 Harmonic chain

A uniform harmonic chain couples neighbouring displacements by quadratic spring energies. With periodic boundaries and spacing $a$, a Fourier transform gives ${\omega }_k=2\sqrt{\kappa /m_0}|\sin (ka/2)|$. Quantization makes each nonzero normal mode a phonon. The zero-frequency uniform displacement is a free centre-of-mass coordinate unless pinned, not an ordinary oscillator.^[ref]