Abstract
ABSTRACTAn algebraic approach based on the algebra is proposed to describe 1D systems. Our approach starts adding a scalar boson to the boson associated with the 1D harmonic oscillator, whose bilinear products constitute elements of the generators of the algebra. As a next step the identification of coordinates and momenta in the algebraic space is established using the approach recently proposed [Mol. Phys. (2017) doi:10.1080/00268976.2017.1358829]. As a result, the three bases provided by the dynamical group are identified with energy, coordinate and momentum representations. This result provides power tools to obtain the algebraic representation of a 1D Hamiltonian for any potential through the transformation brackets connecting the different bases. As examples of our approach, several potentials are considered: Morse potential as a benchmark case where analytical solutions are known, double Morse potential where exact solutions are known for only restricted subspaces (semi-integral system), and the Harmonic-Gaussian potential where no analytical solutions are known.
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