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Computational Methods in Science and Technology

On Fractional Schrödinger Equation

P. Rozmej, Bartosz Bandrowski

Published Jan 1, 2010• 30 citations
Publisher

Abstract

In the note, recent efforts to derive fractional quantum mechanics are recalled.Some applications of a fractional approach to the Schrödinger equation are discussed as well.
DOI
10.12921/cmst.2010.16.02.191-194

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In this paper, the fractional Schrödinger equation that contains the quantum Riesz fractional derivative instead of the Laplace operator is revisited for the case of a particle moving in the infinite potential well. In the recent papers [M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz, “On the nonlocality of the fractional Schrödinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552] and [S. S. Bayin, “On the consistency of the solutions of the space fractional Schrödinger equation,” J. Math. Phys. 53, 042105 (2012)10.1063/1.4705268] published in this journal, controversial opinions regarding solutions to the fractional Schrödinger equation for a particle moving in the infinite potential well that were derived by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] have been given. In this paper, a thorough mathematical treatment of these matters is provided. The problem under consideration is reformulated in terms of three integral equations with the power kernels. Even if the equations look not very complicated, no solution to these equations in explicit form is known. Still, the obtained equations are used to show that the eigenvalues and eigenfunctions of the fractional Schrödinger equation for a particle moving in the infinite potential well given by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] and many other papers by different authors cannot be valid as has been first stated by Jeng et al. [“On the nonlocality of the fractional Schrödinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552].
A time fractional quantum framework has been introduced into quantum mechanics.A new version of the space-time fractional Schrödinger equation has been launched.The introduced space-time fractional Schrödinger equation has a new scale parameter, which is a fractional generalization of Planck's constant in quantum physics.It has been shown that the presence of a time fractional time derivative in the space-time fractional Schrödinger equation significantly impacts quantum mechanical fundamentals.Time fractional quantum mechanical operators of coordinate, momentum and angular momentum were defined and their commutation relationships were established.The pseudo-Hamilton operator was introduced and its Hermicity has been proven.Two new functions related to the Mittag-Leffler function have been introduced to solve the space-time fractional Schrödinger equation.Energy of a time fractional quantum system has been defined and calculated in terms of the newly introduced functions.It has been found that in the framework of time fractional quantum mechanics there are no stationary states, and the eigenvalues of the pseudo-Hamilton operator are not the energy levels of the time fractional quantum system.A free particle solution to the space-time fractional Schrödinger equation was found.A free particle space-time fractional quantum mechanical kernel has been found and expressed in terms of the H-function.Renormalization properties of a free particle solution and the space-time fractional quantum kernel were established.Some particular cases of time fractional quantum mechanics have been analyzed and discussed.