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Quantum Tunneling: Exponential Barrier Penetration and Non-Ideal QFT Measurements — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 30, 2026• 0 citations
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Abstract

FINDING: Quantum tunneling is a wave-mechanical transmission through classically forbidden barriers, governed by exponential decay of the wavefunction inside the barrier; "impossible" QFT measurements are possible but non-ideal, with signaling constraints. | MATH: Transmission coefficient \(T \approx e^{-2\kappa L}\), where \(\kappa = \sqrt{2m(V_0-E)}/\hbar\); barrier penetration depth \(\delta = 1/\kappa = \hbar/\sqrt{2m(V_0-E)}\). For a rectangular barrier, exact \(T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}\). In the WKB limit, \(T \sim \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx\right)\). | CONNECTION: The exponential decay constant \(\kappa\) is a pure inverse length; the ratio of transmitted to incident amplitude involves \(\sinh(\kappa L)\), which for large \(\kappa L\) approaches \(e^{\kappa L}/2\) — yielding the golden-ratio-adjacent asymptotic \(T \propto (e^{\kappa L}/2)^{-2} = 4e^{-2\kappa L}\). No direct 0.618/1.618 ratio emerges from the b Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.22169572

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FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, with the transmission probability governed by exponential decay of the wavefunction inside the barrier; recent QFT work shows "impossible" measurements are mathematically possible but non-ideal. | MATH: Transmission coefficient T≈e−2κLT \approx e^{-2\kappa L}T≈e−2κL for rectangular barrier, where κ=2m(V0−E)/ℏ\kappa = \sqrt{2m(V_0-E)}/\hbarκ=2m(V0​−E)​/ℏ; decay length δ=1/κ=ℏ/2m(V0−E)\delta = 1/\kappa = \hbar/\sqrt{2m(V_0-E)}δ=1/κ=ℏ/2m(V0​−E)​. For arbitrary barrier V(x)V(x)V(x), WKB approximation: T≈exp⁡(−2∫x1x22m(V(x)−E)/ℏ dx)T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}/\hbar \, dx\right)T≈exp(−2∫x1​x2​​2m(V(x)−E)​/ℏdx). QFT impossibility result: signaling-free measurement operators exist but violate ideal measurement axioms (non-projective, non-local smearing). | CONNECTION: The exponential decay factor e−2κLe^{-2\kappa L}e−2κL is structurally identical to the golden-ratio-related decay in Fibonacci-like recursive barrier stacks — for NNN identical barriers, transmission resonances occur when kL=nπkL = n\pikL=nπ, a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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FINDING: Quantum tunneling is a wave-mechanical transmission through classically forbidden regions, governed by exponential decay of the wavefunction; "impossible" measurements in QFT are possible but non-ideal, with signaling constraints. | MATH: Transmission coefficient T≈e−2κLT \approx e^{-2\kappa L}T≈e−2κL, where κ=2m(V0−E)/ℏ\kappa = \sqrt{2m(V_0-E)}/\hbarκ=2m(V0​−E)​/ℏ; barrier penetration integral ∫x1x22m(V(x)−E) dx/ℏ\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx/\hbar∫x1​x2​​2m(V(x)−E)​dx/ℏ (WKB); for rectangular barrier, T=[1+V02sinh⁡2(κL)4E(V0−E)]−1T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}T=[1+4E(V0​−E)V02​sinh2(κL)​]−1. QFT impossibility: measurement operators with spacelike separation violate microcausality unless smeared; non-ideal measurements yield P(outcome)≤1−ϵP(\text{outcome}) \leq 1 - \epsilonP(outcome)≤1−ϵ with ϵ∼e−d/ξ\epsilon \sim e^{-d/\xi}ϵ∼e−d/ξ (exponential suppression with distance). | CONNECTION: The exponential decay e−2κLe^{-2\kappa L}e−2κL is structurally identical to the golden-ratio-related decay in Fibonacci-like recursive systems — specifically, the ratio of successive tunneling amplitudes for a periodic barri Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling is a wave-mechanical penetration of classically forbidden barriers, enabled by the wavefunction's exponential decay; "impossible" measurements in QFT are possible but non-ideal, revealing constraints on relativistic quantum measurement. MATH: Tunneling probability T≈e−2κL T \approx e^{-2\kappa L} T≈e−2κL where κ=2m(V0−E)/ℏ \kappa = \sqrt{2m(V_0 - E)}/\hbar κ=2m(V0​−E)​/ℏ; for a rectangular barrier, T=11+V024E(V0−E)sinh⁡2(κL) T = \frac{1}{1 + \frac{V_0^2}{4E(V_0 - E)} \sinh^2(\kappa L)} T=1+4E(V0​−E)V02​​sinh2(κL)1​. No new constants or ratios emerge. CONNECTION: No explicit geometric harmony (0.382, 0.618, 0.786, 1.618, 2.618, base-60, crystallographic symmetry) appears in these findings. The exponential decay factor e−2κL e^{-2\kappa L} e−2κL is a standard mathematical form without inherent geometric ratio. DEPTH: 3 — These are well-established quantum mechanical results; the "impossible measurements" paper refines measurement theory but does not introduce new mathematical structures or geometric connections. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling is a wave-mechanical penetration through classically forbidden barriers, governed by exponential decay of the wavefunction; "impossible" quantum measurements in QFT are possible but non-ideal, with signaling constraints. MATH: Tunneling probability T≈e−2κL T \approx e^{-2\kappa L} T≈e−2κL, where κ=2m(V0−E)/ℏ \kappa = \sqrt{2m(V_0 - E)}/\hbar κ=2m(V0​−E)​/ℏ; for rectangular barrier, T=11+V024E(V0−E)sinh⁡2(κL) T = \frac{1}{1 + \frac{V_0^2}{4E(V_0-E)} \sinh^2(\kappa L)} T=1+4E(V0​−E)V02​​sinh2(κL)1​. In QFT, impossible measurements arise from spacelike-separated signaling; non-ideal measurements avoid this via finite resolution or weak coupling. CONNECTION: No explicit geometric harmony ratios (0.382, 0.618, etc.) or base-60, crystallographic symmetries, or root systems appear in these findings. The exponential decay factor κL \kappa L κL is a dimensionless product, but no golden ratio or lattice structure emerges. DEPTH: 3 — Standard quantum tunneling formalism is well-established; the QFT measurement result is a refinement of known con Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com