Mathematical tools for physicists

686 Pages·2005·4.817 MB·English

by  George L. Trigg

MathematicalToolsforPhysicists
Editedby
GeorgeL.Trigg
Mathematical Tools for Physicists. Edited by George L. Trigg
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40548-8
Mathematical Tools for Physicists
Edited by
George L. Trigg
Editor: (cid:1) AllbookspublishedbyWiley-VCHare
carefullyproduced.Nevertheless,authors,
Dr.GeorgeL.Trigg editors,andpublisherdonotwarrantthe
NewPaltz,NewYork,USA informationcontainedinthesebooks,
includingthisbook,tobefreeoferrors.
Readersareadvisedtokeepinmindthat
EditorialAdvisor: statements,data,illustrations,procedural
detailsorotheritemsmayinadvertentlybe
inaccurate.
ProfessorStuartP.Smith
CaliforniaStateUniversity,Hayward(CSUH), LibraryofCongressCardNo.:appliedfor
USA
BritishLibraryCataloguing-in-Publication
Data:Acataloguerecordforthisbookis
availablefromtheBritishLibrary.
ThiseditionisbasedonReviewArticles
originallywrittenforthe‘‘Encyclopediaof Bibliographicinformationpublishedby
AppliedPhysics’’,editedbyGeorgeL.Trigg,25 DieDeutscheBibliothek
Volumes,VCHPublishers/Wiley-VCH,New DieDeutscheBibliothekliststhispublication
York/Weinheim,1991–2000. intheDeutscheNationalbibliografie;detailed
bibliographicdataisavailableintheinternet
athttp://dnb.ddb.de.
WILEY-VCHVerlagGmbH&Co.KGaA
Weinheim,2005
CoverPicture Allrightsreserved(includingthoseof
translationintootherlanguages).Nopartof
Soundpressureplotoftheacousticwavesin thisbookmaybereproducedinany
wateraroundanaluminumcylinder.Arrowand form–nortransmittedortranslatedinto
deformationplotshowsthedeformationofthe machinelanguagewithoutwritten
cylinder.Modelcomputedbyusingthe permissionfromthepublishers.Registered
FEMLABStructuralMechanicsModule. names,trademarks,etc.usedinthisbook,
COMSOLAB,Stockholm,Sweden.Printedwith evenwhennotspecificallymarkedassuch
kindpermissionofCOMSOLAB, arenottobeconsideredunprotectedbylaw.
www.comsol.com.
PrintedintheFederalRepublicofGermany.
Printedonacid-freepaper.
Composition:LaserwordsPrivateLtd.,
Chennai,India
Printing:betz-druckgmbh,Darmstadt
Bookbinding:J.Scha¨fferGmbH,Gru¨nstadt
ISBN-13:978-3-527-40548-0
ISBN-10:3-527-40548-8
v
Contents
Preface vii
ListofContributors ix
AlgebraicMethods 1
A.J.Coleman
AnalyticMethods 33
CharlieHarper
FourierandOtherMathematicalTransforms 83
RonaldN.Bracewell
FractalGeometry 109
PaulMeakin
GeometricalMethods 127
V.AlanKostelecky´
Green’sFunctions 159
KazuoOhtaka
GroupTheory 189
M.Hamermesh
MathematicalModeling 213
KennethHartt
Monte-CarloMethods 249
K.Binder
vi Contents
NumericalMethods 281
ChristinaC.ChristaraandKennethR.Jackson
PerturbationMethods 385
JamesMurdock
QuantumComputation 417
SamuelL.Braunstein
QuantumLogic 439
DavidJ.Foulis,RichardJ.Greechie,MariaLouisaDallaChiara,
andRobertoGiuntini
SpecialFunctions 475
CharlieHarper
StochasticProcesses 513
MelvinLax
SymmetriesandConservationLaws 565
GinoSegre`
Topology 587
S.P.Smith
VariationalMethods 619
G.W.F.Drake
Index 657
vii
Preface
Mathematicsisacentralstructureinourknowledge.Therigorofmathematicalproof
placesthe subject ina very specialpositionwith enormousprestige. Forthe potential
user of mathematics this has both advantages as well as disadvantages. On the one
hand,onecanusemathematicswithconfidencethatingeneraltheconcepts,definitions,
procedures, and theorems have been thoroughly examined and tested, but the sheer
amountofmathematicsisoftenveryintimidatingto the non-expert.Sincethe results
of mathematics once proved stay in the structure forever, the subject just gets larger
and larger, and we do not have the luxury of discarding older theories as obsolete.
