Exercises and Problems in Mathematical Methods of Physics PDF
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Undergraduate Lecture Notes in Physics Giampaolo Cicogna Exercises and Problems in Mathematical Methods of Physics Undergraduate Lecture Notes in Physics Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topicsthroughoutpureandappliedphysics.Eachtitleintheseriesissuitableasabasisfor undergraduateinstruction,typicallycontainingpracticeproblems,workedexamples,chapter summaries, andsuggestions for further reading. ULNP titles mustprovide at least oneof thefollowing: (cid:129) Anexceptionally clear andconcise treatment ofastandard undergraduate subject. (cid:129) Asolidundergraduate-levelintroductiontoagraduate,advanced,ornon-standardsubject. (cid:129) Anovel perspective oranunusual approach toteaching asubject. ULNPespeciallyencouragesnew,original,andidiosyncraticapproachestophysicsteaching at theundergraduate level. ThepurposeofULNPistoprovideintriguing,absorbingbooksthatwillcontinuetobethe reader’spreferred reference throughout theiracademic career. Series editors Neil Ashby University of Colorado, Boulder, CO, USA William Brantley Department of Physics, Furman University, Greenville, SC, USA Matthew Deady Physics Program, Bard College, Annandale-on-Hudson, NY, USA Michael Fowler Department of Physics, University of Virginia, Charlottesville, VA, USA Morten Hjorth-Jensen Department of Physics, University of Oslo, Oslo, Norway Michael Inglis Department of Physical Sciences, SUNY Suffolk County Community College, Selden, NY, USA More information about this series at http://www.springer.com/series/8917 Giampaolo Cicogna Exercises and Problems in Mathematical Methods of Physics 123 GiampaoloCicogna Dipartimento di Fisica “EnricoFermi” Universitàdi Pisa Pisa Italy ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate Lecture Notesin Physics ISBN978-3-319-76164-0 ISBN978-3-319-76165-7 (eBook) https://doi.org/10.1007/978-3-319-76165-7 LibraryofCongressControlNumber:2018932991 ©SpringerInternationalPublishingAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,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. 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Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAG partofSpringerNature Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This book is a collection of 350 exercises and problems in Mathematical Methods of Physics: its peculiarity is that exercises and problems are proposed not in a “random” order, but having in mind a precise didactic scope. Each section and subsection starts with exercises based on first definitions, elementary notions and properties, followed by a group of problems devoted to some intermediate situa- tions, and finally by problems which propose gradually more elaborate develop- ments and require some more refined reasoning. Part of the problems is unavoidably “routine”, but several problems point out interesting nontrivial properties, which are often omitted or only marginally men- tionedinthetextbooks.Therearealsosomeproblemsinwhichthereaderisguided to obtain some important results which are usually stated in textbooks without completeproofs:forinstance,theclassical“uncertaintyprinciple”Dt Dx(cid:1)1=2,an introduction to Kramers–Kronig dispersion rules and their relation with causality principles, the symmetry properties of the hydrogen atom, and the harmonic oscillator in Quantum Mechanics. Inthissense,thisbookmaybeusedas(orperhaps,tosomeextent,betterthan)a textbook. Avoiding unnecessary difficulties and excessive formalism, it offers indeedanalternativewaytounderstandthemathematicalnotionsonwhichPhysics is based, proceeding in a carefully structured sequence of exercises and problems. Ibelievethatthereisnoneedtoemphasizethatthebest(orperhapstheunique) waytounderstandcorrectlyMathematicsisthatoffacingandsolvingexercisesand problems.Thisholdsafortioriforthepresentcase,wheremathematicalnotionsand procedures become a fundamental tool for Physics. An example can illustrate perfectly the point. The definition of eigenvectors and eigenvalues of a linear operator needs just two or three lines in a textbook, and the notion is relatively simple and intuitive. But only when one tries to find explicitly eigenvectors and eigenvaluesinconcretecases,thenonerealizesthatalotofdifferentproceduresare requiredandextremelyvarioussituationsoccur.Thisbookoffersafairlyexhaustive description of possible cases. v vi Preface This book covers a wide range of topics useful to Physics: Chap. 1 deals with Hilbert spaces and linear operators. Starting from the crucial concept of complete systemofvectors,manyexercisesaredevotedtothefundamentaltoolprovidedby Fourierexpansions,withseveralexamplesandapplications,includingsometypical Dirichlet and Neumann Problems. The second part of the chapter is devoted to studying the different properties of linear operators between Hilbert spaces: their domains, ranges, norms, boundedness, and closedness, and to examining special classes of operators: adjoint and self-adjoint operators, projections, isometric and unitary operators, functionals,andtime-evolutionoperators. Great attentionispaid to the notion of eigenvalues and eigenvectors, with the various procedures and results encountered in their determination. Another frequently raised question concerns the different notions of convergence of given sequences of operators. Chapter 2 starts with a survey of the basic properties of analytic functions of a complex variable, of their power series expansions (Taylor–Laurent series), and oftheirsingularities,includingbranchpointsandcutlines.Theevaluationofmany types of integrals by complex variable methods is proposed. Some examples of conformal mappings are finally studied, in order to solve Dirichlet