coordinates

Specific forms of the differential operators will be used in the respective chapters on the various coordinate systems. Here we provide a summary of the main expressions needed, with emphasis on orthogonal systems, as general expressions can be written down in terms of a metric. Derivations of the results below can be found in any standard mathematical physics or vector calculus textbook. 2.3.1 Metric

Given two coordinate systems, one can write the line element in both systems as ds 2 D

X

dx i2 D

i

X

g i j dq i dq j ,

(2.4)

ij

where x i (i D 1, 2, 3) represents the Cartesian set and the q i are known as curvilinear coordinates. Then g i j D h 2i j D

@x @x @y @y @z @z C C . @q i @q j @q i @q j @q i @q j

(2.5)

g is known as the metric and, since we are only dealing with Euclidean space in this book, no distinction is made between covariant and contravariant indices (an exception will be in the differential-geometric formulation). For orthogonal systems, g i j D h 2i j D 0 for i ¤ j , and we write h i i D h i . The latter is also known as a scale factor.

(2.6)

2.3 Differential Operators in Curvilinear Coordinates

2.3.2 Gradient

The gradient of a scalar field is given by r ψ (q i ) D e 1

1 @ψ 1 @ψ 1 @ψ C e2 C e3 , h 1 @q 1 h 2 @q 2 h 3 @q 3

(2.7)

where the e i are the unit vectors of the curvilinear coordinates, ei D

1 @r . h i @q i

(2.8)

It is often convenient to express the latter in terms of Cartesian unit vectors since the latter are constant vectors. In this case, one can write
 1 @x @y @z (2.9) ex C ey C ez . ei D h i @q i @q i @q i 2.3.3 Divergence

The divergence of a vector field V (q i ) is   @ 1 @ @ (V1 h 2 h 3 ) C (V2 h 3 h 1 ) C (V3 h 1 h 2 ) . r  V (q i ) D h 1 h 2 h 3 @q 1 @q 2 @q 3

(2.10)

2.3.4 Circulation

The circulation of a vector field V is ˇ ˇe h ˇ 1 1 e2 h 2 1 ˇ @ @ rV D ˇ @q 2 h 1 h 2 h 3 ˇ @q 1 ˇ h 1 V1 h 2 V2

ˇ e 3 h 3 ˇˇ @ ˇ @q 3 ˇ . ˇ h 3 V3 ˇ

(2.11)

2.3.5 Laplacian

The Laplacian of a scalar field is obtained by combining Eqs. (2.7) and (2.10): 


 @ h 2 h 3 @ψ @ h 3 h 1 @ψ @ h 1 h 2 @ψ 1 2 C C . r ψD h 1 h 2 h 3 @q 1 h 1 @q 1 @q 2 h 2 @q 2 @q 3 h 3 @q 3 (2.12) 2.3.5.1 Example As an example, consider the circular cylindrical coordinate system with the following coordinates:

q1 D r ,

q2 D φ ,

q3 D z ,

9

10

2 General Theory

and the relationship to the Cartesian coordinates x D r cos φ ,

y D r sin φ ,

zDz.

Then, we have, using Eq. (2.5), hr D 1 ,

hφ D r ,

hz D 1 ,

and Eqs. (2.7)–(2.12) become r ψ D er

e φ @ψ @ψ @ψ C C ez , @r r @φ @z

e r D cos φ e x C sin φ e y ,

e φ D
sin φ e x C cos φ e y ,

ez D ez ,

1 @ @Vz 1 @Vφ (r Vr ) C C , r @r r @φ @z ˇ ˇ ˇe r e φ e z ˇˇ 1 ˇˇ @r @ @ ˇ r  V D ˇ @r @φ @z ˇˇ , rˇ ˇVr r Vφ Vz ˇ
 @ψ 1 @2 ψ 1 @ @2 ψ 2 r C 2 C . r ψD r @r @r r @φ 2 @z 2 rV D

2.4 Separation of Variables

The fundamental idea of the method of separation of variables is to convert a PDE into a system of ordinary differential equations (ODEs). The best known case is when the solution is written in a pure product form: ψ (q 1 , q 2 , q 3 ) D Q 1 (q 1 ) Q 2 (q 2 ) Q 3 (q 3 ) .

(2.13)

If this allows a complete separation of the variables in the differential equations, then the equation is said to be simply separable. If a separation of the differential equations is achieved using the function ψ (q 1 , q 2 , q 3 ) D

Q 1 (q 1 ) Q 2 (q 2 ) Q 3 (q 3 ) , R (q 1 , q 2 , q 3 )

(2.14)

where R is not a constant, then the PDE is said to be R separable. Over the years, many results on the general theory have been obtained. A few of the ones most relevant to this book are:  The Laplace equation in Euclidean 3-space is separable in additional coordinate systems; furthermore, the Laplace equation is R separable [6].  The Helmholtz equation is separable in 11 orthogonal curvilinear coordinate systems in Euclidean 3-space and in four orthogonal curvilinear coordinate systems in Euclidean 2-space [5].

2.4 Separation of Variables

 The vector Helmholtz equation is only separable in the z variable for the same four cylindrical systems as for the scalar equation, whereas it is only separable in the φ variable for the same rotational systems as for the scalar equation [6].  The Schrödinger equation is separable in 11 orthogonal curvilinear coordinate systems in Euclidean 3-space provided the potential function has specific forms [24]. We now discuss a procedure for searching for separable coordinate systems in two and three Euclidean dimensions and present the general formalism for separating the canonical PDEs. The discussion of the theory follows Morse and Feshbach [5]. 2.4.1 Two Dimensions

The basic approach will be the following:  The most general PDE that we will consider, the time-independent Schrödinger equation, can be directly shown to be separable in Cartesian coordinates.  For the Laplace equation, all coordinate systems related to the rectangular one via a conformal transformation are separable.  For the Schrödinger equation, we will derive the condition to be satisfied by conformal transformations such that the differential equation remains separable.

2.4.1.1 Rectangular Coordinate System Though the separation problem in rectangular 2-space is trivial, it is included here as an initial simple illustration of the method as well as the starting point for generating additional separable systems in 2-space. Thus, the PDE is

@2 ψ @2 ψ C C k2 ψ D 0 . 2 @x @y 2

(2.15)

This is known to be separable if k 2 is a constant (Laplace and Helmholtz cases). Writing ψ(x, y ) D X(x)Y(y ), one gets 1 d2 Y 1 d2 X C C k2 D 0 . X dx 2 Y dy 2

(2.16)

Since the first term of Eq. (2.16) is at most a function of the coordinate x, the second term is at most a function of the coordinate y, and the third term is a constant, this means all the terms must be constants. Then, this gives the separated ODEs 1 d2 X D
α 2 X dx 2

(2.17)

1 d2 Y D
β 2 , Y dy 2

(2.18)

and

11

12

2 General Theory

with α2 C β2 D k 2 .

(2.19)

α and β are known as separation constants (though in this case there is obviously only one independent one). If k 2 is position dependent, then the Schrödinger equation is separable provided the potential function is of the form k 2 (r) D γ C f (x) C g(y ) ,

(2.20)

where γ is a constant, giving   d2 X C α 2 C f (x) X D 0 , 2 dx

(2.21)

  d2 Y C β 2 C g(y ) Y D 0 , 2 dy

(2.22)

with α2 C β2 D γ . 2.4.1.2 Other Coordinate Systems The idea is that one can generate an infinite number of new orthogonal curvilinear systems in 2-space (and via the appropriate transformations, even in 3-space) by applying a conformal transformation to the rectangular system just studied. However, this does not guarantee that the PDE will be separable in the new system. The necessary conditions will now be obtained by rewriting the differential equation in terms of new variables. Given

z D x C iy ,

z D x
iy ,

(2.23)

one obtains @ @ @ D C , @x @z @z

@ Di @y

@2 @2 @2 . D4 2 2 @x @y @z@z




@ @
@z @z

 . (2.24)

The Schrödinger equation becomes 4

@2 ψ C k2 ψ D 0 . @z@z

(2.25)

We now carry out the conformal transformation to new coordinates ξ1 , ξ2 where w D ξ1 C i ξ2 ,

w D ξ1
i ξ2 .

(2.26)

2.4 Separation of Variables

Since w D f (z) is an analytic function, w D g(z) , @ dw @ D , @z dz @w

(2.27) @ dw @ D , @z dz @w

(2.28)

and 4

@2 ψ @2 ψ @2 ψ k2 ψ D C D
ˇ ˇ . ˇ dw ˇ2 @w @w @ξ12 @ξ22 ˇ dz ˇ

(2.29)

This new form of the Schrödinger equation is the starting point for studying the conformal transformations that allow separability. For the Laplace equation, k 2 D 0, and Eq. (2.29) becomes @2 ψ @2 ψ C D0. 2 @ξ1 @ξ22

(2.30)

Thus, all coordinate systems related to the rectangular system by a conformal transformation allow separation. Furthermore, one can write down the general solution for all of the systems as ψ (ξ1 , ξ2 ) D e ˙ξ1 (c 1 sin ξ2 C c 2 cos ξ2 ) .

(2.31)

For the Helmholtz and Schrödinger equations, for the right-hand side to be separable, it must be of the form ˇ ˇ ˇ dz ˇ2 ˇ D f (ξ1 ) C g (ξ2 ) , k 2 ˇˇ dw ˇ

(2.32)

that is, @2 @ξ1 @ξ2

ˇ ˇ ! ˇ dz ˇ2 ˇ ˇ ˇ dw ˇ D 0 .

(2.33)

The latter condition is often rewritten in a different form. Using Eq. (2.26), one gets @2 @2 @2 Di
i . 2 @ξ1 @ξ2 @w @w 2

(2.34)

Then, @2 @ξ1 @ξ2

ˇ ˇ !
2 2  ˇ dz ˇ2 ˇ ˇ Di @
@ ˇ dw ˇ @w 2 @w 2

ˇ ˇ ! ˇ dz ˇ2 ˇ ˇ ˇ dw ˇ D 0 ,

(2.35)

13

14

2 General Theory




dz dw



d2 dw 2




dz dw




D



dz dw

d2 dw 2




dz dw

 .

(2.36)

Rearranging so that each side is a function of w and w only implies each side must then be a constant. This leads to


 
 dz dz d2 d2 dz dz D λ , D λ , (2.37) dw 2 dw dw dw dw 2 dw where λ is a constant. Different choices for the latter lead to different coordinate systems.  Consider first λ D 0. Then,
 dz d2 D0, dw 2 dw dz D β C γw . dw

(2.38)

One further has two cases:  If γ D 0, then z D α C βw .

(2.39)

Writing z D x C i y , α D a C i b, and β D c C i d, we have x D a C c ξ1
d ξ2 ,

y D b C c ξ2 C d ξ1 .

(2.40)

The above transformation corresponds to a rotation, scaling, and translation. Since the system remains rectangular, this is not considered further.  If γ ¤ 0, one can ignore the β term and the subsequent integration constant (together, they are equivalent to the previous case). Since γ is a scaling and rotation factor, one can also now set γ D 1. Then, the new transformation is zD

1 2 w , 2

xD

 1 2 ξ
ξ2 , 2 1

(2.41)

and y D ξ1 ξ2 .

(2.42)

This gives rise to the parabolic coordinates, with confocal parabolas as constant-coordinate curves. Conformal theory also allows for a simple determination of the scale factors: ˇ ˇ q ˇ dz ˇ ˇ D jw j D ξ 2 C ξ 2 . (2.43) h 1 D h 2 D ˇˇ 1 2 ˇ dw

2.4 Separation of Variables

 For λ ¤ 0, one can choose λ D 1 since otherwise a rescaling is involved. Then, d2 dw 2





 dz dz D , dw dw dz D e ˙w , dw z D ae w C b e
w .

(2.44)

As for the previous case, there are again two possibilities:  If a D 1, b D 0, then z D e w D e ξ1 Ci ξ2 ,

(2.45)

x D e ξ1 cos ξ2 ,

(2.46)

and y D e ξ1 sin ξ2 .

If one redefines new variables r D e ξ1 ,

φ D ξ2 D tan
1

y , x

(2.47)

then it is seen to be the polar or circular coordinates.  If a and b are general, rewrite using 1
β 1 1 1 de D e α
β , b D d e β D e αCβ , 2 2 2 2 r p b , d D e α D 4ab , e β D a aD

and zD

1 w
β 1 β
w de C de D d cosh(w
β) D e α cosh (ξ1 C i ξ2
β) , 2 2 (2.48)

and, therefore, x D d cosh(ξ1
β) cos ξ2 ,

y D d sinh(ξ1
β) sin ξ2 .

(2.49)

These are known as the elliptic coordinates. The above exhaust all the possible separable coordinate systems in two dimensions.

2.4.2 Three Dimensions

The separability problem is now more complicated for at least four reasons. There are now two separation constants instead of one. In general, all three of the ODEs

15

16

2 General Theory

could depend upon both separation constants. In addition, the form of ˇ ˇ2 ˇ ˇ 2 ˇ dz ˇ k ˇ dw ˇ for separability could be more general than just f (ξ1 ) C g(ξ2 ). Finally, more complicated solutions could exist such as the R-separability of the Laplace equation. It appears the earliest general study of the separability of the Hamilton–Jacobi equation was by Stäckel [20]. This was extended to the Schrödinger equation by Robertson [21] for the types of metrics that would allow separability. The so-called Robertson condition required for separability is h1 h2 h3 D f 1 (ξ1 ) f 2 (ξ2 ) f 3 (ξ3 ) , S

(2.50)

where S is the determinant of the so-called Stäckel matrix. A derivation will now be provided. 2.4.2.1 Stäckel Matrix We will demonstrate here how a specially constructed matrix, the so-called Stäckel matrix, will allow one to automatically write down the separated ODEs of a PDE. Consider, for now, a square matrix Φ whose elements Φi j are such that the ith row is only a function of the ξi coordinate; that is,

Φi j D Φi j (ξi ) .

(2.51)

These functions Φi j will be used to construct the separable terms in the Helmholtz equation. One can form a determinant, the Stäckel determinant, ˇ ˇ S D ˇ Φi j ˇ . (2.52) One will also need the minors, whereby the first minor of the determinant for Φ is the factor that multiplies Φi j in the determinant. Thus, the minors for the elements of the first column are @S D Φ22 Φ33
Φ23 Φ32 , @Φ11 @S M2 D D Φ13 Φ32
Φ12 Φ33 , @Φ21 @S M3 D D Φ12 Φ23
Φ13 Φ22 . @Φ31

M1 D

(2.53)

They satisfy the following orthogonality relation: 3 X

M i Φi j D S δ i j .

(2.54)

iD1

Note that the characteristics of the Stäckel matrix defined by Eqs. (2.51)–(2.54) leave much freedom in the exact choice of the Stäckel matrix. We will now see what constraints will be imposed on Φ by the separability property.

2.4 Separation of Variables

2.4.2.2 Helmholtz Equation We rewrite the Helmholtz equation as

r 2 ψ C k12 ψ D 0 .

(2.55)

Let the separated equations be of the form     1 @ @X i fi C k12 Φi1 C k22 Φi2 C k32 Φi3 X i D 0 , f i @ξi @ξi

(2.56)

where f i D f i (ξi ) and k22 and k32 are the two separation constants. Multiplying the X 1 equation by M1 X 2 X 3 /S and similarly for the other two equations and adding all three gives X M i @  @ψ  fi C k12 ψ D 0 , (2.57) S f i @ξi @ξi i

where ψ D X 1 X 2 X 3 . Now, using the Laplacian in an arbitrary curvilinear coordinate system, Eq. (2.12), one can rewrite the Laplace equation as   X 1 @ h 1 h 2 h 3 @ψ . (2.58) r2 ψ D h 1 h 2 h 3 @ξi h 2i @ξi i The equivalence of the Laplacian in Eqs. (2.57) and (2.58) restricts the choices of h i and Φi j . For example, comparing inside the square brackets, one requires that h1 h2 h3 D f i gi , h 2i

(2.59)

where g i is a function that does not depend on ξi . For concreteness, consider i D 1; then, we have 1 @ h 1 h 2 h 3 @ξ1



h2 h3 @ h 1 @ξ1



g1 @ D h 1 h 2 h 3 @ξ1



@ f1 @ξ1



@ D 2 h 1 f 1 @ξ1 1



 @ f1 , @ξ1 (2.60)

since g 1 D h 2 h 3 /(h 1 f 1 ) from Eq. (2.59). Comparing with Eq. (2.58) gives 1 Mi D 2 , S fi hi f i or Mi 1 , D S h 2i

(2.61)

from which the Robertson condition follows. This theory has been treated in a few places with differing notations. To help the reader, we give in Table 2.1 the correspondence among those notations.

17

18

2 General Theory Table 2.1 Comparison of notations used in Stäckel theory literature. Morse and Feshbach [25]

Moon and Spencer [6]

Robertson [21]

hi

gii

h D h1 h2 h2 S

g 1/2 S

h
1/2 '

Mi

M i1

' 0 ' 0 i1


1/2

1/2

hi

Eisenhart [22] Hi H

The separated differential equations are then   X dXi 1 d fi C Φi j (ξi ) k 2j X i D 0 , f i dξi dξi

(2.62)

j

where k12 is the wave number (squared). Hence, knowing the Stäckel matrix and the f i functions allows one to write down the separated differential equations instantly. As an illustration of the theory, consider the circular cylinder system. Given ξ1 D r ,

ξ2 D φ ,

x D r cos φ , h1 D 1 ,

ξ3 D z ,

y D r sin φ ,

h2 D r ,

zDz,

h3 D 1 ,

Eq. (2.59) becomes h1 h2 h3 D r D g1 f 1 h 21 h1 h2 h3 1 D D g2 f 2 r h 22 h1 h2 h3 D r D g3 f 3 h 23

H)

g1 D 1 , f 1 D r ,

H)

g2 D

H)

g3 D r , f 3 D 1 ,

1 , f2 D 1 , r

and a possible choice of quantities to satisfy Eq. (2.61) is S D1, S M1 D 2 D 1 , h1

M2 D

S 1 D 2 , r h 22

Hence, a possible Stäckel matrix is 0

0 Φ D @0 1


1/r 2 1 0

1
1 0A . 1

M3 D

S D1. h 23

2.4 Separation of Variables

It is clear that the procedure does not define a unique Stäckel matrix. One can now readily write down the ODEs using Eq. (2.62): !
 k φ2 dR(r) 1 d 2 r
C k z R(r) D 0 , r dr dr r2 d2 Φ (φ) C k φ2 Φ (φ) D 0 , dφ 2  d2 Z(z)  2 C k r C k z2 Z(z) D 0 . dz 2

(2.63)

Familiarity with the circular cylinder system would show that these results are usually given in a slightly different form, specifically with the following change in the separation constants: k r2 C k z2 D k32 ,

k φ2 D k22 ,

k r2 D k12 .

This is a consequence of the nonuniqueness in the choice of the Stäckel matrix. One can also easily separate the Helmholtz equation “by hand,” which leads to the conventional form of the ODEs; of course, the Stäckel theory is an alternative approach with the intent of bypassing all the tedious manipulations once the Stäckel matrix is known. 2.4.2.3 Schrödinger Equation Robertson [21] found the additional condition on the potential energy for the Schrödinger equation to be separable. In particular, the potential energy must be of the form

V (ξ1 , ξ2 , ξ3 ) D

X v i (ξi ) . h 2i i

The separated differential equations are then 2 3   X 1 d dXi fi C4 Φi j (ξi )
j
v i (ξi )5 X i D 0 , f i dξi dξi

(2.64)

(2.65)

j

where
1 is the reduced energy. 2.4.2.4 Separable Coordinate Systems Eisenhart [22–24], using geometry, derived the 11 coordinate systems that are compatible with the constraint obtained by Robertson. Moon and Spencer have derived a number of additional results [6, 25, 26]. Note that, even though Moon and Spencer showed how to generate an infinite number of orthogonal curvilinear coordinate systems (by using all the conformal transformations in 2-space irrespective of whether they lead to separation or not and then either translating or rotating about an axis to generate 3-space equivalents), they also confirmed that none of

19

20

2 General Theory

the additional ones lead to simple separability of either the Laplace equation or the Helmholtz equation. Hence, those other coordinate systems will not be considered in this book. A more modern approach is to use symmetry arguments to establish the separability of equations [27, 28]. As an example, Boyer et al. [29] have shown that the time-dependent 2-space free-particle Schrödinger equation is exactly separable in 26 coordinate systems. Furthermore, different types of separability have been studied, including R-separability [26], P-separability [30], and generalized Stäckel matrices [31].

2.5 Series Solutions

The solutions to the ODEs are generally appropriately developed as power series; these can then be classified as special functions. We present here a brief overview of the relevant results that will be used throughout the book. Most of the applications will deal with series expansion in a real coordinate x, though in a few cases one cannot avoid complex variables. However, as little of complex variable theory will be assumed and used as is possible. 2.5.1 Singularities

The standard second-order linear differential equations we will encounter are of the general form dy d2 y C Q(z)y D 0 . C P(z) 2 dz dz

(2.66)

One defines  An ordinary point: P(z0 ) and Q(z0 ) are both finite and can be expanded as series of nonnegative powers (so-called analytic function).  A singular point: either P(z0 ) or Q(z0 ) is nonanalytic. In this case, a series expansion about z0 has terms with negative powers of (z
z0 ) and is known as a Laurent series.  A regular singular point: (z
z0 )P(z) and (z
z0 )2 Q(z) are analytic and nonsingular solutions exist.  A pole. If a series expansion of the function f (z) D

1 X

a n (z
z0 )n

nD
1

is such that the series has a n D 0 8n <
m and a
m ¤ 0, then we say that z0 is a pole of order m.

2.5 Series Solutions

 An essential singularity: a Laurent series about an essential singularity does not terminate.  A branch point is a singularity which is not isolated.  A point at infinity. To determine the nature of the point at infinity, one uses the transformation z D 1/w , whereby Eq. (2.66) becomes d2 y C dw 2




P(w
1 ) 2
w w2



Q(w
1) dy C y D0, dw w4

(2.67)

and the point is now at w D 0. For the rest of our study, we will assume the existence of a domain in which both P(z) and Q(z) are analytic except for a finite number of poles. 2.5.2 Bôcher Equation

Second-order ODEs without essential singularities can be rewritten in Bôcher [6, 32] form with the coefficients given as   m2 m n
1 1 m1 , (2.68) C C C P(z) D 2 z
a1 z
a2 z
a n
1 # " 1 A0 C A1 C    C A l z l Q(z) D , (2.69) 4 (z
a 1 ) m 1 (z
a 2 ) m 2    (z
a n
1 ) m n
1 and m i , n, and l are either positive or zero. Except for degenerate cases, the poles of highest order usually occur for Q(z); these characterize the singularities of the differential equation. Thus, if there are n singularities where the singularity at z D a i is of order m i , then one labels the equation as of type fm 1 m 2 . . . m n g . Conventionally, the point at infinity, if present, is given last. Furthermore, if this is the only singularity, a first index of 0 is still provided to emphasize that the singularity is at infinity and not in the finite plane. 2.5.2.1 Example The Bessel equation is

z2

  d2 y dy C k2 z2
l2 y D 0 . Cz dz 2 dz

Rewritten as 
d2 y 1 dy l2 2 y D0, C k C
dz 2 z dz z2

(2.70)

21

22

2 General Theory

it is seen to be of Bôcher form with a singularity at z D 0 and m1 D 2 ,

a1 D 0 ,

A0 D
l 2 ,

A2 D k 2 ,

and all other A n D 0. Furthermore, to find the nature of the point at infinity, we transform the Bessel equation using w D 1/z, giving a new differential equation in w with P(w ) D

1 2
, w w

Q(w ) D

 1  2 k
l2w2 . 4 w

This shows that w D 0 of the transformed equation is a singularity of order 4, that is, this is the order of the singularity of the point at infinity of the original Bessel equation. Therefore, the Bôcher type for the Bessel equation is f2 4g. 2.5.3 Frobenius Method

The following theorem is important in establishing the series solution method: Theorem 1 Fuch’s theorem At least one series solution can be obtained at an ordinary or a regular singular point. Even at a regular singular point, there may exist finite solutions. Indeed, Morse and Feshbach [5] established the following results:  At an ordinary point, there exist two analytic series solutions valid to the nearest singularity.  If P(z) has a simple pole and Q(z) is analytic at z D z0 , one solution is analytic and the second solution can be written as the product of a function with a branch point at z D z0 times an analytic function.  If z D z0 is a regular singular point, then this requires P and Q to have the form P(z) D

F(z) , (z
z0 )

Q(z) D

G(z) , (z
z0 )2

where F and G are analytic functions.  If P(z) has a pole of order higher than 1 and Q(z) is analytic at z D z0 , one solution is analytic and the second has an essential singularity at z D z0 .  If z D z0 is an irregular singular point (i.e., either P has a pole of order higher than 1 or Q has a pole of order higher than 2 or both), one solution or both solutions must have an essential singularity.  If Q has a pole of order 1 higher than for P and a term in (z
z0 )
2 , then only one solution has an essential singularity.

2.5 Series Solutions

In general, therefore, one can try the following series solution at either an ordinary point or a regular singular point (assumed at z D 0): y (z) D

1 X

a r z rCσ .

(2.71)

rD0

To find σ and the coefficients a r , one can rewrite Eq. (2.66) as z 2 y 00 C z 2 P(z)y 0 C z 2 Q(z)y D 0 ,

(2.72)

z 2 y 00 C z p (z)y 0 C q(z)y D 0 .

(2.73)

or

Then, at an ordinary point or a regular singular point, p (z) and q(z) can be expanded in terms of power series, p (z) D

1 X

pi zi ,

q(z) D

iD0

1 X

qi zi .

(2.74)

iD0

Substituting Eq. (2.71) into Eq. (2.73) and using Eq. (2.74) gives 1 X

a n (n C σ)(n C σ
1)z nCσ C

nD0

1 X 1 X

  a n p i (n C σ) C q i z nCσCi D 0 ,

nD0 iD0

(2.75) which is equivalent to ( ) 1 n X X   (σ C r)p n
r C q n
r a r z nCσ D 0 , a n (n C σ)(n C σ
1) C nD0

rD0

(2.76) or a n (n C σ)(n C σ
1) C

n X   (σ C r)p n
r C q n
r a r D 0 .

(2.77)

rD0

One can then show that Θ (σ)  σ(σ
1) C σ p 0 C q 0 D 0 ,

(2.78)

which is known as the indicial equation and gives the allowed values of the indices σ, and one has the recurrence relation n   P a n
j (σ C n
j ) p j C q j an D


j D1

Θ (σ C n)

.

(2.79)

General properties of the solutions of various types of differential equations are determined by the number and nature of the singularities. These have been exhaustively studied by Morse and Feshbach [5] and a summary is now provided for completeness.

23

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2 General Theory

2.5.3.1 One Regular Singular Point If the regular singular point is at z D a, then the most general equation of this type can be written as

2 dy d2 y D0, C dz 2 (z
a) dz

(2.80)

with solution y (z) D A C

B . z
a

(2.81)

If the singularity is at infinity, then setting a ! 1 gives d2 y D0, dz 2

(2.82)

with solution y (z) D A C B z .

(2.83)

2.5.3.2 Two Regular Singular Points A differential equation with two regular singular points can generally be written as

λµ(a
c)2 2w C c(λ C µ
1)
a(λ C µ C 1) dy d2 y C C y D0. dw 2 (w
a)(w
c) dw (w
a)2 (w
c)2 (2.84) The coefficients have been chosen such that (1) there is no singularity at infinity, (2) the two singularities are at w D a and w D c, and (3) the roots of the indicial equation are λ and µ for w D a, and
λ and
µ for w D c. The equation is of standard form if the two singularities are at z D 0 and z D 1. This can be achieved by using the following transformation: zD

w
a . w
c

(2.85)

Then, Eq. (2.84) becomes d2 y
dz 2




λCµ
1 z



dy λµ C 2 y D0. dz z

(2.86)

It is easy to see that the solutions to the latter equation are z λ and z µ . 2.5.3.3 One Irregular Singular Point Depending upon the order of the singularity, different equations can be written down. A common example, also considered by Morse and Feshbach, is

dy k2 2 d2 y
C y D0, 2 dw (w
a) dw (w
a)4

(2.87)

2.5 Series Solutions

with the singularity at w D a. It is trivial to show that the solutions are of the form e

k ˙ (w
a)

.

One can transform Eq. (2.87) into the standard form (singularity at infinity) by using zD

1 . w
a

Then, d2 y
k2 y D 0 . dz 2

(2.88)

It can be shown that such an equation arises from the convergence of two regular singular points [5]. 2.5.3.4 Three Regular Singular Points Let the three singularities be at w D a, w D b, and w D c with indicial roots fλ, λ 0 g, fµ, µ 0 g, and fν, ν 0 g, respectively. Then, one can show that a general form of such a differential equation is given by [5]

  µ C µ0
1 ν C ν 0
1 dy d2 y λ C λ0
1 C C
dw 2 (w
a) (w
b) (w
c) dw  0 0 µ µ (b
a)(b
c) λλ (a
b)(a
c) C C (w
a)2 (w
b)(w
c) (w
a)(w
b)2 (w
c)  νν 0 (c
a)(c
b) y D0. C (w
a)(w
b)(w
c)2

(2.89)

The above equation is known as the Papperitz equation [5] or the Riemann equation [33]. The indices satisfy the constraint λ C λ0 C µ C µ0 C ν C ν0 D 1 .

(2.90)

The solution is often symbolically written as a Riemann P symbol: 8
b µ µ0

c ν ν0

9 = z

;

.

(2.91)

The standard form is arrived at with a D 0, b D 1, and c D 1. Then, the differential equation becomes   d2 y µ C µ 0
1 dy λ C λ0
1 C
dz 2 z (z
1) dz   0 0   λλ µµ y

C ν λ C λ0 C µ C µ0 C ν
1 D 0 . (2.92) z (z
1) z(z
1)

25

26

2 General Theory

The corresponding solution in the Riemann notation is 8 <0 y (z) D P λ : 0 λ

1 µ µ0

1 ν 1
λ
λ0
µ
µ0
ν

9 = z

;

.

(2.93)

Finally, if one factors out the singular part for one of the solutions at each of z D 0 and z D 1 and relabels the indices to be 1
c at z D 0, c
a
b at z D 1, and a and b at z D 1, then the equation takes the form z(z
1)

d2 y dy C ab y D 0 . C [(a C b C 1)z
c] dz 2 dz

(2.94)

This is known as the hypergeometric equation (Appendix A). The resulting ODEs and their properties are given in Table 2.2. Note that these depend on the choice of coordinates. For example, for the circular cylindrical system, if ξ2 D φ, then the Bôcher type of the equation is f0 4g (ordinary point at origin and regular singular point at infinity), whereas if ξ2 D cos φ, then the Bôcher type of the equation is f2 2 2g (regular singular points at
1, 1, and 1).

2.6 Boundary-Value Problems 2.6.1 Boundary Conditions

The method of separation of variables gave us ODEs for which general solutions can be written down. For physical applications, it will often be the case that we have to solve a so-called boundary-value problem, that is, to find particular solutions that satisfy the boundary conditions on boundary surfaces. The boundary surfaces could be closed (e.g., a cavity) or open (e.g., a waveguide). We will denote the domain V and the boundary surface S. There are four types of boundary conditions that occur often in physical applications: ψj Si D A

8i (Dirichlet boundary condition) ,

nO  r ψj Si D 0 8i (Neumann boundary condition) , ψj Si D 0 ,

(2.95) (2.96)

nO  r ψj S j ¤i D 0 (Dirichlet–Neumann boundary condition) , (2.97)

( nO  r ψ C γ ψ)j Si D 0 (mixed boundary condition) .

(2.98)

A Dirichlet boundary condition implies specifying the value of the solution on the bounding surface. A Neumann boundary condition implies specifying the value of the normal derivative on the bounding surface. Dirichlet and Neumann boundary conditions imply specifying the value of the solution on part of the bounding surface and of the normal derivative on the remainder of the bounding surface. The

2.6 Boundary-Value Problems Table 2.2 Separable coordinate systems in 2- and 3-space for the Helmholtz equation. Note that the various assignments are for nonzero values of the separation constants. ODE: ordinary differential equation; RSP: regular singular point; ISP: irregular singular point. System

ODE

x y

x y z r φ z

Bessel

u v

Mathieu Modified Mathieu

Singularities

Bôcher type

2D Rectangular 1: ISP f0 4g 1: ISP f0 4g 3D Rectangular 1: ISP f0 4g 1: ISP f0 4g 1: ISP f0 4g Circular cylinder 0: RSP, 1: ISP f2 4g 1: ISP f0 4g 1: ISP f0 4g Elliptic cylinder 0,1: RSP, 1: ISP f1 1 3g f1 1 3g 1: ISP

z µ ν z

Weber Weber

r θ

Bessel Associated Legendre

φ

f0 4g Parabolic cylinder 1: ISP f0 6g 1: ISP f0 6g 1: ISP f0 4g Spherical polar 0: RSP, 1: ISP f2 4g 0, 180ı , 1: RSP f2 2 2g 1: ISP

f0 4g

Functions

Separation constants

Trigonometric Trigonometric

1 1

Trigonometric Trigonometric Trigonometric

1 1 1

Bessel Trigonometric Trigonometric

2 1 1

Mathieu

2 2

Trigonometric

1

Weber Weber Trigonometric

2 2 1

Bessel Associated Legendre Trigonometric

1 2

Bessel Lamé wave

1 2

1

Conical r θ, λ

Bessel Lamé wave

ξ, η φ

Bessel wave

α, β

Spheroidal wave

φ µ, ν, λ

Lamé wave

µ, ν, λ

Baer wave

0: RSP, 1: ISP f2 4g 0, a, b: RSP, 1: ISP f1 1 1 2g Parabolic rotational 0: RSP, 1: ISP f2 6g 1: ISP f0 4g Prolate spheroidal Oblate spheroidal ˙1: RSP, 1: ISP f2 2 4g

Bessel wave Trigonometric

2 1

Spheroidal

2

1: ISP

Trigonometric

1

Lamé wave

2

Baer wave

2

f0 4g Ellipsoidal 0, a, b: RSP, 1: ISP f1 1 1 3g Paraboloidal b, c: RSP, 1: ISP f1 1 4g

27

28

2 General Theory

mixed boundary condition implies specifying a linear combination of the value of the solution and of the normal derivative on the bounding surface. It is worth pointing out at this point that for certain coordinate systems the above boundary conditions might not be sufficient for a well-behaved solution. For example, we will find that for the parabolic cylinder coordinates an additional condition of finiteness of the gradient of the solution is required. The origin of such additional boundary conditions can be traced to degenerate coordinate surfaces. Indeed, many of the coordinate systems can be obtained as degenerate cases of the ellipsoidal one. Quoting from the Bateman manuscript [33], “The degenerate surfaces act as branch-cuts, and the postulate of continuity of a function across these branch-cuts has the character of boundary conditions.” The choice of coordinate system is made typically so as to simplify the description of the bounding surface as this is necessary to make the whole problem separable and not just the differential equation. Typically, this implies that the choice is such that the boundary surface is a one-coordinate surface, q i D S , a constant. For example, for a spherical boundary, the choice of spherical polar coordinates leads to the surface being described by r D R. Nevertheless, not all one-coordinate surfaces lead to a separable problem. We will find one such counterexample for an elliptical boundary and the finite-barrier problem in quantum mechanics. Correspondingly, it is also possible to find separable problems for surfaces that depend upon more than one coordinate. For example, a closed cavity formed by two paraboloids can be described by the constraints on two coordinates of the parabolic rotational coordinate system, ξ D X, η D Y , and can still lead to separable problems. There are also special cases, such as Dirichlet boundary conditions, that can lead to “enhanced” separability (e.g., the infinite-barrier potential problem for a cubic box in three dimensions using Cartesian coordinates when the finite-barrier problem is not separable). 2.6.2 Fourier Expansions

Given the ODE in Eq. (2.66), it can be written in Sturm–Liouville form,   d dZ u(z) C [v (z) C f (λ)w (z)]Z D 0 , dz dz

(2.99)

where λ is an eigenvalue, w (z) is a weighting factor, and u(z) D C e

R

dz P(z)

,

Q(z) D

 1  v (z) C f (λ)w (z) . u(z)

(2.100)

This becomes a Sturm–Liouville theory if the Sturm–Liouville equation is supplemented by the homogeneous boundary conditions (

Z i (a) C h Z i0 (a)

D0,

h Z i0 (b)

D0,

Z i (b) C

(2.101)

2.6 Boundary-Value Problems

where i D 1, 2 (the two independent solutions). If there is only one separation constant, then the eigenvalue λ is only a function of the separation constant and there is one value of λ for each allowed value (by the boundary conditions) of the separation constant; the functions u, v, and w are then independent of the separation constant. If there are two separation constants, then the eigenvalue λ can be a function of either separation constant (the choice is dictated by the boundary condition), and the other is a constant and may appear in v and w. For a Sturm–Liouville problem, the solutions satisfy an orthogonality relation: Zb dz Z m (z)w (z)Z n (z) D 0

for m ¤ n .

(2.102)

a

This allows one to express boundary conditions as Fourier expansions in terms of the solutions to the ODEs, 1 X

f (z) D

A m Z m (z) ,

(2.103)

mD0

from which the expansion coefficients can be determined using the orthogonality of the solutions, Am D

1 Nm

Zb dz w (z)Z(z) f (z) ,

(2.104)

a

and the normalization is Zb Nm D

dz w (z)[Z(z)]2 .

(2.105)

a

For the Laplace problem, it is often useful to appeal to the uniqueness theorem at the end to justify the solution obtained. The study of the Laplace equation with boundary conditions is also known as potential theory. The solution to the Laplace equation is then known as a harmonic solution. As an example, the Dirichlet boundary-value problem for the Laplace equation can be formulated as follows: Given a domain V with boundary S and a function f defined on S, obtain a function ψ such that (1) ψ is a harmonic function in V and continuous in V C S , and (2) Ψ becomes f on S. For a well-defined problem, one requires a closed boundary and the specification of either the Dirichlet or the Neumann condition. Then, from the fact that ψ is real and the first and second partial derivatives are continuous in V and on S, one can show that the solution is unique (up to a constant). For an open boundary, it is necessary to specify the behavior of the solution ψ at infinity.

29

30

2 General Theory

2.7 Physical Applications

We finally provide a self-contained overview of various physical theories that lead to PDEs of the types previously discussed. Readers familiar with the following topics can skip the rest of the chapter. 2.7.1 Electrostatics

In the presence of a static charge density , the electric field E satisfies Gauss’s law: rE D

 . ε0

(2.106)

The electrostatic potential φ then satisfies the Poisson equation: r2 φ D


 . ε0

(2.107)

In regions where the charge density is zero, the corresponding equation is the Laplace equation: r2 φ D 0 .

(2.108)

Such a PDE is of elliptic form and has a unique solution for a closed boundary with either a Dirichlet or a Neumann boundary condition. This result also applies to the Poisson equation. 2.7.2 Photonics

A field related to the previous electrostatic problem (in that both are special cases of the Maxwell equations of electromagnetism) is photonics or integrated optics. In this case, one studies the propagation of electromagnetic waves in nonmagnetic dielectrics. Assuming an isotropic, nonmagnetic, and sourceless dielectric, the Maxwell equations are @B , @t @D rH D , @t

rE D


(2.109) (2.110)

rD D0,

(2.111)

rB D0,

(2.112)

2.7 Physical Applications

and the constitutive equations are D D ε ε0 E ,

(2.113)

B D µ0 H .

(2.114)

From the above equations, one can derive vector wave equations; for example, for the electric field, one has   
ε @2 E rε E . (2.115) D r(r  E)
r  (r  E) C r c 2 @t 2 ε For homogeneous dielectrics, r  E D 0 and r ε D 0, and one has the vector wave equation r2 E C

1 @2 E D0. c 2 @t 2

(2.116)

The latter equation is, generally, not separable but simplifies considerably for systems with cylindrical symmetry. As an example, consider the case of cylindrical dielectric tapers, that is, the medium is homogeneous in the direction of the cylinder (z) axis, proposed by Sakai and Marcatili [34]. For an electromagnetic field with only transverse variations, E D E1 (ξ1 , ξ2 ) eO 1 C E2 (ξ1 , ξ2 ) eO 2 C E3 (ξ1 , ξ2 ) eO 3 ,

(2.117)

with the cylinder axis in the 3-direction, and a phase factor exp[i(ωt
k z)], one obtains the following scalar wave equations:   
  rε (2.118)  Et D 0 , r t2 E3 C ω 2 ε ε 0 µ 0
k 2 E3
i k ε  
  1 @g 22 1 g 22 @g 11 @g 22 r t2 E t C ω 2 ε ε 0 µ 0
k 2 E1 C 
g 11 C 2g @ξ1 2 @ξ1 g @ξ1  
2 2 1 @g 11 @ g 22 g 11 @g 22 @ g 11 @g 11  E1 C
g 22 C C @ξ2 2 @ξ2 g @ξ2 @ξ12 @ξ22 
  1 @g 11 @E2 1 @g 22 @E2 1 @ rε  Et C
Cp p p g 11 g @ξ2 @ξ1 g 22 g @ξ1 @ξ2 g 11 @ξ1 ε D0.

(2.119)

2.7.3 Heat Conduction

The heat diffusion equation is r 2 T(r, t)


1 @T(r, t) D0, K @t

(2.120)

31

32

2 General Theory

where T is the temperature distribution and K is the thermal conductivity. In the steady state, the differential equation transforms into the Laplace equation: r 2 T(r) D 0 .

(2.121)

A number of different kinds of boundary condition can be written down depending upon the physical condition. The three main types are T j S D T(r) , where the temperature is given on the surface (Dirichlet), ˇ @T ˇˇ D0, @n ˇ S where the flux across the surface is zero, and ˇ @T ˇˇ C hT D 0 , @n ˇ S

(2.122)

(2.123)

(2.124)

where there is linear radiation at the surface into the medium at zero temperature. 2.7.4 Newtonian Gravitation

This is another example where the Laplace and Poisson equations arise. Specifically, the gravitational potential outside a body of density  is given in Newtonian mechanics as Z G(r 0 ) V(r) D dV 0 , (2.125) jr
r 0 j where G is the gravitational constant. In analogy to electrostatics, it is clear that this is the solution to the Laplace equation r 2 V(r) D 0 ,

(2.126)

with the boundary condition of a vanishing field at infinity (Dirichlet). For the gravitational potential inside the body, the appropriate equation to solve is the Poisson equation: r 2 V(r) D
4π G(r) .

(2.127)

Often, the problem will then require the finiteness of the solution at the origin and/or continuity of the solution at boundaries.

2.7 Physical Applications

2.7.5 Hydrodynamics

The starting point for hydrodynamics can be taken to be the Navier–Stokes equation [35]:   
@v 1  C (v  r)v D
r p C ηr 2 v C ζ C η rr  v , (2.128) @t 3 where v is the velocity field of the fluid,  the fluid density, p the pressure, η the coefficient of viscosity, and ζ the coefficient of second viscosity. In the above equation, the latter two coefficients have been assumed to be spatially homogeneous. For an incompressible viscous fluid, rv D0. Here, we will only consider examples of steady-state flow in a uniform pipe of arbitrary cross section. In this case, there is, of course, no time dependence and the velocity points along the pipe axis (say, z) and only has x and y components. Then, the Navier–Stokes equation takes the form of an inhomogeneous Laplace equation [35]: r2 v D

1 dp . η dz

(2.129)

The appropriate boundary condition describing the physics is for the velocity field to vanish at the surface: v jS D 0 .

(2.130)

2.7.6 Acoustics

We will study cavity and waveguide problems. The properties of a three-dimensional cavity depend upon its shape, filling medium, and the boundary conditions on its surface [36]. There are three methods for determining the properties: the geometrical, the statistical, and the wave methods. The latter can be further classified into the analytical wave method and the numerical (e.g., using finite-element and boundary-element methods) wave method. We discuss here the former case. The wave method allows one to obtain the spatial distribution of the acoustic field inside the cavity. It is convenient when the cavity dimensions are of the order of the wavelength or smaller. When the cavity dimensions are much larger than the wavelength (e.g., for room acoustics), geometrical acoustics is convenient. On the other hand, if the wavelength is much larger than the dimensions, the wave is pretty much in phase inside the cavity. According to Morse and Ingard [37], wave analysis is most useful when the ratio of the wavelength to the cavity dimension is between 0.3 and 3.

33

34

2 General Theory

A nice discussion of the role of cavity normal modes is given by Morse [38]. Consider the problem of sound in a room. A sound source can set the air in vibration. This vibration has two components. The steady-state vibration has the frequency of the source, whereas the transient free vibration has the frequencies of the normal modes and, as the name implies, dies out with time. Nevertheless, the steadystate vibration can be analyzed, in a Fourier sense, in terms of a linear combination of the normal modes. The amplitudes of the normal-mode components are determined by the source frequency, the impedance of the normal mode, and the position of the source in the room. The transient is required to satisfy the initial boundary conditions and must, in general, consist of a linear combination of the standing waves. When the source is removed, the steady-state vibration will also decay. This can be analyzed in terms of the distinct decay of each normal-mode component; the latter is known as reverberation. Furthermore, the usefulness, in real life, of nonsimple cavity shapes is clearly expressed by Morse and Ingard [37]. A simple shape, such as a rectangular box, would have regularly spaced nodal surfaces. Coincidence of these surfaces would then lead to poor room acoustics since some frequencies might be absent. Procedures for breaking up the regularity are to have either a nonsimple shape or a nonuniform wall impedance. Nevertheless, a theoretical treatment of the complex problem can start with the wave analysis of simple rooms combined with a normal-mode analysis. Consequences of nonsimple shape and nonuniformity of impedance are eigenvalue shifts and damping. Therefore, all of the above discussion justifies the study of normal modes in cavities. The theory of linear acoustics is based upon an approximate form of the Navier– Stokes equation. The approximations are:     

Linearized form of the Navier–Stokes equation. Conservation of fluid in an element. Net longitudinal force is balanced by the inertia of the fluid. Processes in a fluid element are adiabatic. Undisturbed fluid is stationary.

Then, a wave equation can be written down for the acoustic pressure: r2 P


1 @2 P D0, c 2 @t 2

(2.131)

where P is the pressure field and c is the speed of sound, which can be written as ( cD

(γ R T )1/2 c 0 C 0.6Tc ,

(perfect gas)

(2.132)

where γ is the ratio of the specific heats at constant pressure and constant volume, c 0 the speed of sound at T D 0ı C (331.6 m/s), and Tc the temperature in degrees Celcius. The latter expression is valid near room temperature. For a harmonic field,

2.7 Physical Applications

one gets the Helmholtz equation: r2 P C k2 P D 0 .

(2.133)

Appropriate boundary conditions on the surfaces are given by Eqs. (2.95)–(2.97), with ψ D P . Physically, they correspond to the surface being either rigid, that is, the normal surface velocity component vanishes on the boundary, which also gives r P  n D 0, where n is a surface normal vector since vD

1 rP , i ω

(2.134)

or of the pressure-release type, that is, the pressure is zero on the surface (free surface). It should be pointed out that the current acoustic problem is closely related to the problem of vibrating membranes, which can also be treated by a wave equation for the transverse displacement of the membrane. 2.7.7 Quantum Mechanics

Quantum mechanics is the description of matter at length scales when Newtonian physics breaks down. The theory can be formulated in terms of a number of postulates, which have since been verified experimentally. We will not delve into all of the facets of the quantum theory, many of which are still controversial, if only on a philosophical level, but will simply formulate a mathematical description which is relevant to our current focus. Thus, it will be taken for granted that the dynamics of a single quantum particle of mass m in a region in space described by a potential-energy function V(r) in the nonrelativistic limit can be studied by solving the following time-dependent PDE:


„2 2 @Ψ (r, t) r Ψ (r, t) C V(r)Ψ (r, t) D i„ , 2m @t

(2.135)

where „ is the reduced Planck constant h/(2π) and m is the mass of the particle. This equation is known as the time-dependent Schrödinger equation. The first term represents the kinetic energy description of the problem. The wave function Ψ (r, t) has the property that jΨ (r, t)j2 gives the probability density for finding the particle at the space-time point (r, t) (Born postulate). We will restrict ourselves to time-independent potential functions, which allows one to separate out the time dependence; the latter will, henceforth, not be mentioned. The result is known as the time-independent Schrödinger equation:


„2 2 r ψ(r) C V(r)ψ(r) D E ψ(r) , 2m

(2.136)

which can be rewritten as Eq. (2.3). Here E is the energy of the particle. For an infinite domain, the fundamental boundary condition is that the wave function

35

36

2 General Theory

must go to zero at infinity for it to be normalizable (i.e., the probability can be normalized to 1). Hence, the quantum mechanics problem is a Dirichlet one. When the domain consists of distinct regions (e.g., with different potential energies), a common technique for solving the problem is to write down general solutions in each region and then use so-called interfacial conditions (they are often known as boundary conditions as well) to match the solutions at the interface. Indeed, the physical and mathematical constraints on the problem are that the wave function and its derivative both be continuous if V(r) has, at most, a finite discontinuity: ψ(r)j S D continuous ,

r ψ(r)j S D continuous .

(2.137)

However, there are two exceptions. When the discontinuity in the potential is infinite (a so-called infinite-barrier or hard-wall problem), then the internal consistency of the theory requires that the wave function be zero in the regions where the potential is infinite. At such boundaries, the wave function is indeed zero on both sides but the slope of the wave function is discontinuous. In such a case, the only boundary condition to impose is the so-called hard-wall boundary condition: ψ(r)j S D 0 .

(2.138)

In fact, this is equivalent to saying that the boundary condition for the Schrödinger problem is a Dirichlet one. The other exception is when the Schrödinger equation is an effective model of the dynamics of electrons in solids, whereby a generalization of Eq. (2.3) might be warranted; the simplest form of such a generalization (the socalled one-band effective-mass model) leads to an equation in which the mass is no longer a constant but is a function of position as well [39]:
 „2 1
r r ψ(r) C V(r)ψ(r) D E ψ(r) . (2.139) 2 m(r) The boundary conditions are then different: ψ(r) continuous ,

(2.140)

1 r ψ(r) continuous . m(r)

(2.141)

This equation is also known as the Ben Daniel–Duke model.

2.8 Problems

1. Show that for a conformal transformation w D w (z), ˇ ˇ ˇ dz ˇ ˇ . h 1 D h 2 D ˇˇ dw ˇ

(2.142)

Hence, evaluate the scale factors for the circular and elliptic coordinate systems.

2.8 Problems

2. The spherical polar coordinates are defined by ξ1 D r ,

ξ2 D θ ,

ξ3 D φ ,

where x D r sin θ cos φ ,

y D r sin θ sin φ ,

z D r cos θ .

a. Derive the scale factors h i . b. Show that a possible Stäckel matrix is 0

1 @0 0


1/r 2 1 0

1 0
1/ sin2 θ A . 1

(2.143)

3. The toroidal coordinates are defined by ξ1 D η ,

ξ2 D θ ,

ξ3 D ψ ,

where xD

a sinh η cos ψ , cosh η
cos θ

yD

a sinh η sin ψ , cosh η
cos θ

zD

a sinh θ . cosh η
cos θ

a. Verify that the constant-coordinate surfaces are given by toroids 1/2  x 2 C y 2 C z 2 C a 2 D 2a x 2 C y 2 coth η , spherical bowls, x 2 C y 2 C (z
a cot θ )2 D

a2 , sin2 θ

and half planes, tan ψ D y /x . b. Derive the scale factors h i . c. Write down the unit vectors in terms of the Cartesian unit vectors. d. Write the Laplacian in the toroidal coordinate system. e. Show that the function Φ D (cosh η
cos θ )1/2 N(η)Θ (θ )Ψ(ψ) separates the Laplace equation into the ODEs for the N, Θ , and Ψ functions. (This is an example of the so-called R-separability of the Laplace equation.) 4. Derive the general indicial equation, Eq. (2.78), and the recurrence relation, Eq. (2.79). 5. Show that z λ and z µ are solutions to Eq. (2.86).

37

38

2 General Theory

6. For the ODE (t
b)(t
c)

d2 X dX 1 C (α 3 t
α 2 ) X D 0 , C [2t
(b C c)] 2 dt 2 dt

a. Write down the singularities of the equation. b. What is its Bôcher type? c. Obtain the recurrence relation for a series solution about t D 0. 7. a. Show that the relations in Eq. (2.100) transform Eq. (2.66) into the standard Sturm–Liouville equation, Eq. (2.99). b. Derive the orthogonality relations for the Sturm–Liouville problem. 8. The damped wave equation can be written as r2 Ψ


1 @2 Ψ @Ψ D0.
R c 2 @t 2 @t

a. Show that the temporal coordinate can be separated out and write the solution as a product of a spatial function and a temporal function. b. Discuss the nature of the temporal function as a function of the parameter R.

Part Two Two-Dimensional Coordinate Systems

41

3 Rectangular Coordinates 3.1 Introduction

This is one of the three well-known coordinate systems. It provides a nice and simple introduction to how the method of separation of variables is implemented in practice. We show how to separate the Laplace, Helmholtz, and Schrödinger equations. The general solutions of the resulting ordinary differential equations (ODEs) are either trigonometric or hyperbolic functions. The Helmholtz equation leads to two ODEs each with its own separation constant. Hence, if the boundaryvalue problem is separable, it is relatively straightforward to apply the boundary conditions to determine the separation constants independently of each other. The Laplace problem is such that the two ODEs are coupled via a single separation constant. Finally, applications of a boundary-value problem in electrostatics and in quantum mechanics are given.

3.2 Coordinate System 3.2.1 Coordinates (x, y )

The rectangular (also known as Cartesian) coordinates are conventionally denoted by x and y. Both coordinates have an infinite domain:
1 < x ,

y <1.

(3.1)

3.2.2 Constant-Coordinate Curves

The curves of constant coordinates are straight lines orthogonal to each other (Figure 3.1): x D constant ,

y D constant .

(3.2)

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

42

3 Rectangular Coordinates

y

x

Figure 3.1 Rectangular coordinates in two dimensions.

We note that a minimum of four constant-coordinate lines are required to form a closed region.

3.3 Differential Operators

In this section, we will apply the general results given in Section 2.3 to the case of the rectangular system. 3.3.1 Metric

Since @x i D δi j , @x j

(3.3)

then, using Eq. (2.5), that is, h 2x D h 2x x D

@y @y @x @x C D1, @x @x @x @x

the scale factors are hx D hy D 1 .

(3.4)

3.4 Separable Equations

3.3.2 Operators

All of the following operators can be obtained from Eqs. (2.7)–(2.12) by keeping only the x and y coordinates. 3.3.2.1 Gradient The gradient of a scalar field ψ is

r ψ D e1

1 @ψ 1 @ψ @ψ @ψ C ey , C e2 D ex h 1 @q 1 h 2 @q 2 @x @y

(3.5)

where e x and e y are the orthogonal unit vectors. 3.3.2.2 Divergence The divergence of a vector field V is

rV D

@Vy @Vx C . @x @y

(3.6)

3.3.2.3 Laplacian The Laplacian is

r2 D

@2 @2 C . @x 2 @y 2

(3.7)

3.4 Separable Equations

We now show how to separate the Laplace, Helmholtz, and Schrödinger equations in rectangular coordinates. 3.4.1 Laplace Equation

The Laplace equation is r2 ψ D

@2 ψ @2 ψ C D0. @x 2 @y 2

(3.8)

Since this is the first separation problem we are presenting, we give all the basic steps for the method. Assuming that a product solution is valid, ψ(x, y ) D X(x)Y(y ) ,

(3.9)

one gets @2 ( X Y ) d2 X(x) d2 Y(y ) @2 ( X Y ) C DY CX D0, 2 2 2 @x @y dx dy 2

(3.10)

43

44

3 Rectangular Coordinates

or 1 d2 Y(y ) 1 d2 X(x) D
. X dx 2 Y dy 2

(3.11)

One then argues that since each side of the equation is at most a function of one of the variables, the only way they can be equal is if both sides are equal to the (same) constant, say,
k12 . Then, d2 X(x) C k12 X(x) D 0 , dx 2

(3.12)

d2 Y(y )
k12 Y(y ) D 0 dy 2

(3.13)

are the separated ODEs. They are both of the same Bôcher type f04g, with only an irregular singular point of order 4 at infinity. They have trigonometric or hyperbolic solutions. For example, for real k1 , X(x) D A cos k1 x C B sin k1 x ,

(3.14)

Y(y ) D C e k1 y C D e
k1 y .

(3.15)

Note that, even in this simple case, the two differential equations are coupled via the single separation constant. The solutions given in Eqs. (3.14) and (3.15) are known as general solutions since they satisfy the corresponding ODEs independent of the values of the unknown constants A, B, C, D, and k1 . Indeed, owing to the linearity of the differential equations, one can say that even an arbitrary linear combination of such product solutions X(x)Y(y ) remains a solution of the original partial differential equation (PDE). Therefore, one can write the most general solution to the Laplace equation (e.g., for real k1 ) as X ψ(x, y ) D X k1 (x)Yk1 (y ) k1

D

X


 (A k1 cos k1 x C B k1 sin k1 x) C k1 e k1 y C D k1 e
k1 y .

(3.16)

k1

Two comments are in order. First, the nature of the solutions to the ODEs can be interchanged by either letting k1 be pure imaginary or simply changing the sign of the separation constant. Either choice is, in principle, valid and the appropriate one to choose depends upon the boundary conditions. Second, there are only two independent constants in Eq. (3.16) since the Laplace equation is homogeneous. 3.4.2 Helmholtz Equation

The Helmholtz equation is r2 ψ C k2 ψ D

@2 ψ @2 ψ C C k2 ψ D 0 . 2 @x @y 2

(3.17)

3.4 Separable Equations

The approach to separating the Helmholtz equation is very similar to that for the Laplace equation. Writing ψ(x, y ) D X(x)Y(y ) ,

(3.18)

one gets as the ODEs d2 X(x) C k12 X(x) D 0 , dx 2

(3.19)

d2 Y(y ) C k22 Y(y ) D 0 , dy 2

(3.20)

k12 C k22 D k 2 .

(3.21)

with

The ODEs are again Euler equations. In this case, there are two separation constants and, together, they determine the allowed wave numbers k 2 . As for the Laplace equation, one can write down general solutions. For example, if the separation constants k i2 are assumed to be both real and positive, the general solutions to the ODEs can be written as X(x) D A cos k1 x C B sin k1 x ,

(3.22)

Y(y ) D C cos k2 x C D sin k2 x .

(3.23)

3.4.3 Schrödinger Equation

The following form of the potential is separable in rectangular coordinates: V(x, y ) D

 „2  v1 (x) C v2 (y ) . 2m

(3.24)

The Schrödinger equation becomes


„2 2m



  „2  @2 @2 ψ(x, y ) C v1 (x) C v2 (y ) ψ(x, y ) D E ψ(x, y ) . C 2 2 @x @y 2m (3.25)

Let ψ(x, y ) D X(x)Y(y ) .

(3.26)

Then, "

# " # 1 d2 X 1 d2 Y
C v1 (x) C
C v2 (y ) D k 2 , X dx 2 Y dy 2

k2 

2m E , „2

(3.27)

45

46

3 Rectangular Coordinates

and  d2 X(x)  2 C k1
v1 (x) X(x) D 0 , dx 2

(3.28)

 d2 Y(y )  2 C k2
v2 (y ) Y(y ) D 0 , dy 2

(3.29)

k12 C k22 D k 2 .

(3.30)

with

Of course, an explicit potential is needed to solve for the solutions to the ODEs. For example, if v1 D v2 D 0, Eqs. (3.28) and (3.29) reduce to the Helmholtz problem.

3.5 Applications

In the previous sections, we showed how to separate the original PDE into ODEs and gave general solutions where appropriate. For applications, it is necessary to impose boundary conditions. As already discussed, the boundary condition may or may not be separable. For example, the finite-barrier problem in quantum mechanics for a rectangular quantum well is not separable. We will present examples of boundary-value problems where the boundary conditions are also separable and discuss possible constraints on the solutions and separation constants. 3.5.1 Electrostatics: Dirichlet Problem for a Conducting Strip

Consider a thin conducting sheet of width a and semi-infinite length with the electric potential maintained at 0 on the two semi-infinite sides and given by a function f (x) on the side of width a (Figure 3.2). In this case, we have Dirichlet boundary conditions. The potential can be determined from the Laplace equation and the boundary conditions lead to a unique solution. From the general solution, Eq. (3.16),
 X (A n cos k n x C B n sin k n x) C n e k n y C D n e
k n y , ψ(x, y ) D n

to satisfy the boundary conditions at x D 0 and y ! 1, we have An D 0 ,

Cn D 0 ,

(3.31)

the latter so that the potential does not blow up for y at infinity, and the boundary condition at x D a requires kn D

nπ , a

n 2 ZC .

(3.32)

3.5 Applications y

ψ=0

ψ=0

ψ=f(x)

x

a

Figure 3.2 Dirichlet problem for a conducting strip.

Therefore, ψ(x, y ) D

1 X

F n sin

nD1

nπ x
nπ y e a . a

(3.33)

The last boundary condition at y D 0 gives f (x) D ψ(x, 0) D

1 X

F n sin

nD1

nπ x , a

(3.34)

which can be inverted to give all of the Fourier coefficients F n : Fn D

2 a

Za

dx 0 f (x 0 ) sin

nπ x 0 . a

(3.35)

0

Given a specific function f (x), one can then evaluate the F n and the complete solution is given by Eq. (3.33), where one truncates the infinite sum depending upon the desired accuracy. The Fourier series in this case is typical of the expansion of the solution in terms of orthogonal functions. Other examples will be seen throughout the text. 3.5.2 Quantum Mechanics: Dirichlet Problem for a Rectangular Box

The quantum-box problem (Figure 3.3) for a single electron is exactly solvable for a hard-wall potential. For v1 (x) D v2 (y ) D 0 (i.e., inside the box), the solutions are X(x) D a 1 sin k1 x C b 1 cos k1 x , Y(y ) D a 2 sin k2 y C b 2 cos k2 y .

(3.36)

47

48

3 Rectangular Coordinates

y L2

x

L1

0

Figure 3.3 Dirichlet problem for a rectangular quantum box.

For a hard-wall potential, the wave function vanishes outside the box. Consider a rectangular box of dimensions L 1 and L 2 and with the origin at one corner. Then, the boundary conditions on the x D 0 and y D 0 lines result in b 1 D b 2 D 0. The (unnormalized) wave function is ψ(x, y ) D sin k1 x sin k2 y .

(3.37)

The boundary conditions on the x D L 1 and y D L 2 lines result in ki D

π ni , Li

n i 2 ZC .

(3.38)

The constraint on positive n i is because n i D 0 leads to a trivial null solution, whereas negative values only lead to wave functions that differ by a phase factor. The solutions can thus be labeled by a pair of positive integers (n 1 , n 2 ) and the latter are known as quantum numbers. Hence, the energy eigenvalues are given by E (n 1 , n 2 ) D

„2 π 2 „2 k 2 D 2m 0 2m 0



n2 n 21 C 22 2 L1 L2



 D 3.76 meV

n2 n 21 C 22 2 L1 L2

 , (3.39)

where, in the last equality, the lengths are in units of angstroms, and m 0 is the free-electron mass. The lowest eigenvalues for a 100 Å quantum square are given in Table 3.1. The ground state is nondegenerate but the first excited state is doubly degenerate owing to the equal energies for the (n 1 , n 2 ) and (n 2 , n 1 ) states. Note that the problem is not separable for the finite-barrier case. Hence, no exact analytic solution can be written down for the latter case and a numerical solution is necessary. Nevertheless, a common starting point for the finite-barrier problem is to use the infinite-barrier solution as a basis set. This reveals the general usefulness in obtaining analytic solutions, even for a different but solvable problem.

3.6 Problems Table 3.1 Lowest eigenvalues and degeneracies for a 100 Å
100 Å square. (n 1 , n 2 )

E (meV)

Degeneracy

(1,1) (2,1), (1,2)

11.222 28.055

1 2

3.6 Problems

1. For the Helmholtz PDE, write down the explicit solution if the separation constants are exactly zero. 2. Consider a semi-infinite slab of material of thickness 2a in the x direction and infinite in size in the z direction. Let the temperature be zero on the two x-plane surfaces and let it have the functional form T(x, 0, z) D 10 sin

πx a

on the y-plane surface. Obtain the temperature distribution within the slab when thermal equilibrium has been reached. 3. Consider a rectangular (L x  L y ) stretched membrane with a fixed rim. a. Obtain expressions for the normal modes of vibration and for the corresponding natural frequencies. b. For L y D 2L x , sketch the lowest four modes. 4. For a rectangular box, compare the lowest two normal modes of sound waves using Dirichlet boundary conditions and using Neumann boundary conditions. 5. For an electron of effective mass m  D 0.067m 0 in a hard-wall quantum rectangle of sides 50 Å  100 Å, obtain the lowest three allowed energies, giving their degeneracies.

49

51

4 Circular Coordinates 4.1 Introduction

This is the second of the three well-known coordinate systems. Both the Laplace and the Helmholtz equations can be written in terms of one separation constant. The radial equation for the Helmholtz equation leads to the Bessel or modified Bessel equation. Thus, special functions to be encountered in this chapter are the various Bessel functions.

4.2 Coordinate System 4.2.1 Coordinates

The circular coordinates are r and φ, where x D r cos φ ,

y D r sin φ ,

0
r<1,

0
φ < 2π .

(4.1)

with

4.2.2 Constant-Coordinate Curves

From x2 C y 2 D r2 and tan
1 φ D y /x , we have that r D constant are circles and φ D constant are radial lines (Figure 4.1). Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

52

4 Circular Coordinates

y

ϕ=constant

r=constant

x

Figure 4.1 Coordinate curves for a 2-space circular coordinate system.

4.3 Differential Operators

The general expressions are given in Section 2.3. 4.3.1 Metric

Using Eq. (2.5), that is, 
2 @x 2 @y C D cos2 φ C sin2 φ D 1 , @r @r  

  @x 2 @y 2 h 2φ D C D r 2 sin2 φ C cos2 φ D r 2 , @φ @φ


h 2r D h 2r r D

the scale factors are hr D 1 ,

hφ D r .

(4.2)

4.3.2 Operators

These are obtained by substituting Eq. (4.2) into Eqs. (2.7)–(2.12). 4.3.2.1 Gradient The gradient operator is

r D er

@ 1 @ C eφ . @r r @φ

(4.3)

4.4 Separable Equations

4.3.2.2 Divergence The divergence of a vector field V is

rV D

@Vr 1 @Vφ C . @r r @φ

4.3.2.3 Laplacian The Laplacian is

r2 D

1 @ r @r


r

@ @r

 C

(4.4)

1 @2 . r 2 @φ 2

(4.5)

4.4 Separable Equations 4.4.1 Laplace Equation

The Laplace equation is 
  @ 1 @2 1 @ ψ(r, φ) D 0 . r C 2 r2 ψ D r @r @r r @φ 2

(4.6)

Let ψ(r, φ) D R(r)Φ (φ) . Then, 1 d r R(r) dr


r

dR(r) dr

 D

(4.7)

1 1 d2 Φ (φ) l2  2 , 2 2 r Φ (φ) dφ r

(4.8)

giving d2 Φ (φ) C l 2 Φ (φ) D 0 , dφ 2
 dR(r) l2 1 d r  2 R(r) D 0 , r dr dr r

(4.9) (4.10)

where l is the separation constant. Both equations can be directly integrated. The solution to the angular equation can be written as Φ (φ) D Ae i l φ C B e
i l φ .

(4.11)

The radial equation, Eq. (4.10), for l ¤ 0, gives as a general solution R(r) D Ar l C B r
l .

(4.12)

Often (but not always), from the physics, the solutions to the angular equation are required to be periodic. When this is the case, it restricts l to be an integer. However, this is part of the boundary-value problem to be treated later in the chapter.

53

54

4 Circular Coordinates

4.4.2 Helmholtz Equation

The Helmholtz equation, r2 ψ C k2 ψ D 0 , becomes  
 1 @ @ 1 @2 ψ(r, φ) C k 2 ψ(r, φ) D 0 . r C 2 r @r @r r @φ 2

(4.13)

Let ψ(r, φ) D R(r)Φ (φ) .

(4.14)

Then, 1 d r R(r) dr




dR(r) r dr

 C k2 D 

1 1 d2 Φ (φ) l2  2 , 2 2 r Φ (φ) dφ r

(4.15)

giving d2 Φ (φ) C l 2 Φ (φ) D 0 , dφ 2 
  dR(r) l2 1 d r C k 2  2 R(r) D 0 , r dr dr r

(4.16) (4.17)

where l is the separation constant. The angular equation is the same as for the Laplace equation. The radial equation, Eq. (4.17), is now the Bessel equation. It is easily seen to have a regular singular point at zero and an irregular singular point at infinity, and is of Bôcher type f24g. Solutions to the latter are known as Bessel functions (see Appendix C for a summary of properties). It is possible to have physical problems where k 2 is either positive or negative. If it is positive and if l is nonintegral, the independent solutions are the Bessel functions of the first kind of nonintegral order J˙l (k r) and the general solution to the radial equation is R(r) D A J l (k r) C B J
l (k r) .

(4.18)

For integral l, the second solution is the Bessel function of the second kind, also known as the Neumann function. The general solution to the radial equation then becomes R(r) D A J l (k r) C B N l (k r) .

(4.19)

Since the Neumann function is singular at r D 0, it is only present in physical problems whereby the origin has been excluded from the domain.

4.4 Separable Equations

If k 2 < 0, one can write k 2 D q 2 and the radial equation is now known as the modified Bessel equation. The general solution can be written as R(r) D A J l (i q r) C B N l (i q r) D AI l (q r) C B K l (q r) .

(4.20)

The functions I l (q r) and K l (q r) are the modified Bessel functions of the first and the third kind; the latter are also known as MacDonald functions. The modified Bessel functions of the first kind diverge at large r; hence, they are only present for physical problems that are restricted to a finite domain. 4.4.3 Schrödinger Equation

The following form of potential is separable in circular coordinates: V(r, φ) D

  „2 1 v1 (r) C 2 v2 (φ) . 2m r

(4.21)

The Schrödinger equation is 

„2 2m 0



1 @ r @r


r

@ @r

 C

1 r2



@2  v2 (φ) @φ 2



 ψC

 „2 v1 (r)  E ψ D 0 . 2m 0 (4.22)

Let ψ(r, φ) D R(r)Φ (φ) .

