Factorization of the Quantum Fractional Oscillator

F Olivar-Romero,O Rosas-Ortiz

doi:10.1088/1742-6596/698/1/012025

Abstract

The applicability of the factorization method is extended to the case of quantum fractional-differential Hamiltonians. In contrast with the conventional factorization, it is shown that the ‘factorization energy’ is now a fractional-differential operator rather than a constant. As a first example, the energies and wave-functions of a fractional version of the quantum oscillator are determined. Interestingly, the energy eigenvalues are expressed as power-laws of the momentum in terms of the non-integer differential order of the eigenvalue equation.

Highlights

In fractional calculus the orders of integration and derivation are real numbers rather than natural ones
We pay attention to the one-dimensional oscillator and show that the ‘factorization energy’, which is a constant in the conventional factorization, must be replaced by a fractional-differential operator in the extended fractional formulation
We have introduced an algebraic technique to solve the eigenvalue problem of the Laskin time-independent, space-fractional Schrodinger equation [10]

Summary

Introduction

In fractional calculus the orders of integration and derivation are real numbers rather than natural ones. Besides the cases of the quantum oscillator [10] (see [11]) and the hydrogen-like potential [10], the space-fractional Schrodinger equation has been solved for some piece-constant potentials [11, 12], and has been extended to the time-fractional case [13]. We pay attention to the one-dimensional oscillator and show that the ‘factorization energy’, which is a constant in the conventional factorization, must be replaced by a fractional-differential operator in the extended fractional formulation. This last permits the application of algebraic methods to determine the energies and wave-functions of the quantum fractional oscillator. Our method is not restricted to the oscillator-like interactions, it is useful in solving the fractional eigenvalue problem of other one-dimensional potentials

Problem and solution

Spectrum and wavefunctions

Discussion and further applications

Conclusions