Time Fractional Schrödinger Equation; Fox's H-functions and the Effective Potential

Abstract

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 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

References in corpus (8)

• Fractional Quantum Mechanics
• On fractional time quantum dynamics
• Fractional-Time Schrödinger Equation: Fractional Dynamics on a Comb
• On the Consistency of the Solutions of the Space Fractional Schrödinger Equation
• Factorizations of one dimensional classical systems
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• Time Fractional Quantum Mechanics
• On the Consistency of the Solutions of the Space Fractional Schrödinger Equation
• Energy-dependent noncommutative quantum mechanics
• Fractional Schrödinger equation in gravitational optics
• Fractional Schrödinger equation and time dependent potentials
• Complex Path Integrals and the Space of Theories
• Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations
• Fractional Schrödinger Equation with Zero and Linear Potentials
• Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach
• Dispersive estimates for time and space fractional Schrödinger equations
• Fractional relativity
• Model-Driven Applications of Fractional Derivatives and Integrals