(4.23)

Then, 1 d r R(r) dr




dR(r) r dr







1 C k  v1 (r) D  2 r 2

"

# l2 1 d2 Φ (φ)  v2 (φ)  2 , 2 Φ (φ) dφ r (4.24)

giving 1 d r dr


r

dR(r) dr



  l2 C k 2  v1 (r)  2 R(r) D 0 , r

d2 Φ (φ)  v2 (φ)Φ (φ) C l 2 Φ (φ) D 0 , dφ 2 where l is the separation constant.

(4.25) (4.26)

55

56

4 Circular Coordinates

4.5 Applications 4.5.1 Quantum Mechanics: Dirichlet and Neumann Problems for a Disk

The quantum mechanics example is an interesting one here in that there are exact solutions for both the infinite-barrier and the finite-barrier particle-in-a-box problems. Indeed, together with the same problem in spherical polar coordinates, these are known to be the only exact solutions for the finite-barrier problem. Furthermore, we show how the treatment of a wedge rather than the whole disk leads to Bessel functions of nonintegral orders. 4.5.1.1 Infinite-Barrier Solutions Let us study the problem in terms of the effective-mass equation, Eq. (2.139), as a variation on textbook quantum mechanics. In circular coordinates, it becomes



„2 2



1 @ r @r




r @ m(r) @r

 C

 1 @2 ψ(r, φ) C [E  V(r)]ψ(r, φ) D 0 . r 2 m(r) @φ 2 (4.27)

Consider now the problem of an electron of mass m in inside a circular disk of radius R0 and confined by an infinite potential. Then, outside the disk, the wave function is zero and the problem is separable. Inside the circular disk, one can write the wave function as ψ(r, φ) D R(r)Φ (φ) and the potential can be chosen to be zero. Assuming single-valuedness of the angular part of the wave function, this means Φ (φ) is periodic and the appropriate solutions of Eq. (4.26) are Φ (φ) D e ˙i l φ ,

l 2N ,

(4.28)

where N is the set of natural numbers. The radial equation is r

d dr

r

d dr


r

dR(r) dr

r

dR(r) dr




C

 2r 2 m in 2 R(r) D 0 , E  l „2

(4.29)

or




  C k 2 r 2  l 2 R(r) D 0 ,

k2 

2m in E. „2

(4.30)

The energy is positive; hence, the separation constant is real. The solutions to the latter equation are the Bessel functions; since the solution needs to be finite at the origin, this excludes the Neumann functions, Eq. (C18). Thus, they are the Bessel functions of the first kind of (integral) order l, J l (k r). The hard-wall boundary condition translates to J l (k R0 ) D 0 ,

(4.31)

4.5 Applications

and, therefore, it allows one to obtain discrete values of k D k n l for each l from the roots of the Bessel function. For example, from Table C.1, the lowest root is x D k10 R0 D 2.405. Thus, the ground-state energy is given by E10 D

2 „2 k10 2.4052 „2 D . 2m in R 2 2m in

It can be seen that the boundary condition is responsible for the discreteness of the energy spectrum. 4.5.1.2 Finite-Barrier Solutions We now study the case where the potential energy of the electron outside the disk is finite, V0 (Figure 4.2). For simplicity, we will assume the mass to be piecewise constant. Thus, it is m in (m out ) inside (outside) the disk. Inside the disk, the solutions are formally unchanged from the infinite-barrier problem. In particular, the φ solution remains the same. As is known from quantum mechanics, for a bound state (i.e., the wave function goes to zero as r ! 1), the energy must be less than the maximum potential height V0 . Thus, for bound states, V0 > E . Outside, we then have
   dR(r) d 2m out (V0  E ) . r r 
2 r 2 C l 2 R(r) D 0 ,
2  (4.32) dr dr „2

This is known as the modified Bessel equation and the solutions are the modified Bessel functions of the third kind, or MacDonald functions l, K l (
r). The modified Bessel function of the first kind of (integral) order I l (
r) is excluded since it diverges at large r (see Appendix C). For a physical solution in the whole domain, one now needs to apply boundary conditions at the interface of continuity of the wave function and of the massnormalized slope (to ensure current conservation [39]). Note that, in principle, it V(r)

V0

r

0 R0

Figure 4.2 Dirichlet problem for a disk with a finite potential barrier.

57

58

4 Circular Coordinates

is the general solution that must be used. Thus, inside the disk, the general wave function can be written as X A l J l (k r)e i l φ , (4.33) ψin (r, φ) D l

and outside, it is ψout (r, φ) D

X

B m K m (
r)e i m φ .

(4.34)

m

Owing to the orthogonality of the exponential functions for l ¤ m, the two infinite series can only be equal (one of the two boundary conditions) if terms corresponding to l D m are equated. A similar discussion applies to the other boundary condition (see Eq. (2.141)). Thus, they become A l J l (k R0 ) D B l K l (
R0 ) , ˇ ˇ B l dK l (
r) ˇˇ A l d J l (k r) ˇˇ D , m in dr ˇ rDR0 m out dr ˇ rDR0 or, equivalently, they can be combined to give ˇ ˇ 1 d ln K l (
r) ˇˇ 1 d ln J l (k r) ˇˇ D . ˇ ˇ m in dr m out dr rDR 0 rDR 0

(4.35) (4.36)

(4.37)

Roots to Eq. (4.37) give the quantized energies for the bound states. Note that there is only one unknown, that is, k since
is related to k. Using Eqs. (4.30) and (4.32), we find
2 2V0 k2 C D 2 . m in m out „

(4.38)

4.5.1.3 Infinite-Barrier Pie There is also an exact solution when the domain is not the whole circle but is a wedge within it. Consider, therefore, a pie-shaped domain with angular size of φ max (Figure 4.3). Instead of periodicity, we have the new boundary conditions

Φ (0) D Φ (φ max ) D 0 .

(4.39)

Hence, the solutions are Φ (φ) D sin νφ ,

(4.40)

with ν nonintegral and sin νφ max D 0 .

(4.41)

4.6 Problems

y

ϕmax

x

Figure 4.3 Dirichlet problem for a wedge.

This constrains the allowed values of ν to be given by νD

pπ , φ max

p 2 ZC .

(4.42)

From the boundary condition on the radial equation, J ν (k R) D 0 ,

(4.43)

one looks for the zeros of the Bessel function of nonintegral order which gives the k values. From Appendix C, there are no complex roots for ν positive.

4.6 Problems

1. Find the general solution to the radial differential equation of the Laplace equation for the separation constant l D 0. 2. Show that the radial equation of the Helmholtz equation is of Bôcher type f24g. 3. Bessel functions. a. Prove the following results for integral n: 1 P i. J n (x C y ) D J s (x) J n
s (y ). sD
1

ii. J n (x) D (1) n J n (x). b. Show that J v (x) D

1 X sD0

(1) s  x v C2s s!(s C v )! 2

satisfies 2v Jv , x D 2 J v0 .

J v
1 C J v C1 D J v
1  J v C1

4. Consider a circular stretched membrane with a fixed rim of radius R.

59

60

4 Circular Coordinates

a. Obtain expressions for the normal modes of vibration and for the corresponding natural frequencies. b. Give the lowest four natural frequencies in terms of the fundamental frequency. c. Sketch the lowest four modes. d. Write down the expression for an arbitrary displacement y (r, t) of circular symmetry. Compute the average displacement of the nth symmetric normal mode: Z 1 dS y n (r) . hy n i D π R2 S

5. For an electron of effective mass m  D 0.067m 0 in a hard-wall quantum pie of radius 100 Å and wedge angle 10ı , find the ground-state energy. Why is p D 0 not an allowed solution for the Φ equation?

61

5 Elliptic Coordinates 5.1 Introduction

The elliptic coordinate system provides a closed one-coordinate curve for the study of boundary-value problems with an elliptic domain and is a generalization of the circular system. The separated ordinary differential equations are known as Mathieu equations. Although the two coordinates separate in the differential equations, the two separation constants do not. Hence, this poses an additional difficulty in solving the boundary-value problem. For this reason, treatment of the elliptic system is rarely included in introductory textbooks. However, we believe this is an important example to illustrate the problem of coupled separation constants which arises in a few other coordinate systems. In particular, it leads to a nonseparable problem even for separable boundary condition.

5.2 Coordinate System 5.2.1 Coordinates (u, v)

The new coordinates u and v are related to the Cartesian coordinates by x D f cosh u cos v , y D f sinh u sin v ,

(5.1)

with 0
u<1,

0
v < 2π ,

f >0.

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

62

5 Elliptic Coordinates

5.2.2 Constant-Coordinate Curves

If u is constant, cos2 v C sin2 v D

x2 f

2 cosh2

u

y2

C

2 sinh2

f

u

D1.

(5.2)

Thus, u D U is an ellipse with semiaxes a D f cosh U and b D f sinh U. Hence, f 2 D a 2  b 2 ; f is known as the focal length and the foci are at x D ˙ f, y D 0. The curves u D constant are a family of confocal ellipses. The curve u D 0 is a straight line segment connecting the two foci, and increasing u leads to larger ellipses. If v is constant, cosh2 u  sinh2 u D

x2 f

2 cos2

v



y2 D1. f 2 sin2 v

(5.3)

Thus, v D V is a half hyperbola (Figure 5.1) and we have a family of confocal hyperbolas for different V. When v D 0, y D 0 and x D f cosh u  f . This is a straight half line starting at x D f . Thus, increasing v leads to hyperbolas with decreasing curvature until v D π/2, which is the y  0, x D 0 half line. v= π/2

40 35 30 25 20 15

v=0.1π

10 5

u=0.05π u=0

v=π 0

v=0

-5 -10 -15 -20 -25 -30 -35 -40 -40

-30

-20

-10

0

v=3π/2

Figure 5.1 Elliptic coordinates in two dimensions.

10

20

30

40

5.3 Differential Operators

5.3 Differential Operators 5.3.1 Metric

The now familiar approach based on Eq. (2.5) gives the scale factors as
1/2 h u D f sinh2 u C sin2 v ,

hv D hu .

(5.4)

5.3.2 Operators

These are obtained by substituting Eq. (5.4) into Eqs. (2.7)–(2.12). 5.3.2.1 Gradient The gradient operator is

rD

ev @ @ eu C
.
1/2 1/2 @u @v 2 2 f cosh u  cos2 v f cosh u  cos2 v

(5.5)

5.3.2.2 Divergence The divergence of a vector field V is

rV D

1




1/2 f cosh2 u  cos2 v    1/2  1/2  @
@
. cosh2 u  cos2 v cosh2 u  cos2 v  Vu C Vv @u @v (5.6)

5.3.2.3 Laplacian The Laplacian is

r2 D




1

f 2 sinh2 u C sin2 v

 

@2 @2 C 2 2 @u @v

 .

(5.7)

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64

5 Elliptic Coordinates

5.4 Separable Equations 5.4.1 Laplace Equation

The Laplace equation in elliptic coordinates is   2 1 @2 @ 2
 r ψ(u, v ) D C 2 ψ(u, v ) D 0 . @u2 @v f 2 sinh2 u C sin2 v

(5.8)

Thus, @2 ψ(u, v ) @2 ψ(u, v ) C D0, @u2 @v 2

(5.9)

and, if ψ(u, v ) D U(u)V(v ) ,

(5.10)

1 d2 V 1 d2 U D  Dc, U du2 V dv 2

(5.11)

then

and the separated equations are d2 U  cU D 0 , du2

(5.12)

d2 V C cV D 0 . dv 2

(5.13)

Hence, for real c, one set of solutions can be written as trigonometric functions and the other set can be written in terms of exponentials. 5.4.2 Helmholtz Equation

The Helmholtz equation, r2 ψ C k2 ψ D 0 , in elliptic coordinates can be written as   2 @2 @ 1
 ψ(u, v ) C k 2 ψ(u, v ) D 0 , C @u2 @v 2 f 2 sinh2 u C sin2 v

(5.14)

or @2 ψ f 2k2 @2 ψ (cosh 2u  cos 2v )ψ D 0 . C C @u2 @v 2 2

(5.15)

5.4 Separable Equations

Let ψ(u, v ) D F(u)G(v ) .

(5.16)

Then, the separated equations are d2 F  (c  2q cosh 2u) F D 0 , du2

(5.17)

d2 G C (c  2q cos 2v ) G D 0 , dv 2

(5.18)

with q D f 2 k 2 /4. They are the modified Mathieu and Mathieu equations, respectively; they have been shown to have the Bôcher type f1 1 3g [6]. One can write the general solution in terms of the Mathieu functions. Since the ordinary differential equations are even in their coordinates, the solutions can be chosen to be either even or odd. The even solutions are written as F(u) D Ace(i u, q) C Bfe(i u, q) ,

(5.19)

G(v ) D Ace(v , q) C Bfe(v , q) ,

(5.20)

and the odd solutions are written as F(u) D Ase(i u, q) C Bge(i u, q) ,

(5.21)

G(v ) D Ase(v , q) C Bge(v , q) .

(5.22)

The solutions ce(v , q) and se(v , q) are known as Mathieu functions of the first kind. The choice of notations ce and se is because, in the limit when the elliptic system becomes circular, these functions become the cosine and sine functions, respectively. Properties of the resulting Mathieu functions are described in Appendix F. Note that there are two unknowns in each of the separated equations: c and q. Thus, for a boundary-value problem, both equations would have to be solved simultaneously to obtain the two unknowns. The presence of the cos 2v factor in the Mathieu equation allows one to restrict the solution space to Mathieu functions that are either π or 2π periodic; this condition allows one to determine one parameter in terms of the other, for example, c D c(q). Such a dependence is known as the characteristic equation. Then, constraints on the solutions to the Mathieu equation (e.g., boundary conditions on an elliptic boundary) provide a unique solution. 5.4.3 Schrödinger Equation

The separable potential is V(u, v ) D



„2 2 v1 (u) C v2 (v ) . 2 2m f (cosh 2u  cos 2v )

(5.23)

65

66

5 Elliptic Coordinates

The Schrödinger equation becomes







„2 2 2m f 2 (cosh 2u  cos 2v )

C



„2 2 v1 (u) C v2 (v ) ψ(u, v ) D E ψ(u, v ) . 2 2m f (cosh 2u  cos 2v )

@2 @2 C 2 2 @u @v

ψ(u, v ) (5.24)

Let ψ(u, v ) D U(u)V(v ) . Then,

(5.25)

# 1 d2 U  v1 (u) U du2 " # 2 1 d2 V  v (v ) D k2 ,  2 2 f (cosh 2u  cos 2v ) V dv 2 2  2 f (cosh 2u  cos 2v )

"

and the separated equations are   2 d2 U 2 2 f cosh 2u C v  k  k (u) U D0, 1 2 du2 2   2 d2 V 2 2 f cos 2v  v C k  k (v ) V D0. 2 2 dv 2 2

(5.26)

(5.27) (5.28)

5.5 Applications

There are a few applications of the two-dimensional elliptic system. Most likely, the earliest was the work of Mathieu on the vibration of an elliptic membrane [40]. Additional applications are discussed in the book by McLachlan. Recent applications include the bound states in elliptic quantum dots [41–43]. 5.5.1 Quantum Mechanics: Dirichlet Problem for an Ellipse 5.5.1.1 Finite-Barrier Solutions Since we have already seen in Chapter 4 that the finite-barrier problem for a oneparameter circular boundary is separable, one might wonder whether such is the case for the elliptic boundary as well [41]. Unfortunately, this is not the case even though the partial differential equation is separable and the boundary condition is separable. The problem here is that the separation constants are not separable. This difficulty is illustrated below. For the infinite-barrier problem, the energy levels are determined by the boundary condition

F lin (q 1 , U) D 0 ,

(5.29)

5.5 Applications

or Mcin l (q 1 , U) D 0 ,

(5.30)

Msin l (q 1 , U) D 0 ,

(5.31)

where l is a discrete set of separation p constants obtained after imposing periodicity 2 in v, q 1 D f 2 k 2 /4 with f D a 2  b 2 and k 2 D 2m
in E/„ , and u D U D 1 tanh (b/a) is the dot boundary. The above equations are analogous to the hardwall boundary condition for the circle, Eq. (4.31). For a finite barrier, one might expect the boundary condition to be similarly of the form (compare Eqs. (4.35) and (4.36)) ˇ ˇ dF in (q 1 , U) ˇˇ dF out (q 2 , U) ˇˇ 1 1 D . ˇ ˇ m in F in (q 1 , U) du m out F out (q 2 , U) du uDU uDU (5.32) However, the experience from the circular problem is deceiving and the above presumes that the problem is separable (so that the G(v ) functions drop out of the boundary condition as given in Eq. (5.32)). Indeed, separability would imply that the wave functions have the forms (   e u, q 1n,l ce l v , q 1n,l , u
U, N n,l Mcin e l Ψn,l (u, v ) D (5.33)   e out N n,l Mc l u, q 2n,l ce l v , q 2n,l , u > U , and

( o Ψn,l (u, v )

D

  o u, q 1n,l se l v , q 1n,l , Msin N n,l l   o N n,l u, q 2n,l se l v , q 2n,l , Msout l

u
U, u>U.

(5.34)

That the latter wave functions cannot be right can be seen by rewriting the continuity of the wave function on the ellipse boundary. For example, Eq. (5.33) gives ce l (v , q 1n,l ) D

   U, q 2n,l Mcout l  ce l v , q 2n,l  constant  ce l v , q 2n,l . in Mc l U, q 1n,l (5.35)

The latter equation cannot be satisfied for an arbitrary v. If the left-hand and righthand side expressions were the same for all v (apart from a constant factor), then the following equations must be satisfied: d2 cel ( v ,q 1 ) dv 2

ce l (v , q 1 )

D 2q 1 cos 2v  ce l,q 1 D 2q 2 cos 2v  ce l,q 2 D

d2 ce l ( v ,q 2 ) dv 2

. ce l (v , q 2 ) (5.36)

Equation (5.36) can be rewritten as 2 (q 1 C q 2 ) cos 2v D ce l,q 1  ce l,q 2 .

(5.37)

67

68

5 Elliptic Coordinates

The latter equation cannot be fulfilled for all v since q 1 C q 2 ¤ 0. This is an example of where the boundary-value problem on a one-coordinate surface is not separable. Of course, even though expressions such as Eq. (5.33) are incorrect, they can be used to form the basis for a complete solution. For example, the correct wave function for the finite-barrier problem can be written in the form (for the even states as an example) Ψn (u, v ) D

X

a l Mc l (u, q 1 ) ce l (v , q 1 )

(5.38)

l

inside the ellipse, and X b l Mc l (u, q 2 ) ce l (v , q 2) Ψn (u, v ) D

(5.39)

l

outside. The correct boundary conditions are then Ψn (U, v ) D

X

a l Mc l (U, q 1 ) ce l (v , q 1 ) D

l

X

b l Mc l (U, q 2 ) ce l (v , q 2 ) ,

l

(5.40) and ˇ @Mc l (u, q 1 ) ˇˇ 1 X al ce l (v , q 1 ) ˇ m1 @u uDU l ˇ 1 X @Mc l (u, q 2 ) ˇˇ bl ce l (v , q 2 ) . D ˇ m2 @u uDU

(5.41)

l

In addition to the summation, we note that the q should not be functions of l. Equations (5.40) and (5.41) lead to 2L C 1 unknown constants (if the summations are truncated after L terms) a l , b l , and E. However, those two equations are for each v value. Since they are linear equations in the a l and b l , this leads to a transcendental determinant for the energy E. This is, of course, not trivial to solve numerically.

5.6 Problems

1. Show that the scale factors are
1/2 . h u D h v D f sinh2 u C sin2 v 2. a. Show that the Mathieu equation is of Bôcher type f1 1 3g. b. Show that the solutions of the Mathieu equation can be classified as even and odd functions.

5.6 Problems

3. a. Consider two solutions to the modified Mathieu equation, y 00  (a  2q cosh 2u)y D 0 , with the same a but different q. Show that they satisfy the following product integral formula: Zz2 2(q 2  q 1 )



z du y 1 (u)y 2(u) cosh 2u D y 10 (u)y 2(u)  y 20 (u)y 1 (u) z21 .

z1

b. Show that, for a solution of the modified Mathieu equation, Zz 4q

du y 2 sinh 2u D y 02 (z)  y (z)y 00(z) .

Hint: Differentiate the following function: φ(u) D y 02 (u)  (a  2q cosh 2u)y 2(u) . 4. Plot the characteristic curves for ce1 (q, z) and se1 (q, z) for small q. 5. Frequency modulation in radio transmission involves carrying a signal of frequency ω 1 on a carrier wave of frequency ω 0 . In an idealized LC circuit (i.e., zero resistance), this can be achieved by, for example, a direct capacitance modulation, where the capacitance varies with time, C(t) D C0 (1 C
cos 2ω 1 t) , and
 1. If the charge Q in the circuit obeys the differential equation Q d2 Q D0, C 2 dt LC show that, to first-order in the small parameter
, the differential equation becomes the Mathieu equation. 6. Solve for the lowest two eigenmodes of an elliptic membrane with a fixed rim. 7. For an electron of effective mass m
D 0.067 m 0 in a hard-wall quantum ellipse with semiaxes of 10 and 20 Å, find the ground-state energy.

69

71

6 Parabolic Coordinates 6.1 Introduction

The coordinate curves are orthogonal confocal parabolas. In separating the Laplace and Helmholtz equations, the former has ordinary differential equations similar to previously encountered coordinate systems, whereas the latter has the new property that the wave number (as well as the separation constant) appears in both ordinary differential equations. The ordinary differential equations for the Helmholtz equation are of a new type; they can be related to the Bessel wave equation.

6.2 Coordinate System 6.2.1 Coordinates (µ, ν)

The coordinates are µ and ν, where xD

 1
2 µ
ν2 , 2

y D µν ,

(6.1)

and 0µ<1,


1 < ν < C1 .

We have chosen the range of ν to include negative values to cover the y < 0 region. 6.2.2 Constant-Coordinate Curves

One can write
 y 2 D µ 2 µ 2
2x ,
 y 2 D ν 2 ν 2 C 2x .

(6.2) (6.3)

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

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6 Parabolic Coordinates

0.6

υ=constant

μ=constant 0.4

0.2

0.0

-0.2

-0.4

-0.6 -0.4

-0.2

0.0

0.2

0.4

Figure 6.1 Parabolic coordinates.

Both curves are then seen to be parabolic in shape. The constant µ curves cut the x axis at x D µ 2 /2 and open toward the left; µ D 0 is the y D 0, x  0 half line. For fixed ν, the curves are actually half parabolas (either above or below y D 0 depending upon the sign of ν) cutting the x axis at x D
ν 2 /2; ν D 0 is the y D 0, x  0 half line. The two constant-coordinate curves also intersect the y axis at y D ˙µ 2 and sgn(ν)ν 2 . Thus, the two sets of parabolas only intersect on the y axis if µ D jνj; nevertheless, they are confocal parabolas (Figure 6.1).

6.3 Differential Operators 6.3.1 Metric

Using Eq. (2.5), the scale factors are 1/2
, h µ D µ2 C ν2

hν D hµ .

6.3.2 Operators

These are obtained by substituting Eq. (6.4) into Eqs. (2.7)–(2.12).

(6.4)

6.4 Separable Equations

6.3.2.1 Gradient The gradient operator is

rD

eµ (µ 2

C

ν 2 )1/2

eν @ @ C . 1/2 @ν 2 2 @µ (µ C ν )

(6.5)

6.3.2.2 Divergence The divergence of a vector field V is  i i   @ h
2 1 @ h
2 2 1/2 2 1/2 C . (6.6) µ µ C ν V C ν V rV D 2 µ ν (µ C ν 2 ) @µ @ν 6.3.2.3 Laplacian The Laplacian is

r2 D

  2 1 @2 @ . C (µ 2 C ν 2 ) @µ 2 @ν 2

(6.7)

6.4 Separable Equations 6.4.1 Laplace Equation

The Laplace equation is r 2 ψ(µ, ν) D

  2 1 @2 @ ψ(µ, ν) D 0 . C (µ 2 C ν 2 ) @µ 2 @ν 2

(6.8)

Then, as we have seen before for other systems, the separated equations are the Euler equations: d2 M
k22 M D 0 , dµ 2

(6.9)

d2 N C k22 N D 0 . dν 2

(6.10)

We note that, of the separable 2-space systems, the rectangular, elliptic, and parabolic ones formally have the same separated ordinary differential equations for the Laplace problem. 6.4.2 Helmholtz Equation

The Helmholtz equation, r2 ψ C k2 ψ D 0 ,

73

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6 Parabolic Coordinates

becomes 

1 (µ 2 C ν 2 )



@2 @2 C @µ 2 @ν 2

 ψ C k2 ψ D 0 .

(6.11)

Let ψ(µ, ν, z) D M(µ)N(ν) .

(6.12)

1 1 1 d2 M 1 d2 N C D
k 2 , (µ 2 C ν 2 ) M dµ 2 (µ 2 C ν 2 ) N dν 2

(6.13)

Then,

and the separated equations are

d2 M(µ) 2 C k2 C k 2 µ 2 M(µ) D 0 , dµ 2

(6.14)

d2 N(ν) 2
k2
k 2 ν 2 N(ν) D 0 . dν 2

(6.15)

Equations (6.14) and (6.15) have an irregular singular point of order 6 at infinity and are labeled as Bôcher type f0 6g. If k 2 were allowed to be negative, then they would be known as Weber equations [6]. However, for the physical problem of a propagating wave, k 2 > 0. Instead, if we use the transformation M(µ) D

p

µZ(µ) ,

Eq. (6.14) becomes Z 00 C

  1 0 1 Z D0. Z C k22 C k 2 µ 2
µ 4µ 2

(6.16)

Comparing with the Bessel wave equation, Eq. (C71), we note that they are the same with p D 1/2. Thus, the solution can be written in terms of the Bessel wave function: M(µ) D

p

µ J1/2 (k, k2 , µ) .

(6.17)

Similarly, for the other ordinary differential equation, the solution is obtained by changing µ to i ν: N(ν) D

p

ν J1/2 (k, k2 , i ν) .

(6.18)

The boundary-value problem will require the simultaneous solution of both equations as they are coupled by the two unknown parameters (the separation constant and the wave number).

6.5 Applications

6.4.3 Schrödinger Equation

The separable potential is   „2 2 (v1 (µ) C v2 (ν)) . V(µ, ν) D 2m (µ 2 C ν 2 )

(6.19)

The Schrödinger equation becomes


„2 2m



1 (µ 2 C ν 2 )



@2 @2 C @µ 2 @ν 2

 ψC

„2 2m



 2 (v (µ) C v (ν)) ψ D Eψ . 1 2 (µ 2 C ν 2 ) (6.20)

Let ψ(µ, ν) D M(µ)N(ν) .

(6.21)

Then, " # # " 1 d2 N 1 1 d2 M 1
C 2v1 (µ) C 2 C 2v2 (ν) D k 2 ,
(µ 2 C ν 2 ) M dµ 2 (µ C ν 2 ) N dν 2 (6.22) and the separated equations are

d2 M(µ) 2 C k2 C k 2 µ 2
2v1 (µ) M(µ) D 0 , dµ 2

(6.23)

d2 N(ν) 2
k2
k 2 ν 2 C 2v2 (ν) N(ν) D 0 . dν 2

(6.24)

6.5 Applications 6.5.1 Heat Conduction: Dirichlet Problem for the Laplace Equation

Consider a thin metallic wire maintained at zero temperature on the negative x axis and another shaped into a parabola as shown in Figure 6.2 and at a temperature of T0 . Since the wires can be described by constant-coordinate curves in the parabolic coordinate system, we use the latter. The temperature distribution in the space surrounding the wires is given by the Laplace equation. Let the parabolic wire be labeled by µ D µ 0 and the straight wire be labeled by µ D 0. Since the temperature is only a function of µ, the Laplace equation becomes r2 T D

d2 M D0. dµ 2

75

76

6 Parabolic Coordinates y

T=0

x µ=0 µ=µ0 T=T0

Figure 6.2 Temperature distribution near a parabolic wire.

The general solution is M(µ) D A C B µ . Applying the boundary conditions gives M(µ) D

T0 µ. µ0

6.6 Problems

1. Show that the scale factors are 1/2
. h µ D h ν D µ2 C ν2 2. Derive the unit vectors in terms of the Cartesian unit vectors. 3. a. Show that the ordinary differential equations obtained by separating the Helmholtz equation each have an irregular singular point of order 6 at infinity. b. Show that the ordinary differential equation for M, Eq. (6.14), can be transformed into the Bessel wave equation, Eq. (6.16). c. What are the solutions when the separation constant is zero?

6.6 Problems

4. Consider a thin grounded straight conducting wire on the negative x axis and another shaped into a parabola as shown in Figure 6.3 and at a potential of V0 . Find the potential and electric field in the space surrounding the wires. y

V=0

V=V0

Figure 6.3 Problems question 4.

x

77

Part Three Three-Dimensional Coordinate Systems

81

7 Rectangular Coordinates 7.1 Introduction

This is one of the three well-known 3-space coordinate systems. Furthermore, the theory and applications in 3-space are a simple extension of those for the 2-space case treated in Chapter 3. Hence, only new results are presented here.

7.2 Coordinate System 7.2.1 Coordinates (x, y, z)

The coordinates are conventionally denoted as x, y, z. All three coordinates have an infinite domain:
1 < x, y, z < 1 .

(7.1)

7.2.2 Constant-Coordinate Surfaces

The planes of constant coordinates (Figure 7.1) are x D constant ,

y D constant ,

z D constant .

(7.2)

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

82

7 Rectangular Coordinates

y x = x0 z = z0 y = y0

x

z Figure 7.1 Rectangular coordinates.

7.3 Differential Operators 7.3.1 Metric

Using Eq. (2.5), one obtains the scale factors as hx D hy D hz D 1 .

(7.3)

7.3.2 Operators

These are obtained by substituting Eq. (7.3) into Eqs. (2.7)–(2.12). 7.3.2.1 Gradient The gradient operator is

r D ex

@ @ @ C ey C ez . @x @y @z

(7.4)

7.3.2.2 Divergence The divergence of a vector field V is

rV D

@Vz @Vz @Vx C C . @x @y @z

(7.5)

7.4 Separable Equations

7.3.2.3 Circulation The circulation is ˇ ˇ ex ˇ ˇ@ r  V D ˇ @x ˇ ˇVx

ey @ @y

Vy

ˇ e z ˇˇ @ ˇ ˇ . @z ˇ Vz ˇ

(7.6)

7.3.2.4 Laplacian The Laplacian is

r2 D

@2 @2 @2 C 2 C 2 . 2 @x @y @z

(7.7)

7.3.3 Stäckel Matrix

A possible Stäckel matrix is 1 0 1
1
1 Φ D@ 0 1 0 A , 0 0 1

(7.8)

and fi D 1 .

(7.9)

These follow from the equations in Section 2.4.2.1 rather trivially for the rectangular coordinate system since h 1 D h 2 D h 3 D 1 and one can see that the equations are satisfied if fi D 1 ,

gi D 1 ,

S D1,

Mi D 1 .

One can also easily verify that Eq. (7.8) satisfies all of the above conditions.

7.4 Separable Equations

To separate the partial differential equations, one can attempt to do so either via the first-principles technique or by using the Stäckel theory. For the Cartesian coordinates, it is trivial to do the separation “by hand.” 7.4.1 Laplace Equation

The Laplace equation is r2 ψ D

@2 ψ @2 ψ @2 ψ C C D0. 2 2 @x @y @z 2

(7.10)

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84

7 Rectangular Coordinates

Writing ψ(x, y, z) D X(x)Y(y )Z(z) ,

(7.11)

one gets @2 ( X Y Z ) @2 ( X Y Z ) @2 ( X Y Z ) C C 2 2 @x @y @z 2 D YZ

d2 X(x) d2 Y(y ) d2 Z(z) C XZ C XY D0, 2 2 dx dy dz 2

(7.12)

or 1 d2 X(x) 1 d2 Y(y ) 1 d2 Z(z) C D
. 2 2 X dx Y dy Z dz 2

(7.13)

As for the two-dimensional case, one then argues that since the right-hand side of the equation is only a function of the coordinate z (at most) whereas the the lefthand side is independent of z, the only way they can be equal is if both sides are equal to a (the same) constant, say, k32 . Then, 1 d2 Z(z) D
k32 , Z dz 2

(7.14)

and Eq. (7.13) can be rewritten as 1 d2 Y(y ) 1 d2 X(x) 2
k D
. 3 X dx 2 Y dy 2

(7.15)

Repeating the argument for the last equation, one can write 1 d2 Y(y ) D
k22 . Y dy 2

(7.16)

This, finally, leads to the following three ordinary differential equations (ODEs):  d2 X(x)
2
k2 C k32 X(x) D 0 , 2 dx

(7.17)

d2 Y(y ) C k22 Y(y ) D 0 , dy 2

(7.18)

d2 Z(z) C k32 Z(z) D 0 . dz 2

(7.19)

There are, therefore, two separation constants k2 and k3 . For a boundary-value problem, this shows that, for example, the last two differential equations can be solved independently, with the boundary conditions determining the separation constants k2 and k3 . We note that Eqs. (7.17)–(7.19) agree with the choice of the Stäckel matrix, Eq. (7.8), if k1 D 0 (since the latter is only present for the Helmholtz equation).

7.4 Separable Equations

Obviously, the separation technique was implemented to agree with the choice of the Stäckel matrix. One could have carried out the separation in a slightly different (but, of course, equivalent) form. For example, in Eq. (7.13), one could argue that each term is at most a function of one coordinate; but, since the sum is zero, the terms must each be equal to a constant, that is, 1 d2 X(x) D
k12 , X dx 2

1 d2 Y(y ) D
k22 , Y dy 2

1 d2 Z(z) D
k32 , Z dz 2

(7.20)

with the constraint k12 C k22 C k32 D 0 .

(7.21)

The new ODEs are then d2 X(x) C k12 X(x) D 0 , dx 2

(7.22)

d2 Y(y ) C k22 Y(y ) D 0 , dy 2

(7.23)

d2 Z(z) C k32 Z(z) D 0 . dz 2

(7.24)

The latter are clearly equivalent to Eqs. (7.17)–(7.19). Finally, we note that the solutions are, as in the two-dimensional case, either trigonometric or hyperbolic functions. 7.4.2 Helmholtz Equation

The Helmholtz equation is r2 ψ C k2 ψ D

@2 ψ @2 ψ @2 ψ C C C k2 ψ D 0 . @x 2 @y 2 @z 2

(7.25)

Writing ψ(x, y, z) D X(x)Y(y )Z(z) ,

(7.26)

one gets 1 d2 X 1 d2 Y 1 d2 Z C C D
k 2 . X dx 2 Y dy 2 Z dz 2

(7.27)

Then, one can write d2 X(x) C k12 X(x) D 0 , dx 2

(7.28)

d2 Y(y ) C k22 Y(y ) D 0 , dy 2

(7.29)

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86

7 Rectangular Coordinates

d2 Z(z) C k32 Z(z) D 0 , dz 2

(7.30)

k12 C k22 C k32 D k 2 .

(7.31)

with

In this case, there are three independent separation constants and, together, they determine the wave number k. The ODEs are also written in a form that is not directly given by the choice of the Stäckel matrix, Eq. (7.8). Nevertheless, they have trigonometric or hyperbolic functions as solutions. 7.4.3 Schrödinger Equation

The following form of potential is separable in rectangular coordinates: V(x, y, z) D

 „2  v1 (x) C v2 (y ) C v3 (z) . 2m

(7.32)

For example, V(r) D

1 1 m ω 2 r 2 D m ω 2 (x 2 C y 2 C y 2 ) , 2 2

is separable in Cartesian coordinates. The Schrödinger equation is


„2 2m



  „2  @2 @2 @2 ψC v1 (x) C v2 (y ) C v3 (z) ψ D E ψ . C C 2 2 2 @x @y @z 2m (7.33)

Let ψ(x, y, z) D X(x)Y(y )Z(z) .

(7.34)

Then, "


# " # " # 1 d2 X 1 d2 Y 1 d2 Z C v (x) C
C v (y ) C
C v (z) D k2 , 1 2 3 X dx 2 Y dy 2 Z dz 2 (7.35)

where k 2  2m E /„2 and  d2 X(x)  2 C k1
v1 (x) X(x) D 0 , dx 2

(7.36)

 d2 Y(y )  2 C k2
v2 (y ) Y(y ) D 0 , dy 2

(7.37)

7.5 Applications

 d2 Z(z)  2 C k3
v3 (z) Z(z) D 0 , dz 2

(7.38)

with k12 C k22 C k32 D k 2 . Thus, the approach to solving a boundary-value problem is identical to that for the Helmholtz equation.

7.5 Applications 7.5.1 Electrostatics: Dirichlet Problem for a Rectangular Box

The standard electrostatics problem is to find the electrostatic potential inside a box subject to a given potential on the boundary. A common example is the one given in Figure 7.2, whereby the potential is zero on all sides except for z D c. The solutions to Eqs. (7.17)–(7.19) have to vanish at the origin; thus, they have the forms X(x) D a 1 sin k x , Y(y ) D a 2 sin q y , Z(z) D a 3 sinh
z ,

(7.39)

where k, q, and
are real constants. For the potential to be zero at x D a and y D b, mπ , a nπ qn D , b

km D

z c

ψ=V(x,y) b y a

x Figure 7.2 Electrostatics example.

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88

7 Rectangular Coordinates

s
m n D π

n2 m2 C 2 . 2 a b

(7.40)

The general potential satisfying the above boundary conditions is, therefore, ψ(x, y, z) D

1 X

A m n sin k m x sin q n y sinh
m n z .

(7.41)

m,nD1

This is a Fourier-series expansion. Finally, the requirement of ψ(x, y, z D c) D V(x, y ) determines the coefficients A m n to be such that A mn D

4 ab sinh
m n c

Za

Zb dx

0

dy V(x, y ) sin k m x sin k n y .

(7.42)

0

0.01

0.8

0.008

0.6

0.006 0.4 0.004 0.2

0.002 0

0 2

2 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1.5

1

1 0.5 0 0

(a)

0.2

0.4

0.6

0.8

1

(b)

0.01

0.8

0.008

0.6

0.006 0.4 0.004 0.2

0.002 0

0 2

2 1.5 1 0.5 0 0

(c)

0.2

0.4

0.6

0.8

1.5

1

1 0.5 0 0

0.2

0.4

0.6

0.8

1

(d) Figure 7.3 Potential distribution inside a rectangular box subject to zero potential on all sides except for a constant potential of 1 V on the top surface. The plots in (a) and (c) are at z D c/2 and the plots in (b) and (d) are

at z D 19c/20. The plots in (a) and (b) are for five terms, and the plots in (c) and (d) are for 10 terms. The parameters in Eq. (7.44) are a D 1, b D 2, c D 3, V0 D 1.

7.6 Problems

As an explicit example, let V(x, y ) D V0 . Then, A mn D

4V0 (cos k m a
1)(cos k n b
1) , ab k m k n sinh
m n c

(7.43)

and ψ(x, y, z) D

1 4V0 X (cos k m a
1) (cos k n b
1) sin k m x sin q n y sinh
m n z . ab m,nD1 k m k n sinh
m n c

(7.44) The series is convergent. For illustration, the series was computed up to five and 10 terms for the following parameter values: a D 1, b D 2, c D 3, V0 D 1. In Figure 7.3, the potentials at z D c/2 (Figure 7.3a,c) and z D c (Figure 7.3b,d) are plotted.

7.6 Problems

1. Using the Stäckel matrix given, Eq. (7.8), obtain the separated ODEs for the Helmholtz equation. Show that these are identical to the solutions given in the text (Eqs. (7.28)–(7.30)). 2. Consider a rectangular box of dimensions L x , L y , L z with rigid walls. Express the acoustic eigenfrequencies and eigenmodes in terms of the parameters L x , L y , L z , c. 3. Consider a rectangular box as in question 2 where all wall sides but the top side are rigid. The top side is a pressure-release boundary. Determine the eigenfrequencies and eigenmodes in terms of the parameters L x , L y , L z , c. Compare the result with that from question 2. 4. Consider a rectangular waveguide with cross-sectional dimensions L x , L y and rigid walls. Determine the cutoff frequency for a fluid with speed of sound c. a. Determine the acoustic pressure in a semi-infinite waveguide (z D [0, 1[) with rigid walls excited by a forced piston at the open end (z D 0). Assume the acoustic pressure amplitude generated by the piston is 1 bar. b. If the piston occupies a fractional part of the open-end area, determine the acoustic pressure in the waveguide specified by the piston geometry. 5. The source-free electromagnetic fields satisfy the wave equations r2 E


1 @2 E D0, c 2 @t 2

(7.45)

r2 B


1 @2 B D0, c 2 @t 2

(7.46)

where E (B) is the electric (magnetic) field.

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7 Rectangular Coordinates

a. For propagation down a uniform metallic waveguide (with axis along the z direction) and assuming a z and t dependence of e i(k z
ω t), show that the fields obey a Helmholtz equation. b. For a waveguide of rectangular cross section with perfectly conducting walls (i.e., E is purely tangential at the wall), i. Calculate the electric and magnetic fields. ii.Find the cutoff frequency below which no modes propagate.

91

8 Circular Cylinder Coordinates 8.1 Introduction

This is the second of the three well-known coordinate systems. The coordinate choices of Moon and Spencer and Morse and Feshbach differ in the second coordinate. We will follow Moon and Spencer and illustrate a few of the differences in the Morse–Feshbach choice. The radial equation for both the Laplace equation and the Helmholtz equation leads to the Bessel equation.

8.2 Coordinate System 8.2.1 Coordinates (r, φ, z)

The coordinates are denoted ξ1 D r, ξ2 D φ, ξ3 D z and the relationship to the Cartesian coordinates is x D r cos φ , y D r sin φ , zDz,

(8.1)

where 0
r<1,

0
φ < 2π ,

1 < z < 1 .

On the other hand, Morse and Feshbach use ξ1 D r ,

ξ2 D cos φ ,

ξ3 D z ,

x D ξ1 ξ2 , q y D ξ1 1  ξ22 , z D ξ3 ,

(8.2)

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

92

8 Circular Cylinder Coordinates

y r = r0 φ=φ0

x z = z0 z Figure 8.1 Cylindrical circular coordinates.

with 0
ξ1 < 1 ,

1
ξ2
1 ,

1 < ξ3 < 1 .

The first choice is more common and will be used here as well. 8.2.2 Constant-Coordinate Surfaces

The constant-coordinate surfaces are (Figure 8.1) r D constant: cylinders , φ D constant: half planes , z D constant: planes .

8.3 Differential Operators 8.3.1 Metric

Using Eqs. (2.5) and (8.1), one obtains the scale factors as hr D 1 ,

hφ D r ,

hz D 1 .

(8.3)

8.3 Differential Operators

8.3.2 Operators

These are obtained by substituting Eq. (8.3) into Eqs. (2.7)–(2.12). 8.3.2.1 Gradient The gradient operator is

r D er

@ 1 @ @ C eφ C ez . @r r @φ @z

(8.4)

8.3.2.2 Divergence The divergence of a vector field V is

rV D

@Vz 1 @ 1 @Vφ (r Vr ) C C . r @r r @φ @z

8.3.2.3 Circulation The circulation is ˇ ˇe 1 ˇˇ @r r  V D ˇ @r rˇ ˇVr

r eφ @ @φ

r Vφ

ˇ e z ˇˇ @ ˇ @z ˇˇ . Vz ˇ

(8.5)

(8.6)

8.3.2.4 Laplacian The Laplacian is

1 @ r D r @r 2




@ r @r

 C

1 @2 @2 C . r 2 @φ 2 @z 2

(8.7)

8.3.3 Stäckel Theory

From Section 2.4.2.1, a possible Stäckel matrix is found to be 0

0 Φ D @0 1

1/r 2 1 0

1 1 0A . 1

(8.8)

Also, we have f1 D r ,

f2 D 1 ,

f3 D 1 .

(8.9)

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8 Circular Cylinder Coordinates

8.4 Separable Equations 8.4.1 Laplace Equation

The Laplace equation r 2 ψ D 0 becomes 

1 @ r @r


r

@ @r

 C

 1 @2 @2 ψ(r, φ, z) D 0 . C r 2 @φ 2 @z 2

(8.10)

Let ψ(r, φ, z) D R(r)Φ (φ)Z(z) .

(8.11)

Then, 1 d r R(r) dr


r

dR(r) dr

 C

1 d2 Z(z) 1 1 d2 Φ (φ) D , 2 2 r Φ (φ) dφ Z(z) dz 2

(8.12)

giving 1 d r dr


r

dR(r) dr




k2  k32 C 22 R(r) D 0 , r

(8.13)

d2 Φ (φ) C k22 Φ (φ) D 0 , dφ 2

(8.14)

d2 Z(z) C k32 Z(z) D 0 , dz 2

(8.15)

where k2 and k3 are the separation constants. The radial equation, Eq. (8.13), is a Bessel equation, which we have already encountered in Chapter 4. If k32 > 0, the radial equation is the modified Bessel equation. The general solution, for nonintegral k2 , can be written as R(r) D A J k2 (i k3 r) C B J
k2 (i k3 r) D AI k2 (k3 r) C B K k2 (k3 r) .

(8.16)

The functions I k2 (k3 r) and K k2 (k3 r) are the modified Bessel functions of the first and the third kind; the latter are also known as MacDonald functions. If k32 D q 2 < 0, the radial equation is the Bessel equation. The general solution, for nonintegral k2 , can be written as R(r) D A J k2 (q r) C B J
k2 (q r) .

(8.17)

The function J k2 (q r) is the Bessel function of the first kind. If k2 is an integer, the Neumann function should be used instead of J
k2 (q r).

8.4 Separable Equations

8.4.2 Helmholtz Equation

We now use the Stäckel theory to separate the Helmholtz equation r2 ψ C k2 ψ D 0 . Using Eqs. (8.8) and (8.9), the separated equations are (with k12 D k 2 ) 

2 dR(r) 1 d k2 2 r  C k3 R(r) D 0 , r dr dr r2 d2 Φ (φ) C k22 Φ (φ) D 0 , dφ 2  d2 Z(z)  2 C k C k32 Z(z) D 0 . 2 dz

(8.18)

It is more conventional to have the k 2 term in the radial equation; hence, transforming k 2 C k32 ! k32 gives 

1 d dR(r) k2 (8.19) r C k 2  22  k32 R(r) D 0 , r dr dr r d2 Φ (φ) C k22 Φ (φ) D 0 , dφ 2

(8.20)

d2 Z(z) C k32 Z(z) D 0 . dz 2

(8.21)

The radial equation, Eq. (8.19), is the Bessel equation and the solutions are as discussed in Section 8.4.1. 8.4.3 Schrödinger Equation

Using the Stäckel theory, V(r, φ, z) D

  „2 1 „2 X v i (ξi ) v D (r) C v (φ) C v (z) . 1 2 3 2m 2m r2 h 2i i

(8.22)

Then, „2 2m „2 C 2m



  @ 1 @2 @2 ψ(r, φ, z) C 2 C @r r @φ 2 @z 2   1 v1 (r) C 2 v2 (φ) C v3 (z) ψ(r, φ, z) D E ψ(r, φ, z) . r



1 @ r @r




r

(8.23)

Let ψ(r, φ, z) D R(r)Φ (φ)Z(z) .

(8.24)

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8 Circular Cylinder Coordinates

Then,  

1 d r R dr


r

dR dr



" # " #  1 d2 Φ 1 1 d2 Z C v1 (r) C 2  C v (φ) C  C v (z) D k2 . 2 3 r Φ dφ 2 Z dz 2 (8.25)

The separated equations are  d2 Z(z)  2 C k3  v3 (z) Z(z) D 0 , dz 2  d2 Φ (φ)  2 C k2  v2 (φ) Φ (φ) D 0 , dφ 2
    d dR r r C k 2  k32  v1 (r) r 2  k22 R(r) D 0 . dr dr

(8.26) (8.27) (8.28)

8.5 Applications 8.5.1 Heat Conduction: Dirichlet Problem for a Cylinder

Consider a solid conducting cylinder of radius R and height H. Let the temperature on the surfaces be zero on the bottom and circular sides and f (r) on the top surface. In the steady state, the temperature satisfies the Laplace equation. The temperature is clearly a function of the radial coordinate r and the height z but is independent of the angle φ. Thus, the temperature distribution satisfies Eqs. (8.13)–(8.15) with k2 D 0. The appropriate solutions to Eq. (8.15) are hyperbolic functions since the temperature is not periodic; indeed, for T(z D 0) D 0, we have Z(z)  sinh
z , where
D i k3 . The radial equation, Eq. (8.13), reduces to the Bessel equation of zero order; thus, R(r)  J0 (
r) . We exclude the Neumann function since it is singular at r D 0. Therefore, the general solution is T(r, z) D

1 X

A n sinh
n z J0 (
n r) .

(8.29)

nD1

The coefficients A n are determined by fitting to the last boundary condition: T(r, z D H ) D f (r) D

1 X nD1

A n sinh
n H J0 (
n r) .

(8.30)

8.6 Problems

Using the orthogonality between J0 and J1 , An D

2 R 2 [ J1 (
n R)]2 sinh
n H

ZR dr r f (r) J0 (
n r) .

(8.31)

0

8.5.2 Quantum Mechanics: Dirichlet Problem for a Cylinder 8.5.2.1 Infinite Barrier Consider an upright cylinder of height H and radius R. Inside the cylinder, v1 (r) D v2 (φ) D v3 (z) D 0, and the general series solutions can be readily written down:

Z(z) D a 3 sin k3 z C b 3 cos k3 z , Φ (φ) D a 2 sin k2 φ C b 2 cos k2 φ (k2 integer) , R(r) D a 1 J k2 (k1 r) C b 1 N k2 (k1 r) ,

(8.32)

where k12 D k 2  k32 . Note that k 2 > k32 since E > E z ; hence, the Bessel functions (as opposed to the modified ones). One can now simplify by applying the boundary conditions. Outside the cylinder, the wave function is zero. For an upright cylinder with the bottom surface at z D 0, this requires b 3 D 0. At the top surface at z D H , the boundary condition gives k3 D

nπ , H

n 2 ZC .

(8.33)

The solutions to the Φ and R equations are identical to those for the two-dimensional case. The energy is given by E(k) D

„2 k 2 „2 k12 „2 k32 D C . 2m 2m 2m

(8.34)

8.6 Problems

1. Derive the Stäckel matrix 0 0 1/r 2 S D@ 0 1 1 0

1 1 0 A . 1

2. a. Show that an alternative formulation of the circular cylinder system is x D e ξ cos ψ , b. Show that fi D 1 .

y D e ξ sin ψ ,

z D ξ3 .

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98

8 Circular Cylinder Coordinates

c. Show that a possible Stäckel matrix is 0

0 S D@ 0 1

1 1 0

1 e
2ξ A . 0 1

d. Find the separated ordinary differential equations. 3. For the coordinates defined as [5] x D ξ1 ξ2 ,

q y D ξ1 1  ξ22 ,

z D ξ3 ,

q 1  ξ22 ,

f3 D 1 .

a. Show that f 1 D ξ1 ,

f2 D

b. Show that a possible Stäckel matrix is 0

1 S D@ 0 0

1/ξ12 1/(1  ξ22 ) 0

1 1 0 A . 1

c. Find the separated ordinary differential equations. d. Give the Bôcher type for the ξ2 equation. 4. Consider a solid conducting cylinder of radius R and height H. Let the temperature on the top and bottom surfaces be zero and g(z) on the circular side. In the steady state, what is the temperature distribution? Hence, or otherwise, find the temperature distribution when the bottom surface is at zero temperature, the top surface has a temperature distribution of f 1 (r), and the side has a temperature distribution of f 2 (z). 5. Consider a hollow conducting cylinder of inner radius R1 , outer radius R2 , and height H. Let the temperature on bottom and circular surfaces be zero and f (r) on the top side. In the steady state, what is the temperature distribution?

99

9 Elliptic Cylinder Coordinates 9.1 Introduction

This coordinate system is a natural generalization of the circular cylinder one, whereby there exists a one-parameter closed curve of elliptic shape. Both the Laplace equation and the Helmholtz equation lead to the Mathieu and modified Mathieu equations. However, for the Helmholtz problem, the separation constants are now not entirely separated. The standard technique for solving for the separation constants involves a continued-fraction approach, though a matrix approach has also been proposed recently [44]. This will be our first example of a more complicated boundary-value problem in 3-space. The 2-space version was treated in Chapter 5.

9.2 Coordinate System 9.2.1 Coordinates (u, v, z)

The coordinates are ξ1 D u, ξ2 D v , ξ3 D z, and the relationship to the Cartesian coordinates is x D f cosh u cos v , y D f sinh u sin v , zDz,

(9.1)

with 0
u<1,

0
v
2π ,

1 < z < 1 .

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

100

9 Elliptic Cylinder Coordinates

A couple of alternative formulations given by Morse and Feshbach are x D ξ1 ξ2 , q

 yD ξ12  d 2 1  ξ22 , z D ξ3 ,

(9.2)

and 1 (r1 C r2 ) , 2

ξ1 D d cosh µ D ξ2 D cos φ D

1 (r1  r2 ) , 2d

ξ3 D z .

(9.3)

9.2.2 Constant-Coordinate Surfaces

These are cylindrical extensions of the results in Chapter 5 whereby u D constant is an elliptic cylinder (Figure 9.1), 

x 2  y 2 C D f2, cosh u sinh u

(9.4)

and v D constant is a half hyperbolic cylinder,  x 2  y 2  D f2. cos v sin v

(9.5)

Finally, z D constant is a plane. y

v=v0 z =z 0

x u = u0

z Figure 9.1 Elliptic cylinder coordinates.

9.3 Differential Operators

Note that both v D 0 and v D 2π give half planes at y D 0 and for x  f . Also, u D 0 is a degenerate ellipse (covering the line segment  f
x
f twice). Thus, the line u D 0 is a branch cut and the points u D 0, v D v1 and u D 0, v D 2π  v1 are identical.

9.3 Differential Operators 9.3.1 Metric

Using Eqs. (2.5) and (9.1), one obtains the scale factors as  1/2  1/2 D f cosh2 u  cos2 v h u D h v D f sinh2 u C sin2 v D f (cosh 2u  cos 2v )1/2 , hz D 1 .

(9.6)

9.3.2 Operators

These are obtained by substituting Eq. (9.6) into Eqs. (2.7)–(2.12). 9.3.2.1 Gradient The gradient operator is

rD

eu 2

f (cosh u 

cos2

v )1/2

ev @ @ @ C C ez . (9.7) 2 2 1/2 @u @v @z f (cosh u  cos v )

9.3.2.2 Divergence The divergence of a vector field V is

rV D

1 f (cosh2 u  cos2 v )1/2    1/2  1/2  @  @  cosh2 u  cos2 v cosh2 u  cos2 v  Vu C Vv @u @v @Vz C . (9.8) @z

101

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9 Elliptic Cylinder Coordinates

9.3.2.3 Circulation The circulation is

1

rV D 

 cosh u  cos2 v 1/2 ˇ ˇ 2 eu ˇ cosh u  cos2 v ˇ @ ˇ ˇ @u 1/2 ˇ ˇ cosh2 u  cos2 v Vu 2

 

cosh2 u  cos2 v @ @v

cosh2 u  cos2 v

1/2

ev

1/2 Vv

ˇ ˇ ez ˇ ˇ @ ˇ . @z ˇ ˇ 1 ˇ V f z 1 f

(9.9) 9.3.2.4 Laplacian The Laplacian is

  2 @2 2 @2 @ C 2 . C 2 2 2 f (cosh 2u  cos 2v ) @u @v @z

r2 D

(9.10)

9.3.3 Stäckel Matrix

From Section 2.4.2.1, a possible Stäckel matrix is found to be 0

0 Φ D@ 0 1

1 1 0

1  f 2 cosh2 u f 2 cos2 v A . 1

(9.11)

The f functions are fi D 1 .

(9.12)

9.4 Separable Equations

There are small changes to the differential equations compared with the 2-space ones presented in Chapter 5. 9.4.1 Laplace Equation

The Laplace equation r 2 ψ D 0 becomes 2 2 f (cosh 2u  cos 2v )



@2 @2 C 2 2 @u @v

ψ(u, v , z) C

@2 ψ(u, v , z) D 0 . @z 2 (9.13)

9.4 Separable Equations

Let ψ(u, v , z) D U(u)V(v )Z(z) .

(9.14)

Then, 

2 f

2 (cosh 2u

2 1 d2 Z 1 d2 U 1 d2 V  2  D0, 2 2  cos 2v ) U du f (cosh 2u  cos 2v ) V dv Z dz 2 (9.15)

and the separated equations are d2 Z(z) C k32 Z(z) D 0 , dz 2   2 d2 V 2 f 2 cos 2v C k C k 3 2 V D0 , dv 2 2   2 d2 U 2 f 2  k cosh 2u C k 3 2 U D0 . du2 2

(9.16) (9.17) (9.18)

We recall that in the 2-space case the equations were simple harmonic differential equations; now they are Mathieu equations and, furthermore, the U and V equations are coupled via two separation constants. However, in a boundary-value problem, k3 is determined by the Z equation. If one writes 2q D k32

f2 , 2

λ D k22 ,

(9.19)

then Eqs. (9.17) and (9.18) become d2 V C (λ  2q cos 2v ) V D 0 , dv 2

(9.20)

d2 U  (λ  2q cosh 2u) U D 0 . du2

(9.21)

These are the standard forms of the Mathieu and modified Mathieu equations. The even solutions are U(u) D Ace(i u, q) C Bfe(i u, q) ,

(9.22)

V(v ) D Ace(v , q) C Bfe(v , q) ,

(9.23)

and the odd solutions are U(u) D Ase(i u, q) C Bge(i u, q) ,

(9.24)

V(v ) D Ase(v , q) C Bge(v , q) .

(9.25)

The solutions ce(v , q) and se(v , q) are known as Mathieu functions of the first kind. Properties of the Mathieu functions (also known as elliptic cylinder functions) are described in Appendix F.

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9 Elliptic Cylinder Coordinates

9.4.2 Helmholtz Equation

The Helmholtz equation is 

2 f 2 (cosh 2u  cos 2v )



@2 @2 C 2 2 @u @v

ψC

 @2 ψ C k2 ψ D 0 . @z 2

(9.26)

Let ψ(u, v , z) D U(u)V(v )Z(z) .

(9.27)

Then, 2 f

2 (cosh 2u

1 d2 U 1 d2 V 1 d2 Z 2 C 2 C D k 2 , 2  cos 2v ) U du f (cosh 2u  cos 2v ) V dv 2 Z dz 2 (9.28)

and the separated equations are d2 Z(z) C k32 Z(z) D 0 , dz 2    2
2 d2 V 2 f 2 cos 2v C k C k  k 3 2 V D 0, dv 2 2    2
2 d2 U 2 f 2 cosh 2u C k  k  k 3 2 U D 0. du2 2

(9.29) (9.30) (9.31)

For a boundary-value problem, k3 is determined by the Z equation. The two Mathieu equations are solved by using one equation to obtain a characteristic curve k D k(k2 ) and then the boundary condition on the other equation selects the discrete set of solutions. They can, again, be written in conventional form if  f2
, 2q D k 2  k32 2

λ D k22 .

(9.32)

Then, Eqs. (9.30) and (9.31) become d2 V C (λ  2q cos 2v ) V D 0 , dv 2

(9.33)

d2 U  (λ  2q cosh 2u) U D 0 . du2

(9.34)

These are the standard forms of the Mathieu and modified Mathieu equations, and the solutions are as given in Section 9.4.1.

9.5 Applications

9.4.3 Schrödinger Equation

The separable potential is   „2 2 (v V(u, v , z) D (u) C v (v )) C v (z) . 1 2 3 2m f 2 (cosh 2u  cos 2v )

(9.35)

The Schrödinger equation becomes „2  2m C

„2 2m

 

2 f 2 (cosh 2u  cos 2v ) 2 f 2 (cosh 2u  cos 2v )





@2 ψ(u, v , z) C 2 ψ(u, v , z) @z  (v1 (u) C v2 (v )) C v3 (z) D E ψ(u, v , z) . @2 @2 C @u2 @v 2



(9.36) Let ψ(u, v , z) D U(u)V(v )Z(z) .

(9.37)

Then, "

# 1 d2 U  v (u) 1 U du2 " # 2 1 d2 V  v2 (v )  2 f (cosh 2u  cos 2v ) V dv 2 " # 1 d2 Z   v3 (z) D k 2 , Z dz 2 

2 f 2 (cosh 2u  cos 2v )

(9.38)

and the separated equations are d2 Z(z)  2 C k3  v3 (z) Z(z) D 0 , 2 dz

 2
2 d2 V 2 f 2 cos 2v C k C k  k C v (v ) V D0, 2 3 2 dv 2 2

 2
2 d2 U 2 f 2 cosh 2u C k  k  k  v (u) U D0. 1 3 2 du2 2

(9.39) (9.40) (9.41)

9.5 Applications

There have been a few applications of the elliptic cylinder coordinate system. Applications are discussed in the book by McLachlan [40]. More recent examples include the flow-acoustic properties of elliptic cylinder waveguides and enclosures [45].

105

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9 Elliptic Cylinder Coordinates

9.5.1 Hydrodynamics: Dirichlet Problem for an Elliptic Pipe

A very beautiful yet simple problem of the flow of an incompressible, viscous fluid in a pipe of elliptic cross section was discussed by Landau and Lifshitz [35]. As already discussed in Section 2.7.5, the original Navier–Stokes equation reduces to a 2-space Poisson equation for the velocity, r2 v D

1 dp , η dz

(9.42)

for the flow in a pipe, with the boundary condition v D 0 on the pipe itself. Here η is the viscosity, p the pressure, and z the pipe axis. Before giving the solution in elliptic coordinates, Landau and Lifshitz [35] showed that the solution can be readily written down in Cartesian coordinates: v (x, y ) D

∆ p a2 b2 2η L a 2 C b 2

1

x2 y2  2 2 a b

,

(9.43)

where a and b are the semiaxes, and for a constant pressure gradient, ∆p dp D . dz L This solution follows trivially from the need to satisfy the boundary condition on the ellipse x 2 /a 2 Cy 2 /b 2 D 1 and the fact that the solution must be at most quadratic in the coordinates. In elliptic coordinates, the solution to Eq. (9.42) can be separated into the particular solution (giving the inhomogeneous term) and the solution to the Laplace equation. For the latter, we have d2 v D0, du2

(9.44)

v (u) D A C B u .

(9.45)

or

9.5.2 Quantum Mechanics: Dirichlet Problem for an Elliptic Cylinder

For v1 (u) D v2 (v ) D v3 (z) D 0, Eqs. (9.39)–(9.41) become d2 Z(z) C k32 Z(z) D 0 , dz 2    2
2 d2 V 2 f 2 cos 2v C k C k  k 3 2 D 0, dv 2 2

(9.46) (9.47)

9.6 Problems

   2
2 d2 U 2 f 2 cosh 2u C k  k  k 3 2 D0 . du2 2

(9.48)

Writing k22  c ,


k32  k 2

 f2  2q , 2

we get the canonical forms of the Mathieu and modified Mathieu equations: d2 V C (c  2q cos 2v ) V D 0 , dv 2

(9.49)

d2 U  (c  2q cosh 2u) U D 0 . du2

(9.50)

Some of the properties are discussed in Appendix F. The bound-state problem is solved by first solving Eq. (9.46) for the allowed k3 ; in this case, k3 is an integer owing to periodicity. Then, Eqs. (9.47) and (9.48) need to be solved self-consistently for the energy k 2 and the second separation constant k2 . Periodicity for the V(v ) function leads to the characteristic equation c D c(q). For an infinite barrier, the hard-wall boundary condition is M c l (q, u D U) D 0 ,

(9.51)

M s l (q, u D U) D 0 .

(9.52)

9.6 Problems

1. a. Show that

 1/2  1/2 D f cosh2 u  cos2 v h u D h v D f sinh2 u C sin2 v D f (cosh 2u  cos 2v )1/2 .

b. Show that the Laplacian is given by Eq. (9.10). 2. Use the Stäckel theory to separate the Helmholtz equation. Compare the result with that given in this chapter. 3. Find explicit solutions to the Helmholtz equation if the two separation constants are both zero. 4. Derive the orthogonality relations of the Mathieu functions given in Appendix F, Eqs. (F11)–(F13). 5. A capacitor is made of two confocal elliptic cylinders of length L with a dielectric medium of resistivity
. The outer conductor (µ D µ 2 ) is grounded and the inner one (µ D µ 1 ) is at a potential V. a. Derive an expression for the potential at any point between the conductors. b. Find the resistance between them.

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10 Parabolic Cylinder Coordinates 10.1 Introduction

This is the last of the cylinder coordinates that allows full separation. The constant-coordinate surfaces are open surfaces but, taken together, they form a closed and bounded region (e.g., by a four-coordinate surface). The separated differential equations for the Helmholtz problem include the Weber equation and two of the ordinary differential equations remain coupled via the two separation constants. For the boundary-value problem, an additional condition on the solutions beyond those already encountered is required.

10.2 Coordinate System 10.2.1 Coordinates (µ, ν, z)

The orthogonal coordinates are labeled ξ1 D µ, ξ2 D ν, ξ3 D z and can be written in terms of the Cartesian coordinates as follows:  1
2 µ
ν2 , 2 y D µν ,

xD

zDz,

(10.1)

where 0µ<1,


1 < ν < 1 ,


1 < z < 1 .

ν takes on both positive and negative values to reproduce the whole y axis.

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

110

10 Parabolic Cylinder Coordinates

y z =z0 µ = µ0 ν = ν0

x

z

Figure 10.1 Parabolic cylinder coordinates.

10.2.2 Constant-Coordinate Surfaces

One can write xD xD

1 2 1 2

 µ2


y2 µ2

y2
ν2 ν2

 ,

(10.2)

.

(10.3)



Both curves (Figure 10.1) are then seen to be parabolas (for constant ν and µ, respectively); for fixed ν, the curves are actually half parabolas (either above or below y D 0 depending upon the sign of ν). Thus, the planes of constant coordinates are µ D constant: right-convex parabolic cylinders , ν D constant: left-convex half-parabolic cylinders , z D constant: planes .

(10.4)

The parabolas cut the x axis at x D
ν 2 /2 and x D µ 2 /2, and intersect the y axis at y D ˙µ 2 and sgnν 2 . Finally, by equating Eqs. (10.2) and (10.3), one finds that the two parabolas intersect at    1
2 µ
ν 2 , ˙µjνj . 2 In the z D 0 plane, the foci of the parabolas are at the origin.

10.2 Coordinate System

10.2.3 Other Geometrical Parameters

For boundary-value problems, one might need a closed region. The simplest one consistent with the coordinate system is the four-coordinate surface shown in Figure 10.2. The expressions for the various geometrical parameters given in Figure 10.2 are as follows: 1 (y max
y min ) D 2µ 0 jν 0 j , 2  1
2 µ 0 C ν 20 , H D xmax
xmin D 2
2  2 2 V D Lµ 0 ν 0 µ 0 C ν 20 D LW H , 3 3 W D

(10.5) (10.6) (10.7)

where L is the axial length of the bounded region (along the z axis), W is the width (in the y direction), H is the height (in the x direction), and V is the volume of the enclosed region. The volume is obtained by integrating a volume element in parabolic cylinder coordinates: Zz2 Zµ 0 Zν 0 VD




 µ 2 C ν 2 dzdµdν ,

(10.8)

z 1 0
ν 0

and use is made of L D z2
z1 , and the metric is given in Eq. (10.9). y

x v0

L

W

µ0

z H Figure 10.2 Geometry of the region for boundary-value problems (cross-sectional view in the z plane).

111

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10 Parabolic Cylinder Coordinates

10.3 Differential Operators 10.3.1 Metric

Using Eqs. (2.5) and (10.1), one obtains the scale factors as 1/2
, h µ D h ν D µ2 C ν2

hz D 1 .

(10.9)

10.3.2 Operators

These are obtained by substituting Eq. (10.9) into Eqs. (2.7)–(2.12). 10.3.2.1 Gradient The gradient operator is

rD

eµ (µ 2

C

ν 2 )1/2

eν @ @ @ C C ez . 1/2 @ν 2 2 @µ @z (µ C ν )

(10.10)

10.3.2.2 Divergence The divergence of a vector field V is

rV D

 i i @V   1 @ h
2 @ h
2 z 2 1/2 2 1/2 C C µ µ . C ν V C ν V µ ν (µ 2 C ν 2 ) @µ @ν @z (10.11)

10.3.2.3 Circulation The circulation is

ˇ
ˇ µ 2 C ν 2 1/2 e µ ˇ 1 ˇ @ rV D 2 ˇ @µ 2 (µ C ν ) ˇ
2  ˇ µ C ν 2 1/2 Vµ





µ2 C ν2

@ @ν  2 1/2

µ Cν 2

1/2

ev Vv

ˇ e z ˇˇ @ ˇ @z ˇˇ . Vz ˇ

(10.12)

10.3.2.4 Laplacian The Laplacian is

r2 D

  2 @2 1 @2 @ C 2 . C 2 2 2 2 (µ C ν ) @µ @ν @z

(10.13)

10.4 Separable Equations

10.3.3 Stäckel Matrix

From Section 2.4.2.1, a possible Stäckel matrix is found to be 0

0 Φ D@ 0 1


1 1 0

1
µ 2
ν 2 A . 1

(10.14)

The f functions are fi D 1 .

(10.15)

10.4 Separable Equations 10.4.1 Laplace Equation

The Laplace equation is    2  @2 @2 @ 1 2 r ψD C 2 C 2 ψ(µ, ν, z) D 0 . (µ 2 C ν 2 ) @µ 2 @ν @z

(10.16)

Let ψ(µ, ν, z) D M(µ)N(ν)Z(z) .

(10.17)

Then,

(µ 2

1 1 d2 M 1 d2 N 1 1 d2 Z C 2 C D0, 2 2 2 2 (µ C ν ) N dν C ν ) M dµ Z dz 2

(10.18)

and the separated equations are d2 Z(z) C k32 Z(z) D 0 , dz 2

(10.19)

d2 M(µ) 2
k2 C k32 µ 2 M(µ) D 0 , dµ 2

(10.20)

d2 N(ν) 2 C k2
k32 ν 2 N(ν) D 0 . dν 2

(10.21)

In a slightly different notation, the last two equations are referred to as the Weber equations (Appendix H). Specifically, if one makes the substitutions   1 k22 D q 2 p C , 2

k32 D

q4 , 4

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10 Parabolic Cylinder Coordinates

then, for example, Eq. (10.21) becomes     1 q4 ν2 d2 N(ν) 2 C q pC
N(ν) D 0 , dν 2 2 4

(10.22)

which was denoted the Weber equation by Moon and Spencer [6] and the solutions are known as the Weber functions: N(ν) D W(p, q ν) .

