  Solid state physics deals with systems composed of atoms with strongly bound
electrons. The tunneling probability of each electron is determined by
interactions that typically extend to neighboring sites, as their corresponding
wave amplitudes decay rapidly away from an isolated atomic core. This kind of
description is essential to material science, and it rules the electronic
transport properties of metals, insulators and other condensed matter systems.
The corresponding phenomenology is well captured by tight-binding models, where
the electronic band structure emerges from atomic orbitals of isolated atoms
plus their coupling to neighboring sites in a cristal. In this work, a
mechanical system that emulates dynamically a tightly bound electron is built.
This is done by connecting mechanical resonators via locally periodic aluminum
bars acting as couplers. When the frequency of a particular resonator lies
within the frequency gap of a coupler, the vibrational wave amplitude imitates
a bound electron orbital. The localization of the wave at the resonator site
and its exponential decay along the coupler are experimentally verified. The
quantum dynamical tight-binding model and frequency measurements in mechanical
structures show an excellent agreement.
