From: hackbard Date: Wed, 27 Feb 2013 07:45:16 +0000 (+0100) Subject: ice work X-Git-Url: https://hackdaworld.org/gitweb/?a=commitdiff_plain;h=c80e0ec43a9128c94dc6d53f16f7aa03ba5f8454;p=lectures%2Flatex.git ice work --- diff --git a/physics_compact/phys_comp.tex b/physics_compact/phys_comp.tex index 5974619..5f753aa 100644 --- a/physics_compact/phys_comp.tex +++ b/physics_compact/phys_comp.tex @@ -64,6 +64,9 @@ \newcommand{\dista}[1]{\unit[#1]{\AA}{}} \newcommand{\perc}[1]{\unit[#1]{\%}{}} +%\newcommand{\bra}[1]{\left\langle #1 \right|} +%\newcommand{\ket}[1]{\left| #1 \right\rangle} +%\newcommand{\braket}[2]{\left\langle #1 \mid #2 \right\rangle} \newcommand{\bra}[1]{\langle #1 |} \newcommand{\ket}[1]{| #1 \rangle} \newcommand{\braket}[2]{\langle #1 | #2 \rangle} 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}