Time-dependent conserved operators for autonomous systems and quantization of resistance

Abstract

Two systems for a charged particle are studied, the first one when it is under the effect of a constant electric field, and the second one when it is under the effect of a constant electromagnetic field. For both systems, it is possible to find time-dependent conserved operators that can be used to derive time-dependent wave functions to the complete Schr\"odinger equation, such that the time variable is not separable from the space coordinates. At the same time, these conserved operators are used to build up a unitary operators, which define the symmetries of the systems. Then, it is shown that the invariance of the wave function under the action of these unitary operators leads to the quantization of resistance as integer multiples of the Klitzing's constant.