What is physical interpretation of the band structure diagrams $E=f(k)$? What is the actual meaning of the lines given in the band structure diagram? For example, the $E=f(k)$ relation for a simple 1-dimensional periodic lattice is $E(k) = -2\cos(k)$.


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*What is the physical meaning of this graph? It is showing the energy of a particle at a specific k-point. Does it mean that when our particle is moving in the k-vector direction, it has to change its energy?

*In this case, there is only one band (only one line). In some other examples, I have seen there are many more bands. What does more than one band mean?

 A: Electrons in a lattice are subject to a periodic potential $V(\vec{r})=V(\vec{r}+\vec{R})$ which has the periodicity of the lattice. The relation $E(\vec{k})$ comes from solving Schrödinger's equation in this periodic potential. It can be shown (Bloch's theorem) that the eigenstates of the Hamiltonian are of the form
$$\psi_{\vec{k}}(\vec{r})=e^{\displaystyle i\vec{k}\cdot\vec{r}}u_{\vec{k}}(\vec{r})\ .$$
Notice several things:

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*The possible energies of an electron in this lattice will depend on $\vec{k}$, since the energy eigenstates depend on this parameter.
$$\hat{H}\psi_{\vec{k}}=E\psi_{\vec{k}}\quad\implies\quad E=E(\vec{k})$$

*The energies will also depend on a quantum number $n$, as we would expect for bound states. Therefore $E=E_n(\vec{k})$. The number $n$ is what we call the band, each $n$ labels a different one, and the relation in your figure is just the plot of $E(\vec{k})$ for a given band.

*You can think of $\vec{k}$ as a tag for the electron, just as $n$. It is related to the expected value of the momentum $\vec{p}$ of the electron, but it's not exactly that. The quantity that is conserved when the electron interacts with other particles (e.g. photons, phonons, etc) is what is called the quasimomentum $\hbar\vec{k}$. A nice discussion on the meaning of $\vec{k}$ can be found in this other question.

A: It is analogous to a gas of particles, where each particle has a momentum $\mathbf p_k$ and an energy $E_k = \frac{p_k^2}{2m}$.
The first big difference is that for the bands the energy is quantized, so only discrete value of momentum and energy are allowed.
The second difference is that the same value of $k$ means different values of energy and crystal momentum, and that is the meaning of several bands.
