Parametric resonance for pedestrians: From a swing to squeezing the quantum vacuum

Taking the classical pendulum as its starting point, this paper unearths the profound physics hidden within this familiar model and constructs an investigative path that links macroscopic classical mechanics to microscopic quantum mechanics. Using MATLAB, we first perform numerical simulations and visual analyses of the dynamical behavior of ideal, large-angle, damped, and externally driven pendulums. By next studying a pendulum whose pivot is subjected to periodic vertical vibration, we introduce and elucidate the classical phenomenon of parametric resonance. Building on this bridge, we extend the model into the quantum realm and demonstrate that, in the one-dimensional quantum harmonic oscillator, periodic modulation of a system parameter (e.g., the mass) can excite particles from the quantum vacuum—an effect known as “quantum vacuum amplification.” Simulations of particle-number evolution and Wigner functions carried out with Python corroborate the theoretical predictions. This study provides an intuitive yet inspiring case for undergraduate and graduate physics education.