Position and Momentum Information Entropies of the Harmonic Oscillator and Logarithmic Potential of Hermite Polynomials Jorge S Anchez-ruiz

The exact analytical values of the position and momentum information entropies for the stationary states of the one-dimensional quantum harmonic oscillator are only known for the ground (n = 0) and rst excited (n = 1) states. In the general case, the problem of calculating these entropies reduces to the evaluation of the logarithmic potential V n (t) = ? R 1 ?1 (H n (x)) 2 ln jx ? tje ?x 2 dx at t = x n;i (i = 1; : : : ; n), where x n;i is the i-th zero of the Hermite polynomial H n (x). Here, a closed analytical formula for V n (t) in terms of 1 F 1 and 2 F 2 hypergeometric functions is obtained, which in turn provides us with analytical expressions for the entropies when the exact location of the zeros of H n (x) is known. The complete asymp-totic expansion of V n (t) for jtj 1, which is expressed in terms of 2 F 0 and 3 F 1 (divergent) hypergeometric series, is also obtained. Finally, it is shown that the exact formula for V n (t) can be written as an innnite series involving the Gauss (2 F 1) hypergeometric function, which allows the entropies to be expressed in terms of the even-order spectral moments of Hermite polynomials, 2k (n) = n ?1 P n i=1 (x n;i) 2k. The asymptotic (n ! 1) limit of this alternative expression for the entropies is discussed.