Effective potentials from semiclassical truncations

Canonical variables for the Poisson algebra of quantum moments are introduced here, expressing semiclassical quantum mechanics as a canonical dynamical system that extends the classical phase space. New realizations for up to fourth order in moments for a single classical degree of freedom and to second order for a pair of classical degrees of freedom are derived and applied to several model systems. It is shown that these new canonical variables facilitate the derivation of quantum-statistical quantities and effective potentials. Moreover, by formulating quantum dynamics in classical language, these methods result in new heuristic pictures, for instance, of tunneling, that can guide further investigations.