V. Balakrishnan Mathematical Physics Applications and Problems Mathematical Physics V. Balakrishnan Mathematical Physics Applications and Problems 123 V.Balakrishnan Department ofPhysics Indian Institute of Technology Madras Chennai, India ISBN978-3-030-39679-4 ISBN978-3-030-39680-0 (eBook) https://doi.org/10.1007/978-3-030-39680-0 JointlypublishedwithANEBooksIndia TheprinteditionisnotforsaleinSouthAsia(India,Pakistan,SriLanka,Bangladesh,NepalandBhutan) andAfrica.CustomersfromSouthAsiaandAfricacanpleaseordertheprintbookfrom:ANEBooks India. ISBNoftheCo-Publisher’sedition:9789386761118 ©Authors2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublishers,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublishers,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthis book are believed to be true and accurate at the date of publication. Neither the publishers nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To Radha Preface “Thebookofnatureiswritteninthelanguageofmathematics.”Thisistheessence oftheprofoundobservationmadebyGalileoGalileiasfarbackas1623.Evidence insupportofthisinsighthasaccumulatedeversince.Muchhasbeenwrittenabout what has been termed “the unreasonable effectiveness of mathematics” in the descriptionofphysicalphenomena.Ismathematicsinherentinnatureitself,orisita construct of the human mind? This deep question has also been debated intensely among mathematicians, physicists and philosophers of science. Whatever be the answer, it is undeniably true that mathematical structures seem to be embedded deeply in the physical universe. After approximately four hundred years of continuous development, physics is undoubtedlythemost‘mathematized’ofthesciences.Physicsattemptstodescribe nature in precise and logical terms, and it requires a language that has logic built intoit.AsRichardFeynmanputit,“Mathematicsislanguagepluslogic”.Acertain degree offacility in mathematics is therefore not only helpful, but also absolutely necessary, in order to really understand physics and to appreciate its concepts and laws even at an elementary level. But what kind of mathematics does physics entail? Both physics and mathe- matics are very vast domains. The natural question that arises in the mind of a student beginning the study of physics is, “Exactly how much mathematics do I need to study and to understand physics?” There can be no definite or complete answer to this question, because it depends on the level at which one wishes to understand the laws of physics and the structure of the physical universe. The problemsthatphysicsaddressesrequiretheapplicationofmathematicsatalllevels. These range from elementary algebra right up to some of the most advanced state-of-the-art developments in mathematical research. Along the way, there are remarkable instances of an almost uncanny match between a physical context and the specific kind of mathematics needed in that context. The first such pairing was between the dynamics of motion and calculus. Indeed, calculus was developed for that very purpose. Subsequent instances are: electromagnetismandvectorcalculus;generalrelativityandRiemanniangeometry; quantum mechanics and linear vector spaces; symmetries in (condensed matter, vii viii Preface atomic, nuclear, and subnuclear) physics and group theory; Hamiltonian dynamics andsymplecticgeometry;andsoon.Ineveryoneofthesecases,themathematical structure involved seems to have been tailor-made for the physical problem concerned. What is mathematical physics? The fact that physics requires mathematics at all levels makes the very definition of mathematical physics as a subject in the university physics curriculum rather fuzzy. Over the years, however, there has emerged a set of mathematical topics and techniques that are the most useful and widelyapplicableonesinvariouspartsofphysics.Itisthisrepertoireorcollection that constitutes ‘mathematical physics’ as the term is generally understood in its pedagogicalsense.Thisbookhaschaptersdevotedtomostofthetopicsofthiscore set, and considerably more, besides. Further, I have taken the phrase ‘mathematical physics’ literally. As a conse- quence,thisbookisnotanappliedmathematicstextintheconventionalsense.Asa glance at the table of contents will show, it digresses into physics whenever the opportunitypresentsitself.Althoughnumerousmathematicalresultsareintroduced and discussed, hardly any formal, rigorous proofs of theorems are presented. Instead, I have used specific examples and physical applications to illustrate and elaborate upon these results. The aim is to demonstrate how mathematics inter- twines with physics in numerous instances. In my opinion, this is the fundamental justification for the very inclusion of mathematical physics as a subject in the physics curriculum. Towhomisthisbookaddressed?Itismybeliefandhopethatappropriateparts ofthebookwillserveawidespectrumofstudents,rangingfromtheundergraduate right up to the doctoral level. More than one route map can be drawn to navigate through the chapters to form courses at different levels and of different durations. I have not done so because I believe this choice is best left to the user. Likewise, fairly self-contained sets of chapters can be selected to provide short courses of study on specific topics in mathematical physics. Here are some examples of the possibilities in this regard: Vector calculus and applications, Chaps. 5–9. Linear vector spaces, matrices and operators, Chaps. 10–15. Probability, statistics and random processes, Chaps. 19–21. Complex analysis, Chaps. 22–27. Special functions, Chaps. 16, 25, 26. Basic partial differential equations of physics, Chaps. 