Time fractional Schrodinger equation

Abstract

The Schrodinger equation is considered with the first order time derivative changed to a Caputo fractional derivative, the time fractional Schrodinger equation. The resulting Hamiltonian is found to be non-Hermitian and non-local in time. The resulting wave functions are thus not invariant under time reversal. The time fractional Schrodinger equation is solved for a free particle and for a potential well. Probability and the resulting energy levels are found to increase over time to a limiting value depending on the order of the time derivative. New identities for the Mittag-Leffler function are also found and presented in an appendix.

23 pages

References in corpus (1)

• Fractional Quantum Mechanics

Cited by in corpus (59)

• Fractional dynamics of systems with long-range interaction
• Fractional Hamilton formalism within Caputo's derivative
• Fractional dynamics of coupled oscillators with long-range interaction
• Geometry and field theory in multi-fractional spacetime
• Experimental realisations of the fractional Schrödinger equation in the temporal domain
• Diffraction-free beams in fractional Schrödinger equation
• From non-Brownian Functionals to a Fractional Schrödinger Equation
• On fractional time quantum dynamics
• Fractional-Time Schrödinger Equation: Fractional Dynamics on a Comb
• On distributions of functionals of anomalous diffusion paths
• Time Fractional Quantum Mechanics
• Dynamics with Low-Level Fractionality
• Fractional Variations for Dynamical Systems: Hamilton and Lagrange Approaches
• Fractional Hamiltonian analysis of higher order derivatives systems
• Prospecting Black Hole Thermodynamics with Fractional Quantum Mechanics
• On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions
• Time Fractional Schrödinger Equation; Fox's H-functions and the Effective Potential
• Definition of the Riesz Derivative and its Application to Space Fractional Quantum Mechanics
• Tunneling in Fractional Quantum Mechanics
• Fractional Dynamics from Einstein Gravity, General Solutions, and Black Holes
• Fractional Derivative as Fractional Power of Derivative
• Fractional Statistical Mechanics
• Nonholonomic Constraints with Fractional Derivatives
• Magnetohydrodynamics of Fractal Media
• Psi-Series Solution of Fractional Ginzburg-Landau Equation
• Quantum mechanics in fractional and other anomalous spacetimes
• Energy-dependent noncommutative quantum mechanics
• Solutions of a particle with fractional $\delta$-potential in a fractional dimensional space
• Broadening quantum cosmology with a fractional whirl
• Inflation and fractional quantum cosmology
• Comments on the Properties of Mittag-Leffler Function
• Gauge Invariant Fractional Electromagnetic Fields
• Time independent fractional Schrodinger equation for generalized Mie-type potential in higher dimension framed with Jumarie type fractional derivative
• Fractional Almost Kahler - Lagrange Geometry
• Fractional Holographic Dark Energy
• Fractional Schrödinger equation in gravitational optics
• Fedosov Quantization of Fractional Lagrange Spaces
• Factorization of the Quantum Fractional Oscillator
• Fractional Scalar Field Cosmology
• Fractional Schrödinger equation and time dependent potentials
• The DuBois-Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler-Lagrange Equation Involving only Derivatives of Caputo
• Lévy Transport in Slab Geometry of Inhomogeneous Media
• Repulsive Casimir Force from Fractional Neumann Boundary Conditions
• Applications of density matrix in the fractional quantum mechanics
• Unitary evolution for a two-level quantum system in fractional-time scenario
• Finite Temperature Casimir Effect for a Massless Fractional Klein-Gordon field with Fractional Neumann Conditions
• On the fractional damped oscillators and fractional forced oscillators
• Topological symmetry breaking of self--interacting fractional Klein--Gordon field on toroidal spacetime
• Fractional Floquet theory
• Anomalous Cyclic in the Neutrino Oscillations
• Similarity solutions for a class of Fractional Reaction-Diffusion equation
• Van der Corput lemmas for Mittag-Leffler functions. II. $\alpha$-directions
• An efffcient numerical scheme for two-dimensional nonlinear time fractional Schrödinger equation
• Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach
• Non-Markovian Light-Matter Dynamics in the Time Fractional Jaynes-Cummings Model with Modulated Coupling
• Hölder regularity for the time fractional Schrödinger equation
• The Fractionary Schrödinger Equation, Green Functions and Ultradistributions
• Generalised fractional Rabi problem
• Speed of propagation of fractional dispersive waves