Green’s Functions in Classical Physics

272 Pages·2017·1.676 MB·English

by  Tom Rother

Lecture Notes in Physics 938
Tom Rother
Green’s
Functions
in Classical
Physics
Lecture Notes in Physics
Volume 938
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Tom Rother
Green’s Functions
in Classical Physics
123
TomRother
GermanAerospaceCenter
Neustrelitz,Germany
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
ISBN978-3-319-52436-8 ISBN978-3-319-52437-5 (eBook)
DOI10.1007/978-3-319-52437-5
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Preface
This book is singing the praises of Green’s functions. Its concept was formed a
couple of years ago, after finishing a consistent Green’s function formulation of
electromagnetic wave scattering on nonspherical objects. It turned out that such
a consistent Green’s function formulationprovidesa sound mathematicalbasis to
discusstheadvantagesanddisadvantagesofdifferentnumericalapproacheswhich
havebeendevelopedsofartosolvethosescatteringproblems.ButIbecamealready
acquaintedwith similar mathematicalstructuresduringmy earlyPhD activitiesin
the field of quantum statistics. This long-lasting activity with Green’s functions
led me to the issue of the conceptual importance of these functions for physics,
in general. Unfortunately, even today Green’s functions are often considered and
discussed only from the point of view of an appropriate mathematical tool for
solving differentialequations.The works of J. Schwingerand F. Dyson belongto
thefew exceptionswhichemphasizetheconceptualimportanceofthese functions
inquantumstatisticsandquantumfieldtheory.
Startingfromtheseexperiencesandfirstbutmorephilosophicalconsiderations—
where the latter are reflected in the prologue of this book—I tried to apply the
Green’sfunctionformalismto well-knownproblemsof classical physics,someof
whichareusuallysolvednotbythisformalismbutothermethods.Thisactivitywas
aimed at convincingmyself from the conceptual importance of Green’s functions
alsoinclassicalfieldsofphysics.Thepresentbookistheresultofthiseffort.Itis
writtenasanintroductionforthosewhowanttobecomemorefamiliarwithGreen’s
functions and their importance and usage in classical physics. However, a short
outlook regarding the importance of Green’s functions in quantum mechanics as
wellastheircalculationbyuseofthemethodsdiscussedintheforegoingchapters
is providedin the finalchapterof this book.Lookingat physicsfrom the pointof
view formulatedin the prologueprovedto be very helpfulfor me when I tried to
enter new fields of physics. Maybe the reader will also benefit from this point of
view.
Finally, I would like to express my deepest gratitude to my parents Elisabeth
andFritzRother,tomywifeDoreenandtomyteacherProf.Wolf-DietrichKraeft
for their support in manyfold ways and their continuous interest in my scientific
vii
viii Preface
activitiesoverdecades.SpecialthanksgoalsotoMr.J.Duff,heritagedevelopment
officer from the Green’s Windmill and Science Centre, Nottingham (UK), for
providingmewiththecoverpictureofGreen’swindmill.Iwouldalsoliketothank
Dr. C. Ascheron, senior editor at Springer Science and Business Media, for his
continuousinterestandassistanceinpublishingthisbook.
Neustrelitz,Germany TomRother
Autumn,2016
Contents
1 Prologue....................................................................... 1
1.1 Aboutthe“State”ofPhysics ........................................... 1
1.2 BasicStructuralElementsofPhysics .................................. 5
1.3 AboutClassicalPhysicsandQuantumMechanics.................... 9
2 Green’sFunctionsofClassicalParticles................................... 15
2.1 TheSimpleHarmonicOscillator....................................... 16
2.1.1 ClassicalConsideration......................................... 16
2.1.2 Green’s Function, Green’s Theorem,Causality,
andReciprocity................................................. 19
2.1.3 DeterminationoftheGreen’sFunctionbyTrying ............ 24
2.1.4 DeterminationoftheGreen’sFunctionbyApplying
theFourierTransformMethod................................. 27
2.1.5 FirstExamplesofSimpleSources............................. 31
2.2 TheDampedHarmonicOscillator ..................................... 33
2.2.1 DeterminationoftheGreen’sFunctionbyApplying
theFourierTransformMethod................................. 35
2.2.2 ThePeriodicallyExcitedDampedHarmonicOscillator...... 36
2.3 BasicMotionsofaPointMass......................................... 39
2.4 Lippmann-SchwingerEquation ........................................ 41
2.5 TwoSystematicWaystoDeriveGreen’sFunctions................... 46
2.5.1 ClassicalMethodtoDeterminetheGreen’sFunctions ....... 47
2.5.2 Alternative Formulation by Using Cauchy’s
IntegralFormula................................................ 55
2.5.3 Kramers-KronigRelation ...................................... 60
2.6 TemporalBoundaryValueProblemoftheHarmonicOscillator ..... 64
2.7 TwoSimpleInteractionProcessesandHuygens’Principle........... 72
2.7.1 InteractionwithaWall ......................................... 73
2.7.2 TemporaryFriction ............................................. 77
ix
x Contents
2.8 ParticleScatteringonaRigidSphereandKeplerProblem ........... 78
2.8.1 TransformationoftheEquationofMotionintoPolar
Coordinates ..................................................... 78
2.8.2 SourcesoftheScatteringProblems............................ 80
2.8.3 SolvingtheScatteringProblems............................... 83
3 Green’sFunctionsofClassicalFields...................................... 95
3.1 CommentsontheFieldConcept ....................................... 95
3.2 TheElasticString....................................................... 97
3.2.1 One-DimensionalPoissonEquation........................... 98
3.2.2 One-DimensionalWave-,Klein-Gordon-,Telegraphy-,
andDiffusionEquation......................................... 103
3.2.3 Reciprocity and General Solution
oftheOne-DimensionalWaveEquation....................... 111
3.2.4 ExamplesofSimpleSources................................... 114
3.2.5 Reflectionofd’Alembert’sSolutionFrom
aFixedBoundary............................................... 117
3.2.6 ReflectionandTransmissionofd’Alembert’sSolution
ataDiscontinuity............................................... 123
3.3 PoissonEquationsofHigherDimensions ............................. 125
3.3.1 Dirac’s Delta Function and Unit Sources
inPolar-andSphericalCoordinates ........................... 126
3.3.2 Green’sFunctionoftheTwo-Dimensional
PoissonEquation ............................................... 128
3.3.3 Green’sFunctionoftheThree-DimensionalPoisson
Equation......................................................... 129
3.4 WaveEquationsofHigherDimensions................................ 131
3.4.1 Three-DimensionalWaveEquation............................ 133
3.4.2 Two-DimensionalWaveEquation ............................. 134
3.4.3 FourierTransformMethodinInfiniteRegions................ 135
3.4.4 FourierTransformMethodinFiniteRegions ................. 138
3.5 TheScalarHelmholtzEquation........................................ 141
3.5.1 Green’sFunctionsoftheOne-DimensionalHelmholtz
Equation......................................................... 141
3.5.2 Green’sFunctionsoftheTwo-andThree-Dimensional
HelmholtzEquation ............................................ 145
4 Green’sFunctionsandPlaneWaveScattering ........................... 149
4.1 GeneralAspects......................................................... 149
4.2 Double-SlitExperiments................................................ 156
4.2.1 ClassicalDouble-SlitExperiment.............................. 156
4.2.2 InteractionofaLinearlyPolarizedPlaneWavewitha
PolarizingFilter................................................. 163
4.2.3 ModifiedDouble-SlitExperiment ............................. 170
4.3 EigensolutionsoftheThree-DimensionalHelmholtzEquation
inSphericalCoordinates................................................ 172