(10.23)

As usual, there are two independent Weber function solutions for N corresponding to specific p and q values; they are even and odd. Hence, the general solution to Eq. (10.21) can be written as N(ν) D AWe (p, q ν) C B Wo (p, q ν) .

(10.24)

Similarly, for the other equation, the general solution is M(µ) D AWe (p, i q µ) C B Wo (p, i q µ) .

(10.25)

In Appendix H, it is shown that the Weber equation can be expressed in terms of the confluent hypergeometric functions. Thus, using Eq. (H9), one can also write N(ν) D e


q2 ν2 4

  p 1 q2 ν2 . M
, , 2 2 2

(10.26)

For arbitrary separation constants, the Laplace equation has an infinite number of solutions. 10.4.2 Helmholtz Equation

The Helmholtz equation is 

1 (µ 2 C ν 2 )



@2 @2 C @µ 2 @ν 2



 @2 C 2 ψ C k2 ψ D 0 . @z

(10.27)

Let ψ(µ, ν, z) D M(µ)N(ν)Z(z) .

(10.28)

One can use the Stäckel theory to give the separated equations directly:  d2 M

α2 C α3 µ2 M D 0 , 2 dµ

(10.29)


d2 N C α2
α3 ν2 N D 0 , dν 2

(10.30)

10.5 Applications


 d2 Z C k 2 C α3 Z D 0 . 2 dz

(10.31)

There are two separation constants, α 2 and α 3 . For a boundary-value problem, they have to be determined self-consistently in the first two equations. Those two equations are, again, Weber equations and the solutions were discussed in Section 10.4.1. 10.4.3 Schrödinger Equation

The separable potential is   „2 2 (v (µ) C v (ν)) C v (z) . V(µ, ν, z) D 1 2 3 2m (µ 2 C ν 2 ) The Schrödinger equation becomes     2 1 @2 @ „2 @2
C 2 C 2 ψ 2m (µ 2 C ν 2 ) @µ 2 @ν @z   2 „ 2 (v C (µ) C v (ν)) C v (z) ψ D E ψ. 1 2 3 2m (µ 2 C ν 2 )

(10.32)

(10.33)

Let ψ(µ, ν, z) D M(µ)N(ν)Z(z) . Then,

(10.34)

" " # # 1 d2 M 1 d2 N 1 1

C 2v1 (µ) C 2 C 2v2 (ν) (µ 2 C ν 2 ) (µ C ν 2 ) M dµ 2 N dν 2 " # 1 d2 Z C
C v3 (z) D k 2 , (10.35) Z dz 2

and the separated equations are

d2 Z(z) 2 C k3
v3 (z) Z(z) D 0 , dz 2

(10.36)



d2 M(µ) 2
2 C k2 C k
k32 µ 2
2v1 (µ) M(µ) D 0 , dµ 2

(10.37)



d2 N(ν) 2
2
k2
k
k32 ν 2 C 2v2 (ν) N(ν) D 0 . dν 2

(10.38)

10.5 Applications

A few applications of the parabolic cylinder coordinate system in the published literature include the Dirichlet problem in Lebedev [17], and the study of acous-

115

116

10 Parabolic Cylinder Coordinates

tic eigenmodes in parabolic cylindrical enclosures [46] and of electron states in parabolic cylinder quantum dots [47]. 10.5.1 Acoustics: Neumann Problem for a Cavity

The problem to be solved consists of the Helmholtz equation in an acoustic enclosure (Figure 10.2) subject to rigid-wall boundary conditions. We repeat the solution that was published in the Journal of Sound and Vibration [47]. Consider first the differential equation in M, Eq. (10.29):  d2 M

α2 C α3 µ2 M D 0 . dµ 2 It can be converted into a Weber equation (Appendix H) if α 2 and α 3 are given explicit forms. Here, we rely on the series-solution method instead to illustrate the Frobenius method. Since there are no singular points in the domain, two independent and well-behaved series solutions exist for each separated equation. Consider the differential equation in M first and expand M about µ D 0: M(µ) D

1 X

a n µ nCk .

(10.39)

nD0

Inserting Eq. (10.39) in Eq. (10.29) and equating terms proportional to µ k
2 and µ k
1 gives k(k
1)a 0 D 0 ,

(10.40)

k(k C 1)a 1 D 0 .

(10.41)

Thus, k D 0 or k D 1 (consistent with µ D 0 being an ordinary point). The two choices for k will give us two independent series solutions. Applying the identity theorem for power series to terms µ kCn , where n D 0, 1, 2, . . ., gives the recursion formula: α2 , a3 D 0 , 2 α 2 a nC2 C α 3 a n , D (n C 4)(n C 3)

a2 D a nC4

where n D 0, 1, 2, 3, . . . ,

(10.42)

if k D 0. This solution M(µ)  M1 (µ) is even in µ. Similarly, when k D 1, the recursion formula is b0 D 1 , b nC4 D

b1 D 0 ,

b2 D

α 2 b nC2 C α 3 b n , (n C 5)(n C 4)

α2 , 6

b3 D 0 ,

where n D 0, 1, 2, 3, . . . ,

(10.43)

10.5 Applications

and the solution M(µ)  M2 (µ) is an odd function of µ. The general solution to Eq. (10.29) is M(µ) D AM1 (µ) C B M2 (µ) ,

(10.44)

where A and B are arbitrary constants, and M1 (µ) D M1 (α 2 , α 3 I µ) D M2 (µ) D M2 (α 2 , α 3 I µ) D

1 X nD0 1 X

a 2n µ 2n ,

(10.45)

b 2n µ 2nC1 .

(10.46)

nD0

Consider next the differential equation in N, Eq. (10.30), which can be obtained by replacing α 2 by
α 2 . Thus, N(ν) D C N1 (ν) C D N2 (ν) ,

(10.47)

where C and D are arbitrary constants, and N1 (ν) D N1 (α 2 , α 3 I ν) D N2 (ν) D N2 (α 2 , α 3 I ν) D

1 X nD0 1 X

c 2n ν 2n ,

(10.48)

d2n ν 2nC1 .

(10.49)

nD0

The coefficients c n satisfy the recurrence relations c0 D 1 , c nC4 D

c1 D 0 ,

c2 D


α 2 c nC2 C α 3 c n (n C 4)(n C 3)


α 2 , 2

c3 D 0 ,

where n D 0, 1, 2, 3, . . .

(10.50)

For d n , we obtain d0 D 1 , d nC4 D

d1 D 0 ,

d2 D



α 2 d nC2 C α 3 d n , (n C 5)(n C 4)

α2 , 6

d3 D 0 ,

where n D 0, 1, 2, 3, . . .

Finally, the differential equation in Z has the simple general solution hp i hp i Z(z) D E sin k 2 C α 3 z C F cos k 2 C α3 z ,

(10.51)

(10.52)

where E and F are arbitrary constants. Given the general solutions, the next step is to apply the boundary conditions. The rigid-wall boundary condition becomes dM (α 2 , α 3 I µ 0 ) D 0 , dµ

(10.53)

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10 Parabolic Cylinder Coordinates

dN dN (α 2 , α 3 I ν 0 ) D (α 2 , α 3 I
ν 0 ) D 0 , dν dν dZ dZ (k, α 3 I z1¸) D (k, α 3 I z2 ) D 0 , dz dz

(10.54) (10.55)

where (µ 0 , ν 0 , z1 , z2 ) defines the acoustic enclosure boundary. It is an interesting observation that Eqs. (10.53)–(10.55), together with normalization, are not sufficient to solve the eigenvalue problem. This differs from the procedure for solving boundary-value problems in commonly known coordinate systems such as rectangular and spherical (the same difficulty, however, arises for the elliptic system). This is due to the vanishing of the Jacobian for the transformation to parabolic cylinder coordinates. It was shown by Lebedev [17] that one also requires that r P be finite everywhere inside the cavity. Since "   2 #  2 @P 2 @P 1 @P 2 C C , (10.56) (r P ) D 2 2 µ Cν @µ @ν @z this leads to "   2 # @P 2 @P C @µ @ν

D0.

(10.57)

µDνD0

Therefore, for the Helmholtz boundary-value problem in parabolic cylinder coordinates, eigenmodes are found by imposing the four conditions given by Eqs. (10.53)–(10.55) and (10.57). For a detailed discussion of the method of solution, one needs to explain how to evaluate Eq. (10.57) in terms of M and N. Combining the general solution (Eq. (10.47)) with Eq. (10.54) and making use of the fact that N1 and N2 are even and odd, respectively, allows us to write dN dN2 dN (α 2 , α 3 I ν 0 ) C (α 2 , α 3 I
ν 0 ) D 2D (ν 0 ) D 0 . dν dν dν

(10.58)

This condition can be fulfilled if either D D0

(10.59)

dN2 (ν 0 ) D 0 . dν

(10.60)

or

However, if D D 0, then C ¤ 0 for P to be nontrivial (different from zero). Thus, dN1 /dν(ν 0 ) D 0 so as to satisfy Eq. (10.54). Instead, if dN2 /dν(ν 0 ) D 0, we obtain dN/dν(ν 0 ) D C dN1 /dν(ν 0 ) D 0, leaving two possibilities open: C D 0 or dN1 /dν(ν 0 ) D 0. The condition in Eq. (10.57), therefore, can be restated as B 2 C 2 C A2 D 2 D 0 ,

(10.61)

10.5 Applications

since N(0) D C , M(0) D A ,  2 ˇˇ dM ˇ D B2 , ˇ dµ ˇ µD0  2 ˇˇ dN ˇ D D2 . ˇ dν ˇ νD0

One should now consider separately the three possible cases: (a) C ¤ 0, D D 0; (b) D ¤ 0, C D 0; and (c) C ¤ 0, D ¤ 0. 10.5.1.1 Case (a) C ¤ 0 and D D 0. Equation (10.60) implies dN1 /dν(ν 0 ) D 0. The condition that r P is finite at µ D ν D 0 gives B D 0 (refer to Eq. (10.61)). If B D 0, then A ¤ 0 and dM1 /dµ(µ 0 ) D 0 so as to satisfy Eq. (10.53). Thus, case (a) requires

dN1 (ν 0 ) D 0 , dν

dM1 (µ 0 ) D 0 . dµ

(10.62)

10.5.1.2 Case (b) D ¤ 0 and C D 0. Thus, dN2 /dν(ν 0 ) D 0. The condition that r P is finite at µ D ν D 0 now gives A D 0 (refer to Eq. (10.61)). If A D 0, then B ¤ 0 and dM2 /dµ(µ 0 ) D 0 so as to satisfy Eq. (10.53). Thus, case (b) requires

dN2 (ν 0 ) D 0 , dν

dM2 (µ 0 ) D 0 . dµ

(10.63)

10.5.1.3 Case (c) It follows immediately from the discussion following Eqs. (10.59) and (10.60) that when C ¤ 0, D ¤ 0, the two conditions

dN1 (ν 0 ) D 0 , dν

dN2 (ν 0 ) D 0 dν

(10.64)

must be imposed. If dM1 /dµ(µ 0 ) D 0 by accident, then dM2 /dµ(µ 0 ) D 0 so as to satisfy Eqs. (10.53) and (10.61) (the latter forces B D 0 if A D 0, C ¤ 0, and D ¤ 0, which is not possible for a nontrivial solution). Similarly, if dM2 /dµ(µ 0 ) D 0 by accident, then dM1 /dµ(µ 0 ) D 0. These two special cases therefore require four conditions to be fulfilled: dM2 dN1 dN2 dM1 (µ 0 ) D (µ 0 ) D (ν 0 ) D (ν 0 ) D 0 . dµ dµ dν dν If instead both dM1 /dµ(µ 0 ) ¤ 0 and dM2 /dµ(µ 0 ) ¤ 0, it is always possible to find nonzero (complex) coefficients A and B such that Eqs. (10.53) and (10.57) are satisfied simultaneously. In the latter case, the two conditions given by Eq. (10.64) are the only conditions that must be satisfied for an eigenstate to be found.

119

120

10 Parabolic Cylinder Coordinates

10.5.1.4 Relation between k and α 3 Finally, consider next the Z equation (Eq. (10.52)) and the associated boundary conditions (Eq. (10.55)). Two possible cases can occur if we choose the origin of the z axis such that z2 D
z1 D L/2: E D 0 or F D 0. If E D 0, then the following relation between k and α 3 must hold:

k 2 D (2p )2

 π 2 L


α3 ,

where p D 0, 1, 2, 3, . . .

(10.65)

If, instead, F D 0, we find k 2 D
α 3 ,

or  π 2 k 2 D (2p C 1)2
α3 , L

where p D 0, 1, 2, 3, . . .

(10.66)

The eigenfrequencies are given by f D

ck . 2π

(10.67)

One procedure for determining eigenstates is as follows. For each case (a)–(c), solution sets (α 2 , α 3 ) are found by solving Eqs. (10.62), (10.63), and (10.64), respectively. The values found for α 3 can then be inserted in Eqs. (10.65) and (10.66) to get k 2 . 10.5.1.5 Results The results for an example symmetric acoustic enclosure with µ 20 D 1m, ν 20 D 1m is given to illustrate the numerical approach to finding the separation constants. The symmetric acoustic enclosure is obtained when µ 0 D ν 0 , whereas asymmetric enclosures correspond to any acoustic enclosure with parameters µ 0 ¤ ν 0 . In searching for frequencies, we started with α 2 and scanned in α 3 for zeros of dM1 /dµ and dN1 /dν, dM2 /dµ and dN2 /dν, and dN1 /dν and dN2 /dν, corresponding to cases (a), (b), and (c), respectively. This technique is illustrated in Figure 10.3 for the symmetric acoustic enclosure. In Figure 10.3a, (α 2 , α 3 ) values are plotted where dM1 /dµ and dN1 /dν have zeros corresponding to case (a). Line codings for zeros of dM1 /dµ and dN1 /dν are squares and diamonds, respectively. The simultaneous zeros are the intersection points in an (α 2 , α 3 ) plot; the solutions are even about y D 0. In a similar way, Figure 10.3b shows the (α 2 , α 3 ) values where dM2 /dµ and dN2 /dν are zero corresponding to case (b). Again, the simultaneous zeros are the intersection points in an (α 2 , α 3 ) plot; the solutions are odd about y D 0. In Figure 10.3c, zeros of dN1 /dν and dN2 /dν are plotted as a function of (α 2 , α 3 ), corresponding to case (c). Evidently, no simultaneous zeros are found (a close inspection of the curves shows that the dN1 /dν and dN2 /dν curves do not intersect). Thus, there will be no eigenstates of the type considered in case (c). This is to be expected on the basis of symmetry since the solutions must have definite parity about y D 0. As mentioned earlier, it follows from Eqs. (10.29) and (10.30) that

M1 (α 2 , α 3 I γ ) D N1 (
α 2 , α 3 I γ ) ,

(10.68)

0

−200

−200

−400

−400

−600

−600

β [m−2 ]

0

−2

β [m ]

10.5 Applications

−800

−800

−1000

−1000

−1200

−1200 −1400

−1400 −60

−40

−20

0

20

40

60

−60

α [m−1]

(a)

−40

−20

0 α [m−1]

20

40

(b) 0 −200 −400

β [m−2 ]

121

−600 −800 −1000 −1200 −1400 −60

−40

−20

0

20

40

60

α [m−1]

(c) Figure 10.3 The zeros of dM1 /dµ and dN1 /dν (a), dM2 /dµ and dN2 /dν (b), and dN1 /dν and dN2 /dν (c) for a symmetric acoustic enclosure.

M2 (α 2 , α 3 I γ ) D N2 (
α 2 , α 3 I γ )

(10.69)

for any value γ . Thus, M1 (α 2 , α 3 I γ ) N1 (α 2 , α 3 I γ ) D N1 (
α 2 , α 3 I γ ) M1 (
α 2 , α 3 I γ ) ,

(10.70)

M2 (α 2 , α 3 I γ ) N2 (α 2 , α 3 I γ ) D N2 (
α 2 , α 3 I γ ) M2 (
α 2 , α 3 I γ ) .

(10.71)

The latter relation applied to the case of a symmetric acoustic enclosure (γ D µ 0 D ν 0 ) shows that if (α 2 , α 3 ) is a simultaneous zero point for dM1 /dµ and dN1 /dν (or dM2 /dµ and dN2 /dν), then (
α 2 , α 3 ) is also a simultaneous zero point for dM1 /dµ and dN1 /dν (or dM2 /dµ and dN2 /dν). Applied to an asymmetric acoustic enclosure, it shows that the (µ 0 , ν 0 ) dot has the same frequency spectrum as the (ν 0 , µ 0 ) dot, as expected from the mirror-image shapes. Examples of eigenmodes for the symmetric acoustic enclosure are given in Figure 10.4. A trivial solution to the Helmholtz equation with Neumann boundary

60

122

10 Parabolic Cylinder Coordinates

conditions always exists. This is the solution corresponding to case (a) with α 2 D α 3 D 0. In this case, it follows from Eqs. (10.45) and (10.48) and the recursive relations Eqs. (10.42) and (10.50) that M1 (µ) and N1 (ν) are both constant functions and so Neumann boundary conditions are trivially satisfied. Notice also that in this particular case (with α 2 D α 3 D 0), it follows that f Dp

c , 2L

where p D 0, 1, 2, 3, . . .

(10.72)

The first nontrivial solution (with the smallest absolute value of α 3 excluding the trivial solution) for the symmetric acoustic enclosure occurs in case (b) with parameters (α 2 , α 3 ) D (0,
4.4817) found by close inspection of the results shown in Figure 10.3b. The degeneracy of this eigenfrequency is 1 and the corresponding eigenmode is shown in Figure 10.4a. The second and third eigenmodes are 3

0.6

2.5 0.4

0.8

4

2

3

0.6 0.4

0.2

0.2

1.5

2 1

1

0 −0.2

0.5

0

0

−1

−0.4 −0.2

−0.6

0

−2

−0.5

−3 1.5

−0.8 1.5 1

−0.4

1

0.5

−0.5

−1 −1.5

y

−0.6

(a)

−1.5

0

−0.5

−2

−0.5

−1 −1.5

x

−1

0.5

0

0

−0.5

1

0.5

0.5

0

y

−1

1

−1

x

(b) 3 2.5

4

2

3

1.5

2 1

1

0.5

0 −1

0

−2

−0.5

−3 1.5

−1

1

1

0.5

0.5

0

y

−1.5

0

−0.5

−2

−0.5

−1 −1.5

−1

x

(c) Figure 10.4 The first three nontrivial eigenmodes for a symmetric acoustic enclosure: (a) the ground state (α 2 , α 3 ) D (0,
4.4817), and (b) the second and (c) the third eigenmodes with parameters (α 2 , α 3 ) D (˙6.037,
13.47).

10.5 Applications

degenerate states (i.e., they have the same α 3 value) corresponding to case (a) with parameters (α 2 , α 3 ) D (6.037,
13.47) and (α 2 , α 3 ) D (
6.037,
13.47), respectively. The two eigenmodes are shown in Figure 10.4b,c. The symmetry properties of the eigenmodes can be easily obtained. For instance, states corresponding to case (a) where B D D D 0 is imposed satisfy ψ(x, y, z) D ψ(µ, ν, z) D M1 (µ)N1(ν)Z(z) D M1 (µ)N1(
ν)Z(z) D ψ(µ,
ν, z) D ψ(x,
y, z) ,

(10.73)

and (if α 2 D 0) ψ(x, y, z) D ψ(µ, ν, z) D M1 (µ)N1(ν)Z(z) D N1 (µ)M1 (ν)Z(z) D ψ(ν, µ, z) D ψ(
x, y, z) ,

(10.74)

where use has been made of Eq. (10.68) in obtaining the third equality. Similarly, states corresponding to case (b) for which A D C D 0 are antisymmetric with respect to mirror reflections in the y D 0 plane, because ψ(x, y, z) D ψ(µ, ν, z) D M2 (µ)N2(ν)Z(z) D
M2 (µ)N2(
ν)Z(z) D
ψ(µ,
ν, z) D
ψ(x,
y, z) . (10.75) In addition, if α 2 D 0, states corresponding to case (b) will be symmetric with respect to reflections in the x D 0 plane: ψ(x, y, z) D ψ(
x, y, z) ,

(10.76)

following steps analogous to those used in deriving Eq. (10.74). Thus, the first nontrivial solution is symmetric (antisymmetric) with respect to a mirror reflection in the x D 0 (y D 0) plane since (α 2 , α 3 ) D (0,
4.4817) as this state belongs to case (b), in agreement with Figure 10.4a. The second and third nontrivial solutions have α 2 values different from zero (and belong to case (a)). These solutions are therefore symmetric with respect to a mirror reflection in the y D 0 plane but they show no symmetry with respect to a mirror reflection in the x D 0 plane. This is in agreement with Figure 10.4b,c.

123

124

10 Parabolic Cylinder Coordinates

10.6 Problems

1. Derive the Stäckel matrix given in the text. 2. For the M separated equation from the Laplace equation, if k22 D 0, k32 D
q, show that the general solution is  M(µ) D µ

1/2

 AJ1 4

q µ2 2



 C B J
1 4

q µ2 2

 ,

where J ν (x) are the Bessel functions. 3. Show that the Weber equation, Eq. (10.22), becomes the Hermite equation if 2 2 z D e
q ν /2 N(ν). 4. Derive Eq. (10.56) and explain why r P must be finite everywhere. 5. Derive Eqs. (10.61)–(10.64). Assuming z2 D
z1 , derive the eigenfrequency solutions Eqs. (10.65) and (10.66). 6. For the acoustics problem considered in the text, replace the Dirichlet boundary conditions by Neumann boundary conditions at the boundaries z1 and z2 . Find the eigenmodes and associated eigenfrequencies. 7. Determine the first two forward-propagating propagating-wave solutions ψ i (i D 1, 2) for the problem of an acoustics waveguide symmetric in the y–z plane, that is, consider the z range to be ]
1, 1[ (the first two solutions refer to those with the smallest absolute values of α 3 ). Assume Z(0) D 1 and find the solutions ψ1 and ψ2 at z D 2 m. 8. Assume a piston is forced to vibrate at the open end (z D z2 ) of a parabolic cylinder semi-infinite waveguide with frequency ω and an acoustic pressure amplitude of 1 bar. a. If the α i coefficient of the third-order (and higher-order) modes satisfies α i <
ω 2 /c 2 (i D 3, 4, . . .), determine an approximate solution for the vibration at z D 2π10c/ω. b. Assume that the piston spans half of the waveguide opening (say, x  0) and the other half (x < 0) is a free end. Provide an analytical method for determining an approximate solution everywhere in the waveguide.

125

11 Spherical Polar Coordinates 11.1 Introduction

This is the third of the three well-known coordinate systems. The Laplace equation leads to the associated Legendre equation and the Helmholtz equation, in addition, has the spherical Bessel equation as one of the other ordinary differential equations. For the application, we place emphasis on the study of boundary-value problems that are not completely spherically symmetric but where the boundary surfaces are appropriately described in spherical polar coordinates. An important result is the presence of associated Legendre functions (as opposed to the betterknown polynomials).

11.2 Coordinate System 11.2.1 Coordinates (r, θ , φ)

The coordinates are ξ1 D r, ξ2 D θ , ξ3 D φ and the relationship to the Cartesian coordinates is x D r sin θ cos φ , y D r sin θ sin φ , z D r cos θ ,

(11.1)

with 0
r<1,

0
θ
π,

0
φ < 2π .

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

126

11 Spherical Polar Coordinates z θ =θ 0 φ = φ0 y

r = r0

x

Figure 11.1 Spherical polar coordinates.

11.2.2 Constant-Coordinate Surfaces

The orthogonal surfaces are spheres (r D constant), circular cones (θ D constant), and half planes (φ D constant) passing through the z axis (Figure 11.1).

11.3 Differential Operators 11.3.1 Metric

Using Eqs. (2.5) and (11.1), one obtains the scale factors as hr D 1 ,

hθ D r ,

h φ D r sin θ .

(11.2)

11.3.2 Operators

These are obtained by substituting Eq. (11.2) into Eqs. (2.7)–(2.12). 11.3.2.1 Gradient The gradient operator is

r D er

eφ eθ @ @ @ C C . @r r @θ r sin θ @φ

(11.3)

11.4 Separable Equations

11.3.2.2 Divergence The divergence of a vector field V is

rV D

1 1 @
2  1 @Vφ @ (sin θ Vθ ) C r Vr C . r 2 @r r sin θ @θ r sin θ @φ

11.3.2.3 Circulation The circulation is

ˇ ˇe ˇ r 1 ˇ@ rV D 2 ˇ r sin θ ˇ @r ˇVr

r eθ @ @θ

r Vθ

(11.4)

ˇ r sin θ e φ ˇˇ ˇ @ ˇ . @φ ˇ r sin θ Vφ ˇ

(11.5)

11.3.2.4 Laplacian The Laplacian is

r2 D

      @ @ @ 1 @2 1 2 @ . sin θ r C sin θ C r 2 sin θ @r @r @θ @θ sin θ @φ 2 (11.6)

11.3.3 Stäckel Matrix

From Section 2.4.2.1, a possible Stäckel matrix is found to be 1 0 0 1 1/r 2 2 Φ D @0 1 1/ sin θ A . 0 0 1

(11.7)

The f functions are f1 D r2 ,

f 2 D sin θ ,

f3 D 1 .

(11.8)

11.4 Separable Equations 11.4.1 Laplace Equation

The Laplace equation is       1 @ 1 @2 @ @ 1 @ ψD0. r2 ψ D 2 r2 C 2 sin θ C r @r @r r sin θ @θ @θ r 2 sin2 θ @φ 2 (11.9) Let ψ(r, θ , φ) D R(r)Θ (θ )Φ (φ) .

(11.10)

127

128

11 Spherical Polar Coordinates

Using the Stäckel theory, one obtains d2 Φ C k32 Φ D 0 , dφ 2     dΘ k32 1 d Θ D0, sin θ C k22  sin θ dθ dθ sin2 θ   d dR r2  k22 R D 0 . dr dr

(11.11) (11.12) (11.13)

Equation (11.12) is the associated Legendre equation, which has regular singular points at x D cos θ D 1, 1, 1 and has the Bôcher type f222g. If k32 D m 2 is positive and if one lets k22 D l(l C 1), then the general solutions can be written as Φ (φ) D A sin m φ C B cos m φ ,

(11.14)

Θ (θ ) D AP lm (cos θ ) C B Q m l (cos θ ) ,

(11.15)

R(r) D Ar l C B r
(lC1) .

(11.16)

The functions P lm (cos θ ) and Q m l (cos θ ) are known as the associated Legendre functions of the first and the second kind, respectively. Note that Q m l (x) blows up as x ! 1. In general, the separation constants can be complex. Nevertheless, they can be restricted to Rem  0 and Rel  1/2 since the opposite conditions leave m 2 and l(l C 1), respectively, unchanged. 11.4.2 Helmholtz Equation

The Helmholtz equation is 

1 @ r 2 @r



@ r @r 2



1 @ C 2 r sin θ @θ



@ sin θ @θ



 1 @2 ψ D k 2 ψ . C r 2 sin2 θ @φ 2 (11.17)

Let ψ(r, θ , φ) D R(r)Θ (θ )Φ (φ) .

(11.18)

Using the Stäckel theory, one obtains d2 Φ C k32 Φ D 0 , dφ 2     k32 dΘ 1 d Θ D0, sin θ C k22  sin θ dθ dθ sin2 θ

(11.19) (11.20)

11.4 Separable Equations

d dr

 r2

dR dr



  C k 2 r 2  k22 R D 0 .

(11.21)

The last two equations are the associated Legendre and Bessel equations, respectively. If k32 D m 2 is positive and if one rewrites k22 D l(l C 1), then the general solutions can be written as Φ (φ) D A sin m φ C B cos m φ ,

(11.22)

Θ (θ ) D AP lm (cos θ ) C B Q m l (cos θ ) ,

(11.23)

  R(r) D r
1/2 A J lC1/2(k r) C B J
(lC1/2)(k r) .

(11.24)

11.4.3 Schrödinger Equation

The potential is separable if   „2 1 1 V(r, θ , φ) D v1 (r) C 2 v2 (θ ) C (φ) . v 3 2m r r 2 sin2 θ

(11.25)

The Schrödinger equation becomes       1 @ 1 „2 1 @ @2 @ 2 @ ψ(r, θ , φ) r C sin θ C 2m r 2 @r @r r 2 sin θ @θ @θ r 2 sin2 θ @φ 2   „2 1 1 C v1 (r) C 2 v2 (θ ) C 2 2 v3 (φ) ψ(r, θ , φ) D E ψ(r, θ , φ) . 2m r r sin θ (11.26) 

Let ψ(r, θ , φ) D R(r)Θ (θ )Φ (φ) . Then,

 

1 d r 2 R dr "

C 

 r2

dR dr



  C v1 (r) C 

(11.27)

d 1 r 2 sin θ Θ dθ #

1 v3 (φ) d2 Φ C 2 r 2 sin θ Φ dφ 2 r 2 sin2 θ

 sin θ

D k 2 ,

dΘ dθ

 C

1 v2 (θ ) r2



(11.28)

and the separated equations are   d2 Φ C k32  v3 (φ) Φ D 0 , dφ 2     k32 dΘ 1 d Θ D0, sin θ C k22  v2 (θ )  sin θ dθ dθ sin2 θ    
 d dR r2 C k 2  v1 (r) r 2  k22 R D 0 . dr dr

(11.29) (11.30) (11.31)

129

130

11 Spherical Polar Coordinates

11.5 Applications 11.5.1 Quantum Mechanics: Dirichlet Problem

We will consider the problem of solving the Schrödinger equation inside the conical section of a sphere; we call this the quantum ice cream (QIC) problem [48]. The potential function is V(r) D V(θ , r) D V0 H(θ  θm )H(r  R) ,

(11.32)

where H is the Heaviside function. Inside the cone, the potential is zero and the Schrödinger equation formally becomes equivalent to the Helmholtz equation. We can use the separated equations, Eqs. (11.19)–(11.21). For the Φ equation, let k32 D m 2 , m 2 ZC since periodicity is required, and the solution is Φ (φ) D e ˙i m φ .

(11.33)

Next, in Eq. (11.20), one can set k22 D ν(ν C 1). The associated Legendre equation for the θ equation becomes     1 d dΘ m2 Θ D0. (11.34) sin θ C ν(ν C 1)  sin θ dθ dθ sin2 θ The solutions are the associated Legendre functions: Θ (θ ) D P νm (cos θ ). Finally, the radial equation is   d dR r2 C r 2 k 2 R  ν(ν C 1)R D 0 . dr dr

(11.35)

Let R(r) D Z(r)/r 1/2 . Then, dR Z0 1 Z D 1/2  , dr r 2 r 3/2 and

"

d2 R Z 00 Z0 3 Z D  C , dr 2 r 1/2 r 3/2 4 r 5/2 

1 r Z C rZ C k r  ν C 2 2

00

0

2 #

2 2

Z D0.

(11.36)

This is the Bessel equation. We have Z(k r) D J νC1/2 (k r) , )

R(r) D

J νC1/2 (k r)  j ν (k r) , (k r)1/2

the spherical Bessel function.

(11.37) (11.38)

11.5 Applications

The general solution to the Helmholtz problem inside the QIC is, therefore, X ψ(r) D a ν m n j ν (k n ν r) P νm (cos θ )e i m φ . (11.39) νmn

Note that the theory developed, so far, applies to the spherical dot as well as QIC. We, therefore, consider the latter case first before the QIC. 11.5.1.1 Infinite-Barrier Spherical Dot For a problem with spherical symmetry, the requirement for the finiteness of the associated Legendre functions at θ D 0, 180ı leads to ν D l 2 N0 . The boundary condition is

ψ(R , θ , φ) D 08θ , φ

H)

j l (k n R) D 0 .

(11.40)

Thus, for each l, one needs to find the nth root α n l with „2 α 2n l „2 k 2 D . 2m 2m R 2 Example: n D l D 0, R(r)  sin k r/(k r), α 00 D π. k n l D α n l /R

H)

En l D

(11.41)

11.5.1.2 Finite-Barrier Spherical Dot Outside the dot, we have a similar equation:

2m (V  E )ψ(r) D
2 ψ(r) . „2 The corresponding radial equation is   d dR r2  r 2
2 R  l(l C 1)R D 0 . dr dr r 2 ψ(r) D

(11.42)

(11.43)

Note the difference between Eqs. (11.43) and (11.35). The solution to the radial equation is the modified spherical Bessel function: R(r) D k l (
n r) .

(11.44)

The solution which diverges at infinity is excluded. The interfacial condition is

(11.45) ψ (R
, θ , φ) D ψ R C , θ , φ , 8θ , φ , and X lmn

a l m n j l (k n l R
)P lm (cos θ )e i m φ D

X



b l m n k l
n l R C P lm (cos θ )e i m φ ,

lmn

ˇ ˇ X b l m n dk l (
n l r) ˇ X a l m n d j l (k n l r) ˇ ˇ ˇ D . ˇ ˇ m in dr m out dr rDR rDR lmn

lmn

(11.46) The last two conditions apply term by term and can be combined to give ˇ ˇ 1 d ln j l (k n l r) ˇˇ 1 d ln k l (
n l r) ˇˇ D . ˇ ˇ m in dr m out dr rDR rDR

(11.47)

131

132

11 Spherical Polar Coordinates

11.5.1.3 Quantum Ice Cream – Infinite Barrier The quantum dot shapes we will be discussing are shown in Figure 11.2. They are all derived from a spherical dot by further restricting the angular coordinates. There have been a few studies of angular confinement in other areas of physics (e.g., the cone shape in electromagnetism [49, 50]). The potential energy can now be written as

V(r) D V(r, θ , φ) D V0 H (φ  φ 0 ) H (θ  θ0 ) H (r  r0 ) .

(11.48)

The quantum dot boundary is defined by the set of parameters (r0 , θ0 , φ 0 ). We write here the three ordinary differential equations as 1 d2 Φ D µ 2 , Φ dφ 2     dΘ µ2 1 d sin θ C ν(ν C 1)  Θ D0, sin θ dθ dθ sin2 θ   dR d r2 C r 2 k 2 R  ν(ν C 1)R D 0 , dr dr

(11.49) (11.50) (11.51)

with µ and ν as separation constants, and k 2 D 2m  E/„2 . For spherically symmetric problems, µ and ν become integers owing to the imposition on the solutions of periodicity in φ and finiteness at θ D π, respectively. Here, we require neither for, for example, the quantum pyramidal horn (QPH) owing to the restricted φ and θ domains; hence, µ and ν are not necessarily integers. The new requirements (boundary conditions) are Θ (θ0 ) D P νm (cos θ0 ) D 0

(11.52)

for the QIC, sin (µ φ 0 ) D 0

(11.53)

for the quantum apple slice (QAS), and sin (µ φ 0 ) D 0 ,

r0

P νµ (cos θ0 ) D 0

(11.54)

θ0 φ0

(a)

(b)

(c)

Figure 11.2 Quantum dot shapes: quantum ice cream (a), quantum apple slice (b), and quantum pyramidal horn (c).

11.5 Applications

for the QPH. Once the quantum numbers ν and µ have been determined from the above equations, the energy quantization is obtained from the boundary condition on the radial equation: R(r0 ) D j ν (k n ν r0 ) D 0 .

(11.55)

µ

P ν (cos θ ) is known as the associated Legendre function; mathematical properties of these functions can be found in Appendix E. Recall that for the spherical dot, the quantum numbers nl m are restricted to n 2 N , l 2 N , m D l, l C 1, . . . , l  1, l, where N is the set of natural numbers and the level degeneracy is 2l C 1, the degeneracy being a consequence of the full rotational symmetry. One would need to reexamine some of these properties and a few results follow. 11.5.1.4 ν(
µ) D ν(µ) This result can be seen by looking at the differential equations (Eqs. (11.49)– (11.51)). It allows one to only consider a subset of the solutions and is also valid for the conventional problems of spherical dots and the hydrogen atom. Note that it is appropriate to label ν as a function of µ since ν is obtained from Eq. (11.50) and the latter equation has µ as a parameter. 11.5.1.5 E(
µ) D E(µ) Since ν is now a function of µ, the energy can also be considered a function of µ; that is, the (2l C 1) degeneracy of the spherical dot is lowered. However, there is still at least a twofold degeneracy. 11.5.1.6 ν 
1/2 Although ν can take any value, only ν  1/2 are needed since those solutions with ν < 1/2 can be mapped onto those with ν  1/2 by using [51] µ

P
ν
1 (z) D P νµ (z) .

(11.56)

This result has been discussed in other areas [13, 50]. 11.5.1.7 ν  jµj We give the proof for the QIC and QPH shapes first. This more important constraint on the allowed ν values can be obtained by using the representation of the Legendre functions by hypergeometric functions [51] together with the integral representation of the latter [17]:

P νµ (z) D

  z C 1 µ/2 1 Γ (1  µ) z  1   1z ,  F ν, ν C 1I 1  µI 2

jz  1j < 2 ,

(11.57)

133

134

11 Spherical Polar Coordinates

F(α, βI γ I z) D

Γ (γ ) Γ (β)Γ (γ  β) Z1  dt t β
1 (1  t) γ
β
1 (1  t z)
α , 0

Rγ > Rβ > 0 ,

jarg(1  z)j < π ,

z D cos θ . (11.58)

We first require that any solution be finite at θ D 0. From Eq. (11.57), we find
µ that P ν (1), µ > 0 exists. Then, since the gamma function has no zeros for a positive argument, Eq. (11.58) now implies that the hypergeometric function is positive definite if ν < µ. Hence, to satisfy the boundary conditions Eqs. (11.52) and (11.54), one must have ν  µ (µ positive). For the case of the QAS and the spherical dot, the requirement is not that there µ
µ are zeros of P ν (z) but rather that P ν (1) exists (this is the standard requirement of the finiteness of the function for θ D π). Using Eq. (11.57), this requires that the hypergeometric function has a zero for z D 1. Since [13] P ν
µ (1)  F(ν, 1 C νI 1 C µ, 1) D

Γ (1 C µ)Γ (µ) , Γ (1 C µ C ν)Γ (µ  ν)

(11.59)

one finds that there is a zero only if ν D µ C p, p 2 N .

(11.60)

Although this last result is well known for the sphere, this establishes it for the QAS. Note, however, that it does not imply that µ changes in integral steps. Furthermore, the proof indicates that Eq. (11.60) need not hold for the QIC and QPH, a result that is verified by the numerical calculations given below. Additional constraint on the allowed ν values are now shape dependent. 11.5.1.8 Additional Constraints Two additional results for the QIC are

ν ¤ 0 8 θ0 ,

(11.61)

µ D m D 0 only if θ0 ¤ π .

(11.62)

Both results can be derived straightforwardly from Eq. (11.57). We give one example of a numerical result taken from [48]. This is for the QAS structure since it, in general, leads to nonintegral µ and nonintegral ν . The boundary condition given in Eq. (11.53) implies µD

pπ , φ0

p 2 ZC .

(11.63)

11.5 Applications

For an arbitrary φ 0 , µ will be an irrational number. We will discuss a subset of φ 0 values in detail, namely, when φ 0 D t π/s, where s, t 2 ZC . Then, Eq. (11.63) becomes µD

ps . t

(11.64)

Representative allowed values of µ are given in Table 11.1. One can make a number of interesting observations from Table 11.1. First, when φ 0 D π, one recovers the same nl m spectrum as for a spherical dot except for the absence of m D 0 states. The latter result is due to the vanishing boundary condition on the constant φ half plane. Furthermore, for φ 0 D 2π, there are additional fractional-µ states. Indeed, there are many more bound states than for the sphere. The difference between the QAS with φ 0 D 2π and the sphere is that the former has a hardwall half plane, whereas the latter imposes periodicity of the solutions; the latter constraint is a severer one. One also observes that, if one increases t in Eq. (11.64), the number of fractional-µ states between two integer-µ ones also increases. Finally, given Eq. (11.60), there is a ladder of ν values for each µ value. There is thus the possibility of additional degeneracies when ν D µ C p D ν0 D µ0 C p 0 . The chance of this happening increases with the ν value. Since these occurrences can also be changed by changing φ 0 , we refer to this as controllable degeneracies. This is a different result from that for either the one-dimensional quantum well or even the three-dimensional spherical quantum dot, where the change in the confinement length scale (well width and radius, respectively) changes the confinement energies but not the degeneracies. In Figure 11.3, we present a more detailed comparison of the energy spectra of a sphere and a specific QAS. We chose φ 0 D 3π/2. Numerical evaluations of the special functions, when required, were done using the computer routines provided by Zhang and Jin [13]. For all the structures, we set the radius r0 to be 100 Å and the mass m  to be 0.067 m 0 (the effective mass of an electron in GaAs). When ν D l (an integer), the energies are obviously the same. Note, however, that the Table 11.1 Allowed values of µ for the quantum apple slice with various φ 0 [48]. φ0

s

t

µ

π 2π

1 1

1 2

1

2 1

3

5 2









π 3

3 2

4 2

5

1 2

3

1

3

6

9

12

15






2π 3

3

2

3 2

3

9 2

6

15 2

9






4π 3

3

4

3 4

6 4

9 4

3

15 4

18 4






135

136

11 Spherical Polar Coordinates

E

8/3 10/3 2/3 4/3 8/3 2/3

2

8/3

3

4 2

4/3 2 4/3

2/3

2/3

0

1

2

3

4

l

2/3

4/3

5/3

2

7/3

10/3 11/3

4

ν

Figure 11.3 Energy spectra (arbitrary units) of a spherical dot and a 3π/2 angle quantum apple slice as a function of the l and ν values, respectively. The numbers next to the states for the quantum apple slice are the allowed µ values.

degeneracies are not. Indeed, not all the m values for the sphere are present for the QAS. For example, for (n, ν) D (1, 4) the only allowed µ values for the QAS are ˙2, ˙4, whereas all integer m between 4 and +4 are allowed for the sphere. Furthermore, for this choice of the QAS, there are fractional-µ states. If φ 0 D π/s, all the QAS states have only integral µ values, but the latter is only a subset of those for a sphere. Hence, this provides a means of tailoring the energy spectrum to be a subset of that for a sphere. In Figure 11.4, we plot the first two wave functions for the QAS as contour plots in the z x half plane. All the states are symmetric about the z D 0 plane. Both states have a similar φ dependence (since µ is the same) but the second state has a node in the z D 0 plane owing to the higher ν value. 100

100

80

z

60 40 20

60 40 20 0

0

x

x

(a)

0.1200 0.1050 0.09000 0.07500 0.06000 0.04500 0.03000 0.01500 0

80

z

0.5500 0.4812 0.4125 0.3438 0.2750 0.2063 0.1375 0.06875 0

(b) Figure 11.4 First two wave functions of the 3π/2 angle quantum apple slice: ground state (a) and second state (b). Contour plots in the z x half plane for φ D φ 0 /2 [48].

11.6 Problems

11.6 Problems

1. The coordinate system can be changed by rescaling r to e ξ , where ξ is the new coordinate. Write down the new form of the Laplacian in terms of the new coordinates. 2. The associated Legendre equation is


1  x2

   d2 y µ2 dy y D0. C ν(ν C 1)   2x dx 2 dx 1  x2

a. What is the Bôcher type? b. Show that it can be transformed into a hypergeometric equation by the substitutions
 µ/2 y D x2  1 v,

zD

1 1  x. 2 2

3. a. Show that the series solution for the modified Bessel function k l (r) given by Eqs. (C70) and (C39) satisfies Eq. (11.43) and fulfills the boundary condition limr!1 k l (r) D 0. b. Plot the spherical Bessel function j l (x) for l D 0, 1. 4. What is a spherically symmetric solution to (a) the Laplace equation and (b) the Helmholtz equation? 5. If the potential v1 (r) appearing in Eq. (11.25) is of the type r α , where α is a positive constant, determine a series solution to Eq. (11.31) which is finite at r D 0. 6. Prove that, for the QIC problem, ν ¤ 0 8 θ0 , µ D m D 0 only if θ0 ¤ π .

137

139

12 Prolate Spheroidal Coordinates 12.1 Introduction

A spheroid is a common generalization of a sphere whereby the shape is still described by a one-coordinate surface. The Laplace equation, when separated, gives rise to the associated Legendre equation, whereas the resulting ordinary differential equations for the Helmholtz equation include the spheroidal wave equation.

12.2 Coordinate System 12.2.1 Coordinates (α, β, φ and ξ , η, φ)

The coordinate choice of Moon and Spencer is ξ1 D α, ξ2 D β, ξ3 D φ such that x D f sinh α sin β cos φ , y D f sinh α sin β sin φ , z D f cosh α cos β ,

(12.1)

with 0
α<1,

0
β
π,

0
φ < 2π ,

f >0.

Morse and Feshbach chose, instead, ξ1 D ξ , ξ2 D η, ξ3 D φ with 1/2 1/2

1  η2 cos φ , x D f ξ2  1  1/2

2 2 1/2 1η sin φ , y D f ξ 1 z D f ξη ,

(12.2)

and 1
ξ <1,

1
η
1 ,

0
φ < 2π ,

f >0.

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

140

12 Prolate Spheroidal Coordinates z φ=φ0 α =α 0

x y

β = β0

Figure 12.1 Prolate spheroidal coordinates.

12.2.2 Constant-Coordinate Surfaces

Since x2 2

f 2 sinh α

C

y2 2

f 2 sinh α

C

z2 f 2 cosh2 α

D sin2 β cos2 φCsin2 β sin2 φCcos2 β D 1 ,

constant α surfaces are prolate spheroids (circular and elliptic cross sections, Figure 12.1). As cosh α > sinh α, they are elongated along z. Also, 

x2 f

2 sin2

β



y2 z2 D  sinh2 α cos2 φ  sinh2 α sin2 φ C 2 2 f cos2 β f sin β 2

C cosh2 α D 1 gives 2-sheeted hyperboloids. They have mirror symmetry about the x y plane and for x D y D 0 and β finite, z D ˙ f cos β; that is, a finite intercept. The 2-sheeted nature is due to the two negative signs. Finally, tan φ D y /x , that is, half planes. The prolate spheroids have foci at (0, 0, ˙ f ). The hyperboloids are confocal with the spheroids. p For the Morse–Feshbach coordinates, f ξ D a and f ξ 2  1 D b are the semimajor and semiminor axes of the confocal ellipse obtained by projecting the spheroid on the x–z plane, and the eccentricity is given by e D f /a. The interfoci distance is 2 f . The curve ξ D 1 is a line segment connecting the two foci. The curve η D 1 (η D 1) gives the z  f (z
 f ) line (with x D y D 0).

12.3 Differential Operators

12.3 Differential Operators

Since the method of separation will be implemented using the Morse–Feshbach coordinates, all remaining expressions will be so expressed. We leave the alternative formulation in terms of the Moon–Spencer coordinates as an exercise. 12.3.1 Metric

Using Eqs. (2.5) and (12.2), one obtains the scale factors as "

#1/2

 f 2 1  η 2 sin2 φ ξ 2 f 2 1  η 2 cos2 φ ξ 2 2 2 hξ D C C f η (ξ 2  1) (ξ 2  1) 1/2
2 ξ  η2 D f , (ξ 2  1)1/2 1/2
2 ξ  η2 hη D f , (1  η 2 )1/2 h φ D f (ξ 2  1)1/2 (1  η 2 )1/2 .

(12.3)

12.3.2 Operators

These are obtained by substituting Eq. (12.3) into Eqs. (2.7)–(2.12). 12.3.2.1 Gradient The gradient operator is


r D eξ

1/2 ξ2  1

f (ξ 2  η 2 )1/2


1/2 1  η2 eφ @ @ @ Ce η C . 1/2 @η 1/2 1/2 @φ 2 2 2 2 @ξ (ξ (ξ ) (1 ) f η f  1) η (12.4)

12.3.2.2 Divergence The divergence of a vector field V is

rV D

  1/2 2 1 @ 
2 Vξ ξ  η 2 (ξ  1)1/2 2 2 f (ξ  η ) @ξ 1/2
1/2  @ 
2 1  η2 C Vη ξ  η 2 @η   Vφ (ξ 2  η 2 ) @ . C @φ (ξ 2  1)1/2 (1  η 2 )1/2

(12.5)

141

142

12 Prolate Spheroidal Coordinates

12.3.2.3 Circulation The circulation is

rV D

f (ξ 2

1  η2)

ˇ (ξ 2
η 2 )1/2 ˇ eξ ˇ (ξ 2
1)1/2 ˇ ˇ @ ˇ @ξ ˇ ˇ (ξ 2
η 2 )1/2 ˇ 2 1/2 Vξ (ξ
1)

eη

(ξ 2
η 2 )1/2 (1
η 2 )1/2 @ @η

(ξ 2
η 2 )1/2 (1
η 2 )1/2

Vη

ˇ e φ (ξ 2  1)1/2 (1  η 2 )1/2 ˇˇ ˇ ˇ @ ˇ . @φ ˇ ˇ (ξ 2  1)1/2 (1  η 2 )1/2 Vφ ˇ

(12.6)

12.3.2.4 Laplacian The Laplacian is

r2 D

1 f  η2)

    @ @ @ (ξ 2  η 2 ) @2 @ . (ξ 2  1) C (1  η 2 ) C 2  @ξ @ξ @η @η (ξ  1)(1  η 2 ) @φ 2 (12.7) 2 (ξ 2

12.3.3 Stäckel Matrix

From Section 2.4.2.1, a possible Stäckel matrix is found to be 0 2 2 1 f (ξ  1) 1 1/(ξ 2  1) Φ D @ f 2 (1  η 2 ) 1 1/(1  η 2 ) A . 0 0 1 The f functions are
1/2 , f1 D ξ 2  1

1/2
f 2 D 1  η2 ,

f3 D f .

(12.8)

(12.9)

12.4 Separable Equations 12.4.1 Laplace Equation

Using the Stäckel theory and the Morse–Feshbach choice, r 2 ψ D 0 becomes d2 ψ C k32 ψ D 0 , dφ 2      dR k2 d
2 ξ 1  k22 C 2 3 R D0, (ξ  1) dξ dξ

(12.10) (12.11)

12.4 Separable Equations

     dS k32 d
C k22  S D0. 1  η2 (1  η 2 ) dη dη

(12.12)

Equation (12.12) is seen to be identical to the associated Legendre equation, whereas Eq. (12.11) differs by a sign (and the domain of ξ is, of course, different). Thus, the general solutions can be written as ψ(φ) D Ae i m φ C B e
i m φ ,

(12.13)

R(ξ ) D C P ml (ξ ) C D Q lm (ξ ) ,

(12.14)

S(η) D E P ml (η) C F Q lm (η) ,

(12.15)

where we have written k32 D m 2 and k22 D l(l C 1). 12.4.2 Helmholtz Equation

Again, using the Stäckel theory, the Helmholtz equation, r2 ψ C k2 ψ D 0 , becomes d2 ψ C k32 ψ D 0 , dφ 2     dR k32 d 2 2 2 2 2 (ξ  1)  k2  f k ξ C 2 RD0, dξ dξ (ξ  1)     d dS k32 (1  η 2 ) C k22  f 2 k 2 η 2  S D0. dη dη (1  η 2 )

(12.16) (12.17) (12.18)

The last two equations are the spheroidal wave equations; the first one is known as the radial equation and the second is known as the angular equation. Note that the two equations are identical except for different domains. They also differ from the associated Legendre equation by the k 2 term. One can thus formally write the solutions as ψ(φ) D Ae i m φ C B e
i m φ ,

(12.19)

R(ξ ) D C P ml ( f k, ξ ) C D Q lm ( f k, ξ ) ,

(12.20)

S(η) D E P ml ( f k, η) C F Q lm ( f k, η) ,

(12.21)

where we have written k32 D m 2 and k22 D l(l C 1). The functions P ml (k, ξ ) and Q lm (k, ξ ) are known as spheroidal wave functions.

143

144

12 Prolate Spheroidal Coordinates

12.4.3 Schrödinger Equation

Using the Morse–Feshbach choice,

 „2 v3 (φ) 1 V(ξ , η, φ) D v , (ξ ) C v (η) C 1 2 2m f 2 (ξ 2  η 2 ) (ξ 2  1)(1  η 2 ) ψ(ξ , η, φ) D R(ξ )S(η)Φ (φ) . Then,

   dR 1 d
2 ξ C v1 (ξ )  1 f 2 (ξ 2  η 2 ) R dξ dξ    dS 1 d
 2 2 1  η2 C v2 (η) f (ξ  η 2 ) S dη dη " # 1 d2 Φ 1  C v3 (φ) D k 2 . C 2 2 f (ξ  1) (1  η 2 ) Φ dφ 2

(12.22) (12.23)



(12.24)

Let  Then,

1 d2 Φ C v3 (φ) D k32 . Φ dφ 2

     dS  dR 1 d
2 1 d
ξ 1 C v1 (ξ )  1  η2 C v2 (η) R dξ dξ S dη dη

 k 2 ξ 2  η2 D f 2 k 2 ξ 2  η 2  23 (ξ  1) (1  η 2 )  
 1 1 D f 2 k 2 ξ 2  η 2  k32 C . (ξ 2  1) (1  η 2 )

(12.25)



(12.26)

Therefore, the separated equations are  d2 Φ C k32  v3 (φ) Φ D 0 , dφ 2     k2 dR d (ξ 2  1)  k22 C v1 (ξ )  f 2 k 2 ξ 2 C 2 3 RD0, dξ dξ (ξ  1)     k32 d dS (1  η 2 ) C k22  v2 (η)  f 2 k 2 η 2  S D0. dη dη (1  η 2 )

(12.27) (12.28) (12.29)

12.5 Applications

There are many textbook examples of the application of spheroidal coordinates. Some of the applications include the gravitational potential of an astronomical body [52, 53], the solution to Maxwell’s equations in electromagnetism [54], and the study of various problems in quantum mechanics [55–57].

12.5 Applications

12.5.1 Dirichlet Problem for the Laplace Equation

The Dirichlet problem for the Laplace equation for a prolate spheroid is so common that a general discussion is first provided before looking at a specific physical application. We already have the ordinary differential equations, Eqs. (12.10)–(12.12), with general solutions Eqs. (12.13)–(12.15). For the boundary condition, consider the special case whereby both the boundary condition and the solution are independent of the angle φ. Then, m D 0 and S(η) satisfies the Legendre equation. For bounded solutions in the whole interval, S(η) D AP l (η) .

(12.30)

Similarly, R(η) D B P l (ξ ) C C Q l (ξ ) .

(12.31)

Now a prolate spheroid is described by ξ D ξ0 and the interior is given by 1
ξ
ξ0 . Thus, inside, when ξ ! 1, we have P l (ξ ) ! 1, Q l (ξ ) ! 1; hence, we must have C D 0. Therefore, for a given l, the solution inside is given by ψ l (ξ , η) D A l P l (ξ )P l (η) .

(12.32)

For the exterior problem, ξ ! 1, and P l (ξ ) ! 1; hence, we must have B D 0, giving the solution outside as ψ l (ξ , η) D C l Q l (ξ )P l (η) .

(12.33)

Consider, then, an interior problem with Dirichlet boundary condition f D f (η) on the surface ξ D ξ0 . If the function f can be expanded in terms of a Legendre series, f (η) D

1 X

f l P l (η) ,

(12.34)

 Z1  1 fl D l C dη f (η)P l (η) , 2

(12.35)

lD0

where


1

then it is easy to see that a solution to the interior problem can be written as ψ l (ξ , η) D

1 X lD0

fl

P l (ξ ) P l (η) . P l (ξ0 )

(12.36)

Since the latter satisfies the Laplace equation and the boundary condition, by the uniqueness theorem, it is the solution.

145

146

12 Prolate Spheroidal Coordinates

12.5.2 Gravitation: Dirichlet–Neumann Problem

A well-known problem is to find the gravitational potential of a solid homogeneous prolate spheroid. Assume that the solid body is characterized by a mass m, density
, and size ξ0 . One has to solve both the interior problem, which is given by the Poisson equation, r 2 ψ i D 4π
,

(12.37)

and the exterior problem, which is given by the Laplace equation, r2 ψe D 0 ,

(12.38)

subject to the following boundary conditions: ˇ ˇ @ψ i ˇˇ @ψ e ˇˇ D , ψ e j1 D 0 . ψ i jS D ψ e jS , @n ˇ S @n ˇ S

(12.39)

We note that the interior problem can be reduced to the Laplace equation, r2 ψ D 0 ,

(12.40)

as well if ψi D ψ C ψP ,

(12.41)

where ψ P is the particular solution of the Poisson equation: r 2 ψ P D 4π
.

(12.42)

Then, from Eqs. (12.32) and (12.33), we have ψ(ξ , η) D

1 X

ψ e (ξ , η) D

A l P l (ξ )P l (η) ,

lD0 1 X

C l Q l (ξ )P l (η) .

(12.43) (12.44)

lD0

Now, a particular solution that satisfies the Poisson equation can be shown to be  2 ψ P D π
(x 2 C y 2 ) D π
f 2 (ξ 2 1)(1η 2) D  π
f 2 (ξ 2 1) P0 (η)  P2 (η) . 3 (12.45) The boundary conditions become A0 


 2 π
f 2 ξ02  1 D C0 Q 0 (ξ0 ) , 3

(12.46)

12.5 Applications

A 2 P2 (ξ0 ) C


 2 π
f 2 ξ02  1 D C2 Q 2 (ξ0 ) , 3

4  π
f 2 ξ0 D C0 Q 0 0 (ξ0 ) , 3 4 A 2 P2 0 (ξ0 ) C π
f 2 ξ0 D C2 Q 2 0 (ξ0 ) , 3

(12.47) (12.48) (12.49)

and A l P l (ξ0 ) D C l Q l (ξ0 ) ,

l D 1, 3, 4, 5, . . .

A l P l 0 (ξ0 ) D C l Q l 0 (ξ0 ) .

(12.50) (12.51)

The last two conditions can only be satisfied if A l D C l D 0. Finally, one can show that if the mass is written as mD


 4 π
f 3 ξ0 ξ02  1 , 3

(12.52)

then, for example, the exterior solution can be written as ψ e (ξ , η) D

m Q 0 (ξ )  Q 2 (ξ )P2 (η) . f

(12.53)

The interior solution is left as a homework problem. 12.5.3 Quantum Mechanics: Dirichlet Problem

In the following, a quasi-analytical method for calculating stationary energy levels in spheroidal quantum-dot structures is described. Two cases are considered: (a) the infinite-barrier problem and (b) the finite-barrier problem with different particle masses inside and outside the spheroid. 12.5.3.1 Infinite-Barrier Problem Consider a particle with mass m eff confined to a three-dimensional spheroid. The potential V(r) is assumed to be of the form

V(r) D 0 if r is located inside the spheroid , V(r) D 1 if r is located outside the spheroid ,

(12.54)

corresponding to the infinite-barrier problem. Stationary energy levels are found by solving the time-independent Schrödinger equation subject to the boundary condition that ψ(r) D 0 ,

(12.55)

147

148

12 Prolate Spheroidal Coordinates

whenever r is located on the surface of the spheroid. Next, consider the spheroid Ω which corresponds to the product of intervals: Ω

W [ξmin I ξmax ]  [φ min I φ max ]  [η min I η max ] ,

(12.56)

where [ξmin I ξmax ] D [1I ξ0 ] , [φ min I φ max ] D [0I 2π] , [η min I η max ] D [1I 1] .

(12.57)

We rewrite Eqs. (12.27)–(12.29) as (with v i D 0)      d X1 m2 d
2 2 2  λh ξ C 2 X1 D 0 , ξ 1 dξ dξ ξ 1      d X2 d
m2 X2 D 0 , C λ  h2 η2  1  η2 dη dη 1  η2 d2 X 3 C m2 X3 D 0 , dφ 2 where ψ  X 1 (ξ ) X 2(η) X 3 (φ), λ is a separation constant, and r 2m eff hD f E. „2

(12.58) (12.59) (12.60)

(12.61)

The equation for X 3 can be solved immediately to give X 3 D A cos m φ C B sin m φ ,

(12.62)

where m D 0, 1, 2, . . . Following [5, 51], the solution to the differential equation in η (Eq. (12.59)) can be expanded as X m X 2 (η) D d r P rCm (η) , (12.63) r m is an associated Legendre function of the first kind and the coefficients where P rCm d r must obey the recursion formula (derived in Appendix G, Eq. (G14))

(2m C r C 1)(2m C r C 2)h 2 r(r  1)h 2 d r
2 C d rC2 (2m C 2r  1)(2m C 2r  3) (2m C 2r C 3)(2m C 2r C 5)   2(m C r)(m C r C 1)  2m 2  1 2 h dr D 0 . C (m C r)(m C r C 1)  λ C (2m C 2r C 3)(2m C 2r  1) (12.64) P m is obtained for a discrete set of λ values A convergent series X 2 (η) D r d r P rCm by use of the continued-fraction method [5, 51]. For a given m, the lowest value of λ

12.5 Applications

is labeled λ m,m , the next λ m,mC1 , and so on (to simplify the notation, we write λ m n instead of λ m,n in the following). Similarly, the eigenfunction X 2 (η) corresponding to the eigenvalue λ D λ m n is denoted X 2 D X 2m n . The continued-fraction method applied to the recursion formula in Eq. (12.64) leads to a transcendental equation for λ m n . This is derived in Appendix G and we reproduce Eqs. (G27)–(G29) here for convenience: U (λ m n ) D U1 (λ m n ) C U2 (λ m n ) D 0 , m U1 (λ m n ) D γ n
m  λmn 

U2 (λ m n ) D 

(12.65) βm n
m

m γ n
m
2  λmn 

βm n
mC2 m γ n
mC2  λmn 

βm n
m
2 m γ n
m
4
λ m n
...

βm n
mC4 m γ n
mC4
λ m n
...

,

,

(12.66) (12.67)

where the coefficients γ rm and β m r are given by   4m 2  1 h2 1 , (12.68) γ rm D (mCr)(mCr C1)C 2 (2m C 2r C 3)(2m C 2r  1) r(r  1)(2m C r)(2m C r  1)h 4 . (12.69) βm r D (2m C 2r  1)2 (2m C 2r C 1)(2m C 2r  3) The first continued fraction (Eq. (12.66)) terminates with either the term containing γ0m or the term with γ1m , depending on whether n  m is even or odd, whereas the second fraction (Eq. (12.67)) is nonterminating (in principle). The transcendental equation is solved in an iterative manner and convergence is established within a dozen runs. Subsequently, insertion of the solution for λ m n in Eq. (12.64) specifies the coefficients d rm n apart from a constant multiplier. The solution to Eq. (12.58), which is finite at ξ D C1, can now be written as [5]  m/2 X  0 (r C 2m)! (n  m)! ξ 2  1 X 1m n (ξ ) D j rCm (h ξ ) , (12.70) i rCm
n d rm n 2 (n C m)! ξ r! r where j rCm (z) is the spherical Bessel function of the first kind with coefficient (r C m). The prime on the summation sign implies that for (n  m) even, even terms are to be summed over, and for (n  m) odd, odd terms are to be summed over. Thus, having determined the possible values for λ n m and d rn m , one can determine allowed values for h from the expression X 1m n (ξ D ξ0 ) D 0 ,

(12.71)

corresponding to particle confinement within the spheroidal surface ξ D ξ0 . In this way, a set of discrete values h D h 0m n are obtained. Finally, the values E n0 m associated with h 0m n are calculated from Eq. (12.61), that is,  0 2 hnm „2 E n0 m D . (12.72) 2m eff f

149

150

12 Prolate Spheroidal Coordinates Table 12.1 Infinite-barrier case. Calculated 0 ) and associated first three enerλ nm (E nm o (values are in electrongy eigenvalues E nm volts) corresponding to the parameter values

m eff D 0.067m 0 , f D 5.0 nm, and ξ0 D 3.0, where m 0 is the free-electron mass. Values are shown for the six cases: n
2.

n and m/parameter


1  λ nm E nm

1 E nm


2  λ nm E nm

2 E nm


3  λ nm E nm

3 E nm

n D 0, m D 0 n D 1, m D 0

0.376 2

0.108 0

1.307 3.374

0.435 0.214

2.435 5.845

0.983 0.636

n D 1, m D 1

2

0

2.468

0.225

3.259

0.665

n D 2, m D 0 n D 2, m D 1

8.159 6

0.354 0

11.58 7.635

0.878 0.360

16.279 9.870

1.612 0.900

n D 2, m D 2

6

0

6.554

0.372

7.286

0.927

In Table 12.1, tabulated values of λ n m (E n0 m ) and the associated first three eigenvalues E n0 m are shown for the parameter values m eff D 0.067m 0 and f D 2.0 nm, where m 0 is the free-electron mass. 12.5.3.2 Finite-Barrier Problem Consider next the finite-barrier problem. Inside the spheroid (ξ < ξ0 ), the effective mass of the particle is m i and the potential V(r) is 0. Outside the spheroid, the effective mass of the particle is m 0 and the potential is V0 . Stationary energy levels are again found by solving the time-independent Schrödinger equation, Eq. (2.139), and subject to the boundary conditions in Eqs. (2.140) and (2.141). These boundary conditions are fulfilled if

ψj ξ Dξ0
D ψj ξ Dξ C ,

(12.73)

1 1 r ψj ξ Dξ0
D r ψj ξ Dξ C . 0 mi mo

(12.74)

0

Following steps analogous to those presented in the previous subsection, a general solution inside the spheroid is found to be ψ i D cos m φ

1 X

a n X 1m n (h 1 , ξ ) X 2m n (h 1 , η) ,

(12.75)

a n X 1m n (h 1 , ξ ) X 2m n (h 1 , η) ,

(12.76)

nDm

or ψ i D sin m φ

1 X nDm

where (if (n  m) is even) X 1m n (h 1 , ξ ) D

 m/2  (n C m)! ξ 2  1 (n  m)! ξ2 X (2r C 2m)! j 2rCm (h 1 ξ ) ,  i 2rCm
n d2r (h 1 , m n) (2r)! r

(12.77)

12.5 Applications

X 2m n (h 1 , η) D

X

m d2r (h 1 , m n) P2rCm (η) ,

(12.78)

r

or (if (n  m) is odd) X 1m n (h 1 , ξ ) D

m/2  (n C m)! ξ 2  1 (n  m)! ξ2 X (2r C 1 C 2m)! j 2rC1Cm (h 1 ξ ) ,  i 2rC1Cm
n d2rC1 (h 1 , m n) (2r C 1)! r (12.79)

X 2m n (h 1 , η) D

X

m d2rC1 (h 1 , m n) P2rC1Cm (η) ,

(12.80)

r

and r h1 D f

2m i E. „2

(12.81)

The latter expression for h 1 holds for all values of n  m. Outside the spheroid, the general solution becomes 1 X

ψ o D cos m φ

b n X 1m n (h 2 , ξ ) X 2m n (h 2 , η) ,

(12.82)

b n X 1m n (h 2 , ξ ) X 2m n (h 2 , η) ,

(12.83)

nDm

or ψ o D sin m φ

1 X nDm

where (if (n  m) is even) m/2  (n C m)! ξ 2  1 (n  m)! ξ2 X (2r C 2m)! k2rCm (h 2 ξ ) , (12.84)  i 2rCm
n d2r (h 2 , m n) (2r)! r X m X 2m n (h 2 , η) D d2r (h 2 , m n) P2rCm (η) , (12.85) X 1m n (h 2 , ξ ) D

r

or (if n  m is odd) X 1m n (h 2 , ξ ) D

m/2  (n C m)! ξ 2  1 (n  m)! ξ2 X (2r C 1 C 2m)! k2rC1Cm (h 2 ξ ) ,  i 2rC1Cm
n d2rC1 (h 2 , m n) (2r C 1)! r

151

152

12 Prolate Spheroidal Coordinates

(12.86) X 2m n (h 2 , η) D

X

m d2rC1 (h 2 , m n) P2rC1Cm (η) ,

(12.87)

r

and r h2 D f

2m o (V0  E ) . „2

(12.88)

The latter expression for h 2 holds for all values of (n  m). The function k2rCm (k2rC1Cm ) appearing in Eq. (12.84) (Eq. (12.86)) is the modified spherical Bessel function of the second kind with coefficient 2r C m (2r C 1 C m). The coefficients a n and b n are determined using a numerical procedure as will be described next. Assume p terms are included in the sum over n in Eqs. (12.75), (12.76), (12.82), and (12.83), that is, n D m, m C 1, . . . , m C p  1. In other words, all terms for which n  m C p are tacitly discarded. Choose p different η values: η i (i D 0, 1, . . . , p  1) between 1 and 1, for example, η i D 1 C

2 1 C iI p p

i D 0, 1, . . . , p  1 .

(12.89)

The boundary conditions given by Eqs. (12.73) and (12.74) imply mCp
1

X

mCp
1

a n X 1m n (h 1 , ξ0 ) X 2m n (h 1 , η i ) D

nDm

X

b n X 2m n (h 2 , ξ0 ) X 2m n (h 2 , η i )

nDm

(12.90) and ˇ 1 @X 1m n (h 1 , ξ ) ˇˇ X 2m n (h 1 , η i ) ˇ mi @ξ ξ Dξ0 nDm ˇ mCp
1 X 1 @X 1m n (h 2 , ξ ) ˇˇ bn X 2m n (h 2 , η i ) , D ˇ m @ξ o ξ Dξ 0 nDm

mCp
1

X

an

(12.91)

where i D 0, 1, . . . , p 1. The system of equations Eqs. (12.90) and (12.91) contains 2p linear equations in 2p unknowns: a m , a mC1 , . . . , a mCp
1, b m ,

12.5 Applications

b mC1 , . . . , b mCp
1, and can be rewritten in matrix form as
 X 1m n h 1 , ξN X 2m n (h 1 , η 0 ) . . . mn 1 @X 1 ( h 1 ,ξ ) X 2m n (h 1 , η 0 ) . . . mi @ ξN

0

B B B B ... B B ... B  B X m n
h , ξN  X m n
h , η 1 p
1 . . . @ 1 mn1 2
 1 @X 1 ( h 1 ,ξ ) m n X 2 h 1 , η p
1 . . . mi @ ξN 
 X 1m n h 2 , ξN X 2m n (h 2 , η 0 ) . . . @X m n ( h ,ξ )  m1o 1 @ ξN 2 X 2m n (h 2 , η 0 ) . . . ... ...

