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§1.6 Green’s functions; Helmholtz Theorem

§1.6 Green’s functions; Helmholtz Theorem. Christopher Crawford PHY 416 2014-09-26. Outline.

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§1.6 Green’s functions; Helmholtz Theorem

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  1. §1.6Green’s functions;Helmholtz Theorem Christopher Crawford PHY 416 2014-09-26

  2. Outline • Helmholtz theorem – L/T projection2nd derivatives – there’s really only one! Longitudinal / transverse projections of the LaplacianLongitudinal / transverse separation of a vector fieldScalar / Vector potentials • Green’s function G(x,y)Gradient & Laplacian of 1/r potential – point chargeDefinition of Green’sfunction– `tent’ functionExpansion in delta functions – `pole’ functionThe Laplacian as a linear operator – `X-ray’ operatorThe inverse Laplacian operator – `shrink-wrap’ operatorParticular solution of Poisson’s equation

  3. 2nd derivatives: only one! • All combinations of vector derivatives: the differential chain

  4. L/T separation of E&M fields

  5. Scalar and vector potentials • Scalar potential (Flow) conservative orirrotational fieldintegral formulation source: divergence (charge) gauge invariance • Vector potential (Flux) solenoidal orincompressible fieldintegral formulation source: curl (current) gaugeinvariance

  6. Potential and field of a point source • Gradient • Divergence Flux • Planar angle • Solidangle

  7. Green’s function G(r,r’) • The potential of a point-charge • A simple solution to the Poisson’s equation • Zero curvature except infinite at one spot

  8. Multiple poles

  9. Infinitely dense poles

  10. General solution to Poisson’s equation • Expand f(x) as linear combination of delta functions • Invert linearLapacian on each delta function individually

  11. Green’s functions as propagators • Action at a distance: G(r’,r) `carries’ potentialfrom source at r' to field point (force) at r • In quantum field theory, potential is quantizedG(r’,r) represents the photon (particle) that carries the force • How do you measure the `shape’ of the proton?

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