From: hackbard Date: Tue, 26 Feb 2013 07:42:42 +0000 (+0100) Subject: ice chnages X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=4533f8a3f2d7319103d979de2255a43ba2cb9235;p=lectures%2Flatex.git ice chnages --- diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index 5e69a79..bc70eae 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -282,18 +282,26 @@ 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. +(Does this constitute a problem?) + \subsubsection{Excursus: Real space representation within an iterative treatment}