Browsing by Author "Mohammad Hasan"

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A black potential for spin less particles
(Elsevier B.V., 2015) Ananya Ghatak; Mohammad Hasan; Bhabani Prasad Mandal
We consider the most general non-Hermitian Hulthen potential to study the scattering of spin-less relativistic particles. The conditions for CC, SS and CPA are obtained analytically for this potential. We show that almost total absorption occurs for entire range of incidence energy for certain parameter ranges of the potential and hence term this as 'black potential'. Time reversed of the same potential shows perfect emission for the entire range of particle energy. We also present the classical analog of this potential in terms of waveguide cross section. © 2015 Elsevier B.V. All rights reserved.
Critical coupling and coherent perfect absorption for ranges of energies due to a complex gain and loss symmetric system
(2014) Mohammad Hasan; Ananya Ghatak; Bhabani Prasad Mandal
We consider a non-Hermitian medium with a gain and loss symmetric, exponentially damped potential distribution to demonstrate different scattering features analytically. The condition for critical coupling (CC) for unidirectional wave and coherent perfect absorption (CPA) for bidirectional waves are obtained analytically for this system. The energy points at which total absorption occurs are shown to be the spectral singular points for the time reversed system. The possible energies at which CC occurs for left and right incidence are different. We further obtain periodic intervals with increasing periodicity of energy for CC and CPA to occur in this system. © 2014 Elsevier Inc.
General(ized) Hartman effect
(IOP Publishing Ltd, 2021) Mohammad Hasan; Bhabani Prasad Mandal
In this letter we prove explicitly that if the Hartman effect exists for an arbitrary "unit cell"potential, then it also exists for a periodic system constructed using the same "unit cell"potential repeatedly. We further show that if the Hartman effect exists, the tunneling time in the limiting case of a sufficiently thick "unit cell"potential is the same as that of its periodic system. This is true for any arbitrary value of the intervening gap between the consecutive "unit cell"of the periodic system. Thus, the generalized Hartman effect always occurs for a general potential constructed using multiple copies of single potential which shows Hartman effect. © Copyright2021 EPLA.
Hartman effect from layered PT-symmetric system
(Springer Science and Business Media Deutschland GmbH, 2020) Mohammad Hasan; Bhabani Prasad Mandal
The time taken by a wave packet to cross through a finite-layered PT-symmetric system is calculated by stationary phase method. We consider the PT-symmetric system of fix spatial length L consisting of N units of the potential system ‘+iV’ and ‘-iV’ of equal width ‘b’ such that L= 2 Nb. In the limit of large ‘b’, the tunneling time is found to be independent of L and therefore, the layered PT-symmetric system displays the Hartman effect. The interesting limit of N→ ∞ such that L remains finite is investigated analytically. In this limit, the tunneling time matches with the time taken to cross an empty space of length L. The result of this limiting case N→ ∞ also shows the consistency of phase space method of calculating the tunneling time, despite the existence of controversial Hartman effect. The reason of Hartman effect is unknown to present day; however, the other definitions of tunneling time that indicate a delay which depends upon the length of traversing region have been effectively ruled out by recent attosecond measurements. © 2020, Società Italiana di Fisica (SIF) and Springer-Verlag GmbH Germany, part of Springer Nature.
New scattering features in non-Hermitian space fractional quantum mechanics
(Academic Press Inc., 2018) Mohammad Hasan; Bhabani Prasad Mandal
The spectral singularity (SS) and coherent perfect absorption (CPA) have been extensively studied over the last one and half decade for different non-Hermitian potentials in non-Hermitian standard quantum mechanics (SQM) governed by Schrödinger equation. In the present work we explore these scattering features in the domain of non-Hermitian space fractional quantum mechanics (SFQM) governed by fractional Schrödinger equation which is characterized by Levy index α (1<α≤2). We observe that non-Hermitian SFQM systems have more flexibility for SS and CPA and display some new features of scattering. For the delta potential V(x)=iρδ(x−x0), ρ>0, the SS energy, Ess, is blue or red shifted with decreasing α depending on the strength of the potential. For complex rectangular barrier in non-Hermitian SQM, it is known that the reflection and transmission amplitudes are oscillatory near the spectral singular point. It is found that these oscillations eventually develop SS in non-Hermitian SFQM. The similar features are also reported for the case of CPA phenomena from complex rectangular barrier in non-Hermitian SFQM. These observations suggest a deeper relation between scattering features of non-Hermitian SQM and non-Hermitian SFQM. © 2018 Elsevier Inc.
New scattering features of quaternionic point interaction in non-Hermitian quantum mechanics
(American Institute of Physics Inc., 2020) Mohammad Hasan; Bhabani Prasad Mandal
Spectral singularities have been extensively studied over the last one and half decades for different non-Hermitian potentials in non-Hermitian quantum mechanics. The nature of spectral singularities has not been studied for the case of quaternionic potential. In the present work, we perform an analytical study on scattering from a quaternionic point interaction represented by a delta function. New features of spectral singularities are observed, which are different than the case of a complex (non-quaternionic) point interaction. The most notable difference is the occurrence of spectral singularity from the lossy point interaction, which is forbidden in the case of standard non-Hermitian quantum mechanics. © 2020 Author(s).
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.
Role of PT-symmetry in understanding Hartman effect
(Springer Science and Business Media Deutschland GmbH, 2020) Mohammad Hasan; Vibhav Narayan Singh; Bhabani Prasad Mandal
We calculate the tunneling time from a layered non-Hermitian system to examine the effect of PT-symmetry over tunneling time. We explicitly find that for system respecting PT-symmetry, the tunneling time saturates with the thickness of the PT-symmetric barrier and thus shows the existence of Hartman effect. For non-PT-symmetric case, the tunneling time depends upon the thickness of the barrier and Hartman effect is lost. We further consider the limiting case in which the non-Hermitian system reduces to the real barrier to show that the Hartman effect from a real barrier is due to PT-symmetry (of the corresponding non-Hermitian system). © 2020, Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature.
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).
Tunneling time from locally periodic potential in space fractional quantum mechanics
(Springer Science and Business Media Deutschland GmbH, 2020) Mohammad Hasan; Bhabani Prasad Mandal
We calculate the time taken by a wave packet to travel through a classically forbidden locally periodic rectangular potential in space fractional quantum mechanics (SFQM). We obtain the closed-form expression of tunneling time from such a potential by stationary phase method. We show that tunneling time depends upon the width b of the single barrier and separation L between the barriers in the limit b→ ∞ and therefore generalized Hartman effect does not exist in SFQM. We observe that in SFQM, the tunneling time for large b in the case of locally periodic potential is smaller than the tunneling from a single barrier of the same width b. It is further shown that with the increase in barrier numbers, the tunneling time reduces in SFQM in the limit of large b. © 2020, Società Italiana di Fisica (SIF) and Springer-Verlag GmbH Germany, part of Springer Nature.
Tunneling time in space fractional quantum mechanics
(Elsevier B.V., 2018) Mohammad Hasan; Bhabani Prasad Mandal
We calculate the time taken by a wave packet to travel through a classically forbidden region of space in space fractional quantum mechanics. We obtain the close form expression of tunneling time from a rectangular barrier by stationary phase method. We show that tunneling time depends upon the width b of the barrier for b→∞ and therefore Hartman effect doesn't exist in space fractional quantum mechanics. Interestingly we found that the tunneling time monotonically reduces with increasing b. The tunneling time is smaller in space fractional quantum mechanics as compared to the case of standard quantum mechanics. We recover the Hartman effect of standard quantum mechanics as a special case of space fractional quantum mechanics. © 2017 Elsevier B.V.