Quite general, analytical (both exact and approximate) forms for discrete
probability distributions (PD's) that maximize Tsallis entropy for a fixed
variance are here investigated. They apply, for instance, in a wide variety of
scenarios in which the system is characterized by a series of discrete
eigenstates of the Hamiltonian. Using these discrete PD's as "weights" leads to
density operators of a rather general character. The present study allows one
to vividly exhibit the effects of non-extensivity. Varying Tsallis'
non-extensivity index q one is seen to pass from unstable to stable systems
and even to unphysical situations of infinite energy.Comment: 22 page