extra clarity to exercises

docs/11_nearly_free_electron_model.md

@@ -281,7 +281,7 @@ Suppose we have a crystal with lattice vectors $\mathbf{a}_ 1$, $\mathbf{a}_ 2$,
 
 1. What can be said about the symmetry of the Hamiltonian $\hat{H}$ of this crystal?
 
-Now define the translation operator $\hat{T}_{\alpha,\beta,\gamma}$ so that 
+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
 $$
 \hat{T}_{\alpha,\beta,\gamma} \psi(\mathbf{r}) = \psi(\mathbf{r} - \alpha \mathbf{a}_1 - \beta \mathbf{a}_2 - \gamma \mathbf{a}_3), 
 $$
@@ -296,18 +296,20 @@ where $\mathbf{\hat{p}} =-i\hbar\nabla$.
 5. What is $u_{n,\mathbf{k}}(\mathbf{r})$ in case of free electrons? Is your answer consistent with the equation above?
 
 ### Exercise 2: the central equation in 1D
-Let's consider a 1D crystal with a period $a$. Let $k_0$ be any wave number of an electron in the first Brillouin zone.
-1. What $k_n$ are equivalent to $k_0$ in this crystal?
+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.
 
-We will now introduce a weak periodic potential $V(x) = V(x+na)$ in our system. This causes coupling between eigenstates $\left| \phi_n\right>$ in the free electron case. In order to find the right eigenstates of the system with that potential, we need an 'LCAO-like' trial eigenstate given by
+We will now introduce a weak periodic potential $V(x) = V(x+na)$ to the system, which causes coupling between eigenstates $\left| \phi_n\right>$ of the free electrons. 
+In order to find the right eigenstates of the system with that potential, we use an 'LCAO-like' trial eigenstate 
 
 $$
 \left|\psi\right> = \sum_{n=-\infty}^{\infty}C_n \left|\phi_n\right>
 $$
 
-4. Using the trial eigenstate above and the Schrödinger equation, show that $$
+4. Using this trial eigenstate and the Schrödinger equation, show that $$
 E C_m = \varepsilon_m C_m+\sum_{n=-\infty}^{\infty} C_{n}V_{m-n}
 ,$$ 
 where $V_n$ are the Fourier components of the potential defined [above](#physical-meaning-of-w).