Time Fractional Schr\"odinger Equation; Fox's H-functions and the Effective Potential

TL;DR

This paper explores the time fractional Schr"odinger equation using Fox's H-functions, revealing that the wave function's time dependence involves the Mittag-Leffler function, which leads to probability decay and new insights into quantum systems.

Contribution

It introduces a novel analysis of the time fractional Schr"odinger equation with Fox's H-functions, showing probability decay and decomposing the Mittag-Leffler function into oscillating and decaying parts.

Findings

Wave function's time dependence is Mittag-Leffler with an imaginary argument.

Total probability decays over time, contrasting with standard quantum mechanics.

Effective potential approach decomposes Mittag-Leffler function into decay and oscillation.

Abstract

After introducing the formalism of the general space and time fractional Schr\"odinger equation, we concentrate on the time fractional Schr\"odinger 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 Schr\"odinger 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: 03.65.Ca, 02.50.Ey, 02.30.Gp, 03.65.Db