Mean Values and Uncertainty Relations

F. Nicacio, F. T. Falciano

In this paper we determine a lower bound to the mean value of the quantum potential for an arbitrary state. Furthermore, we derive a generalized uncertainty relation that is stronger than the Robertson-Schr\"odinger inequality and hence also stronger than the Heisenberg uncertainty principle. The mean value is then associated to the nonclassical part of the covariances of the momenta operator. This imposes a minimum bound for the nonclassical correlations of momenta and gives a physical characterization of the classical and semiclassical limits of quantum systems. The results obtained primarily for pure states are then generalized for density matrices describing mixed states.

V. Alan Kostelecký, Bogdan Tudose

Uncertainty relations between a bounded coordinate operator and a conjugate momentum operator frequently appear in quantum mechanics. We prove that physically reasonable minimum-uncertainty solutions to such relations have quantized expectation values of the conjugate momentum. This implies, for example, that the mean angular momentum is quantized for any minimum-uncertainty state obtained from any uncertainty relation involving the angular-momentum operator and a conjugate coordinate. Experiments specifically seeking to create minimum-uncertainty states localized in angular coordinates therefore must produce packets with integer angular momentum. \textcopyright{} 1996 The American Physical Society.

Tomáš Opatrný

There exist several definitions of the mean value and the phase uncertainty of an optical field. We show that one of these definitions is especially simple and intuitive because of its mechanical analogy. On this basis, we also derive the summation rule and the Tchebyshev inequality for random angular and phase variables and a number-phase uncertainty relation.

Lubomír Skála, V. Kapsa, Martina Lužová