%% This LaTeX-file was created by <jl> Tue Feb 18 15:16:09 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}

Scalings:

expansion factor is \( \alpha =\frac{e^{2}}{\hbar c}\sim \frac{1}{137} \)= fine structure constant

Lengths:

\begin{itemize}

\item \( r_{e}=\frac{e^{2}}{mc^{2}} \)= classical electron radius

\item \( \lambda =\frac{\hbar }{mc} \)

\item \( a_{0}=\frac{\hbar ^{2}}{me^{2}} \)

\item \( \frac{\lambda }{2}=\frac{\hbar ^{3}c}{me^{4}} \)= rydberg energy

\end{itemize}

\( \beta \sim \alpha  \) so the relativistic corrections are small.

Energy levels:

\begin{itemize}

\item \( Ry=\frac{e^{2}}{2a_{0}}=13.6eV \)

\item \( Z^{2}Ry \) for single electron atoms.

\item fine structure: of order \( \alpha ^{2}Ry \)

\item hyperfine structure of order \( \frac{m_{e}}{M_{n}}\alpha ^{2}Ry \)

\item vibrational energy levels: \( E_{n}=(n+\frac{1}{2})\hbar \omega  \) with \( \omega =(\frac{k}{AM})^{\frac{1}{2}} \) a typical k is \( \frac{e^{2}}{a^{3}} \) where a = seperation
of atoms in molecule. Then \( \hbar \omega \sim (\frac{e^{2}\hbar ^{2}}{a^{3}AM})^{\frac{1}{2}}\sim (\frac{m}{AM})^{\frac{1}{2}}(\frac{a_{0}}{a})^{\frac{3}{2}}Ry \) so a typical energy would be \( \frac{1}{10}eV \).

\item rotational energy levels:\( E_{j}=\frac{j(j+1)\hbar ^{2}}{2I} \).with \( I\sim AMa^{2} \) so \( \frac{\hbar ^{2}}{2I}\simeq (\frac{m}{AM})(\frac{a_{0}}{a})^{2}Ry \) so a typical energy would be
\( 10^{-3}eV \)

\end{itemize}

Dipole radiation:

\begin{itemize}

\item power radiated is: \( P=\frac{2(d'')^{2}}{3c^{3}} \) For a free charge: \( P=\frac{2e^{2}v'^{2}}{3c^{3}} \). Characteristic dipole is
\( d\sim ea \). For electronic and vibrational, this is a transitional moment,
while for rotational this is a permanent moment.

\item spontaneous emission: \( A\sim \frac{P}{\hbar \omega }\sim \frac{e^{2}a^{2}\omega ^{3}}{\hbar c^{3}} \). This is \( 10^{9}/s \) for the lyman alpha. For a rotational
system it would be: \( A\sim 10^{-6}/s \) because of the scalings of energies. \( \frac{A}{\omega }\simeq \frac{e^{6}}{c^{3}\hbar ^{3}}=\alpha ^{3} \). This
means ions may have to wait a long time to become unionized, since
radiative capture chances are so low.

\end{itemize}

\end{document}
