Entropy Of Orthogonal Polynomials With Freud Weights And Information Entropies Of The Harmonic Oscil

The information entropy of the harmonic oscillator potential V(x)=1/2λx2 in both position and momentum spaces can be expressed in terms of the so‐called ‘‘entropy of Hermite polynomials,’’ i.e., the quantity Sn(H):= −∫−∞+∞H2n(x)log H2n(x) e−x2dx. 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 excited states (i.e., for large n) in both position and momentum spaces, to be denoted by Sρ and Sγ, respectively. Briefly, it is shown that, for large values of n, Sρ+1/2logλ≂log(π√2n/e)+o(1) and Sγ−1/2log λ≂log(π√2n/e)+o(1), so that Sρ+Sγ≂log(2π2n/e2)+o(1) in agreement with the generalized indetermination relation of Byalinicki‐Birula and Mycielski [Commun. M...

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