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The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.