# Nearly free electron model: Pauli principle

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?

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.