From 1690bbcc2a8510bb56499ac65a62b8a0fca251d6 Mon Sep 17 00:00:00 2001 From: hackbard Date: Tue, 19 Jun 2012 10:25:55 +0200 Subject: [PATCH] more so --- physics_compact/solid.tex | 36 +++++++++++++++++++++++++++++++++++- 1 file changed, 35 insertions(+), 1 deletion(-) diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index 33fe8a4..507de12 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -283,7 +283,41 @@ 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} -All megnetic states $m=-l,-l+1,\ldots,l-1,l$ contribute to the potential for angular momentum $l$. +and if all megnetic states $m=-l,-l+1,\ldots,l-1,l$ 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''}}= +\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'} +\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''} +\sum_mY^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'}) \text{ ,} +\end{equation} +which can be rewritten as +\begin{equation} +\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}} +\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}= +\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'} +\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''} +\frac{2l+1}{4\pi}P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right) +\end{equation} +using the vector addition theorem +\begin{equation} +P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)= +\frac{4\pi}{2l+1}\sum_mY^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'}) +\end{equation} +In total, the matrix elements of the potential for angular momentum $l$ can be calculated as +\begin{eqnarray} +\bra{\vec{r'}}V^{\text{KB,SO}}\ket{\vec{r''}}&=& +\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'}}) +\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'} +P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\\ +&&\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} +\end{eqnarray} + Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots \begin{equation} \end{equation} -- 2.39.2