Quantum Systems: A Mathematical Journey from Few to Many Particles

On Wigner and Bohmian measures in semi-classical quantum dynamics

Christof Sparber

University of Illinois at Chicago


We report on a series of recent results obtained jointly with: P. Markowich, C. Klein, A. Figalli, and T. Paul. We consider a class of phase-space measures, which naturally arise in the Bohmian interpretation of quantum mechanics. These, so-called, Bohmian measures describe the time-evolution of the quantum mechanical position and velocity densities and are given by the push-forward under the Bohmian flow on phase space. The latter can be seen as a perturbation of the classical Hamiltonian flow. We study the classical limit of this flow and of the associated Bohmian measure, whenever the dimensionless Planck's constant tends to zero. Connections to the, by now, classical theory of Wigner measures are also discussed.