From: hackbard Date: Tue, 19 Jun 2012 14:01:50 +0000 (+0200) Subject: more so X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=62d768c49f54dc3c03f0e3df3d08c37af1815aeb;p=lectures%2Flatex.git more so --- diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index 7fa936f..e8c61e1 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -283,7 +283,7 @@ To obtain a final expression for the matrix elements \eqref{eq:solid:so_me}, the \delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''} Y^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'}) \end{eqnarray} -and if all megnetic states $m=-l,-l+1,\ldots,l-1,l$ that contribute to the potential for angular momentum $l$ are considered +and if all states with magnetic quantum numbers $m=-l,-l+1,\ldots,l-1,l$ that contribute to the potential for angular momentum $l$ are considered \begin{equation} \braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}} \braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}= @@ -317,11 +317,14 @@ P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\nonumber\\ {\int_{r}{\delta V_l^{\text{SO}}}^2(r) u_l^2(r) dr} \, \frac{2l+1}{4\pi}\\ &=& --i\hbar(\vec{r'}\times \nabla_{\vec{r'}}) +-i\hbar \delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'} P'_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\nonumber\\ +&&\left(\frac{\vec{r'}\times\vec{r''}}{r'r''}\right) +\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} \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}) @@ -329,6 +332,15 @@ V(\vec{r})=\sum_{\alpha}\sum_n v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n}) and the SO projectors are likewise centered on atoms, the SO potential contribution reads \begin{equation} \end{equation} +The $E_l^{\text{SO,KB}}$ are given by +\begin{equation} +E_l^{\text{SO,KB}}= +\frac{\braket{\delta V_lu_l}{u_l\delta V_l}} + {\bra{u_l}\delta V_l\ket{u_l}}= +\frac{\int_{r}\delta V^2_l(r)u^2_l(r)}r^2dr + {\int_{r'}\int_{r''}\braket{u_l}{r'}\bra{r'}\delta V_l +\ket{r''}\braket{r''}{u_l}}= +\end{equation} Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots \begin{equation} \end{equation}