Time fractional Schrodinger equation

TL;DR

This paper introduces a time fractional Schrödinger equation using Caputo derivatives, revealing non-Hermitian Hamiltonians, non-locality, and time-reversal asymmetry, with solutions for free particles and potential wells showing evolving probabilities and energy levels.

Contribution

It formulates a novel fractional Schrödinger equation with non-Hermitian Hamiltonian and analyzes its solutions, highlighting new properties and identities related to Mittag-Leffler functions.

Findings

Wave functions are not invariant under time reversal.

Probability and energy levels increase over time to a limit.

New identities for Mittag-Leffler functions are derived.

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.