Abstract
We review supersymmetry (SUSY) in nonrelativistic quantum mechanics emphasizing algebraic aspects. We discuss the Hamiltonian subgroup implementing supersymmetry as well as the corresponding algebra of the SUSY generators. In the SUSYQM framework, the two distinct partner potentials are connected by a superpotential satisfying a Riccati differential equation. A full Hamiltonian operator in an extended Hilbert space is defined in order to render the supersymmetry manifest. As a result, the eigenfunctions of the original potentials are connected by generalized ladder operators. We provide an explicit realization of the abstract supersymmetry group for SUSYQM depending on one real and two Grassmann parameters.
Highlights
In a seminal paper analyzing the dynamical breaking of supersymmetry, published in 1981, Witten has proposed a simple model for supersymmetric quan-Received: August 6, 2014 §Correspondence author c 2014 Academic Publications, Ltd. url: www.acadpubl.euR
We have reviewed supersymmetry in nonrelativistic quantum mechanics by joining two partner one dimensional potentials into a full Hamiltonian enjoying supersymmetry
We have seen that the supersymmetry can be described as a transformation connecting the solutions to the two partner potentials
Summary
In a seminal paper analyzing the dynamical breaking of supersymmetry, published in 1981, Witten has proposed a simple model for supersymmetric quan-. In the relativistic case, SUSYQM has proved to be a suitable framework providing important insights to classical problems [2, 6, 7] When it comes to realize the supersymmetry in terms of the action of an abstract group in the Hilbert space one faces some subtleties having to deal with supergroups and Grassmannian manifolds [8, 9]. We shall show in the following that it is possible to define a new Hamiltonian operator comprising both H− and H+ and realize the supersymmetry in a concrete way via the action of two nilpotent operators Q± The invariance of this complete Hamiltonian under supersymmetry determines a realization of the corresponding supersymmetric group parametrized by real and Grassmann variables acting on the Hilbert space of quantum states.
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