Before revision: However, while this "toy model" is superficially attractive, the Ehrenfest theorem seems to suggest that since the electronic motion in the direction is that of a bound state that confines it to the 2D surface, the time-averaged electric field (i.e., including that of the potential that binds it to the 2D surface) that the electron experiences must be zero !
After revision: However, while this "toy model" is superficially attractive, the Ehrenfest theorem seems to suggest that since the electronic motion in the direction is that of a bound state that confines it to the 2D surface, the space-averagedgiven the connection between the time derivative of spatially averaged momentum, which vanishes as a bound state, and the spatial derivative of potential, which gives the electric field electric field (i.e., including that of the potential that binds it to the 2D surface) that the electron experiences must be zero !