Can I draw this diagram in Tikz?

Question

I need to draw this (likely very simple) diagram shown in the image but I've never used Tikz before.

My problem is basiclly how to draw these vertical dotted lines and mainly arrange the math in blocks like this with these arrows pointing from one to the other - of course I have no problem writing the math, the needed structure is giving me problems. Any help would be very appreciated. Thanks in advance.

So far I was able to do this. I feel I'm almost there, but having problems with alignments. Any help?

\begin{tikzpicture}[square/.style={regular polygon,regular polygon sides=4}]

%Draw dashed lines
\draw  [dashed] (2.5,0) -- (2.5,2);
\draw  [dashed] (5,0) -- (5,2);
\draw  [dashed] (7.5,0) -- (7.5,2);

%Draw nodes of equations
\node at (1.25,1) [square,inner sep=-1.3em, draw] {
    $\begin{aligned}
     \vert \Psi \rangle &= \sum_{i}{c_i \vert a_i \rangle},\\
     &\sum_{i}{\vert a_i \rangle \langle a_i \vert} = \mathds{1}
    \end{aligned}$

};
\node at (3.75,1) [square,inner sep=-1.3em, draw] {};
\node at (6.25,1) [square,inner sep=-1.3em, draw] {};

%Draw title nodes

\node at (1.25,2) [square,inner sep=-1.3em, draw] {$\left( \mathcal{H}, \langle \cdot \vert \cdot \rangle \right)$};
\node at (3.75,2) [square,inner sep=-1.3em, draw] {$\left( \mathcal{H}_{Phys}, \langle \cdot \vert \cdot \rangle _{\eta _{+}} \right)$};
\node at (6.25,2) [square,inner sep=-1.3em, draw] {$\left( \mathcal{H}, \langle \cdot \vert \cdot \rangle \right)$};


\end{tikzpicture}

Also I don't know if this is a good approach to do what I want. Thanks.

Answer 1

With arydshln and a couple of tricks.

\documentclass{article}
\usepackage{amsmath,braket,rsfso,arydshln}

\newcommand{\blank}{{\,{\cdot}\,}}

\begin{document}

\begin{equation*}
\addtolength{\arraycolsep}{0.5em}
\begin{array}{@{}c@{}c:c@{}c:c@{}}
% first column
\begin{gathered}[t]
(\mathcal{H},\braket{\blank|\blank}) \\
\ket{\psi}=\sum_i c_i\ket{a_i} \\
\sum_i \lvert a_i\times a_i\rvert = 1
\end{gathered}
&\makebox[0pt][l]{\raisebox{-2\normalbaselineskip}{$\:\longrightarrow$}}&
%second column
\begin{gathered}[t]
(\mathcal{H}_{\rho hy},\braket{\blank|\blank}_{\eta_+}) \\
\begin{aligned}
\ket{\psi}  &\to \ket{\psi'}=\sum_i c_i\rho^{-1}\ket{a_i} \\
\ket{\psi'} &\to \ket{\psi'(t)}=e^{-iHt/\hbar}\ket{\psi'}
\end{aligned}
\end{gathered}
&\makebox[0pt][l]{\raisebox{-2\normalbaselineskip}{$\:\longrightarrow$}}&
% third column
\begin{gathered}[t]
(\mathcal{H},\braket{\blank|\blank}) \\
\begin{aligned}
& \ket{\psi'} \to \ket{\psi(t)} = \rho\ket{\psi'(t)} \\
& \quad = \rho e^{-iHt/\hbar}\ket{\psi'} \\
& \quad = \rho e^{-iHt/\hbar}\rho^{-1}\ket{\psi} \\
& \quad = e^{-iHt/\hbar}\ket{\psi}
\end{aligned}
\end{gathered}
\end{array}
\end{equation*}

\end{document}