Green's Functions in Physics Version 1, Slides of Solid State Physics

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Green’s Functions in Physics

Version 1

M. Baker, S. Sutlief

Revision:

December 19, 2003

List of Figures

• 1 The Vibrating String
  • 1.1 The String
    • 1.1.1 Forces on the String
    • 1.1.2 Equations of Motion for a Massless String
    • 1.1.3 Equations of Motion for a Massive String
  • 1.2 The Linear Operator Form
  • 1.3 Boundary Conditions
    • 1.3.1 Case 1: A Closed String
    • 1.3.2 Case 2: An Open String
    • 1.3.3 Limiting Cases
    • 1.3.4 Initial Conditions
  • 1.4 Special Cases
    • 1.4.1 No Tension at Boundary
    • 1.4.2 Semi-infinite String
    • 1.4.3 Oscillatory External Force
  • 1.5 Summary
  • 1.6 References
• 2 Green’s Identities
  • 2.1 Green’s 1st and 2nd Identities
  • 2.2 Using G.I. #2 to Satisfy R.B.C.
    • 2.2.1 The Closed String
    • 2.2.2 The Open String
    • 2.2.3 A Note on Hermitian Operators
  • 2.3 Another Boundary Condition
  • 2.4 Physical Interpretations of the G.I.s
    • 2.4.1 The Physics of Green’s 2nd Identity
    • 2.4.2 A Note on Potential Energy ii CONTENTS
    • 2.4.3 The Physics of Green’s 1st Identity
  • 2.5 Summary
  • 2.6 References
• 3 Green’s Functions
  • 3.1 The Principle of Superposition
  • 3.2 The Dirac Delta Function
  • 3.3 Two Conditions
    • 3.3.1 Condition
    • 3.3.2 Condition
    • 3.3.3 Application
  • 3.4 Open String
  • 3.5 The Forced Oscillation Problem
  • 3.6 Free Oscillation
  • 3.7 Summary
  • 3.8 Reference
• 4 Properties of Eigen States
  • 4.1 Eigen Functions and Natural Modes
    • 4.1.1 A Closed String Problem
    • 4.1.2 The Continuum Limit
    • 4.1.3 Schr¨odinger’s Equation
  • 4.2 Natural Frequencies and the Green’s Function
  • 4.3 GF behavior near λ = λn
  • 4.4 Relation between GF & Eig. Fn.
    • 4.4.1 Case 1: λ Nondegenerate
    • 4.4.2 Case 2: λn Double Degenerate
  • 4.5 Solution for a Fixed String
    • 4.5.1 A Non-analytic Solution
    • 4.5.2 The Branch Cut
    • 4.5.3 Analytic Fundamental Solutions and GF
    • 4.5.4 Analytic GF for Fixed String
    • 4.5.5 GF Properties
    • 4.5.6 The GF Near an Eigenvalue
  • 4.6 Derivation of GF form near E.Val.
    • 4.6.1 Reconsider the Gen. Self-Adjoint Problem
    • 4.6.2 Summary, Interp. & Asymptotics CONTENTS iii
  • 4.7 General Solution form of GF
    • 4.7.1 δ-fn Representations & Completeness
  • 4.8 Extension to Continuous Eigenvalues
  • 4.9 Orthogonality for Continuum
  • 4.10 Example: Infinite String
    • 4.10.1 The Green’s Function
    • 4.10.2 Uniqueness
    • 4.10.3 Look at the Wronskian
    • 4.10.4 Solution
    • 4.10.5 Motivation, Origin of Problem
  • 4.11 Summary of the Infinite String
  • 4.12 The Eigen Function Problem Revisited
  • 4.13 Summary
  • 4.14 References
• 5 Steady State Problems
  • 5.1 Oscillating Point Source
  • 5.2 The Klein-Gordon Equation
    • 5.2.1 Continuous Completeness
  • 5.3 The Semi-infinite Problem
    • 5.3.1 A Check on the Solution
  • 5.4 Steady State Semi-infinite Problem
    • 5.4.1 The Fourier-Bessel Transform
