Coherent States & Wigner Function

Quantum harmonic oscillator coherent states |α⟩ orbit classically in phase space. The Wigner quasi-probability distribution visualizes quantum states.

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Coherent state |α⟩: Wigner function W(x,p) = (1/π)exp(−2|β−α|²) (Gaussian, always positive). Squeezed coherent: W(x,p) = (1/π)exp(−2[(x−x₀)²e^(2r)+(p−p₀)²e^(−2r)]). Orbit: α(t) = |α|e^(i(ωt+φ₀)). The Wigner function rotates without distortion — "minimum uncertainty state" following classical trajectory. Heisenberg: ΔxΔp = ℏ/2 (saturated).