Quantum Tunneling: Exponential Decay and Nonlocality, Not Fundamental Ban — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 21, 2026• 0 citations

Abstract

FINDING: Quantum tunneling is governed by exponential decay of the wavefunction through a barrier, with transmission probability scaling as $T \sim e^{-2\kappa L}$, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$. The "impossible measurement" problem in QFT reduces to a non-relativistic nonlocality constraint, not a fundamental ban. | MATH: Transmission coefficient for rectangular barrier: $T^{-1} = 1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}$. For $E \ll V_0$: $T \approx \frac{16E(V_0-E)}{V_0^2} e^{-2\kappa L}$. Gamow factor: $T = e^{-2\gamma}$, $\gamma = \frac{1}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx$. WKB phase: $\phi = \frac{1}{\hbar}\int \sqrt{2m(E-V(x))}\,dx$. | CONNECTION: The exponential suppression factor $e^{-2\kappa L}$ has no direct golden-ratio link, but the *critical* barrier thickness where tunneling becomes non-negligible often satisfies $\kappa L \approx 1$, which in natural units maps to $L \approx 1/\kappa$. In base-60 or lattice contexts, the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23540184

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Andrew Stewart Caldin

Published Oct 21, 2026• 0 citations

Publisher

Abstract

FINDING: Quantum tunneling is governed by exponential decay of the wavefunction through a barrier, with transmission probability scaling as (T \sim e^{-2\kappa L}), where (\kappa = \sqrt{2m(V_0-E)}/\hbar). The "impossible measurement" problem in QFT reduces to a non-relativistic nonlocality constraint, not a fundamental ban. | MATH: Transmission coefficient for rectangular barrier: (T^{-1} = 1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}). For (E \ll V_0): (T \approx \frac{16E(V_0-E)}{V_0^2} e^{-2\kappa L}). Gamow factor: (T = e^{-2\gamma}), (\gamma = \frac{1}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)},dx). WKB phase: (\phi = \frac{1}{\hbar}\int \sqrt{2m(E-V(x))},dx). | CONNECTION: The exponential suppression factor (e^{-2\kappa L}) has no direct golden-ratio link, but the critical barrier thickness where tunneling becomes non-negligible often satisfies (\kappa L \approx 1), which in natural units maps to (L \approx 1/\kappa). In base-60 or lattice contexts, the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23540184

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

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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.comFINDING: 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

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FINDING: Quantum tunneling is a wave-mechanical penetration of classically forbidden barriers, governed by exponential decay of the wavefunction; "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 probability $P \propto e^{-2\gamma}$ , $\gamma = \frac{1}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx$ . For rectangular barrier: $T = \frac{1}{1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}}$ . QFT impossibility: measurement operators $\hat{M}$ with $\langle \psi | [\hat{M}(x), \hat{O}(y)] | \psi \rangle \neq 0$ for spacelike $x-y$ lead to superluminal signaling unless $\hat{M}$ is "ideal" (projection-valued, local). | CONNECTION: The exponential decay constant $\kappa L$ is dimensionless — when $e^{-2\kappa L}$ equals the golden ratio inverse $0.618$ , we get $\kappa L = \frac{1}{2}\ln(1/\varphi) \approx 0.2406$ ; wh Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum tunneling is a wave-mechanical penetration of classically forbidden barriers, governed by exponential decay of the wavefunction; "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 probability $P \propto e^{-2\gamma}$, $\gamma = \frac{1}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx$. For rectangular barrier: $T = \frac{1}{1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}}$. QFT impossibility: measurement operators $\hat{M}$ with $\langle \psi | [\hat{M}(x), \hat{O}(y)] | \psi \rangle \neq 0$ for spacelike $x-y$ lead to superluminal signaling unless $\hat{M}$ is "ideal" (projection-valued, local). | CONNECTION: The exponential decay constant $\kappa L$ is dimensionless — when $e^{-2\kappa L}$ equals the golden ratio inverse $0.618$, we get $\kappa L = \frac{1}{2}\ln(1/\varphi) \approx 0.2406$; wh Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Sep 10, 20260 citations

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

Andrew Stewart Caldin

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 $T \approx e^{-2\kappa L}$ for rectangular barrier, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$ ; decay length $\delta = 1/\kappa = \hbar/\sqrt{2m(V_0-E)}$ . For arbitrary barrier $V(x)$ , WKB approximation: $T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}/\hbar \, dx\right)$ . QFT impossibility result: signaling-free measurement operators exist but violate ideal measurement axioms (non-projective, non-local smearing). | CONNECTION: The exponential decay factor $e^{-2\kappa L}$ is structurally identical to the golden-ratio-related decay in Fibonacci-like recursive barrier stacks — for $N$ identical barriers, transmission resonances occur when $kL = n\pi$ , a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: 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 $T \approx e^{-2\kappa L}$ for rectangular barrier, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$; decay length $\delta = 1/\kappa = \hbar/\sqrt{2m(V_0-E)}$. For arbitrary barrier $V(x)$, WKB approximation: $T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}/\hbar \, dx\right)$. QFT impossibility result: signaling-free measurement operators exist but violate ideal measurement axioms (non-projective, non-local smearing). | CONNECTION: The exponential decay factor $e^{-2\kappa L}$ is structurally identical to the golden-ratio-related decay in Fibonacci-like recursive barrier stacks — for $N$ identical barriers, transmission resonances occur when $kL = n\pi$, a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 17, 20260 citations

