Asymptotic formula for the quantum entropy of position in energy eigenstates

Mark W. Coffey

Fumio Hiai, Dėnes Petz

Fumio Hiai, Dėnes Petz

Jorge Sánchez-Ruiz

General asymptotic formulae are derived by means of the WKB approximation for the continuous and discrete Rényi entropies of position of one-dimensional quantum systems in energy eigenstates, in terms of the corresponding entropies for a microcanonical ensemble of analogous classical systems. These results are checked in the simplest particular case of the infinite potential well, where the asymptotic formula for continuous entropies holds as an exact identity. For the discrete entropies, analytical expressions are obtained from which the asymptotic formulae given for the limiting cases of large and small measurement resolution can both be verified.

Shigeo Akashi

Jorge Sánchez-Ruiz

The problem of calculating the information entropy in both position and momentum spaces for the nth stationary state of the one-dimensional quantum harmonic oscillator reduces to the evaluation of the logarithmic potential Vn(t)=−∫−∞∞(Hn(x))2 ln |x−t|e−x2 dx at the zeros of the Hermite polynomial Hn(x). Here, a closed analytical expression for Vn(t) is obtained, which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of Hn(x) is known. An inequality for the values of Vn(t) at the zeros of Hn(x) is conjectured, which leads to a new, nonvariational, upper bound for the entropies. Finally, the exact formula for Vn(t) is written in an alternative way, which allows the entropies to be expressed in terms of the even-order spectral moments of the Hermite polynomials. The asymptotic (n≫1) limit of this alternative expression for the entropies is discussed, and the conjectured upper bound for the entropies is proved to be asymptotically valid.