So the quadratic formula and the Pythagorean theorem are still useful and valid even
thoughtheyarethousandsofyearsold.Euclid’sElementsisstillusedasatextinsome
classrooms today, and it continues to inspire readers as it did in the past although it
treatsthemathematicsfromthetimeofPlatoover2300yearsago.
Despite the prestige of mathematical proof, most mathematics that we use today
arose without proof. The history of the development of calculus is a good example.
Neither Newton nor Leibniz gave definitions of limits, derivatives, or integrals that
wouldmeetcurrentstandards.Eventherealnumbersystemwasnotrigorouslytreated
untilthesecondhalfofthenineteenthcentury.Inthepast,asinmoderntimes,large
parts of mathematics were initiated and developed by scientists and engineers. The
distinction between mathematicians and scientists was often rather vague. Consider
for example, Newton, Euler, Lagrange, Gauss, Fourier, and Riemann. Although these
mendidimportantworkinmathematics,theywerealsodeeplyinvolvedinthesciences
oftheirtimes.Towardtheendofthe nineteenthcentury asplitting occurredbetween
mathematics and the sciences. Some see it in the development of non-Euclidean
geometryandespeciallyaxiomaticmethodsreminiscentofEuclid.
At this time mathematics appeared to be taking its own path independent of the
sciences. Here are two cases that participated in this division. In the late nineteenth
century Oliver Heaviside developed the Operational Calculus to solve problems in
electrical engineering. Although this calculus gave solutions in agreement with
experiment, the mathematicians of Heaviside’s time could not justify or condone
his procedures. Physicists also found the Dirac delta function and Green’s functions
extremelyusefulanddevelopedanappropriatecalculusfortheiruse,buttheunderlying
mathematical theory was not available. It was not until the early 1950’s that Laurent
Schwartzwasabletogivearigorousmathematicalfoundationforthesemethodswith
viii Preface
his Theory of Distributions. Also, early in the twentieth century the relatively new
subject of Group Theory was seen as being of use in applications to chemistry and
physics, but the few texts available at the time were written in a rather abstract and
rigorousmathematicalstylethatwasnoteasilyaccessibletomostnon-mathematicians.
Thesubjectwasquicklylabeledthe‘‘Gruppenpest’’andignoredbymanyresearchers.
Needlesstosay,todaygrouptheorywithitsapplicationstosymmetryisafundamental
toolinscience.
With the complexity of each field in science and engineering growing so rapidly, a
researcherinthesefieldshaslittletimetostudymathematicsforitsownsake.Eachfield
hasmorematerialthancanpossiblybecoveredinatypicalundergraduateprogram,and
evengraduatestudentsmustquicklypickasub-areaofspecialization.Often,however,
there is a sense that if we just knew more mathematics of the right sort, we could
get a better grasp of the subject at hand. So, if we are still in school, we may take a
mathematicscourse,orifnotinschool,wemaylookatsomemathematicaltexts.Here
somequestionsarise:whichcourseshouldwetake,dowehavethecorrectprerequisites,
what if our mathematics instructor has no knowledge of our field or any applications
thatweareinterestedin,arewereallyintherightcourse?Furthermore,mosttextsin
mathematics are intended for classroom use. They are generally very proof oriented,
andalthoughmanynowincludesomehistoricalremarksandhaveamoreuserfriendly
tone,theymaynotgettothepointfastenoughforthereaderoutsideofaclassroom.
