In the quantum theory of electrical conductivity are considered free and bound electrons.
Lev
Brillouin zone theory considers the free electrons to explain crystal and reciprocal lattices. These free electrons occupy the valence band and conduction band in a conductor or semiconductor.
Perhaps the question should be: are they bound locally or globally, globally meaning by the crystal, usually imagined of infinite extent to facilitate calculations, and locally meaning to an atom. Then if the electron has energy above the fermi level, it's bound globally but if below then locally. For a finite crystal even a globally bound electron is not free because it doesn't have a continuous spectrum.
Bonded to the native atom. If the energy levels will be in the energy gap, then those electrons will be bonded to the impurity atoms and if they are above the energy gap, then they are bonded to the several atoms and finally if they are above that 'free electron' energy space, they are detached from the surface of the body (determined by the ionization energy) or they are trapped inside the bulk and are called 'hot electrons'.
To all of them since electron has got both electric field and magnetic...it accounts to the whole energy particle possesses.
If the electrons are acted on by a strictly periodic infinitely extended external potential, due to the crystal lattice, and if interaction between electrons is neglected, then all electrons in fact have wave functions that are extended over the whole crystal, as follows from Bloch's theorem. The extended wave functions have a quasimomentum k, and are in a sense deformed plane waves. The energy then depends on k. However, for electrons close to the nucleus (e.g. core electrons), there is almost perfect degeneracy and the electron energies are essentially independent of k. This allows to construct localized wavefunctions. In fact, interaction, as well as disorder, will localize such electrons, so that the picture presented here is quite academic in that case. However, when the dependence of the elctron energy on the quasimomentum k is significant, we must treat the electrons as free and consider the wave function as exended over the whole crystal.
This is a rough and ready statement of what I know on the subject. There may well be many more complex issues of which I am not aware.
the wave function is just a mathematical abstraction-approximation according to one model, moreover QM too is just based on a crude mathematical model since in this way it is easier to get some quantities and this means that it is a bad way to look at the real phenomena
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