I. INTRODUCTION One of the most prominent areas of research in mathematical physics in the last 30 years has been that of supersymmetry (SUSY), based on the work of Witten in particle physics.[1] In non-relativistic quantum mechanics (QM), the work of Mielnik[2] has been the basis for the development of isospectral potentials for different quantum problems,[3] and also for higher supersymmetric developments.[4] The best example to introduce SUSY QM is that of the simple harmonic oscillator (SHO), whose hamiltonian possess the eigenfunctions and eigenvalues given by where H$_{n}$( x ) are the Hermite polynomials, satisfying Hermite’s equation Hermite’s equation can be derived from the SHO Schr¨ odinger’s equation by acting the hamil- tonian (1) on the functions (2) and leaving the equation for the functions H$_{n}$( x ) alone. One can find in the literature generalizations of eq.(3), which define generalized Hermite poly- nomials H$_{m}$( u ; t ),[5] H$_{n}$( x ; γ ),[6] and H N n ( x ),[7] or simply define the Hermite functions as h$_{n}$ ( x )=H$_{n}$( x ) e − x $^{2}$/ $^{2}$, the quantum eigenfunctions. The hamiltonian (1) can be factorized using the anhilitation and creation operators Mielnik proposed a different way to factorize the hamiltonian, by introducing new operators[2] 2 β ′ + β 2 = 1 + x $^{2}$. (6) b = 1 √$_{2}$ ( d dx + β ( x ) ) , b ∗ = 1 √$_{2}$ ( − d dx + β ( x ) ) , (5) which, in order to satisfy the factorization bb ∗ = H + 1 $_{2}$, lead to a Riccati equation for the function β ( x ) a = 1 √$_{2}$ ( d dx + x ) , a ∗ = 1 √$_{2}$ ( − d dx + x ) , (4) H ′′ $_{n}$( x $^{)}$− 2 x H $^{′}$$_{n}$( x ) + 2 n H$_{n}$( x ) = 0 . (3) ψ$_{n}$ ( x ) = c$_{n}$ H$_{n}$( x ) e − x $^{2}$/ 2 , E$_{n}$ = n + 1 2 , (2) H = − 1 2 d 2 dx 2 + 1 2 x 2 (1) such that a a ∗ = H + 1 $_{2}$, and a ∗ a = H − 1 $_{2}$.