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Physica Scripta

Quantum tunneling through multiple barriers

Dona Ghosh, Asim Pal, Z Ahmed, Jayshri Bachhav

Published May 26, 2026• 0 citations

Abstract

Abstract Quantum tunneling through potential barriers plays a central role in diverse physical phenomena ranging from nuclear fusion and radioactive decay to semiconductor devices and superconducting technologies. In this work, we calculate tunneling probabilities and wave functions for an arbitrary number of potential barriers with diverse functional forms, including square, Gaussian, parabolic, and Lorentzian profiles, using a general numerical framework. The numerical results are first validated by comparison with exact analytical solutions for double, triple and quadruple-square barrier systems, demonstrating excellent agreement. The results reveal that for a system with N barriers, a single resonance peak splits into ( N − 1 ) distinct peaks, which eventually merge into an energy band as N becomes large, in a manner fully consistent with the existing literature. A key additional result is the incorporation of absorption effects through an imaginary component of the potential, providing a realistic description of particle trapping and leading to a progressive suppression of resonant transmission. For sufficiently strong absorption, the resonant structures are completely washed out. Further, if the barriers become non-uniform or disordered, the regular resonance pattern is also observed to be completely washed out. These findings demonstrate the broad applicability of the approach to complex potential landscapes and offer insights relevant to both fundamental quantum mechanics and the design of next-generation quantum devices.

DOI: 10.1088/1402-4896/ae7319

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A simle numerical method of calculating transmission coefficients across arbitrary potentials is presented by using the combination of a transfer matrix procedure and the solution of the wave functions in arbitrary potentials.This method is applied to a parabolic potential and a double barrier potential in an electric field.The transmission coefficients calculated are in agreement with those obtained using a multistep potential approximation.The wave functions at the resonant states are also given for the two examples.

A quantum particle constrained between two high potential barriers provides a paradigmatic example of a system sustaining quasi-bound (or resonance) states. When the system is prepared in one of such quasi-bound states, the wave function approximately maintains its shape but decays in time in a nearly exponential manner radiating into the surrounding space, the lifetime being of the order of the reciprocal of the width of the resonance peak in the transmission spectrum. Naively, one could think that adding more lateral barriers would preferentially slow down or prevent the quantum decay since tunneling is expected to become less probable and due to quantum backflow induced by multiple scattering processes. However, this is not always the case and in the early stage of the dynamics quantum decay can be accelerated (rather than decelerated) by additional lateral barriers, even when the barrier heights are arbitrarily large. The decay acceleration originates from resonant tunneling effects and is associated to large deviations from an exponential decay law. We discuss such a counterintuitive phenomenon by considering the hopping dynamics of a quantum particle on a tight-binding lattice with on-site potential barriers.

Abstract Quantum tunneling, wherein a quanject has a non-zero probability of tunneling into and then exiting a barrier of finite width and height, is the subject of Chapter 13. The description for the one-dimensional case is extended to the barrier being inverted, which forms an attractive potential well. The first application of this analysis is to the emission of alpha particles from the decay of radioactive nuclei, where the alpha-nucleus attraction is modeled by a potential well and the barrier is the repulsive Coulomb potential. Excellent results are obtained. Ditto for the similar analysis of proton burning in stars and yet a different analysis that explains tunneling through a Josephson junction, the connector between two superconductors. The final application is to the scanning tunneling microscope, a device that allows the microscopic surfaces of solids to be mapped via electrons from the surface molecules tunneling into the tip of the STM probe.