  This manuscript provides a general approach to the investigation of field
quantization in high-curvature geometries. The models and calculations can help
with understanding the elastic and inelastic scattering of photons and
electrons in nanostructures and probe-like metallic domains. The results find
important applications in high-resolution photonic and electronic modalities of
scanning probe microscopy, nano-optics, plasmonics, and quantum sensing.
Quasistatic formulation, leading to nonretarded quantities, is employed and
justified on the basis of the nanoscale, here subwavelength, dimensions of the
considered domains of interest. Within the quasistatic framework, the
nanostructure material domains with frequency-dependent dielectric functions
are presented. Quantities associated with the normal modes of the electronic
systems, the nonretarded plasmon dispersion relations, eigenmodes, and fields
are then calculated for several geometric entities of use in nanoscience and
nanotechnology. From the classical energy of the charge density oscillations in
the modeled nanoparticle, the Hamiltonian of the system, which is used for
quantization, is derived. The quantized plasmon field is obtained and,
employing an interaction Hamiltonian derived from the first-order perturbation
theory within the hydrodynamic model of the electron gas, an analytical
expression for the radiative decay rate of the plasmons could be obtained. The
established treatment could be applied to multiple geometries to investigate
the quantized charge density oscillations on their bounding surfaces.