This book is intended to help students and researchers with this problem. The
eighteen articles included here cover a very wide range of topics in mathematics in a
compact, user oriented way. These articles originally appeared in the Encyclopedia of
AppliedPhysics,amagnificenttwenty-threevolumeseteditedbyGeorgeL.Trigg,with
associateeditorsEduardoS.VeraandWalterGreulichandmanagingeditorEdmundH.
Immergut.ThefullEncyclopediawaspublishedinthe1990’sbyVCH,asubsidiaryof
JohnWiley&Sons,NewYork.Eacharticleinthisvolumecoversapartofmathematics
especiallyrelevanttoapplicationsinscienceandengineering.Thearticlesaredesigned
to give a good overview of the subject in a relatively short space with indications
of applications in applied physics. Suggestions for further reading are provided with
extensivebibliographiesandglossaries.Mostimportantly,thesearticlesareaccessible.
Each article seeks to give a quick review of a large area within mathematics without
lapsingintovaguenessoroverspecialization.
Ofcoursenotallofmathematicscanbecoveredinthisvolume:choicesmustbemade
inordertokeepthesizeoftheworkwithinbounds.Wecanonlyproceedbasedonthose
areas that have been most useful in the past. It is certainly possible that your favorite
questionisnotdiscussedhere,andcertainlythefuturewillbringnewmathematicsand
applicationstoprominence,butwesincerelyexpectthatthearticlesinthisvolumewill
bevaluabletomostreaders.
StuartP.Smith
CSUH–January2005
ix
List of Contributors
K.Binder DavidJ.Foulis
..
InstitutfurPhysik, UniversityofMassachusetts,
..
Johannes-Gutenberg-UniversitatMainz, Amherst,Massachusetts,
Mainz, USA
Germany
RobertoGiuntini
RonaldN.Bracewell
Universita`diFirenze,
ElectricalEngineeringDepartment,
Florence,
StanfordUniversity,
Italy
Stanford,California,
USA
RichardJ.Greechie
SamuelL.Braunstein LouisianaTechUniversity,
SEECS, Ruston,Louisiana,
UniversityofWales, USA
Bangor,
UK
M.Hamermesh
ChristinaC.Christara SchoolofPhysicsandAstronomy,
ComputerScienceDepartment, UniversityofMinnesota,
UniversityofToronto, Minneapolis,Minnesota,
Toronto,Ontario, USA
Canada
CharlieHarper
A.J.Coleman
DepartmentofPhysics,
DepartmentofMathandStatistics,
CaliforniaStateUniversity,
QueensUniversity,
Hayward,California,
Kingston,Ontario,
USA
Canada
G.W.F.Drake KennethHartt
DepartmentofPhysics, PhysicsDepartment,
UniversityofWindsor, UniversityofRhodeIsland,
Windsor,Ontario, Kingston,RhodeIsland,
Canada USA
x ListofContributors
KennethR.Jackson PaulMeakin
ComputerScienceDepartment, DepartmentofPhysics,
UniversityofToronto, UniversityofOslo,
Toronto,Ontario, Oslo,
Canada Norway
V.AlanKostelecky´ JamesMurdock
PhysicsDepartment, IowaStateUniversity,
IndianaUniversity, Ames,
Bloomington,Indiana, USA
USA
KazuoOhtaka
MelvinLax LaboratoryofAppliedPhysics,
PhysicsDepartment, FacultyofEngineering,
CityCollegeoftheCity ChibaUniversity,Chiba-shi,
UniversityofNewYork, Japan
NewYork,
USA GinoSegre`
DepartmentofPhysics,
and
UniversityofPennsylvania,
BellLaboratories, Philadelphia,Pennsylvania,
LucentTechnologies, USA
MurrayHill,NewJersey,
S.P.Smith
USA
DepartmentofMathematicsand
MariaLouisaDallaChiara ComputerScience,
Universita`diFirenze, CaliforniaStateUniversity,
Florence, Hayward,California,
Italy USA