The discussion revolves around the concept of energy bands in solids, particularly focusing on the number of bands, their completeness, and the implications for quantum mechanics. Participants explore the relationship between the bands near the Fermi surface and the overall band structure of solids, including the completeness of eigenwavefunctions.
Participants express differing views on the nature of eigenwavefunctions and the completeness of basis sets in quantum mechanics. There is no consensus on whether using only certain bands can lead to a complete basis, indicating an unresolved debate.
The discussion highlights limitations in understanding the relationship between band structure and quantum mechanics, particularly regarding the definitions and roles of eigenstates and basis functions. Some assumptions about the tight-binding approximation and its implications for band structure are also noted but remain unresolved.
howl said:Usually, we only talk about the band near Fermi surface, but we know that atom could have infinite levels, so, for a solid, does it have infinite levels too? So, if we only talk about the levels near Fermi surface, are the eigenwavefunctions complete?
In your link, there are some bands, for instance,Si. The electrons occupy the lowest bands under Fermi surface. For example, the band marked Gamma25', while Gamma15 and Gamma2' are empty. In QM, we should use up all eigenstates to build up a complete basis, then , if we only use Gamma25', could we build a complete basis?ZapperZ said:I'm not sure what you mean by your question on whether "... the eigenwavefunctions complete". ALL wavefunctions in solids are approximate, because you are dealing with a lot of atoms here.
In any case, I do not know if you are aware of the band structure of solids. For example, are you familiar with the band structure shown in the figures in this link?
http://what-when-how.com/electronic...stals-fundamentals-of-electron-theory-part-3/
Each of those lines are the bands (E vs. k) in a particular solid. The simplified band diagram that most people are familiar with are the ones where the momentum (k) has been integrated out, leaving only the energy values E.
Zz.
howl said:In your link, there are some bands, for instance,Si. The electrons occupy the lowest bands under Fermi surface. For example, the band marked Gamma25', while Gamma15 and Gamma2' are empty. In QM, we should use up all eigenstates to build up a complete basis, then , if we only use Gamma25', could we build a complete basis?
Of course not.howl said:In QM, we should use up all eigenstates to build up a complete basis, then , if we only use Gamma25', could we build a complete basis?