Free to Harmonic Unitary Transformations in Quantum and Koopman Dynamics
arXiv:2207.09515 · doi:10.1088/1751-8121/ac97cf
Abstract
It has long been known that there exists a coordinate transformation which exactly maps the quantum free particle to the quantum harmonic oscillator. Here we extend this result by reformulating it as a unitary operation followed by a time coordinate transformation. We demonstrate that an equivalent transformation can be performed for classical systems in the context of Koopman von-Neumann (KvN) dynamics. We further extend this mapping to dissipative evolutions in both the quantum and classical cases, and show that this mapping imparts an identical time-dependent scaling on the dissipation parameters for both types of dynamics. The derived classical procedure presents a number of opportunities to import squeezing dependent quantum procedures (such as Hamiltonian amplification) into the classical regime.
5 pages, no figures
References in corpus (7)
- Wigner flow reveals topological order in quantum phase space dynamics
- Inverted Oscillator
- Equivalence between focussed paraxial beams and the quantum harmonic oscillator
- Special-case closed form of the Baker-Campbell-Hausdorff formula
- Equivalence between free quantum particles and those in harmonic potentials and its application to instantaneous changes
- Ermakov-Lewis invariant in Koopman-von Neumann mechanics
- Entropy non-conservation and boundary conditions for Hamiltonian dynamical systems