Browsing by Author "Mohammad Umar"

Now showing 1 - 5 of 5

Polyadic Cantor potential of minimum lacunarity: special case of super periodic generalized unified Cantor potential
(Institute of Physics, 2025) Mohammad Umar; Mohammad Hasan; Vibhav Narayan Singh; Bhabani Prasad Mandal
To bridge the fractal and non-fractal potentials we introduce the concept of generalized unified Cantor potential (GUCP) with the key parameter N which represents the potential count at the stage S = 1. This system is characterized by total span L, stages S, scaling parameter ρ and two real numbers µ and ν. Notably, the polyadic Cantor potential system with minimal lacunarity is a specific instance within the GUCP paradigm. Employing the super periodic potential formalism, we formulated a closed-form expression for transmission probability T S ( k , N ) using the q-Pochhammer symbol and investigated the features of non-relativistic quantum tunneling through this potential configuration. We show that GUCP system exhibits sharp transmission resonances, differing from traditional quantum systems. Our analysis reveals saturation in the transmission profile with evolving stages S and establishes a significant scaling relationship between reflection probability and wave vector k through analytical derivations. © 2025 IOP Publishing Ltd. All rights, including for text and data mining, AI training, and similar technologies, are reserved.
Quantum tunneling from a new type of generalized Smith-Volterra-Cantor potential
(American Institute of Physics, 2025) Vibhav Narayan Singh; Mohammad Hasan; Mohammad Umar; Bhabani Prasad Mandal
In this paper, we introduce and analyze the Smith-Volterra-Cantor potential of power n, denoted as SVC ρ , n . Bridging the gap between the general Cantor G C and Smith-Volterra-Cantor S V C systems, this novel potential offers a fresh perspective on Cantor-like potential systems within quantum mechanics that unify fractal and non-fractal potentials. Utilizing the Super Periodic Potential formalism, we derive the close form expression of the transmission probability TG(k). Notably, the system exhibits exceptionally sharp transmission resonances, a characteristic that distinguishes it from other quantum systems. Furthermore, the multifaceted transmission attributes of the SVC ρ , n are found to be critically dependent on both parameters, ρ and n, offering an intricate interplay that warrants deeper exploration. Our findings highlight a pronounced scaling behavior of reflection probability with k, which is underpinned by analytical derivations. © 2025 Author(s).
Quantum tunneling from a new type of Unified Cantor Potential
(Academic Press Inc., 2023) Mohammad Umar; Vibhav Narayan Singh; Mohammad Hasan; Bhabani Prasad Mandal
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 ρ>1, stage G and two real numbers α and β. For α=1, β=0, the UCP system represents general Cantor potential while for α=0, β=1, this system represent general Smith-Volterra-Cantor (SVC) potential. We provide close-form expression of transmission probability from UCP system for arbitrary α and β by using q-Pochhammer symbol. Several new features of scattering are reported for this system. The transmission probability TG(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. © 2023 Elsevier Inc.
Quantum tunneling from family of Cantor potentials in fractional quantum mechanics
(Academic Press Inc., 2023) Vibhav Narayan Singh; Mohammad Umar; Mohammad Hasan; Bhabani Prasad Mandal
We explore the features of non-relativistic quantum tunneling in space fractional quantum mechanics through a family of Cantor potentials. We consider two types of potentials: general Cantor and general Smith–Volterra–Cantor potential. The Cantor potential is an example of fractal potential while the Smith-Volterra–Cantor potential does not belong to the category of a fractal system. The present study brings for the first time, the study of quantum tunneling through fractal potential in fractional quantum mechanics. We report several new features of scattering in the domain of space fractional quantum mechanics including the emergence of energy-band like features from these systems and extremely sharp transmission features. Further the scaling relation of the scattering amplitude with wave vector k is presented analytically for both types of potentials. © 2023 Elsevier Inc.
Tunneling from general Smith-Volterra-Cantor potential
(American Institute of Physics Inc., 2023) Vibhav Narayan Singh; Mohammad Umar; Mohammad Hasan; Bhabani Prasad Mandal
We study the tunneling problem from the general Smith-Volterra-Cantor (SVC) potential of finite length L characterized by the scaling parameter ρ and stage G. We show that the SVC(ρ) potential of stage G is a special case of the super periodic potential (SPP) of order G. By using the SPP formalism developed by us earlier, we provide the closed form expression of the tunneling probability TG(k) with the help of the q-Pochhammer symbol. The profile of TG(k) with wave vector k is found to saturate with increasing stage G. Very sharp transmission resonances are found to occur in this system, which may find applications in the design of sharp transmission filters. © 2023 Author(s).