Time-Dependent Invariants and Green's Functions in the Probability Representation of Quantum Mechanics

摘要

In the probability representation of quantum mechanics, quantum states arerepresented by a classical probability distribution, the marginal distributionfunction (MDF), whose time dependence is governed by a classical evolutionequation. We find and explicitly solve, for a wide class of Hamiltonians, newequations for the Green's function of such an equation, the so-called classicalpropagator. We elucidate the connection of the classical propagator to thequantum propagator for the density matrix and to the Green's function of theSchr\"odinger equation. Within the new description of quantum mechanics we givea definition of coherence solely in terms of properties of the MDF and we testthe new definition recovering well known results. As an application, the forcedparametric oscillator is considered . Its classical and quantum propagator arefound, together with the MDF for coherent and Fock states.