Ermakov-Lewis Invariant in Koopman-von Neumann Mechanics

Dynamical Invariants for Generalized Coherent States via Complex Quantum Hydrodynamics

“…In the following, we use a generalized Gaussian wave packet ansatz (6) for the solution of Equation ( 5) that provides the classical Newtonian equation of motion and the parametric Newton-type equation (Ermakov equation) that was used by Ermakov to eliminate the frequency ω(t) and to obtain the dynamical invariant. We will not follow this procedure, but instead use the wave packet (6) to define complex Bohmian quantities.…”

“…The quantum state corresponding to (6) in position space can be obtained by a Fourier transformation and can be written in the form:…”

“…In this Section, the dynamical invariant associated with a linear potential V = −E X along the x-axis is obtained. For that purpose, recall the quantities ( 11)-( 14) computed for a generalized coherent state (6),…”

“…Recently its mathematical properties are still studied [3], as well as its relation to quantum mechanical dynamics. Indeed, "it turns out that the evolution of the quantum states can be reduced to the solutions of the classical Ermakov-Milne-Pinney (EMP) equation" [4], especially for time-dependent quadratic potentials [5,6] and coupled oscillators [7]. The interest in this invariant is due to the possibility of studying the parametric oscillator in a suitable way.…”

Unitary representation of the Poincaré group for classical relativistic dynamics

“…This old approach is due to Koopman [5] and von Neumann [6]. Whether for derivation of purely classical results or for comparison between quantum and classical mechanics, the Koopman-von Neumann formalism (hereafter abbreviated as KvN) has received increasing attention in the past two decades (see [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]), the possibility of formulating quantum-classical hybrid theories has also increased the interest in this formalism [22,23,24,25]. The existence of the KvN theory raises the question of the classification of the unitary representations of the groups of space-time symmetries in the context of classical mechanics.…”