Quantum tunneling from a new type of Unified Cantor Potential

TL;DR

This paper introduces a new unified potential system combining fractal and non-fractal Cantor potentials, providing analytical expressions for tunneling transmission probabilities and revealing new scattering features.

Contribution

The paper presents the first analytical formulation of tunneling in a unified Cantor potential system that encompasses both fractal and non-fractal potentials.

Findings

Analytical expression for transmission probability using q-Pochhammer symbol.

Scaling behavior of transmission probability with wave number.

Potential for studying transitions from fractal to non-fractal systems.

Abstract

We introduce a new type of potential system that combines the families of general Cantor (fractal system) and general Smith-Volterra-Cantor (non-fractal system) potentials. We call this system as Unified Cantor Potential (UCP) system. The UCP system of total span $L$ is characterized by scaling parameter $\rho >1$, stage $G$ and two real numbers $\alpha$ and $\beta$. For $\alpha=1$, $\beta=0$, the UCP system represents general Cantor potential while for $\alpha=0$, $\beta=1$, this system represent general Smith-Volterra-Cantor (SVC) potential. We provide close-form expression of transmission probability from UCP system for arbitrary $\alpha$ and $\beta$ by using $q$-Pochhammer symbol. Several new features of scattering are reported for this system. The transmission probability $T_{G}(k)$ shows a scaling behavior with $k$ which is derived analytically for this potential. The proposed system also opens up the possibility for further generalization of new potential systems that encompass a large class of fractal and non-fractal systems. The analytical formulation of tunneling from this system would help to study the transmission feature at breaking threshold when a system transit from fractal to non-fractal domain.