Generalized semiconfined harmonic oscillator model with a position-dependent effective mas

Abstract

By using a point canonical transformation starting from the constant-mass Schrödinger equation for the isotonic potential, it is shown that a semiconfined harmonic oscillator model with a position-dependent mass and the same spectrum as the standard harmonic oscillator can be easily constructed and extended to a semiconfined shifted harmonic oscillator, which could result from the presence of a uniform gravitational field. A further generalization is proposed by considering a m-dependent position-dependent mass for 0 < m < 2 and deriving the associated semiconfined potential. This results in a family of position-dependent mass and potential pairs, to which the original pair belongs as it corresponds to m = 1.

On the exactly-solvable semi-infinite quantum well of the non-rectangular step-harmonic profile

“…Analytical definition of the mass of that model differs from the analytical definition (3.4) with the power of the denominator (a + x) that equals to 1. After, [25] generalized the same model to the case when the power of the denominator (a+x) of the position-dependent effective mass is greater than 0, but, less than 2. One deduces from these results that if the power of the denominator (a + x) with less than 2 still allows the quantum system to exhibit only the semiconfinement effect, but, with the power of the denominator that equals to 2 behavior of the quantum system under the potential (3.3) changes drastically and both infinitely high wall for the negative value and finite wall for the positive value of the position appear.…”