X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=physics_compact%2Fsolid.tex;h=6eee6c2361d50eb5868b6afa675d1cd5f566c245;hp=306d23523eecaf24bcfa1e653be492f7b9196177;hb=056165f01af9aa49e9e247c9c9ba43cfc52309be;hpb=f4666008c79f572cbe7cbfa5f9a7e306bfa1637c

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 306d235..6eee6c2 100644
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -229,26 +229,40 @@ which, if used in equation~\eqref{eq:solid:so_bs2}, gives the same (diagonal) ma
 \bar{V}_l(\vec{r})+V^{\text{SO}}_l(\vec{r})
 \frac{1}{2}\left(l(l+1)-j(j+1)-\frac{3}{4}\right) \nonumber\\
 &=&\bar{V}_l(\vec{r})+\frac{1}{2}V^{\text{SO}}_l(\vec{r})
-\left\{\begin{array}{rl}
+\cdot\left\{\begin{array}{cl}
 l & \text{for } j=l+\frac{1}{2}\\
 -(l+1) & \text{for } j=l-\frac{1}{2}
 \end{array}\right. \nonumber\\
 &=&\frac{1}{2l+1}\left(lV_{l,l-\frac{1}{2}}(\vec{r})+
                        (l+1)V_{l,l+\frac{1}{2}}(\vec{r})\right)+\nonumber\\
-&&+\frac{1}{2l+1}\left\{\begin{array}{rl}
-l\left(V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right) &
- \text{for } j=l+\frac{1}{2}\\
--(l+1)\left(V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right) &
- \text{for } j=l-\frac{1}{2}
+&&\frac{1}{2l+1}
+\left(V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right)
+\cdot\left\{\begin{array}{c}
+l \\
+-(l+1)
+\end{array}\right. \nonumber\\
+&=&\left\{\begin{array}{cl}
+V_{l,l+\frac{1}{2}}(\vec{r}) & \text{for } j=l+\frac{1}{2}\\
+V_{l,l-\frac{1}{2}}(\vec{r}) & \text{for } j=l-\frac{1}{2}
 \end{array}\right.
 \end{eqnarray} 
-as equation~\eqref{eq:solid:so_bs1}
-\begin{equation}
-\text{ .}
-\end{equation}
-
+as expected and --- in fact --- obtained from equation~\eqref{eq:solid:so_bs1}.
 \end{proof}
 
+In order to include the spin-orbit interaction into the scalar-relativistic formalism of a normconserving, non-local pseudopotential, scalar-relativistic in contrast to fully relativistic pseudopotential wavefunctions are needed as a basis for the projectors of the spin-orbit potential.
+The transformation
+\begin{equation}
+L\cdot S=L_xS_x+L_yS_y+L_zS_z
+\end{equation}
+using the ladder operators
+\begin{equation}
+L_\pm=L_x\pm iL_y
+\end{equation}
+reads
+\begin{equation}
+\ldots
+\end{equation}
+ 
 
 \subsubsection{Excursus: Real space representation within an iterative treatment}