  We theoretically investigate the stability and linear oscillatory behavior of
a naturally unstable particle whose potential energy is harmonically modulated.
We find this fundamental dynamical system is analogous in time to a quantum
harmonic oscillator. In a certain modulation limit, a.k.a. the Kapitza regime,
the modulated oscillator can behave like an effective classic harmonic
oscillator. But in the overlooked opposite limit, the stable modes of
vibrations are quantized in the modulation parameter space. By analogy with the
statistical interpretation of quantum physics, those modes can be characterized
by the time-energy uncertainty relation of a quantum harmonic oscillator.
Reducing the almost-periodic vibrational modes of the particle to their
periodic eigenfunctions, one can transform the original equation of motion to a
dimensionless Schr\"odinger stationary wave equation with a harmonic potential.
This reduction process introduces two features reminiscent of the quantum
realm: a wave-particle duality and a loss of causality that could legitimate a
statistical interpretation of the computed eigenfunctions. These results shed
new light on periodically time-varying linear dynamical systems and open an
original path in the recently revived field of quantum mechanical analogs.
