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

TL;DR

The work investigates how time-dependent conserved operators can be used to solve the Schrödinger equation for autonomous quantum systems under constant electric or electromagnetic fields, yielding non-separable time–space wavefunctions and revealing underlying symmetries. By applying the Lewis–Riesenfeld invariant framework, it identifies conserved operators that generate unitary transformations leaving the dynamics invariant. A key result is that wavefunction invariance under these unitary symmetries enforces a quantized resistance $R=\frac{h}{q^{2}}\,n$, aligning with the von Klitzing constant and echoing quantization phenomena similar to the quantum Hall effect, even in single-particle, low-dimensional settings. The findings highlight symmetry and conserved quantities as fundamental drivers of quantized transport, connecting time-dependent invariants to Landau-level physics in the magnetic case.

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 Schrödinger 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.