Application of Laskin Fractional Quantum mechanics with a changed fractional differential operator to one-dimensional potentials

We studied the quantum mechanics problem of certain one-dimensional potential functions using Laskin fractional quantum mechanics. We used different representations to describe the kinetic energy operator, including the conformable and Riemann-Liouville-Caputo fractional differential operators. We then compared each approach's energy states and wave function outcomes for single and double rectangular and harmonic potentials. As the fractional index increased, there was a noticeable difference between the excited level energy values resulting from each method. Additionally, we find a noticeable change in the probability density when the system exhibits degenerate states. Our results provide a straightforward and standardized approach for solving the one-dimensional fractional Schrödinger equation numerically.