  This paper studies quantum systems with a finite number of degrees of freedom
in the context of non-extensive thermodynamics. A trial density matrix,
obtained by heuristic methods, is proved to be the equilibrium density matrix.
If the entropic parameter q is larger than 1 then existence of the trial
equilibrium density matrix requires that q is less than some critical value q_c
which depends on the rate by which the eigenvalues of the hamiltonian diverge.
Existence of a unique equilibrium density matrix is proved if in addition q<2
holds. For q between 0 and 1, such that 2<q+q_c, the free energy has at least
one minimum in the set of trial density matrices. If a unique equilibrium
density matrix exists then it is necessarily one of the trial density matrices.
Note that this is a finite rank operator, which means that in equilibrium high
energy levels have zero probability of occupancy.
