From: hackbard Date: Tue, 19 Jun 2012 09:46:45 +0000 (+0200) Subject: even more so X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=9070c77ed62df80818cc6cc0f8e8ad2eee745272;p=lectures%2Flatex.git even more so --- diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index 507de12..7fa936f 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -220,7 +220,7 @@ r\ket{\vec{r'}} & = & r'\ket{\vec{r'}} \end{eqnarray} we get \begin{equation} --i\hbar(r'\times \nabla_{\vec{r'}})\braket{\vec{r'}}{\chi_{lm}} +-i\hbar(\vec{r'}\times \nabla_{\vec{r'}})\braket{\vec{r'}}{\chi_{lm}} E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r''}} \text{ .} \label{eq:solid:so_me} @@ -310,14 +310,26 @@ In total, the matrix elements of the potential for angular momentum $l$ can be c \bra{\vec{r'}}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l \braket{\chi_{lm}}{\vec{r''}}\\ &=& --i\hbar(r'\times \nabla_{\vec{r'}}) +-i\hbar(\vec{r'}\times \nabla_{\vec{r'}}) \delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'} -P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\\ +P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\nonumber\\ &&\times\frac{E^{\text{SO,KB}}_l\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}} {\int_{r}{\delta V_l^{\text{SO}}}^2(r) u_l^2(r) dr} \, -\frac{2l+1}{4\pi} +\frac{2l+1}{4\pi}\\ +&=& +-i\hbar(\vec{r'}\times \nabla_{\vec{r'}}) +\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'} +P'_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\nonumber\\ \end{eqnarray} +If the potential at position $\vec{r}$ is considered a sum of atomic potentials $v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n})$ (atom $n$ of species $\alpha$) +\begin{equation} +V(\vec{r})=\sum_{\alpha}\sum_n v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n}) +\end{equation} +and the SO projectors are likewise centered on atoms, the SO potential contribution reads +\begin{equation} +\end{equation} Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots \begin{equation} \end{equation} +