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

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: \( τ = ∂ φ/∂ E \) (phase time) or complex time \( τ_c = τ + i τₗₒₛₛ \); Bloch wavefunction: \( ψ_k(x) = eⁱᵏˣ u_k(x) \) with attenuation factor \( κ \) in band gap: \( ψ ∝ e-κ x \); hyperbolic tunneling: \( τ ∝ {1}{√V_0 - E} \) (imaginary momentum \( k → iκ \)). | CONNECTION: Tunneling time's hyperbolic dependence mirrors golden ratio geometry: \( κ d = ln(1/φ) ≈ 0.481 \) 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 \( k = π/a \) — crystallographic symmetry (e.g., cubic, hexagonal) de Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com