Abstract

The uncertainty relation in quantum mechanics has been explicated sometimes as a statistical relation and at other times as a relation concerning precision of simultaneous measurements. In the present paper, taking the indefiniteness of individual experiments as represented by diameters of Borel sets in projection-valued measure, we mathematically distinguish four expressions, two statistical and two concerning simultaneous measurements, of the uncertainty relation, study their interrelations, and prove that they are nonequivalent to each other and to the eigenvector condition (EV) in infinite-dimensional Hilbert space