1 6 Greens functions Helmholtz Theorem Christopher Crawford

§ 1. 6 Green’s functions; Helmholtz Theorem Christopher Crawford PHY 416 2014 -09 -26

Outline • Helmholtz theorem – L/T projection 2 nd derivatives – there’s really one! Longitudinal / transverse projections of the Laplacian Longitudinal / transverse separation of a vector field Scalar / Vector potentials • Green’s function G(x, y) Gradient & Laplacian of 1/r potential – point charge Definition of Green’s function Expansion in delta functions – `tent’ function – `pole’ function The Laplacian as a linear operator The inverse Laplacian operator – `X-ray’ operator – `shrink-wrap’ operator Particular solution of Poisson’s equation 2

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

L/T separation of E&M fields 4

Scalar and vector potentials • Scalar potential (Flow) • Vector potential (Flux) conservative or irrotational field solenoidal or incompressible field integral formulation source: divergence (charge) source: curl (current) gauge invariance 5

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

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 7

Multiple poles 8

Infinitely dense poles 9

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

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