From f6bf705da61936c1804f10da87c74552da18b52d Mon Sep 17 00:00:00 2001 From: "T. van der Sar" <t.vandersar@tudelft.nl> Date: Tue, 24 Mar 2020 21:13:23 +0000 Subject: [PATCH] Update 11_nearly_free_electron_model.md - typo --- src/11_nearly_free_electron_model.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/src/11_nearly_free_electron_model.md b/src/11_nearly_free_electron_model.md index c9b7db5d..028c6f33 100644 --- a/src/11_nearly_free_electron_model.md +++ b/src/11_nearly_free_electron_model.md @@ -149,15 +149,15 @@ If we change $\mathbf{k}$ by a reciprocal lattice vector $\mathbf{k} \rightarrow An alternative way to write the Bloch wave is to formulate $u^\alpha(r)$ as a Fourier series: $$ -u^\alpha(r) = \Sum_\mathbf{G} u^{\alpha}_\mathbf{G}e^{i\mathbf{G}\cdot\mathbf{r}} +u^\alpha(r) = \sum_\mathbf{G} u^{\alpha}_\mathbf{G}e^{i\mathbf{G}\cdot\mathbf{r}} $$ where $u^{\alpha}_\mathbf{G}$ are the Fourier coefficients. Substituing into our experssion for the Bloch wave, we get $$ -\psi^\alpha(r) = \Sum_\mathbf{G} u^{\alpha}_\mathbf{G} e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}} +\psi^\alpha(r) = \sum_\mathbf{G} u^{\alpha}_\mathbf{G} e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}} $$ which shows that each eigenstate can be written as a sum over plane waves that differ by a reciprocal lattice vector. -??? question "Does the tight-binding wavefunction $|\psi\rangle = \Sum_n e^{ikna}(\phi_0|n,1\rangle+\psi_0|n,2\rangle)$ (see exercise 2 in Lecture 8) satisfy the Bloch theorem? What part of $|\psi\rangle$ describes $u(x)$ in this case?" +??? question "Does the tight-binding wavefunction $|\psi\rangle = \sum_n e^{ikna}(\phi_0|n,1\rangle+\psi_0|n,2\rangle)$ (see exercise 2 in Lecture 8) satisfy the Bloch theorem? What part of $|\psi\rangle$ describes $u(x)$ in this case? Try to describe in words how this Bloch wave is built up." ### Repeated vs reduced vs extended Brillouin zone -- GitLab