Recent content by Sheng

Sorry, I forgot to mention that the summation on k is restricted to first Brillouin zone only, in that case does ## \sum_{\mathbf{k} \in BZ} \rightarrow \frac{N\Omega}{(2\pi)^3} \int_{BZ} d\mathbf{k} ## apply? I mean directly restricting the integration region to BZ. I have some doubts on this...

Since ## \sum_\mathbf{k}=\sum_\mathbf{n}=\frac{V}{(2\pi)^3} \sum_n \frac{(2\pi)^3}{V}=\frac{V}{(2\pi)^3} \int d\mathbf{n} \frac{(2\pi)^3}{V}=\frac{V}{(2\pi)^3} \int d\mathbf{k} ## we have made a transformation from discrete k to continuous k, then why can't we use dirac delta function?

Ok so now ## \sum_{\mathbf{k}} \rightarrow \frac{N^2\Omega}{(2\pi)^3} \int_{BZ} d\mathbf{k} ## and $$ \phi_{n\mathbf{R}}(\mathbf{r}) = \frac{N^{3/2}\Omega}{(2\pi)^3} \int_{BZ} e^{-i\mathbf{k \cdot R}} \psi_{n\mathbf{k}}(\mathbf{r}) d\mathbf{k} $$ and the final form become $$ \langle...

It is not? It is in accordance with that on Wikipedia ##\Omega## in this case is the Brillouin zone volume. Most of the references I consult automatically limit the integration region to the first Brillouin zone. If not then how do I limit the integration region to only the first Brillouin...

Thanks for your reply. I am tempted to use that one because it is closer to the solution I want. But even then the final form is just $$ \langle \phi_{n\mathbf{R}}(\mathbf{r}) \vert \phi_{m\mathbf{R'}}(\mathbf{r}) \rangle = N \frac{\Omega}{(2\pi)^3} \delta_{mn} \delta_{\mathbf{R,R'}} $$ which...

I have trouble reconciling orthogonality condition for Wannier functions using both continuous and discrete k-space. I am using the definition of Wannier function and Bloch function as provided by Wikipedia (https://en.wikipedia.org/wiki/Wannier_function). Wannier function: Bloch function: I...

I do not understand what you mean. If you mean the equation at the third part, I have edited it: $$ \langle w \vert \mathbf{r} \vert w \rangle = \left( \frac{\Omega}{8\pi^3} \right)^2 \int_{BZ} d\mathbf{k} d\mathbf{k}' i e^{i(\mathbf{k-k'}) \cdot r} \langle u_{\mathbf{k}} \vert \nabla...

You are right. I have edited the post and fixed some typos. Do you mean this? $$ (2\pi)^3 \delta({\mathbf{k-k'}}) = \int^\infty_{-\infty} e^{i(\mathbf{k-k'}) \cdot \mathbf{r}} d\mathbf{r} $$ But I cannot figure how to factorize the term out.

Homework Statement I did not manage to get the final form of the equation. My prefactor in the final form always remain quadratic, whereas the solution shows that it is linear, Homework Equations w refers to wannier function, which relates to the Bloch function ##\mathbf{R}## is this case...