Generalizations of the Heisenberg and Schrödinger uncertainty relations

Abstract

In this paper, releasing the restriction on operators which are self-adjoint, we propose a Heisenberg-type uncertainty relation and a Schrödinger-type uncertainty relation with any pair of operators on a Hilbert space. A generalization of Luo's theorem [S. Luo, “Heisenberg uncertainty relation for mixed states,” Phys. Rev. A 72, 042110 (2005)] is investigated.