Update 11_nearly_free_electron_model_solutions.md - update

src/11_nearly_free_electron_model_solutions.md

@@ -147,7 +147,7 @@ Note that the last term represents the change in energy of the wavefunction $|n\
 $$
 \langle n |H_\overline{n}|n\rangle = -\kappa \lambda \sum_{m \neq 0 }\int e^{-2\kappa|x|}\delta(x-ma)  = -\kappa \lambda \sum_{m \neq 0 } e^{-2\kappa|ma|} = -2\kappa\lambda(\frac{1}{1-e^{-2\kappa a}}-1)
 $$
-Note that the result should not depend on $n$, so we chose $n=0$ for convenience.
+Note that the result should not depend on $n$, so we chose $n=0$ for convenience. 
 
 Similarly, we can calculate 
 $$
@@ -163,6 +163,15 @@ and
 =& -\kappa \lambda \sum_{m \neq 0 } e^{-\kappa a|m-1|} e^{-\kappa a |m|} =-\kappa \lambda(e^{ka}+e^{-ka}) \sum_{m=1}^{m=\infty} e^{-2\kappa a m}
 \end{align}
 
+In the limit $\kappa a \gg 1$ (i.e., where the distance between the delta functions is large compared to the width of $|n\rangle$), the onsite energy becomes
+$$
+\langle n|H|n \rangle = \epsilon_0 - 2\kappa \lambda e^{-2\kappa a}
+$$
+and the hopping becomes
+$$
+\langle n-1|H|n \rangle = (\epsilon_0 \kappa a  - \kappa \lambda) e^{-\kappa a}
+$$
+
 ### Subquestion 4
 
 | .. | Lower Band minimum | Lower Band Width|