Tunneling in Fractional Quantum Mechanics

TL;DR

This paper investigates quantum tunneling phenomena in fractional quantum mechanics, revealing unique behaviors such as zero energy tunneling and deriving explicit transmission coefficients for delta potentials.

Contribution

It provides analytical solutions to the fractional Schrödinger equation for delta potentials and characterizes the behavior of transmission coefficients in fractional quantum mechanics.

Findings

Zero energy tunneling occurs when the fractional derivative order differs from 2.

The zero energy transmission coefficient is given by  = ^2{/} for 1 <   2.

Transmission behavior depends on the order of the fractional derivative.

Abstract

We study the tunneling through delta and double delta potentials in fractional quantum mechanics. After solving the fractional Schr\"odinger equation for these potentials, we calculate the corresponding reflection and transmission coefficients. These coefficients have a very interesting behaviour. In particular, we can have zero energy tunneling when the order of the Riesz fractional derivative is different from 2. For both potentials, the zero energy limit of the transmission coefficient is given by $\mathcal{T}_0 = \cos^2{\pi/\alpha}$, where $\alpha$ is the order of the derivative ($1 < \alpha \leq 2$).