Entropy of Hermite polynomials with application to the harmonic oscillator

We analyse the entropy of Hermite polynomials and orthogonal polynomials for the Freud weights w(x) = exp(−|x|) on R and show how these entropies are related to information entropy of the one-dimensional harmonic oscillator. The physical interest in such entropies comes from a stronger version of the Heisenberg uncertainty principle, due to Bialynicki-Birula and Mycielski, which is expressed as a lower bound for the sum of the information entropies of a quantum-mechanical system in the position space and in the momentum space. 1 The information entropies of the harmonic oscillator The Schrödinger equation in D dimensions is given by ( − 2 ∇ + V ) ψ = Eψ, where the potential V and the wave function ψ are functions of x = (x1, . . . , xD) and E is the energy. The wave function ψ is normalized in such a way that ρ(x) = |ψ(x)| is a probability density in position space. If ψ̂(p) is the Fourier transform of the wave function ψ, then by the Plancherel formula γ(p) = |ψ̂(p)| is also a probability density, but now in the momentum space. The information entropy for the quantummechanical system with potential V is then given by Sρ = − ∫ RD ρ(x) log ρ(x) dx ∗The author is a Senior Research Associate of the Belgian National Fund for Scientific Research (NFWO). This work is supported by INTAS project 93-219. 1991 Mathematics Subject Classification : 33C45.