Abstract
Based on methods of numerical integration and Riemann–Liouville definition of the fractional derivatives, we find a numerical algorithm to find solutions of the time independent fractional Schrödinger equation for Morse potential or the quantum oscillator potential in one dimension, and the iteration formula is applied for multiple values of the fractional parameter of the space dependent fractional Schrödinger equation and multiple values of energy. We define and use the dimensionless form of the space dependent fractional Schrödinger equation of Morse potential. We employ the iteration formula of the time independent fractional Schrödinger equation of Morse potential to find the wave functions in the case of hydrogen chloride and hydrogen fluoride molecules for a certain value of the fractional parameter of the space dependent fractional Schrödinger equation and for many values of the dimensionless energy of each molecule.
Highlights
The space dependent fractional Schrödinger equation is a type of linear Schrödinger equation used in fractional quantum mechanics.1–51 In one dimension, this equation is given by jh ∂ ψ(r, ∂t t) = −Kα(h∇)αψ(r, t) + U(r)ψ(r, t), (1)where his the reduced Planck constant, ψ(r, t) is the wave function of the system in space representation, j is the imaginary unit, Kα is a parameter, U(r) is the interaction potential of the system, α is the fractional parameter of the space dependent fractional Schrödinger equation, and (h∇)α is the space fractional operator
We applied the iteration formula that we found for the dimensionless parameter C∗ = 1 and for multiple values of the fractional parameter of the space dependent fractional Schrödinger equation
We derived a numerical algorithm [Eq (12)] to solve the space dependent fractional Schrödinger equation in the case of Morse potential in the stationary state based on the Riemann–Liouville definition of the fractional derivatives for increasing values
Summary
The space dependent fractional Schrödinger equation is a type of linear Schrödinger equation used in fractional quantum mechanics. In one dimension, this equation is given by jh. The space dependent fractional Schrödinger equation is a type of linear Schrödinger equation used in fractional quantum mechanics.. The space dependent fractional Schrödinger equation is a type of linear Schrödinger equation used in fractional quantum mechanics.1–51 In one dimension, this equation is given by jh. In most of the literature, the term fractional Schrödinger equation refers to the space dependent fractional Schrödinger equation; in this work we use this convention. We apply the Riemann–Liouville definition of the fractional derivatives to find numerical solutions for the time independent fractional Schrödinger equation for Morse potential, which is used in the quantum mechanics as an oscillator potential, and this potential is given by the following formula:. We use the dimensionless form of Morse potential in the fractional Schrödinger equation
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