Uncertainty relations: a small zoo of remarkable inequalities discovered since 1927

TL;DR

This paper reviews a variety of mathematical inequalities related to quantum uncertainty, including traditional, generalized, entropic, and higher-order relations, highlighting their development since 1927.

Contribution

It provides a comprehensive overview of the evolution and diversity of uncertainty inequalities in quantum mechanics since 1927.

Findings

Includes traditional Heisenberg, Schrödinger, and Robertson inequalities.

Discusses entropic and local uncertainty relations.

Presents inequalities for higher order moments and state purity.

Abstract

A concise review of various mathematical formulations of the uncertainty relations in quantum mechanics discovered since 1927 is given. Besides the traditional Heisenberg inequality, the modifications made by Schr\"odinger and Robertson, as well as generalizations to sets of several noncommuting operators, are considered. The "entropic" inequalities and "local" uncertainty relations, together with inequalities which connect the so-called total width and the mean peak width of a wave function, are discussed. Inequalities for the products of higher order moments of the coordinate and momentum are presented. Inequalities making the uncertainty relations more accurate when the "purity" of a quantum state is fixed are demonstrated. Diverse formulations of the energy-time uncertainty relations are considered.