Here, $L^3/a^3$ is the number of unit cells in the system, so we see that a single band has room for 2 electrons per unit cell (the factor 2 comes from the spin).

Here, $L^3/a^3$ is the number of unit cells in the system, so we see that a single band has room for 2 electrons per unit cell (the factor 2 comes from the spin).

We come to the important rule:

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@@ -225,10 +229,10 @@ _(based on exercise 15.4 of the book)_

Suppose we have a square lattice with lattice constant $a$, with a periodic potential given by $V(x,y)=2V_{10}(\cos(2\pi x/a)+\cos(2\pi y/a))+4V_{11}\cos(2 \pi x/a)\cos(2 \pi y/a)$.

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: the states that interact have $k$-vectors $(\pi/a,0)$ and $(-\pi/a,0)$; ($\psi_{+}\sim e^{i\pi x /a}$ ; $\psi_{-}\sim e^{-i\pi x /a}$).

2. Let's now study the more complicated 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.