diff --git a/src/11_nearly_free_electron_model.md b/src/11_nearly_free_electron_model.md
index 52dcbc6bcf32da386cb3cdd996d78e160901d677..71191d35564dfdeefc3f5ae7217a4477d5b49e48 100644
--- a/src/11_nearly_free_electron_model.md
+++ b/src/11_nearly_free_electron_model.md
@@ -87,9 +87,9 @@ $$
 ??? question "calculate $E_0$ and the velocity $v$"
     The edge of the Brilloin zone has $k = \pi/a$. Substituting this in the free electron dispersion $E = \hbar^2 k^2/2m$ we get $E_0 = \hbar^2 \pi^2/2 m a^2$, and $v=\hbar k/m=\hbar \pi/ma$.
 
-Note that this Hamiltonian is diagonal so the eigenenergies are on the diagonal and the eigenfunctions are the $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$ plane waves.
+Note that this Hamiltonian is diagonal so the eigenenergies are on the diagonal and the eigenfunctions are the $|k\rangle$ and $|k'\rangle$ plane waves.
 
-As we will see below, the lattice potential $V(x)$ may couple the states $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$. The coupling strength is given by the matrix element $W=\langle \mathbf{k} |V| |\mathbf{k'}\rangle$. Including the coupling into the Hamiltonian, we get
+As we will see below, the lattice potential $V(x)$ may couple the states $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$. The coupling strength is given by the matrix element $W=\langle \mathbf{k} |V| |\mathbf{k'}\rangle$. Including the coupling into our Hamiltonian, we get
 
 $$
 H\begin{pmatrix}\alpha \\ \beta \end{pmatrix} =
@@ -114,7 +114,7 @@ Check out section 15.1.1 of the book for the details of this calculation.
 
 #### Physical meaning of $W$
 
-Now we expand the definition of $W$:
+Now we expand the definition of $W$, concentrating on 1D:
 
 $$W = \langle \psi_+ | V(x) | \psi_- \rangle = \frac{1}{a}\int_0^{a} dx \left[e^{i\pi x/a}\right]^* V(x) \left[e^{-i\pi x/a}\right] = \frac{1}{a}\int_0^a e^{-2\pi i x /a} V(x) dx = V_1$$