Quantum Tunneling: Wave-Mechanical Barrier Penetration and Relativistic Limits — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 14, 2026• 0 citations

Abstract

FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, governed by exponential decay of the wavefunction inside the barrier; "impossible" QFT measurements are possible but non-ideal, with relativistic causality constraints. MATH: Transmission coefficient $T \approx e^{-2\kappa L}$, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$; barrier penetration probability $P \propto e^{-2\int \kappa(x)dx}$. 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

DOI: 10.5281/zenodo.22748143

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Quantum Tunneling: Wave-Mechanical Barrier Penetration and Relativistic Limits — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 14, 2026• 0 citations

Publisher

Abstract

FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, governed by exponential decay of the wavefunction inside the barrier; "impossible" QFT measurements are possible but non-ideal, with relativistic causality 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\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

DOI: 10.5281/zenodo.22748143

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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 finite potential barriers, with the transmission coefficient governed by exponential decay of the wavefunction inside the classically forbidden region; recent work shows "impossible" relativistic measurements are mathematically possible but non-ideal. MATH: - Transmission coefficient: $T \approx e^{-2\kappa L}$ , where $\kappa = \sqrt{2m(V_0 - E)}/\hbar$ (rectangular barrier, $E < V_0$ ). - For arbitrary barrier $V(x)$ : WKB approximation $T \approx \exp\left[-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx\right]$ . - Decay constant ratio: $\kappa L$ dimensionless; critical ratio when $\kappa L = 1$ → $T \approx e^{-2} \approx 0.1353$ . - QFT impossibility result (arXiv:2311.13644): nonlocal joint measurements in relativistic QFT produce superluminal signaling unless measurement operators are restricted; the "impossible" case corresponds to a mathematical idealization with zero-wi 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 coefficient governed by exponential decay of the wavefunction inside the classically forbidden region; recent work shows "impossible" relativistic measurements are mathematically possible but non-ideal. MATH: - Transmission coefficient: $ T \approx e^{-2\kappa L} $, where $ \kappa = \sqrt{2m(V_0 - E)}/\hbar $ (rectangular barrier, $ E < V_0 $). - For arbitrary barrier $ V(x) $: WKB approximation $ T \approx \exp\left[-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx\right] $. - Decay constant ratio: $ \kappa L $ dimensionless; critical ratio when $ \kappa L = 1 $ → $ T \approx e^{-2} \approx 0.1353 $. - QFT impossibility result (arXiv:2311.13644): nonlocal joint measurements in relativistic QFT produce superluminal signaling unless measurement operators are restricted; the "impossible" case corresponds to a mathematical idealization with zero-wi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Andrew Stewart Caldin

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Andrew Stewart Caldin

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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

(rectangular barrier,

E < V_0

). - For arbitrary barrier

V(x)

: WKB approximation

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

. - Decay constant ratio:

\kappa L

dimensionless; critical ratio when

\kappa L = 1

→

T \approx e^{-2} \approx 0.1353

. - QFT impossibility result (arXiv:2311.13644): nonlocal joint measurements in relativistic QFT produce superluminal signaling unless measurement operators are restricted; the "impossible" case corresponds to a mathematical idealization with zero-wi 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

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