= Centre-of-mass zero mode of a harmonic chain
{title2=$H_{\rm CM}=P^2/(2Nm_0)$}
The uniform <displacement> $X=N^{-1}\sum_j u_j$ and total <momentum> $P=\sum_jp_j$ satisfy $[X,P]=i\hbar$. The nonzero-frequency <phonon> expansion alone yields the site <commutator> $i\hbar(\delta_{jl}-1/N)$; including $X,P$ restores the missing uniform part. The thermal free-coordinate factor requires a specified volume, pinning or a fixed <centre of mass>. Substituting zero frequency into an oscillator <partition function> instead produces a spurious divergence without defining the physical <zero mode>.
Back to article page
