Quantifying momenta through the Fourier transform

Abstract

Integral transforms arising from the separable solutions to the Helmholtz differential equation are presented. Pairs of these integral transforms are related via Plancherel theorem and, ultimately, any of these integral transforms may be calculated using only Fourier transforms. This result is used to evaluate the mean value of momenta associated to the symmetries of the reduced wave equation. As an explicit example, the orbital angular momenta of plane and elliptic-cylindrical waves is presented.