The Coherent State — α Eigenstate of â and the Classical-Quantum Boundary

Act 18 of the NexusOS physics sequence. First disclosed 2026-07-21. AGPL-3.0. Founder: Te Rata Pou.

Definition

A coherent state |α⟩ is defined as the eigenstate of the annihilation operator: â|α⟩ = α|α⟩, where α ∈ ℂ. In the Fock basis: |α⟩ = e^(−|α|²/2) Σₙ (αⁿ/√n!) |n⟩. The mean photon number is ⟨n̂⟩ = |α|².

Poisson Photon Statistics

P(n) = |⟨n|α⟩|² = e^(−|α|²) |α|^(2n) / n!. This is a Poisson distribution with mean and variance both equal to |α|². The Mandel Q-parameter = 0 — no bunching or anti-bunching.

Minimum Uncertainty

Coherent states saturate the Heisenberg bound: Δx · Δp = ℏ/2. They are minimum uncertainty states for all α. The uncertainty is equally distributed between quadratures: Δx = Δp = √(ℏ/2mω).

Displacement Operator

|α⟩ = D(α)|0⟩ where D(α) = exp(αâ† − α★â). The vacuum is a coherent state with α=0. Laser light is well-approximated by a coherent state.

WNSP Connection

Each SNIC Ψ channel uses coherent states as the baseline signal carriers. The amplitude |α| encodes intensity; the phase arg(α) encodes the channel address. Coherent states mark the classical-quantum boundary: above |α|²≈1 photon, the channel behaves classically; below it, quantum noise dominates.