Smoothness of Wave Functions in Thermal Equilibrium

We consider the thermal equilibrium distribution at inverse temperature $\beta$, or canonical ensemble, of the wave function $\Psi$ 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 $\Psi$ has probability zero to be differentiable. However, we show that for relevant Hamiltonians the contrary is the case: with probability one, $\Psi$ is infinitely often differentiable and even analytic. We also show that with probability one, $\Psi$ lies in the domain of the Hamiltonian.