\documentclass{article}
\title{Less Than, Greater Than, Bars and Quantum Mechanics}
\begin{document}

\section{Relations}
Relations: $ a < b < c $

Relations: $ > a $

Relations: $ a > $

\section{Quantum Mechanics Notations}
Expectation: $ <a> $

Bra: $<a|$

Bra: $<a,b,c|$

Bra: $<\phi=0|$

Ket: $|b>$

Braket $<a|b>$

Also $<a|f|b>$

\section{Absolute Values}
abs $|a|$

norm $||a||$

expr $|a|-|b|$

nested $||a|-|b||$

non-nested $a|a|+b|b|+c|c|$

\section{Mixtures that look like QM}

abs and rel: $a<c|c|$

worse $|a|< \min\left|z\pm\left(z^{2}-1\right)^{{1/2}}\right|$

\section{In equation}
\[ |A\rangle = |B\rangle + |C\rangle\]

\end{document}