Pauli's Role in Band Structure: Confusion or Clarity?

[nonequilibrium]

"Band structure a result of Pauli"? Professor hopelessly vague or in essence correct?

Hello,

I'm taking an introductory course on Solid State physics (level: last year undergrad physics) and the professor said one can view the band structure in solids as a result of the Pauli principle, since the valence electrons can occupy the same "place", and hence must have different energies to fullfill the Pauli principle. I objected in class (politely, in the format of a question) that Pauli didn't require different energies, but merely different states and reminded him of degeneracies (although in the back of my mind I was struggling with more of what he said, this was what struck me the most), upon which he responded with something along the lines of "indeed that is also possible, but if one is thinking about free electrons in a box, then if we ignore the spin variable part, different electrons have different energies". I don't really get what he's saying; aren't there degeneracies in the free electron gas? And anyway, he afterwards specified that his intuitive view was related to the tight binding method, also known as LCAO (linear combination of atomic orbitals), however this seems to be a long way from free electron gases (which he used in his argumentation).

In short, I would suspect that it's impossible to get something coherent out of it, but on the other hand I don't want to be rash and although my experimental physics professors are often very vague, I'm also willing to believe that somewhere tucked away in his obscure explanation, is a core of truth. Can anybody find it? Am I right in being confused about what he said? (note: I'm not posting this to vent, at all! My main intention is to simply understand what he meant)

 

[eljose79]

I can understand your confusion and frustration with your professor's explanation. While it is true that the Pauli principle plays a significant role in determining the electronic structure of solids, it is not the sole factor responsible for the band structure.

The Pauli principle states that no two identical fermions (such as electrons) can occupy the same quantum state simultaneously. This means that in a solid, the valence electrons will have to occupy different energy levels or states in order to satisfy this principle. However, as you pointed out, this does not necessarily mean that they will have different energies. In fact, there can be degeneracies in the energy levels of electrons, especially in the case of a free electron gas.

Your professor's explanation may have been oversimplified and may have neglected other important factors that contribute to the band structure, such as the influence of the crystal lattice. The tight binding method, which takes into account the interactions between neighboring atoms, is a more accurate approach to understanding the electronic structure of solids.

In essence, your professor's statement may have some truth to it, but it is not a complete or comprehensive explanation. It is always important to critically evaluate and question scientific explanations, and I encourage you to continue seeking a deeper understanding of the band structure in solids.