Oscilador harmônico unidimensional via transformada unilateral de Fourier e mapeamento do potencial de morse unidimensional nos potenciais pseudo-harmônico e de kratzer tridimensionais - Nogueira, Pedro Henrique Fusco [UNESP] (2015)
The objective of this study is to solve the Schrödinger equation for the one-dimensional quantum harmonic oscillator via unilateral Fourier transform, and three-dimensional pseudoharmonic and Kratzer potentials via mapping onto the one-dimensional Morse potential. These methods are little explored in the literature on quantum mechanics. The unilateral Fourier transform method is interesting because the harmonic oscillator potential is an even potential which implies solutions with well-de ned parities. As the domain of unilateral transformed extends from 0 to 1, we can focus on nding the eigenfunction in the positive axis and, with appropriate boundary conditions at the origin plus the continuity of eigenfunction and its rst derivative, we can nd the proper extension to the eigenfunction on the whole axis. Subsequently we solve the one-dimensional generalized Morse potential, so mapping it onto the three dimensional power potential plus an inverse quadratic term. Manipulating the equation for this potential we show that there are only two particular cases that allow the mapping onto the Morse potential. These possibilities are the pseudoharmonic potential and the Kratzer potential. Substituting appropriate variables in both the equation of the pseudoharmonic potential and the Kratzer potential, we nd similar equations to the Morse potential. Thus we can identify the parameters of these equations with the Morse potential parameters, so that we nd the solution for the pseudoharmonic potential and Kratzer potential, as well as their particular cases, the harmonic oscillator and the potential Coulomb without solving the corresponding equations