Recently, it was pointed out that the solutions found in the literature for the space fractional Schrödinger equation in a piecewise manner are wrong, except the case with the delta potential. We re-analyze this problem and show that an exact and a proper treatment of the relevant integral prove otherwise. We also discuss effective potential approach and present a free particle solution for the space and time fractional Schrödinger equation in general coordinates in terms of Fox's H-functions.
Recently, it was pointed out that the solutions found in the literature for the space fractional Schrödinger equation in a piecewise manner are wrong, except the case with the delta potential. We re-analyze this problem and show that an exact and a proper treatment of the relevant integral prove otherwise. We also discuss effective potential approach and present a free particle solution for the space and time fractional Schrödinger equation in general coordinates in terms of Fox's H-functions.
Laskin introduced the fractional quantum mechanics and several common problems were solved in a piecewise fashion. However, Jeng et al pointed out that it was meaningless to solve a nonlocal equation in a piecewise fashion and that all the solutions in publication were wrong except the solution for the delta function potential, which was obtained in the momentum space. Jeng's critique results in a crisis of fractional quantum mechanics, that is, in mathematics it is quite difficult to find solutions to the fractional Schrodinger equation and in physics there is no realization for the fractional quantum mechanics. In order to eliminate this crisis, this paper reports some analytic solutions to the fractional Schrodinger equation without using piecewise method, and introduces the relativistic Schrodinger equation as a realization of the fractional quantum mechanics. These two sister equations should be studied at the same time.
After introducing the formalism of the general space and time fractional Schrödinger equation, we concentrate on the time fractional Schrödinger equation and present new results via the elegant language of Fox's H-functions. We show that the general time dependent part of the wave function for the separable solutions of the time-fractional Schrödinger equation is the Mittag-Leffler function with an imaginary argument by two different methods. After separating the Mittag-Leffler function into its real and imaginary parts, in contrast to existing works, we show that the total probability is ⩽1 and decays with time. Introducing the effective potential approach, we also write the Mittag-Leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts. In the light of these, we reconsider the simple box problem.
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.
After introducing the formalism of the general space and time fractional Schrodinger equation, we concentrate on the time fractional Schrodinger equation and present new results via the elegant language of Fox's H-functions. We show that the general time dependent part of the wave function for the separable solutions of the time fractional Schrodinger equation is the Mittag-Leffler function with an imaginary argument by two different methods. After separating the Mittag-Leffler function into its real and imaginary parts, in contrast to existing works, we show that the total probability is less than one and decays with time. Introducing the effective potential approach, we also write the Mittag-Leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts. In the light of these, we reconsider the simple box problem. PACS numbers: this http URL, 02.50.Ey, this http URL, 03.65.Db
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].
The space-fractional Schrödinger equation for a local potential is difficult to solve because the fractional Riesz operator in it is nonlocal. In fractional quantum mechanics, the infinite wall is a basic problem and has not been solved yet. In this paper, we consider a free particle in an infinite wall region. Making use of the Lévy path integral method, we derive the Lévy path integral amplitude of the particle and take advantage of it to get the solution of the fractional Schrödinger equation for an infinite wall. Then, the infinite wall potential with a delta-function perturbation is also studied. According to Fox’s H-function, we get an equation of bound state energies of a free particle moving in an infinite wall region perturbed by the δ-function for two situations, E < 0 and E > 0. We give a transcendental equation that determines energy levels when E < 0 and prove that there is no bound states when E > 0. The asymptotic behaviors of the center of the delta function approaching to the origin and the infinite are also shown. The results of this paper include special cases in standard quantum mechanics.