  • 5.5 Summary
  • 5.6 References
• 6 Dynamic Problems
  • 6.1 Advanced and Retarded GF’s
  • 6.2 Physics of a Blow
  • 6.3 Solution using Fourier Transform
  • 6.4 Inverting the Fourier Transform
    • 6.4.1 Summary of the General IVP
  • 6.5 Analyticity and Causality
  • 6.6 The Infinite String Problem
    • 6.6.1 Derivation of Green’s Function
    • 6.6.2 Physical Derivation
  • 6.7 Semi-Infinite String with Fixed End iv CONTENTS
  • 6.8 Semi-Infinite String with Free End
  • 6.9 Elastically Bound Semi-Infinite String
  • 6.10 Relation to the Eigen Fn Problem
    • 6.10.1 Alternative form of the GR Problem
  • 6.11 Comments on Green’s Function
    • 6.11.1 Continuous Spectra
    • 6.11.2 Neumann BC
    • 6.11.3 Zero Net Force
  • 6.12 Summary
  • 6.13 References
• 7 Surface Waves and Membranes
  • 7.1 Introduction
  • 7.2 One Dimensional Surface Waves on Fluids
    • 7.2.1 The Physical Situation
    • 7.2.2 Shallow Water Case
  • 7.3 Two Dimensional Problems
    • 7.3.1 Boundary Conditions
  • 7.4 Example: 2D Surface Waves
  • 7.5 Summary
  • 7.6 References
• 8 Extension to N -dimensions
  • 8.1 Introduction
  • 8.2 Regions of Interest
  • 8.3 Examples of N -dimensional Problems
    • 8.3.1 General Response
    • 8.3.2 Normal Mode Problem
    • 8.3.3 Forced Oscillation Problem
  • 8.4 Green’s Identities
    • 8.4.1 Green’s First Identity
    • 8.4.2 Green’s Second Identity
    • 8.4.3 Criterion for Hermitian L
  • 8.5 The Retarded Problem
    • 8.5.1 General Solution of Retarded Problem
    • 8.5.2 The Retarded Green’s Function in N -Dim.
    • 8.5.3 Reduction to Eigenvalue Problem CONTENTS v
  • 8.6 Region R
    • 8.6.1 Interior
    • 8.6.2 Exterior
  • 8.7 The Method of Images
    • 8.7.1 Eigenfunction Method
    • 8.7.2 Method of Images
  • 8.8 Summary
  • 8.9 References
• 9 Cylindrical Problems
  • 9.1 Introduction
    • 9.1.1 Coordinates
    • 9.1.2 Delta Function
  • 9.2 GF Problem for Cylindrical Sym.
  • 9.3 Expansion in Terms of Eigenfunctions
    • 9.3.1 Partial Expansion
    • 9.3.2 Summary of GF for Cyl. Sym.
  • 9.4 Eigen Value Problem for L
  • 9.5 Uses of the GF Gm(r, r′; λ)
    • 9.5.1 Eigenfunction Problem
    • 9.5.2 Normal Modes/Normal Frequencies
    • 9.5.3 The Steady State Problem
    • 9.5.4 Full Time Dependence
  • 9.6 The Wedge Problem
    • 9.6.1 General Case
    • 9.6.2 Special Case: Fixed Sides
  • 9.7 The Homogeneous Membrane
    • 9.7.1 The Radial Eigenvalues
    • 9.7.2 The Physics
  • 9.8 Summary
  • 9.9 Reference
• 10 Heat Conduction
  • 10.1 Introduction
    • 10.1.1 Conservation of Energy
    • 10.1.2 Boundary Conditions
  • 10.2 The Standard form of the Heat Eq. vi CONTENTS
    • 10.2.1 Correspondence with the Wave Equation
    • 10.2.2 Green’s Function Problem
    • 10.2.3 Laplace Transform
    • 10.2.4 Eigen Function Expansions
  • 10.3 Explicit One Dimensional Calculation
    • 10.3.1 Application of Transform Method
    • 10.3.2 Solution of the Transform Integral
    • 10.3.3 The Physics of the Fundamental Solution
    • 10.3.4 Solution of the General IVP
    • 10.3.5 Special Cases
  • 10.4 Summary
