X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;ds=sidebyside;f=physics_compact%2Fsolid.tex;h=7fa936f465fcae0247413068b6836988fa6515ab;hb=9070c77ed62df80818cc6cc0f8e8ad2eee745272;hp=33fe8a4c970900514004a73f54600c7b26875566;hpb=283813d62a10b6805e95f65eee8e2de5000d8d94;p=lectures%2Flatex.git

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 33fe8a4..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}
@@ -283,7 +283,53 @@ 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(\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\\
+&&\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}\\
+&=&
+-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}
+