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 ca00412..5e69a79 100644 (file)
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -249,6 +249,51 @@ V_{l,l-\frac{1}{2}}(\vec{r}) & \text{for } j=l-\frac{1}{2}
 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}
+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}