  • 10.5 References
• 11 Spherical Symmetry
  • 11.1 Spherical Coordinates
  • 11.2 Discussion of Lθϕ
  • 11.3 Spherical Eigenfunctions
    • 11.3.1 Reduced Eigenvalue Equation
    • 11.3.2 Determination of uml (x)
    • 11.3.3 Orthogonality and Completeness of uml (x)
  • 11.4 Spherical Harmonics
    • 11.4.1 Othonormality and Completeness of Y lm
  • 11.5 GF’s for Spherical Symmetry
    • 11.5.1 GF Differential Equation
    • 11.5.2 Boundary Conditions
    • 11.5.3 GF for the Exterior Problem
  • 11.6 Example: Constant Parameters
    • 11.6.1 Exterior Problem
    • 11.6.2 Free Space Problem
  • 11.7 Summary
  • 11.8 References
• 12 Steady State Scattering
  • 12.1 Spherical Waves
  • 12.2 Plane Waves
  • 12.3 Relation to Potential Theory
  • 12.4 Scattering from a Cylinder CONTENTS vii
  • 12.5 Summary
  • 12.6 References
• 13 Kirchhoff ’s Formula
  • 13.1 References
• 14 Quantum Mechanics
  • 14.1 Quantum Mechanical Scattering
  • 14.2 Plane Wave Approximation
  • 14.3 Quantum Mechanics
  • 14.4 Review
  • 14.5 Spherical Symmetry Degeneracy
  • 14.6 Comparison of Classical and Quantum
  • 14.7 Summary
  • 14.8 References
• 15 Scattering in 3-Dim
  • 15.1 Angular Momentum
  • 15.2 Far-Field Limit
  • 15.3 Relation to the General Propagation Problem
  • 15.4 Simplification of Scattering Problem
  • 15.5 Scattering Amplitude
  • 15.6 Kinematics of Scattered Waves
  • 15.7 Plane Wave Scattering
  • 15.8 Special Cases
    • 15.8.1 Homogeneous Source; Inhomogeneous Observer
    • 15.8.2 Homogeneous Observer; Inhomogeneous Source
    • 15.8.3 Homogeneous Source; Homogeneous Observer
    • 15.8.4 Both Points in Interior Region
    • 15.8.5 Summary
    • 15.8.6 Far Field Observation
    • 15.8.7 Distant Source: r′ → ∞
  • 15.9 The Physical significance of Xl
    • 15.9.1 Calculating δl(k)
  • 15.10Scattering from a Sphere
    • 15.10.1 A Related Problem
  • 15.11Calculation of Phase for a Hard Sphere viii CONTENTS
  • 15.12Experimental Measurement
    • 15.12.1 Cross Section
    • 15.12.2 Notes on Cross Section
    • 15.12.3 Geometrical Limit
  • 15.13Optical Theorem
  • 15.14Conservation of Probability Interpretation:
    • 15.14.1 Hard Sphere
  • 15.15Radiation of Sound Waves
    • 15.15.1 Steady State Solution
    • 15.15.2 Far Field Behavior
    • 15.15.3 Special Case
    • 15.15.4 Energy Flux
    • 15.15.5 Scattering From Plane Waves
    • 15.15.6 Spherical Symmetry
  • 15.16Summary
  • 15.17References
• 16 Heat Conduction in 3D
  • 16.1 General Boundary Value Problem
  • 16.2 Time Dependent Problem
  • 16.3 Evaluation of the Integrals
  • 16.4 Physics of the Heat Problem
    • 16.4.1 The Parameter Θ
  • 16.5 Example: Sphere
    • 16.5.1 Long Times
    • 16.5.2 Interior Case
  • 16.6 Summary
  • 16.7 References
• 17 The Wave Equation
  • 17.1 introduction
  • 17.2 Dimensionality
    • 17.2.1 Odd Dimensions
    • 17.2.2 Even Dimensions
  • 17.3 Physics
    • 17.3.1 Odd Dimensions
    • 17.3.2 Even Dimensions CONTENTS ix
    • 17.3.3 Connection between GF’s in 2 & 3-dim
  • 17.4 Evaluation of G
  • 17.5 Summary
  • 17.6 References
