On a Conjecture regarding Fisher Information

ANGELO PLASTINO; GUIDO BELLOMO; A. R. PLASTINO

Advances in Mathematical Physics

Hindawi Publishing Corporation

Lugar: Cairo; Año: 2015 vol. 2015

1687-9120

Fisher?s information measure I plays a very important role in diverse areas of theoretical physics. The associated measures Ix and Ip, as functionals of quantum probability distributions defined in, respectively, coordinate and momentum spaces, are the protagonists of our present considerations. The product IxIp has been conjectured to exhibit a nontrivial lower bound in Hall (2000). More explicitly, this conjecture says that for any pure state of a particle in one dimension IxIp>4. We show here that such is not the case. This is illustrated, in particular, for pure states that are solutions to the free-particle Schrödinger equation. In fact, we construct a family of counterexamples to the conjecture, corresponding to time-dependent solutions of the free-particle Schrödinger equation. We also conjecture that any normalizable time-dependent solution of this equation verifies IxIp->0 for t->inf.