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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.

BRIEFLY REVISITING THE QUANTUM MECHANICS SUPERSYMMETRY ALGEBRA AND GROUP