Smoothness of Wave Functions in Thermal Equilibrium

Details Smoothness of Wave Functions in Thermal Equilibrium Smoothness of Wave Functions in Thermal Equilibrium

2005-09-13

arxiv.org/abs/math-ph/0509028v1

J. Math. Phys. 46 (2005) 112104

Physics

Mathematical Physics

16 pages LaTeX, no figures

Scientific paper

10.1063/1.2109767

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.

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