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arXiv:2312.10501 · Journal of Mathematical Physics

Quantum tunneling from a new type of generalized Smith-Volterra-Cantor potential

Vibhav Narayan Singh, Mohammad Hasan, Mohammad Umar, Bhabani Prasad Mandal

Published Mar 1, 2025• 3 citations
arXivHTMLPDF

Abstract

In this paper, we introduce and analyze the Smith-Volterra-Cantor potential of power n, denoted as SVCρ,n. Bridging the gap between the general Cantor GC and Smith-Volterra-Cantor SVC 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 SVCρ,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.

DOI
10.1063/5.0203470

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In this paper, we introduce and analyze the Smith-Volterra-Cantor potential of power n, denoted as SVCρ,n. Bridging the gap between the general Cantor GC and Smith-Volterra-Cantor SVC 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 SVCρ,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.
We study the quantum transport of Dirac electrons in graphene subjected to a polyadic Cantor-structured electrostatic potential. Using the superperiodic potential formalism, we obtain a closed-form expression for the transmission probability. As the Cantor stage increases, the transmission spectrum evolves from a sparse set of superlattice resonances to a near-transparent regime, with the polyadic order setting the rate of this evolution. The angular response depends on the doping configuration, showing distinct behavior in the nnn--nnn--nnn, nnn--ppp--nnn, and Dirac-point cases. In the near-transparent regime, the transmission follows double-logarithmic scaling laws with respect to four independent control parameters: the Cantor stage, the potential height, the initiator length, and the angle of incidence. By combining these individual scaling relations, we establish a unified scaling law governing quantum transport. These results show that the hierarchical self-similarity of the potential governs the transport properties of such systems.
We study the spectral statistics and wave-function properties of a one-dimensional quantum system subject to a Cantor-type fractal potential. By analyzing the nearest-neighbor level spacings, inverse participation ratio (IPR), and the scaling behavior of the integrated density of states (IDS), we demonstrate how the self-similar geometry of the potential is imprinted on the quantum spectrum. The energy-resolved level spacings form a hierarchical, filamentary structure, in sharp contrast to those of periodic and random systems. The normalized level-spacing distribution exhibits a bimodal structure, reflecting the deterministic recurrence of spectral gaps. A multifractal analysis of eigenstates reveals critical behavior: the generalized fractal dimensions DqD_qDq​ lie strictly between the limits of extended and localized states, exhibiting a distinct qqq-dependence. Consistently, the IPR indicates the coexistence of quasi-extended and localized features, characteristic of critical wave functions. The IDS shows anomalous power-law scaling at low energies, with an exponent close to the Hausdorff dimension of the underlying Cantor set, indicating that the geometric fractality governs the spectral dimensionality. At higher energies, this scaling crosses over to the semiclassical Weyl law. Our results establish a direct connection between deterministic fractal geometry, hierarchical spectral statistics, and quantum criticality.
The reflection of wave packets from a quantum well bounded by rectangular potential barriers, and which has a tunneling transmission resonance, is discussed. Numeric solutions to Schrödinger’s time-dependent equation show that the shape of the probability function ψψ* for a reflected wave packet can be quite different from that of the incident packet. Either a symmetric double hump or an asymmetric shape are possible for the reflected ψψ*, depending on the sharpness of the resonance. These shapes are explained analytically in terms of the reflection coefficient.
We present a comprehensive analysis of the nonlinear Landau-Zener tunneling. We find characteristic scaling or power laws for the critical behavior that occurs as the nonlinear parameter equals to the gap of avoided crossing energy levels. For the nonlinear parameter larger than the energy gap, a closed-form solution is derived for the nonlinear tunneling probability, which is shown to be a good approximation to the exact solution for a wide range of the parameters. Finally, we discuss the experimental realization of the nonlinear model and possible observation of the scaling or power laws using a Bose-Einstein condensate in an accelerating optical lattice.
In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.
A “one-point” approximation is proposed to investigate the transmission of electrons through the extra thin heterostructures composed of two parallel plane layers. The typical example is the bilayer for which the squeezed potential profile is the derivative of Dirac’s delta function. The Schr¨odinger equation with this singular one-dimensional profile produces a family of contact (point) interactions each of which (called a “distributional” б′-potential) depends on the way of regularization. The discrepancies widely discussed so far in the literature regarding the family of delta derivative potentials are eliminated using a two-scale power-connecting parametrization of the bilayer potential that enables one to extend the family of distributional б′-potentials to a whole class of “generalized” б′-potentials. In a squeezed limit of the bilayer structure to zero thickness, the resonant tunneling through this structure is shown to occur in the form of sharp peaks located on the sets of Lebesgue’s measure zero (called resonance sets). A four-dimensional parameter space is introduced for the representation of these sets. The transmission on the complement sets in the parameter space is shown to be completely opaque.