X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=physics_compact%2Fsolid.tex;h=ccb5c26d43995f8fac6c460a1ee90f38c29581d9;hp=bc70eaea8e7b940fd87783d7e5cd412afaef5392;hb=c80e0ec43a9128c94dc6d53f16f7aa03ba5f8454;hpb=4533f8a3f2d7319103d979de2255a43ba2cb9235

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index bc70eae..ccb5c26 100644
--- 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
@@ -300,8 +302,28 @@ L_zS_z\ket{l,m,\pm}=L_z\ket{l,m}S_z\ket{\pm}=
       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.
-(Does this constitute a problem?)
 
+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}