From: hackbard Date: Mon, 25 Feb 2013 16:48:51 +0000 (+0100) Subject: more on SO LS formalism X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=6a26b9f5593acc0bf19241b2fe79f1acf51fb03e;p=lectures%2Flatex.git more on SO LS formalism --- diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index 6eee6c2..5e69a79 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -256,13 +256,44 @@ 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 +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} -\ldots +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}