author hackbard Thu, 14 Feb 2013 21:42:59 +0000 (22:42 +0100) committer hackbard Thu, 14 Feb 2013 21:42:59 +0000 (22:42 +0100)

index 306d235..ca00412 100644 (file)
@@ -229,24 +229,24 @@ which, if used in equation~\eqref{eq:solid:so_bs2}, gives the same (diagonal) ma
\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) \nonumber\\
&=&\bar{V}_l(\vec{r})+\frac{1}{2}V^{\text{SO}}_l(\vec{r})
-\left\{\begin{array}{rl}
+\cdot\left\{\begin{array}{cl}
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}
+&&\frac{1}{2l+1}
+\left(V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right)
+\cdot\left\{\begin{array}{c}
+l \\
+-(l+1)
+\end{array}\right. \nonumber\\
+&=&\left\{\begin{array}{cl}
+V_{l,l+\frac{1}{2}}(\vec{r}) & \text{for } j=l+\frac{1}{2}\\
+V_{l,l-\frac{1}{2}}(\vec{r}) & \text{for } j=l-\frac{1}{2}
\end{array}\right.
\end{eqnarray}
-as equation~\eqref{eq:solid:so_bs1}
-
-\text{ .}
-
-
+as expected and --- in fact --- obtained from equation~\eqref{eq:solid:so_bs1}.
\end{proof}