Quantization of the Vibrations of a Thin Elastic Plate

Eliot Heinrich, Dennis P. Clougherty

Published 2019-02-28Version 1

Suspended thin films have been successfully used as high-Q mechanical oscillators in hybrid optomechanical systems to study fundamental quantum mechanical effects. Motivated by these experiments, we consider a Hamiltonian description of the vibrations of a clamped, elastic circular plate. The Hamiltonian of this system features a potential energy with two distinct contributions: one that depends on the local mean curvature of the plate, and a second one that depends on its Gaussian curvature. We quantize this model using a complete, orthonormal set of eigenfunctions for the clamped, vibrating plate. The resulting quanta are the flexural phonons of the thin circular plate. As an application, we use this quantized description to calculate the fluctuations in displacement of the plate's center for arbitrary temperature.