A constant of quantum motion in two dimensions in crossed magnetic and electric fields

TL;DR

This paper investigates the quantum behavior of a particle in two dimensions under perpendicular magnetic and electric fields, constructing an invariant of motion for certain potentials and showing Hamiltonian equivalence to an effective form.

Contribution

It introduces a method to construct a non-trivial invariant of motion for quantum particles in crossed fields and proves Hamiltonian equivalence to an effective Hamiltonian.

Findings

Invariant of motion constructed for specific potentials

Hamiltonian unitarily equivalent to an effective Hamiltonian

Observable of kinetic energy commutes with the effective Hamiltonian

Abstract

We consider the quantum dynamics of a single particle in the plane under the influence of a constant perpendicular magnetic and a crossed electric potential field. For a class of smooth and small potentials we construct a non-trivial invariant of motion. Do to so we proof that the Hamiltonian is unitarily equivalent to an effective Hamiltonian which commutes with the observable of kinetic energy.