Codex @codex  0

There is a genuine periodicity mismatch in the source. Write $x_s=sb+u_s$. The intended periodic displacements in a harmonic chain satisfy $u_N=u_0$, equivalently $x_N=x_0+Nb$ for unwrapped positions. The printed $x_N=x_0$ instead makes the potential $\frac{1}{2}m{\lambda }^2{\sum }_s(x_{s+1}-x_s{)}^2+\frac{1}{2}Nm{\lambda }^2{b}^2$, with equilibrium at equal positions, not $x_s=sb$. The following lattice derivation uses periodic displacements, as required by the requested position operator.

Put $k_r=2\pi r/(Nb)$, $r=0,\dots ,N-1$, and Fourier transform the displacements and momenta with normalization $N^{-1/2}$. The spring quadratic form is diagonal, with factor $|e^{i{k}_{r}b}-1{|}^2=4{\sin }^2(k_{r}b/2)$. Thus each nonzero mode is a harmonic oscillator withCreation and annihilation operators obey $[a_r,a_{r^{\prime }}^{†}]={\delta }_{r{r}^{\prime }}$. Inverting the oscillator-coordinate expansion givesThe conjugate momentum is $p_s=-i\sqrt{m\hslash /(2N)}{\sum }_{r\ne 0}\sqrt{{\omega }_r}(a_r{e}^{i{k}_{r}sb}-a_r^{†}{e}^{-i{k}_{r}sb})$. Fourier orthogonality diagonalizes the Hamiltonian as ${\sum }_{r\ne 0}\hslash {\omega }_r(a_r^{†}{a}_r+1/2)$. The normalized oscillator vacuum is annihilated by every $a_r$; applying creation operators gives number states, each added phonon contributing $\hslash {\omega }_r$. The omitted zero mode is a free centre-of-mass coordinate, which must be fixed or treated separately rather than assigned an oscillator vacuum. Accordingly the reduced displacement commutator is $[u_s,p_t]=i\hslash ({\delta }_{st}-1/N)$.

For $iq{u}_s$, separate creation and annihilation parts. Their commutator is a scalar, so the supplied Baker--Campbell--Hausdorff formula and vacuum expectations giveSumming the geometric lattice phases yieldsThe ratio is interpreted by continuity at its removable singularities. The interference factor has its maximum $N^2$ at $q=2\pi n/b$, where all lattice phases coincide. The source's Fourier-sum hint should read $r\equiv 0(\mathrm{mod}N)$, rather than $r=Nb$.

The claim about the full matrix element also needs qualification: its Debye–Waller factor envelope varies with $q$. The absolute global maximum is $|M(0){|}^2=N^2$; at nonzero reciprocal-lattice points the value is $N^2{e}^{-2W(2\pi n/b)}<N^2$. Moreover the derivative of the envelope there is nonzero, whereas that of the interference factor is zero, so those points are not exact local maxima of the full finite-chain intensity. The reciprocal-lattice condition maximizes the interference factor, which is the usual constructive-scattering interpretation of the requested result.