The guiding center role in defining the eigenstates of an electron in the fractional quantum Hall effect regime: ladder operators approach

Abstract

The quantum problem of a two-dimensional electron in a uniform perpendicular magnetic field is considered. Using the formalism of ladder operators, the electron eigenfunctions are derived for all quantum numbers n and m, where n denotes the Landau level and m the eigenvalue of the angular momentum L z , with m taking all possible eigenvalues. We note that existing one-electron orbitals, in most of the known literature on the fractional quantum Hall effect (FQHE), correspond to a restricted range of possible eigenvalues m, some are missing. Similarly detailed calculations using ladder operator techniques show that for a state ∣n, m〉, the quantum number (n − m) represents a precise physical quantity, that is the distance from the origin to the center of the electron orbit. This finding allowed us to obtain, for this known quantum problem, a new set of basis states for which both quantum numbers have a physical meaning namely n and (n − m).