X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=physics_compact%2Fsolid.tex;h=5e69a799a18a5174921aabe05797cc83202675e6;hp=5b79972de4325d530b974865067d61c679ef2100;hb=6a26b9f5593acc0bf19241b2fe79f1acf51fb03e;hpb=cb8c7d0935379dbb747f3cef38ca532f0bbf4f26

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 5b79972..5e69a79 100644
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -208,27 +208,92 @@ This can be shown by rewriting the $LS$ operator
 \begin{equation}
 J=L+S \Leftrightarrow J^2=L^2+S^2+2LS \Leftrightarrow
 LS=\frac{1}{2}\left(J^2-L^2-S^2\right)
-\text{ ,}
 \end{equation}
+and corresponding eigenvalue
+\begin{eqnarray}
+j(j+1)-l(l+1)-s(s+1)&=&
+(l\pm\frac{1}{2})(l\pm\frac{1}{2}+1)-l^2-l-\frac{3}{4} \nonumber\\
+&=&
+l^2\pm\frac{l}{2}+l\pm\frac{l}{2}+\frac{1}{4}\pm\frac{1}{2}-l^2-l-\frac{3}{4}
+\nonumber\\
+&=&\pm(l+\frac{1}{2})-\frac{1}{2}=\left\{\begin{array}{rl}
+l & \text{for } j=l+\frac{1}{2}\\
+-(l+1) & \text{for } j=l-\frac{1}{2}
+\end{array}\right.
+\text{ ,}
+\end{eqnarray}
 which, if used in equation~\eqref{eq:solid:so_bs2}, gives the same (diagonal) matrix elements
-\begin{align}
+\begin{eqnarray}
 \bra{l\pm\frac{1}{2},m\pm\frac{1}{2}}V(\vec{r})
-\ket{l\pm\frac{1}{2},m\pm\frac{1}{2}}&=
+\ket{l\pm\frac{1}{2},m\pm\frac{1}{2}}&=&
 \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) \\
-&= \bar{V}_l(\vec{r})+\frac{1}{2}V^{\text{SO}}_l(\vec{r})
-\left\{\begin{array}{rl}
--\left(l+\frac{3}{2}\right) & \text{ for } j=l+\frac{1}{2}\\
-\left(l-\frac{1}{2}\right) & \text{ for } j=l-\frac{1}{2}
+\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})
+\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(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{align}
-as equation~\eqref{eq:solid:so_bs1}
+\end{eqnarray} 
+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 \text{ and } S_\pm=S_x\pm iS_y
+\text{ ,}
+\end{equation}
+with properties
+\begin{eqnarray}
+L_+S_- & = & (L_x+iL_y)(S_x-iS_y)=L_xS_x-iL_xS_y+iL_yS_x+L_yS_y \\
+L_-S_+ & = & (L_x-iL_y)(S_x+iS_y)=L_xS_x+iL_xS_y-iL_yS_x+L_yS_y 
+\end{eqnarray}
+resulting in
 \begin{equation}
+L_+S_-+L_-S_+=2(L_xS_x+L_yS_y)
+\text{ ,}
+\end{equation}
+reads
+\begin{equation}
+L\cdot S=\frac{1}{2}(L_+S_-+L_-S_+)+L_zS_z
 \text{ .}
 \end{equation}
-
-\end{proof}
-
+The contributions of this operator act differently on $\ket{l,m}$ and --- in fact --- depend on the respectively considered spinor component, which is incorporated by $\ket{l,m,\pm}$.
+\begin{enumerate}
+\item \underline{$L_+S_-$}:
+      Updates spin down component and only acts on spin up component
+\begin{equation}
+L_+S_-\ket{l,m,+}=L_+\ket{l,m}S_-\ket{+}=
+\sqrt{(l-m)(l+m+1)}\hbar\ket{l,m+1}\hbar\ket{-}
+\end{equation}
+\item \underline{$L_-S_+$}:
+      Updates spin up component and only acts on spin down component
+\begin{equation}
+L_+S_-\ket{l,m,-}=L_+\ket{l,m}S_-\ket{+}=
+\sqrt{(l-m)(l+m+1)}\hbar\ket{l,m+1}\hbar\ket{-}
+\end{equation}
+\item \underline{$L_zS_z$}: Acts on both and updates both spinor components
+\begin{equation}
+L_zS_z\ket{l,m,\pm}=L_z\ket{l,m}S_z\ket{\pm}=
+\pm\frac{1}{2}m\hbar^2\ket{l,m,\pm}
+\end{equation}
+\end{enumerate}
 
 \subsubsection{Excursus: Real space representation within an iterative treatment}