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Quantum slowdown of thermal barrier crossing: an exact criterion and a universal thin-barrier threshold of seven

Sabrina Hackfort

Published Oct 30, 2026• 1 citation
Publisher

Abstract

Tunneling speeds up barrier crossing; quantum reflection can slow it down. For any one-dimensional barrier, kBT[κ(T) − 1], with κ the tunneling factor, falls with temperature to A, the signed area between quantum and classical transmission. Hence κ < 1 occurs if and only if A < 0, then above a unique temperature. A is the shift of the mean crossing threshold below the barrier top. For any regular thin barrier with heights Vmin ≤ Vmax, A → (7Vmin − Vmax)/6, as a step raises the mean threshold by 1/6 of its height. Numerically, the Eckart threshold Vmax/Vmin lies between 6.8159 and 7 + 4√3. The upload contains the Letter, the Supplemental Material (proofs, ball-arithmetic certificates, numerical methods, reaction data) and the reproducibility package (Python: mpmath, SymPy, NumPy, SciPy, Matplotlib, python-flint/Arb, ELIAH, formerly NETI) with the LaTeX sources.

DOI
10.5281/zenodo.23794466

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Tunneling speeds up barrier crossing; quantum reflection can slow it down. For any one-dimensional barrier, kBT[κ(T) − 1], with κ the tunneling factor, falls with temperature to A, the signed area between quantum and classical transmission. Hence κ < 1 occurs if and only if A < 0, then above a unique temperature. A is the shift of the mean crossing threshold below the barrier top. For any regular thin barrier with heights Vmin ≤ Vmax, A → (7Vmin − Vmax)/6, as a step raises the mean threshold by 1/6 of its height. Numerically, the Eckart threshold Vmax/Vmin lies between 6.8159 and 7 + 4√3. The upload contains the Letter, the Supplemental Material (proofs, ball-arithmetic certificates, numerical methods, reaction data) and the reproducibility package (Python: mpmath, SymPy, NumPy, SciPy, Matplotlib, python-flint/Arb, ELIAH, formerly NETI) with the LaTeX sources.
This work is a contribution to the theory of the quantum tunneling effect. In the literature at least two barriers are studied, for which the third-order contribution to the phase - integral asymptotic approximation of the transmission coefficient do not contribute. These are the parabolic barrier and the inverse Morse barrier. In the present work we will show that with a proper choice of the so called base function there is at least one more barrier in this category namely the Eckart-Epstein potential. The fact that the third - order contribution vanishes is a good indication that we have found an optimal choice of the base function, and the treatment to find an optimal base function may be possible to generalize to other classes of potential barriers. For particles of a low energy compared to the energy near the top of the barrier we obtain a vary low transmission coefficient, which means that the probability for tunneling to occur is very low. There exist some cases, for example that with a double barrier which is transparent, even for certain relatively low energies but no evidence for this kind of transparency for a single barrier has been found. The present work does not give any such evidence. At the same time there are still speculations on cold fusion like effects, which would demand a higher probability for tunneling through for a single barrier.
Quantum tunneling — the passage of a particle through a potential barrier that it classically lacks the energy to surmount — has been described since Gamow (1928) by the tunneling probability T ≈ exp(−2κd), where κ = √(2m(V₀−E))/ℏ and d is the barrier width. The formula is well-confirmed experimentally but has not been derived from a physical principle more fundamental than the Schrödinger equation. We derive it from the principle of information processing load minimization, using the framework established in the preceding papers of this series. We define the tunneling load as C_tunnel = ∫κ(x)dx — the integral of the spatial attenuation coefficient through the barrier — and show that T = exp(−2·C_tunnel) exactly. This is not a restatement of the WKB approximation; it is an identification of the physical meaning of the integral ∫κ(x)dx as the information processing load of the tunneling path. The tunneling path is selected by load minimization: when C_tunnel < C_classical (the load of the classical over-barrier trajectory), the system tunnels. Three physical systems are verified numerically without free parameters: (i) proton-proton nuclear fusion in the solar core — Gamow load C_Gamow = πη = 11.28, T = 1.60 × 10⁻¹⁰, consistent with solar luminosity; (ii) scanning tunneling microscope (STM) current — C_STM = κd = 5.43 at 0.5 nm, T = 1.91 × 10⁻⁵, and the factor-of-ten current change per 0.1 nm distance change follows directly; (iii) Ra-226 alpha decay — C_Gamow = 79.2, T ~ 10⁻⁶⁹, consistent with the 1,600-year half-life. Tunneling is not a mysterious violation of classical mechanics. It is the selection of the minimum-load propagation path — the same principle that explains wave-particle duality (Paper 7), wave function collapse (Paper 4), and the structure of all fundamental forces (Paper 8).
