Update 11_nearly_free_electron_model_solutions.md - fix

src/11_nearly_free_electron_model_solutions.md

@@ -134,23 +134,31 @@ 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 $H=H_0+H_1$, where$H_0$ describes a particle in one delta-function potential well, and $H_1$ is the perturbation by the other delta functions:
+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 $H_{~n}$ is the perturbation by the other delta functions:
 \begin{align}
-H_0 = & \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} - \V_0\delta(x-na) \\
-H_1 = & - V_0 \sum_{m\neq n}\delta(x-ma)
+H_n = & \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} - V_0\delta(x-na) \\
+H_{~n} = & - V_0 \sum_{m\neq n}\delta(x-ma)
 \end{align}
-such that $H_0|n\rangle = -\epsilon_0|n\rangle = -\hbar^2\kappa^2/2m |n\rangle$ with $\kappa=mV_0/\hbar^2$.
+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 
+$$
+\langle n | H |n \rangle = \epsilon_0 + \langle n |H_{~n}|n\rangle
+$$
+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
+$$
+\langle n |H_{~n}|n\rangle = \kappa \sum_{m \neq 0 }\int e^{-2\kappa|x|}\delta(x-ma)  = \kappa \sum_{m \neq 0 } e^{-2\kappa|ma|} = 2\kappa(\frac{1}{1-e^{-2\kappa a}}-1)
+$$
+Note that the result should not depend on $n$, so we chose $n=0$ for convenience.
 
-It follows that 
+Similarly, we can calculate 
 $$
-\langle n | H |n \rangle = \epsilon_0 + \langle n |H_1|n\rangle
+\langle n-1 | H |n \rangle = \epsilon_0\langle n-1  |n \rangle + \langle n-1 |H_{~n}|n\rangle
 $$
-where
+where $\langle n-1|n\rangle$ is the overlap between two neighbouring wavefunctions, given by
 $$
-\langle n |H_1|n\rangle = \kappa \sum_{m \neq 0 }\int e^{-2\kappa|x|}\delta(x-ma)  = \kappa \sum_{m \neq 0 } e^{-2\kappa|ma|}
+$\langle n-1|n\rangle$ = 2\kappa\int_0^\infty e^{-\kappa|x-a/2|}e^{-\kappa|x+a/2|} = e^{-\kappa a}(1+\kappa a)
 $$
 
-In progress of being updated....
+In progress of being updated......
 
 \begin{equation}
 \varepsilon_0=\braket{n|\hat{H}|n}=\braket{n|\hat{K}|n}+\braket{n|\hat{V}(x)|n}