The coordinate-free approach to spherical harmonics

TL;DR

This paper advocates a coordinate-free approach to spherical harmonics, simplifying derivations of key results and enhancing pedagogical clarity for physics and mathematics students.

Contribution

It introduces a unified, self-contained coordinate-free framework for spherical harmonics, streamlining derivations of fundamental results and improving educational understanding.

Findings

Derivation of recursion relations and addition theorem without coordinates

Efficient computation of surface and solid harmonics

Simplified proofs of rotation matrix and Gaunt's integrals

Abstract

We present in a unified and self-contained manner the coordinate-free approach to spherical harmonics initiated in the mid 19th century by James Clerk Maxwell, William Thomson and Peter Guthrie Tait. We stress the pedagogical advantages of this approach which leads in a natural way to many physically relevant results that students find often difficult to work out using spherical coordinates and associated Legendre functions. It is shown how most physically relevant results of the theory of spherical harmonics - such as recursion relations, Legendre's addition theorem,surface harmonics expansions, the method of images, multipolar charge distributions, partial wave expansions, Hobson's integral theorem, rotation matrix and Gaunt's integrals - can be efficiently derived in a coordinate free fashion from a few basic elements of the theory of solid and surface harmonics discussed in the paper.