added ape

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

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 5e69a79..ccb5c26 100644 (file)
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -249,7 +249,9 @@ 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.
+\subsubsection{Scalar relativistic basis}
+
+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
@@ -282,18 +284,46 @@ The contributions of this operator act differently on $\ket{l,m}$ and --- in fac
 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}
+      Moreover, this part only acts on magnetic quantum numbers
+      $m=-l,\ldots,l-1$ and updates quantum numbers $m=-l+1,\ldots,l$.
 \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{-}
+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}
+      Moreover, this part only acts on magnetic quantum numbers
+      $m=-l+1,\ldots,l$ and updates quantum numbers $m=-l,\ldots,l-1$.
 \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}
+      It acts on all magnetic quantum numbers and updates all of them.
 \end{enumerate}
+Please note that the $\ket{l,m,\pm}$ are not eigenfunctions of the two combinations of ladder operators, i.e.\  the $\ket{l,m,\pm}$ do not diagonalize the spin-orbit part of the Hamiltonian.
+
+These equations can be simplified to read
+\begin{eqnarray}
+\ldots
+\text{ .}
+\end{eqnarray}
+
+\subsubsection{A different basis set}
+
+The above basis is composed of eigenfunctions
+\begin{equation}
+\ket{l,m} \text{, } \ket{\pm} \text{ of operators }
+L^2\text{, } L_z \text{ and } S_z
+\text{.}
+\end{equation}
+These eigenfunctions diagonalize the scalar relativistic Hamiltonian.
+Introducing spin-orbit interaction, however, it is a good idea to chose eigenfunctions that diagonalize the perturbation
+\begin{equation}
+L\cdot S=\frac{1}{2}(J^2-L^2-S^2)
+\text{ ,}
+\end{equation}
+i.e.\  simultaneous eigenfunctions of $J^2$, $L^2$ and $S^2$.
 
 \subsubsection{Excursus: Real space representation within an iterative treatment}