§ Separation of spherical variables: zonal harmonics Christopher Crawford PHY

Outline Review of eigenvalue problem Linear function spaces: Sturm-Liouville theorem Separation of Cartesian variables: exponentials Separation of cylindrical variables Azimuthal or sectoral functions: cylindrical harmonics Bessel functions ; zero curvature limit: planar harmonics General solution to Laplace equation Separation of spherical variables Legendre polynomials & functions: spherical harmonics Spherical Bessel functions ; zero curvature: solid harmonics Azimuthal symmetry: zonal harmonics General solution to Laplace equation Example problem – Griffiths example 3.9 Spherical shell of charge 2

`` Vectors vs. Functions Functions can be added or stretched (pointwise operation) Continuous vs. discrete vector space Components: function value at each point Visualization: graphs, not arrows 3

Vectors vs. Functions 4 ``

Sturm-Liouville Theorem Laplacian (self-adjoint) has orthogonal eigenfunctions – This is true in any orthogonal coordinate system! Sturm-Liouville operator – eigenvalue problem – Theorem: eigenfunctions with different eigenvalues are orthogonal 5

Helmholtz equation: free wave k 2 = curvature of wave; k 2 =0 [Laplacian] 6

Linear wave functions – exponentials 7

Circular waves – Bessel functions 8

Polar waves – Legendre functions 9

Angular waves – spherical harmonics 10

Radial waves – spherical Bessel fn’s 11

Solid harmonics 12

General solutions to Laplace eq’n or: All I really need to know I learned in PHY311 Cartesian coordinates – no general boundary conditions! Cylindrical coordinates – azimuthal continuity Spherical coordinates – azimuthal and polar continuity 13

Example: spherical shell of charge 14

Boundary conditions 15