Quantum Harmonic Chain

Quantum harmonic chain: a 1D lattice of N atoms connected by springs (constant κ, mass m) is exactly solvable by Fourier transformation into 3N independent normal modes (phonons). Each mode k has frequency ω_k = ω₀|sin(ka/2)| where ω₀ = 2√(κ/m). The quantum ground state energy includes the famous zero-point energy ½ħω per mode — the lattice can never be perfectly still. At temperature T, each mode is occupied by ⟨n_k⟩ = 1/(exp(ħω_k/kT)−1) phonons (Bose-Einstein distribution). Click a mode button to see its displacement pattern. The dispersion curve (bottom) shows the characteristic sine shape, with linear "acoustic" behavior near k=0 and saturation at the Brillouin zone edge.