The expression of the Uncertainty Principle of Quantum Mechanics

This talk will start with a brief review of the early history of the expression of Werner Heisenberg’s famous (1927) Uncertainty Principle for position (Q) and momentum (P) in Quantum Mechanics. In his 1927 paper, Heisenberg only gave a qualitative formulation of the Uncertainty Principle, but within a few months, the physical community settled on a mathematical expression of this principle in terms of an inequality first presented by E.H. Kennard, wherein the product of the standard deviations of Q and P is greater or equal than half of the (reduced) Planck constant. This is still the most common and well-known version of an uncertainty relation today. Nevertheless, it will be shown in this talk that Kennard’s inequality, is not strong enough to provide a foundation for most of the common examples used in textbooks as illustrations of the Uncertainty Principle. I will argue, therefore, that conceptually stronger inequalities are needed for the purpose. I will review several approaches to obtain such inequalities: the Landau & Pollak inequalities (1961); the entropic uncertainty inequality (Bialynicki-Birula & Mycielski,1975) and a statistico-geometric approach to the uncertainty relation by (Hilgevoord & Uffink,1991), and discuss their strengths and weaknesses vis-a-vis the common textbook examples.