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This manuscript provides a general approach to the investigation of fieldquantization in high-curvature geometries. The models and calculations can helpwith understanding the elastic and inelastic scattering of photons andelectrons in nanostructures and probe-like metallic domains. The results findimportant applications in high-resolution photonic and electronic modalities ofscanning probe microscopy, nano-optics, plasmonics, and quantum sensing.Quasistatic formulation, leading to nonretarded quantities, is employed andjustified on the basis of the nanoscale, here subwavelength, dimensions of theconsidered domains of interest. Within the quasistatic framework, thenanostructure material domains with frequency-dependent dielectric functionsare presented. Quantities associated with the normal modes of the electronicsystems, the nonretarded plasmon dispersion relations, eigenmodes, and fieldsare then calculated for several geometric entities of use in nanoscience andnanotechnology. From the classical energy of the charge density oscillations inthe modeled nanoparticle, the Hamiltonian of the system, which is used forquantization, is derived. The quantized plasmon field is obtained and,employing an interaction Hamiltonian derived from the first-order perturbationtheory within the hydrodynamic model of the electron gas, an analyticalexpression for the radiative decay rate of the plasmons could be obtained. Theestablished treatment could be applied to multiple geometries to investigatethe quantized charge density oscillations on their bounding surfaces.

General Approach to Study Geometric Effects on Classical & Quantum  Fields & Eigenmodes of Particles with Finite & Infinite Extend for  Applications in Plasmonics & Scanning Probe Microscopy