attempt to fix labeling of exercises

docs/11_nearly_free_electron_model.md

@@ -280,16 +280,13 @@ draw_classic_axes(ax, xlabeloffset=4)
 Suppose we have a crystal with lattice vectors $\mathbf{a}_ 1$, $\mathbf{a}_ 2$, and $\mathbf{a}_ 3$.
 
 1. What can be said about the symmetry of the Hamiltonian $\hat{H}$ of this crystal?
-
-To describe the translation through the crystal in terms of the lattice vectors, we define the translation operator $\hat{T}_{\alpha,\beta,\gamma}$ in such a way that
-$$
-\hat{T}_{\alpha,\beta,\gamma} \psi(\mathbf{r}) = \psi(\mathbf{r} - \alpha \mathbf{a}_1 - \beta \mathbf{a}_2 - \gamma \mathbf{a}_3),
-$$
+2. To describe the translation through the crystal in terms of the lattice vectors, we define the translation operator $\hat{T}_{\alpha,\beta,\gamma}$ in such a way that $$
+\hat{T}_{\alpha,\beta,\gamma} \psi(\mathbf{r}) = \psi(\mathbf{r} - \alpha \mathbf{a}_1 - \beta \mathbf{a}_2 - \gamma \mathbf{a}_3)
+, $$
 where $\alpha$, $\beta$, $\gamma$ are integers.
-
-2. Show that $\hat{T}_{\alpha,\beta,\gamma}$ and $\hat{H}$ commute.
+Show that $\hat{T}_{\alpha,\beta,\gamma}$ and $\hat{H}$ commute.
 3. Show that the Bloch wavefunctions defined in the lecture notes are eigenfunctions of $\hat{T}_{\alpha,\beta,\gamma}$. What are the corresponding eigenvalues? What does this say about the eigenfunctions of $\hat{H}$.
-5. By applying $\hat{H}$ to the Bloch wavefunction, show that the Schrödinger equation can be rewritten as $$
+4. By applying $\hat{H}$ to the Bloch wavefunction, show that the Schrödinger equation can be rewritten as $$
 \left[ \frac{\mathbf{\hat{p}}^2}{2m} + \frac{\hbar}{m} \mathbf{k} \cdot \mathbf{\hat{p}} + \frac{\hbar^2 \mathbf{k}^2}{2m} + V(\mathbf{r}) \right] u_{n,\mathbf{k}}(\mathbf{r}) = E_{n,\mathbf{k}} u_{n,\mathbf{k}}(\mathbf{r})
 , $$
 where $\mathbf{\hat{p}} =-i\hbar\nabla$.
@@ -298,6 +295,7 @@ where $\mathbf{\hat{p}} =-i\hbar\nabla$.
 ### Exercise 2: the central equation in 1D
 Let's consider a 1D crystal with a periodicity of $a$.
 Let $k_0$ be any wave number of an electron in the first Brillouin zone.
+
 1. Which $k_n$ are equivalent to $k_0$ in this crystal?
 2. First, we assume that the electrons with these $k_n$ are free. In that case, what are the wavefunctions $\phi_n(x)$ and energies $E_n$ of these electrons?
 3. Make a sketch of the dispersion relation using a repeated Brillouin zone representation. Indicate some $k_n$ and $E_n$ as well as the first Brillouin zone in your sketch.