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

Abstract

In the probability representation of quantum mechanics, quantum states are represented by a classical probability distribution, the marginal distribution function (MDF), whose time dependence is governed by a classical evolution equation. We find and explicitly solve, for a wide class of Hamiltonians, new equations for the Green's function of such an equation, the so-called classical propagator. We elucidate the connection of the classical propagator to the quantum propagator for the density matrix and to the Green's function of the Schrödinger equation. Within the new description of quantum mechanics we give a definition of coherence solely in terms of properties of the MDF and we test the new definition recovering well known results. As an application, the forced parametric oscillator is considered . Its classical and quantum propagator are found, together with the MDF for coherent and Fock states.

29 pages, RevTex, 6 eps-figures, to appear on Phys. Rev. A

References in corpus (6)

• f-Oscillators and Nonlinear Coherent States
• Symplectic tomography as classical approach to quantum systems
• Classical-Like Description of Quantum Dynamics by Means of Symplectic Tomography
• Reconstructing the density operator by using generalized field quadratures
• Density Matrix From Photon Number Tomography
• Quantum singular oscillator as a model of two-ion trap: an amplification of transition probabilities due to small time variations of the binding potential

Cited by in corpus (11)

• Diffraction in time in terms of Wigner distributions and tomographic probabilities
• The Pauli Equation for Probability Distributions
• Duality symmetry for star products
• Probability representation in quantum field theory
• On the $\star$-product quantization and the Duflo map in three dimensions
• Simulation of tunneling in the quantum tomography approach
• Probability Distributions and Hilbert Spaces: Quantum and Classical Systems
• Classical evolution in quantum systems
• Permutation symmetry for the tomographic probability distribution of a system of identical particles
• Continuous measurements in probability representation of quantum mechanics
• Probability Representation of Quantum Mechanics: Comments and Bibliography