   X 1m n h 2 , ξN X 2m n h 2 , η p
1 . . .
 @X m n ( h ,ξ )  m1o 1 @ ξN 2 X 2m n h 2 , η p
1 . . . 0

1

...
  ...
X 1m n h 1 , ξN X 2m n h 1 , η p
1 mn(
 1 @X 1 h 1 ,ξ ) X 2m n h 1 , η p
1 mi @ ξN 
 X 1m n h 2 , ξN X 2m n (h 2 , η 0 ) @X m n ( h ,ξ )  m1o 1 @ ξN 2 X 2m n (h 2 , η 0 )

1

C C C C ... C C ... C

  mn mn N  X 1 h 2 , ξ X 2 h 2 , η p
1 C
 A @X m n ( h ,ξ )  m1o 1 @ξ0 2 X 2m n h 2 , η p
1

0

1 0 C B. . .C B C B C B C B C B B a mCp
1C B 0 C B CDB C , B b1 C B 0 C C B C B @ . . . A @. . .A b mCp
1 0 am ...


 X 1m n h 1 , ξN X 2m n (h 1 , η 0 ) mn 1 @X 1 ( h 1 ,ξ ) X 2m n (h 1 , η 0 ) mi @ ξN

(12.92)

where ξN  ξ0 and @/@ ξN  @/@ξ j ξ Dξ0 . An eigenvalue E is found whenever the secular equation corresponding to Eq. (12.92) becomes zero (note that h 1 and h 2 appearing in the matrix entries depend on the energy E). The eigenvector (a m , . . . , a mCp
1, b m , . . . , b mCp
1) corresponding to an eigenvalue E now completely specifies the wave function by use of Eqs. (12.75), (12.76), (12.82), and (12.83). Example computed eigenvalues are given in Tables 12.2 and 12.3. i (values are in electronvolts) Table 12.2 Finite-barrier case. Calculated energy eigenvalues E m corresponding to the parameter values: m i D 0.023 m 0 , m o D 0.067 m 0 , f D 5.0 nm, and ξ0 D 3.0, where m 0 is the free-electron mass.

m/parameter

1 Em

2 Em

3 Em

mD0

0.158

0.328

0.543

mD1 mD2

0.183 0.118

0.340 0.564

0.549 –

153

154

12 Prolate Spheroidal Coordinates i (values are in electronvolts) Table 12.3 Finite-barrier case. Calculated energy eigenvalues E m corresponding to the parameter values m i D 0.023 m 0 , m o D 0.067 m 0 , f D 2.0 nm, and ξ0 D 3.0, where m 0 is the free-electron mass.

m/parameter

1 Em

mD0

0.475

mD1

0.047

mD2

–

12.6 Problems

1. Show that the coordinates of Moon and Spencer can be transformed into those of Morse and Feshbach. 2. Using the Moon–Spencer coordinate system, a. Derive the following quantities: metric, gradient, divergence, circulation, Laplacian, and Stäckel matrix. b. Separate the Laplace equation. c. Show that the solutions to the angular equations can be written in terms of associated Legendre functions. 3. Find solutions to the Laplace equation with k2 D k3 D 0. 4. For the gravitational potential of a solid homogeneous prolate spheroid, show that the exterior potential is given by Eq. (12.53). Also obtain the interior potential.

155

13 Oblate Spheroidal Coordinates 13.1 Introduction

The oblate spheroidal coordinate system is closely related to the prolate spheroidal one. It is also related to the two-dimensional elliptic coordinate system via a rotation of the latter to create a three-dimensional system.

13.2 Coordinate System 13.2.1 Coordinates (α, β, ' and ξ , η, ')

The choice of Moon and Spencer is with ξ1 D α, ξ2 D β, ξ3 D φ such that x D f cosh α sin β cos φ , y D f cosh α sin β sin φ , z D f sinh α cos β ,

(13.1)

and 0
α<1,

0
β
π,

0
φ < 2π ,

f >0.

Morse and Feshbach give, instead, ξ1 D ξ , ξ2 D η , ξ3 D φ , 1/2 1/2

1  η2 cos φ , x D f ξ2 C 1 
2 1/2
2 1/2 1η y D f ξ C1 sin φ , z D f ξη ,

(13.2)

with 0
ξ <1,

1
η
1 ,

0
φ < 2π ,

f >0.

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

156

13 Oblate Spheroidal Coordinates

z β= β0 φ = φ0

α =α0 x

y

Figure 13.1 Oblate spheroidal coordinates.

13.2.2 Constant-Coordinate Surfaces

Since 

x2 f

2 cosh2

α

C

y2 2

f cosh α 2

C

z2 f

2 sinh2

α

D1,

constant α surfaces are oblate spheroids (circular and elliptic cross sections, Figure 13.1). As cosh α > sinh α, they are flattened along z. Similarly, 

x2 f

2 sin2

β

C

y2 f

2 sin2

β



z2 f

2 cos2

β

D cosh2 α cos2 φ C cosh2 α sin2 φ  cosh2 α D 1 ,

that is, 1-sheeted hyperboloids. They have mirror symmetry about the x–y plane and for x D y D 0 and β finite, there is no solution for z; that is, there is no z-intercept. Finally, tan φ D y /x, gives half planes. This coordinate system is also related to the two-dimensional elliptic coordinate system (Chapter 5) by a rotation of the latter about the perpendicular bisector of the focal points.

13.3 Differential Operators

13.3 Differential Operators 13.3.1 Metric

Using Eqs. (2.5) and (13.1), one obtains the scale factors as h

f 2 cosh2 α sin2 β cos2 φ C f 2 cosh2 α sin2 β sin2 φ C f 2 cosh2 α cos2 β  1/2  1/2 D f cosh2 α C cos2 β D f cosh2 α  sin2 β ,

hα D

i1/2

hβ D hα , h φ D f cosh α sin β ,

(13.3)

or, with the Morse–Feshbach choice, #1/2

 f 2 1  η 2 sin2 φ ξ 2 f 2 1  η 2 cos2 φ ξ 2 2 2 hξ D C C f η (ξ 2 C 1) (ξ 2 C 1) 1/2
2 ξ C η2 D f , (ξ 2 C 1)1/2 1/2
2 ξ C η2 hη D f , (1  η 2 )1/2 1/2
1/2
1  η2 . hφ D f ξ 2 C 1 "

(13.4)

13.3.2 Operators

These are obtained by substituting Eq. (13.4) (we will, henceforth, only give the results in the Morse–Feshbach coordinates) into Eqs. (2.7)–(2.12). 13.3.2.1 Gradient The gradient operator is


r D eξ

1/2 ξ2 C 1

f (ξ 2 C η 2 )1/2


1/2 1  η2 eφ @ @ @ Ce η C . 1/2 @η 1/2 1/2 @φ 2 2 2 2 @ξ (ξ (ξ ) (1 ) f Cη f C 1) η (13.5)

157

158

13 Oblate Spheroidal Coordinates

13.3.2.2 Divergence The divergence of a vector field V is

1 rV D 2 (ξ f C η 2)

(

@ @ξ

"


ξ 2 C η2

(ξ 2 C 1)1/2

1

C

f

(ξ 2

C 1)

1/2

(1 

η 2 )1/2

#

1/2 Vξ

@ C @η

"


ξ 2 C η2

#)

1/2

(1  η 2 )1/2

@Vφ . @φ

Vη (13.6)

13.3.2.3 Circulation The circulation is

rV D

1 f (ξ 2 C η 2 )

ˇ 2 2 1/2 ˇ ( ξ Cη ) ˇ 2 1/2 e ξ ˇ ( ξ C1) ˇ ˇ @ ˇ @ξ ˇ ˇ ( ξ 2 Cη 2 )1/2 ˇ ˇ ( ξ 2 C1)1/2 Vξ

( ξ 2 Cη 2 )1/2 e (1
η 2 )1/2 η @ @η 1/2 ξ 2 Cη 2

( ) V (1
η 2 )1/2 η

1/2 ˇˇ 1/2
ξ2 C 1 1  η2 eφ ˇ ˇ ˇ ˇ @ ˇ . @φ ˇ
2 1/2 ˇˇ 1/2
ξ C1 1  η2 Vφ ˇ


(13.7)

13.3.2.4 Laplacian The Laplacian is

(

     @  @ @
@
2 2 ξ C1 C 1η r D 2 2 f (ξ C η 2 ) @ξ @ξ @η @η )
2  ξ C η2 @2 . C 2 (ξ C 1) (1  η 2 ) @φ 2 1

2

(13.8)

13.3.3 Stäckel Matrix

From Section 2.4.2.1, a possible Stäckel matrix is found to be
 f 2
ξ 2 C 1 Φ D @ f 2 1  η2 0 0

1 1 0


 1 1/
ξ 2 C 1 1/ 1  η 2 A . 1

(13.9)

The f functions are
1/2 f1 D ξ 2 C 1 ,

1/2
f 2 D 1  η2 ,

f3 D f .

(13.10)

13.4 Separable Equations

13.4 Separable Equations 13.4.1 Laplace Equation

Using the Stäckel theory and the Morse–Feshbach choice, the Laplace equation r 2 ψ D 0 becomes d2 Φ C k32 Φ D 0 , dφ 2      dX k32 d
2 ξ C1  k2  2 X D0, (ξ C 1) dξ dξ      dN d
k32 1  η2 C k2  N D0. (1  η 2 ) dη dη

(13.11) (13.12) (13.13)

Equation (13.13) is seen to be identical to the associated Legendre equation, whereas Eq. (13.12) can be made the same by changing the coordinate by i (and the domain of ξ is, of course, different). Therefore, one can write the general solutions as Φ (φ) D Ae i m φ C B e
i m φ ,

(13.14)

X(ξ ) D C P ml (i ξ ) C D Q lm (i ξ ) ,

(13.15)

N(η) D E P ml (η) C F Q lm (η) ,

(13.16)

where we have written k32 D m 2 and k22 D l(l C 1). 13.4.2 Helmholtz Equation

Again, using the Stäckel theory, the Helmholtz equation, r2 ψ C k2 ψ D 0 , becomes d2 Φ C k32 Φ D 0 , dφ 2      dX k2 d
2  k2  f 2 k 2 ξ 2  2 3 X D0, ξ C1 (ξ C 1) dξ dξ      dN k32 d
C k2 C f 2 k 2 ξ 2  1  η2 N D0. (1  η 2 ) dη dη

(13.17) (13.18) (13.19)

These are the spheroidal wave equations; the first one is known as the radial equation and the second is known as the angular equation. Note that the two equations

159

160

13 Oblate Spheroidal Coordinates

are identical except for different domains. They also differ from the associated Legendre equation by the k 2 term. One can thus formally write the solutions as Φ (φ) D Ae i m φ C B e
i m φ ,

(13.20)

X(ξ ) D C P ml ( f k, i ξ ) C D Q lm ( f k, i ξ ) ,

(13.21)

N(η) D E P ml ( f k, η) C F Q lm ( f k, η) ,

(13.22)

where we have written k32 D m 2 and k22 D l(l C 1). The functions P ml (k, ξ ) and Q lm (k, ξ ) are known as spheroidal wave functions. 13.4.3 Schrödinger Equation

The separable potential is 

 1 v3 (φ) „2 (ξ ) C v (η) C v , (13.23) V(ξ , η, φ) D 1 2 (ξ 2 C 1) (1  η 2 ) 2m f 2 (ξ 2 C η 2 ) ψ(ξ , η, φ) D X(ξ )N(η)Φ (φ) . Then,

   dX d
2 ξ C1 C v1 (ξ ) dξ dξ    dN d
1  η2 C v2 (η) dη dη " # 1 1 d2 Φ C 2 2  C v3 (φ) D k 2 . f (ξ C 1) (1  η 2 ) Φ dφ 2

(13.24)

1 f 2 (ξ 2 C η 2 ) X 1  2 2 f (ξ C η 2 ) N 

(13.25)

Let  Then,

1 d2 Φ C v3 (φ) D k32 . Φ dφ 2

(13.26)

     dN  dX 1 d
2 1 d
ξ C1 C v1 (ξ )  1  η2 C v2 (η) X dξ dξ N dη dη
 k 2 (ξ 2 C η 2 ) D f 2 k2 ξ 2 C η2  2 3 (ξ C 1) (1  η 2 )  
2  1 1 2 2 2 2  . (13.27) D f k ξ C η C k3 (ξ 2 C 1) (1  η 2 )



Therefore, the separated equations are  d2 Φ C k32  v3 (φ) Φ D 0 , 2 dφ

(13.28)

13.5 Applications

     dX k2 d
2  k2 C v1 (ξ )  f 2 k 2 ξ 2  2 3 X D0, ξ C1 dξ dξ (ξ C 1)      dN d
k32 C k2  v2 (η) C f 2 k 2 η 2  N D0. 1  η2 (1  η 2 ) dη dη

(13.29) (13.30)

13.5 Applications

We give one example to illustrate a difference between the prolate and oblate coordinate systems (in terms of the boundary conditions necessary to solve the problem) and another example to illustrate asymptotic solutions which, in the present context of solutions to Maxwell’s equations, reveals a connection to the paraxial approximation in optics. 13.5.1 Dirichlet Problem for the Laplace Equation

The Dirichlet problem for the Laplace equation for a prolate spheroid was discussed in Chapter 12. We already have the ordinary differential equations, Eqs. (13.11)– (13.13), with general solutions, Eqs. (13.14)–(13.16). For the boundary condition, consider the special case whereby both the boundary condition and the solution are independent of the angle φ. Then, m D 0 and N(η) satisfies the Legendre equation. For bounded solutions in the whole interval, N(η) D AP l (η) .

(13.31)

Similarly, X(ξ ) D B P l (i ξ ) C C Q l (i ξ ) .

(13.32)

Hence, the solution for the case of rotational symmetry can be written as ψ(ξ , η) D [B l P l (i ξ ) C C l Q l (i ξ )]P l (η) .

(13.33)

The oblate spheroid is described by ξ D ξ0 and the interior is given by 1
ξ
ξ0 ; or, in terms of the Moon–Spencer coordinates, 0
α
α 0 , just as for the prolate spheroid. However, in this case, Q l (i ξ ) D Q l (i sinh α) , which is finite inside the spheroid. Thus, it is not apparent that the function Q l (i ξ ) can be dropped inside the spheroid. Nevertheless, that it remains true follows from an argument provided by Lebedev [17]. Thus, it is also necessary for r ψ to be wellbehaved inside the spheroid. Consider, therefore, "  #   @ψ 2  @ψ 2

2 1 2 2 . (13.34) C 1η ξ C1 (r ψ) D 2 2 f (ξ C η 2 ) @ξ @η

161

162

13 Oblate Spheroidal Coordinates

In particular, for the point ξ D η D 0, it would appear that (r ψ)2 diverges, unless the quantity in the square brackets has the same (ξ 2 C η 2 ) factor as in the denominator. Using the explicit form of ψ given in Eq. (13.33), the inside of the square brackets becomes, at ξ D η D 0 and assuming C l D 0, " #    2  2  2  2  @ψ 2 @ψ 2 D B l2  P l 0 (i ξ ) P l (η) C P l (i ξ ) P l 0 (η) C @ξ @η which equals 0 if η D i ξ ; that is, it has a factor of (ξ 2 C η 2 ). Therefore, for a given l, the solution inside is given by ψ l (ξ , η) D A l P l (i ξ )P l (η) .

(13.35)

For the exterior problem, ξ ! 1, and P l (ξ ) ! 1; hence, we must have B D 0, giving the solution outside as ψ l (ξ , η) D C l Q l (i ξ )P l (η) .

(13.36)

13.5.2 Asymptotic Solutions

There are many problems where an exact solution is not necessary but rather some approximate solution in the limit of a large or small parameter. One such example is the paraxial approximation in optics, which can be described as the shortwavelength limit of the wave equation in the oblate spheroidal coordinate system. Having such an approximate analytical theory also allows one to compute corrections. We give a brief review of the basic approach to illustrate an application of the spheroidal coordinates and the technique of asymptotic expansion. The material is taken from a paper by Zeppenfeld [54]. The problem at hand is to study the solutions to Eqs. (13.18) and (13.19) in the limit of small wavelength (i.e., large frequency). To connect to the notation in [54], we rewrite Eqs. (13.18) and (13.19) as   d m2 ξ2 C 1  λ m ν  c2 ξ 2  2 R m ν (ξ ) D 0 , (ξ C 1) dξ    d m2 d
1  η2 C λ m ν C c2 ξ 2  S m ν (η) D 0 . (1  η 2 ) dη dη



d dξ






(13.37) (13.38)

In the limit of large c, one can expand the solutions as a series in 1/c. For the angular equation, Eq. (13.37), writing the solution as m/2
c(1
η)
S m (η) D 1  η 2 e s m (x) ,

(13.39)

13.6 Problems

with x D 2c(1 η), gives a differential equation to be satisfied by the new functions sm: " x

d2 mC1 d c2 C λ m ν  C C (m C 1  x) 2 dx dx 2 4c

# x 2 d2 x (m C 1)  (m 2 C m) x 2  2x (m C 1) d  C C s m (x) D 0 . 4c dx 2 4c dx 4c (13.40) In the limit of large c, the second line in Eq. (13.40) goes to zero and the remaining equation is the associated Laguerre equation. Thus, the solutions to Eq. (13.40) can be written as (m)

s m (x) D L ν (x) C O(1/ c) , and, from the theory of Laguerre functions, the eigenvalues are given by  λ m ν D c 2 C 2(m C 1) C 4ν c C O(1) .

(13.41)

(13.42)

The result, Eq. (13.41), motivates one to expand the exact solution s m (x) in terms of Laguerre polynomials: s m (x) D

1 X

(m)

A rm L r (x) .

(13.43)

rD0

Ways for computing the expansion coefficients A rm are described in, for example, [54]. A similar analysis can be carried out for the radial equation.

13.6 Problems

1. a. Show how the coordinates of Moon and Spencer can be transformed into those of Morse and Feshbach. b. Give the form of the following in the Moon–Spencer coordinates: gradient, divergence, circulation, Laplacian, and Stäckel matrix. 2. Separate the Schrödinger equation in the Moon–Spencer coordinates. 3. Derive the following recursion relations for the Laguerre polynomials: (x)  L αC1 L αn (x) D L αC1 n n
1 (x) , x L αn (x) D (n C α)L αn
1(x) C (2n C α C 1)L αn (x)  (n C 1)L αnC1 (x) , x L αn (x) D (n C α)L α
1 (x)  (n C 1)L α
1 n nC1 (x) , x

d α L (x) D nL αn (x)  (n C m)L αn
1(x) . dx n

163

164

13 Oblate Spheroidal Coordinates

4. Obtain the gravitational potential of a solid homogeneous oblate spheroid. 5. a. Derive Eq. (13.40) starting from Eqs. (13.38) and (13.39). b. Use the theory developed in [54] to obtain the O(1) correction to the lowest eigenvalue for the spheroidal wave equation for c D 100.

165

14 Parabolic Rotational Coordinates 14.1 Introduction

The parabolic rotational coordinate system allows one to study boundary-value problems with parabolic surfaces. A bounded region can be formed by specifying a parabolic surface for each of two coordinates. The Laplace equation leads to the Bessel and modified Bessel equations, whereas for the Helmholtz equation, the partial differential equation reduces to ordinary differential equations which include the Bessel wave equation. Two of the ordinary differential equations are coupled via the two separation constants.

14.2 Coordinate System 14.2.1 Coordinates (ξ , η, φ)

The coordinates are ξ1 D ξ , ξ2 D η, ξ3 D φ and are related to the Cartesian ones via the following equations: x D ξ η cos φ , y D ξ η sin φ ,  1
2 zD η
ξ2 , 2

(14.1)

where 0ξ ,

η<1,

0  φ < 2π .

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

166

14 Parabolic Rotational Coordinates z ξ= ξ0 φ =φ0

η=η0

x

y

Figure 14.1 Parabolic rotational coordinates.

14.2.2 Constant-Coordinate Surfaces

Since



 x 2 C y 2 D ξ 2 η 2 D ξ 2 2z C ξ 2 D η 2 η 2
2z , tan φ D y /x ,

the surfaces of constant coordinates are (Figure 14.1) ξ D constant: paraboloid, Cz axis , η D constant: paraboloid,
z axis , φ D constant: half planes . ξ D 0 (η D 0) corresponds to the positive (negative) z half line. Note that ξ and η have the same domain; this is different from the prolate spheroidal coordinates. 14.2.3 Other Geometrical Parameters

The relationships of various geometrical parameters are given as follows (Figure 14.2): R D ξ0 η 0 ,

(14.2)

ξ02 C η 20 , (14.3) 2
 π H V D ξ02 η 20 ξ02 C η 20 D π R 2 , (14.4) 4 2 where R, H, and V denote the radius, full height, and volume, respectively. The volume is obtained by integrating a volume element in parabolic rotational coordinates (using the scale factors given in Eq. (14.5)): HD

Z2π V D

Zη 0 dφ

0

Zξ0 dη

0

0


dξ ξ 2 C η 2 ξ η .

14.3 Differential Operators

z

(x',z')

ξ0

x

H

L η0 R Figure 14.2 Geometrical parameters of the closed region in parabolic rotational coordinates. Note that this is a y-plane view and the region is obtained by rotating about the z axis.

The two parabolas intersect at (x 0 , z 0 ) in the x–z plane, where x 0 D R D ξ0 η 0 ,

z0 D

ξ02
η 20 . 2

14.3 Differential Operators 14.3.1 Metric

Using Eqs. (2.5) and (14.1), one obtains the scale factors as 1/2
2 1/2  h ξ D η 2 cos2 φ C η 2 sin2 φ C ξ 2 D η C ξ2 , hη D hξ ,  1/2 D ηξ . h φ D ξ 2 η 2 sin2 φ C ξ 2 η 2 cos2 φ

(14.5)

14.3.2 Operators

These are obtained by substituting Eq. (14.5) into Eqs. (2.7)–(2.12). 14.3.2.1 Gradient The gradient operator is

rD

eξ (ξ 2

C

η 2 )1/2

eη eφ @ @ @ C 2 C . @ξ (ξ C η 2 )1/2 @η ξ η @φ

(14.6)

167

168

14 Parabolic Rotational Coordinates

14.3.2.2 Divergence The divergence of a vector field V is  i 1 @ h
i   1 1 @ h
2 2 1/2 2 2 1/2 C rV D 2 ξ ξ η ξ C η V C η V ξ ν (ξ C η 2 ) ξ @ξ η @η 1 @Vφ C . (14.7) ξ η @φ 14.3.2.3 Circulation The circulation is

ˇ
ˇ ξ 2 C η 2 1/2 e ξ ˇ ˇ 1 ˇ @ rV D ˇ @ξ ξ η (ξ 2 C η 2 ) ˇ ˇ
2  ˇ ξ C η 2 1/2 Vξ







ξ 2 C η2 @ @η

ξ 2 C η2

1/2

1/2

eη

Vη

ˇ ξ η e φ ˇˇ ˇ ˇ @ . @φ ˇˇ ˇ ξ ηVφ ˇ (14.8)

14.3.2.4 Laplacian The Laplacian is

r2 D

1 (ξ 2 C η 2 )



1 @ ξ @ξ

 ξ

@ @ξ

C

1 @ η @η

 η

@ @η



C

1 ξ 2 η2

@2 . @φ 2

(14.9)

14.3.3 Stäckel Matrix

From Section 2.4.2.1, a possible Stäckel matrix is found to be 0


1 1 0

ξ2 Φ D @ η2 0

1
1/ξ 2
1/η 2 A . 1

(14.10)

The f functions are f1 D ξ ,

f2 D η ,

, f3 D 1 .

(14.11)

14.4 Separable Equations 14.4.1 Laplace Equation

The Laplace equation is  

 @ψ 1 @ @ψ 1 @2 ψ 1 1 @ ξ C η C 2 2 D 0 . (14.12) r2 ψ D 2 2 (ξ C η ) ξ @ξ @ξ η @η @η ξ η @φ 2

14.4 Separable Equations

Let ψ(ξ , η, φ) D M(ξ )N(η)Φ (φ) .

(14.13)

Then, 1 2 (ξ C η 2 )



1 d ξ M dξ

 ξ

dM dξ

C

1 d η N dη

 η

dN dη



C

1 1 d2 Φ D0. ξ 2 η 2 Φ dφ 2 (14.14)

Let


1 d2 Φ D k32 . Φ dφ 2

(14.15)

Then, the separated equations are d2 Φ C k32 Φ D 0 , dφ 2

  k2 1 d dM ξ C k22
32 M D 0 , ξ dξ dξ ξ   2

k 1 d dN η
k22 C 32 N D 0 . η dη dη η

(14.16) (14.17) (14.18)

The last two equations are the Bessel equation and the modified Bessel equation, respectively. Hence, the general solutions are Φ (φ) D A sin k3 φ C B cos k3 φ ,

(14.19)

M(ξ ) D A J k3 (k2 ξ ) C B J
k3 (k2 ξ ) ,

(14.20)

N(η) D A J k3 (i k2 η) C B J
k3 (i k2 η) .

(14.21)

If k3 is an integer, then the Bessel function J
k3 is replaced by the Neumann function N k3 . 14.4.2 Helmholtz Equation

The Helmholtz equation r2 ψ C k2 ψ D 0 , becomes 1 (ξ 2 C η 2 )



1 @ ξ @ξ



@ψ ξ @ξ



1 @ C η @η



@ψ η @η



C

1 @2 ψ D
k 2 ψ . ξ 2 η 2 @φ 2

(14.22)

169

170

14 Parabolic Rotational Coordinates

Let ψ(ξ , η, φ) D M(ξ )N(η)Φ (φ) .

(14.23)

Then, 1 (ξ 2 C η 2 )



1 d ξ M dξ

 ξ

dM dξ

C

1 d η N dη

 η

dN dη



C

1 ξ 2 η2

1 d2 Φ D
k 2 . Φ dφ 2 (14.24)

Let


1 d2 Φ D k32 . Φ dφ 2

(14.25)

Then, the separated equations are d2 Φ C k32 Φ D 0 , dφ 2

  dM k2 1 d ξ C k22 C k 2 ξ 2
32 M D 0 , ξ dξ dξ ξ

  1 d dN k32 2 2 2 η
k2
k η C 2 N D 0 . η dη dη η

(14.26) (14.27) (14.28)

The last two equations are the Bessel wave equations (Appendix C). They have a regular singular point at the origin and an irregular singular point at infinity; the Bôcher type is f26g. The general solutions can be written as Φ (φ) D A sin k3 φ C B cos k3 φ ,

(14.29)

M(ξ ) D A J k3 (k, k2 ξ ) C B J
k3 (k, k2 ξ ) ,

(14.30)

N(η) D A J k3 (k, i k2 η) C B J
k3 (k, i k2 η) .

(14.31)

If k3 is an integer, then the Bessel wave function J k,
k3 is replaced by the Neumann wave function N k,k3 . 14.4.3 Schrödinger Equation

The separable potential is     1 „2 2 v (ξ ) C v (η) C v (z) . V(ξ , η, φ) D 1 2 3 2m (ξ 2 C η 2 ) ξ 2 η2

(14.32)

Then, the Schrödinger equation becomes  

 @ψ @ψ 1 @ 1 @ 1 ξ C 2v1 (ξ )ψ
η C 2v2 (η)ψ
(ξ 2 C η 2 ) ξ @ξ @ξ η @η @η  2

@ ψ 1 (14.33) C 2 2
2 C v3 (φ)ψ D k 2 ψ . ξ η @φ

14.5 Applications

Let ψ(ξ , η, φ) D M(ξ )N(η)Φ (φ) .

(14.34)

Then,

   1 d dM dN 1 d 1 (ξ )
(η) ξ C 2v η C 2v
1 2 (ξ 2 C η 2 ) ξ M dξ dξ η N dη dη " # 1 d2 Φ 1 C 2 2
C v3 (φ) D k 2 . ξ η Φ dφ 2 (14.35) Let

"

# 1 d2 Φ
C v3 (φ) D k32 . Φ dφ 2

(14.36)

Then,   1 dM 2v1 (ξ ) dN d 1 d
2 ξ C 2 η
(ξ C η 2 ) ξ M dξ (ξ C η 2 ) (ξ 2 C η 2 ) η N dη dξ dη 2 k 2v2 (η) D k2
23 2 , C 2 (ξ C η 2 ) ξ η  1 d dM k2
ξ C 2v1 (ξ )
k 2 ξ 2 C 32 ξ M dξ dξ ξ  2 k dN 1 d η
2v2 (η) C k 2 η 2
32  k22 . D η N dη dη η Hence, the separated equations are   d2 Φ C k32
v3 (φ) Φ D 0 , dφ 2

  k2 dM 1 d ξ C k22
2v1 (ξ ) C k 2 ξ 2
32 M D 0 , ξ dξ dξ ξ

  k2 1 d dN η
k22
2v2 (η)
k 2 η 2 C 32 N D 0 . η dη dη η

(14.37) (14.38) (14.39)

14.5 Applications

There are many applications of the parabolic rotational coordinate system. These include, for example, the study of parabolic reflectors in acoustics [58, 59] and electromagnetism [60], plasmonics [61], heat conduction [2], and for various problems in quantum mechanics [62]. Recent applications include wave problems in quantum mechanics and acoustics [63–66].

171

172

14 Parabolic Rotational Coordinates

14.5.1 Heat Conduction: Boundary-Value Problem for the Laplace Equation

A very detailed analysis of the temperature distribution for arbitrary parabolic rotational (which many call paraboloidal) systems subject to various boundary conditions was carried out by Bauer [2]. We present a brief summary of a few of the results obtained by him to illustrate the solution to the boundary-value problem for the Laplace equation in parabolic rotational coordinates. In the process, we also study the use of Fourier–Bessel expansions. The solutions to the ordinary differential equations obtained from the Laplace equation were given in Eqs. (14.19)–(14.21). Hence, the general solution to the Laplace equation is given by T(ξ , η, φ) D T0 C

1 X

[A m sin m φ C B m cos m φ]

mD1

    C m J m (k ξ ) C D m J
m (k ξ ) E m I m (k η) C F m I
m (k η) , (14.40) where, for example, I m (k η) is used instead of J m (i k η). Consider now the application of various boundary conditions. 14.5.1.1 Dirichlet For example, given a paraboloidal solid body with the following temperature distribution on the surfaces

T D T0 at ξ D ξ0 , T D F(ξ , φ) at η D η 0 ,

(14.41)

the finiteness for ξ D η D 0 implies D m D F m D 0. Then, the boundary condition at ξ D ξ0 defines
m n such that J m (
m n ) D 0 , and the boundary condition at η D η 0 gives F(ξ , φ) D

1 X 1 X

(α m n cos m φ C β m n sin m φ) J m (
m n ξ /ξ0 ) ,

(14.42)

mD0 nD1

where

αmn D

βmn D

2 π

2 π

R2π 0

dφ cos m φ

Rξ0 0

 dξ ξ J m
m n ξξ0 F(ξ , φ)

2 (
m n ) ξ02 J mC1  ξ 2π 0 R R dφ sin m φ dξ ξ J m
m n ξξ0 F (ξ , φ) 0

0

2 (
m n ) ξ02 J mC1

,

(14.43)

.

(14.44)

14.5 Applications

The last two equations were obtained by using the orthonormality relation, Eq. (C83). Finally, the temperature distribution is then T(ξ , η, φ) D

  η ξ 1 X 1 (α X m n cos m φ C β m n sin m φ) J m
m n ξ0 I m
m n ξ0  T0 C . I m
m n ηξ00 mD0 nD1 (14.45)

14.5.2 Quantum Mechanics: Interior Dirichlet Problem

For an infinite-barrier problem (Figure 14.3), the interior Schrödinger problem (Eqs. (14.37)–(14.39)) reduces to the Helmholtz one (Eqs. (14.26)–(14.28)), where k 2 D 2m  E/„2 . Imposing periodic boundary conditions on the first equation, Eq. (14.26), gives the solutions as e ˙i m φ , m 2 ZC ; hence, k3 D m. We now consider two methods for solving Eqs. (14.27) and (14.28). The first method, suggested by Zhang and Jin [13], involves a change of variables. First, let us rewrite the separation constants k22 D q 2 and k32 D p 2 . Then, Eq. (14.27) becomes

  dM p2 1 d (14.46) ξ C q2 C k2 ξ 2
2 M D 0 . ξ dξ dξ ξ Now, let V(v ) M(ξ ) D p , v

ξ2 D v .

(14.47)

z

η0 h

x

R

y

ξ0

Figure 14.3 Lens-shaped quantum dot in parabolic rotational coordinates.

173

174

14 Parabolic Rotational Coordinates

Equation (14.46) becomes d2 V C dv 2

"

q2 k2 C C 4 4v

1 4

2


p4 v2

# V D0.

With a further substitution v D z/(i k), Eq. (14.48) becomes # " 1
ξ2 d2 V  1 4 V D0, C
C C dz 2 4 z z2

(14.48)

(14.49)

where D

q2 , 4ik

ξ D

p , 2

z D ikv D ikξ2 .

(14.50)

Equation (14.49) is known as the Whittaker equation [51]. Its solutions are known as the Whittaker functions [51]:  z 1 1 C ξ
, 1 C 2ξ , z , (14.51) M,ξ (z) D e
2 z ξ C 2 M 2 where M(a, b, z) is the confluent hypergeometric function (Eq. (A23). There is also a second solution that diverges logarithmically near the origin; we reject the latter solution on physical grounds. Finally, we can write down the solution to Eq. (14.46) as  1Cp p q2 1 ik 2 M(ξ ) D (i k) 2 ξ p e
2 ξ M (14.52) C
, 1 C p, i k ξ 2 . 2 2 4ik Equation (14.52) is convenient since it relates the solution to a standard function, the confluent hypergeometric function. However, it poses some computational difficulties since the arguments are both real and complex. For instance, the useful computer routines provided by Zhang and Jin [13] can only handle real arguments. The series expansion of the Bessel wave function given in Appendix C, Eq. (C77), is reproduced here for convenience:  M(ξ ) D

qξ 2

p

Γ (p C 1) 8 9  2  4 qξ qξ ˆ >  < = 2

2 2 4(p C 1)k 1
C    C 1
ˆ > 1!(p C 1) 2!(p C 1)(p C 2) q4 : ;

 J p (k, q, ξ ) .

(14.53)

This series solution is easily amenable to numerical work. Furthermore, note that Eq. (14.53) is completely real. Our two solutions, Eqs. (14.52) and (14.53), allow

14.5 Applications

us to obtain a relationship between the Bessel wave function and the Whittaker function. Rewriting Eq. (14.52) in series form, we get 2  ( (1Cp ) 1Cp 
4qi k
ikξ2 k2 ξ 4 p 2 2 1


 1 C ikξ2 M(ξ ) D ξ (i k) 2 8 1C p h ih i 2 2 ) (1Cp ) (3Cp )
4qi k
4qi k
 2 2 2 4
k ξ
 2(1 C p )(2 C p )   1Cp q2 ξ 2
 , (14.54) D ξ p (i k) 2 1
4(1 C p ) and comparing the latter with Eq. (14.53) gives
q p J p (k, q, ξ ) D

2

(i k)

1Cp 2

Γ (p C 1)

M q2

p 4ik , 2

(ξ ) .

(14.55)

This relation does not appear in the standard references on special functions [33, 51, 67]. The solution to Eq. (14.27) is obtained in a fashion similar to that for M(ξ ). Indeed, the recurrence relation, Eq. (C76), applies with the change q 2 !
q 2 :
q 2 a n
2 C k 2 a n
4 , (14.56) (p C n)2
p 2
η p (
η 2
η 4

 q2 q2 q2 4(p C 1)k 2 N(η) D 1C C 1
Γ (p C 1) 1!Γ (p C 1) 2!Γ (p C 1)(p C 2) q4 )

an D


C 

.

(14.57)

The hard-wall boundary condition now becomes M (k, q, p, ξ0 ) D N (k, q, p, η 0 ) D 0 ,

(14.58)

where (ξ0 , η 0 ) defines the quantum-dot boundary. Equation (14.58) is solved by first specifying p (an integer). For each p, there remain two unknowns, k (related to the energy) and q a separation constant. A simple procedure is to scan in k and find the q values for each function M and N to have zeros; the boundary condition is satisfied when the two q values are identical. In [66], two types of structures were studied: symmetric and asymmetric quantum dots. Symmetric quantum dots are obtained when ξ0 D η 0 , when the plane of intersection of the two paraboloids is the z D 0 plane. When ξ0 ¤ η 0 , the two surfaces have different curvatures. We repeat the results obtained for the symmetric quantum dot here. For the symmetric quantum dot, the radius and height were both chosen to be 70 Å. The quantum dot has some basic spatial symmetries. It has rotational and mirror symmetry about the z axis; in addition, the symmetric quantum dot has a reflection plane (z D 0). The symmetry about the z axis allows

175

14 Parabolic Rotational Coordinates

for twofold degeneracies. The reflection symmetry guarantees that the states of the symmetric quantum dots can be classified as even or odd about the z D 0 plane. From Eqs. (14.53) and (14.56), one can deduce the following properties of the wave functions: M(k D 0, q D 0, ξ ) D N(k D 0, q D 0, η) D 1 (p D 0) ,

(14.59)

ψ(ξ D 0, η D 0, φ) D 0 8 p ¤ 0 .

(14.60)

The second equation implies that all the wave functions for nonzero p have at least one node (at the origin). In searching for the energies, one approach is to start with q 2 and scan in k for each of the two functions M and N that give zeros. The simultaneous zeros are the intersection points in a (q 2 , k) plot (Figure 14.4). Some actual values obtained for the symmetric quantum dot are given in Table 14.1. A given bound state is, therefore, distinguished by three labels: p, q (or q 2 ), and k. p is equivalent to the azimuthal quantum number often labeled by m. Also, conventionally, instead of k, one uses the counting index of k and labels it by n, the principal quantum number; we will also do so here. However, q does not have a direct analog. Furthermore, it is nonintegral and complex. Hence, we have decided 0.4 0.35 0.3 0.25 k [1/Å]

176

0.2 0.15 0.1 0.05 −3

−2

−1

0 q2

Figure 14.4 k versus

q2

1

2

3

[1/Å]

for a symmetric quantum dot (ξ02 D 70Å; η 20 D 70 Å).

Table 14.1 Lowest (q 2 , k) values for a symmetric quantum dot (ξ02 D η 20 D 70 Å). p

q 2 (Å
1 )

k (Å
1 )

0

0.0

0.0687, 0.1577, 0.2473

0.191

0.1122

0.0




0.0898

1

14.5 Applications

to fold it into n; that is, n counts the number of distinct (q 2 , k) pairs in increasing order of energy (for each p). 14.5.2.1 Numerical Results The lowest calculated energies for two structures are given in Table 14.2. For all the calculations, we used the effective mass of GaAs, m  D 0.067m 0 . The ordering of levels obtained is (n, p ) D (1, 0) < (1, 1) < (1, 2) < (2, 0) < . . . The meanings of the quantum numbers n and p are similar to, for example, those for the elliptic dot [68] (see Chapter 5). Thus, they relate to the nodal structure in the radial and φ directions. This is evident in the wave functions plotted in the x y plane at z D 0 (Figure 14.5). It was verified numerically that the absolute value of the wave function at the rim of the quantum dot is less than 10
7 as compared with the its maximum value for all the states shown. The contour plot of the (2,0) and (4,0) states in a φ plane is shown in Figure 14.6. The number of features is a reflection of the excited nature of these states. The interesting difference is that the (4,0) state is symmetric about the z D 0 plane but the (2,0) state has no such symmetry. Indeed, the only states that display such a symmetry must have q D 0, which also implies that these states are either even or odd. One can, in fact, prove that only the even states are allowed. To demonstrate this, we need the inverse of the coordinate equations in Eq. (14.1). In the z x plane, we have i1/2 h p , (14.61) ξ D
z C z 2 C x 2

i1/2 h p . η D z C z2 C x 2

(14.62)

Note that ξ and η go into each other under the z !
z transformation. Looking at Eqs. (14.53) and (14.56), one notes that the solutions can therefore be rewritten in the form in p X h M(x, 0, z) D a n
z C z 2 C x 2 , (14.63) n

Table 14.2 Energies (eV) for a symmetric lens-shaped quantum dot (using m  D 0.067m 0 ). The q D 0 solutions are given in bold. ξ02 D 70 Å, η 20 D 70 Å p D0 p D1 p D2 nD1 nD2

0.268 0.712

0.458 1.01

0.682 1.34

nD3

1.37

1.78

2.20

nD4 nD5

1.43 2.23

1.83 2.72

2.28 3.26

177

178

14 Parabolic Rotational Coordinates

wavefunc

0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 2

wavefunc

1.2 1 0.8 0.6 0.4 0.2 0 −0.2 2

1

1 x 10 2 Å

x 10 2Å

0 −1 y

(a)

−2

−2

−1.5 −1

−0.5 0 x

0.5 1

1.5 2 x 10 2 Å

−1 −2

−2

−1.5 −1

−0.5 0 x

−2

−1.5 −1

−0.5 0 x

0.5 1

1.5 2 x 10 2 Å

1 0.5 wavefunc

wavefunc

y

(b)

0.1 0.05 0

0

−0.5

−0.05

−1

−1.5

−0.1 2

−2 2 1

x 10 2 Å

1 x 10 2 Å

0 −1 y

(c)

0

−2

−1.5 −1

−2

−0.5 0 x

0.5 1

1.5 2 x 10 2Å

0 −1

(d)

y

−2

0.5 1

1.5 2 x 10 2 Å

Figure 14.5 The first (a), second (b), third (c), and fourth (d) wave functions in the x y plane (for z D 0) for the symmetric quantum dot. 30

2.000

30

20

1.663

10

0.9875

10

0.6500

0

z

0

0.3125

-10

-0.02500 -0.3625

-20

-0.7000

0.8125 0.6250 0.4375 0.2500 0.06250

-10

-0.1250 -0.3125

-20

-0.5000

-30

-30

(a)

Z

1.325

1.000

20

(b)

x

x

Figure 14.6 Contour plot in the z x plane (for y D 0) of the (2,0) (a) and (4,0) (b) states for the symmetric quantum dot.

N(x, 0, z) D

X

h in p b n z C z2 C x 2 ,

(14.64)

n

with the a n only equal to b n for all n if q D 0. When that is the case,

 ψ(x, 0,
z) D M ξ 0 N η 0 D N(η)M(ξ ) D ψ(x, 0, z) .

(14.65)

14.6 Problems

Hence, the state is even. When q ¤ 0, no such symmetry exists in our solutions. However, this is due to the double degeneracy of these states (with respect to the sign of q 2 , as is evident in Figure 14.4). Indeed, we know already that M and N are interchanged upon sign change of q 2 and the two solutions we have obtained are mirror images of each other with respect to the z D 0 plane. This also explains why the q D 0 states are all even. Recalling the degeneracy with respect to the sign of p, we therefore have the following degeneracies for the first few states of the symmetric lens-shaped quantum dot (in order of increasing energy): 1 (ground state), 2, 2, 2, 4, . . . Note that there are other states beyond the ground state that are nondegenerate. For example, the (4,0) state we have already studied is nondegenerate. On the other hand, no degeneracies higher than 4 are expected.

14.6 Problems

1. Derive Eq. (14.48) from Eq. (14.46). 2. Derive Eq. (14.49) from Eq. (14.48). 3. Show that Eq. (14.46) corresponds to a single singularity at ξ0 D 0 in the finite complex plane. 4. Show that a solution to Eq. (14.46) is M(ξ ) D

X

a m ξ mCσ ,

(14.66)


q 2 a m
2 C k 2 a m
4 , (p C m)2
p 2

(14.67)

m

with σ D ˙p , am D

where p is an integer. 5. Derive Eq. (14.55) by comparing Eqs. (14.53) and (14.52). 6. Consider three parabolic rotational enclosures of volume V D 8 m3 having ξ02 D 1 m, ξ02 D 2 m, and ξ02 D 5 m, respectively. a. Determine the three corresponding η 0 values. b. Find the acoustic eigenfrequencies subject to rigid-wall boundary conditions. c. Find the eigenmodes and plot the first three solutions in a Cartesian coordinate system. 7. For the problem considered in (6), replace the rigid boundary condition corresponding to a fixed ξ0 value by a pressure-release condition. Repeat steps (b) and (c) for (6).

179

181

15 Conical Coordinates 15.1 Introduction

The conical coordinate system has the useful feature that an elliptic cone can be described by a one-parameter surface. It is related to the spherical one in that they both describe a sphere by a one-coordinate surface. Both the Laplace equation and the Helmholtz equation lead to the Lamé equation when separated, whereas the Schrödinger equation leads to the Lamé wave equation. A related coordinate system is the spheroconical one. It is not clear that the conical coordinate system has been used for the full solution to a three-dimensional boundary-value problem.

15.2 Coordinate System 15.2.1 Coordinates (r, θ , λ)

The coordinates are ξ1 D r, ξ2 D θ , ξ3 D λ and are related to the Cartesian coordinates via the following equations: rθ λ , bc p r (θ 2
b 2 ) (b 2
λ 2 ) p , yD b (c 2
b 2 ) p r (c 2
θ 2 ) (c 2
λ 2 ) zD p , c (c 2
b 2 )

xD

(15.1)

where 0r<1,

b2 < θ 2 < c2 ,

0 < λ2 < b2 .

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

182

15 Conical Coordinates

z θ=θ 0

λ =λ 0 r= r0

y x

Figure 15.1 Conical coordinates.

15.2.2 Constant-Coordinate Surfaces

Since

 r 2 θ 2
b2 b2
λ2 r 2 θ 2 λ2 C b2 c2 b 2 (c 2
b 2 )
2 
2  2 2 r c
θ c
λ2 C D r2 , c 2 (c 2
b 2 )

 r 2 b2
λ2 r 2 c2
λ2 y2 r 2 λ2 z2 x2 C D C D D , (c 2
θ 2 ) θ2 λ2
b2 b2 c2 b 2 (c 2
b 2 ) c 2 (c 2
b 2 ) (
)  
θ 2
b2 c2
θ 2 z2 x2 r2 θ 2 y2 2 C D r C D 2 2 D 2 , 2 2 2 2 2 2 2 2 2 2 (c ) (c ) b
λ c
λ b
b b
b b c λ x 2 C y 2 C z2 D

then (Figure 15.1) r D constant: sphere, θ D constant: elliptic cone, z axis, λ D constant: elliptic cone, x axis.

15.3 Differential Operators

15.3 Differential Operators 15.3.1 Metric

Using Eqs. (2.5) and (15.1), one obtains the scale factors as "


 
 #1/2
2
2 θ
b2 b2
λ2 c
θ 2 c2
λ2 r2 θ 2 hr D C D1, C b2 c2 b 2 (c 2
b 2 ) c 2 (c 2
b 2 ) #1/2 "

 r 2 c2
λ2 θ 2 r 2 b2
λ2 θ 2 r 2 λ2 hθ D C 2 2 C 2 2 b2 c2 b (c
b 2 ) (θ 2
b 2 ) c (c
b 2 ) (c 2
θ 2 ) " #1/2
 r 2 θ 2
λ2 D , (θ 2
b 2 ) (c 2
θ 2 ) "
 #1/2 r 2 θ 2
λ2 hλ D . (15.2) (b 2
λ 2 ) (c 2
λ 2 ) 15.3.2 Operators

These are obtained by substituting Eq. (15.2) into Eqs. (2.7)–(2.12). 15.3.2.1 Gradient The gradient operator is


2 1/2
2 1/2
2 1/2
2 1/2 θ
b2 c
θ2 b
λ2 c
λ2 @ @ @ r D er C eθ C eλ . 1/2 1/2 2 2 2 2 @r @θ @λ r (θ
λ ) r (θ
λ ) (15.3) 15.3.2.2 Divergence The divergence of a vector field V is

rV D


2 
 1/2 i θ
b 2 c 2
θ 2 @ h
2 1 @
2  C r θ
λ2 V Vθ r 2 2 2 r @r r (θ
λ ) @θ
2 
 i  b
λ 2 c 2
λ 2 @ h
2 2 1/2 C . θ
λ V λ r (θ 2
λ 2 ) @λ

(15.4)

183

184

15 Conical Coordinates

15.3.2.3 Circulation The circulation is 


1/2 
2 θ
b2 c2
θ 2 b2
λ2 c2
λ2 rV D (θ 2
λ 2 ) ˇ ˇ 2 2 ˇ er ˇ ( θ
λ )1/2 (θ 2
λ 2 )1/2 ˇr ˇ e e θ λ 1/2 1/2 1/2 1/2 ( θ 2
b 2 ) ( c 2
θ 2 ) ( b 2
λ 2 ) ( c 2
λ 2 ) ˇ ˇ ˇ@ ˇ @ @ ˇ ˇ .  ˇ @r @θ @λ ˇ ˇ ˇ 1/2 1/2 ˇ Vr ˇ ( θ 2
λ 2 ) (θ 2
λ 2 ) ˇr 1/2 2 1/2 Vθ 1/2 2 1/2 Vλ ˇ 2 2 2 2 2 2 ( ) ( ) ( ) ( ) θ
b

c
θ

b
λ

(15.5)

c
λ

15.3.2.4 Laplacian The Laplacian is   p 2   (θ
b 2 ) (c 2
θ 2 ) @ p 2 1 @ @ @ 2 ) (c 2
θ 2 ) (θ r2 D 2 r2 C
b r @r @r r 2 (θ 2
λ 2 ) @θ @θ p p  2 2 2 2 (b
λ ) (c
λ ) @ @ (b 2
λ 2 ) (c 2
λ 2 ) . (15.6) C r 2 (θ 2
λ 2 ) @λ @λ 15.3.3 Stäckel Theory

From Section 2.4.2.1, a possible Stäckel matrix is found to be 1 0 0
r12 0 B C 2 C 0 ( θ 2
b 2θ)( c 2
θ 2 ) ( θ 2
b 2
1 Φ DB )( c 2
θ 2 ) A . @ 0


λ 2 ( b 2
λ 2 )( c 2
λ 2 )

(15.7)

1 ( b 2
λ 2 )( c 2
λ 2 )

We also have f1 D r2 ,

f2 D




θ 2
b2




c2
θ 2

1/2

,

f3 D




b2
λ2




c2
λ2

1/2

.

(15.8)

15.4 Separable Equations 15.4.1 Laplace Equation

Using the Stäckel theory and Eqs. (15.7) and (15.8), we get, dropping the k12 term which is only present for the Helmholtz problem,   k2 1 d 2 dR r
22 R D 0 , (15.9) 2 r dr dr r

15.4 Separable Equations

    k22 θ 2
k32 θ d2 Θ θ dΘ C C
Θ D0, (θ 2
b 2 ) (c 2
θ 2 ) dθ (θ 2
b 2 ) (c 2
θ 2 ) dθ 2     λ k22 λ 2
k32 λ dΛ d2 Λ C
C ΛD0. (b 2
λ 2 ) (c 2
λ 2 ) dλ (b 2
λ 2 ) (c 2
λ 2 ) dλ 2

(15.10) (15.11)

The Euler equation, Eq. (15.9), had been encountered before (e.g., in Chapter 4). If k22 D l(l C 1) ¤ 0, then the solutions are R(r) D Ar l C B r
l .

(15.12)

The other two equations are known as the Lamé equation (Appendix D). We note that they have five regular singular points, at ˙b, ˙c, and infinity. If one writes k22 D l(l C 1), k32 D q(b 2 C c 2 ), the solutions can be given as follows: q

q

Θ (θ ) D AE l (θ ) C B F l (θ ) , q

q

Λ(λ) D AE l (λ) C B F l (λ) , q

(15.13) (15.14)

q

where E l (θ ) and F l (θ ) are known as Lamé functions of the first and second kind, respectively [6], of degree l and order q. 15.4.2 Helmholtz Equation

Similarly, using the Stäckel theory, one gets 1 d r 2 dr

 r2

dR dr



  k2 C k12
22 R D 0 , r

(15.15)

    k22 θ 2
k32 θ d2 Θ dΘ θ C C Θ D0,
(θ 2
b 2 ) (c 2
θ 2 ) dθ (θ 2
b 2 ) (c 2
θ 2 ) dθ 2

(15.16)

    k22 λ 2
k32 d2 Λ λ λ dΛ
C C ΛD0. (b 2
λ 2 ) (c 2
λ 2 ) dλ (b 2
λ 2 ) (c 2
λ 2 ) dλ 2

(15.17)

The radial equation is identical to the radial equation for the spherical polar coordinate system, that is, it is the Bessel equation. The other two equations are again the Lamé equations. If one writes k22 D l(l C 1), k32 D q(b 2 C c 2 ), the solutions can be given as follows: R(r) D j l (k r) ,

(15.18)

q

q

Θ (θ ) D AE l (θ ) C B F l (θ ) , q

q

Λ(λ) D AE l (λ) C B F l (λ) .

(15.19) (15.20)

185

186

15 Conical Coordinates

15.4.3 Schrödinger Equation

The separable potential is V(r, θ , λ) D

  „2 1 (v v1 (r) C 2 2 (θ ) C v (λ)) . 2 3 2m r (θ
λ 2 )

(15.21)

Then, the Schrödinger equation becomes "   p 2   (θ
b 2 ) (c 2
θ 2 ) @ p 2 „2 @ 1 @ 2 @ 2 ) (c 2
θ 2 ) (θ r C

b 2m r 2 @r @r r 2 (θ 2
λ 2 ) @θ @θ # p   (b 2
λ 2 ) (c 2
λ 2 ) @ p 2 @ (b
λ 2 ) (c 2
λ 2 ) C ψ(r, θ , λ) 2 2 2 r (θ
λ ) @λ @λ   „2 1 (v (15.22) C v1 (r) C 2 2 2 (θ ) C v3 (λ)) D E ψ(r, θ , λ) . 2m r (θ
λ 2 ) Let ψ(r, θ , λ) D R(r)Θ (θ )Λ(λ) .

(15.23)

Then,     1 d dR
2 r2 C v1 (r) r R dr dr # " p   2 (θ
b 2 )(c 2
θ 2 ) d p 2 v2 (θ ) dΘ 2 ) (c 2
θ 2 ) (θ C
b C
r 2 (θ 2
λ 2 ) Θ dθ dθ r 2 (θ 2
λ 2 ) # " p   (b 2
λ 2 ) (c 2
λ 2 ) d p 2 v dΛ (λ) 3 (b
λ 2 ) (c 2
λ 2 ) C 2 2 C
r 2 (θ 2
λ 2 ) Λ dλ dλ r (θ
λ 2 ) D k2 .

(15.24)

Let p

p  v2 (θ ) dΘ (θ 2
b 2 ) (c 2
θ 2 ) C 2 (θ
λ 2 ) dθ p   (b 2
λ 2 ) (c 2
λ 2 ) d p 2 v3 (λ) dΛ 2 ) (c 2
λ 2 ) (b C 2
λ
(θ 2
λ 2 ) Λ (θ
λ 2 ) dλ dλ   1 d dR r2
v1 (r)r 2 C k 2 r 2 D R dr dr




(θ 2
b 2 ) (c 2
θ 2 ) d (θ 2
λ 2 ) Θ dθ

 k2 .

15.5 Applications

Therefore,    ˚ d dR r2 C k 2
v1 (r) r 2
k2 R D 0 , dr dr   p d p 2 dΘ (θ 2
b 2 ) (c 2
θ 2 ) (θ
b 2 ) (c 2
θ 2 ) dθ dθ   2 C k2 θ
v2 (θ )
k3 Θ D 0 ,   p d p 2 dΛ (b 2
λ 2 ) (c 2
λ 2 ) (b
λ 2 ) (c 2
λ 2 ) dλ dλ  
k2 λ 2
v3 (λ)
k3 Λ D 0 .

(15.25)

(15.26)

(15.27)

15.5 Applications

The present system has been used for studying the wave propagation from elliptic cones [69] and the electrostatic potential near sharp corners [70] and wedges [71]. We give here an example using the related spheroconical coordinates. 15.5.1 Electrostatics: Dirichlet and Neumann Problems on a Plane Angular Sector

There are a few solved problems that rely exclusively on the conical coordinates. One of those considered the Laplace equation satisfying Dirichlet or Neumann boundary conditions on the surface of a plane angular sector [72]. A modified form of the conical coordinates is actually used. This is known as the spheroconical coordinates and these coordinates are related to the Cartesian coordinates as follows: p (15.28) x D r cos θ 1

02 cos2 ' , y D r sin θ p sin ' , (15.29) (15.30) z D r cos ' 1

2 cos2 θ , p where
0 D 1

2 ,
D cos β/2 (with β the wedge angle of the plane angular sector) and r0,

0θ π,

0  '  2π .

The Laplace equation separates into a radial equation,   d 2 dR r
ν(ν C 1)R D 0 , dr dr

(15.31)

and two angular equations,   p d p dΘ 2 2 2 2 C [ν(ν C 1)
2 sin2 θ C µ]Θ 1

cos θ 1

cos θ dθ dθ D0,

(15.32)

187

188

15 Conical Coordinates

  p d p dΦ C [ν(ν C 1)
02 sin2 '
µ]Φ 1

02 cos2 ' 1

02 cos2 ' d' d' D0,

(15.33)

where µ and ν are the two separation constants. The two angular equations are clearly coupled via the two separation constants. As we have seen a few times before, the solution to the radial equation is simply R(r) D Ar ν C B r
(νC1) ,

(ν > 0) .

For the boundary-value problem of interest, r is measured from the tip of the plane angular sector. Thus, for the solution to be well behaved, only the first term in the general solution should be retained. Hence, the complete solution to the Laplace equation can be written as V(r, θ , ') D r ν Y(θ , ') .

(15.34)

This surprisingly painless result tells us that the r dependence of the potential near the tip of the plane angular sector depends upon the value of the separation constant ν. In turn, the latter is determined by the wedge angle β and the boundary condition on the plane angular sector. Two sets of boundary conditions were studied in [72]:  Even Dirichlet Θ e0 (0) D 0 ,

Θ e (π) D 0 ,

Φe0 (0) D 0 ,

Φe0 (π) D 0 .

(15.35)

 Odd Neumann Θ o (0) D 0 ,

Θ o0 (π) D 0 ,

Φo (0) D 0 ,

Φo (π) D 0 .

(15.36)

Consider, for example, the even Dirichlet boundary condition. The Dirichlet part on the boundary surface means Θ (π) D 0. Even means Θ 0 (0) D 0. Regarding the function Φ , it is 2π periodic and, coupled with the fact that we are looking for the even solution (i.e., Φ 0 (0) D 0), one can show that Φe0 (0) D 0. Hence, the solution to the plane angular sector boundary-value problem is as follows. The eigenvalues (µ, ν) are to satisfy the periodicity condition on Φ and the appropriate boundary condition on Θ . The numerical approach adopted in [72] was to start with a value of ν and compute the corresponding µ values to satisfy each of the two angular equations, and then to vary ν until the two values of µ do not differ within the tolerance criterion. A few of the eigenvalues that were obtained for a plane angular sector of wedge angle β D 60ı and with a Dirichlet boundary condition are reproduced in Table 15.1.

15.6 Problems Table 15.1 Eigenvalues of the planar angular sector Laplace problem with a Dirichlet boundary condition and β D 60ı . Reproduced from [72]. ν

µ

0.240 100 1.061 291

0.036 081
0.738 682

1.347 988

0.404 089

Finally, we note that there are some similarities between this problem of the plane angular sector in spheroconical coordinates and the quantum ice cream problem in spherical polar coordinates discussed in Chapter 11. One might wonder about the relevance of the study of a plane angular sector. Abawi et al. [72] showed that it is related to the problem of finding the eigenvalues of the Laplacian on a sphere with a boundary condition specified on a segment of a great circle. Maybe more interestingly, one also notes that the plane angular sector geometry is a degenerate case of an elliptic cone when the angle θ is π. Thus, a similar treatment for an arbitrary angle θ D θ0 should allow one to study boundary-value problems on an elliptic cone.

15.6 Problems

1. Obtain the Stäckel matrix given in the text. 2. Obtain the Bôcher type of the Lamé equation. 3. Spheroconical coordinates. a. Show that the conical coordinates (r, θ , λ) rθ λ , bc p r (θ 2
b 2 ) (b 2
λ 2 ) yD p , b (c 2
b 2 ) p r (c 2
θ 2 ) (c 2
λ 2 ) zD p , c (c 2
b 2 )

xD

0r<1,

b2 < θ 2 < c2 ,

0 < λ2 < b2 ,

can be transformed into the spheroconical coordinates x D r sin θ sin ' , p y D r cos ' 1

2 cos2 θ , p z D r cos θ 1

02 cos2 ' ,

189

190

15 Conical Coordinates

where
0 D

p 1

2 and

r0,

0θ π,

0  '  2π .

b. Find the scale factors of the spheroconical coordinate system. 4. Show how the Helmholtz equation is separated using the Stäckel theory (i.e., derive Eqs. (15.15)–(15.17)). 5. For the Laplace equation, find the solution to the radial equation for k2 D 0. 6. For the Helmholtz equation, find the solution independent of θ and λ. Verify that the radial solution is identical to that in the spherical polar coordinate case. 7. Derive the conditions on the Θ and Φ functions for the odd Neumann boundary condition of the Laplace problem on a plane angular sector.

191

16 Ellipsoidal Coordinates 16.1 Introduction

The previous nine coordinate systems can all be viewed as degenerate cases of a more general system known as the (confocal) ellipsoidal coordinates. The separation of the Laplace equation leads to three identical ordinary equations, the Lamé equation, all coupled by the two separation constants. The ellipsoidal or Lamé wave equation results when one separates the Helmholtz equation in the ellipsoidal coordinate system [6, 33, 73]. The solutions of the Lamé wave equation are very complicated and the reasons for the difficulty are discussed in the paper by Arscott et al. [74]. Formally, this is related to the presence of three regular singularities and one irregular singularity at infinity which originates from the confluence of two regular singularities. Computationally, series solutions involve up to a five-term recursion relation. Furthermore, the ordinary differential equations, though being separated in their coordinates, are completely coupled via the two separation constants and the wave number. Not surprisingly, therefore, very few attempts have been made at a complete solution. Formal solutions were briefly discussed in the Bateman manuscript, Vol. 3 [33]. Some basic properties of the equation (in the Jacobian elliptic form) were worked out by Arscott in 1959 [75] and methods for evaluating a separation constant were briefly discussed but not applied; this paper also contains a short review of earlier work. In 1965, Arscott obtained the recurrence relations [76] of one of the eight types of ellipsoidal wave functions. However, it was not until 1983 that Arscott et al. [74] evaluated the separation constants explicitly and, even then, only again for one of the eight types of ellipsoidal wave functions. Given the paucity of work on the single ordinary differential equation, it is of course not surprising that the full solution to the three-dimensional problem is even more lacking. One exception is a series of papers by a Russian group [77, 78] that gave the angular and radial Lamé wave functions, the latter being eigenfunctions of the continuous spectrum of the Helmholtz equation in a semi-infinite interval. Therefore, it appears discrete eigensolutions to the interior Dirichlet problem had not been given until our recent work [79]. The Lamé wave equation has been reviewed [80].

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

192

16 Ellipsoidal Coordinates

16.2 Coordinate System

The ellipsoidal coordinates are defined by three families of orthogonal confocal quadric surfaces. (Quadric surfaces are defined by an equation of second degree in the three coordinates.) The latter are defined by the equation

ξ2

x2 y2 z2 C 2 C 2 D1, 2 2
a ξ
b ξ
c2

abc0,

(16.1)

where different types of surfaces are obtained for different values of the parameter ξ . Those three types of surfaces are obtained as follows [5]:  ξ  ξ1 > a All three terms on the left-hand side of Eq. (16.1) are positive and the equation describes an ellipsoidal surface. The surface traces ellipses on the Cartesian planes (Figure 16.1) with the Cartesian axes intercepts being at x D ˙(ξ 2
a 2 )1/2 , y D ˙(ξ 2
b 2 )1/2 , z D ˙(ξ 2
c 2 )1/2 . As ξ ! a C , the x intercepts both go to zero. Hence, the ellipsoidal surface goes into the inside of the ellipse in the y z plane with major axis 2(a 2
c 2 )1/2 and minor axis 2(a 2
b 2 )1/2 .  a > ξ  ξ2 > b The first term on the left-hand side of Eq. (16.1) is negative. The quadric surfaces are then a family of confocal hyperboloids of one sheet. The cross sections on the Cartesian planes are ellipses on y z, and hyperbolas on x y and x z. Note that the surface does not include the origin; indeed, a projection of the hyperboloid onto the y z plane is a region away from the origin. Hence, when ξ ! a
, the surface becomes the exterior of the ellipse with major axis 2(a 2
c 2 )1/2 and minor axis 2(a 2
b 2 )1/2 .

z

y

z (ξ2–c2) ½

(ξ2–c2) ½ (ξ2–b2) ½

x (ξ –a ) ½ 2

2

y

x (ξ –a ) ½ 2

2

Figure 16.1 Quadric surface with ξ > a.

(ξ –b ) ½ 2

2

16.2 Coordinate System x,y

x,y

y(x=0)

x(y=0)

(ξ2–c2)½

z

x

y

z

Figure 16.2 Quadric surface with b > ξ > c.

 b > ξ  ξ3 > c The first two terms on the left-hand side of Eq. (16.1) are negative. The quadric surfaces are confocal hyperboloids of two sheets. In both cases, the hyperbolas intersect the z axis at (ξ 2
c 2 )1/2 when x D y D 0 (Figure 16.2). It is also clear from the figure that when ξ ! c
, the hyperbolas become the x y plane.

Since all ξi are positive, from Eq. (16.1), one of each of the three types of quadrics passes through each point (x, y, z) for which x y z ¤ 0 (there are three roots since the equation is cubic in ξ 2 ). 16.2.1 Coordinates (ξ1 , ξ2 , ξ3 )

Since each value of ξ3 leads to a surface, one can use them to define a new orthogonal curvilinear coordinate system (Figure 16.3). The relationship of the ellipsoidal coordinates ξ1 , ξ2 , ξ3 to the Cartesian ones is [5, 6, 81] 


2 ξ1
a 2 ξ22
a 2 ξ32
a 2 2 , x D (a 2
c 2 ) (a 2
b 2 ) 


2 ξ
b 2 ξ22
b 2 ξ32
b 2 y2 D 1 , (b 2
c 2 ) (b 2
a 2 ) 


2 ξ
c 2 ξ22
c 2 ξ32
c 2 z2 D 1 , (16.2) (a 2
c 2 ) (b 2
c 2 ) with ξ1 > a > ξ2 > b > ξ3 > c .

(16.3)

193

194

16 Ellipsoidal Coordinates (0)

z

ξ2 = ξ2

(0)

ξ1 = ξ 1

x y (0) ξ3= ξ 3

Figure 16.3 Ellipsoidal coordinates.

This transformation can be verified by, for example, showing that Eqs. (16.2) satisfy Eq. (16.1). Since Eqs. (16.2) are symmetric in ξ1 , ξ2 , ξ3 , the same quadric equation is satisfied by all three surfaces. Without loss of generality, one can set c D 0; however, we will carry c along as far as possible. In Eqs. (16.2), note the interchange symmetry: interchanging a ! b (b ! c) leads from the x equation to the y equation (y equation to z equation). Also, one set of ξ1 , ξ2 , ξ3 corresponds to eight Cartesian points. Finally, all the quantities appear as squared. The above transformation is known as the algebraic representation. Two other ways of defining the ellipsoidal coordinates are via the Weierstrassian and Jacobian elliptic functions [5, 33, 73]. 16.2.2 Ellipsoid

In general, a, b, c can take on any values subject to the constraint given in Eq. (16.1); the description of an ellipsoid fixes their values. If the Cartesian coordinates of the points of intersection of the ellipsoid with the Cartesian axes are ˙x0 , ˙y 0 , ˙z0 , then, Eq. (16.1) for ξ D ξ1 relates x0 , y 0 , z0 to a, b, c (see also Figure 16.1):



2 ξ10
a2 2 ξ10





b

2 ξ10
c

1/2

 2 1/2  2 1/2

D x0 , D y0 , D z0 ,

(16.4)

where ξ10 is the value of ξ1 on the ellipsoidal surface. We observe that the ordering a > b > c implies z0 > y 0 > x0 . There are four unknown on the left-hand side of Eqs. (16.4), and three constants are given on the right-hand side. There is, therefore, the freedom in setting one of the unknowns; it is common to choose

16.3 Differential Operators

c D 0. Then, ξ10 D z0 , 1/2
2
x02 , a D ξ10
2  2 1/2 b D ξ10
y 0 .

(16.5)

All three constants are positive by definition.

