Spherical Harmonics in p Dimensions PDF
Preview Spherical Harmonics in p Dimensions
Spherical Harmonics in p Dimensions 9134_9789814596695_tp.indd 1 21/1/14 11:29 am May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Spherical Harmonics in p Dimensions Costas Efthimiou University of Central Florida, USA Christopher Frye Harvard University, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 9134_9789814596695_tp.indd 2 21/1/14 11:29 am Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Efthimiou, Costas, author. Spherical harmonics in p dimensions / by Costas Efthimiou (University of Central Florida, USA) & Christopher Frye (Harvard University, USA). pages cm Includes bibliographical references and index. ISBN 978-9814596695 (hardcover : alk. paper) 1. Spherical harmonics. 2. Spherical functions. 3. Legendre's polynomials. 4. Mathematical physics. I. Frye, Christopher, author. II. Title. QC20.7.S645E38 2014 515'.785--dc23 2014000203 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Cover image: 3 spherical balls on cover: Adapted from ‘Spherical harmonics in 3D’, Wikipedia Commons. This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore December12,2013 WorldScientificBook-9inx6in MAIN-SphHarm pagev Contents Preface vii Acknowledgements ix List of Symbols xi 1 Introduction and Motivation 1 1.1 Separation of Variables . . . . . . . . . . . . . . . . . . 2 1.2 Quantum Mechanical Angular Momentum . . . . . . . 9 2 Working in p Dimensions 13 2.1 Rotations in Ep . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Spherical Coordinates in p Dimensions . . . . . . . . . 15 2.3 The Sphere in Higher Dimensions . . . . . . . . . . . . 20 2.4 Arc Length in Spherical Coordinates . . . . . . . . . . 24 2.5 The Divergence Theorem in Ep . . . . . . . . . . . . . 26 2.6 ∆ in Spherical Coordinates . . . . . . . . . . . . . . . 30 p 2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Orthogonal Polynomials 39 3.1 Orthogonality and Expansions . . . . . . . . . . . . . 39 3.2 The Recurrence Formula . . . . . . . . . . . . . . . . . 42 3.3 The Rodrigues Formula . . . . . . . . . . . . . . . . . 45 3.4 Approximations by Polynomials . . . . . . . . . . . . . 48 3.5 Hilbert Space and Completeness . . . . . . . . . . . . 57 3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 62 v December12,2013 WorldScientificBook-9inx6in MAIN-SphHarm pagevi vi CONTENTS ................................................................... 4 Spherical Harmonics in p Dimensions 63 4.1 Harmonic Homogeneous Polynomials . . . . . . . . . . 63 4.2 Spherical Harmonics and Orthogonality . . . . . . . . 71 4.3 Legendre Polynomials . . . . . . . . . . . . . . . . . . 78 4.4 Boundary Value Problems . . . . . . . . . . . . . . . . 105 4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 116 5 Solutions to Problems 119 5.1 Solutions to Problems of Chapter 2 . . . . . . . . . . . 119 5.2 Solutions to Problems of Chapter 3 . . . . . . . . . . . 127 5.3 Solutions to Problems of Chapter 4 . . . . . . . . . . . 131 Bibliography 139 Index 141 December12,2013 WorldScientificBook-9inx6in MAIN-SphHarm pagevii Preface We prepared the following book in order to make several useful top- ics from the theory of special functions, in particular the spherical harmonics and Legendre polynomials of Rp, available to undergrad- uates studying physics or mathematics. With this audience in mind, nearly all details of the calculations and proofs are written out, and extensive background material is covered before beginning the main subject matter. The reader is assumed to have knowledge of multi- variable calculus and linear algebra (especially inner product spaces) as well as some level of comfort with reading proofs. Literature in this area is scant, and for the undergraduate it is virtually nonexistent. To find the development of the spherical har- monics that arise in R3, physics students can look in almost any text on mathematical methods, electrodynamics, or quantum me- chanics (see [3], [5], [10], [12], [17], for example), and math students can search any book on boundary value problems, PDEs, or special functions (see [7], [20], for example). However, the undergraduate will have a very difficult time finding accessible material on the cor- responding topics in arbitrary Rp. WehavebeengreatlyinfluencedbyHochstadt’sThe Functions of MathematicalPhysics [9]. Whenthisbookwaspreparedandreleased as notes, Hochstadt’s book was out of print. Fortunately, Dover reprinted it in the Spring of 2012. However, this book contains a more expanded and detailed point of view than in Hochstadt’s book, as well additional information and a chapter with the solutions to all problems. There are several additional references (see [2], [6], [13], [21], [22], for example) where the reader can search for information vii December12,2013 WorldScientificBook-9inx6in MAIN-SphHarm pageviii viii PREFACE ................................................................... on the topic of this book, but either the coverage is brief or the level of difficulty is considerably higher. In addition, the point of view is very different from the one adopted in this work. Therefore, we hope thatthecurrentbookwillbecomeausefulsupplementforanyreader interested in special functions and mathematical physics, especially students who learn the topic. December12,2013 WorldScientificBook-9inx6in MAIN-SphHarm pageix Acknowledgements Christopher Frye worked on this book as an undergraduate and is very grateful to Costas Efthimiou for his guidance and assistance during this project. Chris would also like to thank Professors Maxim ZinchenkoandAlexanderKatsevichfortheirvaluablecommentsand suggestions on the presentation of the ideas. In addition, he thanks his friends Jie Liang, Byron Conley, and Brent Perreault for their feedback after proofreading portions of this manuscript. He is also grateful to Kyle Anderson for assistance with one of the figures. ThisworkhasbeensupportedinpartbyaNationalScienceFoun- dation Award DUE 0963146 as well as UCF SMART and RAMP Awards. ChrisFryeisgratefultotheBurnettHonorsCollege(BHC), the Office of Undergraduate Research (OUR), and the Research and MentoringProgram(RAMP)atUCFfortheirgeneroussupportdur- ing his studies. He would like to thank especially Alvin Wang, Dean ofBHC;MartinDupuis, AssistantDeanofBHC;KimSchneider, Di- rector of OUR; and Michael Aldarondo-Jeffries, Director of RAMP. Last but not least, he thanks Paul Steidle for providing him a quiet office to work in while writing. ix
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