Quantum mechanical model for charge excitation: Surface binding and dispersion

TL;DR

This paper develops an idealized quantum mechanical model to describe surface charge excitations and dispersion of electromagnetic waves near a plane, linking microscopic scales to collective phenomena like surface plasmons.

Contribution

It introduces an exact analytical approach to derive dispersion relations for charge excitations using a linearized Hartree-type equation and Mittag-Leffler series expansion.

Findings

Derived exact dispersion relation for surface charge excitations.

Established asymptotic expansion matching classical hydrodynamic predictions.

Formulated a functional equation for the scattering amplitude in a 3D quantum model.

Abstract

By an idealized quantum mechanical model, we formally describe the dispersion of nonretarded electromagnetic waves that express charge density oscillations near a fixed plane in three spatial dimensions (3D) at zero temperature. Our goal is to capture the interplay of microscopic scales that include a confinement length in the emergence of the surface plasmon, a collective low-energy charge excitation in the vicinity of the plane. We start with a time-dependent Hartree-type equation in 3D. This model accounts for particle binding to the plane and the repulsive Coulomb interaction associated with the induced charge density relative to the ground state. By linearizing the equation of motion, we formulate a homogeneous integral equation for the scattering amplitude of the particle wave function in the (z-) coordinate vertical to the plane. For a binding potential proportional to a negative delta function and symmetric-in-z wave function, we apply the Laplace transform with respect to positive z and convert the integral equation into a functional equation that involves several values of the transformed solution. The scattering amplitude and dispersion relation are derived exactly in terms of rapidly convergent series via the Mittag-Leffler theorem. In the semiclassical regime, our result furnishes an asymptotic expansion for the energy excitation spectrum. The leading-order term is found in agreement with the prediction of a classical hydrodynamic model based on a projected-Euler-Poisson system.