Ground State Quantum Oscillator and Lack of Information?

  The spatial and momentum wavefunctions of a ground state quantum oscillator are Gausians in x and p respectively. A Gaussian is also the distribution which maximizes Shannon’s entropy - Sum over i P(i) ln(P(i)) for a fixed variance. Usually one would use average energy as a constraint, but for the nonrelativistic oscillator KE = pp/2m and PE=.5kxx so such a constraint is linked to the variance of x and p. 

   In this note we consider the following questions? Why is the ground state of the quantum oscillator (i.e this form) the only quantum solution which matches a maximum entropy case (for a fixed variance)? Is there a link to a lack of information? Why do higher oscillator states represent non-maximum distributions for fixed variances? Can this also be linked to information? Here we apply the term  information to actual information in the system not ln(P(i)).