Foldy–Wouthuysen Transformation and Structured States of a Graphene Electron in External Fields and Free (2 + 1)-Space

Abstract

The relativistic Foldy–Wouthuysen transformation is used for an advanced description of a free and interacting planar graphene electron. The exact Foldy–Wouthuysen Hamiltonian of a graphene electron in a uniform and a nonuniform magnetic field is derived. The exact energy spectrum agreeing with experimental data and exact Foldy–Wouthuysen wave eigenfunctions are obtained. These eigenfunctions describe structured states in (2 + 1)-space. It is proven that the Hermite–Gauss beams exist even in the free space. In the structured Hermite–Gauss states, graphene electrons acquire nonzero effective masses dependent on a quantum number and move with group velocities which are less than the Fermi velocity.