Table Of ContentUndergraduate Lecture Notes in Physics
Giampaolo Cicogna
Exercises and
Problems in
Mathematical
Methods of Physics
Undergraduate Lecture Notes in Physics
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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)
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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
