Free electron square lattice

[aaaa202]

Homework Statement

Show for a simple square lattice that the kinetic energy of a free electron is higher at the corner of the first zone than at the midpoint a side face by a factor of 2.

Homework Equations

Simple geometry.

The Attempt at a Solution

I think I know how to solve, but I am not completely sure I understand the question. The position of the electron refers to the position of its k-vector in k-space right? Well this is not real space, and I can't really see how this has anything to do with the lattice being square in real space.

 

[Physics Monkey]

I suspect the hidden assumption here is that you should use a tight binding model with nearest neighbor hopping on the lattice given. So in your case the square lattice is specifying the shape of the Brillouin zone and telling you the energy spectrum.

Does this help?