Refinement of Robertson-type uncertainty principles with geometric interpretation

A generalization of the classical covariance for quantum mechanical observables has previously been presented by Gibilisco et al. Gibilisco and Isola has proved that the usual quantum covariance gives the sharpest inequalities for the determinants of covariance matrices. We introduce a new generalization of the classical covariance which gives better inequalities than the classical one, furthermore it has a direct geometric interpretation.

SHANNON ENTROPY OF THE WIGNER FUNCTION AND POSITION-MOMENTUM CORRELATION IN MODEL SYSTEMS

Shannon entropies of the Wigner function are calculated for ground and excited stationary states of the Particle-In-A-Box and Harmonic Oscillator model systems and examined as a measure of the localization of the phase-space distribution. We show that their behavior is consistent with that of the sum of the position and momentum space entropies as a function of quantum number. Position-momentum correlation is then analyzed in these systems by defining mutual information between position and momentum variables. This mutual information yields non-zero values, in contrast to the quantum covariance, and increases with quantum number.