## Bloch's theorem and Born-von Karman boundary condition ### Bloch's theorem if $V(r)$ is the [[Periodic boundary condition#^e466f3|periodic potential]], and for Schrödinger equation $-\frac{\hbar^2}{2m}\nabla^2\Psi_i+V(r)\Psi_i=E_i\Psi_i$ Then $\Psi$ would have the form $\Psi_{nk}(\vec{r}) = u_{nk}(r)e^{ikr}$ as stated in [[Reciprocal lattice and Bragg's law#^98c7c6|previous section]]. The proof and detailed discussion of this is given in other lectures (quantum transport and semiconductor nanostructures). Here, $\hbar k$ is called crystal momentum of the electron, and $u_{nk}(r)$ is a periodic function with the same periodically as the crystal. Any function follows the form of Bloch's theorem is called a *Bloch function*. ^73ff80 >[!Note] >And in above Schrödinger equation, the set of $E_n(k)$ gives the band structure. If we also apply the periodic condition to the wave function, then we have Born-von Karman boundary condition $\Psi(r+N_ia_i)=\Psi(r)$ in this way, allowed $k$ vectors would only be those satisfy $k=\sum_{i=1}^3 \frac{m}{N_i}b_i$ ^156416 >[!Note] >In above expression, $N_i$ is some fixed integer number, determined by the B-vK BC we selected. If we wanna the periodic structure has $N_x$ replica of unit cell, then $N_i = N_x$. >[!Notice] >In the allowed $k$ expression, $k$ is the translational vector, not wavevector$m_i$ are integer and $m \leq N_i$. >A problem with B-vK BC (or general PBC) is, they cannot deal with size-effect or surface-related problems. >[!Info] >The derivation for Bloch function and Bloch theorem is stated in [[Bloch function and Bloch theorem]]. It is always good to review if not clear.