A 1D chain of quantum oscillators coupled by springs. Normal modes are phonons — quantized lattice vibrations. The dispersion relation ω(k) = 2ω₀|sin(ka/2)| shows linear (acoustic) behavior near k=0. Zero-point motion persists even at T=0.
Phonons are quantized normal modes. Energy Eₙ = ℏω(k)(n+½). The zero-point energy ½ℏω is a quantum signature. At higher T, higher modes become populated (Bose-Einstein statistics).