Supersymmetric Quantum Mechanics in Three-Dimensional Space. I: One-Particle System with Spin-Orbit Potential

Abstract

As a direct generalization of the model of supersymmetric quantum mechanics by Witten, which describes the motion of a spin one-half particle in the one-dimensional space, we construct a model of the supersymmetiric quantum mechanics in the three-dimensional space, which describes the motion of a spin one-half particle in central and spin-orbit potentials in the context of the nonrelativistic quantum mechanics. With the simplest choice of the (super) potential, this model is shown to reduce to the model of the harmonic oscillator plus constant spin-orbit potential of unit strength of both positive and negative signs, which was studied in detail in our recent paper in connection with “accidental degeneracy” as well as the “graded groups”. This simplest model is discussed in some detail as an example of the three-dimensional supersymmetric quantum mechanical system, where the supersymmetry is an exact symmetry of the system. More general choice of a polynomial superpotential is also discussed. It is shown that the supersymmetry cannot be spontaneously broken for any polynomial superpotential in our three-dimensional model; this result is contrasted to the corresponding one in the one-dimensional model.