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If I look at the bandstructure in the nearly free electron model in a diagram with energy over wave vector $\bf{k}$, there are some parts of parabolas in my first Brillouin zone. I understand, why they are there. But I am not sure, how the fact that there are several different energys possible for one $\bf{k}$ (and therefore for one state $\psi_{\bf{k}}(r)$) doesn't contradict with the Pauli principle, because there could be some electrons with the same position probability density in the solid.

Is it because the energy is a quantum number here, and the Pauli principle says that the states must have different quantum numbers?

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As an analogy, recall the structure of a single multi-electron atom in the shell model. Nitrogen atom, for example, has 5 electrons with principal quantum number $n=2$. So you might as well wonder: how doesn't the fact that there are several different energies possible for one $n$ contradict the Pauli principle? Because principal quantum number $n$ doesn't completely describe a state. There are also orbital angular momentum quantum number $\ell$ and magnetic quantum number $m_\ell$, as well as spin quantum number $m_s$.

Similarly with electron in a crystal: quasiwavevector $\vec k$ doesn't completely describe a state. The full description of the state of an electron in crystal should additionally include energy and spin.

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