# Phonon as Fourier transform

I am learning about Debye's treatment of solids. In particular, he considers the vibrations throughout the whole solid and quantizes them. My question is threefold.

1. First: why do phonons have the harmonic oscillator energies $E = \hbar \omega(n + \frac{1}{2})$?

2. Second: Why do the Boltzmann factors apply to phonons, as if they are particles? It seems to me that the temperature of the material is defined relative to the particle energies (i.e. the $\sigma$ in $\partial \sigma / \partial U$ is the multiplicity of the energy as the atom's fill out their orbitals; it seems unrelated to the phonons).

3. Third: Can the treatment itself be conceptualized as a sort of Fourier transform of the atoms motion? If so, I would love a thoughtful explanation of this.

In the following, I will use the 1D atomic chain as a canonical illustration (3D generalization being inessensial for the discussion).

1. Why do phonons have the harmonic oscillator energies $E=\hbar\omega(n+1/2)$?

Atoms in a crystal interact with each other through a given bonding potential $U(x)$ which features a minimum. Such minimum settle the position of each atoms at zero temperature such that it forms a lattice where atoms are equally spaced by a distance $a$. If you are interested in the low energy properties of such system, each individual potential $U$ can be expand into quadratic form around its minimum : $$U(x)\sim \frac{\kappa}{2}x^2,\quad\text{with}\;\kappa>0$$ Let $\{x_j(t),\dot{x}_j(t)\}$ be the position and the velocity of the $j$th atom. Then the low energy lagrangian describing the system of $N$ atoms interacting with their closest neighbours reads : $$L=\sum_{j=1}^N \frac{m}{2}\dot{x}_j^2-\frac{\kappa}{2}(x_{j+1}-(x_j+a))^2$$ Defining $\phi_j$ as the atomic displacement away from the equilibrim position such that $x_j(t)=\phi_j(t)+ja$, the lagrangian reads now : $$L=\sum_{j=1}^N \frac{m}{2}\dot{\phi}_j^2-\frac{\kappa}{2}(\phi_{j+1}-\phi_j)^2$$ In the continuum limit where $|\phi_{j+1}-\phi_j|\ll a$, it is appropriate to define an atomic classical field $\phi(x,t)$ such that : $$L=\int_0^L\mathrm{d}x\;\mathcal{L}(\phi,\partial_x\phi,\dot{\phi})=\int_0^L\mathrm{d}x\;\frac{m}{2}\dot{\phi}^2-\frac{\kappa a^2}{2}(\partial_x\phi)^2$$ where $L=Na$ is the length of the chain and $\mathcal{L}$ the lagrangian density. At this point, it maybe interesting to note that the Langrange equation of motion $\partial_\phi\mathcal{L}-\partial_x(\partial_{\partial_x\phi}\mathcal{L})-\partial_t(\partial_{\dot{\phi}}\mathcal{L})=0$ gives you a wave equation : $$(m\partial_t^2-\kappa a^2\partial_x^2)\phi(x,t)=0$$ meaning that we know that the solution of the equation of motion of 1D atomic chain are sound waves, i.e. classical collectives excitations of the atoms motion with the dispersion relation $\omega_k=\sqrt{\frac{a^2\kappa}{m}}k$.

By Lengendre transform, in a way which is analogous to the classical point mechanics, we can calculate the associated hamiltonian $H$ : $$H=\int_0^L\mathrm{d}x\;\mathcal{H}(\phi,\partial_x\phi,\pi)=\int_0^L\mathrm{d}x\;\frac{\pi^2}{2m}-\frac{\kappa a^2}{2}(\partial_x\phi)^2$$ where $\pi=\partial\mathcal{L}/{\partial\dot{\phi}}$ is the canonical momentum associted to $\phi$, and $\mathcal{H}$ is the hamiltonian density.

From now, all the work has been done in the framework of classical mechanics. Getting into the quantum version the 1D atomic chain can be done by a quantization procedure of the classical hamiltonian $H$. It consist in replacing the classical field conjugate coordinates $\{\phi(x),\pi(x')\}=\delta(x-x')$ by quantum conjugate coordinates, said non-commutating operators $\left[\hat{\phi}(x),\hat{\pi}(x')\right]=\mathrm{i}\hbar\,\delta(x-x')$.

Then, the hamiltonian density describing the quantum 1D atomic chain reads : $$\mathcal{H}=\frac{\hat{\pi}^2}{2m}-\frac{\kappa a^2}{2}(\partial_x\hat{\phi})^2$$ We are almost done ; at this point we can take the calculations in the Fourier space for the operators, using the notation $\hat{A}=\frac{1}{\sqrt{L}}\sum_k \hat{A}_k e^{\mathrm{i}kx}$ : $$\mathcal{H}=\sum_k \frac{1}{2m}\hat{\pi}_k\hat{\pi}_{-k}+\frac{m\omega_k^2}{2}\hat{\phi}_k\hat{\phi}_{-k}$$ where $\omega_k^2=\frac{a^2k^2\kappa}{m}$.

The hamiltonian density almost identitical to the usual quantum harmonic oscillator one. That's why without any loss of generality, one can define the associated creation/annihilation operators $\hat{a}^\dagger_k$ and $\hat{a}_k$ and find that : $$\mathcal{H}=\sum_k \hbar\omega_k\left(\hat{a}^\dagger_k\hat{a}_k+\frac{1}{2}\right)$$ What we get here is that the hamiltonian of a quantum 1D atomic chain describes the system as a superposition of independent harmonic oscillators. What we call phonon is the quantum collective excitation explaining the low-energy behavior of the quantum 1D chain (in the same way that sound waves explain the low-energy dynamics of the classical system). The only thing that identify a phonon as a "particle" is its energy, or rather its dispersion relation $\hbar\omega_k$. Determine how much energy there is in the system is thus equivalent to count how much phonon modes $\omega_k$ are populated through the counting operators $\hat{a}^\dagger_k\hat{a}_k$. It is important to note that phonon have no real objective existence, they are just convenient tools for us to understand how works the quantum 1D atomic chain.

1. Why do the Boltzmann factors apply to phonons, as if they are particles?

Boltzmann distribution only apply for phonon "at high temperature". Since there can be any number $n_k$ of phonons populating an energy state $\omega_k$, it suggests that phonons have a bosonic nature, so that their energy distribution at thermal equilibrium is actually a Bose-Einstein distribution $f_B(\omega_k)$. However, you can see that in the limit $\omega_k\ll k_BT$, $f_B(\omega_k)\sim\exp(-\hbar\omega_k/k_BT)$.

1. Can the treatment itself be conceptualized as a sort of Fourier transform of the atoms motion?

As seen previously, the hamiltonian of the quantum 1D chain indentifies as a superposition of independant harmonic oscillators. It is very important here not to confuse the motion of individual atoms which is oscillatory (but coupled to their neighbours through $U$) with the collective motion of the atoms, which is also oscillatory where these modes $\omega_k$ are not coupled to each other. Actually, since the moment we took the limit $|\phi_{j+1}-\phi_j|\ll a$ we admit that the individual motion of each motion was not relevant and it was more convenient to understand the atomic motion as collective field $\phi$.