16.3 Differential Operators 16.3.1 Metric

Using Eqs. (2.5) and (16.2), one obtains the scale factors as

 ξ12 ξ12
ξ22 ξ12
ξ32 2 

 , h1 D
2 ξ1
a 2 ξ12
b 2 ξ12
c 2

 ξ2 ξ2
ξ2 ξ2
ξ2 h 22 D
2 2 22
2 1 22
2 3 2  , ξ2
a ξ2
b ξ2
c

 ξ2 ξ2
ξ2 ξ2
ξ2 h 23 D
2 3 23
2 1 23
2 2 2  . ξ3
a ξ3
b ξ3
c

(16.6)

16.3.2 Operators

In the following, without loss of generality, we will set a D 0 (following Moon and Spencer). The differential operators are obtained by substituting Eq. (16.6) into Eqs. (2.7)–(2.12). 16.3.2.1 Gradient The gradient operator is

"
rD




ξ12
b 2

ξ12

"
C





ξ22





ξ22
b

ξ12
c 2 ξ12


2


2

ξ22
ξ3




 #1/2

ξ32



e ξ1

c 2
ξ22

 #1/2

 2

ξ12
ξ2

@ @ξ1

e ξ2

"

 #1/2 b 2
ξ32 c 2
ξ32 @ @ 
 C
2 e ξ3 . @ξ2 @ξ3 ξ1
ξ32 ξ22
ξ32 (16.7)

195

196

16 Ellipsoidal Coordinates

16.3.2.2 Divergence The divergence of a vector field V is

1/2
2 1/2 i 
 ξ1
c 2 ξ12
b 2 @ h
2 1/2 2 1/2 
 ξ ξ
ξ
ξ V rV D
2 1 1 ξ 1 2 3 @ξ1 ξ1
ξ22 ξ12
ξ32
2 
 1/2 1/2 i 
2  c 2
ξ22 b
ξ2 @ h
2 2 1/2 2 1/2  ξ ξ C
2 2 2
2
ξ
ξ V ξ 2 1 2 2 3 @ξ2 ξ1
ξ2 ξ3
ξ22
2 
 1/2 1/2 i 1/2
2 1/2 b
ξ2 ξ 2
c2 @ h
2  ξ2
ξ32 ξ1
ξ32 C
2 3 2 
32 Vξ3 . (16.8) 2 @ξ3 ξ1
ξ3 ξ3
ξ2


16.3.2.3 Circulation The circulation is








1/2 ξ12
c 2 ξ22
b 2 c 2
ξ22 b 2
ξ32 c 2
ξ32
2 

 ξ1
ξ22 ξ12
ξ32 ξ22
ξ32 ˇ h 2 2 2 2 i1/2 ˇ h 2 2 2 2 i1/2 h 2 2 2 2 i1/2 ˇ ( ξ1
ξ2 )( ξ1
ξ3 ) ˇ (ξ2
ξ3 )( ξ1
ξ2 ) ( ξ1
ξ3 )(ξ2
ξ3 ) ˇ e e e ξ1 ξ2 ξ3 ˇ ( ξ22
b 2 )(c 2
ξ22 ) ( b 2
ξ32 )( c 2
ξ32 ) ˇ ( ξ12
b 2 )( ξ12
c 2 ) ˇ ˇ ˇ @ @ @ ˇ ˇ . @ξ @ξ @ξ 1 2 3 ˇh 2 2 2 2 i1/2 ˇ h h i i 1/2 1/2 ˇ ( ξ1
ξ2 )( ξ1
ξ3 ) ˇ (ξ22
ξ32 )( ξ12
ξ22 ) ( ξ12
ξ32 )(ξ22
ξ32 ) ˇ 2 2 2 2 ˇ V V V ξ ξ ξ 2 2 2 2 1 2 3 2 2 2 2 ( ξ
b )( ξ
c ) ( ξ
b )(c
ξ ) ( b
ξ )( c
ξ )

rV D

1

ξ12
b 2




1

2

2

3

3

(16.9) 16.3.2.4 Laplacian The Laplacian is readily obtained:

q
r2 D q


2 q
3 

 

 ξ22
a 2 ξ22
b 2 ξ22
c 2 @ 6 ξ22
a 2 ξ22
b 2 ξ22
c 2 @ 7

 4 5 @ξ2 ξ2 @ξ2 ξ2 ξ22
ξ12 ξ22
ξ32

q


2 q
3 

 

 ξ32
a 2 ξ32
b 2 ξ32
c 2 @ 6 ξ32
a 2 ξ32
b 2 ξ32
c 2 @ 7

 4 5. @ξ3 ξ3 @ξ3 ξ3 ξ32
ξ12 ξ32
ξ22

C

C

2 q
3 

 

 ξ12
a 2 ξ12
b 2 ξ12
c 2 @ 6 ξ12
a 2 ξ12
b 2 ξ12
c 2 @ 7

 4 5 @ξ1 ξ1 @ξ1 ξ1 ξ12
ξ22 ξ12
ξ32

(16.10) Not surprisingly, the Laplacian has coordinate symmetry. However, since some of the square roots are negative as currently written, they will be converted into the

16.4 Separable Equations

positive forms. For c D 0, q



  ξ12
a 2 ξ12
b 2 @ q

2  @ 2 2 2 
 r D
2 ξ ξ
a
b 1 1 @ξ1 ξ1
ξ22 ξ12
ξ32 @ξ1 q

  a 2
ξ22 ξ22
b 2 @ q

2  @ 2
ξ2 2 

b C
2 a ξ 2 2 @ξ2 ξ1
ξ22 ξ22
ξ32 @ξ2 q

  a 2
ξ32 b 2
ξ32 @ q

 @ 2
ξ2 2
ξ2 
 . C
2 a b 3 3 @ξ3 ξ1
ξ32 ξ22
ξ32 @ξ3 2

(16.11)

16.4 Separable Equations 16.4.1 Laplace Equation

Let ψ D F (ξ1 ) G (ξ2 ) H (ξ3 ) .

(16.12)

The Laplace equation r 2 ψ D 0 becomes    1 d q
2 
 dF ξ1
a 2 ξ12
b 2 F dξ1 dξ1 q  q




 dG d 1 2 2 a 2
ξ22 ξ22
b 2 ξ1
ξ3 a 2
ξ22 ξ22
b 2 C G dξ2 dξ2 q  q




 1 dH d a 2
ξ32 b 2
ξ32 ξ12
ξ22 a 2
ξ32 b 2
ξ32 D0. C H dξ3 dξ3 (16.13)

q


ξ12
a 2




ξ12
b 2




ξ22
ξ32

If we now let q


 d dξ1 q

 d a 2
ξ22 ξ22
b 2 dξ2 q

 d a 2
ξ32 b 2
ξ32 dξ3 ξ12
a 2




ξ12
b 2

q
q


ξ12
a 2





2

ξ12
b 2

 dF



dξ1   dG


 D α 2 ξ12

F ,


 D
α 2 ξ22

G , dξ2 q 

 dH
 a 2
ξ32 b 2
ξ32 D α 2 ξ32

H , dξ3 (16.14) a 2
ξ2

ξ22
b 2

we can show by substituting Eqs. (16.14) into Eq. (16.13) that the latter is satisfied:


ξ22
ξ32








 α 2 ξ12


ξ12
ξ32 α 2 ξ22

C ξ12
ξ22 α 2 ξ32

D 0 .

197

198

16 Ellipsoidal Coordinates

Hence, Eqs. (16.14) are the separated ordinary differential equations. Equations (16.14) all have similar form. One can write a generic one as     z

α2 z 2 d2 F dF z C C F D 0 , (16.15) C (z 2
a 2 ) (z 2
b 2 ) dz (z 2
a 2 ) (z 2
b 2 ) dz 2 or, equivalently, 

   

 z 4
a 2 C b 2 C a 2 b 2 F 00 C z 2z 2
a 2 C b 2 F 0 C

α 2 z 2 F D 0 . (16.16)

This is known as the Lamé equation (Appendix D). From Eq. (16.15), we see that the equation has regular singular points at z D ˙b, ˙a, 1. It is shown in Appendix D that, if the following substitutions are made, 
a 2 ! b , b 2 ! c ,
! b 2 C c 2 q , α 2 ! p (p C 1) , the general solution can be written as F(z) D AE pq (z) C B F pq (z) , q

(16.17)

q

where E p (θ ) and F p (θ ) are known as Lamé functions of the first and second kind, respectively [6], of degree p and order q. It is possible to simplify the form of the Lamé equation via a coordinate transformation too. Here we follow the definition of the elliptic functions (see also Appendix I) from Morse and Feshbach [5]: Zξ1 λ (ξ1 ) D a a

Za µ (ξ2 ) D a ξ2

Zξ3 ν (ξ3 ) D a 0

dt , p 2 (t
a 2 ) (t 2
b 2 ) dt , p (a 2
t 2 ) (t 2
b 2 ) dt . p (a 2
t 2 ) (b 2
t 2 )

(16.18)

Equation (16.14) becomes a2

  d2 F C

m(m C 1)ξ12 F D 0 , 2 dλ

(16.19)

with similar equations for the other coordinates: a2

 d2 G 


m(m C 1)ξ22 G D 0 , 2 dµ

(16.20)

a2

  d2 H C

m(m C 1)ξ32 H D 0 . 2 dν

(16.21)

16.4 Separable Equations

16.4.2 Helmholtz Equation

The Laplacian is obviously the same as before. Let ψ D F (ξ1 ) G (ξ2 ) H (ξ3 ) .

(16.22)

The Helmholtz equation r ψ C k ψ D 0 becomes   q


 1 d q
2 
 dF ξ12
a 2 ξ12
b 2 ξ22
ξ32 ξ1
a 2 ξ12
b 2 F dξ1 dξ1 q  q




 dG 1 d a 2
ξ22 ξ22
b 2 ξ12
ξ32 a 2
ξ22 ξ22
b 2 C G dξ2 dξ2 q  q




 d 1 dH a 2
ξ32 b 2
ξ32 ξ12
ξ22 a 2
ξ32 b 2
ξ32 C H dξ3 dξ3
2 
2 
2  2 2 2 2 (16.23) C k ξ1
ξ2 ξ2
ξ3 ξ1
ξ3 D 0 . 2

2

If we now let   q

 d q
2 
 dF
 ξ12
a 2 ξ12
b 2 ξ1
a 2 ξ12
b 2 D
k 2 ξ14 C α 2 ξ12

F , dξ1 dξ1 q  q




 d dG a 2
ξ22 ξ22
b 2 a 2
ξ22 ξ22
b 2 D

k 2 ξ24 C α 2 ξ22

G, dξ2 dξ2   q

 d q

 dH
 a 2
ξ32 b 2
ξ32 a 2
ξ32 b 2
ξ32 D
k 2 ξ34 C α 2 ξ32

H , dξ3 dξ3 (16.24) just as for the Laplace equation, we can show by substituting Eqs. (16.24) into Eq. (16.23) that the latter is satisfied. We only need to demonstrate this for the new terms proportional to k 2 :
2    




ξ2
ξ32 ξ14
ξ12
ξ32 ξ24 C ξ12
ξ22 ξ34 C ξ12
ξ22 ξ22
ξ32 ξ12
ξ32 D 0 . Hence, Eqs. (16.24) are the separated ordinary differential equations. All three equations are of the form


ξ12
a 2




ξ12
b 2

 d2 F   dF  2 4
 Cξ1 2ξ12
a 2 C b 2 C k ξ1
α 2 ξ12 C
F D 0 . 2 dξ1 dξ1 (16.25)

The latter is known as the Lamé wave equation (Appendix D). It is shown in Appendix D that, if the following substitutions are made, a2 ! b ,

b2 ! c ,


! (b 2 C c 2 )q ,

α 2 ! p (p C 1) ,

the general solution can be written as F(z) D AE pq (z) C B F pq (z) , q E p (θ )

q F p (θ )

(16.26)

and are known as Lamé wave functions of the first and second where kind, respectively [6], of degree p and order q.

199

200

16 Ellipsoidal Coordinates

16.5 Applications

There have been limited applications of the ellipsoidal coordinates, almost all for the Laplace problem. Here, the main use has been in boundary-value problems whereby the domain of interest is bounded by an ellipsoidal surface and the solution is expressed in terms of ellipsoidal harmonics. Applications have been made to, for example, electrostatics [82–84], geodesy [85], gravitational theory [1], and a four-shell ellipsoidal model of the human head [3]. Applications for the Helmholtz equation include for modeling electron states [86] and phonon states [87] in triaxial ellipsoidal quantum dots, and of eigenmodes of triaxial ellipsoidal acoustical cavities [88]. A study of the scalar wave equation for application in optics has also been carried out [34]. Integrable systems in ellipsoidal coordinates have been studied by Gadella et al. [89]. 16.5.1 Interior Problem for the Laplace Equation

We now explain how to find series solutions to the Laplace equation that are finite. The following discussion follows Morse and Feshbach [5]. Note that there are three ordinary differential equations (in fact, all Lamé equations), each with its own domain. Nevertheless, they share the same two separation constants. Thus, all three solutions must be well behaved for given α 2 and
. This implies we need solutions that are well behaved for the whole domain 0 < z < 1. The recurrence relation obtained from a power-series solution, X F(z) D dn z n , n

is derived in Appendix D and is repeated here for convenience:
     a 2 b 2 n(n
1)d n D (n
2)2 a 2 C b 2

d n
2 C α 2
(n
3)(n
4) d n
4 . (16.27) We now see if convergent infinite-series solutions can be found. Using the ratio test [5], ) (
 (n
2)2 a 2 C b 2

a nC1 α 2
(n
3) (n
4) d n
4 z2 dn z2 . C D D an d n
2 a 2 b 2 n(n
1) a 2 b 2 n(n
1) d n
2 (16.28) Let γ n  d n /d n
2 . Then,


 (n
2)2 a 2 C b 2

α 2
(n
3)(n
4) 1 C , a 2 b 2 n(n
1) a 2 b 2 n(n
1) γ n
2
2  2 2 (n
3)(n
4) n!1 (n
2) a C b . D γ n
2
2 2 n(n
1)a 2 b 2 a b n(n
1)

γn D H)

γ n γ n
2

16.5 Applications

Now, (n
2)2 D 1
n(n
1) (n
3)(n
4) D 1
n(n
1)

3n
4 n(n
1) 6 n
12 n(n
1)

3 , n 6 n!1 D 1
, n n!1

D

1


giving    
2  a C b2 3 6 1 γ n γ n
2
1
γ C 1
D0. n
2 n a2 b2 n a2 b2

(16.29)

In the limit n ! 1, γ n
2 ! γ n ! γ and Eq. (16.29) becomes a quadratic in γ with roots 8 31/2 9  2  !2   
2 2
2  = a C b2 a C b2 1< 3 6 3 1 5 ˙4 1

4 1
1
γD 2 2 2 2 2 2 ; 2: n a b n a b n a b 8 9  "  !#1/2 =  
2 
2 a C b2 a
b2 3 1< 3 1
 ˙ 1
; 2: n a2 b2 n a2 b2     3 1 3 1 D 1
, 1
. (16.30) n a2 n b2 Thus, jz 2 γ j < 1 if z < a or z < b. An infinite series will only be convergent up to z D b or z D a. Note that this result is independent of the values of
and α 2 . Therefore, there will be two convergent infinite series for all
and α 2 , but not three since the third solution is defined for the z > a domain. Indeed, expanding about any point gives the same result [5]: there are only convergent infinite series up to only two of the five singular points. However, for specific values of
and α 2 , one of the series terminates giving a polynomial solution finite at four of the five regular singular points. Thus, to build finite solutions to the Laplace equation, polynomials will have to be used. Polynomial solutions of the Lamé equation about the ordinary point z D 0 are derived in Appendix D. Since all three ordinary differential equations are identical in form and have the same two separation constants, this implies that the product solution of the partial differential equation involves the same three Lamé functions. Such product functions are known as ellipsoidal harmonics. 16.5.1.1 Ellipsoidal Harmonic of the First Species Starting with the Lamé function

E00 (z) D 1 , a possible solution of the Laplace equation is, therefore, E00 (ξ1 ) E00 (ξ2 ) E00 (ξ3 ) . This is known as an ellipsoidal harmonic of the first species [5].

(16.31)

201

202

16 Ellipsoidal Coordinates

16.5.1.2 Ellipsoidal Harmonic of the Second Species If one chooses

E10 (z) D z , a possible solution of the Laplace equation is, therefore, AE10 (ξ1 ) E10 (ξ2 ) E10 (ξ3 ) D Aξ1 ξ2 ξ3 D Aab z . This is an ellipsoidal harmonic of the second species [5]. Or, if the Lamé function is p E11 (z) D z 2
a 2 ,

(16.32)

(16.33)

the corresponding polynomial solution to the Laplace equation is 1/2
1/2
1/2
ξ2
a 2 ξ3
a 2 C E11 (ξ1 ) E11 (ξ2 ) E11 (ξ3 ) D C ξ1
a 2 1/2
D C a a2
b2 x.

(16.34)

This is also an ellipsoidal harmonic of the second species. 16.5.1.3 Ellipsoidal Harmonic of the Third Species If the Lamé function is p E22 (z) D z z 2
a 2 ,

(16.35)

the corresponding polynomial solution to the Laplace equation is 1/2
1/2
1/2
ξ2
a 2 ξ3
a 2 ξ1 ξ2 ξ3 D E22 (ξ1 ) E22 (ξ2 ) E22 (ξ3 ) D D ξ1
a 2  
2 2 1/2 D D ab a a
b xz . (16.36) This is an ellipsoidal harmonic of the third species. Or, if the Lamé function is p p E42 (z) D z 2
a 2 z 2
b 2 ,

(16.37)

the corresponding polynomial solution to the Laplace equation is E42 (ξ1 ) E42 (ξ2 ) E42 (ξ3 ) 1/2
1/2
1/2
1/2
1/2
1/2
ξ2
a 2 ξ3
a 2 ξ1
b 2 ξ2
b 2 ξ3
b 2 D ξ1
a 2 1/2
2 1/2
b
a2 D ab a 2
b 2 xy . (16.38) This is also an ellipsoidal harmonic of the third species.

16.5 Applications

16.5.1.4 Ellipsoidal Harmonic of the Fourth Species If the Lamé function is p p E(z) D z z 2
a 2 z 2
b 2 ,

(16.39)

the corresponding polynomial solution to the Laplace equation is E (ξ1 ) E42 (ξ2 ) E42 (ξ3 )
1/2
1/2
1/2 D ξ1 ξ2 ξ3 ξ1
a 2 ξ2
a 2 ξ3
a 2 1/2
1/2
1/2
ξ2
b 2 ξ3
b 2  ξ1
b 2
1/2
2 1/2 b
a2 D a2 b2 a2
b2 xyz .

(16.40)

This is an ellipsoidal harmonic of the fourth species. 16.5.2 Elliptic Functions

One can write the Laplace equation directly in terms of elliptic functions. In Perram and Stiles [82] and Romain and Jean-Pierre [1], the elliptic functions are written as Zξ1 λ 1 (ξ1 ) D a

Zξ2 λ 2 (ξ2 ) D b

Zξ3 λ 3 (ξ3 ) D 0

dt , p 2 (t
a 2 ) (t 2
b 2 ) dt , p 2 (a
t 2 ) (t 2
b 2 ) dt . p 2 (a
t 2 ) (b 2
t 2 )

(16.41)

Then, dλ 1 1 D q

 , dξ1 2 ξ1
a 2 ξ12
b 2 dψ dψ dλ 1 dψ , D D q

 dξ1 dλ dξ1 ξ12
a 2 ξ12
b 2 dλ and similarly for the others. Equation (16.13) becomes


ξ22
ξ32

 @2 ψ
2  @2 ψ
2  @2 ψ C ξ1
ξ32 C ξ1
ξ22 D0. 2 2 @λ 1 @λ 2 @λ 23

(16.42)

We note that there is a typographical error in the analogous equation in Romain and Jean-Pierre [1]. One can now proceed to separate it. Even though we have already considered this problem of separation, we reproduce the approach in terms

203

204

16 Ellipsoidal Coordinates

of elliptic functions because it is both simple and also demonstrates another way of achieving the result that all three functions have to be the same. To start, let ψ D E1 (ξ1 ) E2 (ξ2 ) E3 (ξ3 ) .

(16.43)

Then, Eq. (16.42) becomes


  
2
2 ξ1
ξ32 d2 E2 (ξ2 ) ξ1
ξ22 d2 E3 (ξ3 ) ξ22
ξ32 d2 E1 (ξ1 ) C C D0. E1 (ξ1 ) E2 (ξ2 ) E3 (ξ3 ) dλ 21 dλ 22 dλ 23 (16.44)

Following the same argument as before, the solutions must be valid for all values of the coordinates. In particular, for ξ1 D ξ2 , we have E1 D
E2 . Together we find E1 (ξ1 ) D
E2 (ξ2 ) D E3 (ξ3 ) .

(16.45)

If we now let φ (ξi ) D

1 d2 E (ξi ) , E (ξi ) dλ 2i

Eq. (16.44) can be rewritten as   
2

ξ2
ξ32 φ (ξ1 ) C ξ12
ξ32 φ (ξ2 ) C ξ12
ξ22 φ (ξ3 ) D 0 .

(16.46)

(16.47)

We finally achieve separation by choosing the coordinates to be zero in turn. For example, if ξ1 D 0, φ (ξ2 ) C φ(0) φ (ξ3 )
φ(0) D
q. ξ32 ξ22 By also setting ξ3 D 0 and writing φ(0) D
p , we get the following three ordinary differential equations: φ (ξ1 ) D

1 d2 E (ξ1 ) D q ξ12
p , E (ξ1 ) dλ 21

φ (ξ2 ) D

1 d2 E (ξ2 ) D
q ξ22 C p , E (ξ2 ) dλ 22

φ (ξ3 ) D

1 d2 E (ξ3 ) D q ξ32
p . E (ξ3 ) dλ 23

(16.48)

16.5.3 Dirichlet Problem for the Helmholtz Equation: ATZ Algorithm

A computational algorithm for solving the interior Dirichlet problem for the Helmholtz equation was first proposed by Arscott et al. [74] (ATZ algorithm) and extended by Willatzen and Lew Yan Voon [79]. For completeness, this algorithm is reproduced here in its entirety.

16.5 Applications

16.5.3.1 First Solution to the Ellipsoidal Wave Equation The goal is to find solutions to the Helmholtz equation that are finite and smooth everywhere within an ellipsoid defined uniquely by its center (assumed to be located at (x, y, z) D (0, 0, 0)) and ellipsoid intersections ˙x0 , ˙y 0 , and ˙z0 with the x, y, and z axes, respectively (Figure 16.1). The three separated ordinary differential equations (e.g., Eq. (16.25)) are rewritten as

t(t
1)(t
c)

  dX
1 d2 X C 3t 2
2(1 C c)t C c C λ C µ t C γ t2 X D 0 . 2 dt 2 dt (16.49)

It is always possible to write X in the general form [74] X(t) D t /2 (t
1) σ/2 (t
c) τ/2 F(t) ,

(16.50)

where , σ, and τ are either 0 or 1, that is, eight different types of X are possible, and F is an integral function of t. Note that since t / ξ 2 , this expression for X(t) guarantees that any solution is still one of the four forms of solutions to each of the equations in Eq. (16.24). The function F must be found numerically as follows. Inserting Eq. (16.50) into Eq. (16.49) leads to the differential equation [74]: t(t
1)(t
c) D0,

 dF
 d2 F 1
A 2 t 2
2A 1 t C A 0 C λ
λ 0 C (µ C µ 0 ) t C γ t 2 F C dt 2 2 dt (16.51)

where  1 ( C τ)2 C ( C σ)2 c , 4 1 µ 0 D ( C σ C τ) ( C σ C τ C 1) , 4 A 0 D (2 C 1)c , λ0 D

A 1 D (1 C )(1 C c) C τ C σ c , A 2 D 2( C σ C τ) C 3 .

(16.52)

Series solutions will now be found for Eq. (16.51): F(t) D

1 X

a r (t
t0 ) r ,

(16.53)

rD0

where t0 is a constant. Inserting Eq. (16.53) in Eq. (16.51) and employing the identity theorem for power series yields the recursion expression  2  1 0 C N rC1 (r C 1) C N rC1 r(r C 1) a rC1 N r0 a r C N rC1  2  1 0 C N rC2 C N rC2 (r C 2) C N rC2 (r C 2)(r C 1) a rC2  0  1 0 C N rC3 (r C 3) C N rC3 (r C 3)(r C 2) a rC3 C N rC4 a rC4 D 0 , (16.54)

205

206

16 Ellipsoidal Coordinates

with N r0 D γ , 1 2 A 2 , N rC1 D µ C µ 0 C 2γ t0 , 2 1 2 D 3t0
1
c , N rC2 D A 2 t0
A 1 , N rC2 D λ
λ 0 C (µ C µ 0 ) t0 C γ t02 , 1 1 1 D (2t0
1) (t0 C c) C t02
t0 , N rC3 D A 2 t02
A 1 t0 C A 0 , 2 2
 D t02
t0 (t0
c) . (16.55)

0 N rC1 D1, 0 N rC2 0 N rC3 0 N rC4

1 N rC1 D

Following Arscott et al. [74] for the case  D σ D τ D 0, one can choose the expansion parameter t0 D 1 for all eight combinations of , σ, and τ so as to avoid convergence problems with increasing n and m values (for the definition of n and m refer to the discussion in the next paragraph). With this choice of t0 , the term involving a rC4 in Eq. (16.54) disappears and the general five-term recursion formula simplifies to a four-term recursion formula. Equation (16.54) forms the starting point of the numerical approach. Firstly, one needs to specify the possible (discrete) sets of µ and λ at γ D 0 corresponding to the Laplacian problem (the Lamé equation). It follows from Eq. (16.54) that the coefficient of a rC1 for r D n
1 becomes zero whenever µ D
µ 0
n(n
1)


1 A2n . 2

(16.56)

In this case, a finite polynomial solution for F can be found by employing the condition on λ that the determinant of the first (n C 1) equations vanishes, that is, an equation of degree (n C 1) in λ is obtained. Hence, (n C 1) λ solutions for each value of n (denoted λ m n , where m D 0, 1, 2, . . . , n) are determined; the indexing is the one used for the Lamé functions. All the roots are real [74, 79]. These solutions give initial values at γ D 0 for the subsequent computation of the separation constants µ(γ ) and λ(γ ) at finite γ values. Values of µ(γ ) and λ(γ ) are obtained next using Newton’s method, which is known to be locally convergent [74] when employing sufficiently small steps in γ . In [79], ∆γ was chosen to be 0.01 starting from γ D 0. Newton’s local convergence method can be employed as follows. Once the values of µ(γ ) and λ(γ ) have been computed at a given value of γ , first-order estimates of new µ and λ values (µ C ∆µ and λ C ∆λ, respectively) corresponding to γ C ∆γ are defined as µ(γ C ∆γ ) D µ(γ ) C ∆µ , λ(γ C ∆γ ) D λ(γ ) C ∆λ ,

(16.57)

where ∆µ and ∆γ are given by ˇ ˇ @G ˇˇ @G ˇˇ C ∆λ C ∆γ ∆µ @µ ˇ[µ(γ),λ(γ),γ] @λ ˇ[µ(γ),λ(γ),γ] ˇ ˇ @H ˇˇ @H ˇˇ C ∆λ C ∆γ ∆µ @µ ˇ @λ ˇ [µ(γ),λ(γ),γ]

[µ(γ),λ(γ),γ]

ˇ @G ˇˇ D0, @γ ˇ[µ(γ),λ(γ),γ] ˇ @H ˇˇ D0, @γ ˇ [µ(γ),λ(γ),γ]

(16.58)

16.5 Applications

and  2  1 0 G(µ, λ, γ ) D N r0 a r C N rC1 C N rC1 (r C 1) C N rC1 r(r C 1) a rC1  2  1 0 C N rC2 C N rC2 (r C 2) C N rC2 (r C 2)(r C 1) a rC2  0  1 0 C N rC3 (r C 3) C N rC3 (r C 3)(r C 2) a rC3 C N rC4 a rC4 I r D M
2,  2  1 0 C N rC1 (r C 1) C N rC1 r(r C 1) a rC1 H(µ, λ, γ ) D N r0 a r C N rC1  2  1 0 C N rC2 C N rC2 (r C 2) C N rC2 (r C 2)(r C 1) a rC2  0  1 0 C N rC3 (r C 3) C N rC3 (r C 3)(r C 2) a rC3 C N rC4 a rC4 I r D M
1.

(16.59)

The parameter M is chosen as an integer value close to N/2 in the numerical implementation. It is necessary to specify how the values a r , r D 0, 1, . . . , N , for a convergent power series F are determined before the partial derivatives @G/@µ, @G/@λ, @G/@γ , @H/@µ, @H/@λ, and @H/@γ can be calculated. This is done in the following way [13]. The a r are calculated by first taking a N D 1, a N C1 D a N C2 D a N C3 D a N C4 D 0 and then computing a r backwards to a M using Eq. (16.54) with r D N
1, N
2, . . . , M . Since the solution found here satisfies a r D 0 8 r  N C 1, it is convergent by construction. Next, Eq. (16.54) is employed with r D 0, 1, 2, . . . , M
3 so as to calculate a 0 forwards to a M . The starting value of a 0 is fixed such that the two a M values are equal. The two equations left unused in the determination of the a r values are the equations for G and H in Eq. (16.59) which must be used in the application of Newton’s method (Eq. (16.57)). The coefficients @G/@µ and so on can now be found from the recursion relations N r0

 2  @a rC1 @a r 1 0 C N rC1 C N rC1 (r C 1) C N rC1 r(r C 1) @µ @µ  2  @a rC2 1 0 C N rC2 C N rC2 (r C 2) C N rC2 (r C 2)(r C 1) @µ  0  @a rC4 @a rC3 1 0 C N rC3(r C 3) C N rC3 C N rC4 (r C 3)(r C 2) @µ @µ

D
a rC1
t0 a rC2 ,  2  @a rC1 @a r 1 0 N r0 C N rC1 C N rC1 (r C 1) C N rC1 r(r C 1) @λ @λ  @a rC2  2 1 0 (r C 2) C N rC2 (r C 2)(r C 1) C N rC2 C N rC2 @λ   0 @a rC4 @a rC3 1 0 C N rC4 (r C 3)(r C 2) C N rC3(r C 3) C N rC3 @λ @λ D
a rC2 ,

207

208

16 Ellipsoidal Coordinates

N r0

 2  @a rC1 @a r 1 0 C N rC1 C N rC1 (r C 1) C N rC1 r(r C 1) @γ @γ  @a rC2  2 1 0 (r C 2) C N rC2 (r C 2)(r C 1) C N rC2 C N rC2 @γ   0 @a rC4 @a rC3 1 0 C N rC4 (r C 3)(r C 2) C N rC3(r C 3) C N rC3 @γ @γ D
a r
2t0 a rC1
t02 a rC2 ,

(16.60)

obtained by explicit differentiation of Eq. (16.59). This completes the derivation of first-order estimates of separation constants µ C ∆µ and λ C ∆λ at γ C ∆γ . The iterative procedure that follows in the search of converged values for µ C ∆µ and λ C ∆λ corresponding to the updated γ C ∆γ value is obtained by employing Newtons’ method (Eq. (16.58)) now with ∆γ D 0 until the values of µ C ∆µ and λ C ∆λ at iteration steps i and i C 1 are the same. 16.5.3.2 Second Solution to the Ellipsoidal Wave Equation Once a set of solution parameters γ , µ, and λ has been obtained, the total eigenfunction within one octant of the ellipsoid becomes

Ψ (ξ1 , ξ2 , ξ3 ) D X (ξ1 ) X (ξ2 ) X (ξ3 ) ,

(16.61)

where X satisfies the ellipsoidal wave equation (Eq. (16.49)). Note that the functional form for all three coordinates is the same; a proof of this important result can be found in, for example, Morse and Feshbach [5]. In principle, two independent solutions exist for each coordinate ξi . The numerical method described above leads to one well-behaved (finite) solution everywhere within the ellipsoid. We will next show that a second independent solution is not finite everywhere within the ellipsoid (so it will not represent a physically allowable solution for our application). A second solution Y to Eq. (16.49) is [81]   u R Zξ exp
P(v )dv Y(ξ ) D X(ξ ) du , (16.62) X(u)2 where


 v 2v 2
a 2 C b 2 , P(v ) D (v 2
a 2 ) (v 2
b 2 )

(16.63)

and X is the solution found above using the numerical method of Arscott et al. [74]. The above integral can be easily simplified so as to obtain Zξ Y(ξ ) D X(ξ )

X(u)2

p

1 (u2


b 2 ) (u2
a 2 )

du .

(16.64)

The latter result shows that lim jY(ξ )j D lim jY(ξ )j D 1 .

ξ !a

ξ !b

(16.65)

16.5 Applications

This conclusion, of course, applies for any of the eight octants. Consider next two octants O A and O B with a common plane of definition in x y z space, for example, the x y plane. The solution in octant O A (O B ) is Ψ A (Ψ B ) and the fact that only one independent allowable solution exists in both octants leads to r Ψ A (ξ1 , ξ2 , ξ3 ) D αr Ψ B (ξ1 , ξ2 , ξ3 ) , Ψ A (ξ1 , ξ2 , ξ3 ) D α Ψ B (ξ1 , ξ2 , ξ3 ) ,

(16.66)

where (ξ1 , ξ2 , ξ3 ) in Eq. (16.66) is a point in the common plane of definition and α is a constant. The above requirement determines k uniquely since Ψ A (Ψ B ) and r Ψ A (r Ψ B ) cannot both be zero everywhere in the common plane of definition for the following reason. Assume, for example, that both Ψ A and r Ψ A are zero in the common plane of definition. Then, k can be any constant and infinitely (uncountably) many solutions would exist to the Helmholtz equation having the same eigenvalue, which is impossible. Hence, after a solution in one octant has been determined, the whole solution for the ellipsoid (all eight octants) is specified uniquely. 16.5.3.3 Ellipsoidal Domain Given an ellipsoid specified via x0 , y 0 , z0 , we have from Eq. (16.5), ξ10 , a, b. For example, for x0 D 1, y 0 D 1.5, z0 D 2 (in arbitrary units),

ξ10 D z0 D 2.0 a.u. , 1/2
2
x02 D 1.732 05 a.u. , a D ξ10 
2 2 1/2 b D ξ10
y 0 D 1.322 88 a.u. The method of Arscott et al. [74] requires the specification of c and t0 . Remember we now have two choices:  cD

b2 a2

1 2 ξ D 1.333 33 , a 2 10

y 02 1
2 2 b z0

D 0.583 33 . cD 2 D x02 a 1


t0 D

z 02

Note that c is dimensionless. Upon solving the differential equation, one obtains γ , λ, µ; γ is also dimensionless. Hence, for a given ellipsoid, one only need solve the differential equation once for either the quantum-dot or the acousticenclosure problem. We now have k2 D

1 γ. a2

(16.67)

For the quantum-dot problem, the energy is given by ED

„2 k 2 „2 3.8097 D 4γ D  2 γ D 18.957γ ,  2m 2m 0 m  a 2 m a

(16.68)

209

210

16 Ellipsoidal Coordinates

with a in angstroms, m  in units of free-electron mass, and the energy in electronvolts. The last equality is for m  D 0.067m 0 . For the acoustic-enclosure problem, the frequency is given by f D

ω 54.5901 p c p D 2 γD 2 γ, 2π 2π a a

(16.69)

with a in meters, f in hertz, and for air at room temperature (c D 343 m/s). 2  c D ab 2 1 2 t0 D 2 ξ10 D 2.285 70 , b

x2 1
z02 a2 0 cD 2 D

D 1.714 28 , y2 b 1
z02 0

and ED

„2 k 2 „2 3.8097 D 4γ D  2 4γ D 129.968γ . 2m  2m 0 m  b 2 m b

(16.70)

16.5.4 Quantum Mechanics: Interior Dirichlet Problem for an Ellipsoid

An example of the solution to the Helmholtz problem using the ATZ algorithm will now be given. The physical problem is to find the bound states of an electron in an infinite-barrier ellipsoidal quantum dot [79]. The results to be presented are all from our paper. The ellipsoid studied is given in Figure 16.4.

2

0

−2 2 1 0 y

0 −2 −1

x

Figure 16.4 Triaxial ellipsoid with intersections x0 D ˙1 a.u., y 0 D ˙1.5 a.u., and z0 D ˙2 a.u. (where a.u. stands for arbitrary units).

16.5 Applications

211

The basic quantities of interest in quantum mechanics are the energies and wave functions. The energies are related to the eigenvalues γ via ED

2„2 „2 k 2 D γ, 2m 0 m0 b2

(16.71)

(see Eq. (16.67)) where b is defined in Eq. (16.5) and m 0 is the free-electron mass (or, for semiconductor quantum-dot applications, the effective mass). The wave functions are the eigenfunctions Ψ . For concreteness, the energies will be given for arbitrary units equal to 100 Å, that is„ the absolute semiaxes are (x0 , y 0 , z0 ) D (100, 150, 200) Å. To summarize, the solution of the Helmholtz equation using the ATZ algorithm consists of three steps. First, one determines the characteristic curves (i.e., µ and λ versus γ ). Second, one subjects the ξ1 equation to the boundary condition on the ellipsoidal surface. Third, one forms the eigenfunctions for the three-dimensional problem from the product of the three Lamé wave functions of the same type. 16.5.4.1 Characteristic Curves In Table 16.1, values of µ and λ are given corresponding to γ D 0 for n  1, 0  m  n for each of the eight possible combinations of , σ, and τ values in the case of an ellipsoid with x0 D 1 a.u., y 0 D 1.5 a.u., and z0 D 2 a.u., corresponding to c D 1.71 428, using Eq. (16.4). Here arbitrary units of length are used as owing to the scaling property of the Helmholtz equation [64], no explicit units need be used. Indeed, it is worth pointing out that both c and t in Eq. (16.49) are dimensionless.

In Figure 16.5, µ and λ are plotted as a function of γ in the range 0  γ  20 for the case  D σ D τ D 0, n D m D 0. Negative values of γ could similarly be considered; however, they do not arise for application to the Helmholtz equation where γ k 2 (e.g., see Eq. (16.67)). As mentioned above, it is important to keep ∆γ small so as to ensure that discontinuous jumps in µ and λ do not occur in the Table 16.1 Computed values of µ and λ at γ D 0 for an ellipsoid with x0 D 1 a.u., y 0 D 1.5 a.u., and z0 D 2 a.u. for the eight possible combinations of , σ, and τ. Reproduced from [64]. n, m

Case 1
D1

Case 2
D1

Case 3
D1

Case 4
D0

Case 5
D1

Case 6
D0

Case 7
D0

Case 8
D0

σD1

σD1

σD0

σD1

σD0

σD1

σ D0

σD0

τD1

τD0

τD1

τD1

τD0

τD0

τD1

τD0

0, 0

µ D
3.0

µ D
1.5

µ D
1.5

µ D
1.5

µ D
0.5

µ D
0.5

µ D
0.5

µ D 0.0

1, 0

λ D 2.71 µ D
7.5

λ D 1.96 µ D
5.0

λ D 1.43 µ D
5.0

λ D 6.79 µ D
5.0

λ D 6.79 µ D
3.0

λ D 4.29 µ D
3.0

λ D 0.25 µ D
3.0

λ D 0.00 µ D
1.5

λ D 4.55

λ D 3.60

λ D 3.49

λ D 1.30

λ D 2.42

λ D 0.98

λ D 0.96

λ D 0.61

µ D
7.5 λ D 9.02

µ D
5.0 λ D 7.47

µ D
5.0 λ D 5.80

µ D
5.0 λ D 5.48

µ D
3.0 λ D 4.36

µ D
3.0 λ D 4.30

µ D
3.0 λ D 3.25

µ D
1.5 λ D 2.10

1, 1

16 Ellipsoidal Coordinates 3 2.5

λ

2 1.5 1 0.5 0

0

5

10

0

5

10

γ

15

20

25

15

20

25

(a) 0 −5 −10 µ

212

−15 −20 −25

γ

(b) Figure 16.5 Computed values of λ (a) and µ (b) as a function of γ for the ellipsoid with intersections x0 D 1 a.u., y 0 D 1.5 a.u., and z0 D 2 a.u. The results correspond to the case with  D σ D τ D 0 and n D m D 0.

course of iterations. This process of obtaining the characteristic curves is extremely fast. 16.5.4.2 Determination of γ Eigenvalues The previous infinite set of solutions for the separation constants and the γ parameter all satisfy the ordinary differential equation. We are specifically interested in the subset of solutions that satisfies the Dirichlet boundary condition on the surface of a given ellipsoid. The restricted values of γ can be found once the separation constants µ(γ ) and λ(γ ) have been determined as a function of γ , as will be next explained. The ellipsoidal surface corresponds to ξ1 D z0 . By inspection, in the process of increasing γ from zero and upwards in small steps, the solutions µ(γ ) and λ(γ ) corresponding to a certain set of parameters n, m, , σ, and τ are first determined. The insertion of all these parameters into the recursive relations for the coefficients a r in F (Eq. (16.54)) allows the total wave function Ψ at ξ1 D z0 to be evaluated from Eq. (16.61) for each value of γ . Whenever γ is such that Ψ D 0 at ξ1 D z0 , γ is an eigenvalue. As an example, in Figure 16.6, Ψ evaluated at ξ1 D z0 as a function of γ in the range 0  γ  20 is shown corresponding to n D m D 0 and  D σ D τ D 0. Two eigenvalues were found at γ D 2.405 and γ D 12.654. It is important to point out that this step of matching the boundary condition is distinct from the previous step of obtaining the characteristic curves. Hence, the characteristic curves obtained above can be reused for other boundary conditions.

16.5 Applications

0.02 0.01

wave function

0 −0.01 −0.02 −0.03 −0.04 −0.05 −0.06 −0.07 2

4

6

8

10

12

14

16

18

20

y Figure 16.6 Computed wave function at ξ1 D z0 as a function of γ corresponding to  D σ D τ D 0 and n D m D 0. Zero crossings determine the eigenvalues γ .

16.5.4.3 Lamé Wave Functions There is an infinite number of Lamé wave functions given a continuum set of γ values. This set is made finite once boundary conditions are imposed. In Figure 16.7, a few corresponding to the lowest γ values and consistent with the Dirichlet boundary condition on the X(ξ1 ) function are given. The X(ξ3 ) function is shown since its domain starts from ξ3 D 0.

Lamé wave function

5

4

3

2

1

0 0

0.5

ξ3

1

Figure 16.7 Lamé wave functions corresponding to the lowest three values of γ . The solid curve is the lowest (n, m, , σ, τ) D (0, 0, 0, 0, 0), the dash-dotted curve the sec-

1.5 ond lowest (n, m, , σ, τ) D (0, 0, 1, 0, 0), and the dashed curve the third lowest (n, m, , σ, τ) D (0, 0, 0, 1, 0).

213

214

16 Ellipsoidal Coordinates

In Figure 16.8, the ground-state wave function in one octant (x  0, y  0, z  0) is shown as a function of x, y in the z D 0 plane, x, z in the y D 0 plane, and y, z in the x D 0 plane. The corresponding energy is 2.094 meV (γ D 2.405) and the indices n, m, , σ, τ are all 0 (Table 16.1). This state has no nodes along the three planes and peaks at the center of the ellipsoid ((x, y, z) D (0, 0, 0)); as expected, it resembles the ground state of a spherical and of a prolate spheroidal dot [56]. Since the solution is obtained as a series solution, it is smooth and differentiable. This contrasts with solutions obtained via purely numerical techniques such as finite difference and finite element that only give the wave functions at the grid or nodal points and, owing to the severe memory requirements of a threedimensional problem, are often not very smooth. In Figure 16.9, similar plots are shown for the first excited state along the three planes. The associated energy is 3.301 meV (γ D 3.791) and the indices n, m, , σ, τ are 0, 0, 1, 0, 0, respectively. The wave function is zero in the z D 0 plane because ξ3 D 0 when z D 0, and Ψ (ξ1 , ξ2 , ξ3 ) D

ξ1 ξ2 ξ3 F (ξ1 ) F (ξ2 ) F (ξ3 ) , b3

(16.72)

where F is the solution to Eq. (16.51) (note that t D ξi2 /b 2 ). However, Ψ is nonzero when plotted in the y D 0 and x D 0 planes corresponding to ξ2 D b > 0 (or

1 0.5 0 0

1 1 y

2 0

1

1 1.5 0.5 0 0

1 1

0.5 x

2 z

3 0

1.5 0.5 0 0

0.5 x

1

1 y

(b)

(a)

3 2 z

2 0

(c)

Figure 16.8 The ground-state wave function for the ellipsoid with intersections x0 D 1 a.u., y 0 D 1.5 a.u., and z0 D 2 a.u.: x y plane (a), x z plane (b), and y z plane (c).

1 0

1.5

−1 0

1 1 y

(a)

0.5

0.5

2 0

3

1.5 0 0

1 1

0.5 x

2 z

(b)

3 0

0 0

0.5 x

2 1

1 y

2 0

z

(c)

Figure 16.9 The first excited state for the ellipsoid with intersections x0 D 1 a.u., y 0 D 1.5 a.u., and z0 D 2 a.u.: x y plane (a), x z plane (b), and y z plane (c).

16.6 Problems

ξ3 D b > 0) and ξ1 D a > 0 (or ξ2 D a > 0), respectively. In fact, the firstexcited-state peaks at x D 0, z D z0 /2 and z D 0, y D y 0 /2 in the y D 0 and z D 0 planes, respectively. In addition, the wave function is nodeless in both planes. This solution resembles the first excited state of the sphere in having a nodal plane. However, it (in fact, all solutions) is found to be nondegenerate, in contrast to the spherical and spheroidal cases (the latter for the (111) state). For example, for the sphere, this state would be threefold degenerate (i.e., m degeneracy for l D 1). The lowered symmetry of the triaxial ellipsoid is responsible for the removal of the degeneracy. Additional plots of wave functions are given in [79].

16.6 Problems

1. a. By substituting x, y, z from Eq. (16.2), show that Eq. (16.1) is satisfied. b. Show that one of each of the three types of quadrics passes through each point (x, y, z) for which x y z ¤ 0. 2. Derive the scale factors. 3. Invert the coordinate transformations in Eq. (16.18) to show that [5] ξ1 D a

dn(λ, k) , ξ2 D adn(µ, k 0 ), ξ3 D bsn(ν, k) , cn(λ, k)

where p p k D b/a, k 0 D (1/a) a 2
b 2 D 1
k 2 . 4. a. Implement the ATZ algorithm described in the text, verifying the results presented. b. Obtain the lowest two modes for an ellipsoidal acoustic cavity defined by (x0 , y 0 , z0 ) D (1.0, 1.5, 2.0) m subject to (a) the pressure-release boundary condition, and (b) the rigid-wall boundary condition.

215

217

17 Paraboloidal Coordinates 17.1 Introduction

This coordinate system is best characterized by the presence of elliptic paraboloidal surfaces as one-coordinate surfaces. In this respect, it can be viewed as a generalization of the parabolic rotational surfaces. Such paraboloidal surfaces do show up in applications, for example, in the study of the failure of transversely isotropic materials [90]. There have also been calculations of the free vibrations of shallow shells [91]. Separation of the Laplace equation leads to the same three ordinary differential equations, the Baer equation, and all coupled by the two separation constants. The Helmholtz equation leads to the so-called Baer wave equation [6]. Given that a series solution leads to a five-term recurrence relation and all three ordinary differential equations are completely coupled via the two separation constants, the corresponding Baer wave functions [6] do not appear to have been previously studied.

17.2 Coordinate System 17.2.1 Coordinates (µ, ν, λ)

The relationship of the paraboloidal coordinates µ, ν, λ to the Cartesian ones is [6] 4 (µ
b)(b
ν)(b
λ) , (b
c) 4 (µ
c)(c
ν)(λ
c) , y2 D (b
c)

x2 D

z DµCνCλ
b
c,

(17.1)

with µ>b>λ>c>ν>0.

(17.2)

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

218

17 Paraboloidal Coordinates

z ν = ν0 λ= λ 0

y x

µ = µ0

Figure 17.1 Paraboloidal coordinates.

17.2.2 Constant-Coordinate Surfaces

The equation y2 x2 C D
4(z
µ) µ
b µ
c

(17.3)

with µ constant is of the form x2 y2 C 2 D 2d z , 2 a b which represents a so-called elliptic paraboloid (Figure 17.1), whereby the surface intersects a z D z0 plane in an ellipse. For d < 0, it is downward opening, intersecting the z axis at z D µ. Similarly, y2 x2 C D 4(z
ν) b
ν c
ν

(17.4)

for ν constant is an elliptic paraboloid, upward opening and intersecting the z axis at z D ν. Finally, x2 y2
D 4(z
λ) b
λ λ
c

(17.5)

with λ constant represents a hyperbolic paraboloid: z planes generate hyperbolas and x and y planes generate parabolas. The surfaces µ D µ 0 and ν D ν 0 intersect at most at four symmetrically disposed points for a given value of z D z0 , whose x and y coordinates are obtained from

17.3 Differential Operators

the first two equations of Eqs. (17.1) using the value of λ given by the last of that set of equations: λ D z
µ
ν C b C c.

17.3 Differential Operators 17.3.1 Metric

Using Eqs. (2.5) and (17.1), one obtains the scale factors as
hµ D
hν D
hλ D

(µ
ν)(µ
λ) (µ
b)(µ
c) (µ
ν)(λ
ν) (b
ν)(c
ν) (λ
ν)(µ
λ) (b
λ)(λ
c)

1/2 , 1/2 , 1/2 .

(17.6)

17.3.2 Operators

These are obtained by substituting Eq. (17.6) into Eqs. (2.7)–(2.12). Gradient The gradient operator is



rD

 (µ
b)(µ
c) 1/2 @ eµ (µ
ν)(µ
λ) @µ     (b
ν)(c
ν) 1/2 @ @ (b
λ)(λ
c) 1/2 C C . eν eλ (µ
ν)(λ
ν) @ν (λ
ν)(µ
λ) @λ

(17.7)

Divergence The divergence of a vector field V is

rV D

 (µ
b)1/2 (µ
c)1/2 @  (µ
b)1/2 (µ
c)1/2 Vµ (µ
ν)(µ
λ) @µ  (b
ν)1/2 (c
ν)1/2 @  C (b
ν)1/2 (c
ν)1/2 Vν (µ
ν)(λ
ν) @ν  (b
λ)1/2 (λ
c)1/2 @  C (b
λ)1/2 (λ
c)1/2 Vλ . (µ
λ)(λ
ν) @λ

(17.8)

219

220

17 Paraboloidal Coordinates

Circulation The circulation is

rV D

 1/2 (µ
b)(µ
c)(b
ν)(c
ν)(b
λ)(λ
c) (µ
ν)(µ
λ)(λ
ν) ˇh h i1/2 h i1/2 ˇˇ ˇ (µ
b)(µ
c) i1/2 (b
ν)(c
ν) (b
λ)(λ
c) ˇ (µ
ν)(µ
λ) e e e λ ˇˇ µ ν (µ
ν)(λ
ν) (λ
ν)(µ
λ) ˇ ˇ ˇ @ @ @ ˇ ˇ . @µ @ν @λ ˇh i1/2 h i1/2 h i1/2 ˇ ˇ ˇ (µ
b)(µ
c) (b
ν)(c
ν) (b
λ)(λ
c) Vµ Vν Vλ ˇ ˇ (µ
ν)(µ
λ) (µ
ν)(λ
ν) (λ
ν)(µ
λ) (17.9)

Laplacian The Laplacian is

   @ (µ
b)(µ
c) 1/2 @ (µ
b)1/2 (µ
c)1/2 (µ
ν)(µ
λ) @µ @µ 1/2    @ @ (b
ν)(c
ν) (b
ν)1/2 (c
ν)1/2 C (µ
ν)(λ
ν) @ν @ν 1/2    @ @ (b
λ)(λ
c) (b
λ)1/2 (λ
c)1/2 . C (λ
ν)(µ
λ) @λ @λ 

r2 D

(17.10)

17.3.3 Stäckel Matrix

From Section2.4.2.1, a possible Stäckel matrix is found to be 1 0 2 µ

B (µ
b)(µ
c) ν2 Φ DB @ (b
ν)(c
ν)
λ 2 (b
λ)(λ
c)


1 (µ
b)(µ
c)
1 (b
ν)(c
ν) 1 (b
λ)(λ
c)

µ (µ
b)(µ
c) C ν C (b
ν)(c
ν) A
λ (b
λ)(λ
c)

.

(17.11)

The f functions are f 1 D (µ
b)1/2 (µ
c)1/2 , f 2 D (b
ν)1/2 (c
ν)1/2 , f 3 D (b
λ)1/2 (λ
c)1/2 .

(17.12)

17.4 Separable Equations

17.4 Separable Equations 17.4.1 Laplace Equation 17.4.1.1 Let

Separation of Variables

ψ D M(µ)N(ν)Λ(λ) .

(17.13)

Then, 

   (µ
b)(µ
c) 1/2 1 d dM (µ
b)1/2 (µ
c)1/2 (µ
ν)(µ
λ) M dµ dµ 1/2    (b
ν)(c
ν) 1 d dN (b
ν)1/2 (c
ν)1/2 C (µ
ν)(λ
ν) N dν dν 1/2    1 d dΛ (b
λ)(λ
c) (b
λ)1/2 (λ
c)1/2 D0. C (λ
ν)(µ
λ) Λ dλ dλ

(17.14)

The separated equations are (µ
b)(µ
c)

dM d2 M 1 C [α 3 µ
α 2 ]M D 0 , C [2µ
(b C c)] 2 dµ 2 dµ

(b
ν)(c
ν) (b
λ)(λ
c)

dN 1 d2 N C [α 3 ν
α 2 ]N D 0 , C [2ν
(b C c)] dν 2 2 dν

(17.15)

dΛ d2 Λ 1
[α 3 λ
α 2 ] Λ D 0 .
[2λ
(b C c)] dλ 2 2 dλ

They are known as the Baer equations [6]. They are of Bôcher type f113g. There are regular singular points at b and c, and an irregular singular point at infinity. If one rewrites the separation constants as α 2 D (b C c)q and α 3 D
p (p C 1), then the equation takes the standard form of the Baer equation with solutions [6] B(z) D AB pq (z) C B C pq (z) .

(17.16)

Though this does not introduce any constraint on α 2 , it requires α 3 < 1/4 for a real index p. 17.4.1.2 Series Solutions Each ordinary differential equation is of the form

(t
b)(t
c)

dX d2 X 1 C (α 3 t
α 2 ) X D 0 . C [2t
(b C c)] dt 2 2 dt

(17.17)

This equation is known as the Baer equation. It has regular singular points at t D b and t D c, and an irregular singular point of order 3 at t D 1. Hence, the Baer equation is of Bôcher type f1 1 3g. In particular, t D 0 is an ordinary point.

221

222

17 Paraboloidal Coordinates

Consider first a power series about t D 0: X(t) D

1 X

Cn t n .

(17.18)

nD0

Substituting into Eq. (17.17) gives (  1 X (b C c) C nC2 (n C 1)(n C 2)b c
C nC1 (n C 1)
α 2 Cn t n 2 nD0   C
C nC2 (n C 1)(n C 2)(b C c) C C nC1 (n C 1) C α 3 C n t nC1 ) C C nC2 (n C 1)(n C 2)t nC2 D 0 .

(17.19)

Equating the coefficients of each power of t gives the ancillary and recurrence relations: (b C c) C1
α 2 C0 D 0 , 2b c C2
2 6b c C3
3(b C c)C2 C (1
α 2 ) C1 C α 3 C0 D 0 ,   (b C c) C nC3
α 2
(n C 2)2 C nC2 (n C 3)(n C 4)b c C nC4
(2n C 5)(n C 3) 2 C α 3 C nC1 D 0 . (17.20) Using the same procedure as for the Lamé wave equation to test for series convergence, one can write   α 2
(n C 2)2 C nC2 C nC4 (2n C 5)(n C 3) (b C c) C D C nC3 (n C 3)(n C 4) 2b c (n C 3)(n C 4)b c C nC3 C nC1 α3
. (17.21) (n C 3)(n C 4)b c C nC3 Writing this ratio as γ n and, in the limit n ! 1, γ n ! γ , giving     3 (b C c) 5 1 γ D 1

1
, n bc n bcγ or 1 1 γD , . c b For jz γ j < 1, then one requires jzj < c or jzj < b. Thus, two of the series will converge in µ but not the third series. For the case α 2 D α 3 D 0, there are simple solutions [6, 12]. Equations (17.15) reduce to (µ
b)(µ
c)

d2 M dM 1 D0, C [2µ
(b C c)] dµ 2 2 dµ

d2 N dN 1 D0, C [2ν
(b C c)] dν 2 2 dν dΛ d2 Λ 1 (b
λ)(λ
c) 2 C [2λ
(b C c)] D0, dλ 2 dλ

(b
ν)(c
ν)

(17.22)

17.4 Separable Equations

2.0 1.8

N (υ)

1.6 1.4 1.2 1.0 0.8 0.6 0

0.0

0.2

0.4

υ

0.6

0.8

1.0 10

Figure 17.2 Baer function for α 2 D α 3 D 0 (with b D 3, c D 1).

which can be rewritten as  dM d  (µ
b)1/2 (µ
c)1/2 D0, dµ dµ  dN d  D0, (ν
b)1/2 (ν
c)1/2 dν dν  dΛ d  (b
λ)1/2 (λ
c)1/2 D0. dΛ dλ

(17.23) (17.24) (17.25)

The solutions are i h p p M(µ) D A C B ln 2µ
b
c C 2 µ
b µ
c , h i p p N(ν) D C C D ln 2ν
b
c C 2 ν
b ν
c ,   2λ
(b C c) Λ(λ) D E C F sin
1 . b
c

(17.26) (17.27) (17.28)

The function N(ν) is plotted in Figure 17.2. 17.4.1.3 Polynomial Solutions One can also find polynomial solutions. It is trivial to show that a constant function is a solution. Setting α 2 D 1, α 3 D 0 in Eq. (17.20) gives the linear solution

X(t) D t
(b C c) .

(17.29)

One can develop additional polynomial solutions by expanding X(t) about t D b. We will start with a more general expansion. Let X(t) D (t
b)
(t
c) σ F(t) ,

(17.30)

223

224

17 Paraboloidal Coordinates

where F is an infinite series. We have X 0 D (t
b)
(t
c) σ F 0 C
(t
b)

1 (t
c) σ F C σ(t
b)
(t
c) σ
1 F , X 00 D (t
b)
(t
c) σ F 00 C 2
(t
b)

1 (t
c) σ F 0 C 2σ(t
b)
(t
c) σ
1 F 0 C
(

1)(t
b)

2 (t
c) σ F C σ(σ
1)(t
b)
(t
c) σ
2 F C 2
σ(

1)(t
b)

1 (t
c) σ
1 F , and
  1 (t
b)
(t
c) σ (t
b)(t
c)F 00 C 2
(t
c) C 2σ(t
b) C (2t
(b C c)) F 0 2  (t
b) (t
c) C σ(σ
1) C 2
σ C
(

1) (t
b) (t
c)     1 σ
C (2t
(b C c)) C C (α 3 t
α 2 ) F D 0 . 2 (t
b) (t
c) (17.31) Now,

  (t
b) 1 σ
(t
c) C σ(σ
1) C (2t
(b C c)) C
(

1) (t
b) (t
c) 2 (t
b) (t
c)     1 (t
b)
σ 1 (t
c) D


Cσ σ
C C 2 (t
b) 2 (t
c) 2 2

or, if
, σ D 0 or 1/2, the first two terms vanish for both cases and    1 (t
b)(t
c)F 00 C (2
C 2σ C 1)t
2
c C 2σ b C (b C c) F 0 2 C [α 3 t
α 2 C 2
σ C
C σ] F D 0 .

(17.32)

It is useful to rescale t to t/b, giving    c 00 1 c

c (t
1) t
F C (2
C 2σ C 1)t
2
C 2σ C 1C F0 b b 2 b C (α 3 b t
α 2 C 2
σ C
C σ) F D 0 .

(17.33)

Rewrite this as  (t
1)(t
a)F 00 C (A 1 t
A 0 ) F 0 C α 03 t C α 02 F D 0 ,

(17.34)

where c , b A 1 D 2
C 2σ C 1 , 1 A 0 D 2
a C 2σ C (1 C a) , 2 α 02 D
α 2 C (2
σ C
C σ) , aD

α 03 D α 3 b .

(17.35)

17.4 Separable Equations

Finally, let F(t) D

1 X

a r (t
t0 ) r .

(17.36)

rD0

Then, F0 D

1 X

a r r (t
t0 ) r
1 ,

F 00 D

rD0

1 X

a r r(r
1) (t
t0 ) r
2 ,

rD0

giving 1 X

(

  a r r(r
1) t 2
(1 C a)t C a (t
t0 ) r
2 C r (A 1 t
A 0 ) (t
t0 )r
1

rD0

C D

1 X



α 03 t

C

α 02



) (t
t0 )

r

(

  a r r(r
1) (t
t0 C t0 )2
(1 C a) (t
t0 C t0 ) C a (t
t0 ) r
2

rD0

C r (A 1 (t
t0 C t0 )
A 0 ) (t
t0 ) r
1 )   0 r 0 C α 3 (t
t0 C t0 ) C α 2 (t
t0 ) D

D

(

1 X

   a r r(r
1) (t
t0 ) r C 2t0
(1 C a) (t
t0 )r
1

rD0

   C t02
(1 C a)t0 C a (t
t0 ) r
2   A1 (t
t0 )r C (A 1 t0
A 0 ) (t
t0 )r
1 Cr 2 ) h i  C α 03 (t
t0 ) rC1 C α 03 t0 C α 02 (t
t0 )r

1 X

( (t
t0 ) rC2 [(r C 1) (r C 2) a rC2 ]

rD0



  2t0
(1 C a) (r C 1)(r C 2)a rC2  A1 (r C 1)a rC1 C α 03 a r C 2  C (t
t0 ) r (t0
1) (t0
a) (r C 1)(r C 2)a rC2 C (A 1 t0
A 0 ) (r C 1)a rC1 )   0 0 D0. C α 3 t0 C α 2 a r C (t
t0 )

rC1

225

226

17 Paraboloidal Coordinates

Equating the coefficients of (t
t0 ) r to zero gives

 2 (t0
1) (t0
a) a 2 C (A 1 t0
A 0 ) a 1 C α 03 t0 C α 02 a 0 D 0 ,   6 (t0
1) (t0
a) a 3 C 2 (A 1 t0
A 0 ) C 2 (2t0
(1 C a)) a 2   A1 C α 03 t0 C α 02 a 1 C α 03 a 0 D 0 , (17.37) C 2

and

 2  2 1 0 N rC1 a rC1 C N rC2 C N rC2 (r C 2) C N rC2 (r C 2)(r C 1) a rC2  1  0 C N rC3 (r C 3) C N rC3 (r C 3)(r C 2) a rC3 0 C N rC4 (r C 3)(r C 4)a rC4 D 0 ,

(17.38)

with 2 N rC1 D α 03 ,

A1 2 , N rC2 D α 03 t0 C α 02 , 2 1 D 2t0
(1 C a) , N rC3 D (A 1 t0
A 0 ) ,

0 N rC2 D1, 0 N rC3

1 N rC2 D

0 D (t0
1) (t0
a) . N rC4

If we now choose t0 D 1, then the recurrence relation becomes   α 03 a rC1 C α 02 C α 03 C A 1 (r C 2) C (r C 1)(r C 2) a rC2   C (1
a)(r C 2) C (A 1
A 0 ) (r C 3)a rC3 D 0 .

(17.39)

It follows immediately from Eq. (17.34) that polynomial solutions in t require the term α 03 t to vanish, that is, α 03 D 0. Then, the infinite and divergent series in Eq. (17.39) can be terminated after the a N term if α 02 D
A 1 N
N(N
1) ,

N 2,

and the polynomial coefficients become h i 0 2 1 N rC2 C N rC2 (r C 2) C N rC2 (r C 2)(r C 1) h i a rC3 D
a rC2 I 0 1 N rC3 (r C 3) C N rC3 (r C 3)(r C 2)

(17.40)

rC3 N . (17.41)

In particular, we have α 02 a0 , A1
A0 A 1 C α 02  a1 . a2 D
 2 (A 1
A 0 ) C (1
a) a1 D


For example, for N D 2 and
D σ D 0, α 02 D
4 and
 8 40 2 . tC t X 2 (t) D a 0 1
1
a 3(1
a)2

(17.42)

(17.43)

17.5 Applications

17.4.2 Helmholtz Equation 17.4.2.1 Let

Separation of Variables

ψ D M(µ)N(ν)Λ(λ) .

(17.44)

Then, 

   dM (µ
b)(µ
c) 1/2 1 d (µ
b)1/2 (µ
c)1/2 (µ
ν)(µ
λ) M dµ dµ     dN (b
ν)(c
ν) 1/2 1 d (b
ν)1/2 (c
ν)1/2 C (µ
ν)(λ
ν) N dν dν     dΛ (b
λ)(λ
c) 1/2 1 d (b
λ)1/2 (λ
c)1/2 D
k 2 . (17.45) C (λ
ν)(µ
λ) Λ dλ dλ

The separated equations are   dM d2 M 1 C k 2 µ2 C α3 µ C α2 M D 0 , C [2µ
(b C c)] 2 dµ 2 dµ   d2 N dN 1 (b
ν)(c
ν) 2 C [2ν
(b C c)] C k 2 ν2 C α3 ν C α2 N D 0 , dν 2 dν  dΛ  2 2 d2 Λ 1 (b
λ)(λ
c) 2
[2λ
(b C c)]
k λ C α3 λ
α2 Λ D 0 . dλ 2 dλ (17.46)

(µ
b)(µ
c)

They are known as the Baer wave equations [6]. They are of Bôcher type f1 1 4g. There are regular singular points at b and c, and an irregular singular point at infinity.

17.5 Applications

We now give an application of paraboloidal coordinates to a (paraboloidal) conductor problem (solution to the Laplace equation). 17.5.1 Electrostatics: Dirichlet Problem for a Paraboloid

Consider the Laplace problem of a conducting paraboloid defined by µ D constant. This problem is unbounded in space and a simple solution for the potential Ψ can be found by imposing Ψ D U on the conducting paraboloid, where U is (nonzero and) independent of both ν and λ. Indeed, we can deduce that the potential is

227

17 Paraboloidal Coordinates

independent of ν and λ everywhere in space. From Eq. (17.15) we find that only if α 2 D α 3 D 0 is it possible to ensure that both N and Λ have a constant solution. The latter statement follows immediately from Eqs. (17.18) and (17.19) by imposing C0 ¤ 0 and C n D 0 if n  1. The preceding result now dictates that Ψ is given by Eqs. (17.26)–(17.28) with D D F D 0. We next obtain: i

h p p (17.47) Ψ (µ, ν, λ) / A C B ln 2µ
b
c C 2 µ
b µ
c , and the electric field E is easily obtained from Eq. (17.7) using E D
r Ψ . In Figure 17.3, the electric field is plotted versus the x, y, and z axes along with the equipotential surface: µ D 3.4. Other parameters used are A D 0, B D
0.7, b D 3, and c D 1. We notice that, though the potential diverges as µ ! 1, the electric field approaches zero in the same limit as expected.

10 8 6 z

228

4 2 0 1

5 2

4

3 4

3 5

2

6 x

y 1

7 8

0

Figure 17.3 Electric potential and electric field near paraboloidal conductor. The parameters are µ D 3.4, b D 3, and c D 1.

17.6 Problems

17.6 Problems

1. 2. 3. 4.

Obtain the scale factors for the paraboloidal coordinate system. Use the Stäckel theory to separate the Laplace equation. Show that the Baer wave equation is of Bôcher type f114g. Show that the infinite series given by the recurrence relation in Eq. (17.39) is divergent. 5. Using paraboloidal coordinates, compute the most general third-order and fourth-order polynomial solution to the Laplace equation.

229

Part Four

Advanced Formulations

233

18 Differential-Geometric Formulation 18.1 Introduction

We have seen many times in earlier chapters when the physics is confined to a one-coordinate surface or in the three-dimensional neighborhood of a (parameterizable) curve, the natural mathematical framework for treating such problems uses differential geometry. Such a formalism is now presented and we will recast the study of fields into it over the next three chapters.

18.2 Review of Differential Geometry

For completeness, we provide a brief review of differential geometry [81]. Note that the formalism presented applies to nonorthogonal systems as well. 18.2.1 Curvilinear Coordinates

In the following, some basic rules and theorems using curvilinear coordinates will be given and proven to some extent. First, we recall the result from basic linear algebra that if three vectors e 1 , e 2 , e 3 satisfy the relation e1
e2  e3 ¤ 0 ,

(18.1)

then a unique set of scalars v 1 , v 2 , v 3 exists such that for any vector v we may write v D v i ei ,

(18.2)

where repeated indices are to be summed over (Einstein notation). Definition 1 Given a basis e i (i D 1, 2, 3), then a basis e i is called a dual basis if e i  e j D δ ij .

(18.3)

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

234

18 Differential-Geometric Formulation

Theorem 1 If e i is a basis, then e i exists and is a unique basis assuming Eq. (18.3) holds. The proof is trivial. Definition 2 Define the metric tensor gi j  ei  e j ,

gi j  ei  e j .

(18.4)

Theorem 2 The following expressions hold: gi j e j D ei , gi j e j D ei , g i j g j k D δ ik ,
 det g i j > 0 ,
 det g i j > 0 .

(18.5)

Consider an open set D  R 3 and let M W D ! D M  R 3 belong to the class of twice-differentiable continuous functions (C 2 ) and be one to one. If x 2 D, we may write M(x) D M k (x)i k , where i k are the standard Cartesian unit basis vectors of R 3 . We then let e i (x) 

@M d (M (x C t i i ))j tD0 . (x)  @x i dt

For e i to be a basis, we must require   @M ¤0. det @x i

(18.6)

(18.7)

Define y i to be the usual coordinates with respect to i i , that is, y i D M i (x) .

(18.8)

Then it follows from the inverse function theorem that the inverse exists, x i D x i (y) ,

(18.9)

and is a unique C 2 function. Hence, rxk D

@x k (y) ii , @y i

(18.10)

and r x i (y)  e j D

@x i @y l @x i @y k ik  il D D δ ij , j k @x @y @y k @x j

(18.11)

18.2 Review of Differential Geometry

from the chain rule. In other words, the dual basis is given by e i (x) D r x i (y) .

(18.12)

Definition 3 Let F W U ! R n be differentiable. We call F a vector field (the deriva@F i tive @x j is then also a vector field). Now, scalars F, j and F i, j exist such that @F D F,ij e i (x) D F i, j e i (x) . @x j

(18.13)

Theorem 3 If x and z are curvilinear coordinates, then F,kl (x) D F,ij (z) F k,l (x)

@x l @z j , @z i @x k

(18.14)

@x k @x l D F i, j (z) . @z i @z j

(18.15)

Proof: We will give the proof of Eq. (18.14). By direct use of the above rules, we have F,kl e l (x) 

@F(x) @F (z) @z j @z j @x l @z j i i D D F e (z) D F e (z) e l (x) , i i , j , j @x k @z j @x k @x k @z i @x k (18.16)

and the first result is obtained. 






k The Christoffel symbol of the second kind (also written as Γ ki j ) is defined ij by   @e i (x) k D (18.17) e k (x) , ij @x j and satisfies the following relations:   @e i (x) i D  e k (x) , kj @x j     k k D , ij ji     @g k j @g i k g j m @g i j m . C  D ik 2 @x k @x i @x j

(18.18) (18.19) (18.20)

We will prove Eq. (18.20). Use of the above definitions and theorems easily shows that     @g i j l l D g C g . (18.21) l j ik j k li @x k

235

236

18 Differential-Geometric Formulation

Switching i and k and using Eq. (18.19) yields     @g k j l l D C g glk . l j ik ji @x i Next, switching j and k in Eq. (18.21), we find     @g i k l l g g . D C i j lk j k li @x j Adding Eq. (18.21) to Eq. (18.22) and subtracting Eq. (18.23) yields   @g i j @g k j @g i k l C  D 2 g . ik l j @x k @x i @x j

(18.22)

(18.23)

(18.24)

Finally, multiplying both sides by g j m and using g l j g j m D δ m l , we obtain Eq. (18.20). 18.2.2 Gradient, Divergence, and Laplacian

Let y k be the standard coordinates with respect to the usual basis vectors i k (k D 1, 2, 3). Thus, y D ykik , e k (y) D i k D e k (y) .

