Quantum mechanics on phase space : geometry and motion of the Wigner distribution

Abstract

We study the Wigner phase space formulation of quantum mechanics and compare it to the Hamiltonian picture of classical mechanics. In this comparison we focus on the differences in initial conditions available to each theory as well as the differences in dynamics. First we derive new necessary conditions for the admissibility of Wigner functions and interpret their physical meaning. One advantage of these conditions is that they have a natural, geometric interpretation as integrals over polygons in phase space. Furthermore, they hint at what is required beyond the uncertainty principle in order for a Wigner function to be valid. Next we design and implement numerical methods to propagate Wigner functions via the quantum Liouville equation. Using these methods we study the quantum mechanical phenomena of reflection, interference, and tunnelling and explain how these phenomena arise in phase space as a direct consequence of the first quantum correction to classical mechanics.