Time Fractional Schrodinger Equation Revisited

Abstract

The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a "free particle" obeying fractional dynamics, obtained by replacing the integer order derivatives with respect to time by those of fractional order. The equations of motion contain quantities which have "fractional" dimensions, chosen such that the "energy" has the correct dimension [ 2 / 2 ]. The action is defined as a fractional time integral of the Lagrangian, and a "fractional Planck constant" is introduced. The TFSE corresponds to a "subdiffusion" equation with an imaginary fractional diffusion constant and reproduces the regular Schrodinger equation in the limit of integer order. The present work corrects a number of errors in Naber's work. The correct continuity equation for the probability density is derived and a Green function solution for the case of a "free particle" is obtained. The total probability for a "free" particle is shown to go to zero in the limit of infinite time, in contrast with Naber's result of a total probability greater than unity. A generalization to the case of a particle moving in a potential is also given.

Feynman path formula for the time fractional Schrödinger equation

“…This expression is announced in [1] and the justification is given by taking the limit α → 1, since Φ 1/2 is nothing but the Gaussian (see [13]), hence by replacing α = 1 in (5.5), we obtain (5.1).…”

“…Historically, there are two approaches to study the Feynman path integral, either we take the definition of Feynman path integral as it is presented in [8] and deduce the Schrödinger picture from that, or starting from Schrödinger equation, we try to end up at Feynman path integral (see [9,10,11,12]). In [1], we can find the first approach and see how one can derive the Schrödinger equation from the Feynman path integral. For the reverse approach, let us denote by Ω x the set of all paths 3392 HASSAN EMAMIRAD AND ARNAUD ROUGIREL (continuous function) ω :…”

Efficient Numerical Computation of Time-Fractional Nonlinear Schrödinger Equations in Unbounded Domain

“…Naber [39] built the Time Fractional Schrödinger Equation (TFSE) in analogy with the fractional Fokker-Planck equation as well as with the application of the time Wich rotation. Related works have next been developed [10,15,32,35,40,45], including e.g. the generalization of the TFSE to a full space and some new results on the correct continuity equation for the probability density.…”

“…The fractional nonlinear Schrödinger equation is used to describe the nonlocal quantum phenomena in quantum physics and explore the quantum behaviors of either long-range interactions or multi-scale time-dependent processes. Many possible applications explain why this new direction in quantum physics is rapidly emerging [26,28,29,35,39,40,46]. Some analytical and approximate solutions have first been considered for the TFSE [23,41].…”

On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross–Pitaevskii equations

“…The development of fractional partial differential equations (PDEs) has grown impressively during the last few years, because of the huge potential of emerging applications in science. In particular, some important impacts concern fractional quantum dynamics based on the space or/and time Fractional Schrödinger equation and its nonlinear version (SFNLSE or TFNLSE) [1,13,24,29,31,33,34,58,64,66,79]. The fractional nonlinear Schrödinger equation is used to describe the nonlocal phenomena in quantum physics and to explore the quantum behaviors of either long-range interactions or time-dependent processes with many scales [1, 45, 46, 48-50, 58, 64, 74, 81, 82].…”