= Harmonic chain
{title2=$H=\sum_j[p_j^2/(2m_0)+\kappa(u_{j+1}-u_j)^2/2]$}
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.<Applications of Quantum Mechanics by David Tong>[§§4.1.1 and4.1.4, monatomic chain and quantized vibrations]{ref}
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