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We investigate field quantization in high-curvature geometries. The modelsand calculations can help with understanding the elastic and inelasticscattering of photons and electrons in nanostructures and probe-like metallicdomains. The results find important applications in high-resolution photonicand electronic modalities of scanning probe microscopy, nano-optics,plasmonics, and quantum sensing. Quasistatic formulation, leading tononretarded quantities, is employed and justified on the basis of thenanoscale, here subwavelength, dimensions of the considered domains ofinterest. Within the quasistatic framework, we represent the nanostructurematerial domains with frequency-dependent dielectric functions. Quantitiesassociated with the normal modes of the electronic systems, the nonretardedplasmon dispersion relations, eigenmodes, and fields are then calculated forseveral geometric entities of use in nanoscience and nanotechnology. From theclassical energy of the charge density oscillations in the modelednanoparticle, we then derive the Hamiltonian of the system, which is used forquantization. The quantized plasmon field is obtained and, employing aninteraction Hamiltonian derived from the first-order perturbation theory withinthe hydrodynamic model of the electron gas, we obtain an analytical expressionfor the radiative decay rate of the plasmons. The established treatment isapplied to multiple geometries to investigate the quantized charge densityoscillations on their bounding surfaces. Specifically, using one sheet of atwo-sheeted hyperboloid of revolution, paraboloid of revolution, andcylindrical domains, all with one infinite dimension, and the finite spheroidaland toroidal domains are treated. ...

Field Quantization for Radiative Decay of Plasmons in Finite and Infinite Geometries