  Plasmonics has attracted much attention not only because it has useful
properties such as strong field enhancement, but also because it reveals the
quantum nature of matter. To handle quantum plasmonics effects, ab initio
packages or empirical Feibelman d-parameters have been used to explore the
quantum correction of plasmonic resonances. However, most of these methods are
formulated within the quasi-static framework. The self-consistent hydrodynamics
model offers a reliable approach to study quantum plasmonics because it can
incorporate the quantum effect of the electron gas into classical
electrodynamics in a consistent manner. Instead of the standard scattering
method, we formulate the self-consistent hydrodynamics method as an eigenvalue
problem to study quantum plasmonics with electrons and photons treated on the
same footing. We find that the eigenvalue approach must involve a global
operator, which originates from the energy functional of the electron gas. This
manifests the intrinsic nonlocality of the response of quantum plasmonic
resonances. Our model gives the analytical forms of quantum corrections to
plasmonic modes, incorporating quantum electron spill-out effects and
electrodynamical retardation. We apply our method to study the quantum surface
plasmon polariton for a single flat interface.
