2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/95968We consider the thermal equilibrium distribution at inverse temperature $β$, or canonical ensemble, of the wave function $Ψ$ of a quantum system. Since $L^2$ spaces contain more nondifferentiable than differentiable functions, and since the thermal equilibrium distribution is very spread-out, one might expect that $Ψ$ has probability zero to be differentiable. However, we show that for relevant Hamiltonians the contrary is the case: with probability one, $Ψ$ is infinitely often differentiable and even analytic. We also show that with probability one, $Ψ$ lies in the domain of the Hamiltonian.16 pages LaTeX, no figuresMathematical PhysicsQuantum Physics82B10; 60G15; 60G17text