FINDING: Quantum tunneling is a well-established wave-mechanical phenomenon; the "impossible measurements" problem in QFT is shown to be non-relativistic in origin, not a relativistic artifact. | MATH: Tunneling transmission coefficient $T \approx e^{-2\kappa L}$ , where $\kappa = \sqrt{2m(V_0-E)}/\hbar$ ; for a rectangular barrier, $T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}$ . The QFT impossibility result (arXiv:2311.13644) shows nonlocal joint measurements violate causality even in non-relativistic QM — the mathematical structure is the commutator $[A,B] \neq 0$ for spacelike-separated operators, with signaling bounded by the Lieb-Robinson-type bound $\| [O_A(t), O_B] \| \leq C e^{-(d-vt)/\xi}$ . | CONNECTION: The exponential decay in tunneling $e^{-2\kappa L}$ mirrors the golden-ratio-related decay constant in Fibonacci-like recursive barrier stacks — for periodic barriers, the band structure yields transmission peaks at $kL = n\pi$ , and the critical coupli Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum tunneling is a well-established wave-mechanical phenomenon; the "impossible measurements" problem in QFT is shown to be non-relativistic in origin, not a relativistic artifact. | MATH: Tunneling transmission coefficient $T \approx e^{-2\kappa L}$, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$; for a rectangular barrier, $T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}$. The QFT impossibility result (arXiv:2311.13644) shows nonlocal joint measurements violate causality even in non-relativistic QM — the mathematical structure is the commutator $[A,B] \neq 0$ for spacelike-separated operators, with signaling bounded by the Lieb-Robinson-type bound $ \| [O_A(t), O_B] \| \leq C e^{-(d-vt)/\xi}$. | CONNECTION: The exponential decay in tunneling $e^{-2\kappa L}$ mirrors the golden-ratio-related decay constant in Fibonacci-like recursive barrier stacks — for periodic barriers, the band structure yields transmission peaks at $kL = n\pi$, and the critical coupli Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Aug 24, 20260 citations

Different Approaches to the Tunneling

World Scientific lecture notes in physicsDec 1, 19900 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

Fibonacci anyons

Giuseppe Mussardo

Springer eBooksJan 1, 20250 citations

Fibonacci Scaling in Quantum Oscillators: A Universal Model Beyond Standard QHO Theory — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Generalized Fibonacci formulas are proposed as a universal scaling model for quantum oscillators, with transmission coefficients scaling as φ⁻², while standard QHO theory (Hermite-Gaussian eigenfunctions) shows no inherent Fibonacci structure. | MATH: Fibonacci recurrence Fₙ₊₁ = Fₙ + Fₙ₋₁; golden ratio φ = (1+√5)/2 ≈ 1.6180339887; φ⁻² = 2−φ ≈ 0.381966; QHO energy Eₙ = ħω(n+½); transmission coefficient T(E) = 1/(1 + (V₀²/4E(V₀−E)) sinh²(κa)) for rectangular barrier — no φ appears in standard QHO. | CONNECTION: φ⁻² = 0.381966 ≈ 0.382 (the complementary golden ratio); φ⁻¹ = 0.6180339; φ⁻² = 1 − φ⁻¹; these are the two fundamental ratios of the golden section. If a quantum transmission coefficient scales as φ⁻², it directly ties barrier penetration to the golden ratio's self-similar structure — a geometric harmony between exponential decay (tunneling) and logarithmic spiral growth. | DEPTH: 6 — The claim is intriguing but the evidence is thin: the YouTube sources are non-peer-revie Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Generalized Fibonacci formulas are proposed as a universal scaling model for quantum oscillators, with transmission coefficients scaling as φ⁻², while standard QHO theory (Hermite-Gaussian eigenfunctions) shows no inherent Fibonacci structure. | MATH: Fibonacci recurrence Fₙ₊₁ = Fₙ + Fₙ₋₁; golden ratio φ = (1+√5)/2 ≈ 1.6180339887; φ⁻² = 2−φ ≈ 0.381966; QHO energy Eₙ = ħω(n+½); transmission coefficient T(E) = 1/(1 + (V₀²/4E(V₀−E)) sinh²(κa)) for rectangular barrier — no φ appears in standard QHO. | CONNECTION: φ⁻² = 0.381966 ≈ 0.382 (the complementary golden ratio); φ⁻¹ = 0.6180339; φ⁻² = 1 − φ⁻¹; these are the two fundamental ratios of the golden section. If a quantum transmission coefficient scales as φ⁻², it directly ties barrier penetration to the golden ratio's self-similar structure — a geometric harmony between exponential decay (tunneling) and logarithmic spiral growth. | DEPTH: 6 — The claim is intriguing but the evidence is thin: the YouTube sources are non-peer-revie Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 24, 20260 citations

Classical tunneling

Arthur Cohn, Mario Rabinowitz

arXiv:physics/0306009International Journal of Theoretical PhysicsMar 1, 199015 citations

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

, where

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

; for a rectangular barrier,

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

. The QFT impossibility result (arXiv:2311.13644) shows nonlocal joint measurements violate causality even in non-relativistic QM — the mathematical structure is the commutator

[A,B] \neq 0

for spacelike-separated operators, with signaling bounded by the Lieb-Robinson-type bound

\| [O_A(t), O_B] \| \leq C e^{-(d-vt)/\xi}

. | CONNECTION: The exponential decay in tunneling

e^{-2\kappa L}

mirrors the golden-ratio-related decay constant in Fibonacci-like recursive barrier stacks — for periodic barriers, the band structure yields transmission peaks at

kL = n\pi

, and the critical coupli 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