The quantum-mechanical movement of an electron in crossed uniform electric and magnetic fields

Abstract

The quantum-mechanical task on the electron movement in external crossed electric and magnetic fields is considered. The solution of the Schr¨odinger equation has been found for the electron wave function in such electromagnetic field configuration. A part of the electron wave function describing the electron movement in a plane normal to the magnetic field is expressed by the well-known Hermite function. The electron moves as a free particle along the magnetic field. The electron drift movement along the axis normal to the plane formed by the electric and magnetic field vectors arises in the crossed electric and magnetic fields. The possible electron energy spectrum has been found. It consists of four parts, namely: a quantified electron energy spectrum related to the electron movement energy in the plane normal to the magnetic field direction; the electron movement energy along the magnetic field; the energy of the electron drift movement defined by the electric and magnetic field strengths; and the potential energy of an electron in the external electric field.