Matrix with rows and columns labeled with equations

Question

I want to label the rows and columns with quantum states as $\ket{n_1l_1m_1}$. To save space I tried to rotate these states by using

\rotatebox{80}{\ket{n_1l_1m_1}}

but this is not possible - how can I rotate math equations?

To make it more clear I want to replace the H (and also the Y and D) in the image by 90 degree rotatet quantum ket states

Answer 1

With nicematrix:

\documentclass{article}
\usepackage{nicematrix}

\begin{document}


$\begin{pNiceMatrix}[first-row,first-col,code-for-first-row=\rotate]
& |n_1l_1m_1\rangle& |n_1l_1m_1\rangle& |n_1l_1m_1\rangle\\ 
|n_1l_1m_1\rangle & 0.8 & 0.2 & 0 \\
|n_1l_1m_1\rangle& 0.3 & 0.4 & 0.3 \\
|n_1l_1m_1\rangle& 0.2 & 0.1 & 0.7
\end{pNiceMatrix}$

\end{document}

By default, Overleaf uses an old version of nicematrix. However, you can provide it the newest version of nicematrix.sty. Go here. At the top of the page, you have the latest version of nicematrix. It's the revision 56256. You do a right click on download and you save the file under the name nicematrix.sty. Put that file next to your file in your Overleaf project (buttom import next to new file).

Answer 2

Another alternative could be using blkarray with a combination of braket packages to define the command \ket and the graphicx package to use the rotation of \ket{n_1l_1m_1} with the macro \def\rotbra#1{\rotatebox{90}{$#1$}}.

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath}
\usepackage{blkarray}
\usepackage{braket}
\usepackage{graphicx}
\def\rotbra#1{\rotatebox{90}{$#1$}}

\begin{document}

\[
\begin{blockarray}{cccc}
 & \rotbra{\ket{n_1l_1m_1}} & \rotbra{\ket{n_1l_1m_1}} & \rotbra{\ket{n_1l_1m_1}}\\
\begin{block}{c(ccc)}
  \ket{n_1l_1m_1} & .8  & .2 & 0 &\\
  \ket{n_1l_1m_1} & .3 & .4 & .3 &\\
  \ket{n_1l_1m_1} & .2  & .1 & .7 &\\
\end{block}
\end{blockarray}
 \]
\end{document}

This is the output of the code: