## Bloch function and Bloch theorem First consider spinless and non-interacting electrons in crystals. The Schrödinger equation is, $\left[ -\frac{\hbar^2}{2m_e} \Delta + V(\mathbf{r}) \right] \psi(\mathbf{r}) = E \psi(\mathbf{r})$ $V(\mathbf{r}) = V(\mathbf{r} + \mathbf{R})$, it's periodic potential, the $\mathbf{R}$ is real space. translational vector. And place it on the reciprocal space, we have $V(\mathbf{r}) = \sum_{\mathbf{G}} V_{\mathbf{G}} e^{i \mathbf{G} \cdot \mathbf{r}}$, and $\mathbf{G} \cdot \mathbf{R} = 2\pi n$, namely $e^{i \mathbf{G} \cdot \mathbf{R}} = 1$. Now we propose the idea of Brillouin zone (BZ), i.e., $\mathbf{G}$ vector space if still give the potential the periodic form and place it in the Schrödinger equation, we will obtain, $\left[ -\frac{\hbar^2}{2m_e} \Delta + \sum_{\mathbf{G}} V_{\mathbf{G}} e^{i \mathbf{G} \cdot \mathbf{r}} \right] \psi(\mathbf{r}) = E \psi(\mathbf{r})$ A natural idea is to write the wave function also in Fourier form, after expansion and simplification, we will have the reciprocal space vector $\mathbf{k}$ in $\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i \mathbf{k} \cdot \mathbf{r}} u_{n\mathbf{k}}(\mathbf{r})$ This $u_{n\mathbf{k}}(\mathbf{r})$ is the **Bloch function**, also have the translational periodicity, namely, $u_{n\mathbf{k}}(\mathbf{R}+\mathbf{r})=u_{n\mathbf{k}}(\mathbf{r})$ And the above relation is called the **Bloch theorem**. If we put more factors into our consideration, when calculating the potential, we may use the effective potential, or other models like tight binding model. But, if we specifically wanna consider the cases where we neglect the "core states" of the VB, and only the "extended states" of CB, we may use the "[[Pseudopotential|pseudopotential]]," by constructing states that are orthogonal to the core states.