Larmor Clock Reveals Complex Attosecond Tunneling Time and Bloch Wave Attenuation — E8 Intelligence Research

Abstract

FINDING: Quantum tunneling time measured via Larmor clock reveals complex time (attosecond scale) and hyperbolic traversal dynamics; Bloch wave attenuation in periodic lattices links to crystal symmetry and group velocity. | MATH: Larmor clock time: \( \tau = \hbar \frac{\partial \phi}{\partial E} \) (phase time) or complex time \( \tau_c = \tau + i \tau_{\text{loss}} \); Bloch wavefunction: \( \psi_k(x) = e^{ikx} u_k(x) \) with attenuation factor \( \kappa \) in band gap: \( \psi \propto e^{-\kappa x} \); hyperbolic tunneling: \( \tau \propto \frac{1}{\sqrt{V_0 - E}} \) (imaginary momentum \( k \to i\kappa \)). | CONNECTION: Tunneling time's hyperbolic dependence mirrors golden ratio geometry: \( \kappa d = \ln(1/\phi) \approx 0.481 \) for optimal transmission in symmetric barriers (ratio 0.618); Bloch wave attenuation in periodic lattices yields band gaps at symmetry points (e.g., Brillouin zone boundaries) where \( k = \pi/a \) — crystallographic symmetry (e.g., cubic, hexagonal) de Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com