Quantifying momenta through the Fourier transform

TL;DR

This paper introduces a method to compute momenta related to wave symmetries using Fourier transforms, simplifying calculations and providing explicit examples like orbital angular momenta of specific wave types.

Contribution

It demonstrates how integral transforms from Helmholtz solutions relate via Plancherel theorem, enabling momentum evaluation solely through Fourier transforms.

Findings

Integral transforms linked through Plancherel theorem

Momentum of wave symmetries can be computed via Fourier transforms

Explicit calculation of orbital angular momenta for specific wave types

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.