(18.25)

Also, r φ(y) D

@φ(y) e k (y) . @y k

(18.26)

Expressing the gradient in a different set of coordinates x, r φ(x) D

@φ(x) @x i @y j @φ(x) i @φ(x) e l (x) D δ el D ei . @x i @y j @x l @x i l @x i

(18.27)

The latter expression reveals that the covariant components of r φ(x) are (r φ(x)) i D

@φ(x) . @x i

(18.28)

The contravariant components are found in the usual way by raising the index: (r φ(x)) i D g i j (r φ(x)) j D g i j

@φ(x) . @x j

(18.29)

We now turn to the divergence of a vector defined by @F i (y) D F,ii (y) . @y i

(18.30)

18.2 Review of Differential Geometry

Then, F,ij (y)

D

  ! l @F k @x @y i k m (x) C F (x) (x) l ml @x @y i @x k !   @F k k m (x) C F (x) (x) δ lk ml @x l   ! @F k k m (x) C F (x) (x) . mk @x k

@x l F,lk (x) j

D D

@y i D @y @x k

(18.31)

Use of Eq. (18.20) allows us to write   g j i @g i j i D . ik 2 @x k

(18.32)

We next use


1


1 D A i j det g i j , g i j D A j i det g i j

(18.33)

where A j i is the j ith cofactor of the matrix (g k l ). Further, recall that for any j (not summing over j in the following expression!), g D det(g i j ) D

3 X

g i j Ai j .

(18.34)

iD1

Thus, Ai j D

@g , @g i j

(18.35)

and Eq. (18.32) gives   1 g j i @g i j i j 1 @g g j i @g i j i D A D . D k ik 2 @x 2g 2 @x k 2g @x k

(18.36)

With this result, employing Eq. (18.31) yields the final expression for the divergence of a vector in curvilinear coordinates: r F D

@F i (x) 1 @g(x) 1 @ j p F (x) g(x) . (18.37) C F i (x) D p j i i @x 2g(x) @x g(x) @x

The Laplacian is r2 φ D r  r φ , so r 2 φ(x) D p

@ g(x) @x i 1

(18.38)  g i k (x)

 @φ(x) p g(x) . @x k

(18.39)

237

238

18 Differential-Geometric Formulation

18.2.3 Curl and Cross Products

Firstly, we define the permutation symbol ε m q p that satisfies ε 123 D 1, ε 213 D 1, and so on and ε m q p D 0 if any two of the indices m, q, p are equal. Further, we introduce 1
i j k (x)  ε i j k p . (18.40) g(x) It now follows that  p l @z l @z m @z n @z @z m @z n @x i jk D ε det i j k q @x @x @x @y @x i @x j @x k  i  p  i @z @x @z lmn det D ε 
l m n (z) . D ε l m n det det q k @y @x @y q (18.41)


i j k (x)

Now,
i j k (y) D ε i j k ,

(18.42)

and for a vector field F , r
F D
l m n (x)

@y i @y j @y k @x p @x q @x r F p ,q (x) k er l m n @x @x @x @y @y j @y i

D
l m n δ r l δ q m δ p n F p ,q (x)e r (x) D
r q p (x)F p ,q (x)e r (x) . We also have F p ,q (x) D

  @F p (x) r ,  F r pq @x q

but the second term equals zero since     r r D , pq qp and


m q p (x)

     r r r D
m p q (x) D 
m q p (x) . pq qp pq

(18.43)

(18.44)

(18.45)

(18.46)

Thus, @F p (x) e r (x). @x q From the definition of the cross product, r
F D
r q p (x)

(18.47)

F
G D
i j k (y)F j (y)G k (y)e i (y) ,

(18.48)

we can, following steps similar to those used in deriving the expression for the curl, find the cross product in curvilinear coordinates: F
G D
i j k (x)F j (x)G k (x)e i (x) .

(18.49)

18.3 Problems

18.2.4 Vector Calculus Expressions in General Coordinates

Consider a scalar function φ and a vector function F with components F i along the general coordinate directions e i , that is„ F D F j e j D F j e j . The central expressions derived above for the gradient, divergence, Laplacian, curl, and cross products of vectors are @φ(x) , @x r @φ(x) (r φ(x))r D g r k (x) , @x k

1 @ i p F (x) g(x) , rF D p i g(x) @x   1 @φ(x) p @ ik g (x) g(x) r 2 φ(x) D p @x k g(x) @x i (r φ(x))r D

@g

@ ik @ @φ(x) k C g i k @x C g Dg i k @x @x 2g @x k @F p (x) e m (x) , r
F D
m p q (x) @x q ik

F
G D
l p q (x)F p (x)G q (x)e l (x) ,

(18.50) (18.51) (18.52)

!

@φ(x) , @x i

(18.53) (18.54) (18.55) j

where g i k are the metric tensor entries, g D det(g i j ), g i j g j k D δ ik , e i  e j D δ i , 1 , ε m q p is the usual permutation symbol (ε 123 D 1, ε 213 D
m q p (x) D ε m q p p g(x) 1, ε 112 D 0, etc.), G is a vector function, and δ ik D 1 ,

if k D i ,

D0,

if k ¤ i .

δ ik

(18.56)

The Laplacian is often called the Laplace–Beltrami operator. In the following, we will frequently use the compact notation @ i D @x@ i .

18.3 Problems

1. Show that, for an orthogonal curvilinear coordinate system, the Laplace equation can be written as  1/2  g @ @ r 2 ψ (q i ) D g
1/2 . @q i g i i @q i Also show that, if the potential is only a function of one coordinate, the Laplace equation can be integrated once to give dψ A D , dq i fi

239

240

18 Differential-Geometric Formulation

where A is a constant and f i are the f functions in the Stäckel theory. 2. Prove a.     k k D ij ji b. Equation (18.15): F k,l (x)

@x k @x l D F i, j (z) . @z i @z j

241

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves 19.1 Introduction

In this chapter, we develop the theory for wave propagation in confined geometries that are parameterized by a curve, for example, electromagnetic waveguides and nanowires. We provide a number of example applications from quantum mechanics. They are particularly interesting examples because such systems have been realized in the laboratory; for example, a carbon nanotube ring [92], a Möbius strip single crystal [93], tubular graphite cones [94], and InGaAs toroidal nanorings [95]. This chapter also draws from papers that have studied the localization of electronic states near a surface and near bent nanowires [56, 96–99]; an important result has been the localization of the electron near regions of high curvature. 1)

19.2 Laplacian in a Tubular Neighborhood of a Curve – Arc-Length Parameterization

Having introduced the differential-geometric formulation to be used in this book, we now use it to reformulate the Laplacian in a tubular neighborhood of a curve. This formulation is important for handling the quantum-wire problem and wave propagation in electromagnetic or acoustic waveguides. 19.2.1 Arc-Length Parameterization

Consider a nanowire structure where the axis is given as a curve r(s) parameterized by arc length s. The tangent vector t(s) D r 0 (s) D dr/ds is a unit vector field along the curve and we can augment it with vector fields p (s) and q(s) along the curve such that t(s), p (s), q(s) constitutes an orthonormal frame at each point r(s) along

1) The work described in this chapter is partially based on notes developed in conjunction with Jens Gravesen. Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

242

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

the axis. Differentiation of the identities, t
tD1,

p
p D1,

q
qD1,

t
p D0,

t
qD0,

p
qD0,

yields dt
t D0, ds dt dp
pCt
D0, ds ds 2

dp dq
p D0, 2
qD0, ds ds dt dq dp dq
qCt
D0,
qCp
D0. ds ds ds ds

2

If we now let a(s) D

dt
p, ds

b(s) D

dt
q, ds

c(s) D

dp
q, ds

then we obtain the following equation: 2 3 2 32 3 t 0 a b t d 4 5 4 p D a 0 c 5 4p5 . ds q b c 0 q

(19.1)

Observe that the curvature
of the axis is ˇ ˇ ˇ ˇ p
(s) D ˇ r 00 (s)ˇ D ˇ t 0 (s)ˇ D a 2 C b 2 .

(19.2)

One way of obtaining vector fields p , q is to let p be the principal normal n D t 0 /
and let q be the binormal b D t  n. In this case Eq. (19.1) becomes the Frenet– Serret equations, where a D
, b D 0, and c D τ, the torsion of the axis. Let U  R 2 be a fixed domain in the uw plane and let h W I ! R C be a positive function. We will consider the Laplace operator on the tubular neighborhood Ω D fx(s) C h(s) (u 2 p (s) C u 3 q(s)) js 2 I ^ (u 2 , u 3 ) 2 Ug ,

(19.3)

where I is an open domain in R 1 . A different choice of h(s) gives another scaling of the cross sections of Ω and a different choice of the frame p , q gives a different orientation of the cross sections, but they all have the same shape U. The parameter ranges of u 2 and u 3 are, in general, different to allow for a rectangular cross section. The domain Ω can obviously be parameterized by xQ (s, u 2 , u 3 ) D x(s) C h(s) (u 2 p (s) C u 3 q(s)) ,

(s, u 2 , u 3 ) 2 I  U , (19.4)

where coordinates u 2 , and u 3 refer to a local rectangular frame perpendicular to t and spanned by vectors p and q We then have 
@ xQ D x 0 C h 0 (u 2 p C u 3 q) C h u 2 p 0 C u 3 q 0 @s 

D (1  h (au 2 C b u 3 )) t C h 0 u 2  h c u 3 p C h 0 u 3 C h c u 2 q , (19.5)

x1 D

19.2 Laplacian in a Tubular Neighborhood of a Curve – Arc-Length Parameterization

@ xQ D hp , @u 2 @ xQ D hq . x3 D @u 3

x2 D

(19.6) (19.7)

The metric tensor (G i j D x i
x j ) is then 

Gi j





(1  h (au 2 C b u 3))2 C h 02 C h 2 c 2 (u 2 )2 C (u 3 )2 D4 h (h 0 u 2  h c u 3 ) h (h 0 u 3 C h c u 2) 3 h (h 0 u 2  h c u 3 ) h (h 0 u 3 C h c u 2) 5 . h2 0 2 0 h 2

(19.8)

The determinant is G D h 4 (1  h (au 2 C b u 3 ))2 ,

(19.9)

and the inverse is 

G

 ij

2 0 1 4 D 2 0 h 0

2 3 0 h2 1 4 5 (u h 2 h 0  u 3 h c) 0 C 2 (1  h (au C b u ))2 h 2 3 h (u 3 h 0 C u 2 h c) 1 3 h (u 3 h 0 C u 2 h c) h (u 2 h 0  u 3 h c) 2 (u 2 h 0  u 3 h c) (u 2 h 0  u 3 h c) (u 3 h 0 C u 2 h c)5 . 0 0 (u 2 h  u 3 h c) (u 3 h C u 2 h c) (u 3 h 0 C u 2 h c)2 0 1 0

(19.10) 19.2.1.1 Minimal Rotating Frame It is possible to develop a simplifying representation by choosing a specific frame of local orthogonal eigenvectors t, p , and q. This is always possible as for a given local centerline tangent vector an infinite set of orthogonal choices of p and q vectors can be chosen to span the local cross section (all varying to within a rotation). We shall consider one such frame, the so-called minimal rotating frame. This will be found useful in simplifying the expression for the Laplacian. Consider a space curve and let t, n, b be the Frenet–Serret frame where
and τ are the curvature and torsion, respectively. We can write the frame p , q as

p D αn C βb ,

q D β n C α b ,

(19.11)

where α 2 C β 2 D 1. Then, p 0 D α 0 n C α(
t C τ b) C β 0 b  β τ n D α
t C (α 0  β τ)n C (β 0 C α τ)b , (19.12)

243

244

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves



 q 0 D β 0 n  β(
t C τ b)C α 0 b  α τ n D β
t  β 0 C α τ n C α 0  β τ b , (19.13) and a D t
p 0 D α
,

b D t
q 0 D β
,

c D q
p 0 D α β0  β α0 C τ . (19.14)

As α α 0 C β β 0 D 0, we see that c D 0 if and only if α0 D τ β ,

β 0 D τ α ,

(19.15)

so if T 0 (s) D τ(s), then α(s) D cos(T(s)) ,

β(s) D sin(T(s)) .

(19.16)

In that case, we have a D
cos T ,

b D 
sin T,

a 0 D
0 cos T C
τ sin T ,

b 0 D 
0 sin T C
τ cos T ,

(19.17) (19.18)

and 

a 00 D
00 
τ 2 cos T C 2
0 τ C
τ 0 sin T ,

(19.19)



b 00 D 
00 
τ 2 sin T C 2
0 τ C
τ 0 cos T .

(19.20)

In other words, choosing α(s) D cos(T(s)), β(s) D sin(T(s)), where T 0 (s) D τ(s), we obtain a D
cos T, b D 
sin T, c D 0. This frame is the so-called minimal rotating frame.

19.2 Laplacian in a Tubular Neighborhood of a Curve – Arc-Length Parameterization

19.2.2 Laplacian

The Laplace operator can now be calculated in (s, u 2 , u 3 ) coordinates:   @j G r R2 3 D G i j @ i @ j C G i j C @ j G i j @i 2G     1 1 @2 @ @2 @  D 2 C C b a h h (1  h (au 2 C b u 3 )) @u 2 @u 3 @ (u 2 )2 @ (u 3 )2 ( 1 h (a 0 u 2 C b 0 u 3 C c (au 3  b u 2 )) @ @2 C C 2 2 (1  h(au 2 C b u 3 )) @s 1  h (au 2 C b u 3 ) @s  0    h @ @ @ @ @ u2 2 C c u2 C u3  u3 h @u 2 @u 3 @u 3 @u 2 @s 2     @2 h0 @2 h0 h0  u C u C 2 u c u c C u2  u3c 2 3 3 2 2 h h h @u 2 @u 3 @ (u 2 )  2 0 2 h @ C u3 C u2 c h @(u 3 )2 !   0 2 c 2 C (a 0 u 2 C b 0 u 3 ) h 0 h 00 @ @ h  u2 C 2  C u3 h h 1  h (au 2 C b u 3 ) @u 2 @u 3     c h
1 h 0 C c (a 0 u 2 C b 0 u 3 ) h c h0 @ @ C  u2 u3 C c0  h 1  h (au 2 C b u 3 ) @u 2 @u 3
   c 2 (u 2 )2 C (u 3 )2 h @ @ Cb a C 1  h (au 2 C b u 3 ) @u 2 @u 3
  ) c (u 2 )2 C (u 3 )2 h 0 @ @ C . (19.21) b a 1  h (au 2 C b u 3 ) @u 2 @u 3 It is clear that the expression for the Laplacian is extremely complex. Hence, and for the rest of the chapter, we will make use of the minimal rotating frame approximation derived in the previous subsection. If c D 0, the Laplacian in Eq. (19.21) can be rewritten as     1 @ 1 @2 @ @2 r2 D 2  a C C b h h @u 2 @u 3 @ (u 2 )2 @ (u 3 )2   @2 @ @ @ @ C 2 , (19.22)  a2 u 2 C b2 u 3 C ab u 2 C ab u 3 @u 2 @u 3 @u 3 @u 2 @s where terms proportional to @2 /@(u 2 )2 , @/@u 2 , and u 2 @/@u 2 are kept (order zero or negative power in u 2 ) assuming small cross-sectional dimensions. Note that, e. g., the two terms: u 3 @/@u 3 and u 2 @/@u 3 are of order (u 3 )0 and (u 2 )1 (u 3 )
1 , respectively. Hence both terms are of zeroth order in smallness and must be kept in the perturbative analysis. Note that @/@u 2 and @/@u 3 are of order (u 2 )
1 and (u 3 )
1 , respectively, and so on. Thus, a term proportional to (u 2 )2 @/@u 2 is of order u 2 and,

245

246

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

hence, is negligible for curved structures with large aspect ratios (small cross-sectional dimensions compared with the length dimension). Observe that the L2 -norm on Ω is given by Z “ p kψk2 D jψj2 G du 2 du 3 ds I

U

Z “ p Z “ D j F ψj2 du 2 du 3 ds D jχj2 du 2 du 3 ds , p

I

I

U

(19.23)

U

F ψ and p F D G D h 2 (1  h (au 2 C b u 3 )) . (19.24) p That is, the map ψ 7! χ D F ψ is an isometry L2 (Ω ) ! L2 (I  U). Using Maple, we transform Eq. (19.21) to   p 2 χ 1 @2 χ @2 @2 1 χC F rR3 p D 2 C 2 2 2 h (1  h (au 2 C b u 3 )) @s 2 @ (u 2 ) @ (u 3 ) F 0   2h @ @ @χ h u  C u 2 3 2 @u @u @s (1  h (au 2 C b u 3 )) 2 3 ! 0 2h @χ 2h (a 0 u 2 C b 0 u 3 ) h  C (1  h (au 2 C b u 3 ))3 (1  h (au 2 C b u 3 ))2 @s 1 0  0 2 00 0 (a 0 0 4 hh  hh u 2 C b u 3) C 2h B C@  A (1  h (au 2 C b u 3 ))2 (1  h (au 2 C b u 3 ))3   @ @  u2 χ C u3 @u 2 @u 3  0 2 where χ D

h h

C

(1  h (au 2 C b u 3))2   @2 @2 @2 2 (u ) χ  (u 2 )2 C 2u u C 2 3 3 @u 2 @u 3 @ (u 2 )2 @ (u 3 )2  0 2 00
2 C 2 hh  hh 4 C χ (1  h (au 2 C b u 3))2 h (a 00 u 2 C b 00 u 3 )  4h 0 (a 0 u 2 C b 0 u 3 ) χ C 2 (1  h (au 2 C b u 3 ))3 C

5h 2 (a 0 u 2 C b 0 u 3 )2 χ. 4 (1  h (au 2 C b u 3))4

(19.25)

In the case where h D 1 (constant cross-sectional area as a function of s) and neglecting terms of zero or positive order in u 2 and u 3 , we have   p 2 χ
2 @2 @2 @2 χC χ. (19.26) Fr R 3 p D C C 2 2 2 4 @ (u 1 ) @ (u 2 ) @ (u 3 ) F

19.2 Laplacian in a Tubular Neighborhood of a Curve – Arc-Length Parameterization

19.2.3 Circular Cross Section

Previously, we considered a local Cartesian coordinate system spanned by t, p , and q. This choice is convenient in the case of a rectangular wire cross section; however, if the cross section is circular, we instead use polar coordinates , θ defined by u 2 D  cos θ ,

(19.27)

u 3 D  sin θ ,

(19.28)

sin θ @ @ @  , D cos θ @u 2 @  @θ @ cos θ @ @ C , D sin θ @u 3 @  @θ

(19.29) (19.30)

sin2 θ @2 @2 2 cos θ sin θ @2 @2 2 C D cos θ  @(u 2 )2 @2  @@θ 2 @θ 2 C

2 cos θ sin θ @ sin2 θ @ C ,  @ 2 @θ

(19.31)

cos θ sin θ @2 @2 cos2 θ  sin2 θ @2 @2  D cos θ sin θ 2 C @u 2 @u 3 @  @@θ 2 @θ 2 

cos2 θ  sin2 θ @ cos θ sin θ @  ,  @ 2 @θ

(19.32)

cos2 θ @2 @2 @2 2 cos θ sin θ @2 C D sin2 θ 2 C 2 @(u 3 ) @  @@θ 2 @θ 2 C

2 cos θ sin θ @ cos2 θ @  ,  @ 2 @θ

(19.33)

and @2 @2 @2 1 @2 1 @ , C D C C 2 2 2 @θ 2 @(u 2 )2 @   @ (u ) @ 3 @2 @2 @2 @2 (u 2 )2 C 2u 2 u 3 C (u 3 )2 D 2 2 , 2 2 @u 2 @u 3 @ @ (u 2 ) @ (u 3 ) @ @ @ . u2 C u3 D @u 2 @u 3 @

(19.34) (19.35) (19.36)

In Section 19.2.1.1, we obtained a D
cos T ,

b D 
sin T ,

(19.37)

247

248

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

a 0 D
0 cos T C
τ sin T , b 0 D 
0 sin T C
τ cos T , 

a 00 D
00 
τ 2 cos T C 2
0 τ C
τ 0 sin T ,

 b 00 D 
00 
τ 2 sin T C 2
0 τ C
τ 0 cos T ,

(19.38) (19.39) (19.40)

and, hence, au 2 C b u 3 D 
cos(T C θ ) , 0

0

(19.41)

0

(19.42) a u 2 C b u 3 D 
cos(T C θ )  
τ sin(T C θ ) , 
0 
00 00 00 2 0 a u 2 Cb u 3 D 

τ cos(T Cθ ) 2
τ C
τ sin(T Cθ ) . (19.43) So Eq. (19.25) can be written as  2  p 2 χ 1 1 @2 1 @ @ F rR3 p D 2 C C χ h @2 2 @θ 2  @ F 0

2 hh 1 @2 χ @ @χ  C (1  h
cos(T C θ ))2 @s 2 (1  h
cos(T C θ ))2 @ @s

! 2h 0 2h(
0 cos(T C θ ) 
τ sin(T C θ )) @χ h C  (1  h
cos(T C θ ))3 (1  h
cos(T C θ ))2 @s  1 0   2 h0 h 00   4 h h B 2h 0 2 (
0 cos(T C θ ) 
τ sin(T C θ )) C C @χ CB A @ @ (1  h
cos(T C θ ))2  (1  h
cos(T C θ ))3  0 2 00
2 C 2 hh  hh @2 χ 4 C C χ (1  h
cos(T C θ ))2 (1  h
cos (T C θ ))2 @2 (h
00  4h 0
0  h
τ 2 ) cos(T C θ )  (2h
0 τ C h
τ 0  4h 0
τ) sin(T C θ ) C χ 2(1  h
cos(T C θ ))3 2

C

 0 2 h h

5h 2 2 (
0 cos(T C θ ) 
τ sin(T C θ ))2 χ. 4(1  h
cos(T C θ ))4

(19.44)

19.3 Application to the Schrödinger Equation

With the determination of the Laplace operator in the curved coordinates u i (i D 1, 2, 3), the Schrödinger equation for a quantum-mechanical particle of mass m and energy E reads (applies to zeroth order in u 1 and u 2 )   „2
2 2 2 2 @1 χ C @2 χ C @3 χ C χ C V (u 1 , u 2 , u 3 ) χ D E χ , (19.45) 2m 4 and the potential V satisfies V (u 1 , u 2 , u 3 ) D 0 ,

(19.46)

19.3 Application to the Schrödinger Equation

if (u 1 , u 2 , u 3 ) is a point within the nanowire structure, that is, the domain 2  u 2  2 and 3  u 3  3 in the case of a nanowire having a rectangular cross section. Similarly, the potential V is assumed infinite outside the nanowire structure. As the curvature
is a function of u 1 only, it is immediately apparent that a separable solution χ D χ 1 (u 1 ) χ 2 (u 2 )χ 3 (u 3 ) can be sought. Insertion into Eq. (19.45) gives   2
 λ  µ χ1 D 0 , @21 χ 1 C (19.47) 4 @22 χ 2 C ν 2 χ 2 D 0 , 
@23 χ 3 C µ  ν 2 χ 3 D 0 ,

(19.48) (19.49)

with λ D 2m E/„2 and µ and ν are separation constants. We also note the interesting fact that the form of the three ordinary differential equations (Eqs. (19.47)– (19.49)) as well as the associated boundary conditions (i.e., energies and eigenstates) are unchanged if the nanowire cross section rotates with varying u 1 in the present approximation. This is so as varying directions of vectors p and q with u 1 do not influence the form of Eqs. (19.47)–(19.49) or the boundary conditions. 19.3.1 Solutions to the χ 2 and χ 3 Equations

Consider the cross-sectional domain of the quantum wire to be parameterized by 2  u 2  2 and 3  u 3  3 . The equations in χ 2 and χ 3 can be solved immediately. Firstly, the general solution to Eq. (19.48) in χ 2 is χ 2 (u 2 ) D sin (νu 2 C φ 2 ) ,

(19.50)

where c and φ 2 are constants determined by the Dirichlet boundary conditions imposed, that is, χ 2 (2 ) D sin (ν2 C φ 2 ) D 0 , χ 2 (2 ) D sin (ν2 C φ 2 ) D 0 .

(19.51)

These conditions require νD

mπ , 22

(19.52)

where m is an integer different from zero. The other constant, the phase φ 2 , is next chosen such that φ 2 D ν2 ,

(19.53)

and both Dirichlet conditions in Eq. (19.51) are now fulfilled. When m is even (and different from zero), χ 2 becomes   mπ (19.54) χ 2 (u 2 ) D sin u2 , 22

249

250

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

whereas for m odd,  χ 2 (u 2 ) D cos

mπ u2 22

 .

(19.55)

Similar arguments applied to Eq. (19.49) show that  µν D 2

nπ 23

2 ,

(19.56)

where n is an integer different from zero. The eigenfunction χ 3 is (when n is even (and different from zero))   nπ χ 3 (u 3 ) D sin u3 , (19.57) 23 whereas for n odd, 

nπ χ 3 (u 3 ) D cos u3 23

 .

(19.58)

Now, combining Eqs. (19.52) and (19.56) allows us to specify µ:  µD

mπ 22

2

 C

nπ 23

2 ,

(19.59)

with m D ˙1, ˙2, ˙3 and n D ˙1, ˙2, ˙3. The possible values of the particle energy E are finally found from the χ 1 eigenvalue equation (Eq. (19.47)) by imposing appropriate boundary conditions given the value of µ (obtained from Eq. (19.59)).

19.4 Schrödinger Equation in a Tubular Neighborhood of a Curve – General Parameterization

In general, it is difficult to find an explicit arc-length parameterization r(s). Hence, we need to account for a general parameterization r(t) with t D t(s) and jr 0 (t)j ¤ 1 as follows:  
1 dt d d ds d D D , (19.60) ds ds dt dt dt  
1 dχ 1 dχ 1 ds D , (19.61) ds dt dt !  
1  
1 ds ds d2 χ 1 d dχ 1 D ds 2 dt dt dt dt  
3 2  
2 2 ds ds d χ1 d s dχ 1 . (19.62)  D dt dt 2 dt dt 2 dt

19.5 Applications

Now, ˇ ˇ p ds D ˇ r 0 (t)ˇ D r 0
r 0 , dt

(19.63)

r 0
r 00 d2 s D p , dt 2 r0
r0

(19.64)


2 D

jr 0  r 00 j2 jr 0 j6

D

(r 0  r 00 )
(r 0  r 00 ) jr 0 j2 jr 00 j2  (r 0
r 00 )2 D . (r 0
r 0 )3 (r 0
r 0 )3

(19.65)

In other words, in terms of an arbitrary parameterization of the curve, Eq. (19.47) becomes χ 001 

r 0
r 00 0 χ C r0
r0 1



 (r 0
r 0 )(r 00
r 00 )  (r 0
r 00 )2 0 0  (λ C µ)(r
r ) χ1 D 0 . 4(r 0
r 0 )2 (19.66)

For completeness, we now give the full set of ordinary differential equations in the three curvilinear coordinates for a general parameterization: χ 001 

r 0
r 00 0 χ C r0
r0 1



 (r 0
r 0 )(r 00
r 00 )  (r 0
r 00 )2 0 0  (λ C µ)(r
r ) χ1 D 0 . 4(r 0
r 0 )2 (19.67)

@22 χ 2 C ν 2 χ 2 D 0 ,

(19.68)


@23 χ 3 C µ  ν 2 χ 3 D 0 ,

(19.69)

χ 01 D @1 χ 1 ,

(19.70)

where

and λD

2m eff E . „2

(19.71)

We note that the equations for χ 2 and χ 3 are the same as before and the solutions presented in Section 19.3.1 still apply.

19.5 Applications

In this section, the solution χ 1 to Eq. (19.67), or Eq. (19.47) in cases with arc-length parameterizations, for three cases of nanowire axes – the straight-line axis, the circular axis, and the sinusoidal axis – is derived. Clearly, the cases chosen here are arbitrary and similar calculations can be made for any shape of the nanowire axis.

251

252

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

The determination of χ 1 also implies specification of the full eigenstate χ D χ 1 χ 2 χ 3 and its energy E (through λ) using the analytical expressions for χ 2 and χ 3 when µ is given. Example 1 Straight-line nanowire axis In the case of a straight-line nanowire axis, the following parameterization applies: r (u 1 ) D (u 1 , 0, 0) ,

(19.72)

ˇ 0 ˇ ˇ r (u 1 )ˇ D 1 ,

(19.73)

giving ˇ 00 ˇ ˇ r (u 1 )ˇ D 0 ,

that is, the parameterization is a parameterization by arc length. Hence, we may use the simpler Eq. (19.47) so as to determine χ 1 . Since jr 00 (u 1 ) j D 0, the curvature is zero and Eq. (19.47) reads χ 001  (λ C µ) χ 1 D 0 ,

(19.74)

with the general solution  p λ  µu 1 C φ 1 , χ 1 D sin

(19.75)

with φ 1 an arbitrary phase. Next, employing the boundary conditions χ 1 (u 1 D 0) D χ 1 (u 1 D L) D 0 , the solution becomes   lπ u1 , χ 1 (u 1 ) D sin L

l D ˙1, ˙2, ˙3, . . . ,

and the associated energy eigenvalue is "   0 2  0 2 # „2 λ „2 lπ 2 m π nπ . E D D C C 2m 2m L 22 23

(19.76)

(19.77)

(19.78)

This result is the exact three-dimensional result for energies of a particle confined to a straight nanowire structure, that is, perfect agreement is found using the present differential-geometry model approximation where small values of u 2 and u 3 have been assumed. This conclusion is not surprising. In fact, if, for example, a nanowire cross section does not rotate with varying u 1 (equivalent to having p and q constant with u 1 ), then the coordinates (u 1 , u 2 , u 3 ) are simply the Cartesian coordinates.

19.5 Applications

Example 2 Circular nanowire axis Consider next the circular nanowire axis with parameterization u    u  1 1 r (u 1 ) D R cos , R sin ,0 , R R

(19.79)

where ˇ ˇ 0 ˇ r (u 1 )ˇ D 1 ,

ˇ 00 ˇ ˇ r (u 1 )ˇ D 1 . R2

(19.80)

This corresponds to a circular nanowire axis of radius R parameterized by arc length, that is, again we may use the simpler Eq. (19.47) to determine χ 1 :   1 χ1 D 0 , χ 001  λ C µ  (19.81) 4R 2 with the general solution r χ 1 D sin

1 λ  µ C u 1 C φ1 4R 2

! ,

(19.82)

with φ 1 an arbitrary phase. Next, employing the boundary conditions, χ 1 (u 1 D 0) D χ 1 (u 1 D L) D 0 , corresponding to an open circular nanowire structure, gives   lπ u 1 , l D ˙1, ˙2, ˙3, . . . , χ 1 (u 1 ) D sin L and the associated energy eigenvalue is # "   0 2  0 2 „2 λ „2 1 lπ 2 m π n π . E D D C C  2m 2m L 22 23 4R 2

(19.83)

(19.84)

(19.85)

The case with a closed circular nanowire structure is less restrictive in that χ 1 (u 1 ) D χ 1 (u 1 C 2π R) . In this case, we obtain the condition r 1  µ  λ D 2l π , l D 0, ˙1, ˙2, ˙3, . . . , 2π R 4R 2 giving the following eigenstate solutions (with L D 2π R)   2l π u 1 C φ 1 , l D 0, ˙1, ˙2, ˙3, . . . , χ 1 (u 1 ) D sin L

(19.86)

(19.87)

(19.88)

253

254

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

and the energy spectrum „2 „2 λ E D D 2m 2m

"

2l π L

2

 C

m0 π 22

2

 C

n0 π 23

2

1  4R 2

# .

(19.89)

Note that l D 0 is possible here since no restrictions on the phase φ 1 are imposed for the closed circular nanowire axis. This result is exactly the same as is found for the cylinder surface of revolution in [100] when noting that L in [100] equals the present 22 . It was concluded in [100] that the energy expression (Eq. (19.89)) is excellent whenever the cylinder thickness is less than approximately 10 % of the radius R.

Example 3 Helical quantum-wire axis Next, let us consider the simple but relevant case of a helical quantum wire which can be parameterized as an arc-length parameterization (shown below). We emphasize that a similar procedure can be carried out for a general parameterization using Eq. (19.67) instead of Eq. (19.47). A helical quantum-wire axis parameterization is       b u1 u1 u1 , a sin p ,p , (19.90) r (u 1 ) D a cos p a2 C b2 a2 C b2 a2 C b2 where a and b are the helix radius and pitch length, respectively. Thus, ˇ 0 ˇ ˇ r (u 1 )ˇ D 1 ,

ˇ 00 ˇ ˇ r (u 1 )ˇ D

a2

a , C b2

(19.91)

and an arc-length parameterization is obtained such that Eq. (19.47) applies:   a2 χ 001  λ C µ  χ1 D 0 . (19.92) 4 (a 2 C b 2 )2 Evidently, the general solution is χ 1 D A n m sin (α n m u 1 ) C B n m cos (α n m u 1 ) , where A n m and B n m are constants, and s a2 αnm D  λ  µnm . 2 4 (a C b 2 )2

(19.93)

(19.94)

Observe that the cases of a straight quantum wire and a circular quantum wire correspond to a D 0, b ¤ 0 and a ¤ 0, b D 0, respectively. Hence, these cases are solved simultaneously. For an incoming electron with energy E, λ and α n m are known immediately for specific n, m cross-sectional indices, that is, specific µ n m values.

19.5 Applications

Example 4 Sinusoidal nanowire axis Consider a sinusoidal nanowire axis parameterized as u    1 r (u 1 ) D u 1 , R sin ,0 , a

(19.95)

corresponding to a circular nanowire axis variation of “amplitude” R and “period” 2π a. The derivatives r 0 and r 00 become  u   R 1 0 (u ) cos ,0 , r 1 D 1, a a  u   R 1 ,0 , (19.96) r 00 (u 1 ) D 0,  2 sin a a that is, the parameterization is not by arc length and use of the general parameterization equation is necessary. One finds u  u  R2 1 1 sin , cos 3 a a a   2 u1 R , D 1 C 2 cos2 a a   2 R u1 , D 4 sin2 a a

r 0
r 00 D  ˇ 0 ˇ2 ˇr ˇ ˇ 00 ˇ2 ˇr ˇ

(19.97)

and the general u 1 equation reads (to second order in R) χ 001



 R 2 cos ua1 sin ua1 0  C
 χ 1 2 a 3 1 C Ra 2 cos2 ua1 # "
    sin2 ua1 R2 R2 2 u1 χ1 D 0 . C
  (λ C µ) 1 C 2 cos 4a 4 1 C R22 cos2 u 1 a a a

a

(19.98) Imposing the boundary conditions χ 1 (u 1 D 0) D χ 1 (u 1 D L) D 0

(19.99)

allows determination of the eigenvalues λ (or E) and the corresponding eigenstates χ1. Consider the case where L is an integer number of half periods: L D π N a, with N an integer. In Figure 19.1, we show the first three eigenstates of Eq. (19.98) for the parameter values R D 20 nm, a D 10 nm, and N D 2. The corresponding eigenvalues E are listed in Table 19.1.

255

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

256 0.08

0.08

0.06

0.06

0.04

0.04

0.02

0.02

χ1

χ1

0

0

−0.02

−0.02

−0.04

−0.04 −0.06

−0.06

−0.08

−0.08 0

0.1

0.2

0.3

0.4

(a)

0.5

0.6

0.7

0.8

0

0.9

0.1

0.2

0.3

0.4

(b)

u1/L

0.5

0.6

0.7

0.8

0.9

u1/L

0.1

0.05

χ

1

0

−0.05

−0.1

0

0.1

0.2

0.3

0.4

(c)

0.5

0.6

0.7

0.8

0.9

u1/L

Figure 19.1 The first (a), second (b), and third (c) χ 1 eigenstates for a sinusoidal nanowire with parameters R D 20 nm, a D 10 nm, and N D 2. Table 19.1 The first three energy eigenvalues for a sinusoidal nanowire with parameters R D 20 nm, a D 10 nm, and N D 2. Parameter values

E(1)


Case 1


0.057

„2 2m

µ (meV)

E(2)

0.017

„2 2m

µ (meV)

E(3)


„2 2m

µ (meV)

0.27

Example 5 Straight-axis nanowire with two subsequent 90° bends Consider now a straight-axis nanowire with two subsequent 90ı bends parameterized as   u  u  A   u  u   A 1 A 1 B 1  tanh C 1 C tanh ,0 . r (u 1 ) D u 1 , 2 δ 2 δ (19.100) The two bends are located at u 1 D u A and u 1 D u B . In Table 19.2, the first three eigenvalues are shown (relative to µ) for a nanowire with bends at 1.5 and 3.0 nm with δ D 0.5 nm, A D 2 nm, and u 1 parameter range of 5 nm.

19.5 Applications

257

0.12 0.1 0.1 0.05

1

0

χ

χ1

0.08

0.06

0.04

−0.05

0.02

−0.1

0

0.1

0.2

0.3

0.4

(a)

0.5

0.6

0.7

0.8

0.9

0

0.1

0.2

0.3

(b)

u1/L

0.4

0.5

0.6

0.7

0.8

u1/L

0.1

χ1

0.05

0

−0.05

−0.1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

u1/L

(c)

Figure 19.2 The first (a), second (b), and third (c) χ 1 eigenstates for a straight-axis nanowire with two subsequent 90ı bends. The nanowire parameters are A D 2 nm, δ D 0.5 nm, u A D 1.5 nm, u B D 3.0 nm, and the u 1 range is 5 nm. Table 19.2 First three energy eigenvalues for a straight-axis nanowire with two subsequent 90ı bends. The nanowire parameters are A D 2 nm, δ D 0.5 nm, u A D 1.5 nm, u B D 3.0 nm, and the u 1 range is 5 nm. Energy

E(1)
22.5

„2 2m

µ (meV)

E(2)
93.2

„2 2m

µ (meV)

E(3)


„2 2m

µ (meV)

191

The corresponding eigenstates can be seen in Figure 19.2.

Example 6 Elliptic nanowire axis Consider finally an elliptic nanowire parameterized as  u   u    1 1 , R2 sin ,0 . r (u 1 ) D R1 cos 2π R 2π R

(19.101)

0.9

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

258

0.1

0.09

0.08

0.08

0.06 0.07

0.04

0.06

1

0.02

χ1

χ

0.05

0

0.04

−0.02

0.03

−0.04 −0.06

0.02

−0.08 0.01 −0.1 0

0.1

0.2

0.3

0.4

0.5

(a)

0.6

0.7

0.8

0.9

0

0.1

0.2

0.3

(b)

u1/L

0.4

0.5

0.6

0.7

0.8

0.9

u1/L

0.1 0.08 0.06 0.04

χ

1

0.02 0

−0.02 −0.04 −0.06 −0.08 −0.1 0

0.1

0.2

0.3

0.4

(c)

0.5

0.6

0.7

0.8

0.9

u1/L

Figure 19.3 The first (a), second (b) and third (c) three eigenstates for a closed elliptic nanowire. The nanowire parameters are R1 D 5 nm, R2 D 8 nm, and 2πR D 1 nm. Table 19.3 First three energy eigenvalues for a closed elliptic nanowire. The nanowire parameters are R1 D 5 nm, R2 D 8 nm, and 2πR D 1 nm. Energy

E(1)

0.06

„2 2m

µ (meV)

E(2)
0.50

„2 2m

µ (meV)

E(3)


„2 2m

µ (meV)

1.71

In Table 19.3, the first three eigenvalues are shown (relative to µ) for a nanowire with R1 D 5 nm, R2 D 8 nm, and 2π R D 1 nm. The three associated eigenstates can be seen in Figure 19.3.

19.6 Perturbation Theory Applied to the Curved-Structure Problem

19.6 Perturbation Theory Applied to the Curved-Structure Problem

We can now derive a simple expression for eigenvalue and eigenstate changes using first- and second-order perturbation theory in terms involving the curvature and/or torque. Since eigenstates for the unperturbed problem are parity eigenstates, terms changing parity in one or more coordinates such as a@/@u 2 (changes parity in u 2 ) do not contribute in first-order perturbation theory but they do contribute in second-order perturbation theory. For example, a@/@u 2 gives a contribution proportional to a 2 in second-order perturbation theory, whereas a 2 u 2 @/@u 2 is parity conserving in u 2 and contributes to order a 2 using first-order perturbation theory. Eigenvalue contributions to lower order in a (or b) cannot occur and contributions to order higher than a 2 and so on are too small and will be discarded henceforth. Mixed terms in Eq. (19.22) contribute to order a 2 b 2 using secondorder perturbation theory as they are parity changing in both u 2 and u 3 . Terms proportional to a 2 b 2 are, however, negligibly small. Summarizing, the additional terms in the Laplacian involving a or b lead to eigenvalue changes ∆λ n l m in the unperturbed eigenstate jnl mi given by the expression   @ @ jnl mi C b2 u 3 ∆λ n l m D hnl mj  a 2 u 2 @u 2 @u 3 X hnl mjV jn 0 l 0 m 0 ihn 0 l 0 m 0 jV jnl mi C , (19.102) λ 0n l m  λ 0n 0 l 0 m 0 0 0 0 jn l m i

where V D

1 h

 a

@ @ Cb @u 2 @u 3

 ,

(19.103)

and E n0 l m is the unperturbed eigenvalue associated with eigenstate jnl mi, and n, l, m are the quantum indices associated with the u 1 , u 2 , u 3 coordinates, respectively. We emphasize, here and in the following, that jnl mi is assumed to be a nondegenerate unperturbed state and that no energy crossings with other (unperturbed) states take place owing to bending effects. If this assumption is violated, a degenerate perturbation theory analysis is necessary. It is, however, still possible to employ Eq. (19.102) if degenerate unperturbed states are not mixed owing to perturbative effects. 19.6.1 Dirichlet Unperturbed Eigenstates

A normalized unperturbed Dirichlet eigenstate jnl mi is 3 , jnl mi  ψ n l m D ψ n1 ψ l2 ψ m

(19.104)

259

260

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

where ZL

ˇ 1 ˇ ˇ ψ (u 1 )ˇ2 du 1 D 1 , n

(19.105)

0

and ψ n1 (u 1 D 0) D ψ n1 (u 1 D L) D 0 . Further, ψ l2 D p

  1 lπ cos u2 2h2 h2

(19.106)

(19.107)

if l is odd, whereas ψ l2 D p

  1 lπ sin u2 2h2 h2

if l is even. Similarly, 3 ψm D p

1 h3

if m is odd, and 3 ψm



cos

mπ u3 2h3



mπ D p sin u3 2h 3 h3 1

(19.108)

 (19.109)

 (19.110)

if m is even. 19.6.2 Evaluation of ∆ λ n in the Case with Dirichlet Boundary Conditions

In the following, we choose the scaling parameter h D 1. Using Eqs. (19.105)– (19.110) for the unperturbed eigenstates in Eq. (19.104) yields for the first-order perturbation contributions proportional to a (if l is odd) (1a) ∆λ n l m

1 D  cos l π 2 D

1 2

ZL ψ n1 (u 1 ) a (u 1 )2 ψ n1 (u 1 ) du 1 0

ZL

ψ n1 (u 1 ) a (u 1 )2 ψ n1 (u 1 ) du 1 ,

(19.111)

0

since hlju 2 @u@ 2 jli D (1a) ∆λ n l m

1 2

cos(l π), whereas

1 D cos l π 2 D

1 2

ZL

ZL ψ n1 (u 1 ) a (u 1 )2 ψ n1 (u 1 ) du 1 0

ψ n1 (u 1 ) a (u 1 )2 ψ n1 (u 1 ) du 1 , 0

(19.112)

19.6 Perturbation Theory Applied to the Curved-Structure Problem

if l is even. A similar first-order perturbation contribution due to b exists (if m is odd): (1b) ∆λ n l m

1 D  cos m π 2 D

1 2

ZL ψ n1 (u 1 ) b (u 1 )2 ψ n1 (u 1 ) du 1 0

ZL

ψ n1 (u 1 ) b (u 1 )2 ψ n1 (u 1 ) du 1 ,

(19.113)

0

and (1b)

∆λ n l m D

1 cos m π 2

1 D 2

ZL ψ n1 (u 1 ) b (u 1 )2 ψ n1 (u 1 ) du 1 0

ZL

ψ n1 (u 1 ) b (u 1 )2 ψ n1 (u 1 ) du 1

(19.114)

0

if m is even. Using second-order perturbation theory, the change in eigenvalues due to terms involving a becomes (if l is odd) (2a) ∆λ n l m

ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ 1 1 D 2 ψ n 0 (u 1 ) a (u 1 ) ψ n (u 1 ) du 1 ˇˇ 2 n 0 ˇˇ ˇ 0 " #    0  2 0 l 0 sin



X l 02 π 2 (22 )2

l 0 D2,4,...

(l
l)π 2 l
l 0

l 2 π2 (22 )2



l 0 sin

C C

(lCl )π 2 lCl 0

n 02 π 2 L2

n2 π2 L2



ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ 1 1 ˇ (u ) (u ) (u ) D ψ a ψ du 1 1 1 1ˇ . n n0 ˇ 4 0 ˇ ˇ n

(19.115)

0

In the case where l is even, we have (2a)

∆λ n l m

ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ 1 1 ˇ (u ) (u ) (u ) D 2 ψ a ψ du 0 1 1 1 1 n n ˇ ˇ 2 n 0 ˇ ˇ 0 "  0  #  2  0 l sin



(l
l )π 2

l 0
l

X l 02 π 2 l 0 D1,3,... (22 )2



l 2 π2 (22 )2

C C

l sin

(l Cl)π 2

l 0 Cl n 02 π 2 L2



n2 π2 L2

ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ 1 1 D ψ n 0 (u 1 ) a (u 1 ) ψ n (u 1 ) du 1 ˇˇ . 4 0 ˇˇ ˇ n

0

(19.116)

261

262

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

An additional contribution using second-order perturbation theory must be added involving b instead of a (if m is odd): (2b)

∆λ n l m

ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ 1 1 D 2 ψ n 0 (u 1 ) b (u 1 ) ψ n (u 1 ) du 1 ˇˇ 3 n 0 ˇˇ ˇ 0 " #   0  0  2 (m
m)π 2 m
m 0

m 0 sin



X

m 02 π 2 (23 )2

m 0 D2,4,...



(mCm )π 2 mCm 0

m 0 sin

C

m2 π2 (23 )2

C

n2 π2 L2

n 02 π 2 L2



ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ 1 1 (u ) (u ) (u ) D ψ n 0 1 b 1 ψ n 1 du 1 ˇˇ . 4 0 ˇˇ ˇ n

(19.117)

0

In the case where m is even, we have for the b contribution ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ (2b) 1 1 ˇ (u ) (u ) (u ) ∆λ n l m D  2 ψ b ψ du 1 1 1 1ˇ n0 n ˇ 3 n 0 ˇ ˇ 0

"



  (m
m 0 )π m sin 2 m 0
m

X m 0 D1,3,...

m 02 π 2 (23 )2



C

m2 π2 (23 )2

 0  #2 (m Cm)π m sin 2

C

m 0 Cm

n2 π2 L2



n 02 π 2 L2

ˇ L ˇ2 ˇZ ˇ X ˇ ˇ 1 ˇ ψ 1 0 (u 1 ) b (u 1 ) ψ 1 (u 1 ) du 1 ˇ . D n n ˇ ˇ 4 0 ˇ ˇ n

(19.118)

0

In the p case of a circular-bent or helical-bent structure with constant curvature
D a 2 C b 2 , an exact analytical evaluation of the four perturbative contributions, Eqs. (19.111)–(19.118), gives the following result independent of quantum indices n, l, and m (neglecting small terms proportional to τ 2 a 2 and so on in the helix case): (1a)

(1b)

(2a)

(2b)

∆λ n l m D ∆λ n l m C ∆λ n l m C ∆λ n l m C ∆λ n l m D


2 . 4

(19.119)

For a general bent structure, a simple analytical result as Eq. (19.119) cannot be found but the fast computation of eigenstate and eigenvalue changes for bent structures based on Eq. (19.102) still applies. This result obtained perturbatively agrees with the result obtained in previous works [96, 98, 99, 101]. We emphasize that Eq. (19.102) applies to the case with general boundary conditions (e.g., Dirichlet, Neumann, Robin [102]), whereas previous works [96, 98, 99, 101] required Dirichlet conditions. The possibility to address Neumann boundary conditions (refer to later discussions) allows, for example, curved-structure rigid-wall acoustic problems to be solved using the present computationally effective method.

19.6 Perturbation Theory Applied to the Curved-Structure Problem

19.6.3 Eigenstate Perturbations

A first-order perturbative change in eigenstates due to bending involving a linear integral expression in a results from the term a 2 u 2 @/@u 2 in Eq. (19.22). For the odd l states, the expression is RL (a) ∆ψ n l m

1 X D 2 0

0

ψ n1 0 (u 1 ) a (u 1 )2 ψ n1 (u 1 ) du 1  n Lπ2 C 2 2

n ¤n

n 02 π 2 L2

ψ n0 l m .

(19.120)

There is an additional contribution from the term b 2 u 3 @/@u 3 (odd m states), RL (b) ∆ψ n l m

1 X D 2 0

0

ψ n1 0 (u 1 ) b (u 1 )2 ψ n1 (u 1 ) du 1  n Lπ2 C 2

n ¤n

2

n 02 π 2 L2

ψ n0 l m .

(19.121)

ψ n0 l m ,

(19.122)

ψ n0 l m .

(19.123)

Similarly, for the even l states, RL (a) ∆ψ n l m

1 X D 2 0

0

ψ n1 0 (u 1 ) a (u 1 )2 ψ n1 (u 1 ) du 1  n Lπ2 C 2 2

n ¤n

n 02 π 2 L2

and for even m states, RL (b) ∆ψ n l m

1 X D 2 0

0

ψ n1 0 (u 1 ) b (u 1 )2 ψ n1 (u 1 ) du 1

n ¤n

 n Lπ2 C 2

2

n 02 π 2 L2

It should be stressed that the first-order perturbative changes in eigenstates due to the terms a(1/ h)(@/@u 2 ) and b(1/ h)(@/@u 3 ) are proportional to 2 /L and 3 /L, respectively, and hence are negligibly small. 19.6.4 Neumann Unperturbed Eigenstates

A normalized unperturbed Neumann eigenstate jnl mi is 3 , jnl mi  ψ n l m D ψ n1 ψ l2 ψ m

(19.124)

where ZL jψ n1 (u 1 ) j2 du 1 D 1 , 0

(19.125)

263

264

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

and @ψ n1 @ψ n1 (u 1 D 0) D (u 1 D L) D 0 . @u 1 @u 1

(19.126)

Further, ψ l2



lπ D p cos u2 2h2 h2 1

 (19.127)

if l is even, whereas ψ l2

  1 lπ D p sin u2 2h2 h2

(19.128)

if l is odd. Similarly, 3 ψm D p



1 h3

cos

mπ u3 2h3

 (19.129)

if m is even, and 3 ψm D p



1 h3

sin

mπ u3 2h3

 (19.130)

if m is odd. 19.6.5 Evaluation of ∆ λ n in the Case with Neumann Boundary Conditions

We choose again the scaling parameter h D 1. Using Eqs. (19.125)–(19.130) for the unperturbed eigenstates in Eq. (19.102) yields for the first-order perturbation contributions proportional to a (if l is odd) (1a) ∆λ n l m

1 D cos l π 2 D

1 2

ZL

ZL ψ n1 (u 1 ) a (u 1 )2 ψ n1 (u 1 ) du 1 0

ψ n1 (u 1 ) a (u 1 )2 ψ n1 (u 1 ) du 1 ,

(19.131)

0

whereas 1 (1a) ∆λ n l m D  cos l π 2 1 D 2

ZL

ZL ψ n1 (u 1 ) a (u 1 )2 ψ n1 (u 1 ) du 1 0

ψ n1 (u 1 ) a (u 1 )2 ψ n1 (u 1 ) du 1 0

(19.132)

19.6 Perturbation Theory Applied to the Curved-Structure Problem

if l D 2, 4, . . .. In the special case l D 0, we obtain (1a)

∆λ n0m D 0 .

(19.133)

A similar first-order perturbation contribution due to b exists (if m is odd): (1b) ∆λ n l m

1 D cos m π 2 D

1 2

ZL ψ n1 (u 1 ) b (u 1 )2 ψ n1 (u 1 ) du 1 0

ZL

ψ n1 (u 1 ) b (u 1 )2 ψ n1 (u 1 ) du 1 ,

(19.134)

0

and 1 (1b) ∆λ n l m D  cos m π 2 1 D 2

ZL ψ n1 (u 1 ) b (u 1 )2 ψ n1 (u 1 ) du 1 0

ZL

ψ n1 (u 1 ) b (u 1 )2 ψ n1 (u 1 ) du 1

(19.135)

0

if m D 2, 4, . . .. Again, if m D 0, we have (1b)

∆λ n l0 D 0 .

(19.136)

Using second-order perturbation theory, the change in eigenvalues due to terms involving a becomes (if l is even and nonzero)

(2a)

∆λ n l m

ˇ L ˇ2 ˇZ ˇ X ˇ ˇ 1 ˇ ψ 1 0 (u 1 ) a (u 1 ) ψ 1 (u 1 ) du 1 ˇ D 2 n n ˇ ˇ 2 n 0 ˇ ˇ 0 " #   0  0  2 l 0 sin



(l
l)π 2

l
l 0

X l 02 π 2 (22 )2

l 0 D1,3,...



l 2 π2 (22 )2

C C

l 0 sin

(lCl )π 2

lCl 0 n 02 π 2 L2



n2 π2 L2

ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ 1 1 ˇ . (u ) (u ) (u ) D ψ a ψ du 0 1 1 1 1 n n ˇ ˇ 4 0 ˇ ˇ n

0

(19.137)

265

266

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

In the case where l is odd, we have (2a)

∆λ n l m

ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ 1 1 ˇ (u ) (u ) (u ) D 2 ψ a ψ du 0 1 1 1 1 n n ˇ ˇ 2 n 0 ˇ ˇ 0 "  0  #  0  2 l sin



X l 02 π 2 l 0 D2,4,... (22 )2

(l
l )π 2 0 l
l

C

l 2 π2 (22 )2



(l Cl)π 2 0 l Cl

l sin

n 02 π 2 L2

C

n2 π2 L2



ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ 1 1 D ψ n 0 (u 1 ) a (u 1 ) ψ n (u 1 ) du 1 ˇˇ . 4 0 ˇˇ ˇ n

(19.138)

0

When l D 0, the second-order perturbative contribution stemming from a vanishes: (2a)

∆λ n l m D 0 .

(19.139)

An additional contribution using second-order perturbation theory must be added involving b instead of a (if m is even and nonzero):

(2b) ∆λ n l m

ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ 1 1 (u ) (u ) (u ) D 2 ψ n 0 1 b 1 ψ n 1 du 1 ˇˇ 3 n 0 ˇˇ ˇ 0 " #   0  0  2 (m
m)π 2 m
m 0

m 0 sin



X m 0 D1,3,...

m 02 π 2 (23 )2



(mCm )π 2 mCm 0

m 0 sin

C

m2 π2 (23 )2

C

n2 π2 L2

n 02 π 2 L2



ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ 1 1 ˇ (u ) (u ) (u ) D ψ b ψ du 1 1 1 1ˇ . n n0 ˇ 4 0 ˇ ˇ n

(19.140)

0

In the case where m is odd, we have for the b contribution, (2b)

∆λ n l m

ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ 1 1 ˇ (u ) (u ) (u ) D 2 ψ b ψ du 0 1 1 1 1 n n ˇ ˇ 3 n 0 ˇ ˇ 0 "  #2   0 0  m sin



(m
m )π 2

m 0
m

X m 0 D2,4,...

m 02 π 2 (23 )2



C

m2 π2 (23 )2

m sin

C

(m Cm)π 2

m 0 Cm n2 π2 L2



n 02 π 2 L2

ˇ L ˇ2 ˇZ ˇ ˇ 1 X ˇˇ 1 1 D ψ n 0 (u 1 ) b (u 1 ) ψ n (u 1 ) du 1 ˇˇ . 4 0 ˇˇ ˇ n

0

(19.141)

19.6 Perturbation Theory Applied to the Curved-Structure Problem

When m D 0, the second-order perturbative contribution stemming from b vanishes (2b)

∆λ n l m D 0 .

(19.142)

In p the case of a circular-bent or helical-bent structure with constant curvature
D a 2 C b 2 , an exact analytical evaluation of the four perturbative contributions (Eqs. (19.131)–(19.141)) gives the following result (neglecting small terms proportional to τ 2 a 2 , etc. in the helix case): (1a)

(1b)

(2a)

(2b)

∆λ n l m D ∆λ n l m C ∆λ n l m C ∆λ n l m C ∆λ n l m  3 D  (1  δ l0) a 2 C (1  δ m0 ) b 2 . 4

(19.143)

Again, for a general bent structure, a simple analytical result as Eq. (19.143) cannot be found but fast computation of eigenstate and eigenvalue changes for bent structures based on Eq. (19.102) is possible. 19.6.6 Perturbation Theory in the General Parameterization Case

It follows from Eqs. (19.60)–(19.66) that for a general parameterization given by t as compared with an arc-length parameterization given by s,  
2 2  
3 2 1 @2 ds ds r 0
r 00 @ @2 @ d s @ D 0 0 2 0 02 . (19.144) D  2 2 2 @s dt @t dt dt @t r
r @t (r
r ) @t Hence, @2 χ 1 @2 χ 1 D C 2 @s @t 2



 2 r 0
r 00 @χ 1 1 @ χ1 ,  1  0 02 0 0 2 r
r @t (r
r ) @t

(19.145)

where the last two terms can be treated using perturbation theory as we shall show next. In the splitting of terms in Eq. (19.145), it is implicitly assumed that the deviation of r 0
r 0 from 1 is solely due to structure bending. gen The change in eigenvalues provided by the last two terms in Eq. (19.145), ∆λ n l m , accounting for both first-order and second-order perturbation theory can be written as gen

∆λ n l m D hnjV jni C

0 X hnjV jn 0 i hn 0 j V † jni n0

where

 VD

λ 0n l m  λ 0n 0 l m

 2 1 r 0
r 00 @ @ ,  1  0 02 0 0 2 r
r @t (r
r ) @t

,

(19.146)

(19.147)

and jni is a set of unperturbed eigenstates corresponding to the case of a straight structure which can be solved immediately. Thus, it is possible to obtain the influence of bending effects on eigenvalues (and eigenstates) of a general bent structure by solving (computationally fast) integrals in known unperturbed eigenstates.

267

268

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

19.6.7 Comparison between Analytical Results and Perturbation Theory for Circular-Bent Rectangular Domains in Two Dimensions – Dirichlet Boundary Conditions

In this section, we will check the perturbative results for Dirichlet boundary conditions using the simplest possible nontrivial case study. 19.6.7.1 Rectangular Domain – No Bending Consider a two-dimensional rectangular domain with thickness  and length L. The length and thickness directions are the x and y directions, respectively. Solving the Helmholtz equation,  2  @ @2 ψ C k2 ψ D 0 , C (19.148) @x 2 @y 2

using the separation of variables method gives  nπ 2  m π 2 k 2  k n2 m,1 D C , L  where n and m are positive integers.

(19.149)

19.6.7.2 Rectangular Domain – With Bending Consider the rectangular domain bent into a circle with radius of curvature R. We use polar coordinates r, θ and a parameterization of the bent domain is R
D R  /2  r  R C /2 D RC and 0  θ  L/R. In terms of the quantum index,

nπ R , (19.150) L where n is a positive integer, eigenvalues for k (indexed k l m ) are found by imposing Dirichlet conditions on the general solution to the resulting Bessel equation of order l:   (19.151) ψ l m (r, θ ) D sin(l θ ) A l m J l (k l m r) C B l m Yl (k l m r) . lD

The corresponding secular equation specifying the possible k l m values reads J l (k l m R
) Yl (k l m RC )  J l (k l m RC ) Yl (k l m R
) D 0 .

(19.152)

The difference in (squared-k) eigenvalues between Eqs. (19.149)–(19.152), 1 , 4R 2 agrees with the perturbative expression given by Eq. (19.119). ∆λ 1a n a D k n2 m,1  k l2m D

(19.153)

19.6.8 Comparison between Analytical Results and Perturbation Theory for Circular-Bent Rectangular Domains in Two Dimensions – Neumann Boundary Conditions

In this section, we will check the perturbative results for Neumann boundary conditions using the simplest possible nontrivial case study.

19.7 Problems

19.6.8.1 Rectangular Domain – No Bending Consider a two-dimensional rectangular domain with thickness  and length L. The length and thickness directions are the x and y directions, respectively. Solving the Helmholtz equation,  2  @2 @ C 2 ψ C k2 ψ D 0 , (19.154) @x 2 @y

using the separation of variables method to the Neumann boundary problem gives  nπ 2  m π 2 k 2  k n2 m,2 D C , (19.155) L  where n and m are positive integers. Note that the eigenvalue spectrum is identical to that in the Dirichlet case. 19.6.9 Rectangular Domain – With Bending

Consider the rectangular domain bent into a circle with radius of curvature R. We use polar coordinates r, θ and a parameterization of the bent domain is R
D R  /2  r  R C /2 D RC and 0  θ  RL . In terms of the quantum index, lD

nπ R , L

(19.156)

where n is an integer (n D 0 is possible), eigenvalues for k (indexed k l m ) are found by imposing Neumann conditions on the general solution to the resulting Bessel equation of order l: ψ l m (r, θ ) D cos (l θ ) [A l m J l (k l m r) C B l m Yl (k l m r)] .

(19.157)

The corresponding secular equation specifying the possible k l m values reads   k l2m J l0 (k l m R
) Yl0 (k l m RC )  J l0 (k l m RC ) Yl0 (k l m R
) D 0 . (19.158) The difference in (squared-k) eigenvalues between Eqs. (19.155)–(19.158), ∆λ 2a n a D k n2 m,2  k l2m D 

 3
1  δ klm0 , 4R 2

(19.159)

agrees with the perturbative expression given by Eq. (19.143).

19.7 Problems

In the following, it is implicitly assumed that wire structures have a large aspect ratio defined as the wire length divided by a cross-sectional dimensional parameter (radius for a circular cross section or largest side length for a rectangular cross section).

269

270

19 Quantum-Mechanical Particle Confined to the Neighborhood of Curves

1. Consider a helical nanowire structure with the centerline parameterization  r (u 1 ) D

     u1 b u1 u1 , a sin p ,p , a cos p a2 C b2 a2 C b2 a2 C b2 (19.160)

where a and b are constants and 0  u 1  L. Write down eigenstate energy expressions assuming a rectangular cross section with side lengths L 2 and L 3 along the u 2 and u 3 coordinates, respectively, for a particle with mass m. Infinite barriers are considered. (Hint: Use the separated equations in χ 1 , χ 2 , χ 3 .) 2. Consider a bent nanowire structure parameterized by u 1 , u 2 , u 3 . Transform the Schrödinger equation in χ in terms of coordinates u 1 , r, and θ , where r and θ are the polar coordinates of the cross-sectional plane perpendicular to the local tangent vector t(u 1 ). Compute energy eigenstates for a helical nanowire with a circular cross section of radius R assuming infinite barriers. 3. Show that it is not possible to separate the differential equation in χ for Neumann boundary conditions: @ n ψ D 0, where @ n denote the normal derivative at the nanowire cross-sectional rim. (Hint: Remember that ψ is the ”true” wave p function and χ D F ψ, where p (19.161) F D G D h 2 (1  h (au 2 C b u 3 )) , in the case with a rectangular nanowire cross section.) 4. Using perturbation theory, compute energy eigenstates of a helical nanowire with rectangular cross sections assuming infinite barriers. Compute the ground state using perturbation theory and compare that ground state with the ground state of a straight nanowire of the same length. 5. Using perturbation theory, compute energy eigenstates of a helical nanowire with rectangular cross sections assuming Neumann boundary conditions. Compute the ground state using perturbation theory and compare that ground state with the ground state of a straight nanowire of the same length. 6. a. Consider an acoustic cavity shaped as a helix with parameters a, b, L as above. Compute eigen(wave)vectors k for the rigid-wall case. (Hint: Note that the acoustic pressure p satisfies the Helmholtz equation: r 2 p C k 2 p D 0.) b. Assume that the acoustic cavity contains air characterized by a sound speed c D 340 m/s. Calculate the eigenfrequencies.

271

20 Quantum-Mechanical Particle Confined to Surfaces of Revolution 20.1 Introduction

In this chapter, we develop the theory of a quantum-mechanical particle confined to a surface of revolution using differential-geometry methods including the derivation of a general set of three ordinary differential equations in the spatial coordinates. Case studies involve the computation of eigenstates and energy eigenvalues to the Schrödinger problem of a quantum-mechanical particle confined to a truncated cone, a cylinder, and three elliptic tori (with the semimajor axis perpendicular to or parallel to the plane of the generating circle of the torus, respectively, and the circular torus).

20.2 Laplacian in Curved Coordinates

We already have the Laplacian in general curvilinear coordinates, Eq. (18.53). We now consider a tubular neighborhood of a surface Σ embedded in R 3 [98]. In the following, let k, ` 2 f1, 2, 3g, α, β 2 f1, 2g, and x W U ! R 3 be a parameterization of Σ with unit normal vector n, metric tensor g α β D x α
x β , and second fundamental form b α β D x α β
n D x α
n β ,

(20.1)

where x α denotes differentiation of x with respect to u α . We also need the third fundamental form n α
n β D 2M b α β  K g α β ,

(20.2)

where M is the mean curvature and K is the Gaussian curvature of Σ ; see [103]. They are functions of u 1 and u 2 only. Consider the usual parameterization of a tubular neighborhood of Σ : x W (u 1 , u 2 , u 3 ) 7! x(u 1 , u 2 ) C u 3 n(u 1 , u 2 ) . Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

272

20 Quantum-Mechanical Particle Confined to Surfaces of Revolution

With X α D x α C u 3 N α and X 3 D N, the metric tensor G α β D X α
X β is given by
 G α β D g α β  2u 3 b α β C (u 3 )2 2M b α β  K g α β ,

(20.3)

G i,3 D G3,i D δ i3 ,

(20.4)

(i D 1, 2, 3) .

This gives   2 D gF2 , G D det [G k` ] D det G α β D G11 G22  G12

(20.5)

F D 1  2M u 3 C K(u 3)2 .

(20.6)

where

Denote bQ 11 D b 22 /g, bQ 12 D b 12 /g, and bQ 22 D b 11 /g, then G



 g α β 1  K (u 3 )2  2 bQ α β u 3  M (u 3 )2 D . F2

αβ

We can now express the usual Laplacian in R 3 using these tubular coordinates (or Gaussian normal coordinates):  r R2 3 D G i j @ i @ j C D G α β @α @β C „

Gi j @j G C @j Gi j 2 G

 @i !

@3 F G α β @β G C @ β G α β @ α C @23 C @3 . 2 G „ ƒ‚F … ƒ‚ … 2 r t2

(20.7)

rn

This gives the splitting of the Laplacian in the tangential and the normal part r R2 3 D r t2 C r n2 . Use of the above expression for F and the Laplacian yields to zeroth order in u 3 
r R2 3 D @23  2M @3 ψ C r Σ2 . Next, χ D

p

F ψ, then,   1 1 @3 F @3  χ, @3 ψ D p 2 F F

  ! 3 @3 F 2 @3 F 1 @23 F χ, @3  C  F 2 F 4 F !   1 @3 F 2 1 1 @23 F r n2 ψ D p @23  χ C 2 F 4 F F ! (M  K u 3 )2 1 K @23 C χ. D p  F2 F F @23 ψ

(20.8)

1 D p F

@23

(20.9) (20.10) (20.11) (20.12)

20.2 Laplacian in Curved Coordinates

As F depends on u 3 only, we have r t2 ψ D F
1/2 r t2 χ. Hence, p

Fr R2 3 ψ

p
 (M  K u 3 )2 K D F r t2 C r n2 ψ D r t2 C @23 C  F2 F

! χ

If we expand the right-hand side in u 3 we obtain p 2
 F r R 3 ψ D r02 C u 3 r12 C (u 3 )2 r22 C . . . χ C @23 χ ,

(20.13)

(20.14)

where r k2 , k D 0, 1, 2, . . . , are operators on Σ . As F(u 3 D 0) D 1 and G α β (u 3 D 0) D g α β , we have that r t2 (u 3 D 0) D r Σ2 , the Laplace–Beltrami operator on Σ . In particular, the following applies: r02 D r Σ2 C (M 2  K ) .

(20.15)

We now specialize to a surface of revolution. It can be parameterized as x (u 1 , u 2 ) D (r (u 1 ) cos u 2 , r (u 1 ) sin u 2 , z (u 1 )) ,

(20.16)

where (r(u 1 ), z(u 1 )) is a curve in the x z plane. The metric tensor is given by g 11 D r 0 (u 1 )2 C z 0 (u 1 )2 ,

g 22 D r(u 1 )2 ,

g 12 D g 21 D 0 ,

(20.17)

and the principal curvatures are [103]
1 D
(u 1 ) D

r 0 z 00  z 0 r 00 (r 02 C

z 02 )3/2

,

z0
2 D p . r r 02 C z 02

(20.18)

The mean and Gaussian curvatures are MD

1 (
1 C
2 ) , K D
1
2 , 2

(20.19)

that is, M2  K D

1 (
1 
2 )2 , 4

and r02 D

  @21 @22 r 0 r 00 C z 0 z 00 r0 @1 C C  r 02 C z 02 r2 r (r 02 C z 02 ) (r 02 C z 02 )2  0 00  r z  z 0 r 00 1 1 z0 2 C  . 4 r 02 C z 02 r 02 C z 02 r

(20.20)

273

274

20 Quantum-Mechanical Particle Confined to Surfaces of Revolution

20.3 The Schrödinger Equation in Curved Coordinates

The Schrödinger equation in curved coordinates for a particle confined to the surface (u 1 , u 2 , u 3 D 0) is given by  „2
2 r0 C @23 χ (u 1 , u 2 , u 3 ) C V (u 1 , u 2 , u 3 ) χ (u 1 , u 2 , u 3 ) D E χ (u 1 , u 2 , u 3 ) , 2m (20.21) where the Laplacian is as given by Eq. (20.15) and V is a potential confining the particle to the surface. If the potential V is a function of u 3 only, then the separation-ofvariables method can be used to solve Eq. (20.21). Hence, assuming χ(u 1 , u 2 , u 3 ) D χ 1 (u 1 )χ 2 (u 2 )χ 3 (u 3 ), inserting into Eq. (20.21) and using the expression for r02 in Eq. (20.20), we obtain   0 r 0 r 00 C z 0 z 00 r 2 @1 χ 1  @1 χ 1 C r r 02 C z 02 !    z0 2  c 2 
02 1 r 0 z 00  z 0 r 00 02 χ1 D 0 , C   c1 C 2 r C z (20.22) 4 r 02 C z 02 r r @22 χ 2 C c 2 χ 2 D 0 ,   2m (E  V (u 3 )) @23 χ 3 C C c 1 χ3 D 0 . „2

(20.23) (20.24)

20.4 Applications

In this section, we compare results for eigenvalues and eigenstates of electrons confined to two surfaces of revolution – the truncated cone and the elliptic torus – by employing differential form methods using curvilinear coordinates (u 1 , u 2 , u 3 ) with u 3 nearly zero in the three-dimensional structure. 20.4.1 Truncated Cone

Consider a quantum-mechanical particle confined to the surface of a truncated cone (Figure 20.1) parameterized as follows: (r (u 1 ) , z (u 1 )) D (R o C α u 1 , u 1 ) ,

0  u1  L I

0  u 2 < 2π ,

(20.25)

where R o is the smallest cone radius and α is the slope of the cone along the axis (being zero in the case of a cylinder). Hence, r0 D α ,

r 00 D 0 ,

z0 D 1 ,

z 00 D 0 ,

(20.26)

20.4 Applications Truncated cone Ro+αL Ro

L

0

u1

Figure 20.1 The truncated cone.

and Eqs. (20.22)–(20.24) read α @1 χ 1 Ro C α u1


2  1 c2 1 α C 1 χ1 D 0 , C  c1  4 (R o C α u 1 )2 (R o C α u 1 )2

@21 χ 1 

@22 χ 2 C c 2 χ 2 D 0 ,   2m (E  V (u 3 )) @23 χ 3 C χ3 D 0 , C c 1 „2

(20.27) (20.28) (20.29)

where c 1 and c 2 are separation constants. Equation (20.28) can be solved immediately so as to give c2 D l 2 ,

(20.30)

where l is an integer and χ 2 D exp(i l u 2) ,

(20.31)

by employing the boundary condition χ 2 (0) D χ 2 (2π) .

