%% This LaTeX-file was created by <jl> Thu Feb  6 12:30:55 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[10pt,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}

Rotation operators are matrices constructed from generators.

\( e^{i\theta \widehat{n}*\overrightarrow{j}} \)in a spinorial basis \(|\)jm\(>\) (2j+1 dimensional)

A 3-d representation of of spinors is a family of cones.

This satisfies:

\begin{itemize}

\item \( <J_{x}>=<J_{y}>=0 \) 

\item \( <J_{z}>=m\hbar  \)

\item \( <J_{x}^{2}>=<J_{y}^{2}>=\frac{1}{2}(j(j+1)-m^{2}) \)

\end{itemize}

The cone has an x-y length of \( (j(j+1)-m^{2})^{.5} \)and a hypoteneuse length of 

\( (j(j+1))^{.5} \)

Relation to \( Y_{lm} \)in coordinate space:

You can represent \( L_{z}L^{2},L_{\pm } \)in \( \theta  \)and \( \phi  \)

Since \( L_{+}Y_{ll}=0\Rightarrow (\partial _{\theta }-lcot(\theta ))f(\theta )=0 \) Then use lowering operator to get other \( Y_{lm} \).

For rotation of an wavefunction, \( R\Psi (\overrightarrow{r})=\Psi (R^{-1}\overrightarrow{r}) \).


\section{rotations in s=\protect\( \frac{1}{2}\protect \)space}

Rotations generated by total angular momentum: \( J_{i}=L_{i}+S_{i}=-i\hbar \epsilon _{ijk}r_{j}s_{k}Id+\frac{\hbar }{2}s_{i} \) for \( s_{i} \) the pauli spin
matrices. This means \( J_{z}=\hbar *\begin{array}{cc}
-i\frac{\partial }{\partial \phi }+\frac{1}{2} & 0\\
0 & -i\frac{\partial }{\partial \phi }-\frac{1}{2}
\end{array}
 \). For electrons \( S_{i} \), \( S^{2} \)commute with everything,
so CSCO is\( H_{o},L^{2},L_{z},S^{2},S_{z} \). This means a general eigenfunction is \( |n,l,m_{l},s,m_{s}> \). and \( \psi (r_{i},s)=\begin{array}{c}
\psi _{+}(r_{i})\\
\psi _{-}(r_{i})
\end{array}
 \). \( R(n,\theta )=e^{-in_{i}s_{i}\theta /2}=cos(\theta /2)Id-in_{i}s_{i}sin(\theta /2) \). rotating
\( |\frac{1}{2},\frac{1}{2}> \)about \( x \) you get \( \begin{array}{c}
cos(\theta /2)\\
-isin(\theta /2)
\end{array}
 \)

\end{document}
