KvN mechanics approach to the time-dependent frequency harmonic oscillator

Abstract

Using the Ermakov-Lewis invariants appearing in KvN mechanics, the time-dependent frequency harmonic oscillator is studied. The analysis builds upon the operational dynamical model, from which it is possible to infer quantum or classical dynamics; thus, the mathematical structure governing the evolution will be the same in both cases. The Liouville operator associated with the time-dependent frequency harmonic oscillator can be transformed using an Ermakov-Lewis invariant, which is also time dependent and commutes with itself at any time. Finally, because the solution of the Ermakov equation is involved in the evolution of the classical state vector, we explore some analytical and numerical solutions.

Koopman wavefunctions and classical–quantum correlation dynamics

“…Therein, Sudarshan proposed to couple classical and quantum dynamics by exploiting the Koopman-von Neumann (KvN) formulation of classical dynamics in terms of classical wavefunctions [25,26]. Rediscovered in several instances [27,28], this reformulation of classical mechanics has been attracting increasing attention [29][30][31][32][33][34] (Wilczek F 2015, unpublished data). See also [35] with quantum degrees of freedom by invoking special superselection rules to enforce physical consistency [24].…”

Light propagation in inhomogeneous media, coupled quantum harmonic oscillators and phase transitions

“…where ψ nx (x) and ψ ny (y) are Hermite-Gauss functions; i.e., eigenfunctions of the harmonic oscillators given in (11). This allows us to write the initial transformed field ( 16) as E(x, y, z = 0) = ψ nx (x)ψ ny (y) and therefore write the solution of Equation ( 11) as…”

“…The existence of analogies between quantum and classical mechanics has been applied for many years, particularly in the generation of mathematical tools to provide solutions of optical problems and vice versa [1][2][3][4][5][6][7][8][9][10][11]. The reason is that the optical paraxial wave equation is mathematically equivalent to the stationary Schrödinger equation, and on the other hand, Helmholtz equation is isomorphic to the time-independent Schrödinger equation.…”

Entropy nonconservation and boundary conditions for Hamiltonian dynamical systems

“…These path integral formulations may be usefully applied with classical many-body diagrammatic methods [30] and for the derivation and analysis of generalized Langevin equations [31]. KvN has been used to study specific phenomena, including examinations of dissipative behavior [32], linear representations of nonlinear dynamics [33], analysis of the time-dependent harmonic oscillator [34], and other industrial applications [35,36].…”