Atomic supersymmetry and the Stark effect

Abstract

A search is conducted for physical quantum-mechanical supersymmetries involving the hydrogen atom. In all coordinate systems admitting a direct separation of the Schrödinger equation, the structure of the separated equations is examined for possible supersymmetric extensions. In addition to the known supersymmetry involving the radial equation for spherical coordinates, we uncover a related supersymmetry involving the radial equation for conical coordinates and a pair of supersymmetries involving parabolic coordinates. The associated spectra and possible physical import of the latter are discussed. They connect certain eigenfunctions of the hydrogen and lithium atoms in the unbroken-symmetry limit. Following the established procedure for the case of spherical coordinates, the breaking of these parabolic supersymmetries is incorporated in a model constructed using notions of quantum-defect theory. The model yields analytical wave functions in parabolic coordinates for the valence electron of alkali-metal atoms, while correctly reproducing the eigenvalue spectra. These ideas are applied to the study of the Stark effect in alkali-metal atoms. Using supersymmetry-based quantum-defect eigenfunctions, we obtain Stark maps for lithium and sodium. The spherical case shows striking agreement with experiment.