Quantum Harmonic Lattice

1D phonon chain: dispersion relation, zero-point energy, Bose-Einstein occupation

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Zero-point E/site (ℏω₀)

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⟨n⟩ at k=0

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k_BT / ℏω₀

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ωmax (Debye)

N atoms 40

Spring κ 1.00

Temperature k_BT/ℏω₀ 2.0

Animation speed 1.0

1D monatomic chain: ω(k)=2√(κ/m)|sin(ka/2)|. Quantum occupation: ⟨n_k⟩=1/(e^{ℏω_k/k_BT}−1) (Bose-Einstein). Zero-point energy: ½ℏω per mode, persists at T=0. The lattice animation shows each mode as a classical wave weighted by √(⟨n_k⟩+½) — quantum amplitude. Blue=quantum zero-point contribution, orange=thermal.