  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.
