Smoothness of Wave Functions in Thermal Equilibrium

TL;DR

This paper proves that wave functions in thermal equilibrium are almost surely infinitely differentiable and analytic, contrary to expectations based on their spread-out distribution.

Contribution

It demonstrates that for relevant Hamiltonians, thermal equilibrium wave functions are almost surely smooth and lie within the Hamiltonian's domain.

Findings

Wave functions are infinitely differentiable with probability one.

Wave functions are analytic with probability one.

Wave functions lie in the domain of the Hamiltonian with probability one.

Abstract

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.