Quantum mechanics serves as the fundamental cornerstone for numerous areas of research. Finding simple and analytical quantum transmission coefficients for potentials is a rare occurrence. Notably, we have discovered a readily provable yet unreported symmetry tunneling law. For an infinite parabolic potential, it can be proved analytically that the sum of the one-dimensional tunneling probabilities at incident energies equal to the barrier height plus or minus any energy deviation is consistently 1. Based on this relationship, the original WKB approximation and the Kemble's formula can be generalized to the above-barrier energy region in a simplified but more accurate manner, referred to as the symmetric WKB approach in this context. For the realistic potential between two nuclei, numerical results demonstrate that the symmetry relationship at the above-barrier energy region and the sub-barrier energy is also well satisfied, as well as in the cases of the multichannel WKB method. The symmetry tunneling law has been effectively extended to the Eckart potential as well. Such symmetry in tunneling may have universal characteristics but its underlying reason remains to be uncovered, which may help us to further understand the quantum mechanisms.
For two-body s-wave collisions subject to tunnelling though an Eckart potential barrier, cross-sections and rate coefficients in the low-energy regime can be evaluated in analytic form. These provide criteria for approach to the Wigner limit and a generic plot applicable to ultracold collisions (<1 mK). The Eckart barrier shape is found to fit fairly well accurate potential curves available for F(2P) + F(2P) and He(1S) + He*(1S). Also discussed are tunnelling-dominated ultracold chemical reactions, exemplified by F + H2. The very slow approach of the reactants allows averaging over rovibrational motions, so the effective adiabatic (‘dressed’) potential surface differs substantially from the Born–Oppenheimer (‘bare’) surface obtained from electronic structure calculations. The dressed barrier is more than two-fold lower and also thinner. For it the Eckart model gives an ultracold rate coefficient for forming HF in its v′ = 2, j′ = 0 rovibrational state (6.5 × 10−14 cm3/s) that is about 200-fold higher than for the bare barrier and of the same magnitude found from a full-scale 3-D quantum scattering calculation.
Reproducibility package for the paper "Number-projected coherent-state kernels for variational Bose-field dynamics" submitted to New Journal of Physics (2026). Contains the LaTeX manuscript source, Python benchmark scripts, numerical data (CSV), and publication-quality figures. Scripts reproduce all results (Figs. 1–5) for two-, three-, and four-mode Bose-Hubbard models using number-projected coherent states with fixed-basis Galerkin and moving-basis TDVP methods. Requirements: Python 3.8+, NumPy, SciPy, Matplotlib.
Attosecond ionization experiments have not resolved the question "What is the tunneling time?". Different definitions of tunneling time lead to different results. Second, a zero tunneling time for a material particle suggests that the nonrelativistic theory includes speeds greater than the speed of light. Chemical reactions, occurring via tunneling, should then not be considered in terms of a nonrelativistic quantum theory calling into question quantum dynamics computations on tunneling reactions. To answer these questions, we define a new experimentally measurable paradigm, the tunneling flight time, and show that it vanishes for scattering through an Eckart or a square barrier, irrespective of barrier length or height, generalizing the Hartman effect. We explain why this result does not lead to experimental measurement of speeds greater than the speed of light. We show that this tunneling is an incoherent process by comparing a classical Wigner theory with exact quantum mechanical computations.