LOGARITHMIC POTENTIAL OF HERMITE-POLYNOMIALS AND INFORMATION ENTROPIES OF THE HARMONIC-OSCILLATOR EIGENSTATES

The problem of calculating the information entropy in both position an d momentum spaces for the nth stationary state of the one-dimensional quantum harmonic oscillator reduces to the evaluation of the logarithm ic potential V-n(t) = -integral(-infinity)(infinity) (H-n(x))(2) ln \x -t\e(-x2) dx at the zeros of the Hermite polynomial H-n(x). Here, a cl osed analytical expression for V-n(t) is obtained, which in turn yield s an exact analytical expression for the entropies when the exact loca tion of the zeros of H-n(x) is known. An inequality for the values of V-n(t) at the zeros of H-n(x) is conjectured, which leads to a new, no nvariational, upper bound for the entropies. Finally, the exact formul a for V-n(t) is written in an alternative way, which allows the entrop ies to be expressed in terms of the even-order spectral moments of the Hermite polynomials. The asymptotic (n greater than or equal to 1) li mit of this alternative expression for the entropies is discussed, and the conjectured upper bound for the entropies is proved to be asympto tically valid. (C) 1997 American Institute of Physics.