diff --git a/src/11_nearly_free_electron_model_solutions.md - typo b/src/11_nearly_free_electron_model_solutions.md - typo index 036072a22207b85277e950dbf25f11d1f9a080a7..020da00e0aa1aedb4ba2fdaad376c5d0ef824730 100644 --- a/src/11_nearly_free_electron_model_solutions.md - typo +++ b/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......