diff --git a/src/11_nearly_free_electron_model_solutions.md b/src/11_nearly_free_electron_model_solutions.md index e05d80cb72de113e34ae72b94adfa7f833ad9689..53c9170cf317c7d37604ce70c13b666460356e1a 100644 --- a/src/11_nearly_free_electron_model_solutions.md +++ b/src/11_nearly_free_electron_model_solutions.md @@ -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 $H=H_n+H_{\hat{n}}$, 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: +We split the Hamiltonian into two parts $H=H_n+H_{\overline{n}}$, 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_\hat{n} = & - V_0 \sum_{m\neq n}\delta(x-ma) +H_n = & \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} - \lambda\delta(x-na) \\ +H_\overline{n} = & - \lambda \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 +such that $H_n|n\rangle = \epsilon_0|n\rangle = -\hbar^2\kappa^2/2m |n\rangle$ with $\kappa=m\lambda/\hbar^2$. We can now calculate $$ -\langle n | H |n \rangle = \epsilon_0 + \langle n |H_\hat{n}|n\rangle +\langle n | H |n \rangle = \epsilon_0 + \langle n |H_\overline{n}|n\rangle $$ 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 $$ -\langle n |H_\hat{n}|n\rangle = \kappa \sum_{m \neq 0 }\int e^{-2\kappa|x|}\delta(x-ma) = \kappa \sum_{m \neq 0 } e^{-2\kappa|ma|} = 2\kappa(\frac{1}{1-e^{-2\kappa a}}-1) +\langle n |H_\overline{n}|n\rangle = -\kappa \lambda \sum_{m \neq 0 }\int e^{-2\kappa|x|}\delta(x-ma) = -\kappa \lambda \sum_{m \neq 0 } e^{-2\kappa|ma|} = -2\kappa\lambda(\frac{1}{1-e^{-2\kappa a}}-1) $$ Note that the result should not depend on $n$, so we chose $n=0$ for convenience. Similarly, we can calculate $$ -\langle n-1 | H |n \rangle = \epsilon_0\langle n-1 |n \rangle + \langle n-1 |H_\hat{n}|n\rangle +\langle n-1 | H |n \rangle = \epsilon_0\langle n-1 |n \rangle + \langle n-1 |H_\overline{n}|n\rangle $$ where $\langle n-1|n\rangle$ is the overlap between two neighbouring wavefunctions: $$ @@ -159,8 +159,8 @@ $$ $$ and \begin{align} -\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} +\langle n-1|H_\overline{n}|n\rangle = & -\kappa \lambda \sum_{m \neq 0 }\int e^{-\kappa|x-a|} \delta(x-ma) e^{-\kappa|x|} \\ +=& -\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} ### Subquestion 4