  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.
