I'm currently trying to figure out how electron bands form in solids. The Pauli exclusion principle states that:
two or more identical fermions (particles with half-integer spin) cannot occupy the same quantum state within a quantum system simultaneously
Here's another excerpt from the book University Physics with modern Physics:
Because of the electrical interactions and the exclusion principle, the wave functions begin to distort, especially those of the outer, or valence, electrons. The corresponding energies also shift, some upward and some downward, by varying amounts, as the valence electron wave functions become less localized and extend over more and more atoms.
As single atoms, the gaps are quite large between each subsequent energy level $E_n$. How come that the energy levels are so close to each other in solids, and what is the formula for determining the 2nd or 3rd atom's energy level?