Gaussians as Solutions of the Schrodinger Equation and Maximum Entropy Calculations

Recently, maximum entropy calculations have become associated with solutions of the single particle Schrodinger equation. In some cases, the single particle Schrodinger equation has been altered to represent a single quantum particle with a temperature-Shannon entropy term added. In other cases, Tsalis entropy has been maximized subject to constraints to obtain the ground state single particle wavefunction for T (temperature)=0. In both cases, Gaussian wavefunctions have played a key role. The purpose of this note, is to examine the link between the Schrodinger equation and maximum entropy for the case of Gaussian wavefunctions (and hence densities). In particular, we show that a Shannon entropy term is included in the single particle Schrodinger equation for a Gaussian wavefunction and an r2 type of potential. The quantum density in such a case also maximizes Shannon´s entropy subject to a constraint on the integral of  r2  times density. We also point out that the solution of the time independent Schrodinger equation represents a resonance exp(iEt) where E is the energy which does not seem to appear in statistical mechanics. If there is a link between the Schrodinger equation and maximization of entropy, perhaps classical statistical systems can also exhibit a time resonance.