TITLE: Claim that DeBroglie relation doesn't work in crystal
QUESTION [2 upvotes]: In this Wikipedia article on Position and Momentum Space, https://en.wikipedia.org/wiki/Position_and_momentum_space
there is a claim that "the de Broglie relation is not true in a crystal" in the sentence before the content box.
Is this claim valid? If so, why? What is the implications for quasi-particles (e.g. plasmons and polaritons) in materials?

REPLY [2 votes]: In a crystal, $\vec p$ does not necessarily have the same direction than $\vec k$. So, I suppose that it's indeed true that the de Broglie relation ($\vec p = \hbar \vec k$) does not always hold in a crystal.
If we take a perfect crystal, then the wavefunction of an electron can be written as the Bloch electron wavefunction $\Psi = u(\vec r) e^{i\vec k \cdot \vec r}$ where $u(\vec r)$ is a periodic function whose periodicity matches the lattice's. By applying the momentum operator $\hat p =-i\hbar \nabla_\vec r$ to that wavefunction, one finds that it's equal to $\hbar \vec k \Psi + \text{something not proportional to } \Psi$ (nor to $\vec k$ for that matter.). Here, $\hbar \vec k$ is called the crystal momentum and does not match the electron's momentum. See Ashcroft and Mermin pages 139 and 219 for a detailed discussion about that.
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