• 18 The Method of Steepest Descent
  • 18.1 Review of Complex Variables
  • 18.2 Specification of Steepest Descent
  • 18.3 Inverting a Series
  • 18.4 Example 1: Expansion of Γ–function
    • 18.4.1 Transforming the Integral
    • 18.4.2 The Curve of Steepest Descent
  • 18.5 Example 2: Asymptotic Hankel Function
  • 18.6 Summary
  • 18.7 References
• 19 High Energy Scattering
  • 19.1 Fundamental Integral Equation of Scattering
  • 19.2 Formal Scattering Theory
    • 19.2.1 A short digression on operators
  • 19.3 Summary of Operator Method
    • 19.3.1 Derivation of G = (E − H)−
    • 19.3.2 Born Approximation
  • 19.4 Physical Interest
    • 19.4.1 Satisfying the Scattering Condition
  • 19.5 Physical Interpretation
  • 19.6 Probability Amplitude
  • 19.7 Review
  • 19.8 The Born Approximation
    • 19.8.1 Geometry
    • 19.8.2 Spherically Symmetric Case
    • 19.8.3 Coulomb Case
  • 19.9 Scattering Approximation
  • 19.10Perturbation Expansion
    • 19.10.1 Perturbation Expansion
    • 19.10.2 Use of the T -Matrix
• 1.1 A string with mass points attached to springs.
• 1.2 A closed string, where a and b are connected.
• 1.3 An open string, where the endpoints a and b are free.
• 3.1 The pointed string
• 4.1 The closed string with discrete mass points.
• 4.2 Negative energy levels
• 4.3 The θ-convention
• 4.4 The contour of integration
• 4.5 Circle around a singularity.
• 4.6 Division of contour.
• 4.7 λ near the branch cut.
• 4.8 θ specification.
• 4.9 Geometry in λ-plane
• 6.1 The contour L in the λ-plane.
• 6.2 Contour LC 1 = L + LU HP closed in UH λ-plane.
• 6.3 Contour closed in the lower half λ-plane.
• 6.4 An illustration of the retarded Green’s Function.
• 6.5 GR at t 1 = t′ + 12 x′/c and at t 2 = t′ + 32 x′/c.
• 7.1 Water waves moving in channels.
• 7.2 The rectangular membrane.
• 9.1 The region R as a circle with radius a.
• 9.2 The wedge.
• 10.1 Rotation of contour in complex plane.
• 10.2 Contour closed in left half s-plane. xii LIST OF FIGURES
• 10.3 A contour with Branch cut.
• 11.1 Spherical Coordinates.
• 11.2 The general boundary for spherical symmetry.
• 12.1 Waves scattering from an obstacle.
• 12.2 Definition of γ and θ..
• 13.1 A screen with a hole in it.
• 13.2 The source and image source.
• 13.3 Configurations for the G’s.
• 14.1 An attractive potential.
• 14.2 The complex energy plane.
• 15.1 The schematic representation of a scattering experiment.
• 15.2 The geometry defining γ and θ.
• 15.3 Phase shift due to potential.
• 15.4 A repulsive potential.
• 15.5 The potential V and Veff for a particular example.
• 15.6 An infinite potential wall.
• 15.7 Scattering with a strong forward peak.
• 16.1 Closed contour around branch cut.
• 17.1 Radial part of the 2-dimensional Green’s function.
• 17.2 A line source in 3-dimensions.
• 18.1 Contour C & deformation C 0 with point z
• 18.2 Gradients of u and v.
• 18.3 f (z) near a saddle-point.
• 18.4 Defining Contour for the Hankel function.
• 18.5 Deformed contour for the Hankel function.
• 18.6 Hankel function contours.
• 19.1 Geometry of the scattered wave vectors.

