Quantum tunneling in a capacitor Ampere's law of Maxwell's equation includes displacement current as the Maxwell's correction.
Suppose consider a capacitor with a thin distance of separation. For an applied voltage do they posses some tunneling current?. If so, Do we have to include these tunneling current in the Maxwell's equation which will be again a correction for the maxwells equation(inclusion of tunneling current)? 
 A: Quantum tunneling is defined when there are wavefucntions describing the particles in potential wells. The potential wells are defined in the solid plates of the capacitors and any effect of tunneling can only happen over distances where a potential well can be modeled accross the capacitor gap.
Quantum mechanical effects obey Heisenberg's uncertainty principle, and are bounded by it . 
$ΔxΔp>h/{2π}$
When h/{2π}$  is effectively zero, the problem is well described by the classical models. Lets make an order of magnitude estimate:
$h/{2π}$ is of order $10^{-34}$ Joule.second=$kilogram Meter^2/second^2$
The smallest distance between two metal plates so as to have a capacitor might be a few microns .   micron= $10^{-6}$ meters, 
The drift velocity of electrons in a circuit is of the order ~$10^6$meters/second 
The mass of the electron is ~$10^{-30}$kilograms
All these order of magnitude, and we get the HUP satisfied : $10^{-30}>10^{-34}$
That is why for nanotechnology quantum mechanical effects are important. For micron plate separation a tunneling model might give a barely measurable change in the AC currents . For smaller distances there would no longer be a capacitor to model.
With this estimate, I expect that  the classical theory is unaffected for millimeter plate separation distances . DC currents will not exist, after transience, and AC currents will be  well modeled  by the classical maxwell equations.
p.s. if you google "nanotechnology capacitors and quantum tunneling effects" a list comes up , and the problem is being considered in those dimensions.
