Comments on employing the Riesz-Feller derivative in the Schrödinger equation
- 446 Downloads
In this paper, we deal with a fractional Schrödinger equation that contains the quantum Riesz-Feller derivative instead of the Laplace operator in the case of a particle moving in a potential field. In particular, this equation is solved for a free particle in terms of the Fox H-function. On the other hand, we show that from physical viewpoint, the fractional Schrödinger equation with the quantum Riesz-Feller derivative of order α, 0 < α ≤ 2 and skewness θ makes sense only if it reduces to the Laplace operator (α = 2) or to the quantum Riesz fractional derivative (θ = 0). The reason is that the quantum Riesz-Feller derivative is a Hermitian operator and possesses real eigenvalues only when α = 2 or θ = 0. We then focus on the time-independent one-dimensional fractional Schrödinger equation with the quantum Riesz derivative in the case of a particle moving in an infinite potential well. In particular, we show that the explicit formulas for the eigenvalues and eigenfunctions of the time-independent fractional Schrödinger equation that some authors recently claimed to receive cannot be valid. The problem to find right formulas is still open.
KeywordsEuropean Physical Journal Special Topic Laplace Operator Fractional Derivative Free Particle Quantum Particle
Unable to display preview. Download preview PDF.
- 2.W. Feller, On a generalization of Marcel Riesz’ potentials and the semi-groups generated by them, Meddelanden Lunds Universitets Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund), Tome suppl. dédié à M. Riesz, Lund, 73 (1952)Google Scholar
- 3.R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965)Google Scholar
- 9.N. Laskin, Fractals Quantum Mech. Chaos 10, 780 (2000)Google Scholar
- 12.N. Laskin, Principles of fractional quantum mechanics, edited by J. Klafter, S.C. Lim and R. Metzler, Fractional Dynamics: Recent Advances (World Scientific, Singapore, 2012), p. 393Google Scholar
- 15.A.M. Mathai, R.K. Saxena, H.J. Haubold, The H-function. Theory and Applications (Berlin, Springer, 2010)Google Scholar
- 16.I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999)Google Scholar
- 17.S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publ., London, 1993)Google Scholar
- 19.G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics (Oxford University Press, Oxford, 2005)Google Scholar