xiv LIST OF FIGURES

not necessarily used for the development of the original lectures. Books marked with an asterisk are are more supplemental. Com- ments on the books listed are given above.

• Index The index was composed by skimming through the text and picking out places where ideas were introduced or elaborated upon. No attempt was made to locate all relevant discussions for each idea.

A Note About Copying: These notes are in a state of rapid transition and are provided so as to be of benefit to those who have recently taken the class. Therefore, please do not photocopy these notes.

Contacting the Authors: A list of phone numbers and email addresses will be maintained of those who wish to be notified when revisions become available. If you would like to be on this list, please send email to

[email protected]

before 1996. Otherwise, call Marshall Baker at 206-543-2898.

Acknowledgements: This manuscript benefits greatly from the excellent set of notes taken by Steve Griffies. Richard Horn contributed many corrections and suggestions. Special thanks go to the students of Physics 425- at the University of Washington during 1988 and 1993. This first revision contains corrections only. No additional material has been added since Version 0.

Steve Sutlief Seattle, Washington 16 June, 1993 4 January, 1994

Chapter 1

The Vibrating String

4 Jan p Chapter Goals: p1prv.yr.

• Construct the wave equation for a string by identi- fying forces and using Newton’s second law.
• Determine boundary conditions appropriate for a closed string, an open string, and an elastically bound string.
• Determine the wave equation for a string subject to an external force with harmonic time dependence.

The central topic under consideration is the branch of differential equa- tion theory containing boundary value problems. First we look at an pr:bvp example of the application of Newton’s second law to small vibrations: transverse vibrations on a string. Physical problems such as this and those involving sound, surface waves, heat conduction, electromagnetic waves, and gravitational waves, for example, can be solved using the mathematical theory of boundary value problems.

Consider the problem of a string embedded in a medium with a pr:string restoring force V (x) and an external force F (x, t). This problem covers (^) pr:V most of the physical interpretations of small vibrations. In this chapter pr:F we will investigate the mathematics of this problem by determining the equations of motion.

1

1.1. THE STRING 3

force in the u-direction, which is τi+1 sin θ. From the figure we see that pr:theta θ ≈ (ui+1 − ui)/a for small angles and we can thus write

F (^) iyτi +1= τi+

(ui+1 − ui) a and pr:Fiyt F (^) iyτi = −τi

(ui − ui− 1 ) a

Note that the equations agree with dimensional analysis: Grif’s uses Taylor exp pr:m pr:l pr:t

F (^) iyτi = dim(m · l/t^2 ), τi = dim(m · l/t^2 ), ui = dim(l), and a = dim(l).

Elastic Force pr:elastic We add an elastic force with spring constant ki: (^) pr:ki

F (^) ielastic = −kiui,

where dim(ki) = (m/t^2 ). This situation can be visualized by imagining pr:Fel vertical springs attached to each mass point, as depicted in figure 1.1. A small value of ki corresponds to an elastic spring, while a large value of ki corresponds to a rigid spring.

External Force

We add the external force F (^) iext. This force depends on the nature of pr:ExtForce the physical problem under consideration. For example, it may be a pr:Fext transverse force at the end points.

1.1.2 Equations of Motion for a Massless String

The problem thus far has concerned a massless string with mass points attached. By summing the above forces and applying Newton’s second law, we have pr:Newton pr:t Ftot = τi+

(ui+1 − ui) a

− τi

(ui − ui− 1 ) a

− kiui + F (^) iext = mi

d^2 dt^2

ui. (1.1)

This gives us N coupled inhomogeneous linear ordinary differential eq1force equations where each ui is a function of time. In the case that F (^) iext pr:diffeq is zero we have free vibration, otherwise we have forced vibration. (^) pr:FreeVib

pr:ForcedVib

4 CHAPTER 1. THE VIBRATING STRING

1.1.3 Equations of Motion for a Massive String

4 Jan p For a string with continuous mass density, the equidistant mass points on the string are replaced by a continuum. First we take a, the sep- aration distance between mass points, to be small and redefine it as pr:deltax1 a = ∆x. We correspondingly write ui − ui− 1 = ∆u. This allows us to

pr:deltau1 write^ (u i −^ ui− 1 ) a

( (^) ∆u

∆x

)

i

The equations of motion become (after dividing both sides by ∆x)

1 ∆x

[ τi+

( (^) ∆u

∆x

)

i+

− τi

( (^) ∆u

∆x

)

i

] −

ki ∆x

ui +

F (^) iext ∆x

mi ∆x

d^2 ui dt^2

eq1deltf In the limit we take a → 0, N → ∞, and define their product to be

a^ lim→ 0 N →∞

N a ≡ L. (1.4)

The limiting case allows us to redefine the terms of the equations of pr:sigmax1 motion as follows:

mi → (^0) ∆mxi → σ(xi) ≡ (^) lengthmass = mass density; ki → (^0) ∆kix → V (xi) = coefficient of elasticity of the media; F (^) iext → 0 F^

ext ∆x = (^

mi ∆x ·^

F ext mi )^ →^ σ(xi)f^ (xi) (1.5) where f (xi) =

F ext mi

external force mass

Since

xi = x xi− 1 = x − ∆x xi+1 = x + ∆x

pr:x1 we have ( ∆u ∆x

)

i

ui − ui− 1 xi − xi− 1

∂u(x, t) ∂x