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<H2><A NAME="SECTION00052000000000000000">
A..2 Fully relativistic case</A>
</H2>
<P>
<EM>The relativistic KS equations are
Dirac-like equations for a spinor with a ``large'' <!-- MATH
 $R_{nlj}(r)$
 -->
<I>R</I><SUB>nlj</SUB>(<I>r</I>)
<tex2html_verbatim_mark> and
a ``small'' <!-- MATH
 $S_{nlj}(r)$
 -->
<I>S</I><SUB>nlj</SUB>(<I>r</I>)
<tex2html_verbatim_mark> component:
</EM>
<BR>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{eqnarray}
c\left({d \over dr} + {\kappa\over r}\right)R_{nlj}(r) & = &
       \left(2mc^2 - V(r) + \epsilon \right)S_{nlj}(r)\\
c\left({d \over dr} - {\kappa\over r}\right)S_{nlj}(r) & = & 
       \left( V(r) + \epsilon \right)       R_{nlj}(r)
\end{eqnarray}
 -->
<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>c</I><IMG
 WIDTH="20" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img69.png"
 ALT="$\displaystyle \left(\vphantom{{d \over dr} + {\kappa\over r}}\right.$"><IMG
 WIDTH="27" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="img70.png"
 ALT="$\displaystyle {d \over dr}$"> + <IMG
 WIDTH="19" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="img71.png"
 ALT="$\displaystyle {\kappa\over r}$"><IMG
 WIDTH="20" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img72.png"
 ALT="$\displaystyle \left.\vphantom{{d \over dr} + {\kappa\over r}}\right)$"><I>R</I><SUB>nlj</SUB>(<I>r</I>)</TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><IMG
 WIDTH="15" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img73.png"
 ALT="$\displaystyle \left(\vphantom{2mc^2 - V(r) + \epsilon }\right.$">2<I>mc</I><SUP>2</SUP> - <I>V</I>(<I>r</I>) + <IMG
 WIDTH="12" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img18.png"
 ALT="$\displaystyle \epsilon$"><IMG
 WIDTH="15" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img74.png"
 ALT="$\displaystyle \left.\vphantom{2mc^2 - V(r) + \epsilon }\right)$"><I>S</I><SUB>nlj</SUB>(<I>r</I>)</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(9)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>c</I><IMG
 WIDTH="20" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img75.png"
 ALT="$\displaystyle \left(\vphantom{{d \over dr} - {\kappa\over r}}\right.$"><IMG
 WIDTH="27" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="img70.png"
 ALT="$\displaystyle {d \over dr}$"> - <IMG
 WIDTH="19" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="img71.png"
 ALT="$\displaystyle {\kappa\over r}$"><IMG
 WIDTH="20" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img76.png"
 ALT="$\displaystyle \left.\vphantom{{d \over dr} - {\kappa\over r}}\right)$"><I>S</I><SUB>nlj</SUB>(<I>r</I>)</TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><IMG
 WIDTH="13" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img77.png"
 ALT="$\displaystyle \left(\vphantom{ V(r) + \epsilon }\right.$"><I>V</I>(<I>r</I>) + <IMG
 WIDTH="12" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img18.png"
 ALT="$\displaystyle \epsilon$"><IMG
 WIDTH="13" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img78.png"
 ALT="$\displaystyle \left.\vphantom{ V(r) + \epsilon }\right)$"><I>R</I><SUB>nlj</SUB>(<I>r</I>)</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(10)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL">
<EM>
where <I>j</I>
<tex2html_verbatim_mark> is the total angular momentum (<I>j</I> = 1/2
<tex2html_verbatim_mark> if <I>l</I> = 0
<tex2html_verbatim_mark>, 
<!-- MATH
 $j=l+1/2,l-1/2$
 -->
<I>j</I> = <I>l</I> + 1/2, <I>l</I> - 1/2
<tex2html_verbatim_mark> otherwise); <!-- MATH
 $\kappa=-2(j-l)(j+1/2)$
 -->
<IMG
 WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img79.png"
 ALT="$ \kappa$"> = - 2(<I>j</I> - <I>l</I> )(<I>j</I> + 1/2)
<tex2html_verbatim_mark> is the Dirac 
quantum number (<IMG
 WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img79.png"
 ALT="$ \kappa$"> = - 1
<tex2html_verbatim_mark> is <I>l</I> = 0
<tex2html_verbatim_mark>, <!-- MATH
 $\kappa=-l-1,l$
 -->
<IMG
 WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img79.png"
 ALT="$ \kappa$"> = - <I>l</I> - 1, <I>l</I>
<tex2html_verbatim_mark> otherwise);
and the charge density is given by
</EM>
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{equation}
n(r) = \sum_{nlj} \Theta_{nlj}{R^2_{nlj}(r)+S^2_{nlj}(r)\over 4\pi r^2}.
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>n</I>(<I>r</I>) = <IMG
 WIDTH="32" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img80.png"
 ALT="$\displaystyle \sum_{{nlj}}^{}$"><IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img81.png"
 ALT="$\displaystyle \Theta_{{nlj}}^{}$"><IMG
 WIDTH="143" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="img82.png"
 ALT="$\displaystyle {R^2_{nlj}(r)+S^2_{nlj}(r)\over 4\pi r^2}$">.
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(11)</TD></TR>
</TABLE>
<tex2html_verbatim_mark>
</DIV>
<P>
<BR><HR>
<ADDRESS>
Layla Martin-Samos Colomer
2012-11-21
</ADDRESS>
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