29–32. The chapters listed in each case do not stand in complete isolation from the rest ofthebook,ofcourse.Inframingshortspecific-topiccourses,itwouldnaturallybe helpful to include appropriate sections from other chapters, as needed. Exercisesandproblemscompriseanindispensablecomponentofanybookon mathematicalmethods,andthisbookisnoexception.Thereare370oftheseinthis book,manyofthemwithseveralpartsandsubparts.Most(butnotall)ofthemare problems, rather than exercises of the drillwork type. They form an integral part Preface ix of the text. In many cases, they require the reader to complete, or verify, or work out, or extend the details described in the text, in order to acquire a better under- standing of the subject matter. In other cases, they explore sidelights and inter- connections between different aspects. For this reason, I have made the problems contiguous with the text, indicating the beginning of each one with the symbol H. Solutions: Student readers are best served if a book containing problems also providessolutions.Attheendofeachchapter,solutionstotheproblemsthereinare given either in outline orin detail,except in those cases inwhich thesolutions are obviously straightforward extensions of the text. The end of each solution is indicatedbythesymbolI.Ireiteratethatworkingouttheproblemsineachchapter isofparamountimportance.Onlyafteranhonestattempthasbeenmadeshouldthe reader consult the solutions provided. Thetableofcontentslistsnotonlythechaptersandsectionsofeachchapter,as is customary, but also the subsections, which is somewhat less common. This has beendoneinorder toprovide thereader with aconvenientlydetailed list ofallthe topicsdiscussedinthebook,avoidingtheneedtohuntfortheseinthebodyofthe text. Together with the Index, the table of contents should make it easy for you to navigate from place to place, back and forth, through this rather lengthy book. Theindexattheendofthebookrunstothirteenpages.Ithasintentionallybeen made rather extensive, because I believe that the reader should be able to refer quickly to any topic or theme, and the different contexts in which it occurs in the book, based on just a keyword or phrase. Cross-Referencing: As one might expect, numerous topics, themes and equa- tions appear more than once in the book. I have tried as far possible to cross-reference these with chapter, section and equation number, so that recall becomes easier. In a few necessary instances, equations have been repeated for ready reference. Above all, it has been my intention to make this book as comprehensive, self-contained and amenable to self-study as possible. Naturally, there are many omissions in the book. Several important topics that I should like to have touched upon have had to be left out. Examples that stand out include the calculus of variations,functionalintegration,theelementsofdifferentialgeometry,andamore systematic account of group theory in physics. But the need to keep the length of the book within a manageable limit necessitated these omissions. Finally, I must point out that all the mathematics involved in this book is ‘classical’. By and large, more modern and/or abstract parts of mathematics have not asyet become part of the standard repertoire referred to earlier, although some areas such as topology and differential geometry increasingly find application in various parts of physics such as quantum field theory, general relativity and con- densed matter physics. This book has been several years in the writing, having grown out of various coursesandsetsoflecturesgivenbymeoveraconsiderablenumberofyears.Iowe adebtofgratitudetoallthestudentswhoattendedthelectures,askedquestionsthat set me thinking, and enabled me get a better understanding of the subject matter. x Preface I thank Suresh Govindarajan for his generous help and valuable assistance, and Ashok Velayutham for drawing all the figures in the book. As always, my wife Radhahasbeenapillarofsupportandencouragement,andherassistancehasbeen invaluable. I dedicate this book to her with affection. Chennai, India V. Balakrishnan Contents 1 Warming Up: Functions of a Real Variable . . . . . . . . . . . . . . . . . . 1 1.1 Sketching Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Features of Interest in a Function . . . . . . . . . . . . . . . 1 1.1.2 Powers of x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.3 A Family of Ovals . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.4 A Family of Spirals . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Maps of the Unit Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Gaussian Integrals, Stirling’s Formula, and Some Integrals . . . . . . 9 2.1 Gaussian Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 The Basic Gaussian Integral . . . . . . . . . . . . . . . . . . . 9 2.1.2 A Couple of Higher Dimensional Examples. . . . . . . . 10 2.2 Stirling’s Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 The Dirichlet Integral and Its Descendants . . . . . . . . . . . . . . . . 13 2.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Some More Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 Functions Represented by Integrals . . . . . . . . . . . . . . . . . . . . . 19 3.1.1 Differentiation Under the Integral Sign . . . . . . . . . . . 20 3.1.2 The Error Function. . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1.3 Fresnel Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.4 The Gamma Function. . . . . . . . . . . . . . . . . . . . . . . . 22 3.1.5 Connection to Gaussian Integrals. . . . . . . . . . . . . . . . 23 3.2 Interchange of the Order of Integration . . . . . . . . . . . . . . . . . . 25 3.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Generalized Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1 The Step Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 The Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.1 Defining Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.2 Sequences of Functions Tending to the d-Function. . . 33 xi