more on SO LS formalism

[lectures/latex.git] / physics_compact / solid.tex

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 306d235..5e69a79 100644 (file)
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -229,26 +229,71 @@ 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}
+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}