From: hackbard Date: Thu, 14 Feb 2013 15:49:44 +0000 (+0100) Subject: some changes X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=f4666008c79f572cbe7cbfa5f9a7e306bfa1637c;p=lectures%2Flatex.git some changes --- diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index 5b79972..306d235 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -208,20 +208,40 @@ This can be shown by rewriting the $LS$ operator \begin{equation} J=L+S \Leftrightarrow J^2=L^2+S^2+2LS \Leftrightarrow LS=\frac{1}{2}\left(J^2-L^2-S^2\right) -\text{ ,} \end{equation} +and corresponding eigenvalue +\begin{eqnarray} +j(j+1)-l(l+1)-s(s+1)&=& +(l\pm\frac{1}{2})(l\pm\frac{1}{2}+1)-l^2-l-\frac{3}{4} \nonumber\\ +&=& +l^2\pm\frac{l}{2}+l\pm\frac{l}{2}+\frac{1}{4}\pm\frac{1}{2}-l^2-l-\frac{3}{4} +\nonumber\\ +&=&\pm(l+\frac{1}{2})-\frac{1}{2}=\left\{\begin{array}{rl} +l & \text{for } j=l+\frac{1}{2}\\ +-(l+1) & \text{for } j=l-\frac{1}{2} +\end{array}\right. +\text{ ,} +\end{eqnarray} which, if used in equation~\eqref{eq:solid:so_bs2}, gives the same (diagonal) matrix elements -\begin{align} +\begin{eqnarray} \bra{l\pm\frac{1}{2},m\pm\frac{1}{2}}V(\vec{r}) -\ket{l\pm\frac{1}{2},m\pm\frac{1}{2}}&= +\ket{l\pm\frac{1}{2},m\pm\frac{1}{2}}&=& \bar{V}_l(\vec{r})+V^{\text{SO}}_l(\vec{r}) -\frac{1}{2}\left(l(l+1)-j(j+1)-\frac{3}{4}\right) \\ -&= \bar{V}_l(\vec{r})+\frac{1}{2}V^{\text{SO}}_l(\vec{r}) +\frac{1}{2}\left(l(l+1)-j(j+1)-\frac{3}{4}\right) \nonumber\\ +&=&\bar{V}_l(\vec{r})+\frac{1}{2}V^{\text{SO}}_l(\vec{r}) \left\{\begin{array}{rl} --\left(l+\frac{3}{2}\right) & \text{ for } j=l+\frac{1}{2}\\ -\left(l-\frac{1}{2}\right) & \text{ for } j=l-\frac{1}{2} +l & \text{for } j=l+\frac{1}{2}\\ +-(l+1) & \text{for } j=l-\frac{1}{2} +\end{array}\right. \nonumber\\ +&=&\frac{1}{2l+1}\left(lV_{l,l-\frac{1}{2}}(\vec{r})+ + (l+1)V_{l,l+\frac{1}{2}}(\vec{r})\right)+\nonumber\\ +&&+\frac{1}{2l+1}\left\{\begin{array}{rl} +l\left(V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right) & + \text{for } j=l+\frac{1}{2}\\ +-(l+1)\left(V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right) & + \text{for } j=l-\frac{1}{2} \end{array}\right. -\end{align} +\end{eqnarray} as equation~\eqref{eq:solid:so_bs1} \begin{equation} \text{ .}