Measures of spread for periodic distributions and the associated uncertainty relations

A physicist's intuition in Fourier theory is generally established from the parallels between Fourier series and transforms. Remarkably, one element of this theory that is especially significant in physics, namely the uncertainty principle, is never treated for Fourier series. We resolve this by first showing that a natural measure of spread for a periodic distribution follows simply upon regarding the distribution as a mass density on a ring. Even though the centroid of this ring is expressed in terms of just a first moment, its distance from the geometric center gives a close analog of variance. We then derive direct analogs of the uncertainty principle for both the Fourier series of a continuous periodic function as well as the fast Fourier transform of discrete data. The: results have similar applications to those of the standard uncertainty principle. (C) 2001 American Association of Physics Teachers.