Quantization of resistivity as consequence of symmetry invariance

Abstract

The Schr\"odinger equation for an electron under the influence of an electromagnetic field is analyzed based on the conserved operators of the system when the magnetic field is described by Landau's gauge. It is shown that the Lorentz force can be recovered only if two conserved generalized momentum operators are considered: one along the $x$-axis and the second one along $y$-axis; otherwise, the system cannot be fully described. Based on the general solution found, a ground state is built which has the characteristic of having quantized resistivity proportional to integer multiples of the von Klitzing's constant when it is invariant under a unitary transform.