Quantum mechanical version of the classical Liouville theorem

Abstract

In terms of the coherent state evolution in phase space, we present a quantum mechanical version of the classical Liouville theorem. The evolution of the coherent state from |z〉 to |sz − rz*〉 corresponds to the motion from a point z (q,p) to another point sz − rz* with |s|2 − |r|2 = 1. The evolution is governed by the so-called Fresnel operator U(s,r) that was recently proposed in quantum optics theory, which classically corresponds to the matrix optics law and the optical Fresnel transformation, and obeys group product rules. In other words, we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space, which seems to be a combination of quantum statistics and quantum optics.