@@ -223,12 +223,12 @@ Consider a single orbital per atom and only nearest-neighbour interactions.
### Exercise 3: Nearly-free Electron model in 2D
_(based on exercise 15.4 of the book)_
Suppose we have a square lattice with lattice constant $a$, with a periodic potential given by $V(x,y)=2V_{10}(cos(2\pi x/a)+cos(2\pi y/a))+4V_{11}cos(2 \pi x/a)cos(2 \pi y/a)$.
Suppose we have a square lattice with lattice constant $a$, with a periodic potential given by $V(x,y)=2V_{10}(\cos(2\pi x/a)+\cos(2\pi y/a))+4V_{11}\cos(2 \pi x/a)\cos(2 \pi y/a)$.
1. Use the Nearly-free electron model to find the energy of state $\mathbf{q}=(\pi/a, 0)$.
??? hint
This is analogous to the 1D case, as the interacting states are $(\pi/a,0)$ and $(-\pi/a,0)$; (\psi_{+}~e^{i\pi x /a} and \psi_{-}~e^{-i\pi x /a})$.
This is analogous to the 1D case: the states that interact have $k$-vectors $(\pi/a,0)$ and $(-\pi/a,0)$; (\psi_{+}\sim e^{i\pi x /a} and \psi_{-}\sim e^{-i\pi x /a})$.
2. Let's now study the more complicate case of state $\mathbf{q}=(\pi/a,\pi/a)$. How many $k$-points have the same energy? Which ones?
3. Write down the nearly free electron model Hamiltonian near this point.
