%% This LaTeX-file was created by <jl> Tue Feb 18 11:19:45 1997
%% LyX 0.10 (C) 1995 1996 by Matthias Ettrich and the LyX Team

%% Don't edit this file unless you are sure what you are doing.
\documentclass[12pt,letterpaper,oneside,onecolumn]{article}
\usepackage[]{fontenc}
\usepackage[dvips]{epsfig}

%%
%% BEGIN The lyx specific LaTeX commands.
%%

\makeatletter
\def\LyX{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\spacefactor1000}%
\newcommand{\lyxtitle}[1] {\thispagestyle{empty}
\global\@topnum\z@
\section*{\LARGE \centering \sffamily \bfseries \protect#1 }
}
\newcommand{\lyxline}[1]{
{#1 \vspace{1ex} \hrule width \columnwidth \vspace{1ex}}
}
\newenvironment{lyxbibliography}
{
\begin{thebibliography}{99}}
{\end{thebibliography}}
\def\lxq{"}
\newenvironment{lyxcode}
{\list{}{
\rightmargin\leftmargin
\raggedright
\itemsep 0pt
\parsep 0pt
\ttfamily
}%
\item[]
}
{\endlist}
\newcommand{\lyxlabel}[1]{#1 \hfill}
\newenvironment{lyxlist}[1]
{\begin{list}{}
{\settowidth{\labelwidth}{#1}
\setlength{\leftmargin}{\labelwidth}
\addtolength{\leftmargin}{\labelsep}
\renewcommand{\makelabel}{\lyxlabel}}}
{\end{list}}
\newcommand{\lyxletterstyle}{
\setlength\parskip{0.7em}
\setlength\parindent{0pt}
}
\newcommand{\lyxaddress}[1]{
\par {\raggedright #1 
\vspace{1.4em}
\noindent\par}
}
\newcommand{\lyxrightaddress}[1]{
\par {\raggedleft \begin{tabular}{l}\ignorespaces
#1
\end{tabular}
\vspace{1.4em}
\par}
}
\newcommand{\lyxformula}[1]{
\begin{eqnarray*}
#1
\end{eqnarray*}
}
\newcommand{\lyxnumberedformula}[1]{
\begin{eqnarray}
#1
\end{eqnarray}
}
\makeatother

%%
%% END The lyx specific LaTeX commands.
%%

\pagestyle{plain}
\setcounter{secnumdepth}{3}
\setcounter{tocdepth}{3}
\begin{document}

Adding nuclear spin doubles the degeneracy. 

Hyperfine splittings are of order \( \frac{m_{e}}{m_{N}}\alpha ^{2} \).

\( H_{hf}=\frac{1}{2}(\frac{g_{p}e\hbar }{2m_{p}c})(\frac{g_{e}e\hbar }{2m_{e}c})[(\frac{1}{r}\frac{d}{dr}(\frac{1}{r}))[\frac{2}{g_{e}}\frac{L^{i}I^{i}}{\hbar ^{2}}+\frac{1}{\hbar ^{2}}[3S^{i}r^{i}I^{j}r^{j}-S^{i}I^{i}]] \)

\( +\frac{\nabla ^{2}(\frac{1}{r})}{3}S^{i}I^{i}a] \)

The first term is spin angular momentum of the nucleus. For S-wave
electrons only the last term is nonzero. This is the contact term.

\( E_{contact}=\frac{2}{3}g_{e}\mu _{e}g_{p}\mu _{B}4\pi |\Psi (0)|^{2}<S^{i}I^{i}>/\hbar ^{2} \) 

The \( 1s_{\frac{1}{2}} \)is 4-fold degenerate with the hyperfine considerations. This
is split into 3 and 1 by the hyperfine structure. \( \triangle E=5.9*10^{-6}eV \) Which corresponds
to \( \nu =1420MHz \) and \( \lambda =21cm \). This is really the first excited state of Hyrdorgen.

for l\(>\)0 states, the fine structure are coupled \( J^{i}=L^{i}+S^{i} \). To calculate
\( S^{i}I^{i} \) must decouple \( L^{i} \), \( S^{i} \) and recouple \( S^{i} \), \( I^{i} \). This gives:

\( |nlsjm_{j}>|Im_{I}>=\sum _{m_{l},m_{s},m_{k}}|nlsIkm_{k}>|lm_{l}><km_{k}|sIm_{s}m_{I}><lsm_{l}m_{s}|jm_{j}> \).m

\end{document}
