Supersymmetry in quantum mechanics

Abstract

We give some illustrations and interpretations of supersymmetry in quantum mechanics in simple models. We show that the value of 2 for the g factor of the electron expresses the presence of supersymmetry in the Hamiltonian for an electron in a uniform magnetic field. The problem is considered both in the Schrödinger and Dirac formulations. We also show that the radial Coulomb problem with orbital angular momentum l, nuclear charge Z, and principal quantum number n, is supersymmetrically linked to the similar problem with charge Z(1−1/n) and quantum number n−1. Thereby the dependence of Coulomb energies only on the combination Z/n is seen as a manifestation of the supersymmetry in the radial Coulomb problem. Other examples of supersymmetry we consider are the Morse potential, the three-dimensional isotropic oscillator, the states of the helium atom and those of the hydrogen atom in an extremely strong magnetic field.