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Quantum tunnelling for thousands potential barriers

Domenico Oricchio

Published Feb 20, 2026• 0 citations
Publisher

Abstract

I use the Laplace transform to obtain the transmission coefficient for each number of potential barriers.There are a little number of close-form solution of Schröginger equation, and this is another exact solution.It is possible to obtain the tunnelling for electronic components (resonant-tunnelling diode), or neutronand proton diffusion in astrophysics (stellar nucleosyntheis) and nuclear physics (nuclear fission and fusion),or chemical tunnelling for two molecules reaction (that I solve in a generic case), or to obtain a quantumfilter for the energy spectrum (that I solve for some coating layers).

DOI
10.5281/zenodo.18716987

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I use the Laplace transform to obtain the transmission coefficient for each number of potential barriers.There are a little number of close-form solution of Schröginger equation, and this is another exact solution.It is possible to obtain the tunnelling for electronic components (resonant-tunnelling diode), or neutronand proton diffusion in astrophysics (stellar nucleosyntheis) and nuclear physics (nuclear fission and fusion),or chemical tunnelling for two molecules reaction (that I solve in a generic case), or to obtain a quantumfilter for the energy spectrum (that I solve for some coating layers).
Transmission coefficients for arbitrary potential barriers are obtained using a spreadsheet approach to solve the pertinent Schrödinger equation for the corresponding dispersion eigenfunctions in one dimension. Potentials should be of finite range, otherwise arbitrary. The method of implementation in personal computers is presented here, and it is also shown that the graphical capabilities of the software make it more useful in teaching. This approach reproduces exact solutions for square step barriers and it is illustrated with Gaussian potential energy functions. As a practical example, we simulate a chemical reaction whose rate shows an Arrhenius-like behavior.
Quantum tunnelling represents one of the most fundamental conse-quences of quantum mechanics. The phenomenon allows particles to pen-etrate potential barriers even when their kinetic energy is insufficient ac-cording to classical mechanics. Although initially discovered while study-ing nuclear decay, tunnelling has since emerged as a unifying mechanism across multiple areas of physics. In astrophysics, tunnelling enables nu-clear fusion inside stellar cores by allowing positively charged nuclei to penetrate the electrostatic Coulomb barrier. This process provides the primary energy source for stars and enables the cosmic synthesis of heavy elements necessary for planetary systems and biological life. The proba-bility of such reactions is governed by the Gamow factor and the resulting Gamow peak in stellar energy distributions. Beyond astrophysics, tun-nelling phenomena play a central role in modern quantum technologies, including scanning tunnelling microscopy, semiconductor devices, and su-perconducting Josephson junctions used in quantum computing architec-tures. In theoretical physics, tunnelling also appears in models of Hawk-ing radiation, providing a quantum interpretation for particle emission from black holes. This work presents a unified theoretical exploration of quantum tunnelling across multiple physical scales. Beginning with the Schrödinger equation and barrier penetration theory, we examine tun-nelling in nuclear astrophysics, stellar reaction rates, quantum devices, and gravitational systems. Finally, we discuss emerging informational perspectives in modern physics, suggesting that probabilistic quantum processes may reveal deeper informational structures underlying physical reality.
Exact solutions are obtained for the diffusion in single- and square-well potentials, by using the Laplace transform method. We compare transmission of a classical particle through one barrier of given length and through two barriers with a well between them of the same overall length. It turns out that the existence of a well facilitates the transmission of a classical particle for intermediate times, which bears some resemblance to resonances tunneling through quantum barriers. \textcopyright{} 1996 The American Physical Society.