Why & How do Electrons Behave in Crystaline Structures?

[zrs_12]

*Why are electrons confined to bands where they can posses a partial quantum of energy (right?) as opposed to discrete energy levels in quantum dots?
*Why do electrons behave differently in crystaline structures?
*How do electrons behave in crystaline structure? (like bouncing back and forth) and why?
*How and why are the energy levels changed when spatial restriction occurs?
*Why does spatial restriction occur?
*What coordinate systems are used for the space and time coordinates of the Schrödinger equation?

 

[genneth]

The basic issue with electrons in lattices is that the usual translational symmetry of space is broken. You cannot translate (or rotate) by any arbitrary amount, but only by a lattice vector. This basic fact is formally called Bloch's theorem, which gives a constraint on the form of the wavefunction for an electron in a periodic potential. How electrons "behave" is not a particularly well-posed problem. Like any quantum particle, trying to visualise their behaviour as a billiard ball bouncing back and forth is going to run into problems. The bands come from the fact that you have a macroscopic (so effectively infinite) single electron states with the same energy --- one from each atom --- which mix together to produce a macroscopic number of multi-electron states, each slightly differing in energy. Thus you get a continuum of energies, occasionally separated by finite energy gaps.

Hope that's not too confusing.

 

[Gokul43201]

*Why are electrons confined to bands where they can posses a partial quantum of energy (right?)

Not right.

as opposed to discrete energy levels in quantum dots?

Why should they behave the same way is 2 different environments?

*Why do electrons behave differently in crystaline structures?

Deifferently from what?

*How do electrons behave in crystaline structure? (like bouncing back and forth) and why?

Read genneth's post. The so-called behavior is encoded in the electron wavefunction. You need to learn how the wavefunction of an electron in a crystal lattice is figured, from Bloch's theorem to nearly free electron approximations to the tight binding model (to the mott-hubbard model).

*How and why are the energy levels changed when spatial restriction occurs?

Hand waving answer: spatial restriction affects the momenta through the Uncertainty relation. Alternatively, spatial restriction affects the wave vector of bound states, which in turn affects the energy through the dispersion.

*Why does spatial restriction occur?

Because you put in a confining potential barrier.

*What coordinate systems are used for the space and time coordinates of the Schrödinger equation?

You choose a co-ordinate system depending on the geometry of the physical system; e.g., for a cubical quantum dot, you will use Cartesian co-ordinates.

 

[zrs_12]

According to Wikipedia:

When a large number of atoms (of order 10^20 or more) are brought together to form a solid, the number of orbitals becomes exceedingly large, and the difference in energy between them becomes very small, so the levels may be considered to form continuous bands of energy rather than the discrete energy levels of the atoms in isolation.

So why do bands occur in quantum dots containing a hundred to a few thousand atoms?

 

[genneth]

The key is the symmetry. In quantum dots you arrange the structure such that you don't have a regular lattice. There's also the issue of temperature...