(20.32)

Next, the expression specifying the relation of the energy E to the separation constant c 1 is found by imposing the boundary condition χ (u 1 , u 2 , u 3 D ˙3 ) D 0 on Eq. (20.29) with ( 0 V(r) D 1,

if R i (u 1 )  r  R o (u 1 ) , otherwise ,

(20.33)

(20.34)

275

276

20 Quantum-Mechanical Particle Confined to Surfaces of Revolution

where R i (u 1 ) and R o (u 1 ) are the cone inner and outer radii corresponding to u 1 , respectively. This yields   nπ (u 3 C 3 ) n D 1, 2, 3, . . . , (20.35) χ 3 (u 3 ) D sin 23 and the energy can be written as # "  „2 nπ 2 ED  c1 , 2m 23

(20.36)

where n D 1, 2, 3, . . . (and c 1 ) are quantum indices. The remaining expression giving c 1 in terms of c 2 is determined by numerically solving Eq. (20.27) and imposing the boundary conditions χ 1 (u 1 D 0) D χ 1 (u 1 D L) D 0 ,

(20.37)

for example, by use of the finite-difference method. The first few eigenvalues c 1 for c 2 D 0, 1, 4 corresponding to the case of a truncated cone with R o D 1 nm, α D 0.1, and L D 2 nm are given in Table 20.1. We also computed the case of a cylinder corresponding to the following parameter values: R o D 1 nm, α D 0, and L D 2 nm. Values are listed in Table 20.2. The c 1 values obtained agree well with those obtained analytically [100]: c1 D 

 m π 2 L



c2 1 C , R2 4R 2

(20.38)

where m is an integer (column 2,3, and 4 in Table 20.2 correspond to m D 1, m D 2, and m D 3, respectively) and R D (R i C R o )/2. The analytical values are also given in Table 20.2. In Figure 20.2, we show χ 1 as a function of u 1 for the first three c 1 values in Table 20.1 corresponding to c 2 D 0. Evidently, the first three eigenstates have zero, one, and two nodes, respectively, with respect to the u 1 coordinate as one would intuitively expect. However, since inversion symmetry with respect to u 1 D L/2 is broken for the cone with α ¤ 0, eigenstates are not even or odd with respect to u 1 D L/2 (this conclusion is reached by close inspection of Figure 20.2 and Table 20.1 The first three values of the separation constant c 1 (m
2 ) computed using the finite-difference method for the truncated cone with parameter values R o D 1 nm, α D 0.1, c 1 (first)

c 1 (second)

and L D 2 nm in cases with c 2 D 0, 1, 4. The energy contribution stemming from the particle motion in u 1 and u 2 coordinate space is
„2 c 1 /(2m) (refer to Eq. (20.36)).

c 1 (third)

c2 D 0


2.24  1018


9.56  1018


21.78  1018

c2 D 1 c2 D 4


3.07  1018
5.55  1018


10.40  1018
12.90  1018


22.61  1018
25.11  1018

20.4 Applications The energy contribution stemming from the particle motion in u 1 and u 2 coordinate space is
„2 c 1 /(2m) (refer to Eq. (20.36)). The corresponding analytical values – obtained from Eq. (20.38) – are also given.

Table 20.2 The first three values of the separation constant c 1 (m
2 ) computed using the finite-difference method for the cylinder with parameter values R o D 1 nm, α D 0, and L D 2 nm in cases with c 2 D 0, 1, 4. c 1 (first)

c 1 (second)

c 1 (third)

c2 D 0


2.22  1018


9.62  1018


21.95  1018

c 2 D 0 (analytical)


2.22  1018


9.62  1018


21.96  1018

c2 D 1 c 2 D 1 (analytical)


3.22  1018
3.22  1018


10.62  1018
10.62  1018


22.95  1018
22.96  1018

c2 D 4


6.22  1018


13.62  1018


25.95  1018

c 2 D 4 (analytical)


6.22  1018


13.62  1018


25.96  1018

0.05

0.05 χ1

0.1

χ1

0.1

χ1

0.1 0.05 0

0

0

−0.05

−0.05

−0.05

−0.1

−0.1

−0.1

(a)

0

1/4

1/2 u1/L

3/4

0

(b)

1/4

1/2 u1/L

3/4

0

1/4

(c)

1/2 u1/L

Figure 20.2 The first (a), second (b), and third (c) eigenstates: χ 1 as a function of u 1 corresponding to c 2 D 0 for the truncated cone with parameters R o D 1 nm, α D 0.1, and L D 2 nm.

becomes more apparent as the slope α increases). For the cylinder with α D 0, inversion symmetry with respect to u 1 D L/2 applies as the boundary conditions at u 1 D 0 and u 1 D L are the same and the geometry is obviously left unchanged under inversion with respect to u 1 D L/2. 20.4.2 Elliptic Torus

Next, consider a particle confined to the surface of an elliptic torus (Figure 20.3) parameterized as (r (u 1 ) , z (u 1 )) D (R1 cos (u 1 ) C a, R2 sin (u 1 )) ,

277

0  u 1  2π I

0  u 2 < 2π ,

(20.39)

where a, R1 , and R2 are the radius of the generating circle of the torus, the semiaxis of the torus elliptic cross section in the plane of the generating circle, and the semiaxis of the torus elliptic cross section perpendicular to the plane of the generating circle, respectively (note that the well-known circular torus corresponds to the case

3/4

278

20 Quantum-Mechanical Particle Confined to Surfaces of Revolution

R2

a

R1

Figure 20.3 The elliptic torus.

where R1 D R2 ). Hence, r 0 D R1 sin u 1 , 0

z D R2 cos u 1 ,

r 00 D R1 cos u 1 , 00

z D R2 sin u 1 ,

, r 000 D R1 sin u 1 , z 000 D R2 cos u 1 ,

(20.40)

and Eq. (20.22) reads @21 χ 1

! R1 sin(u 1 ) 1 R12 sin (2u 1 )  R22 sin (2u 1 ) @1 χ 1 C  C R1 cos (u 1 ) C a 2 R12 sin2 (u 1 ) C R22 cos2 (u 1 ) 2 !2 R2 cos (u 1 ) R1 R2 41   4 R12 sin2 (u 1 ) C R22 cos2 (u 1 ) R1 cos (u 1 ) C a

  
2 2 c2 2 2 R1 sin (u 1 ) C R2 cos (u 1 ) χ 1 D 0 , (20.41) c1 C (R1 cos (u 1 ) C a)2

where c 1 and c 2 are separation constants. The corresponding two ordinary differential equations in u 2 , u 3 are the same as in Eqs. (20.23)and (20.24) (in fact, the two equations in u 2 , u 3 are the same for all surfaces of revolution). We also note that the expression for energy eigenvalues, Eq. (20.36), applies to a general surface of revolution. Again, Eq. (20.41) is solved using the finite-difference method. The only change in boundary conditions is for the χ 1 function, where χ 1 (0) D χ 1 (2π) applies in the elliptic torus case. In Table 20.3, we show the computed first three values for c 1 as obtained for an elliptic torus with a D 5 nm, R1 D 2 nm, and R2 D 0.5 nm in the cases with c 2 D 0, 1, 4. In Table 20.4, a similar set of computed c 1 values is given for an elliptic torus with parameters a D 5 nm, R1 D 0.5 nm, and R2 D 2 nm. Apparently, the values in Tables 20.3 and 20.4 are almost the same (but not identical as the two geometries are slightly different!). Finally, we give in Table 20.5 c 1 values for the circular torus with the same crosssectional area as the two elliptic tori mentioned above, that is, corresponding to parameter values a D 5 nm, R1 D 1 nm, and R2 D 1 nm.

20.4 Applications Table 20.3 First three values of the separation constant c 1 (m
2 ) computed using the finitedifference method for the elliptic torus with parameter values R1 D 2 nm, R2 D 0.5 nm,

and a D 5 nm in cases with c 2 D 0, 1, 4. The energy contribution stemming from the particle motion in u 1 and u 2 coordinate space is
„2 c 1 /(2m) (refer to Eq. (20.36)).

c 1 (first)

c 1 (second)

c 1 (third)

c2 D 0 c2 D 1


0.51  1018
0.55  1018


11.33  1018
11.77  1018


20.87  1018
21.34  1018

c2 D 4


0.68  1018


13.09  1018


22.76  1018

Table 20.4 First three values of the separation constant c 1 (m
2 ) computed using the finitedifference method for the elliptic torus with parameter values R1 D 0.5 nm, R2 D 2 nm,

and a D 5 nm in cases with c 2 D 0, 1, 4. The energy contribution stemming from the particle motion in u 1 and u 2 coordinate space is
„2 c 1 /(2m) (refer to Eq. (20.36)).

c 1 (first)

c 1 (second)

c 1 (third)

c2 D 0


0.51  1018


11.33  1018


20.88  1018

c2 D 1


0.55  1018


11.74  1018


21.29  1018

c2 D 4


0.67  1018


12.98  1018


22.52  1018

Table 20.5 First three values of the separation constant c 1 (m
2 ) computed using the finitedifference method for the circular torus with parameter values R1 D 1 nm, R2 D 1 nm,

and a D 5 nm in cases with c 2 D 0, 1, 4. The energy contribution stemming from the particle motion in u 1 and u 2 coordinate space is
„2 c 1 /(2m) (refer to Eq. (20.36)).

c 1 (first)

c 1 (second)

c 1 (third)

c2 D 0

C0.2606  1018


0.738 71  1018


0.739 35  1018

c2 D 1 c2 D 4

C0.2182  1018 C0.0925  1018


0.7805  1018
0.9040  1018


0.7825  1018
0.9160  1018

In Figure 20.4, we show χ 1 as a function of u 1 for the first three c 1 values in Table 20.3, respectively, corresponding to c 2 D 0 and the elliptic torus parameter values a D 5 nm, R1 D 2 nm, and R2 D 0.5 nm. The differential equation in u 1 for the general elliptic torus (Eq. (20.41)) is invariant under the inversion operator, since @1 ! @1 and sin(u 1 ) !  sin(u 1 ), cos(u 1 ) ! cos(u 1 ) when u 1 ! u 1 . Hence, given the boundary conditions χ 1 (0) D χ 1 (2π) D χ 1 (2π), all eigenstates are even or odd with respect to inversion in u 1 , that is, χ 1 (π C x) D ˙χ 1 (π  x) D ˙χ 1 (π  x C 2π) D ˙χ 1 (π  x) , (20.42) such that all states are either even or odd with respect to inversion around u 1 D π. This conclusion (even and odd states) is also reached by close inspection of Figures 20.4–20.6.

279

280

20 Quantum-Mechanical Particle Confined to Surfaces of Revolution

0.1

0.1

0.05

0.05

0.05

0

0.15

χ1

χ1

0.15

0.1

χ1

0.15

0

−0.05

−0.05

0

−0.05

−0.1

−0.1

−0.1

−0.15 0

−0.15

−0.15

1/4

1/2

3/4

1

u1/(2π)

(a)

0

1/4

1/2

3/4

1

0

u1/(2π)

(b)

1/4

1/2

3/4

1

u1/(2π)

(c)

Figure 20.4 The first (a), second (b), and third (c) eigenstates: χ 1 as a function of u 1 corresponding to c 2 D 0 for the elliptic torus with parameters R1 D 2 nm, R2 D 0.5 nm, and a D 5 nm. 0.15

0.1

0.1

0.1

0.05

0.05

0.05

χ1

χ1

0

−0.05

0

−0.05

−0.1 −0.15 0

χ1

0.15

0.15

1/4

1/2

3/4

−0.1

−0.1

−0.15

−0.15 0

0

1

u1/(2π)

1/4

1/2

3/4

1

u1/(2π)

(b)

(a)

0

−0.05

1/4

1/2

3/4

1

u1/(2π)

(c)

Figure 20.5 The first (a), second (b), and third (c) eigenstates: χ 1 as a function of u 1 corresponding to c 2 D 0 for the elliptic torus with parameters R1 D 0.5 nm, R2 D 2 nm, and a D 5 nm. 0.15

0.1

0.1

0.1

0.05

0.05

0

χ1

χ1

χ1

0.05

0

−0.05

−0.05

−0.05 −0.1 −0.1 0

(a)

1/4

1/2 u1/(2π)

3/4

1

−0.15 0

(b)

0

−0.1 1/4

1/2 u1/(2π)

3/4

1

0

1/4

1/2 u1/(2π)

3/4

1

(c)

Figure 20.6 The first (a), second (b), and third( c) eigenstates: χ 1 as a function of u 1 corresponding to c 2 D 0 for the elliptic torus with parameters R1 D 1 nm, R2 D 1 nm, and a D 5 nm.

In Figure 20.5, we show χ 1 as a function of u 1 for the first three c 1 values in Table 20.4, respectively, corresponding to c 2 D 0 and the elliptic torus parameter values a D 5 nm, R1 D 0.5 nm, and R2 D 2 nm. The same conclusion with respect to symmetry around u 1 D π is apparent.

20.5 Problems

In Figure 20.6, we show χ 1 as a function of u 1 for the first three c 1 values, respectively, in Table 20.5 corresponding to c 2 D 0 and the circular torus parameter values a D 5 nm, R1 D 0.5 nm, and R2 D 0.5 nm. Again, symmetry with respect to inversion about u 1 D π is apparent.

20.5 Problems

1. Derive Eq. (20.13) employing Eqs. (20.8) and (20.12). 2. By use of the preceding result, show that Eq. (20.15) applies. 3. Derive Eq. (20.20) by use of the expressions 1 (
1 C
2 ) , 2 K D
1
2 .

MD

(20.43)

4. Derive the set of ordinary differential equations (Eqs. (20.22)–(20.24)) using the Schrödinger equation (Eq. (20.21)) assuming the potential to be a function of u 3 only.

281

283

21 Boundary Perturbation Theory In this chapter, we give a theory for how to calculate the shift in eigenvalues of the Schrödinger equation due to geometrical perturbations of the boundary. We firstly derive the shift for the case with nondegenerate unperturbed eigenstates and then extend the analysis to the case with degenerate unperturbed eigenstates.

21.1 Nondegenerate States

Consider the Schrödinger equation for a three-dimensional quantum-mechanical particle moving in a potential U(r) defined on a domain V0 :


„2 2 r ψ0 (r) C U(r)ψ0 (r) D E0 ψ0 (r) , 2m

(21.1)

where „, m, ψ0 , E0 , and r are Planck’s constant divided by 2π, the particle mass, the wave function, the eigenenergy, and the position coordinate, respectively. Equation (21.1) can be rewritten as 
r 2 ψ0 (r) C U ren (r) C γ02 ψ0 (r) D 0 ,

(21.2)

with 2m U(r) , „2 2m γ02 D 2 E0 . „

U ren (r) D


(21.3)

Assume that we know the solutions ψ0 and E0 to Eq. (21.2) for the case with hardwall boundary conditions (ψ0 j S0 D 0) on the domain boundary: @V0 D S0 (the unperturbed problem). Our task is to determine the eigenvalue shifts E
E0 due to domain-boundary perturbations, that is, to consider the perturbed Schrödinger problem 
r 2 ψ(r) C U ren (r) C γ 2 ψ(r) D 0 ,

(21.4)

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

284

21 Boundary Perturbation Theory

with γ2 D

2m E. „2

(21.5)

Equation (21.4) must be solved for E on a perturbed domain V with boundary @V D S again imposing hard-wall boundary conditions (ψj S D 0). Applying Green’s second identity, Z Z 
(21.6) φr 2 ψ
ψr 2 φ dV D (φr ψ
ψr φ)  ndS , V

S

with n a unit normal vector to the surface S bounding the domain V (directed outward from inside the volume V), and using Eqs. (21.2), (21.4) allows the following identity to be obtained: Z Z

2 
2
 2
ψ0 r ψ
ψr ψ0 dV D
γ ψ0 ψ C γ02 ψ ψ0
dV V0




D γ02
γ

 2

Z V0

V0

ψ0
ψdV D

Z 

ψ0


S0

@ψ
@ψ
ψ 0 @n @n

 dS0 .

(21.7)

Here, and in the following, we assume that ψ D ψ0 C O(δ), r ψ D r ψ0 C O(δ), and r 2 ψ D r 2 ψ0 C O(δ), where O(δ) is a (small) quantity of order δ, the latter being a measure of the normal distance from the unperturbed surface S0 to the perturbed surface S (note that δ is a function of the position on the surface). In obtaining the second equality, we made use of the fact that two terms involving the potential U ren cancel out exactly, and the notation @ f /@n is equivalent to r f  n for any function f. Since we are considering the effect of small geometrical perturbations, we may use first-order approximations in small quantities: @ψ jS , 0 D ψj S  ψj S0 C δ(x, y ) @n 0
 
2 γ0
γ 2 ψ0
ψ  γ02
γ 2 jψ0 j2 ,

(21.8)

as γ02
γ 2 is a small quantity. Equation (21.7) can now be recast as R ˇˇ @ψ0 ˇˇ2 ˇ @n ˇ δ(x, y )dS0 S0 R , γ02
γ 2  jψ0 j2 dV0

(21.9)

V0

where ψj S D ψ0 j S0 D 0 has been used. Equation (21.9) allows the shifts in eigenvalues to be obtained immediately once the unperturbed eigenvalues and eigenstates are known. We will give two examples in this chapter and determine the eigenvalue shifts for (a) a cylindrical nanodot system with infinite quantum-confinement potential and sinusoidal perturbation of the quantum-dot radius as a

21.2 Degenerate States

function of the nanodot axial coordinate (we also prove that the present perturbation theory agrees to first order in δ with analytical results if the nanodot radius is increased from a value r1 to r2 independent of the axial coordinate), and (b) a cylindrical nanodot subject to an electric field along the axial coordinate and sinusoidal perturbation of the nanodot radius with nanodot axial coordinate.

21.2 Degenerate States

Consider next the case with degenerate states for the unperturbed geometry (domain V0 and boundary S0 ), that is, a general eigenstate corresponding to the eigenvalue E0 can be written as ψD

N X

(i)

α i ψ0 ,

(21.10)

iD1 (i)

for some coefficients fα i g where ψ0 satisfies  (i)
(i) r 2 ψ0 (r) C U ren (r) C γ02 ψ0 (r) D 0 , (i) ψ0 j S0

i D 1, 2, 3, . . . , N ,

D0.

(21.11)

Here, N is the level of degeneracy and a i are coefficients. Similarly, the perturbed problem (to first order in the eigenvalue but zeroth order in the eigenstate) can be formulated as (domain V and boundary S) 
r 2 ψ (k) (r) C U ren (r) C γ k2 ψ (k) (r) D 0 , ψ (k) j S D 0 , ψ (k) D

N X

(i)

a ki ψ0 .

(21.12)

iD1

A similar derivation as in Eq. (21.7) now gives Z 

( j )


ψ0

( j )


r 2 ψ (k)
ψ (k) r 2 ψ0

V0

Z


 D a kj γ02
γ k2 Z S0


 dV D γ02
γ k2

( j)

jψ0 j2 dS0 D

( j )
@ψ

(k)

@n

Z

( j )


ψ0 V0

N X


 a ki δ i j γ02
γ k2

iD1

V0

ψ0

D




ψ

( j )
(k) @ψ0

@n

!

dS0 D

N X iD1

Z

ψ (k) dV

Z ˇ ˇ ˇ ( j ) ˇ2 ˇψ0 ˇ dS0 V0 ( j )


δ(x, y )

ai S0

@ψ0 @n

(i)

@ψ0 dS0 , @n (21.13)

285

286

21 Boundary Perturbation Theory

where δ i j is a Kronecker delta (1 if i D j , otherwise 0). In deriving the last equality, we have made use of the first-order Taylor expansion, @ψ (k) jS D 0 , (21.14) @n 0 and neglected terms to second or higher order in the perturbation δ(x, y ). Equation (21.13) can be rewritten as an N  N matrix equation in the coefficients a i as follows: ψ (k) j S D ψ (k) j S0 C δ(x, y )

N X 


  γ k2
γ02 N j δ i j C ∆ i j a i D 0 ,

j D 1, 2, 3, . . . , N ,

(21.15)

iD1

with Nj D

Z ˇ ˇ ˇ ( j ) ˇ2 ˇψ0 ˇ dS0 , V0

Z

∆i j D

( j )


δ(x, y ) S0

@ψ0 @n

(i)

@ψ0 dS0 . @n

(21.16)

This matrix system can be solved so as to give N values of γ k corresponding to the perturbed eigenvalues and N sets of fa i g parameters specifying the coefficients of the associated eigenstates ψ k to zeroth order in the perturbation. Note that each ψ k usually corresponds to an eigenvalue with less or an equal level of degeneracy than N following the onset of the geometrical perturbation due to symmetry breaking (lifting of degeneracies).

21.3 Applications

In this section, we present a few problems and compare results obtained using perturbation theory versus analytical results whenever possible. The first example is the case of a two-dimensional particle confined to a circular region. The particle is assumed to be in a nondegenerate state in the unperturbed case. We consider as a first corollary also the case of a three-dimensional particle confined to a circular cylinder. As a second corollary to Example 1, we consider the case where the circular cylinder (constant radius) is perturbed into a cylinder with a sinusoidally varying radius as a function of the axial distance. This case cannot be solved analytically. Example 2 discusses the problem of an electron confined to a circular cylinder subject to an electric field along the axial direction, that is, a spatially dependent potential in the confinement domain. The change in energy of the ground state due to changing the radius of the cylinder into a sinusoidally varying radius is determined. Finally, in Example 3, we analyze the change of eigenvalues for a set of unperturbed degenerate states confined to a square and show that perturbing the square into a rectangle leads to splitting of the degeneracy in agreement with the analytical result to first order in the perturbation.

21.3 Applications

Example 1 Eigenvalues Of a Perturbed Circular Or Cylindrical System In a circular domain (Figure 21.1a), the Laplacian is conveniently written in polar coordinates. Thus, we have for angular-symmetric states (the ground state),  2  @ 1 @ ψ0 C γ02 ψ0 D 0 , C (21.17) @r 2 r @r with ψ0 (r D r1 ) D 0, where r1 is the radius of the circle. The first solution to this problem is ψ0 D J0 (γ0 r) ,

γ0 D

j 01 , r1

(21.18)

where j 01 is the first zero of the Bessel function J0 . This is the solution for the unperturbed eigenvalue γ0 . Assume that we perturb the radius from r1 to r2 , then the perturbed eigenvalue becomes γD

j 01 , r2

(21.19)

and the change in eigenvalue becomes to first order in the perturbation γ02
γ 2 D

2 j 01 j2 2 j 2 (r2
r1 )
01  01 3 . 2 2 r1 r2 r1

(21.20)

Dirichlet

ri

(a) r Dirichlet

ri Dirichlet

0

L

Dirichlet z

L

Dirichlet z

Dirichlet

(b) r Dirichlet

r1 Dirichlet

(c)

0 Dirichlet

Figure 21.1 The circle geometry (a), the circular cylinder (unperturbed geometry) (b), and the cylinder with a sinusoidally varying radius (perturbed geometry) (c) and associated boundary conditions considered in this work.

287

288

21 Boundary Perturbation Theory

For the numerator of Eq. (21.9), we have ˇ 2  Z ˇ ˇ @ψ0 ˇ2 ˇ δ(x, y )dS0 D 2π r1 (r2
r1 ) j 01 ˇ J12 ( j 01) , ˇ @n ˇ r1

(21.21)

S0

where J1 is the Bessel function of the first kind of order 1. For the denominator of Eq. (21.9), we have 

Zr1

Z 2

jψ0 j dV0 D 2π

r J02 0

V0

 j 01 r dr D π r12 J12 ( j 01 ) , r1

(21.22)

and Eq. (21.9) reads γ02
γ 2 

2 (r 2 j 01 2
r1 ) , r13

(21.23)

in agreement with Eq. (21.20) as expected.

Corollary 1 to Example 1

Consider next the case with an electron confined to a circular cylinder of length L and radius r1 (Figure 21.1b). We now have boundary conditions ψ0 (r D r1 , z) D ψ0 (r, z D 0) D ψ0 (r, z D L) D 0 , and the ground-state solution to the unperturbed problem is   π j 01 ψ0 (r, z) D J0 , r sin r1 L   j 01 2  π 2 C . γ02 D r1 L

(21.24)

(21.25) (21.26)

For the numerator of Eq. (21.9), we have this time ˇ 2  Z ˇ ZL π  ˇ @ψ0 ˇ2 2 ˇ δ(x, y )dS0 D 2π r1 (r2
r1 ) j 01 ˇ z dz . (21.27) J ( j )  sin2 1 01 ˇ @n ˇ r1 L 0

S0

Similarly, for the denominator of Eq. (21.9), Z



Zr1 jψ0 j dV0 D 2π 2

r

J02

0

V0

 ZL π  j 01 z dz , r dr  sin2 r1 L

(21.28)

0

and Eq. (21.9) reads γ02
γ 2 

2 (r 2 j 01 2
r1 ) , r13

(21.29)

in agreement with Eq. (21.20), which also applies to the three-dimensional circular cylinder case.

21.3 Applications

Corollary 2 to Example 1

Let us now consider the perturbed nanowire geometry case with a sinusoidally varying radius (this problem cannot be solved analytically):   Mπ r1 sin z , (21.30) δ(x, y, z) D N L where M/2 is a measure of the number of corrugation periods in the axial direction (Figure 21.1c). Assume M D 1. The numerator of Eq. (21.9) now becomes ˇ 2 ZL Z ˇ π  2  ˇ @ψ0 ˇ2 2 ˇ ˇ δ(x, y )dS0 D 2π r1 j 01 z dz , (21.31) J1 ( j 01)  sin3 ˇ @n ˇ N r1 L

S0

0

whereas the denominator is as given in Eq. (21.28). The integral over z in Eq. (21.31) is 4L/(3π); thus, the eigenvalue shift for the ground state reads (by use of Eq. (21.9) RL and 0 sin2 (π/L)zdz D L/2)   16 1 j 01 2 , (21.32) γ02
γ 2  3 N r1 and the eigenvalue for the ground state of the perturbed geometry is   j2 16 1 π2 γ 2  2 C 01 1
. L 3 N r12

(21.33)

Evidently, the energy of the ground state, usually of significant importance in applications (such as transport and optical applications of quantum dots), is modified by a term proportional to 1/N , being a measure of the change in radius due to, for example, imprecise geometry control in the quantum-dot growth process. The sinusoidal radius modulation with axial distance is responsible for the prefactor of 16/3 appearing in last term in Eq. (21.33), that is, other functional variations around the radius r1 with z will generally lead to the same 1/N dependence with a modified prefactor. Example 2 Stark Effect in Perturbed Cylindrical Nanowires The Schrödinger equation for an electron confined to a cylindrical nanowire subject to a DC electric field F along the nanowire axis is   „2 2
r C e Fz ψ0 D E0 ψ0 , (21.34) 2m where
e is the electron charge. This is the unperturbed problem. We wish to determine the influence of a sinusoidally varying perturbation of the radius as a function of z on the nondegenerate ground-state energy, that is, we assume δ D r1 /N sin(M π/L)z. Equation (21.34) can be separated into equations   d 1 d r f C c1 f D 0 , (21.35) r dr dr

289

290

21 Boundary Perturbation Theory

2m d2 g C 2 (E0
e Fz) g
c 1 g D 0 , dz 2 „

(21.36)

where ψ0 D f (r)g(z) assuming as usual the ground state to be cylindrically symmetric, and c 1 is a separation constant to be determined. Equation (21.35) is the Bessel equation with the first solution to the hard-wall unperturbed problem:   j 01 f (r) D J0 r , (21.37) r1 2 where r1 is the radius of the unperturbed cylindrical nanowire and c 1 D j 01 /r12 . The second equation can be transformed into the Airy equation,

d2 g
ηg D 0 , d η2

(21.38)

by performing the coordinate transformation zD

1 eF




  1/3 „2 „2 c 1 C E0 C η. 2m 2m e F

(21.39)

The solution to Eq. (21.38) is g(η) D AAi(η) C BBi(η) ,

(21.40)

where Ai and Bi are the Airy functions and A and B are coefficients. The possible values of η are determined by requiring that g(z D 0) D g(z D L) D 0 ,

(21.41)

where z D 0 and z D L are the ends of the cylinder along the axial direction. By solving numerically the secular equation, expressed as a 22 matrix problem in the coefficients A and B based on Eqs. (21.40) and (21.41), one finds the ground state ψ0 and energy E0 . For the case with parameters m D 0.067m 0 (GaAs), L D 10 nm, F D 5  105 V/m, r1 D 5 nm, N D 10, and M D 1, where m 0 is the free-electron mass and j 01 D 2.40, the ground-state energy is found to be E0 D 189.6 meV. The ground-state dependence with z [g(z)] is shown in Figure 21.2. With the determination of the unperturbed ground state and its energy, we can easily numerically evaluate the ground-state energy shift due to the perturbed geometry: RL „ E
E0 D
2m 2

2 2 j 01 0 N r12

sin


Mπ  z jg(z)j2 dz L RL


22.2 meV .

(21.42)

jg(z)j2 dz

0

As expected, we observe that the radius modulation with axial distance proportional to 1/N , also in this case with a DC electric field applied, leads to a change in

21.3 Applications 0 −0.05 −0.1

g(z)

−0.15 −0.2 −0.25 −0.3 −0.35 −0.4 −0.45 0

1

2

3

4

5 6 z [nm]

7

8

9

10

Figure 21.2 Ground-state dependence g(z) with the axial coordinate z for the Stark problem considered in Example 2.

the ground-state energy proportional to 1/N . Moreover, if the sinusoidal modulation has a large (small) overlap with the ground-state function, then the associated energy change becomes correspondingly large (small). The case of a DC electric field (or sufficiently slowly varying AC voltages) applied to quantum-dot structures as described in this subsection is important in relation to quantum-dot device transport applications.

Example 3 Splitting of Degenerate States by Perturbatively Deforming a Quadratic Region into A Rectangular Region Let us finally determine the splitting of degenerate states, due to deforming a quadratic domain into a rectangular domain (Figure 21.3), for a particle obeying y

y

Dirichlet

L Dirichlet 0

(a)

Dirichlet

L

Dirichlet Dirichlet

0

Dirichlet

L

0

x

Dirichlet

0

Dirichlet

L+δ

x

(b)

Figure 21.3 The square region (unperturbed geometry) (a) and the rectangular geometry (perturbed geometry) (b). Associated boundary conditions are indicated.

291

292

21 Boundary Perturbation Theory

the Schrödinger equation. In this case, we have (two dimensions)


„2 2m



@2 @2 C 2 2 @x @y

 ψ 0 D E0 ψ 0 ,

(21.43)

and considering the unperturbed quadratic region to have a side length L, we find the eigenstates for the hard-wall boundary problem,  mπ   nπ  x sin y , L L

„2  nπ 2  m π 2 , E0 D C 2m L L ψ0 (x, y ) D sin

(21.44) (21.45)

where n and m can take any of the integer values: 1, 2, 3, . . . . It is immediately apparent that the ground state with n D m D 1 is nondegenerate, whereas the firstexcited state is degenerate, corresponding to the two possible values: n D 2, m D 1 and n D 1, m D 2. Consider now a perturbation of the x D L boundary to x D L C δ. The exact solution to this rectangular problem is simply    mπ  nπ ψ(x, y ) D sin x sin y , (21.46) LCδ L # "  „2 nπ 2  m π 2 ED . (21.47) C 2m LCδ L Hence, the energies of the second and third eigenstates of the rectangular problem are  2  2 5π 2 8π 2 δ 2π π E(n D 2, m D 1) D C  2
, LCδ L L L3  2  2 5π 2 2π 2 δ 1π 2π E(n D 1, m D 2) D C  2
, (21.48) LCδ L L L3 where the last two approximate results hold to first order in the perturbation δ. As expected and required, these two states are not degenerate. In other words, the degeneracy is split by the deformation of the quadratic domain into a rectangular domain. Let us now employ perturbation theory. Since we wish to evaluate the shifts in energies of degenerate unperturbed eigenstates due to the deformation, we must apply degenerate theory as described in the previous section. For the coefficients N i and ∆ i j we find easily (in the following, subscripts 1 and 2 refer to the eigenstates

21.4 Problems

with n D 2, m D 1 and n D 1, m D 2, respectively) 

Z L ZL N1 D

sin2 0

 π  L2 2π x sin2 y dxdy D , L L 4

0

L2 N2 D , 4  2 Z L  2   2π 2π 2 π δ sin y dy D δ, ∆ 11 D L L L 0  2 π δ, ∆ 22 D 2L ∆ 12 D ∆ 21 D 0 ,

(21.49)

and the matrix in Eq. (21.15) becomes diagonal. Evaluation of the corresponding secular equation yields 5π 2 8π 2 δ
, L2 L3 5π 2 2π 2 δ E(n D 1, m D 2)  2
, L L3

E(n D 2, m D 1) 

(21.50)

which agrees to first order with the exact result in Eq. (21.48). We point out that we could have chosen as unperturbed eigenstates any linear combination of the two states corresponding to n D 2, m D 1 and n D 1, m D 2 in Eq. (21.44), respectively, and repeated the calculation above. In this case, a nondiagonal matrix is obtained but the eigenvalues given in Eq. (21.50) still apply and the eigenstates to zero order in the perturbation δ would be exactly those in Eq. (21.44) with n D 2, m D 1 and n D 1, m D 2, respectively.

21.4 Problems

1. Consider a circular region of radius R. a. Solve the Helmholtz equation in two dimensions in terms of eigen(wave)vectors. R cos(c θ ), b. Next, assume the circular rim is distorted such that δ(θ ) D 10 where c is a constant. Compute the change in eigen(wave)vectors using geometry perturbation theory. Compare the result with the exact result when c D 0.

293

294

21 Boundary Perturbation Theory

2. a. Solve the Stark effect eigenvalue problem for a cylindrical nanowire where the rim perturbation is given by 2
R z
L2 δD , L2 10

(21.51)

4

and R and L are the unperturbed nanowire radius and length, respectively. b. Assume instead 2 !
z
L2 R . (21.52) 1
δD L2 10 4

Compute the eigenvalues.

295

Appendix A Hypergeometric Functions A.1 Introduction

There exists a general formalism for dealing with functions which are solutions to arbitrary second-order linear ordinary differential equations. These are based upon the solution of the hypergeometric equation. These so-called hypergeometric functions are generalized versions of functions such as the Legendre and Bessel functions. The reason is because nearly all special functions of mathematical physics arise from second-order linear differential equations having three or fewer regular singularities; the hypergeometric functions are generalized solutions of such equations. After introducing the hypergeometric functions, we discuss briefly a few other functions that are related to them and that are used in this book.

A.2 Hypergeometric Equation

The hypergeometric (or Gauss) equation is x (1
x)

d2 y (x) dy (x)
ab y (x) D 0 . C [c
(a C b C 1)] 2 dx dx

(A1)

This is the canonical form of a second-order linear differential equation with regular singularities at x D 0, 1, 1. The constants a, b, c can be complex. All differential equations with at most three singularities can be transformed into the hypergeometric equation, hence its importance.

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

296

Appendix A Hypergeometric Functions

A.3 Hypergeometric Functions

One can obtain series solutions about the origin by using the Frobenius method. Writing 1 X

y (x) D

a n x nCσ ,

(A2)

nD0

and substituting into Eq. (A1), the indicial equation gives σ D 0 or 1
c, and the recurrence relation is a nC1 D

(n C σ)(n C σ C a C b) C ab an . (n C σ C 1)(n C σ C c)

(A3)

A.3.1 First Solution

One solution (for σ D 0) is ab x a(a C 1)b(b C 1) x 2 C C    , 8c ¤ 0,
1,
2, . . . c 1! c(c C 1) 2! 1 X a(a C 1)    (a C n
1)b(b C 1)    (b C n
1) n D1C x n!c(c C 1)    (c C n
1) nD1

y (x) D 1 C

 2 F1 (a, bI cI x)  F(a, bI cI x) .

(A4) (A5)

This is the hypergeometric function. If one introduces the Pochhammer symbol, whereby (a) n  a(a C 1)(a C 2)    (a C n
1) D

(a C n
1)! , (a
1)!

(a)0 D 1 ,

(A6) (A7)

then the hypergeometric function can be written as 2 F1 (a,

b, cI x) D

1 X (a) n (b) n x n . (c) n n! nD0

(A8)

 If c 2 Z
, writing c D
n, y (x) D x nC1

1 X (a C n C 1) m (b C n C 1) m x m (n C 2) m m! mD0

D z nC1 2 F1 (a C n C 1, b C n C 1I n C 2I x) .

(A9)

A.3 Hypergeometric Functions

 If c D
n and a D
m or b D
m, where n, m 2 N , 8P n (
m) l (b) l x l ˆ D 2 F1 (
m, bI
nI x) , ˆ < (
n) l l! lD0 y (x) D n P ˆ (a) l (
m) l x l ˆ : D 2 F1 (a,
mI
nI x) . (
n) l l!

(A10)

lD0

Thus, if a or b 2 Z
, the hypergeometric function becomes a polynomial. The series Eq. (A4) if a or b 2 Z
or Eq. (A9) if c 2 Z
, converges for all jxj < 1 and also either for x D 18c > a C b, or for x D
18c > a C b
1. We will write the hypergeometric function simply as F from now on. A.3.1.1

Examples

ln(1 C x) D x F(1, 1I 2I
x) ,

(A11)

(1
x) s D F(
s, bI bI x) ,
 a a 1 2 cos ax D F . ,
I I (sin x) 2 2 2

(A12) (A13)

A.3.1.2 Properties A few properties of the hypergeometric function are: [33]

F(a, b, cI x) D F(b, a, cI x) ,

(A14)

(a) n (b) n dn F(a, b, cI x) D F(a C n, b C n, c C nI x) , dx n (c) n Γ (c)Γ (c
a
b) , F(a, b, cI 1) D Γ (c
a)Γ (c
b) 8c ¤ 0,
1,
2, . . . , Rec > Re(a C b) .

(A15)

(A16)

A.3.2 Second Solution

The second solution (for σ D 1
c) has the recurrence relation a nC1 D

[n C (2
c)
1][n C (2
c)
1 C a C b] C ab an . (n C 1)(n C (2
c))

(A17)

With the transformations c0 D 2
c ,

a0 D 1 C a
c ,

b0 D 1 C b
c ,

one gets a nC1 D

(n C a 0 ) (n C b 0 ) an . (n C 1) (n C c 0 )

(A18)

297

298

Appendix A Hypergeometric Functions

This expression is similar to the one for the first solution, which allows one to write the second solution down immediately: y (x) D x 1
c 2 F1 (a C 1
c, b C 1
c, 2
cI x) ,

c ¤ 2, 3, 4, . . .

(A19)

A.4 Confluent Hypergeometric Equation

The confluent hypergeometric (or Kummer) equation is x

d2 y (x) dy (x) C (c
x)
a y (x) D 0 . 2 dx dx

(A20)

This is a second-order linear differential equation with a regular singularity at x D 0 and an irregular singular point at x D 1.

A.5 Confluent Hypergeometric Functions A.5.1 First Solution

One solution is y (x) D 1 F1 (aI cI x) D M(a, c, x) D1C D

a(a C 1) x 2 a x C C    , c ¤ 0,
1,
2, . . . . c 1! c(c C 1) 2!

1 X (a) n x n . (c) n n! nD0

(A21) (A22) (A23)

This is the confluent hypergeometric function. In the Bateman manuscript [33], the Humbert symbol Φ (a, cI x) is also used. A.5.1.1 Examples


2 1 3 2 , ,
x 2 , dt e
t D p x M 2 2 π 0 
 z ν e
i z 1 J ν (z) D 1 F1 ν C I 2ν C 1I 2i z 2 Γ (ν C 1) 2

2 erf(x) D p π

Zx

The confluent hypergeometric function converges for all x.

(A24)

(A25)

A.6 Whittaker Functions

A.5.2 Second Solution

y (x) D x 1
c M(a C 1
c, b C 2
cI x) ,

c ¤ 2, 3, 4, . . . .

(A26)

A.5.3 Properties

M(a, c, x) D e x M(c
a, c,
x) (Kummer transformation) ,

(A27)

M(a, a, x) D e ,

(A28)

x

n

(a) n d M(a, c, x) D M(a C n, c C n, x) . n dx (c) n

(A29)

A.6 Whittaker Functions A.6.1 Whittaker Equation

The Whittaker equation is given as [51] # " 1
µ2
1 d2 M 4 M D0, C
C C dx 2 4 x x2

(A30)

It can be obtained from the confluent hypergeometric equation by the transformation y D x
c/2 e x/2 M ,

aD

1

Cµ , 2

c D 1 C 2µ .

A.6.2 Whittaker Functions

The solutions to Eq. (A30) are known as the Whittaker functions [51]: 
1 C µ

, 1 C 2µ, x . M
,µ (x) D e
x/2 x µC1/2 M 2 There is also a second solution that diverges logarithmically near the origin.

(A31)

299

300

Appendix A Hypergeometric Functions

A.7 Associated Laguerre Functions A.7.1 Associated Laguerre Equation

The associated Laguerre differential equation is given by x y 00 C (α C 1
x)y 0 C n y D 0 .

(A32)

A.7.2 Associated Laguerre Function

The solution to Eq. (A32) is known as the associated Laguerre function and is denoted by L αn (x). A.7.3 Laguerre Equation

The Laguerre differential equation is given by x y 00 C (1
x)y 0 C n y D 0 . A.7.3.1 Alternative Representation e x dn L n (x) D (x n e
x ) . n! dx n The Schaefli integral representation is I ex z n e
z . L n (x) D dz 2π i (z
x) nC1

(A33)

(A34)

(A35)

A.7.4 Generalized Laguerre Polynomials

A few properties are provided:  n
X n C α (
x) m α L n (x) D , n
m m! mD0
 (α C 1) n nCα L αn (0) D , D n n!
 nCα L αn (x) D M(
n, α C 1, x) , n L α0 (x) D 1 ,

L α1 (x) D α C 1
x ,

L αn (x) D L αC1 (x)
L αC1 n n
1 (x) ,

(A36) (A37) (A38) (A39) (A40)

A.8 Hermite Polynomial

x L αn (x) D
(n C α)L αn
1(x) C (2n C α C 1)L αn (x)
(n C 1)L αnC1(x) ,

(A41)

x L αn (x) D (n C α)L α
1 (x)
(n C 1)L α
1 n nC1 (x) , d α L (x) D nL αn (x)
(n C α)L αn
1(x) , dx n d α L (x) D
L αC1 n
1 (x) . dx n

x

(A42) (A43) (A44)

A.8 Hermite Polynomial A.8.1 Hermite Equation

The Hermite differential equation is H n00
2x H n0 C 2nH n D 0 .

(A45)

The solutions are known as Hermite polynomials. If one sets Hn D φ n e x

2 /2

,

(A46)

then Eq. (A45) becomes φ 00n (x) C (2n C 1
x 2 )φ n (x) D 0 .

(A47)

which is the parabolic cylinder equation [17] (see Chapter 10). A.8.2 Hermite Polynomials

The generating function for the Hermite polynomials is g(x, t) D e
t

2 C2t x



1 X

H n (x)

nD0

tn . n!

(A48)

Expanding the generating function, e
t

2 C2t x

  1 2 2 1  2 3
t C 2t x C
t C 2t x C    D 1 C
t 2 C 2t x C 2 6     D 1 C (2x)t C
1 C 2x 2 t 2 C
2x C x 3 t 3 C    

1 X nD0

H n (x)

tn H2 2 H3 3 D H0 C H1 t C t C t C  n! 2 6

301

302

Appendix A Hypergeometric Functions Table A.1 A few of the Hermite polynomials. H0 (x )

=

1

H1 (x ) H2 (x )

= =

2x 4x 2
2

H3 (x )

=

8x 3
12x

gives the Hermite polynomials (Table A.1). In general, one can write X (
1) m (2x) n
2m , m!(n
2m)! mD0 [n/2]

H n (x) D n!

(A49)

where [n/2] is n/2 if n is even or (n
1)/2 if n is odd. A.8.3 Properties

H n (
x) D (
1) n H n (x) ,

(A50)

H nC1 (x) D 2x H n (x)
2nH n
1 (x) ,

(A51)

H n0 (x) D 2nH n
1 (x) ,

(A52)

H2n (x) D (
1) n 22n n!L
1/2 (x 2 ) , n

(A53)

2 H2nC1 (x) D (
1) n 22nC1 n!x L1/2 n (x ) ,

(A54)

H n (x) D 2 n/2 e x

2 /2

D n (21/2 x) , 
1 (
1) n (2n)! M
n, , x 2 , H2n (x) D n! 2
 3 (
1) n (2n C 1)! 2x M
n, , x 2 , H2nC1 (x) D n! 2

(A55)

H2n (0) D (
1) n (2n)!/n!, H2nC1 (0) D 0.

(A58)

(A56) (A57)

The L αn (x) are Laguerre polynomials and the D ν (x) are Weber functions.

A.9 Airy Functions

Solutions to the one-particle Schrödinger equation for an electron moving in a potential due to a constant electric field can be written in terms of Airy functions. Since this is an important problem (refer to Eq. (21.38)) we give some properties of the Airy functions.

A.9 Airy Functions

A.9.1 Airy Equation

The Airy equation is [51] d2 y (z) C z y (z) D 0 , dz 2

(A59)

and two independent solutions are the Ai and B i functions, which can be written as two independent combinations of infinite series, Ai(z) D 3
2/3 B i(z) D 3
1/6

1 X

1 X z 3n z 3nC1
4/3   ,  
3 9 n n!Γ n C 23 9 n n!Γ n C 43 nD0 nD0 1 X

1 X z 3n z 3nC1
5/6   .   C 3 2 n n 9 n!Γ n C 3 9 n!Γ n C 43 nD0 nD0

(A60) (A61)

A.9.2 Properties

We now list some of the properties of the Airy functions: 3
2/3 B i(0) , Ai(0) D p D Γ ( 23 ) 3 3
1/3 B i 0(0) Ai 0(0) D
p D
1 , Γ (3) 3 Ai(z) C ωAi(ωz) C ω 2 Ai(ω 2 z) D 0 , B i(z)
i ωAi(ωz) C i ω 2 Ai(ω 2 z) D 0 , where ω D exp(
2i π/3). Relations to the modified Bessel functions are sometimes useful,
3/2  p 2z , Ai(z) D π
1 z/3K1/3 3
3/2 
3/2   p 2z 2z B i(z) D z/3 I
1/3 C I1/3 , 3 3 as are the integral representations, Ai(z) D

B i(z) D

1 π 1 π

 Z1
1 3 t C x t dt , cos 3 0

Z1


exp

0



t 3 1 3 C x t C sin t C x t dt . 3 3

(A62)

(A63)

303

305

Appendix B Baer Functions B.1 Introduction

The Baer (wave) functions are solutions to the ordinary differential equations arising in the separation of the Laplace (Helmholtz) equation in paraboloidal coordinates. There does not appear to be any reference to these two functions apart from the brief discussion in the book by Moon and Spencer [6].

B.2 Baer Equation

The Baer equation is defined to be [6] d2 B 1 C 2 dz 2




1 1 C z
b z
c




 dB p (p C 1)z C q(b C c)
B D0. dz (z
b)(z
c)

(B1)

It is of Bôcher type f1 1 3g. There are regular singular points at b and c, and an irregular singular point at infinity.

B.3 Baer Functions q

q

The Baer functions B p (z) and C p (z) are defined to be the series solutions about the origin (and up to the nearest singularity at z D b) of the Baer equation such that B pq (0) D 0 ,

C pq (0) D 1 ,

(B2)

and one writes the general solution to the Baer equation as ψ(z) D AB pq (z) C B C pq (z) .

(B3)

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

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Appendix B Baer Functions

B.4 Baer Wave Equation

Moon and Spencer give the Baer wave equation as follows: 1 d2 B C dz 2 2




1 1 C z
b z
c




2 2  dB k z
p (p C 1)z
q(b C c) C B D0. dz (z
b)(z
c) (B4)

It is of Bôcher type f1 1 4g. There are regular singular points at b and c, and an irregular singular point at infinity. The Baer wave equation arises in the separation of the Helmholtz equation in paraboloidal coordinates. There is very little work on this differential equation and the resulting solutions. Hence, we follow Moon and Spencer [6].

B.5 Baer Wave Functions q

q

The Baer wave functions B p (k, z) and C p (k, z) are defined to be the series solutions about the origin (and up to the nearest singularity at z D b) of the Baer wave equation such that B qp (k, 0) D 0 ,

C pq (k, 0) D 1 ,

(B5)

and one writes the general solution to the Baer wave equation as B(z) D AB qp (k, z) C B C pq (k, z) .

(B6)

The two series solutions are B qp (k, z) D z

1 X (
1) m ∆ m (1)z m , m!(m C 1)! mD0

C pq (k, z) D 1 C

1 X (
1) m ∆ 0m (0)z mC1 , m!(m C 1)! mD1

(B7) (B8)

where ∆ m (1) and ∆ 0m (0) are defined in Table B.1. B.5.1 Orthogonality

Since the Baer wave equation has two separation constants, two orthogonality relations can be written down depending upon which of the two is treated as the eigenvalue (the choice depends upon the boundary conditions [6]). A general orthogonality can be written down for both cases: 1 Nm

Zb dz w (z)B qp m (k, z)B qp n (k, z) D δ m n , a

(B9)

B.5 Baer Wave Functions Table B.1 Series solution parameters for the Baer wave functions.

∆ m (1)

=

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

A1 A 2 C B2 A 3 C B3




2
1 2A 1 2A 2 C B2

A m C Bm 2
1 B2 B3 2A 1 B4 2A 2 C B2 ...

∆ 0m (0)

=

A0

=

B mC1 0,

A1

=



A2 Am

= =

B0

=

B1 D 0 ,

B2

=



B3

=

B4

=

Bm

=

(bCc) 2bc , (b 2 Cc 2 )  2b2 c 2 ,  1 bc A m
1 (b

0 3
2 3A 1

0 3
2 3A 1

0 0 4
3

0 0 4
3












0 0 0












ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ m(m  1) ˇˇ ˇ mA 1 ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0 ˇ, ˇ ˇ m(m  1) ˇˇ ˇ mA 1 0 0 0

 C c)  A m
2 for m > 2 ,

q(bCc) , bc h i (bCc)2 1  bc p (p C 1) C q bc , h h (bCc) (bCc) (bCc)2 1 2 1  bc bc k  p (p C 1) bc C q bc   1 bc B m
1 (b C c)  B m
2 for m > 4 .

ii ,

Table B.2 Parameters for orthogonality relations for Baer wave functions. λ

f (λ)

u

v

p

λ(λ C 1)

[(z  b)(z  c)]1/2

1 2 2 u [k z

 q(b C c)]

q

λ

[(z  b)(z  c)]1/2

1 2 2 u [k z

 p (p C 1)z]

w (z) z u (bCc) u

where the appropriate quantities are defined in Table B.2. One can now define a Fourier–Baer series expansion, f (z) D

1 X

A m B qp m (k, z) ,

(B10)

mD0

with the coefficients given by Am

1 D Nm Zb

Nm D a

Zb dz w (z)B qp m (k, z) f (z) ,

(B11)

a

h i2 dz w (z) B qp m (k, z) .

(B12)

307

309

Appendix C Bessel Functions C.1 Introduction

Bessel functions are among the most widely occurring higher transcendental functions. They are solutions to the ordinary differential equations arising in the separation of the Helmholtz equation in two dimensions using circular (polar) coordinates, and of the Laplace and Helmholtz equations using cylindrical and rotational coordinate systems. The three main types of Bessel functions to be considered are the Bessel functions, the spherical Bessel functions, and the Bessel wave functions.

C.2 Bessel Equations

The Bessel equation of order ν is dy d2 y C (x 2
ν 2 )y D 0 . Cx (C1) 2 dx dx It is easily seen to have a regular singular point at zero and an irregular singular point at infinity, and is of Bôcher type f2 4g. We also consider here the modified Bessel equation: x2

 dy
2 d2 y (C2)
x C ν2 y D 0 . Cx 2 dx dx In the above equation, both the argument x and the order ν can be complex. The modified Bessel equation can be converted into the Bessel equation by the transformation x ! i x. The above two Bessel equations arise in cylindrical problems. For problems in spherical polar coordinates, we come across the spherical Bessel equation: x2

  dy d2 y C x 2
n(n C 1) y D 0 . C 2x (C3) 2 dx dx Again, this equation can be converted into the Bessel equation (of order (n C 1/2)) p by writing y D J/ x. x2

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

310

Appendix C Bessel Functions

One also gets the modified spherical Bessel equation: x2

 dy  2 d2 y
x C n(n C 1) y D 0 . C 2x dx 2 dx

(C4)

It can be converted into the modified Bessel equation (of order (n C1/2)) by writing p y D J/ x.

C.3 Bessel Functions

There are a number of Bessel functions. C.3.1 ν Nonintegral

One can define two linearly independent solutions to the Bessel equation:  z 2mCν (
1) m , m!Γ (m C ν C 1) 2 mD0 1  z 2m
ν X (
1) m J
ν (z) D , m!Γ (m
ν C 1) 2 mD0

J ν (z) D

1 X

(C5) (C6)

where the gamma function can be defined by the Euler integral of the second kind: Z1 Γ (z) D dt t z
1 e
t ,

Re(z) > 0 .

(C7)

0

The above solutions are known as Bessel functions of the first kind and order ν and
ν. They are analytic in the complex plane except for a cut along the negative real axis. C.3.2 ν Integral

Writing ν D n, where n 2 N , J n and J
n are no longer independent since one can easily show that [81] J
n D (
1) n J n .

(C8)

A conventional second solution is the Bessel function of the second kind and of integral order, defined as N n (z)  lim

ν!n

cos νπ J ν (z)
J
ν (z) . sin νπ

(C9)

C.3 Bessel Functions 1.2 1.0

J0 0.8 0.6

J1

Jn(x)

0.4

J2

0.2 0.0

J3 -0.2 -0.4 -0.6 0

2

4

6

8

10

x Figure C.1 Bessel functions of integral order and real argument.

This is also known as the Neumann function. We have N
n D (
1) n N n .

(C10)

Infinite series can be written down for the two Bessel functions of integral order [13]:  z 2mCn (
1) m , (C11) m!(m C n)! 2 mD0 " 1 ! mCn m  z 2mCn X 1 X z 1 X (
1) m 1 2 ln C 2γ

N n (z) D π mD0 m!(m C n)! 2 2 l l lD1 lD1 # n
1 X (n
m
1)!  z 2m
n , (C12)
m! 2 mD0

J n (z) D

1 X

where γ D 0.5772 is the Euler constant. J n is analytic over the whole complex plane, and N n is analytic over the whole complex plane excluding the negative real axis. A few Bessel functions are sketched in Figure C.1. C.3.3 Properties

Some useful properties are now given:

311

312

Appendix C Bessel Functions

 Complex conjugate J ν (z) D J ν (z) ,

(C13)

N ν (z) D N ν (z) .

(C14)

 For x, ν  0, J ν (x) is real and bounded [17]. The same holds for N ν (x) if x D 0 is excluded.  For z ! 0 [13], J0 (z)  1 ,

(C15)

(z/2) ν , (ν ¤
1,
2, . . .) , Γ (ν C 1) 2 N0 (z)  ln z , π Γ (ν)  z 
ν N ν (z) 
, Re(ν) > 0 . π 2 J ν (z) 

(C16) (C17) (C18)

 For jzj ! 1 [13], r

 νπ π 2 cos z

, (jargzj < π) , πz 2 4 r  νπ π 2 N ν (z)  sin z

, (jargzj < π) . πz 2 4 J ν (z) 

(C19) (C20)

 For integral order [81], j J0 (z)j  1 ,

1 j J n (z)j  p , 2

n 2 ZC ,

(C21)

J
n D (
1) n J n ,

(C22)

N
n D (
1) n N n .

(C23)

 For nonnegative integral order or for arbitrary nonintegral order [17], d dz d dz 2ν z



 z ν J ν (z) D z ν J ν
1 (z) ,

(C24)



 z
ν J ν (z) D
z
ν J νC1 (z) ,

(C25)

J ν (z) D J ν
1 (z) C J νC1(z) ,

(C26)

2 J ν0 (z) D J ν
1 (z)
J νC1 (z) ,

(C27)

J00 (z) D
J1 .

(C28)

Equations (C26) and (C27) are also satisfied by the Neumann functions. They show that all the functions of integral order can be computed starting from the n D 0 and n D 1 ones.

C.3 Bessel Functions Table C.1 A few zeros of Bessel functions J n of integral order and real argument. n

1st zero

2nd zero

3rd zero

0 1

2.405 0

5.520 3.832

8.654 7.016

2

0

5.136

8.417

 Wronskian [33] 2 W [ J ν , J
ν ] D
sin νπ , π 2 z . W [ Jν , Nν ] D πz  Zeros [17]

(C29) (C30)

 J n (z) has an infinite number of real zeros and no complex ones (Table C.1).  J ν (z) has an infinite number of positive real zeros and 2N(ν) conjugate complex zeros, where N(ν) D 0 if ν >
1 or ν D
1,
2, . . . , N(ν) D m if
(m C 1) < ν <
m ,

m D 1, 2, . . . .

C.3.4 Hankel Functions

One also defines Bessel functions of the third kind, or Hankel functions of the first and second kinds: (1)

C.3.4.1

H ν (z) D J ν (z) C i Yν (z) ,

(C31)

(2) H ν (z)

(C32)

D J ν (z)
i Yν (z) .

Properties

H
ν (z) D e i ν π H ν (z) ,

(C33)

H
ν (z) D e
i ν π H ν (z) ,

(C34)

(1)

(1)

(2)

(1)

(2)

(1)

H ν (z) D H ν (z) , i h 2i (1,2) D˙ , W Jν , Hν πz i h 4i (1) (2) W Hν , Hν D
. πz

(C35) (C36) (C37)

313

314

Appendix C Bessel Functions

C.4 Modified Bessel Functions

These functions are solutions to the modified Bessel equation, Eq. (C2). For a domain in the complex plane with a cut on the negative real axis, the solutions are  z 2mCν 1 , Γ (m C 1)Γ (m C ν C 1) 2 mD0 π I
ν (z)
I ν (z) K ν (z) D , ν ¤ 0, ˙1, ˙2, . . . , 2 sin νπ I ν (z) D

1 X

K n (z) D lim K ν (z) .

(C38) (C39) (C40)

ν!n

They are known as the modified Bessel function of the first kind and the modified Bessel function of the third kind or MacDonald function (or Basset function), respectively. C.4.1 Properties

Some useful properties are now given:  For x > 0, ν  0, I ν (x) (K ν (x)) is positive and increases (decreases) monotonically [17].  For x ! 0 [17], I ν (x) 

xν 2 ν Γ (ν

K0 (x)  ln K ν (x) 

C 1)

2 , x

2 ν
1 Γ (ν) . xν

,

(C41) (C42) (C43)

 For x ! 1 [17], ex I ν (x)  p , 2π x r π
x K ν (x)  e . 2x

(C44) (C45)

 For integral order [17], I
n D (
1) n I n ,

(C46)

I00 (z) D I1 (z) ,

(C47)

K00 (z) D
K1 (z) .

(C48)

C.5 Spherical Bessel Functions

 For nonintegral order [17, 33], I ν (z) D e
i ν π/2 J ν (z e i π/2) ,

(C49)

2ν I ν (z) D I ν
1 (z)
I νC1 (z) , z

(C50)

2I ν0 (z) D I ν
1 (z) C I νC1(z) ,

(C51)

K
ν (z) D K ν (z) ,

(C52)

K ν (z) D

i π i ν π/2 (1) i π/2 iπ (2) e H ν (z e ) D
e
i ν π/2 H ν (z e
i π/2) , 2 2

(C53)

2ν K ν (z) D K νC1 (z)
K ν
1 (z) , z

(C54)

2K ν0 (z) D
K ν
1 (z)
K νC1 (z) ,

(C55)

2 sin νπ , πz 1 W [I ν , K ν ] D
. z W [I ν , I
ν ] D


(C56) (C57)

 Zeros [17]  I ν (z) only has imaginary zeros if ν >
1.  For real ν, K ν (z) has no zeros for jargzj < π/2 and a finite number of zeros in the rest of the domain.  K ν (z) is real when ν is real and z is positive.

C.5 Spherical Bessel Functions

It has already been mentioned that the spherical Bessel functions can be related to the ordinary Bessel functions. The exact definitions are [13] p π j n (z) D J nC1/2 (z) , (C58) 2z p π YnC1/2 (z) , y n (z) D (C59) 2z and j n and y n are known as the spherical Bessel functions of the first and second kinds of order n, respectively. One can also define spherical Hankel functions: p π (1) (1) h n (z) D j n (z) C i y n (z) D H (z) , (C60) 2z nC1/2

315

316

Appendix C Bessel Functions

p (2)

h n (z) D j n (z)
i y n (z) D

π (2) H (z) . 2z nC1/2

(C61)

C.5.1 Properties

A few useful properties are now given [13]: sin z , z sin z cos z , j 1 (z) D
z2 z cos z y 0 (z) D
, z cos z sin z y 1 (z) D
2
, z z   1 d n sin z j n (z) D z n
, z dz z  n cos z 1 d y n (z) D
z n
. z dz z j 0 (z) D

(C62) (C63) (C64) (C65) (C66) (C67)

C.6 Modified Spherical Bessel Functions

One can define two linearly independent modified spherical Bessel functions as follows: [13] p π I nC1/2 (z) , (C68) i n (z) D 2z p π I
(nC1/2) (z) . (C69) i
(nC1) (z) D 2z i n and i
(nC1) are known as the modified spherical Bessel functions of the first kind of order n and
(n C 1), respectively. One defines modified spherical Bessel functions of the second kind as follows: p π K nC1/2 (z) . (C70) k n (z) D 2z C.7 Bessel Wave Functions

The Bessel wave equation is   1 dZ p2 d2 Z 2 2 2 Z D0. C q C C k z
dz 2 z dz z2

(C71)

C.7 Bessel Wave Functions

For p nonintegral, the general solution can be written as Z(z) D AJ p (k, q, z) C B J
p (k, q, z) ,

(C72)

where J p (k, q, z) is known as the Bessel wave function. If q 2 !
q 2 , then the solution can be written as Z(z) D AJ p (k, q, i z) C B J
p (k, q, i z) .

(C73)

These are the Bessel wave functions with nonintegral order. For integral p, J p (k, q, z) and J
p (k, q, z) are no longer independent (just as for the Bessel functions) and, hence, one defines the Bessel wave function of the second kind, N p (k, q, z), analogously to the Bessel function of the second kind N n (k, z). C.7.1 Series Solution

We now give a series solution to Eq. (C71) for integral p. The only singularity of Eq. (C71) in the finite complex plane is a regular singular point at z D 0. Expanding the solution about z D 0, one writes M(z) D

1 X

a m z mCσ .

(C74)

mD0

If the differential equation is x 2 y 00 C x p (x)y 0 C q(x)y D 0 , with p (x) D

1 X

q(x) D

pm x m ,

mD0

1 X

qm x m ,

mD0

the indicial equation is Θ (σ)  σ(σ
1) C p 0 σ C q 0 D 0 , and the recurrence relation is n P

an D


  a n
m (σ C n
m)p m C q m

mD1

Θ (σ C n)

.

In our case, we have σ D ˙p , an D


q 2 a n
2 C k 2 a n
4 . (p C n)2
p 2

(C75) (C76)

317

318

Appendix C Bessel Functions

Here, p is an integer; hence, the indicial equation only leads to one independent solution (will choose σ D Cp ). Choosing a 0 D (q/2) p /Γ (p C 1), one obtains the series expansion of the Bessel wave function [6] )

4(p C 1)k 2 C 1
C 1
Z(z) D Γ (p C 1) 1!(p C 1) 2!(p C 1)(p C 2) q4 (


qz p 2


q z 2


q z 4

2

2

 J p (k, q, z) .



(C77)

This series solution is easily amenable to numerical work. Furthermore, note that Eq. (C77) is completely real. C.7.2 Orthogonality

The Bessel wave functions are orthogonal with respect to the weighting factor z: Zb dz J p (
, q m , z) z J p (
, q n , z) D δ m n ,

(C78)

a

with the boundary condition J p (
, q m , z) C h J p0 (
, q m , z) D 0 ,

(C79)

at z D a, b. If a function f (z) is expanded in terms of the Bessel wave functions, 1 X

f (z) D

A m J p (
, q m , z) ,

(C80)

mD0

then Am D

1 Nm

Zb dz J p (
, q m , z) z f (z) ,

(C81)

a

and Zb Nm D a

 2 dz J p (
, q m , z) z .

(C82)

C.7 Bessel Wave Functions

Specifically, 

Zb dz z J m a



Zb dz z J m a

z m n z0 z  0m n z0



 Jm



 Jm

z m ν z0 z  0m ν z0

(

 D  D

for n ¤ ν

0 z 02 2

8 <0 :

2 ( m n ) J mC1

z 02 22m n

for n D ν

,

 
0  2 m n  0 2m n
m 2 J mC1

(C83) for n ¤ ν for n D ν

,

(C84) where J m ( m n ) D 0 ,


 J m0  0m n D 0 .

(C85)

319

321

Appendix D Lamé Functions The Lamé and Lamé wave functions arise in the study of the Laplace and Helmholtz equations in the conical, ellipsoidal, and paraboloidal coordinate systems. There is not a lot of account of these functions. Most of the information in this appendix was gathered from the book by Arscott [73], the Bateman manuscript [33], the book by Morse and Feshbach [5], and the book by Todhunter [104].

D.1 Lamé Equations

The Lamé equation can be written as
 
Ä
α2 ξ 2 ξ dF ξ d2 F C C F D 0 , (D1) C (ξ 2
a 2 ) (ξ 2
b 2 ) dξ (ξ 2
a 2 ) (ξ 2
b 2 ) dξ 2 or, equivalently,  4  2        ξ
a C b 2 C a 2 b 2 F 00 C ξ 2ξ 2
a 2 C b 2 F 0 C Ä
α 2 ξ 2 F D 0 . (D2) The equation has five regular singular points at z D ˙b, ˙a, 1. Alternative forms can be written down. If one makes the substitutions   z D ξ 2 , a 2 ! b , b 2 ! c , Ä ! b 2 C c 2 q , α 2 ! p (p C 1) , one gets d2 F 1 C 2 dz 2




1 1 1 C C z z
b z
c



dF 1 C dz 4

"

#  b 2 C c 2 q
p (p C 1)z F D0. z(z
b)(z
c) (D3)

In the Bateman manuscript [33], the Lamé equation is given in terms of elliptic functions:   d2 F  C h
n(n C 1) ksn(z, k)2 F D 0 , 2 dz where k is a (geometrical) parameter and h and n are separation constants.

(D4)

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

322

Appendix D Lamé Functions

D.2 Lamé Functions D.2.1 First Kind

We now explain how to find series solutions to the Lamé equation that are finite. The discussion follows Morse and Feshbach [5]. There are four ways to develop solutions and, hence, the solutions can be classified into four classes [84]: 8 ˆ K(z) D F(z) , ˆ ˆ ˆ < L(z) D p z 2
a 2 B(z) , p E(z) D ˆ M(z) D z 2
b 2 B(z) , ˆ ˆ p p ˆ : N(z) D z 2
a 2 z 2
b 2 B(z) ,

.

(D5)

where B(z) is a polynomial and F(z) D

X

dn z n .

(D6)

n

D.2.1.1 F(z) We start with K(z); note that z D 0 is an ordinary point and, hence, one expects the existence of series solutions. Equation (D2) becomes

X ˚ nD0

     n(n C 1)
α 2 d n C Ä
a 2 C b 2 (n C 2) d nC2

     Ca 2 b 2 (n C 3)(n C 4)d nC4 C Ä d0 C 2a 2 b 2 d2 C Ä d1 C 6a 2 b 2 d3 D 0 . (D7)

Equating the coefficients of d n to zero gives the recurrence relation       a 2 b 2 n(n
1)d n D (n
2)2 a 2 C b 2
Ä d n
2 C α 2
(n
3)(n
4) d n
4 . (D8) For finite solutions to the interior potential problem, it was shown in Chapter 16 that the Lamé functions have to be polynomials. Hence, we will focus on these polynomial solutions. The recurrence relation is a three-term one which keeps the even and odd power series distinct. For a terminating series, two consecutive coefficients must be zero. We will work through a few examples in detail before writing down the general rules. We will often make use of the explicit forms of the recurrence relation given in Table D.1.

D.2 Lamé Functions Table D.1 First few recurrence relations for Lamé equation. Even: 2a 2 b 2 d2 12a 2 b 2 d4

= =


Ä d0 , [4(a 2 C b 2 )
Ä]d2 C α 2 d0 ,

30a 2 b 2 d6

=

[16(a 2 C b 2 )
Ä]d4 C (α 2
6)d2 ,

56a 2 b 2 d8 90a 2 b 2 d10

= =

[36(a 2 C b 2 )
Ä]d6 C (α 2
20)d4 , [64(a 2 C b 2 )
Ä]d8 C (α 2
42)d6 .

Odd: 6a 2 b 2 d3

=

[(a 2 C b 2 )
Ä]d1 ,

20a 2 b 2 d5

=

[9(a 2 C b 2 )
Ä]d3 C (α 2
2)d1 ,

42a 2 b 2 d7

=

[25(a 2 C b 2 )
Ä]d5 C (α 2
12)d3 .

