Constants of motion and quantum non-relativistic motion of a charged particle on a flat surface with transversal magnetic field

TL;DR

This paper investigates the classical and quantum motion of a charged particle on a flat surface under a magnetic field, identifying constants of motion and deriving solutions in different gauges, revealing flux quantization from symmetry invariances.

Contribution

It introduces new non-separable solutions for the symmetric gauge and links flux quantization to invariance under unitary transformations.

Findings

Derived Landau's solution for magnetic gauge

Found non-separable solutions in symmetric gauge

Connected flux quantization to invariance of solutions

Abstract

The motion of a charged particle moving on a flat surface is studied through the constants of motion associated to the system, given the magnetic gauge. The usual Landau' solution and the non separable solution for the Landau's gauge are found, and new non separable solution for the symmetric gauge is given. As a consequence of this, the quantization of the magnetic flux results from the invariance of the solutions under the unitary transformations which arise from the operators constants of motion.