A NUMERICAL STUDY ON THE LOWER BOUNDS OF THE CLASSICAL AND REFINED HEISENBERG UNCERTAINTY PRINCIPLES

Abstract

In Fourier analysis and quantum mechanics, the Heisenberg uncertainty principle stands as a fundamental result. Its classical version provides a product lower bound for the variances of position and frequency. More recently, for vector-valued functions, Dang, Deng and Qian introduced an improved inequality featuring an extra covariance term derived from the current density. The present work performs a one-dimensional numerical comparison of the two lower bounds using four test functions: a pure Gaussian, an asymmetric sum of two Gaussians, a linear chirp, and a vector-valued signal with spatially separated orthogonal components. For the Gaussian, the covariance term vanishes up to machine precision, making the two bounds identical. For the asymmetric sum, the refined bound exceeds the classical one by about three percent. The chirp signal yields a dramatically larger refined bound, approximately sixteen times above the classical one. For the vector-valued case with each component a real Gaussian, the refined bound is about eight times higher. These numerical outcomes confirm the validity of the refined inequality and highlight the sensitivity of the covariance term to phase structure and vectorial coupling.