= Solution
Take a uniform nearest-neighbour <harmonic chain> with <displacement> $u_j$, conjugate <momentum> $p_j$, mass $m_0$, <spring> stiffness $\kappa$, lattice spacing $a$ and periodic length $\ell=Na$. These specify a standard model for the requested notes, not additional numerical exam data. Its <Lagrangian> and <Hamiltonian> are
$$
L=\sum_j\left[\frac{m_0}{2}\dot u_j^2-\frac\kappa2(u_{j+1}-u_j)^2\right],\qquad H=\sum_j\left[\frac{p_j^2}{2m_0}+\frac\kappa2(u_{j+1}-u_j)^2\right].
$$
Variation gives $m_0\ddot u_j=\kappa(u_{j+1}-2u_j+u_{j-1})$. For a mode $e^{i(kja-\omega t)}$, periodicity requires $k=2\pi n/\ell$ and
$$
\boxed{\omega_k=2\sqrt{\kappa/m_0}\,|\sin(ka/2)|.}
$$
The lattice makes $k$ periodic modulo $2\pi/a$, so choose one <Brillouin zone>. At long wavelength $\omega_k\simeq v|k|$, where $v=a\sqrt{\kappa/m_0}$ is the <sound speed>. The <continuum limit of a harmonic chain> expands $u_{j+1}-u_j\simeq a\partial_xu$ and replaces $\sum_j$ by $a^{-1}\int dx$. Defining <mass density> $\rho=m_0/a$ and one-dimensional elastic coefficient $K=\kappa a$ gives
$$
L=\int_0^\ell dx\left[\frac\rho2(\partial_tu)^2-\frac K2(\partial_xu)^2\right],\qquad \rho\partial_t^2u-K\partial_x^2u=0,\qquad v^2=K/\rho.
$$
The <momentum> density is $\Pi=\rho\partial_tu$, and the <Hamiltonian> density is $\Pi^2/(2\rho)+K(\partial_xu)^2/2$. Translational invariance of $u$ makes total <momentum> conserved. The continuum approximation describes acoustic wavelengths large compared with $a$; it is not a justification for extending the lattice dispersion and zero-point sums to arbitrarily large <wavevector>.<Applications of Quantum Mechanics by David Tong>[chapter 4, §4.1.1, printed pp.103–105: periodic monatomic chain and acoustic dispersion]{ref}
<canonical quantization> promotes the coordinates and momenta to operators with $[u_j,p_l]=i\hbar\delta_{jl}$ and the other equal-time <commutators> zero. In the continuum, $[u(x),\Pi(x')]=i\hbar\delta_{\rm per}(x-x')$. The nonzero <normal modes> can be represented by independent <phonon> operators obeying $[b_k,b_q^\dagger]=\delta_{kq}$. A useful lattice expansion is
$$
u_j=X+\frac1{\sqrt N}\sum_{k\ne0}\sqrt{\frac{\hbar}{2m_0\omega_k}}\left(b_ke^{ikja}+b_k^\dagger e^{-ikja}\right),
$$
where $X=N^{-1}\sum_j u_j$ is the centre-of-mass <displacement>. The corresponding <momentum> is
$$
p_j=\frac PN-\frac i{\sqrt N}\sum_{k\ne0}\sqrt{\frac{m_0\hbar\omega_k}{2}}\left(b_ke^{ikja}-b_k^\dagger e^{-ikja}\right),\qquad [X,P]=i\hbar.
$$
Including the centre-of-mass term restores the full site <commutator>: the nonzero-mode sum alone gives $i\hbar(\delta_{jl}-1/N)$. Substitution diagonalizes the <Hamiltonian>,
$$
\boxed{H=\frac{P^2}{2Nm_0}+\sum_{k\ne0}\hbar\omega_k(b_k^\dagger b_k+\tfrac12).}
$$
A <phonon> is one quantum of a collective <normal mode>, carrying energy $\hbar\omega_k$ and lattice <momentum> $\hbar k$ modulo a reciprocal-lattice <vector>. Reality of $u_j$ relates its Fourier coordinates by $u_k^\dagger=u_{-k}$; it does not remove the independent oppositely travelling <phonon> modes. Using one complete <Brillouin zone> avoids double-counting real standing-wave coordinates.<Applications of Quantum Mechanics by David Tong>[§4.1.4, quantization of lattice vibrations]{ref}
The free <centre of mass> has no restoring force and must not be treated as an oscillator of zero frequency. Fixing it, weakly pinning it, or specifying a finite centre-of-mass volume defines the corresponding zero-mode factor. With a fixed <centre of mass>, the internal thermal <partition function> and occupations are
$$
Z_{\rm int}=\prod_{k\ne0}\frac1{2\sinh(\beta\hbar\omega_k/2)},\qquad \langle n_k\rangle=\frac1{e^{\beta\hbar\omega_k}-1}.
$$
The internal energy is $\sum_{k\ne0}\hbar\omega_k(\langle n_k\rangle+1/2)$; its high-temperature <heat capacity> approaches $(N-1)k_B$. For a large one-dimensional chain at <temperatures> below the lattice cutoff but above its finite-size mode spacing,
$$
\frac{U_T}{\ell}=\int\frac{dk}{2\pi}\frac{\hbar v|k|}{e^{\beta\hbar v|k|}-1}=\frac{\pi(k_BT)^2}{6\hbar v},\qquad \frac C\ell=\frac{\pi k_B^2T}{3\hbar v}.
$$
This <one-dimensional acoustic phonon heat capacity> shows that \b[quantization turns classical normal waves into bosonic excitations with a linear low-temperature <heat capacity> in one dimension]. The Euclidean field action $S_E=\tfrac12\int_0^{\beta\hbar}d\tau\,dx\,[\rho(\partial_\tau u)^2+K(\partial_xu)^2]$ is Gaussian, with periodic imaginary-time boundary conditions; its Fourier modes reproduce these oscillators. Absolute <displacement> correlations require the zero-mode convention, and the lattice regulates short-distance fluctuations.
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