Update 11_nearly_free_electron_model_solutions.md - bug fix

src/11_nearly_free_electron_model_solutions.md - typo

@@ -134,24 +134,24 @@ E_-(k) = -\frac{\lambda}{a}+\frac{\hbar^2}{4m}\left[k^2+\left(k-\frac{2\pi}{a}\r
 See the lecture notes!
 
 ### Subquestion 3
-We split the Hamiltonian into two parts $H=H_n+H_{~n}$, where $H_n$ describes a particle in a single delta-function potential well, and $H_\overline{n}$ is the perturbation by the other delta functions:
+We split the Hamiltonian into two parts $H=H_n+H_{\hat{n}}$, where $H_n$ describes a particle in a single delta-function potential well, and $H_\hat{n}$ is the perturbation by the other delta functions:
 \begin{align}
 H_n = & \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} - V_0\delta(x-na) \\
-H_\overline{n} = & - V_0 \sum_{m\neq n}\delta(x-ma)
+H_\hat{n} = & - V_0 \sum_{m\neq n}\delta(x-ma)
 \end{align}
 such that $H_n|n\rangle = -\epsilon_0|n\rangle = -\hbar^2\kappa^2/2m |n\rangle$ with $\kappa=mV_0/\hbar^2$. We can now calculate 
 $$
-\langle n | H |n \rangle = \epsilon_0 + \langle n |H_\overline{n}|n\rangle
+\langle n | H |n \rangle = \epsilon_0 + \langle n |H_\hat{n}|n\rangle
 $$
 Note that the last term represents the change in energy of the wavefunction $|n\rangle$ that is centered at the $n$-th delta function caused by the presence of the other delta functions. This term yields
 $$
-\langle n |H_\overline{n}|n\rangle = \kappa \sum_{m \neq 0 }\int e^{-2\kappa|x|}\delta(x-ma)  = \kappa \sum_{m \neq 0 } e^{-2\kappa|ma|} = 2\kappa(\frac{1}{1-e^{-2\kappa a}}-1)
+\langle n |H_\hat{n}|n\rangle = \kappa \sum_{m \neq 0 }\int e^{-2\kappa|x|}\delta(x-ma)  = \kappa \sum_{m \neq 0 } e^{-2\kappa|ma|} = 2\kappa(\frac{1}{1-e^{-2\kappa a}}-1)
 $$
 Note that the result should not depend on $n$, so we chose $n=0$ for convenience.
 
 Similarly, we can calculate 
 $$
-\langle n-1 | H |n \rangle = \epsilon_0\langle n-1  |n \rangle + \langle n-1 |H_\overline{n}|n\rangle
+\langle n-1 | H |n \rangle = \epsilon_0\langle n-1  |n \rangle + \langle n-1 |H_\hat{n}|n\rangle
 $$
 where $\langle n-1|n\rangle$ is the overlap between two neighbouring wavefunctions:
 $$
@@ -159,7 +159,7 @@ $$
 $$
 and
 $$
-\langle n-1|H_\overline{n}|n\rangle$ = \kappa \sum_{m \neq 0 }\int e^{-\kappa|x-a|} \delta(x-ma)  e^{-\kappa|x|} =\kappa \sum_{m \neq 0 } e^{-\kappa a|m-1|} e^{-\kappa a |m|} =\kappa(e^{ka+e^{-ka}}) \sum_{m=1}^{m=\infty} e^{-2\kappa a m}
+\langle n-1|H_\hat{n}|n\rangle$ = \kappa \sum_{m \neq 0 }\int e^{-\kappa|x-a|} \delta(x-ma)  e^{-\kappa|x|} =\kappa \sum_{m \neq 0 } e^{-\kappa a|m-1|} e^{-\kappa a |m|} =\kappa(e^{ka+e^{-ka}}) \sum_{m=1}^{m=\infty} e^{-2\kappa a m}
 $$
 
 Update in progress......