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.