Problems; the results are compared with those obtained in other chapters with different methods, with a discussion about the uniqueness of the solutions. The problems in Chap. 3 concern Fourier and Laplace transforms with their differentapplications.ThephysicalmeaningoftheFouriertransformas“frequency analysis” is carefully presented. The Fourier transform is extended to the space of tempereddistributionsS0,whichincludetheDiracdelta,theCauchyprincipalpart, and other related distributions. Applications concern ordinary and partial differen- tialequations(inparticulartheheat,d’Alembert,andLaplaceequations,includinga discussion about the uniqueness of solutions), and general linear systems. The importantnotionofGreenfunctionisconsideredinmanydetails,togetherwiththe notion of causality. Various examples and applications of Laplace transform are proposed, also in comparison with Fourier transform. ThefirstproblemsinChap.4deal with basic propertiesofgroups andofgroup representations. Fundamental results following from Schur lemma are introduced sincethebeginninginthecaseoffinitegroups,withasimpleapplicationofcharacter theory, in the study of vibrational levels of symmetric systems. Other problems concernthenotionandthepropertiesofLiegroupsandLiealgebras,mainlyoriented tophysicalexamples:rotationgroupsSO ;SO ;SU ,translations,Euclideangroup, 2 3 2 Lorentztransformations,dilations,Heisenberggroup,andSU ,withtheirphysically 3 relevantrepresentations.Thelastsectionstartswithsomeexamplesandapplications of symmetry properties of differential equations, and then provides a group- theoretical description of some problems in quantum mechanics: the Zeeman and Stark effects, the Schrödinger equation of thehydrogen atom (the group SO ), and 4 thethree-dimensional harmonic oscillator (the group U ). 3 Preface vii At the end of the book, there are the solutions to almost all problems. In par- ticular, there is a complete solution of the more significant or difficult problems. This book is the result of my lectures during several decades at the Department ofPhysicsoftheUniversityofPisa.Iwouldliketoacknowledgeallmycolleagues whohelpedmeintheorganizationofthedidacticactivity,inthepreparationofthe problemsandfortheirassistanceintheexaminationsofmystudents.Specialthanks areduetoProf.GiovanniMorchio,forhisconstantinvaluablesupport:manyofthe problems, specially in Sect. 2 of Chap. 1, have been written with his precious collaboration. I am also grateful to Prof. Giuseppe Gaeta for his encouragement to write this book, which follows my previous lecture notes (in Italian) Metodi Matematici della Fisica, published by Springer-Verlag Italia in 2008 (second edi- tion in 2015). Finally,Iwouldthankinadvancethereadersfortheircomments,andinparticular those readers who will suggest improvements and amendments to all possible misprints,inaccuracies,andinadvertentmistakes(hopefully,nottooserious)inthis book, including also errorsand imperfections in myEnglish. Pisa, Italy Giampaolo Cicogna January 2018 Contents 1 Hilbert Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Complete Sets, Fourier Expansions . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Preliminary Notions, Subspaces, and Complete Sets . . . . . 1 1.1.2 Fourier Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 Harmonic Functions: Dirichlet and Neumann Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2 Linear Operators in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.1 Linear Operators Defined Giving Ten ¼ vn, and Related Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.2 Operators oPf the Form Tx¼vðw;xÞ and Tx¼ nVnðwn;xÞ . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.2.3 Operators of the Form TfðxÞ ¼ uðxÞfðxÞ . . . . . . . . . . . . 31 1.2.4 Problems Involving Differential Operators. . . . . . . . . . . . . 33 1.2.5 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.2.6 Time-Evolution Problems: Heat Equation . . . . . . . . . . . . . 43 1.2.7 Miscellaneous Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 48 2 Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.1 Basic Properties of Analytic Functions. . . . . . . . . . . . . . . . . . . . . 57 2.2 Evaluation of Integrals by Complex Variable Methods . . . . . . . . . 62 2.3 Harmonic Functions and Conformal Mappings. . . . . . . . . . . . . . . 69 3 Fourier and Laplace Transforms. Distributions . . . . . . . . . . . . . . . . 73 3.1 Fourier Transform in L1ðRÞ and L2ðRÞ. . . . . . . . . . . . . . . . . . . . . 73 3.1.1 Basic Properties and Applications. . . . . . . . . . . . . . . . . . . 75 3.1.2 Fourier Transform and Linear Operators in L2ðRÞ . . . . . . . 82 3.2 Tempered Distributions and Fourier Transforms. . . . . . . . . . . . . . 84 3.2.1 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.2.2 Fourier Transform, Distributions, and Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 ix x Contents 3.2.3 Applications to ODEs and Related Green Functions . . . . . 102 3.2.4 Applications to General Linear Systems and Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.2.5 Applications to PDE’s . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.3 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4 Groups, Lie Algebras, Symmetries in Physics. . . . . . . . . . . . . . . . . . 127 4.1 Basic Properties of Groups and of Group Representations. . . . . . . 127 4.2 Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.3 The Groups SO ; SU ; SU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3 2 3 4.4 Other Relevant Applications of Symmetries to Physics. . . . . . . . . 138 Answers and Solutions .. .... ..... .... .... .... .... .... ..... .... 145 Bibliography .. .... .... .... ..... .... .... .... .... .... ..... .... 181
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