Quantum information entropies and orthogonal polynomials

摘要

This is a survey of the present knowledge on the analytical determination of the Shannon information entropies for simple quantum systems: single-particle systems in central potentials. Emphasis is made on D-dimensional harmonic oscillator and Coulombian potentials in both position and momentum spaces. First of all, these quantities are explicitly shown to be controlled by the entropic integrals of some classical orthogonal polynomials (Hermite, Laguerre and Gegenbauer). Then, the connection of these integrals with more common mathematical objects, such as the logarithmic potential, energy and Lp-norms of orthogonal polynomials, is briefly described. Third, its asymptotic behaviour is discussed for both general and varying weights. The explicit computation of these integrals is carried out for the Chebyshev and Gegenbauer polynomials, which have a bounded orthogonality interval, as well as for Hermite polynomials to illustrate the difficulties encountered when the interval is unbounded. These results have allowed us to find the position and momentum entropies of the ground and excited states of the physical systems mentioned above.