First, consider whether a solution exists with α 2 D Ä D 0. Let us look for an even solution: d0 ¤ 0, d1 D 0. Then, from Table D.1,

H)

d2 D 0 ,   4 a2 C b2 d4 D d2 D 0 , 12a 2 b 2 d2n D 08n ¤ 0 .

Hence, we have a polynomial solution: E(z) D d0 . We can rewrite this as E00 (z) D 1 .

(D9)

This is the first Lamé function. Next, it is easy to see from Table D.1 that we must have simultaneously α 2 D 0 and Ä D 0 for the series to terminate at d0 . One might wonder whether with only one of the two constants being zero, the series might terminate at a higher order. It turns out not to be the case. For example, let Ä D 0. Then, d2 D 0 , α2 d0 ¤ 0 , d4 D 2a 2b 2  16 a 2 C b 2 d6 D d4 ¤ 0 , 30a 2 b 2 # "  2 96 a 2 C b 2 1 d8 D C (α 2
20) d4 . 56a 2 b 2 5a 2 b 2 Thus, if α 2 ¤ 0, the next highest coefficient that can be zero is d8 if 2  96 a 2 C b 2 α 2 D 20
. 5a 2 b 2

323

324

Appendix D Lamé Functions

Now,   90a 2 b 2 d10 D 64 a 2 C b 2 d8 C (α 2
42) d6 , and for d10 D 0 too, we require 2  96 a 2 C b 2 D0, α 2
42 D
22
5a 2 b 2 which is impossible. Hence, the only even solution with Ä D 0 requires α 2 D 0. It is clear from Table D.1 that a rule for constructing polynomial solutions is to set the factor involving α 2 to be zero and to solve for Ä in the same equation such that the other d coefficient is zero. From Eq. (D8), the factor involving α 2 is zero if    α 2 D n 0
3 n 0
4  m(m C 1) .

(D10)

We will assume this form for α 2 from now on. It can be shown that the index m is the degree of the expression in the coordinate z and, for each m, there are (2m C 1) different solutions to be labeled by p. Thus, the Lamé functions will be denoted by p E m (z). Our earlier finding of α 2 D 0 for a solution corresponds to m D 0. The next higher solution can be expected to arise from m D 1 (this is obvious from Table D.1). Then, d5 D 0 if d3 D 0, which leads to Ä D (a 2 C b 2 ). The solution is E(z) D d1 z . We can rewrite this as E10 (z) D z .

(D11)

Note that both solutions so far are entire functions. Higher polynomial solutions are found by increasing m and solving the corresponding equation for Ä. We give one more illustration of the latter process. For m D 2, we can get an even solution by requiring d4 D 0, n   o   Ä d4 D
4 a 2 C b 2
Ä C 6 d0 D 0 , 2a 2 b 2 giving
ÄD2



 a2 C b2 ˙

q

 (a 2 C b 2 )2
3a 2 b 2

p D.2.1.2 F(z) D z 2
a 2 B(z) We now consider solutions of the form p F(z) D z 2
a 2 B(z) .

.

(D12)

D.2 Lamé Functions

Substituting into Eq. (D2), we find that the function B satisfies the new ordinary differential equation 

      z 4
a 2 C b 2 C a 2 b 2 B 00 C z 4z 2
a 2 C 3b 2 B 0   C Ä
b 2
(m
1)(m C 2)z 2 B D 0 .

(D13)

One could derive the new recurrence relation. Since we are not interested in listing all the solutions, we will take a heuristic approach. We are again looking for polynomial solutions. It is clear that B D 1 is a solution of Eq. (D13) if   Ä
b 2
(m
1)(m C 2)z 2 D 0 . This is satisfied if m D 1 and Ä D b 2 . The Lamé function is then p E11 (z) D z 2
a 2 .

(D14)

Similarly, B D z will be a solution if      z 4z 2
a 2 C 3b 2 C Ä
b 2
(m
1)(m C 2)z 2 z D 0 . This is satisfied if m D 2 and Ä D a 2 C 4b 2 . The Lamé function is then p E22 (z) D z z 2
a 2 .

(D15)

Of course, solutions similar to those above are obtained by writing p F(z) D z 2
b 2 B(z) instead. We will not repeat them. p p D.2.1.3 F(z) D z 2
a 2 z 2
b 2 B(z) We now consider solutions of the form p p F(z) D z 2
a 2 z 2
b 2 B(z) .

(D16)

Then, 0

F D

p




p

"p

a2

z2




b2 B 0

Cz p

z 2
a2

p

z 2
b2

#

B, Cp z 2
b2 z 2
a2 # "p p p p z 2
a2 z 2
b2 00 00 2 2 2 2 F D z
a z
b B C 2z p B0 Cp z 2
b2 z 2
a2 (p p z 2
a2 z 2
b2 C p Cp z 2
b2 z 2
a2 " 1/2  2 1/2 #)  2 z
a2 z
b2 2 2 B. C z p p

(z 2
b 2 )3/2 (z 2
a 2 )3/2 z 2
a2 z 2
b2 z2

325

326

Appendix D Lamé Functions

Substituting into Eq. (D2), we find that the function B satisfies the new ordinary differential equation       4  2 z
a C b 2 C a 2 b 2 B 00 C 3z 2z 2
a 2 C b 2 B 0   (D17) C Ä
a 2
b 2 C [6
m(m C 1)]z 2 B D 0 . Again, let us see if B D 1 is a solution. This would require   Ä
a 2
b 2 C [6
m(m C 1)]z 2 D 0 , that is, Ä D a 2 C b 2 and m D 2. The Lamé function is then p p E42 (z) D z 2
a 2 z 2
b 2 .

(D18)

Now if B D z is a solution, we would require      3z 2z 2
a 2 C b 2 C Ä
a 2
b 2 C [6
m(m C 1)]z 2 z D 0 , that is, Ä D 3(a 2 C b 2 ) and m D 3. The Lamé function is then p p E(z) D z z 2
a 2 z 2
b 2 .

(D19)

D.2.2 Second Kind

The previous solutions have zeros at z D 0, a, or b (except for E00 ) and behave as z m for large z. They are not the only solutions. Remember that a second-order linear differential equation has two independent solutions. If y 1 (x) is a solution of dy d2 y C q(x)y D 0 , C p (x) dx 2 dx then a second solution is given by [5] Z R x0 1 00 00 y 2 (x) D B y 1(x) dx 0 2 0 e
dx p (x ) . y 1 (x ) Morse and Feshbach give Z1 F mp (z) D (2m C 1)E mp (z)

p z

dx p  p 2 . x 2
a 2 x 2
b 2 E m (x)

(D20)

D.3 Lamé Wave Equation

The Lamé wave equation can be written as 

ξ12
a 2



ξ12
b 2

 d2 F   dF  2 4   C ξ1 2ξ12
a 2 C b 2 C k ξ1
α 2 ξ12 C Ä F D 0 . dξ1 dξ12 (D21)

D.3 Lamé Wave Equation

D.3.1 Moon–Spencer Form

With a transformation and a change in notation, this is identical to the equation in Moon and Spencer [6]. We make the transformation tQ D ξ12 . Then, dF dF d tQ dF D D 2ξ1 , Q dξ1 dξ dt d tQ 1 2 dF d2 F 2d F D 2 , C 4ξ 1 d tQ d tQ2 dξ12

and Eq. (D21) becomes   d2 F   dF   1 2 tQ tQ
a 2 tQ
b 2 C 3 tQ
2 a 2 C b 2 tQ C a 2 b 2 2 Q 2 dt d tQ  1 C Ä
α 2 tQ C k 2 tQ2 F D 0 . 4

(D22)

If we now make the substitutions a2 ! b ,

b2 ! c ,

  Ä ! b2 C c2 q ,

α 2 ! p (p C 1) ,

tQ ! z ,

we get  1 1 1 dF C C z z
b z
c dz " #  b 2 C c 2 q
p (p C 1)z C k 2 z 2 1 C F D0, 4 z(z
b)(z
c)

1 d2 F C dz 2 2




(D23)

which is Eq. (7.89) of Moon and Spencer [6]. D.3.2 Arscott’s Algebraic Form

To get Arscott’s algebraic form [74], t(t
1)(t
c)

  dF  d2 F 1 2 3t
2(1 C c)t C c C λ C µ t C γ t2 F D 0 , C 2 dt 2 dt (D24)

we start from Eq. (D22) and set tQ D a 2 t, giving  
 b 2 dF b 2 d2 F 1 b2 2 t C t(t
1) t
2 3t C
2 1 C a dt 2 2 a2 a 2 dt i h 1 Ä
α2 t C k 2 a2 t 2 F D 0 . C 4 a2

(D25)

327

328

Appendix D Lamé Functions

Comparing Eq. (D25) with Eq. (D24), we have b2 Dc, a2

Ä D 4a 2 λ ,

α 2 D
4µ ,

k2 D

4 γ. a2

(D26)

Alternatively, we can instead set tQ D b 2 t, giving  
 a 2 dF a 2 d2 F 1 a2 2 t C t(t
1) t
2 3t C
2 1 C b dt 2 2 b2 b 2 dt i 1hÄ
α2 t C k 2 b2 t 2 F D 0 . C 4 b2

(D27)

Comparing Eq. (D27) with Eq. (D24), we have a2 Dc, b2

Ä D 4b 2 λ ,

α 2 D
4µ ,

k2 D

4 γ. b2

(D28)

329

Appendix E Legendre Functions E.1 Introduction

The Legendre and associated Legendre equations arise, for example, in potential problems in spheroidal and toroidal coordinates and in potential and wave problems in spherical polar coordinates. Domains with complete spherical symmetry give rise to the polynomial solutions, whereas restricted domains (such as a cone) allow for infinite-series solutions.

E.2 Legendre Equation

The Legendre equation is ν(ν C 1) 2x dy d2 y C
y D0. 2 2 dx 1
x dx 1
x2

(E1)

It has regular singular points at x D ˙1 and x D 1; hence, it is of Bôcher type f2 2 2g. In general, ν is an arbitrary complex number.

E.3 Series Solutions E.3.1 Recurrence Relation

About an ordinary point, one can write the series solution as y (x) D

1 X

bn x n .

(E2)

nD0

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

330

Appendix E Legendre Functions

Substituting into Eq. (E1) gives 1 X

b n n(n
1)x n
2


nD0

1 X

b n fn(n C 1)
ν(ν C 1)g x n D 0 ,

(E3)

nD0

and one has the recurrence relation b nC2 D


ν(ν C 1)
n(n C 1) bn . (n C 1)(n C 2)

(E4)

This is a two-term recurrence relation, so there are two independent solutions. If ν D l is integral, then b n D 0 8 n  l, leading to a polynomial solution and an infinite-series solution. There are two independent solutions since b 0 and b 1 are arbitrary. For the general case of a nonintegral ν, the two independent series solutions are known as the Legendre function of the first and second kind and are denoted as P ν (x) and Q ν (x), respectively. E.3.2 Convergence

The series solution does not always converge. Applying the Raabe test,

 un 1 P D lim n
1 D n
1 , n!1 u nC1 x2 the series is seen to be convergent for all jxj < 1. Applying the Gauss test, 
u2 j b2 j 1 1 1 lim D D 1 C , n!1 u 2 j C2 b 2 j C2 x 2 j x2 one finds that, for x D ˙1, the series is divergent.

E.4 Legendre Polynomials E.4.1 Normalization

The polynomials, given appropriate normalization, are known as the Legendre polynomials. The normalization is P l (1) D 1 .

(E5)

Owing to the definite parity of the polynomials, P l (
1) D (
1) l . The first few Legendre polynomials are given in Table E.1.

(E6)

E.4 Legendre Polynomials Table E.1 Legendre polynomials. P0 (x )

=

1

P1 (x ) P2 (x )

= =

x

P3 (x )

=

1 2 2 (3x 1 3 (5x 2


1)
3x )

...

E.4.2 Representations

There are a number of useful representations of the Legendre polynomials. E.4.2.1 Hypergeometric Function For n D 0, 1, 2, 3, . . . ,
 1 1 π 1/2 1  F
n, n C I I x 2 P2n (x) D 2 2 n!Γ 2
n 
n (
1) (2n)! 1 1 2 D 2n I I x , F
n, n C 2 (n!)2 2 2 
2π 1/2 x 3 3  1  F
n, n C I I x 2 P2nC1 (x) D
2 2 n!Γ
2
n 
(
1) n (2n C 1)! 3 3 2 . I I x x F
n, n C D 22n (n!)2 2 2 E.4.2.2

E.4.2.3

E.4.2.4

Rodrigue’s Formula l 1 dl  2 x
1 . P l (x) D l 2 l! dx l Generating Function 1 X 1 G(x, h) D D P l (x)h l . (1
2x h C h 2 )1/2 lD0 Schaefli Integral Representation I (t 2
1) n 1 dn 2 1 n dt P n (z) D n (z
1) D , n n 2 n! d z 2π i2 (t
z) nC1

with the contour enclosing t D z.

(E7)

(E8)

(E9)

(E10)

(E11)

331

332

Appendix E Legendre Functions

E.4.3 Special Values

One can use the generating function (or the series representation) to obtain special values of the Legendre polynomials: P2l (0) D (
1) l

(
1) l (2l)! 1  3    (2l
1) (2l
1)!! D 2l 2 , D (
1) l l (2l)!! 2 l! 2 (l!)

(E12)

P2lC1(0) D 0 ,

(E13)

jP l (cos θ )j  1 .

(E14)

E.4.4 Orthogonality

The Legendre equation is self-adjoint. Given appropriate boundary conditions, the polynomials will be orthogonal: Z1 dx P n (x)P m (x) D 0 (m ¤ n) .

(E15)


1

E.4.5 Expansions

The Legendre polynomials also form a complete set; hence, they can be used to expand functions in the interval [
1, 1]: f (x) D

1 X

a n P n (x) .

(E16)

nD0

This is known as a Legendre series. One then finds that

am

2m C 1 D 2

Z1 dx f (x)P m (x) ,

(E17)


1

and 0 1 1 Z 1 X 2n C 1 @ dt f (t)P n (t)A P n (x) . f (x) D 2 nD0

(E18)


1

Equation (E17) is also known as the Legendre transform, and Eq. (E18) is known as the inverse Legendre transform.

E.5 Legendre Function

E.5 Legendre Function

Legendre functions are to be distinguished from Legendre polynomials, whereby the former can be represented by an infinite series. E.5.1 Hypergeometric Representation

The Legendre functions can be represented in terms of the hypergeometric function. Following Lebedev [17], 
1
z P ν (z) D F
ν, ν C 1I 1I , jz
1j < 2 , (E19) 2 p 
πΓ (ν C 1) ν 1 3 1 ν , C 1, C I ν C I Q ν (z) D F Γ (ν C 3/2)(2z) νC1 2 2 2 2 z2 jzj > 1 ,

jargzj < π ,

ν ¤
1,
2, . . .

(E20)

E.5.2 Properties

 Normalization P ν (x D 1) D 1

(E21)

P
ν
1 (z) D P ν (z)

(E22)



 Integrals Z1 dx P ν (x)Q σ (x) D 1

1 , (σ
ν)(σ C ν C 1)

Reσ > Reν

(E23)

E.6 Associated Legendre Functions E.6.1 Associated Legendre Equation

The associated Legendre equation (of degree ν and order µ) is     d2 y µ2 dy 1
x2 y D0. C ν(ν C 1)

2x dx 2 dx 1
x2

(E24)

333

334

Appendix E Legendre Functions

It can be related to the hypergeometric equation, Eq. (A1), by setting in the latter aDµ
ν,

b D µCνC1,

c D µC1.

(E25)

E.6.2 Associated Legendre Functions

If one transforms Eq. (E24) using  µ/2  v, y D x2
1

zD

1 1
x, 2 2

and comparing with Eqs. (E25) and (A4), one can immediately write down a solution to the associated Legendre equation as y (x) D P νµ (x) D

1 Γ (1
µ)




x C1 x
1

 µ/2


1 1 F
ν, ν C 1I 1
µI
x . 2 2 (E26)

If, instead, one sets z D x2 , one gets 4z(1
z)

dZ d2 Z
(µ
ν)(µ C ν C 1)Z D 0 . C [2
(4µ C 6)z] dz 2 dz

(E27)

This is also a hypergeometric equation with a D 1/2(µ C ν C1), b D 1/2(µ
ν), c D 1/2 and, hence, the solution can be written down as [33] Γ (ν C µ C 1)
ν
µ
1 2  x  (x
1) µ/2 Γ ν C 32 
1 1 1 1 3 1 ν C µ C 1, ν C µ C I ν C I x
2 . (E28) F 2 2 2 2 2 2

y (x) D Q µν (x) D e µ i π 2
ν
1 π 1/2

For µ D 0, Eqs. (E26) and (E28) agree with Eqs. (E19) and (E20). For µ D m integral, the two independent solutions, P νm (x) and Q m ν (x), can be written in terms of the Legendre functions [17]:   m/2 d m P ν , P νm (x) D x 2
1 dx m m  2  m/2 d Q ν Qm , ν (x) D x
1 dx m where m 2 N .

(E29) (E30)

E.6 Associated Legendre Functions Table E.2 Associated Legendre polynomials. P11 (x )

=

(1
x 2 )1/2 D sin θ

P21 (x ) P22 (x )

= =

P31 (x )

=

(1
x 2 )1/2 3x D 3 cos θ sin θ 3(1
x 2 ) D 3 sin2 θ    2   3 2 1/2 D 3 5 cos2 θ
1 sin θ 2 5x
1 1
x 2

P32 (x ) P33 (x )

= =

15x (1
x 2 ) D 15 cos θ sin2 θ 15(1
x 2 )3/2 D 15 sin3 θ

...

E.6.2.1

Properties µ P νµ (z) D P
ν
1 (z) ,





Γ (ν C µ C 1) 2 P ν
µ (z) C e
i µ π sin(µ π)Q µν (z) , Γ (ν
µ C 1) π   Γ (ν
µ C 1) 2 P νµ (z)
e
i µ π sin(µ π)Q µν (z) , P ν
µ (z) D Γ (ν C µ C 1) π Γ (ν C m C 1)
m m P ν (z) D P (z) , Γ (ν
m C 1) ν 2 P νµ (
z) D e ν π i P νµ (z)
e
i µ π sin(µ C ν)π Q µν (z) , π P νµ (z) D

Q µν (
z) D
e ˙i ν π Q µν (z) .

(E31) (E32) (E33) (E34) (E35) (E36)

E.6.3 Associated Legendre Polynomials

The first few associated Legendre polynomials are given in Table E.2. E.6.4 Generating Function

G(m, x, t) D

(2m)!(1
x 2 ) m/2 2 m m! (1


2t x C

t 2 ) mC1/2

D

1 X

m P sCm (x)t s .

(E37)

sD0

E.6.5 Recurrence Relations m m (2n C 1)x P nm D (n C m)P n
1 C (n
m C 1)P nC1 ,

P nmC1 D

2m x P nm
[n(n C 1)
m(m
1)]P nm
1 . (1
x 2 )1/2

(E38) (E39)

335

336

Appendix E Legendre Functions

E.6.6 Parity

P nm (
x) D (
1) nCm P nm (x) .

(E40)

P nm (˙1) D 0 8m ¤ 0 .

(E41)

Also,

E.6.7 Orthogonality

Z1 dx P pm (x)P qm (x) D
1

2 (q C m)! δ p ,q . 2q C 1 (q
m)!

(E42)

E.7 Spherical Harmonics

They can be defined to be the eigensolutions of the angular part of the Laplace operator. Note that Lebedev defines them to be the solutions to the associated Legendre equation. E.7.1 Definition

s Ynm (θ , ')

 (
1)

m

2n C 1 (n
m)! m P (cos θ )e i m' . 4π (n C m)! n

(E43)

The phase factor is known as the Condon–Shortley phase. The first few spherical harmonics are given in Table E.3. Table E.3 Spherical harmonics. Y00 (θ , ')

=

Y1˙1 (θ , ')

=

Y20 (θ , ')

=

Y2˙1 (θ , ')

=

Y2˙2 (θ , ')

=

...

p1 4π

q 3  8π sin θ e ˙i' q   5 3 1 2 4π 2 cos θ
2 q 5  24π 3 sin θ cos θ e ˙i' q 5 ˙2i' 96π 3 sin θ cos θ e

E.7 Spherical Harmonics

E.7.2 Orthogonality

The functions are orthonormal: Z2π Zπ 'D0 θ D0

sin θ dθ d' Ynm1 1  (θ , ')Ynm2 2 (θ , ') D δ n 1 ,n 2 δ m 1 ,m 2 .

(E44)

337

339

Appendix F Mathieu Functions F.1 Introduction

The Mathieu and modified Mathieu functions are solutions to the ordinary differential equations arising in the separation of the Helmholtz equation in two dimensions using elliptic coordinates, and of the Laplace and Helmholtz equations using elliptic cylinder coordinates. A very comprehensive discussion of the properties and applications of these functions can be found in the book by McLachlan [40]. Additional discussion can be found in, for example, Abramowitch and Stegun [51], Morse and Feshbach [5], Moon and Spencer [6], and Zhang and Jin [13].

F.2 Mathieu Equation

The Mathieu functions, or elliptic cylinder functions, are the solutions to the Mathieu equation: d2 y C (λ
2q cos 2v ) y D 0 . dv 2

(F1)

The Mathieu equation often arises in the separation of a partial differential equation (e.g., using elliptic cylinder coordinates), indeed from the angular equation, in which case λ is also a separation constant. The Mathieu equation has been shown to be a special case of the Baer equation [6]: " #
 d2 y 1 1 dy 1 1 A0 C A1 z C C y D0, (F2) C dz 2 2 z
b z
c dz 4 (z
b)(z
c) if b D 0, c D 1, A0 D
(2q C λ), A1 D 4q, and z D cos2 v . In particular, the Bôcher type of the Mathieu equation is the same as that for the Baer equation, that is, f1 1 3g.

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

340

Appendix F Mathieu Functions

One also comes across the modified Mathieu equation: d2 y
(λ
2q cosh 2u) y D 0 . du2

(F3)

The two types of the Mathieu equation are, in fact, related by changing the independent variable into an imaginary one. In the following, we will assume λ and q are real unless stated otherwise.

F.3 Mathieu Function

The solutions ce(v , q) and se(v , q) to the Mathieu equation, Eq. (F1), are known as Mathieu functions of the first kind. The solutions of Eq. (F3) are conventionally written as Mc(λ, q, u) and Ms(λ, q, u). Comparing Eq. (F3) with Eq. (F1), one sees that the modified Mathieu functions can be related to the Mathieu functions if one allows for imaginary numbers: Mc(q, u) D ce(q, i u) ,

Ms(q, u) D se(q, i u) .

(F4)

Because of the cos 2v factor, the solutions can have periods of π or 2π. We will consider both in the following. We will assume q to be a given parameter and λ to be an eigenvalue or a characteristic value to be determined from the imposition of periodicity. This typically leads to a relation between q and λ known as the characteristic equation and written as λ(q). There is a trivial solution when q D 0 as the Mathieu equation then becomes d2 y C λy D 0 . dv 2

(F5)

This is nothing but the harmonic equation with trigonometric solutions y 0 (x) D cos m v or sin m v ,

(F6)

where λ(0) D m 2 . Thus, for q ¤ 0, one expects two solutions such that they reduce to y 0 (v ) when q D 0. They are, therefore, written as ce p (v , q) and se p (v , q), whereby lim ce p (v , q) D cos p v ,

(F7)

lim se p (v , q) D sin p v .

(F8)

q!0

q!0

This explains the origin of the notations ce and se, which basically stand for “cosine elliptic” and “sine elliptic,” respectively. Note that, in general, p is not necessarily an integer. If it is not an integer, then the solutions ce p (v , q) and se p (v , q) are independent. For integral p D m, the second solution is the nonperiodic one, called the Mathieu function of the second kind, and denoted by fe m (v , q) and ge m (v , q).

F.3 Mathieu Function

F.3.1 Properties

A few useful properties are listed below [13]:  For m even, ce m (v , q) and se m (v , q) have period π.  For m odd, ce m (v , q) and se m (v , q) have period 2π.  The Mathieu functions are either even or odd: ce m (
v , q) D ce m (v , q) ,

(F9)

se m (
v , q) D
se m (v , q) .

(F10)

 Within [0, π], ce m (v , q) and se m (v , q) have the same number of zeros as cos m v and sin m v , respectively. The periodicity means that the Mathieu functions can be expanded in a Fourier series. Also, for q ¤ 0, the characteristic equation depends upon the periodicity and parity. Hence, one can classify the Mathieu functions into four types; in fact, each satisfies a different Mathieu equation. One observation is that they do not (together) form the independent solutions of the second-order differential equation. Indeed, the second independent solution is a nonperiodic one. F.3.2 Orthogonality

The property of orthogonality is a bit tricky for the periodic Mathieu functions of integral order since different functions satisfy different differential equations. Nevertheless, the following can be established [40]: Z2π dv ce m (v , q)ce n (v , q) D 0 (m ¤ n) ,

(F11)

dv se m (v , q)se n (v , q) D 0 (m ¤ n) ,

(F12)

dv ce m (v , q)se n (v , q) D 0 ,

(F13)

0

Z2π 0

Z2π 0 Z2π

1 h i i h X (2n) 2 (2n) 2 A 2r dv ce22n (v , q) D 2π A 0 Cπ ,

dv ce22nC1 (v , q) D π 0

(F14)

rD1

0 Z2π

1 h X rD0

i (2nC1) 2

A 2rC1

,

(F15)

341

342

Appendix F Mathieu Functions

Z2π dv se22nC1 (v , q) D π

1 h X

i (2nC1) 2

B2rC1

,

(F16)

,

(F17)

rD0

0 Z2π

dv se22nC2 (v , q) D π

1 h X

i (2nC2) 2

B2rC2

rD0

0

given that ce2n (v , q) D

1 X

(2n)

A 2r cos 2r v ,

rD0 1 X

ce2nC1 (v , q) D se2nC1 (v , q) D se2nC2 (v , q) D

rD0 1 X rD0 1 X

(F18)

(2nC1)

(F19)

(2nC1)

(F20)

(2nC2)

(F21)

A 2rC1 cos(2r C 1)v , B2rC1 sin(2r C 1)v , B2rC2 sin(2r C 2)v .

rD0

Also, ZU

Z2π dv (cosh 2u
cos 2v )ψ n,m ψ p ,r D 0 (p ¤ n or p D n , r ¤ m) ,

du 0

0

(F22) where ψ n,m D Mc n,m ce n,m or Ms n,m se n,m , and ZU

(

Z2π dv (cosh 2u
cos 2v )

du 0

0

Mc2n,m (u)ce2n,m (v ) Ms2n,m (u)se2n,m (v )

¤0.

(F23)

The following integrals are also useful [40]: 1 π 1 π

Z2π

(2n)

(2n)

dv ce22n (v ) cos 2v D A 0 A 2

C

(2n)

(2n)

A 2r A 2rC2  Θ2n ,

(F24)

rD0

0

Z2π dv ce22nC1 (v ) cos 2v D 0

1 X

1 1 (2nC1) 2 X (2nC1) (2nC1) [A 1 ] C A 2rC1 A 2rC3 2 rD0

 Θ2nC1 ,

(F25)

F.4 Characteristic Equation

1 π

1 π

Z2π dv se22nC1 (v ) cos 2v D
0

1 1 h (2nC1) i2 X (2nC1) (2nC1) B1 C B2rC1 B2rC3 2 rD0

 Ψ2nC1 ,

Z2π

dv se22nC2 (v ) cos 2v D

1 X

(2nC2)

(F26) (2nC2)

B2rC2 A 2rC4  Ψ2nC2 .

(F27)

rD0

0

F.3.3 Periodic Solution for Small q

One approach to finding periodic solutions to the Mathieu equation is discussed in McLachlan [40]. This involves expanding the solutions and characteristic equation in a power series in q. Thus, if one writes λ(q) D m 2 C α 1 q C α 2 q 2 C α 3 q 3 C    ,

(F28)

y (v ) D cos m v C q c 1 (v ) C q 2 c 2 (v ) C q 3 c 3 (v ) C    ,

(F29)

substitutes them into Eq. (F1), and equates coefficients in terms of the same powers in q, one could, in principle, obtain power-series solutions – although the procedure is rather tedious. For ce1 , McLachlan gives the result   1 1 2 1 q
cos 3v C cos 5v ce1 (v , q) D cos v
q cos 3v C 8 64 3   4 1 1 3 1 q cos 3v
cos 5v C cos 7v C O(q 4) ,
(F30) 512 3 9 18 1 1 3 1 4 q
q C O(q 5 ) , λ(q) D 1 C q
q 2
(F31) 8 64 1536 and, for se1 ,

  1 1 2 1 q sin 3v C q sin 3v C sin 5v 8 64 3   4 1 1 3 1 q sin 3v C sin 5v C sin 7v C O(q 4 ) ,
512 3 9 18 1 2 1 3 1 4 λ(q) D 1
q
q C q
q C O(q 5 ) . 8 64 1536 se1 (v , q) D sin v


(F32) (F33)

F.4 Characteristic Equation

We now describe how the characteristic equation is obtained. The following discussion is partly based upon Abramowitz and Stegun [51]. Since the solutions are periodic, one can expand them in a Fourier series: y (v ) D

1 X mD0

(A m cos m v C B m sin m v ) .

(F34)

343

344

Appendix F Mathieu Functions

The goal is to find the characteristic values λ and the Fourier coefficients A m , B m . Substituting Eq. (F34) into Eq. (F1) gives 1 X 

1 X  [A m cos m v C B m sin m v ]
A m m 2 cos m v
B m m 2 sin m v C λ

mD0

mD0 1 X


2q

[A m cos m v cos 2v C B m sin m v cos 2v ] D 0 ,

(F35)

mD0

or 1 X

"



  
A m m 2 C λA m cos m v C
B m m 2 C λB m sin m v

mD0 1 X


q

#

" A m cos(m C 2)v C A m cos(m
2)v

mD0

# C B m sin(m C 2)v C B m sin(m
2)v D 0 .

(F36)

Now, 1 X mD0 1 X

A m cos(m C 2)v D A m cos(m
2)v D

mD0

1 X

A m
2 cos m v 

mD2 1 X

1 X

A m
2 cos m v ,

mD
2

A mC2 cos m v ,

mD
2

if A
m D 0 8 m > 0. A similar treatment for the sine terms gives 1 X 

  λ
m 2 A m
q (A m
2 C A mC2) cos m v

mD
2

C

1 X 

  λ
m 2 B m
q (B m
2 C B mC2) sin m v D 0 .

(F37)

mD
2

We can start the second summation from m D
1 if B0 is chosen to be zero. Equating the coefficients to zero leads to three-term recurrence relations. We do this separately for even/odd and π/2π periodic solutions. F.4.1 Recurrence Relations F.4.1.1 (Even, π) Solutions

y 0 (v ) 

1 X nD0

A 2n cos 2nv ,

(F38)

F.4 Characteristic Equation

that is, m is even in Eq. (F37). For m D 0, λA 0
q (A
2 C A 2 ) D 0 , H)

λA 0
q A 2 D 0 .

(F39)

For m D 2 and m D
2, 

   (λ
4)A 2
q (A 0 C A 4 ) cos 2v C (λ
4)A
2
q(A
4 C A 0 ) cos 2v D 0 I

hence, (λ
4)A 2
q (2A 0 C A 4 ) D 0 .

(F40)

For m  4, 

 λ
m 2 A m
q (A m
2 C A mC2) D 0 .

F.4.1.2 (Even, 2π) Solutions 1 X y 0 (v )  A 2nC1 cos(2n C 1)v .

(F41)

(F42)

nD0

For the cos v terms (i.e., n D 0), set m D 1 and m D
1 in Eq. (F37):     (λ
1)A 1
q (A
1 C A 3 ) cos v C (λ
1)A
1
q (A
3 C A 1 ) cos v D 0 I hence, (λ
1)A 1
q (A 1 C A 3 ) D 0 .

(F43)

Equation (F41) still applies. F.4.1.3 (Odd, π) Solutions 1 X y 1 (v )  B2n sin 2nv ,

(F44)

nD0

that is, m is even in Eq. (F37). For m D
2 and m D 2 (obviously no m D 0 term),     (λ
4)B2
q (B0 C B4 ) sin 2v C (λ
4)B
2
q (B
4 C B0 ) sin 2v D 0 I hence, (λ
4)B2
q B4 D 0 .

(F45)

For m  4, (λ
m 2 )B m
q (B m
2 C B mC2 ) D 0 .

(F46)

345

346

Appendix F Mathieu Functions

F.4.1.4 (Odd, 2π) Solutions 1 X y 1 (v )  B2nC1 sin(2n C 1)v .

(F47)

nD0

For the sin v terms (i.e., n D 0), set m D 1 and m D
1 in Eq. (F37):     (λ
1)B1
q (B
1 C B3 ) sin v
(λ
1)B
1
q (B
3 C B1 ) sin v D 0 I hence, (λ
1)B1 C q (B1
B3 ) D 0 .

(F48)

Equation (F46) still applies. One can solve the above three-term recurrence relations by the continued fraction method. First, let us rewrite the recurrence relations in a more compact notation. Let Ge m D

Am , A m
2

Go m D

Bm , B m
2

(F49)

and let G m stand for Ge m or Go m . Also, let Vm D

λ
m2 . q

(F50)

F.4.1.5 (Even, π) Solutions For example, Eq. (F39) becomes

A2 λ D , A0 q H)

Ge2 D V0 .

(F51)

Equation (F40) becomes

H)

A4 A0 λ
4
2 D A2 q A2 2 Ge4 D V2
. G e2

(F52)

Finally, Eq. (F41) becomes A m
2 λ
m2 A mC2 D C , q Am Am 1 1 C , H) Vm D Gm G mC2 1 . F Gm D Vm
G mC2 Note that the latter equation is valid for all solutions.

(F53)

F.4 Characteristic Equation

F.4.1.6 (Even, 2π) Solutions Equation (F43) becomes

A3 λ
1 D 1C , q A1 H)

V1 D 1 C Ge3 .

(F54)

F.4.1.7 (Odd, π) Solutions Equation (F45) becomes

B4 λ
4 , D q B2 H)

V2 D Go4 .

(F55)

F.4.1.8 (Odd, 2π) Solutions Equation (F48) becomes

B3 λ
1 D
1 , q B1 H)

V1 D Go3
1 .

(F56)

F.4.2 Continued Fraction Solution

We start with the most general equation, Eq. (F53): Gm D D

1 D Vm
G mC2 Vm
1 Vm


1 VmC2
V

1 1 VmC2
G mC4

,

(F57)

1 mC4


or G mC2 D Vm
D Vm


1 1 D Vm
Gm Vm
2


1 G m
2

1 . Vm
2
Vm
41


(F58)

Equation (F58) terminates. We will develop it first, for each of the four possible solutions. From now on, we will use the following notation for a continued fraction: G mC2 D Vm


1 1 1  .  Vm
Vm
2
Vm
4
Vm
2
Vm
41


347

348

Appendix F Mathieu Functions

F.4.2.1 (Even, π) Solutions Given Eqs. (F51) and (F52), we have

1 1 1  Vm
2
Vm
4
Ge4 1 1 1  D Vm
Vm
2
Vm
4
V2


Ge mC2 D Vm


2 V0

.

(F59)

F.4.2.2 (Even, 2π) Solutions Given Eq. (F54),

1 Vm
2
Vm
4
Ge3 1 1 1  . D Vm
Vm
2
Vm
4
V1
1 1

1

Ge mC2 D Vm




(F60)

F.4.2.3 (Odd, π) Solutions Given Eq. (F55),

1 1 1  Vm
2
Vm
4
Go4 1 1 1  D Vm
. Vm
2
Vm
4
V2

Go mC2 D Vm


(F61)

F.4.2.4 (Odd, 2π) Solutions Given Eq. (F56),

1 1 1  Vm
2
Vm
4
Go3 1 1 1  . D Vm
Vm
2
Vm
4
V1 C 1

Go mC2 D Vm


(F62)

All of the above four continued fractions can be written as a single equation: G mC2 D Vm


1

1

Vm
2
Vm
4




'0 , V0Cd C '1

(m  4)

(F63)

where the values of ' i and d are given in Table F.1. One can use Eq. (F57) to obtain characteristic equations: Gm D

1 Vm


1 VmC2
V

.

1 mC4


F.4.2.5 (Even, π) Solutions Given Eq. (F51),

Ge2 D V0 D V2


1 1    ! A 2r (q) . V4
V6


(F64)

F.5 Mathieu Functions of Fractional Order Table F.1 Parameters for the continued fraction of Eq. (F63). '0

'1

d

Even, π Even, 2π

2 1

0
1

0 1

Odd, π

0

0

0

Odd, 2π

1

1

1

F.4.2.6 (Even, 2π) Solutions Given Eq. (F54),

V1
1 D Ge3 D

1 1    ! A 2rC1(q) . V3
V5


(F65)

F.4.2.7 (Odd, π) Solutions Given Eq. (F55),

V2 D Go4 D

1 1    ! B2r (q) . V4
V6


(F66)

F.4.2.8 (Odd, 2π) Solutions Given Eq. (F56),

V1 C 1 D Go3 D

1 1    ! B2rC1(q) . V3
V5


(F67)

F.5 Mathieu Functions of Fractional Order

Mathieu functions of fractional order have been discussed by McLachlan [40]. The definition is that the function satisfies the Mathieu equation and reduces to cos νz and sin νz when q D 0. Thus, if one writes λ(q) D ν 2 C

1 X

αr qr ,

rD1

ce ν (z, q) D cos νz C se ν (z, q) D sin νz C

1 X rD1 1 X rD1

(F68) cr qr ,

(F69)

sr qr .

(F70)

349

350

Appendix F Mathieu Functions

McLachlan gives the following results:
 cos(ν C 2)z cos(ν
2)z 1
ce ν (z, q) D cos νz
q 4 (ν C 1) (ν
1)
 1 2 cos(ν C 4)z cos(ν
4)z C q C 32 (ν C 1)(ν C 2) (ν
1)(ν
2) "    2 2 ν ν
4ν C 7 cos(ν
2)z C 4ν C 7 cos(ν C 2)z 1 3
q
128 (ν
1)(ν C 1)3 (ν C 2) (ν C 1)(ν
1)3 (ν
2)  cos(ν C 6)z cos(ν
6)z C
C , (F71) 3(ν C 1)(ν C 2)(ν C 3) 3(ν
1)(ν
2)(ν
3)
 sin(ν C 2)z sin(ν
2)z 1
se ν (z, q) D sin νz
q 4 (ν C 1) (ν
1)
 1 2 sin(ν C 4)z sin(ν
4)z C q C C  (F72) 32 (ν C 1)(ν C 2) (ν
1)(ν
2)  2  5ν C 7 1 λ(q) D ν 2 C q2 C q4 2 (ν 2
1) 32 (ν 2
1)3 (ν 2
4) C

9ν 4 C 58ν 2 C 29 64 (ν 2
1)5 (ν 2
4) (ν 2
9)

q6

C  .

(F73)

These formulae are valid for q /2(ν
1) << ν . Note that for nonintegral ν, ce ν (z, q) and se ν (z, q) are independent solutions to the same Mathieu equation (hence the same characteristic equation). 2

2

2

F.6 Nonperiodic Second Solutions

There is much less work on the nonperiodic second solutions. We give one example of such a function, fe1 (z, q) which is associated with ce1 (z, q):
 3 3 3 4 31 5 6 q
q C q C O(q ) zce1 (z, q) fe1 (z, q) D q
64 256 36 864 "   1 1 2 1 q 5 sin 3z C sin 5z C sin z
q sin 3z C 8 64 3   35 1 3 8 1
q
sin 3z C sin 5z C sin 7z 512 3 3 18   17 1 4 343 61 1 C q
sin 3z
sin 5z C sin 7z C sin 9z 4096 3 54 108 180 #


.

(F74)

Note that if ν D m C β and β D p /s is a rational fraction, then the solution has a periodicity of 2s π.

351

Appendix G Spheroidal Wave Functions G.1 Introduction

The spheroidal wave functions are solutions to the ordinary differential equations obtained by separating the Helmholtz equation in prolate and oblate spheroidal coordinates. One of the standard texts on the spheroidal wave functions is the book by Flammer [105]. Additional discussion can be found in, for example, Abramowitch and Stegun [51], Morse and Feshbach [5], and Zhang and Jin [13].

G.2 Spheroidal Wave Equation

The spheroidal wave equation is obtained by separating the Helmholtz equation in spheroidal coordinates: 

 dS(c, η) d  m2 S(c, η) D 0 . 1
η2 C λ
c2 η2
dη dη 1
η2

(G1)

The ordinary differential equation has regular singular points at ξ , η D ˙1. Requiring finiteness of the functions at the latter points leads to discrete values of λ D λ m n (the so-called eigenvalues). Hence, we will write the two types of spheroidal wave equations as

  dR m n (c, ξ ) m2 d  2 ξ
1
λ m n
c2 ξ 2 C 2 R m n (c, ξ ) D 0 , (G2) dξ dξ ξ
1 

 dS m n (c, η) m2 d  S m n (c, η) D 0 . (G3) 1
η2 C λ m n
c2 η2
dη dη 1
η2 We will, for now, assume that m 2 N .

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

352

Appendix G Spheroidal Wave Functions

G.3 Spheroidal Wave Functions

Our discussion follows Abramowitz and Stegun [51] and Flammer [105]. G.3.1 Prolate Angular Functions

Values of λ m n leading to finite solutions at η D ˙1 are eigenvalues. The eigen(1) functions S m n (c, η) are the prolate spheroidal angle functions of the first kind, of (2) order m and degree n. Angle functions of the second kind, S m n (c, η), are singular at η D ˙1. G.3.1.1 Recurrence Relation If c D 0, Eq. (G3) becomes the associated Legendre equation; hence, (1)

S m n (0, η) D P nm (η) , λ m n (0) D n(n C 1) ,

(G4) nm.

(G5)

Therefore, one can do an expansion in terms of the associated Legendre functions. If c ¤ 0, let X m d rm n (c)P rCm (η) , (G6) S m n (c, η) D r m where P rCm is an associated Legendre function of the first kind. (The limits on r will later, from the recurrence relations, be found to start either from 0 or from 1 for the two independent solutions and range over either even or odd values only.) One can now derive a recursion relation for the coefficients d rm n . Substituting Eq. (G6) into Eq. (G3), we have

X

 d rm n (c)

r



m  d P mCr m2 d  2 2 2 m P mCr 1
η C λmn
c η
D0. dη dη 1
η2 (G7)

One can get rid of the differential operators by using the associated Legendre equation: 
  m2 1
η 2 P nm 00
2η P nm 0 C n(n C 1)
P nm D 0 , (G8) 1
η2 giving X r

X   m m d rm n λ m n
(m C r)(m C r C 1) P mCr D c2 d rm n η 2 P mCr . r

(G9)

G.3 Spheroidal Wave Functions

Using the recurrence relation given in Eq. (E38), (n C m) m (n
m C 1) m P n
1 C P nC1 , (2n C 1) (2n C 1) (n C m) (n
m C 1) m D xPm C x P nC1 (2n C 1) n
1 (2n C 1)

x P nm D

H)

x 2 P nm

D

(G10) 




(n C m) (n
1 C m) m (n
m) m P n
2 C P (2n C 1) (2n
1) (2n
1) n 
(n
m C 1) (n C 1 C m) m (n
m C 2) m Pn C P nC2 . C (2n C 1) (2n C 3) (2n C 3) (G11)

Substituting Eq. (G11) into Eq. (G9), X

X   m m d rm n λ m n
(m C r)(m C r C 1) P mCr D c2 d rm n η 2 P mCr

r

D c2

X r

( d rm n

r

(2m C r)(2m C r
1) Pm (2m C 2r C 1)(2m C 2r
1) mCr
2

 (r C 1)(2m C r C 1) (2m C r)(r) m C P mCr (2m C 2r C 1)(2m C 2r
1) (2m C 2r C 1)(2m C 2r C 3) ) (r C 1)(r C 2) m . (G12) P C (2m C 2r C 1)(2m C 2r C 3) mCrC2


C

m The coefficient of the P mCr term on the right-hand-side of Eq. (G12) is

r(2m C r)(2m C 2r C 3) C (r C 1)(2m C r C 1)(2m C 2r
1) , (2m C 2r C 1)(2m C 2r
1)(2m C 2r C 3) the numerator of which is (2m C 2r C 1)[r(2m C r)] C (2m C 2r C 1)[(r C 1)(2m C r C 1)] C 2r(2m C r)
2(r C 1)(2m C r C 1) D (2m C 2r C 1)[r(2m C r) C (r C 1)(2m C r C 1)
2]   D (2m C 2r C 1) 2(m C r)(m C r C 1)
2m 2
1 .

353

354

Appendix G Spheroidal Wave Functions

Hence, Eq. (G12) becomes
 2(m C r)(m C r C 1)
2m 2
1 m d rm n λ m n
(m C r)(m C r C 1)
P mCr (2m C 2r
1)(2m C 2r C 3) r " X (2m C r 0 )(2m C r 0
1) 2 mn Pm Dc dr0 0 C 1) (2m C 2r 0
1) mCr 0
2 (2m C 2r r 0 D0,1 # (r 0 C 1)(r 0 C 2) m P C 0 (2m C 2r 0 C 1)(2m C 2r 0 C 3) mCr C2 X
m n (2m C r C 2)(2m C r C 1) d rC2 D c2 (2m C 2r C 3)(2m C 2r C 5) r0  r(r
1) mn m P mCr . (G13) Cd r
2 (2m C 2r
1)(2m C 2r
3)

X

The limits on the summation on the right-hand side of Eq. (G13) are valid since, for the first summation, terms with 0  r 0  1 (i.e., r D
2,
1) do not contribute m since P mCr D 0 and, for the second summation, terms with 0  r < 2 do not contribute since the coefficient is zero. We, therefore, have a three-term recurrence relation: (2m C r C 1)(2m C r C 2)c 2 m n r(r
1)c 2 mn d r
2 d C (2m C 2r
1)(2m C 2r
3) (2m C 2r C 3)(2m C 2r C 5) rC2 " C (m C r)(m C r C 1)
λ m n C

# 2(m C r)(m C r C 1)
2m 2
1 2 m n c dr D 0 . (2m C 2r C 3)(2m C 2r
1)

(r  0)

(G14)

G.3.1.2 Eigenvalue Problem One can rewrite it as [51] mn mn C (β r
λ m n ) d rm n C γ r d r
2 D0. α r d rC2

(G15)

This is a linear homogeneous difference equation of second order. It can be rewritten as an eigenvalue equation: 2

β0 6 γ0 6 4   

α2 β2 γ2

 α4 β4

α6

32 3 2 3 d0 d0 76 7 7 7 4 d2 5 D λ m n 6 4 d2 5 5 . .. .. .

!

λ m n (c), d rm n (c) . (G16)

The eigenvalues follow from the convergence of the series expansion (cf. the Mathieu equation where we have periodicity).

G.3 Spheroidal Wave Functions

G.3.1.3 Continued Fractions One can rewrite Eq. (G15) as

αr

mn d rC2

d rm n

C (β r
λ m n ) C γ r

mn d r
2 D0. d rm n

Let (2m C r)(2m C r
1)c 2 d rm n d rm n m m n  Nr D
mn , d r
2 (2m C 2r
1)(2m C 2r C 1) d r
2 α r
2 γ r βm d mn r
γ r r
2 D  , d rm n N rm N rm


α r
2

(G17) (G18)

where βm r 

r(r
1)(2m C r)(2m C r
1)c 4 . (2m C 2r
1)2 (2m C 2r C 1)(2m C 2r
3)

(G19)

Equation (G15) becomes m
N rC2 C (β r
λ m n )


βm r D0, N rm

(G20)

or N rm D

βm r , m β r
λ m n
N rC2

(G21)

where 2(m C r)(m C r C 1)
2m 2
1 2 c (2m C 2r C 3)(2m C 2r
1)
 4m 2
1 c2 1
D (m C r)(m C r C 1) C 2 (2m C 2r C 3)(2m C 2r
1)

β r D (m C r)(m C r C 1) C

 γ rm .

(G22)

Equation (G20) becomes m N rC2 D γ rm
λ m n


βm r , N rm

(r  2)

(G23)

with N2m D γ0m
λ m n ,

N3m D γ1m
λ m n .

(G24)

Note that the N in Flammer [105] and Abramowitz and Stegun [51] are missing a negative sign. However, they are correct in Zhang and Jin [13].

355

356

Appendix G Spheroidal Wave Functions

One can now derive two continued fractions corresponding to the two recurrence relations Eqs. (G21) and (G23). Starting from Eq. (G21), we have m D N rC2

D

D

βm rC2 m γ rC2

m
λ m n
N rC4

βm rC2 m γ rC2
λmn


βm rC4 m m γ rC4
λ m n
N rC6

βm rC4

βm rC2 m γ rC2


λmn


m γ rC4

βm rC6 m γ rC6


λ mn



λmn


 .

(G25)

The continued fraction is convergent if one chooses the solution such that limr!1 N rm D 0. From Eq. (G23), we have m D γ rm
λ m n
N rC2

D γ rm
λ m n
D γ rm
λ m n


βm r N rm βm r m γ r
2
λmn


m γ r
2

βm r
2 m N r
2

βm βm βm r r
2 r
4  . m m
λ m n
γ r
4
λ m n
γ r
6
λ m n
(G26)

The continued fraction terminates with either N2m  γ0m
λ m n or N3m  γ1m
λ m n . Equating Eqs. (G25) and (G26) leads to a transcendental equation: U (λ m n ) D U1 (λ m n ) C U2 (λ m n ) D 0 , m U1 (λ m n ) D γ n
m
λmn


U2 (λ m n ) D


(G27) βm n
m

m γ n
m
2
λmn


βm n
mC2 m γ n
mC2
λmn


βm n
m
2 m γ n
m
4
λ m n
...

βm n
mC4 m γ n
mC4
λ m n
...

.

,

(G28) (G29)

The transcendental equation is solved in an iterative manner and convergence is usually established within a dozen runs. The roots are λ m n (c 2 ) for n D m, m C 1, m C 2, . . . . An application is given in the text.

357

Appendix H Weber Functions The Weber functions are related to the solutions obtained when separating the Laplace or the Helmholtz equation in parabolic cylinder coordinates. Hence, they are also known as the parabolic cylinder functions.

H.1 Weber Equation

The Weber equation is given in a number of forms. We use the definition of Moon and Spencer [6] and that given in the Bateman manuscript [33]: 
1 1 2 d2 D ν (z) C ν C
z D ν (z) D 0 . (H1) dz 2 2 4 The solution D ν (z) is called a parabolic cylinder or Weber–Hermite function. Moon and Spencer labeled the solution as W(ν, z). We now give a few of the other forms. Lebedev [17] calls  d2 A  C 2ν C 1
t 2 A D 0 2 dt

(H2)

the parabolic cylinder equation. It clearly follows from Eq. (H1) if we set p z D 2t . Thus, one can write p  2t . A ν (t) D D ν Alternately, Zhang and Jin [13] write the Weber equation as 
d2 y 1 2 y D0, z
a C dz 2 4

(H3)

which is equivalent to Eq. (H1) if
 1 aD
pC . 2 Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

358

Appendix H Weber Functions

Equation (H3) is also one of the two forms given by Abramowitch and Stegun [51]. The other form (also studied in [13]) is 
1 2 d2 y y D0. (H4) z
a
dz 2 4 Of course, these are all special cases of the more general differential equation [51]   d2 y C az 2 C b z C c y D 0 . 2 dz

(H5)

We have seen that, for example, by separating the Laplace or the Helmholtz equation in parabolic cylinder coordinates (Chapter 10), one ends up with ordinary differential equations of the type   d2 N C α2
α3 x 2 N D 0 . dx 2

(H6)

This can be reduced to the standard forms for the Weber equation. Thus, replacing
 1 q4 2 α2 D q p C , α3 D 2 4 gives   
q4 x 2 1 d2 N(x) 2
N(x) D 0 , p C C q dx 2 2 4

(H7)

which is similar to Eq. (H1).

H.2 Weber Functions

The solutions to the above Weber equations can be related to the confluent hypergeometric functions. For example, in Eq. (H3), if one makes the replacements y (z) D e


z2 4

u(z) ,

tD

z2 , 2

one gets t

d2 u C dt 2




1
t 2



du
dt




1 1 aC 2 4

 uD0.

(H8)

From a comparison with the confluent hypergeometric equation with α D a/2 C 1/4, γ D 1/2, Eq. (H3) has solutions 
2 1 1 z2 a
z4 y 1 (z) D e C , , , (H9) M 2 4 2 2

H.2 Weber Functions

y 2 (z) D e




2


z4

M

3 3 z2 a C , , 2 4 2 2

 .

(H10)

To demonstrate the nonuniqueness of the expressions, one can also obtain another relationship if one chooses y (z) D e


z2 4

z2 , 2

tD


u(z) ,

giving a confluent hypergeometric equation with α D
a/2 C 1/4, γ D 1/2, and the solutions become 
z2 1 1 z2 a y 1 (z) D e 4 M
C , ,
, (H11) 2 4 2 2
 a 3 3 z2 z2 y 2 (z) D e 4 M
C , ,
. (H12) 2 4 2 2 One also has [33] " D ν (z) D 2

2

ν/2
z4

e

1


 ν 1 z2  M
, , 2 2 2 Γ 2
ν2  1
# z Γ
2 1 ν 3 z2
, , . C 1/2  ν  M 2 Γ
2 2 2 2 2 Γ 1

2

(H13)

Alternative forms of the Weber functions are provided by Zhang and Jin [13]. H.2.1 Properties

 The Weber functions can be chosen as even or odd.  For p an even integer, one finds [6] p !
(q z)2/4 p /2 2 e H p (q z) , We (p, q z) D (
2) p!

(H14)

and if p is an odd integer,  Wo (p, q z) D (
2)( p
1)/2

p
1 2

 !

p!

e
(q z) /4 H p (q z) , 2

(H15)

where H p (q z) are Hermite polynomials.  The Weber functions have the following special values: We (p, 0) D 1 ,

Wo (p, 0) D 0 ,

We,o (p, ˙1) D 0 .

(H16)

359

361

Appendix I Elliptic Integrals and Functions In the text, the elliptic functions show up mainly as an alternative treatment of the Lamé equation. Though not essential for most of the problems, we have included it here for completeness. The results summarized have been compiled from four main sources: Whittaker and Watson [106], Arscott [73], Abramowitz and Stegun [51], and Zhang and Jin [13].

I.1 Elliptic Integrals

In Abramowitz and Stegun [51], an elliptic integral is defined as one whose integrand is a rational function of x and y and where y 2 is equal to a cubic or quartic polynomial in x: Z R(x, y )dx . (I1) According to Abramowitz and Stegun [51], there are three canonical forms of elliptic integrals. I.1.1 Elliptic Integral of the First Kind

Z' F(k, ')
F('nα)
F('jk) D 0

Zx D 0




dt [(1  t 2 ) (1  k 2 t 2 )]1/2

dθ 1

k 2 sin2

θ

1/2

(I2) (I3)

Zu dw D u ,

D

(I4)

0

Separable Boundary-Value Problems in Physics, First Edition. Morten Willatzen and Lok C. Lew Yan Voon © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA

362

Appendix I Elliptic Integrals and Functions

where t D sin θ ,

(I5)

k D sin α ,

(I6)

x D sin ' D snu .

(I7)

α is known as the modular angle. In Abramowitz and Stegun [51], k 2 is replaced by m. The elliptic function snu will be discussed in more detail in Section I.2. I.1.2 Elliptic Integral of the Second Kind

Z' E(k, ')
E('nα)
E(ujk) D Zx
D 0

1  k2 t2

1/2

(1  t 2 )1/2

1  k 2 sin2 θ

1/2

dθ

(I8)

0

dt

Zu

(I9) Zu

2

D




dn w dw D m 1 u C m 0

cn2 w dw ,

(I10)

0

where m C m1 D 1 ,

(I11)

cos ' D cnu ,

(I12)




1  k 2 sin2 '

1/2

D dnu D ∆(') ,

' D sin
1 (snu) D amu ,

delta amplitude ,

the amplitude .

(I13) (I14)

I.1.3 Elliptic Integral of the Third Kind

Z' Π (k, n, ')
Π (nI 'nα) D Zx D 0


0

dθ 
1/2 1  n sin θ 1  k 2 sin2 θ

dt (1  nt 2 ) [(1  t 2 ) (1  k 2 t 2 )]1/2

2

D Π (k, n, x) .

(I15) (I16)

I.2 Jacobian Elliptic Functions

I.1.4 Complete Elliptic Integrals

Arscott [73] defines complete elliptic integrals K, E, K 0 , E 0 as Zπ/2

dθ
1/2 D F(k, ' D π/2) , 2 1  k sin2 θ

(I17)


1/2 dθ 1  k 2 sin2 θ D E(k, ' D π/2) ,

(I18)

K(k) D 0 Zπ/2

E(k) D 0

K 0 D K(k 0 ) ,

(I19)

E 0 D E(k 0 ) ,

(I20)

where k is a parameter known as the modulus, and the complementary modulus k 0 D (1  k 2 )1/2 . k will be chosen real and 0  k  1 for now. For jkj < 1, the integrand in Eq. (I17) may be expanded in powers of k and is uniformly convergent with respect to θ . Thus, one can integrate term by term, giving   1 1 π (I21) , I 1I k 2 . KD F 2 2 2 I.1.4.1 Limiting Values For k ! 0,

K(0) D E(0) D π/2 ,

K0 D 1 ,

E0 D 1 .

(I22)

K 0 D E 0 D π/2 .

(I23)

For k ! 1, K(1) D 1 ,

E(0) D 1 ,

I.2 Jacobian Elliptic Functions

To introduce the elliptic functions, consider the following integral (which is actually the same as in Eq. (I3), that is, an elliptic integral of the first kind): Zy zD 0

dt . p 2 (1  t ) (1  k 2 t 2 )

(I24)

One can view y as a function of z and k: y D sn(z, k) .

(I25)

363

364

Appendix I Elliptic Integrals and Functions

We have thus defined the function sn. Indeed, by differentiating Eq. (I24), one obtains  2 dy D (1  y 2 )(1  k 2 y 2 ) , dz and the solution of this differential equation is y D y (z) , and which we call sn(z, k). If we now write the inverse of Eq. (I25),

sn


1

Zy yDzD

p 0

dt (1 

t 2 )(1

 k 2 t 2)

.

(I26)

Define elliptic functions sn(z, k) D snz, cn(z, k) D cnz, dn(z, k) D dnz, and am(z, k) D amz by Zsnz zD

p 0

(1  t 2 )(1  k 2 t 2 )

Z1 p

D dn z

Z1

dt

(1 

cn z

dt p (1  t 2 )(k 02 C k 2 t 2 )

Zamz

dt t 2 )(t 2

D



k 02 )

D 0

(1 

dθ . sin2 θ )1/2

k2

(I27)

I.2.1 Notation

The current notation sn, cn, and dn is due to Gudermann. Jacobi used sinam, cosam, ∆amu instead. For the reciprocals and quotients of Jacobian elliptic functions, one uses Glaisher’s notation. The reciprocals are written with the letters of the function reversed: nsz D

1 , snz

ncz D

1 , cnz

ndz D

1 . dnz

Quotients are written using the first letter of the two functions: sn z , cn z cn z cs z D , sn z

sc z D

sn z , dn z dn z ds z D , sn z sd z D

cn z , dn z dn z dc z D . cn z cd z D

Glaisher also wrote s, c, d D snz, cnz, dnz ,

S, C, D D sn2z, cn2z, dn2z .

I.2 Jacobian Elliptic Functions

I.2.2 Degeneracy I.2.2.1

I.2.2.2

k!0 sn(z, k) ! sin z ,

cn(z, k) ! cos z ,

k!1 sn(z, k) ! tanh z ,

cn(z, k) ,

dn(z, k) ! 1 .

dn(z, k) ! sechz .

(I28)

(I29)

I.2.3 Relations

sn z D sin am z ,

cn z D cos am z ,

dn z D

d am z , dz

(I30)

sn2 z C cn2 z D 1 ,

(I31)

k 2 sn2 z C dn2 z D 1 .

(I32)

I.2.4 Derivatives

d snz D cnzdnz , dz

d cnz D snzdnz , dz

d dnz D k 2 snzcnz . dz (I33)

I.2.5 Parity

sn(z) D snz ,

cn(z) D cnz ,

dn(z) D dnz .

(I34)

I.2.6 Addition Theorems I.2.6.1

snz

sn(u C v ) D

s 1 c 2 d2 C s 2 c 1 d1 snucnv dnv C snv cnudnu
. 2 2 2 1  k sn usn v 1  k 2 s 21 s 22

The latter notation is due to Glaisher. I.2.6.2 cnz We now obtain the addition formula for cn by using the one for sn:
2 2 
 1  k 2 s 21 s 22 cn2 (u C v ) D 1  k 2 s 21 s 22 1  sn2 (u C v ) 2
D 1  k 2 s 21 s 22  (s 1 c 2 d2 C s 2 c 1 d1 )2

D (c 1 c 2  s 1 s 2 d1 d2 )2 ,

(I35)

365

366

Appendix I Elliptic Integrals and Functions

giving cn(u C v ) D ˙

c 1 c 2  s 1 s 2 d1 d2 . 1  k 2 s 21 s 22

The expansions on both sides are single valued and from analytic continuation must be well defined. With u D 0, we find +1. Hence, cn(u C v ) D I.2.6.3 dnz

dn(u C v ) D

c 1 c 2  s 1 s 2 d1 d2 . 1  k 2 s 21 s 22

(I36)

d1 d2  k 2 s 1 s 2 c 1 c 2 . 1  k 2 s 21 s 22

(I37)

I.2.7 K0

K0 D

Zπ/2 0

dθ D (1  k 2 sin2 θ )1/2

Z1

π dt D F [(1  t 2 )(1  k 02 t 2 )]1/2 2



1 1 , I 1I k 02 2 2

 .

0

(I38) The c 0
k 02 plane is cut from 1 to 1 (i.e., c plane from 0 to 1). I.2.8 Special Values


sn K C i K 0 D 1/ k , 
cn K C i K 0 D i k 0/ k ,

sn 0 D 0 ,

sn K D 1 ,

cn 0 D 1 ,

cn K D 0 ,

dn 0 D 1 ,

dn K D k 0 ,

(I39) (I40)

dn (K C i K 0 ) D 0 .

(I41)

sn (z C 2i K 0 ) D sn z ,

(I42)

I.2.9 Period

sn (z C 2K ) D sn z ,

0

cn (z C 2i K ) D cn z ,

cn (z C 2K ) D cn z , dn (z C 2K ) D dn z ,

dn (z C 2i K 0 ) D dn z .

I.2.10 Behavior near the Origin and i K 0

One can analyze the behavior by performing a Maclaurin expansion.

(I43) (I44)

I.2 Jacobian Elliptic Functions

I.2.10.1

Near the Origin d sn z D cn z dn z, dz 2 d sn z D k 2 sn z cn2 z  sn z dn2 z, dz 2   d3 sn z D 4k 2 sn2 z cn z dn z  cn z dn z dn2 z C k 2 cn2 z , 3 dz d z 3 d3 H) sn z D z sn z C sn z C    dz 6 dz 3 1 D z  (1 C k 2 )z 3 C O(z 5 ) . 6

Similarly, 1 2 z C O(z 4 ) , 1 1 dn z D 1  k 2 z 2 C O(z 4 ) . 2 cn z D 1 

I.2.10.2

Near i K 0




1 1 1 1 2 2 1  (1 C k )z C    sn (z C i K ) D ns z D k kz 6 2 1Ck 1 C z C O(z 3 ) , D kz 6k 2k 2  1 i C i z C O(z 3 ) , cn (z C i K 0 ) D  kz 6k 2  k2 i i z C O(z 3 ) . dn (z C i K 0 ) D  C z 6 0

367

369

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References 103 Lipshutz, M. (1969) Theory and Problems of Differential Geometry, McGraw-Hill, New York. 104 Todhunter, I. (1875) An Elementary Treatise on Laplace’s Functions, Lamé’s Functions, and Bessel’s Functions, MacMillan, London.

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Index a acoustics 33 Airy equation 290, 303 Airy function 290, 303 analytic function 13 applications – acoustics 116 – electrostatics 46, 87, 187, 227 – gravitation 146 – heat conduction 75, 96, 172 – hydrodynamics 106 – quantum mechanics 47, 56, 66, 97, 106, 130, 147, 173, 210 associated Laguerre equation 163, 300 associated Laguerre function 300 associated Legendre equation 128–130, 143, 159, 333, 352 associated Legendre function 130, 133, 148, 334 – first kind 128, 352 – second kind 128 associated Legendre polynomial 335 asymptotic expansion 162 b Baer equation 221, 305, 339 Baer function 305 Baer wave equation 227, 306 Baer wave function 306 Basset function 314 Ben Daniel–Duke model 36 Bessel equation 21, 54, 94–95, 129– 130, 169, 185, 268–269, 290, 309 – modified 57 Bessel function 97, 169, 268–269, 287 – first kind 54, 56, 94 – second kind 54 – third kind 94, 313 Bessel wave equation 74, 170, 316

Bessel wave function 74, 170, 316 – second kind 317 binormal 242 Bocher equation 21 boundary-value problem 26 branch point 21 c characteristic curve 104 characteristic equation 65, 340, 343 Christoffel symbol – second kind 235 circular coordinates 15, 51 circular cylinder coordinates 91 circular cylindrical coordinates 9, 18, 26 circulation 9 confluent hypergeometric equation 298 confluent hypergeometric function 114, 174, 298, 358 conformal transformation 12 conical coordinates 181 continued fraction 148, 355 contravariant component 236 covariant component 236 cross product 239 curl 239 curvature 242 curvilinear coordinates 8 d differential geometry 233 differential operators 8 diffusion equation 8, 31 Dirichlet boundary condition 26 Dirichlet–Neumann boundary condition 26 divergence 9, 236, 239 dual basis 233 e eigenvalue 28

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Index electric field 30, 289 electrostatic potential 30 electrostatics 7, 30 ellipsoidal coordinates 191 – algebraic representation 194 ellipsoidal harmonics 201 elliptic coordinates 15, 61 elliptic cylinder coordinates 99 elliptic cylinder function 103, 339 elliptic function 198 – Jacobian 363 elliptic functions 203 elliptic integral – complete 363 – first kind 362–363 – second kind 362 – third kind 362 elliptic paraboloid 218 essential singularity 21 Euler constant 311 Euler equation 185 f Fourier–Baer series 307 Fourier expansion 28 Fourier series 47, 88, 343 Frenet–Serret equations 242 Frenet–Serret frame 243 Frobenius method 22, 116 Fuch’s theorem 22 g gamma function 134, 310 Gauss equation 295 Gaussian curvature 271 Gaussian normal coordinates 272 Gauss’s law 30 gradient 9, 236, 239 gravitation 7, 32 Green’s second identity 284 h Hankel function – first kind 313 – second kind 313 hard-wall boundary condition 36 hard-wall potential 47 harmonic solution 29 heat conduction 31 Helmholtz equation 10 Hermite equation 301 Hermite polynomial 301, 359 Humbert symbol 298 hydrodynamics 33

hypergeometric equation 26, 295 hypergeometric function 133, 296, 331, 333 i indicial equation 23 isometry 246 j Jacobian elliptic function 363 k Kummer equation 298 l Laguerre equation 300 Laguerre polynomial 300, 302 Lamé equation 185, 198, 321 Lamé function 322 – first kind 185, 198, 322 – second kind 185, 198, 326 Lamé wave equation 199, 326 Lamé wave function – first kind 199 – second kind 199 Laplace–Beltrami operator 239, 273 Laplace equation 10, 30 Laplacian 9, 237, 239 Laurent series 20 Legendre equation 329 Legendre function 333 – first kind 330 – second kind 330 Legendre polynomial 330 – generating function 331 – hypergeometric function 331 – Rodrigue’s formula 331 – Schaefli integral representation 331 Legendre series 145, 332 Legendre transform 332 line element 8 m MacDonald function 55, 57, 94, 314 Mathieu equation 65, 103–104, 107, 339 Mathieu function 103, 339 – first kind 65, 103, 340 – second kind 340 Maxwell equations 30 mean curvature 271 metric 8, 234 minimal rotating frame 243 mixed boundary condition 28 modified Bessel equation 55, 94, 169, 309

Index modified Bessel function 303, 314 – first kind 55, 57, 94, 314 – third kind 55, 57, 94, 314 modified Mathieu equation 65, 103–104 modified spherical Bessel equation 310 modified spherical Bessel function 131, 316 – first kind 316 – second kind 152, 316

r R-separability 10, 20 rectangular coordinates 11, 41, 81 recurrence relation 23 regular singular point 20 Riemann equation 25 Riemann P symbol 25 Robertson condition 16

n Navier–Stokes equation 33 Neumann boundary condition 26 Neumann function 54, 94, 169, 311 Neumann wave function 170

s scale factor 8 Schrödinger equation 11, 19 second fundamental form 271 separation constant 12, 29 series solution 20 simply separable 10 singular point 20 spherical Bessel equation 309 spherical Bessel function 130, 149, 315 – first kind 315 – second kind 315 spherical Hankel function 315 spherical harmonics 336 spherical polar coordinates 125 spheroconical coordinates 187 spheroidal wave equation 143, 159, 351 spheroidal wave function 143, 160, 352 Stark effect 289 Sturm–Liouville theory 28 Stäckel determinant 16 Stäckel matrix 16

o oblate spheroid 156 oblate spheroidal coordinates 155 ordinary point 20 orthogonality – Baer wave function 306 – Bessel wave function 318 – Legendre polynomial 332 – Mathieu function 341 orthogonality relation 29 p P-separability 20 Papperitz equation 25 parabolic coordinates 14 parabolic cylinder coordinates 109 parabolic cylinder equation 301, 357 parabolic cylinder function 357 parabolic rotational coordinates 165 paraboloidal coordinates 217 photonics 30 Pochhammer symbol 296 point at infinity 21 Poisson equation 30 polar coordinates 15 pole 20 potential theory 29 principal normal 242 prolate angular function 352 prolate spheroid 140 prolate spheroidal angle function – first kind 352 – second kind 352 prolate spheroidal coordinates 139 q quadric surface 192 quantum mechanics 35 quantum numbers 48

t third fundamental form 271 torsion 242 u uniqueness theorem 29 unit vector 9 v vector field 235, 238 vector Helmholtz equation 11 vector wave equation 31 w wave equation 7 Weber equation 74, 113–114, 357 Weber function 114, 302, 358 Weber–Hermite function 357 weighting factor 28 Whittaker equation 174, 299 Whittaker function 174, 299

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