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arXiv · 2308.08180

Quantum tunneling from a new type of Unified Cantor Potential

Mohammad Umar·Vibhav Narayan Singh·Mohammad Hasan·Bhabani Prasad Mandal
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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.

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Mohammad Umar, Vibhav Narayan Singh, Mohammad Hasan, Bhabani Prasad Mandal. 2023-08-16. Quantum tunneling from a new type of Unified Cantor Potential. https://arxiv.org/abs/2308.08180

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