Structural Origin of Square-Root Mass Scaling inQuantum Tunnelling

Krishna Kingkar Pathak

Published Feb 11, 2026• 0 citations

Abstract

Abstract Quantum tunnelling phenomena across physics and chemistry are governed by an exponential sensitivity to the effective mass associated with the tunnelling coordinate. Within semiclassical WKB and instanton formalisms, this dependence takes a characteristic form, $\ln \mathcal{R} \propto -\sqrt{M_{\mathrm{eff}}}$, arising from the factorisation of the Euclidean action into mass-dependent and geometric contributions. Although this scaling is well established at the level of theory, its status as a unifying structural principle across distinct physical regimes has not been systematically clarified. In this work, we examine the extent to which the square-root effective-mass dependence functions as a universal semiclassical descriptor of tunnelling across radically different physical platforms. By reanalysing representative experimental datasets from molecular isotopologues, collective tunnelling in ultracold atomic systems, and macroscopic quantum escape in Josephson circuits, we demonstrate that the leading semiclassical trend persists over more than twenty orders of magnitude in tunnelling rate. When expressed in terms of $\ln \mathcal{R}$ versus $\sqrt{M_{\mathrm{eff}}}$, all platforms exhibit near-linear behaviour, indicating that the dominant contribution to the tunnelling exponent is controlled by a common Euclidean-action structure rather than system-specific microscopic details. We further identify the physical mechanisms—finite temperature, dissipation, many-body effects, and mass-dependent barrier reshaping—that delimit the regime of validity of this leading-order scaling. Taken together, the results support the view that the $\sqrt{M_{\mathrm{eff}}}$ dependence reflects a form of mechanical similarity in imaginary time, providing an ontologically neutral and cross-domain organising principle for quantum tunnelling phenomena from molecular systems to engineered macroscopic quantum devices.

DOI: 10.21203/rs.3.rs-8813754/v1

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Structural Origin of Square-Root Mass Scaling inQuantum Tunnelling

Krishna Kingkar Pathak

Published Feb 11, 2026• 0 citations

Publisher

Abstract

\ln \mathcal{R} \propto -\sqrt{M_{\mathrm{eff}}}

, arising from the factorisation of the Euclidean action into mass-dependent and geometric contributions. Although this scaling is well established at the level of theory, its status as a unifying structural principle across distinct physical regimes has not been systematically clarified. In this work, we examine the extent to which the square-root effective-mass dependence functions as a universal semiclassical descriptor of tunnelling across radically different physical platforms. By reanalysing representative experimental datasets from molecular isotopologues, collective tunnelling in ultracold atomic systems, and macroscopic quantum escape in Josephson circuits, we demonstrate that the leading semiclassical trend persists over more than twenty orders of magnitude in tunnelling rate. When expressed in terms of

\ln \mathcal{R}

versus

\sqrt{M_{\mathrm{eff}}}

, all platforms exhibit near-linear behaviour, indicating that the dominant contribution to the tunnelling exponent is controlled by a common Euclidean-action structure rather than system-specific microscopic details. We further identify the physical mechanisms—finite temperature, dissipation, many-body effects, and mass-dependent barrier reshaping—that delimit the regime of validity of this leading-order scaling. Taken together, the results support the view that the

\sqrt{M_{\mathrm{eff}}}

dependence reflects a form of mechanical similarity in imaginary time, providing an ontologically neutral and cross-domain organising principle for quantum tunnelling phenomena from molecular systems to engineered macroscopic quantum devices.

DOI: 10.21203/rs.3.rs-8813754/v1

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Journal of Nanotechnology and Innovative MaterialsNov 19, 20250 citations

\ln \mathcal{R} \propto -\sqrt{M_{\mathrm{eff}}}

\ln \mathcal{R}

versus

\sqrt{M_{\mathrm{eff}}}

\sqrt{M_{\mathrm{eff}}}

\Gamma = A \exp(-S_{\text{inst}}/\hbar)

, where

S_{\text{inst}}

is the Euclidean action. For a cubic potential

V(q) = \frac{1}{2}m\omega_0^2 q^2 - \lambda q^3

, the decay length

\ell = \hbar / \sqrt{2m\Delta U}

(where

\Delta U

is barrier height) enters as

S_{\text{inst}} = \frac{36}{5} \frac{\Delta U}{\omega_0}

. The qubit's Josephson energy ratio

E_J/E_C

sets the effective mass; the crossover from quantum to thermal regime occurs at

T^* = \hbar \omega_0 / (2\pi k_B)

. | CONNECTION: The decay length ratio

\ell / \Delta q_{\text{zero-point}}

(zero-point width

\Delta q_0 = \sqrt{\hbar/(2m\omega_0)}

) yields \( \ell/\Delta q_0 = \sqrt{\Delta U / (\hbar \o Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\Gamma \sim \exp(-S_{\text{eff}}/\hbar)

, where

S_{\text{eff}} = S_0 \cdot N^{\alpha} \cdot f(\gamma)

. Here

N

is particle number,

\alpha

is a universal exponent (typically

\alpha = 1

for mean-field,

\alpha = 2/3

for certain correlated regimes), and

f(\gamma)

is a universal function of the interaction strength

\gamma = gN/\hbar v

(Lieb-Liniger parameter). The crossover between exponential suppression and enhancement is marked by a critical ratio

\gamma_c \approx 2.618

(i.e.,

1+\varphi^2

, where

\varphi = 1.618...

). | CONNECTION: The critical interaction ratio

\gamma_c \approx 2.618

is exactly

\varphi^2 + 1 = 2.618

, li Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

χ/2π= 361\pm 13\,\mathrm{MHz}

, agreeing with the independently measured value of

366\pm 0.5\,\mathrm{MHz}

\cite{Ye2025} to within 1.4\%. A formal error-propagation analysis establishes that the dominant uncertainty is the Josephson-energy extraction uncertainty (

\sim\!2\%

), not the geometric projection factor. Cross-platform validation against five independent experiments spanning eight orders of magnitude in coupling strength confirms the universality of the factorization to within reported experimental uncertainties. The ghost-sector spectral correction introduced in earlier versions is reframed as an optional phenomenological self-energy model with no mandatory role in the validated scaling law.