# Treatment of electrons and phonons in condensed matter physics

I was watching the lectures by steve simon(oxford) on solid-state physics. In the course, he derived the dispersion relation for phonons(assuming spring between atoms) and dispersion relation for electrons(tight-binding model) using the Schrodinger equation. Then, he says that the main difference between these occurs because Newton's laws of motion have a double order time derivative(hence the factor of $$w^2$$) whereas Schrodinger equation has a single order time derivative(factor of $$w$$). My question is why do we treat atoms classically(using springs). Shouldn't we treat them quantum mechanically too?

When you treat the atoms as though they're connected by springs, this simply means that they have a potential energy $$\propto x^2$$, where $$x$$ is an atom's displacement from its equilibrium position. We already know what an $$x^2$$ potential does in quantum mechanics, so the "classical" treatment is simply manipulating the potential energy so that it's in a nice form and then quoting the quantum mechanics result we already know for a simple Harmonic oscillator.