\documentstyle{article}
\parskip=1ex
\parindent=0ex
\pagestyle{plain}
\begin{document}
\newcommand{\qi}{{\bf i}}
\newcommand{\qj}{{\bf j}}
\newcommand{\qk}{{\bf k}}
\newcommand{\rx}{\mbox{$\bf \hat{x}$}}
\newcommand{\ry}{\mbox{$\bf \hat{y}$}}
\newcommand{\rz}{\mbox{$\bf \hat{z}$}}
\newcommand{\id}{{\bf 1}}
\newcommand{\zero}{{\bf 0}}
\newcommand{\tr}{{\rm tr}}
\newcommand{\reals}{{\bf R}}
\newcommand{\cmplx}{{\bf C}}
\newcommand{\up}{|{\rm up}\rangle}
\newcommand{\down}{|{\rm down}\rangle}
\title{Lie Groups and Quantum Mechanics}
\author{Michael Weiss}
\date{}
\maketitle
\section{Introduction}
These notes attempt to develop some intuition about Lie groups, Lie algebras,
spin in quantum mechanics, and a network of related ideas. The level is
rather elementary--- linear algebra, a little topology, a little physics.
I don't see any point in copying proofs or formal definitions that can be
had from a shelf full of standard texts. I focus on a couple of concrete
examples, at the expense of precision, generality, and elegance. See the
first paragraph on Lie groups to get the flavor of my ``definitions''. I
state many facts without proof. Verification may involve anything from
routine calculation to a deep theorem. Phrases like ``Fact:'' or ``it
turns out that'' give warning that an assertion is not meant to be obvious.
A quote from the Russian mathematician V. I. Arnol'd:
\begin{quotation}
It is almost impossible for me to read contemporary mathematicians who,
instead of saying ``Petya washed his hands,'' write simply: ``There is a
$t_1<0$ such that the image of $t_1$ under the natural mapping
$t_1 \mapsto {\rm Petya}(t_1)$ belongs to the set of dirty hands, and a
$t_2$, $t_1