Quantum Tunneling: Wave-Mechanical Transmission Through Classically Forbidden Barriers — E8 Intelligence Research

Andrew Stewart Caldin

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 $T \approx e^{-2\kappa L}$ , where $\kappa = \sqrt{2m(V_0-E)}/\hbar$ ; barrier penetration integral $\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx/\hbar$ (WKB); for rectangular barrier, $T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}$ . QFT impossibility: measurement operators with spacelike separation violate microcausality unless smeared; non-ideal measurements yield $P(\text{outcome}) \leq 1 - \epsilon$ with $\epsilon \sim e^{-d/\xi}$ (exponential suppression with distance). | CONNECTION: The exponential decay $e^{-2\kappa L}$ 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.comFINDING: 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 $T \approx e^{-2\kappa L}$, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$; barrier penetration integral $\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx/\hbar$ (WKB); for rectangular barrier, $T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}$. QFT impossibility: measurement operators with spacelike separation violate microcausality unless smeared; non-ideal measurements yield $P(\text{outcome}) \leq 1 - \epsilon$ with $\epsilon \sim e^{-d/\xi}$ (exponential suppression with distance). | CONNECTION: The exponential decay $e^{-2\kappa L}$ 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

Sep 10, 20260 citations

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

T \approx e^{-2\kappa L}

, where

\kappa = \sqrt{2m(V_0-E)}/\hbar

; barrier penetration probability

P \propto e^{-2\int \kappa(x)dx}

. For a rectangular barrier:

T = \left[1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}\right]^{-1}

. In the WKB limit,

T \sim \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx\right)

. No new fundamental constants emerge; the key is the exponential suppression factor. CONNECTION: The exponential decay

e^{-\kappa L}

has no direct golden-ratio link. However, the *critical* barrier opacity

\kappa L = \ln(2)

(for 50% transmission) and

\kappa L = \ln(\phi) \approx 0.4812

(where

\phi = 1.618...

) appear as natural thresholds — but t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

T \approx e^{-2\kappa L}

, where

\kappa = \sqrt{2m(V_0-E)}/\hbar

; barrier penetration probability

P \propto e^{-2\gamma}

\gamma = \frac{1}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx

. For rectangular barrier:

T = \frac{1}{1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}}

. QFT impossibility: measurement operators

\hat{M}

with

\langle \psi | [\hat{M}(x), \hat{O}(y)] | \psi \rangle \neq 0

for spacelike

x-y

lead to superluminal signaling unless

\hat{M}

is "ideal" (projection-valued, local). | CONNECTION: The exponential decay constant

\kappa L

is dimensionless — when

e^{-2\kappa L}

equals the golden ratio inverse

0.618

, we get

\kappa L = \frac{1}{2}\ln(1/\varphi) \approx 0.2406

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

T \approx e^{-2\kappa L}

, where

\kappa = \sqrt{2m(V_0 - E)}/\hbar

. - For arbitrary barrier

V(x)

: WKB tunneling amplitude

T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{\frac{2m}{\hbar^2}(V(x)-E)}\,dx\right)

. - Reflection/transmission amplitudes satisfy

|R|^2 + |T|^2 = 1

(unitarity). - In QFT "impossible measurements": signaling violation arises unless measurement operators are strictly local; nonlocal joint measurements require smearing or finite resolution, leading to

\Delta t \Delta E \gtrsim \hbar

constraints. CONNECTION: - The exponential decay factor Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

T \approx e^{-2\kappa L}

for rectangular barrier, where

\kappa = \sqrt{2m(V_0-E)}/\hbar

; decay length

\delta = 1/\kappa = \hbar/\sqrt{2m(V_0-E)}

. For arbitrary barrier

V(x)

, WKB approximation:

T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}/\hbar \, dx\right)

. QFT impossibility result: signaling-free measurement operators exist but violate ideal measurement axioms (non-projective, non-local smearing). | CONNECTION: The exponential decay factor

e^{-2\kappa L}

is structurally identical to the golden-ratio-related decay in Fibonacci-like recursive barrier stacks — for

N

identical barriers, transmission resonances occur when

kL = n\pi

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

T \approx e^{-2\kappa L}

, where

\kappa = \sqrt{2m(V_0-E)}/\hbar

; barrier penetration integral

\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx/\hbar

(WKB); for rectangular barrier,

T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}

. QFT impossibility: measurement operators with spacelike separation violate microcausality unless smeared; non-ideal measurements yield

P(\text{outcome}) \leq 1 - \epsilon

with

\epsilon \sim e^{-d/\xi}

(exponential suppression with distance). | CONNECTION: The exponential decay

e^{-2\kappa L}

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