Kinetic energy of a free electron in a lattice

[Dixanadu]

Homework Statement

Show that for a simple square lattice (in 2-D) with the lattice spacing = a, the kinetic energy of a free electron at a corner (point A in the figure below) of the first Brillouin zone is higher than that of an electron at the midpoint of a side of the zone (point B in the figure below) by a factor of 2.
http://imageshack.com/a/img856/2000/zuci.jpg

Homework Equations

No clue

The Attempt at a Solution

So...like I have no clue about this...has anyone got a hint for me? I don't even know which equations to use...thanks guys...

 

[maajdl]

I guess this is quantum mechanics applied to solid physics?
In QM, electrons can't be totally at rest, which gives some sense to the question ...
You need first to remember everything about what a Brillouin zone is!
Then you simply need to translate "corner" and "midpoint" to answer this question.

This question contains little very physical meaning.
It is more a check that you know what a Brillouin zone is.
Therefore, I also suggest you the question yourself about why this concept (Brillouin zone) emerged and what its utility is.
This might help you for the next questions, maybe?