Journal of mathematical physics vol:36 issue:8 pages:4106-4118

Abstract:

The information entropy of the harmonic oscillator potential V(x) = 1/2 lambda x(2) in both position and momentum spaces can be expressed in terms of the so-called ''entropy of Hermite polynomials,'' i.e., the quantity S-n(H):= -integral(-infinity)(+infinity)H(n)(2)(x)log H-n(2)(x)e(-x2) dx. These polynomials are instances of the polynomials orthogonal with respect to the Freud weights w(x) = exp(-\x\(m)), m>0. Here, a very precise and general result of the entropy of Freud polynomials recently established by Aptekarev et al. [J. Math. Phys: 35, 4423-4428 (1994)], specialized to the Hermite kernel (case m = 2), leads to an important refined asymptotic expression for the information entropies of very elicited states (i.e., for large n) in both position and momentum spaces, to be denoted by S-rho and S-gamma, respectively. Briefly, it is shown that, for large values of n, S-rho + 1/2log lambda similar or equal to log(pi root 2n/e) + o(1) and S-gamma - 1/2log lambda similar or equal to log(pi root 2n/e)+ o(1), so that S-rho + S-gamma similar or equal to log(2 pi(2)n/e(2))+o(1) in agreement with the generalized indetermination relation of Byalinicki-Birula and Mycielski [Commun. Math. Phys. 44, 129-132 (1975)]. Finally, the rate of convergence of these two information entropies is numerically analyzed. In addition, using a Rakhmanov result, we describe a totally new proof of the leading term of the entropy of Freud polynomials which, naturally, is just a weak version of the aforementioned general result. (C) 1995 American Institute of Physics.