Basic law of Gauss, Maxwell: The area density of charge in $As/m^2$ at the plane surface of a metallic conductor is the definition of the field D. The Maxell equation $\text{div} D (A s / m^3) = \rho (A s /m^3)$ degenerates to $D(x>0) - D(x<0) = \sigma(x=0)$, the surface density.

Division by $\varepsilon_0$ yields the electric force field $E (W s/(A s m = V/m)) =\frac{1}{\varepsilon_0 } \ D( A s/m^2)$

It follows, by the thermodynamic basic laws of minimizing field energy, external plus the thin electron surface layer, that exactly as many electrons needed to shield the inner space of the conductor completely from the external field move to surface states. The density of the surface states falls off again so slow, that the always positive quantum concentration energy measured by $|\partial_x \psi|^2 dx$.

An internal field cannot live inside a conductor, it decays by its induced a current $j=\sigma E $, that rerdiributes charges until $E=0$.

Basic law of Gauss, Maxwell: The area density of charge in $As/m^2$ at the plane surface of a metallic conductor is the definition of the field D. The Maxell equation $\text{div} D (A s / m^3) = \rho (A s /m^3)$ degenerates to $D(x>0) - D(x<0) = \sigma(x=0)$, the surface density.

Division by $\varepsilon_0$ yields the electric force field $E (W s/(A s m = V/m)) =\frac{1}{\varepsilon_0 } \ D( A s/m^2)$

It follows, by the thermodynamic basic laws of minimizing field energy, external plus the thin electron surface layer, that exactly as many electrons needed to shield the inner space of the conductor completely from the external field move to surface states. The density of the surface states falls off again so slow, that the always positive quantum concentration energy measured by $|\partial_x \psi|^2 dx$ does not grow to large.

An internal field cannot live inside a conductor, it decays by its induced a current $j=\sigma E $, that rerdiributes charges until $E=0$.