I am studying from the Aschroft & Mermim. In the chapter 8 they prove the Bloch theorem and introduce the concept of band. they say that k is a quantum number that charactrize the eigenfunction and can be chosen to be in the first brillouin zone. then they say that for a given k there are discrete eigenvalue which corrispond to the different eigenfunction. So they label them by an index n, the band index. But what does it represent? Then when they try to construc the ground state they say that some band can be partially or totally filled and Others empty. (And thus we can have a band gap.) But how this is possible to have some band filled and Others not.

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    $\begingroup$ In an atom, how is it possible to have some levels filled and others not? $\endgroup$ – Jon Custer Nov 23 '18 at 20:53

It's a general result of Quantum Mechanics that energy eigenstates may be labeled not only by a single quantum number, like in one-dimensional problems, but also by an additional set of quantum numbers, i.e. numbers directly related to the eigenvalues of all the operators commuting with the hamiltonian.

In an infinite crystalline solid, or in a finite solid system with periodic boundary conditions, the hamiltonian commutes with all the translations by Bravais lattice vectors. As a consequence, energy levels can be labeled by an integer (the band index $n$) as well as by the three component of a wavevector k, directly connected to the way the wavefunction changes after a Bravais lattice translation (which is the Bloch's theorem content) .

So, n and k uniquely select a single spacial eigenstate of the hamiltonian. Once one has the one-electron states, this information is only saying which are the possible quantum states of one electron. In order to find the ground state of a system of N electrons, one has to decide how to organize the N electrons among all the possible states in such a way to minimize the energy. Taking into account that electrons are fermions, the antisymmetry of the wave-function by any permutation implies that one spacial state cannot be used for more than 2 electrons, one per value of its spin projection along one direction. Thus the ground state corresponds to occupy the lowest $N/2$ spatial states.

As a consequence, depending on the shape of the bands as function of k and on the number of electrons, each band (all the states with a fixed value of $n$) can be entirely filled, completely empty or partly filled. The presence or not of a gap of energy between the highest occupied level and the lowest empty level directly controls most of the electronic properties of a material.


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