# Quantum and thermal fluctuations of a thin elastic plate

## TL;DR

This paper develops a quantum mechanical model of a thin elastic circular plate, quantizes its vibrations into flexural phonons, and analyzes displacement fluctuations at various temperatures, comparing them to elastic membranes.

## Contribution

It introduces a Hamiltonian framework for quantizing vibrations of a thin elastic plate and calculates temperature-dependent displacement fluctuations.

## Key findings

- Displacement fluctuation profiles differ between plates and membranes, especially at low temperatures.
- At higher temperatures, the fluctuation profile of the plate develops additional extrema.
- The model provides insights into quantum and thermal effects on elastic plate vibrations.

## Abstract

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 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 for arbitrary temperature. We compare the fluctuation profile with that from an elastic membrane under tension. At low temperature, we find that while both profiles have a circular ring of local maxima, the ring in the membrane profile is much more pronounced and sharper. We also note that with increasing temperature the plate profile develops two additional rings of extrema.