Chain of curly arrows above and underneath each other

Question

I'm trying to draw a chain of arrows going under and above each other, as depicted:

The issue isn't labelling the arrows, I can easily do that, but I don't know how to get them to curl, have the arrow head in the middle (ideally, though not necessary), and have one on top of the other. Any help is appreciated. If a solution exists without TikZ, I would prefer it.

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For those interested, the above is an eigenvector of the Cartan generators of a Lie algebra acting on a state as a raising or lowering operator, used in the roots and weights method to classify all semi-simple compact Lie algebras.

Answer 1

Taking all the best from the prior answers, I've got your desired result here:

% arara: pdflatex

\documentclass{article}
\usepackage{tikz-cd}
\usetikzlibrary{decorations.markings}
\tikzset{middleArrowHead/.style={decoration={markings, mark= at position #1 with \arrow{>}}, postaction=decorate}}
\usepackage{braket} % defines \ket and \bra

\begin{document}    
    \[\begin{tikzcd}[every arrow/.append style={bend left, dash, middleArrowHead=.55}] % change the last value until the arrow position pleases you.
    \ket{\vec{\mu}-\vec{\alpha}}
    \arrow[start anchor={[yshift=1.5pt]east},end anchor={[yshift=1.5pt]west}]{r}{E_{+}} 
    & \ket{\vec{\mu}}
    \arrow[start anchor={[yshift=-1.5pt]west},end anchor={[yshift=-1.5pt]east}]{l}{E_{-}}
    \arrow[start anchor={[yshift=1.5pt]east},end anchor={[yshift=1.5pt]west}]{r}{E_{+}} 
    & \ket{\vec{\mu}+\vec{\alpha}}
    \arrow[start anchor={[yshift=-1.5pt]west},end anchor={[yshift=-1.5pt]east}]{l}{E_{-}}
    \end{tikzcd}\]
\end{document}

Answer 2

A solution with tikz-cd

\documentclass{article}

\usepackage{tikz-cd}

\newcommand{\ket}[1]{\mathopen{|}#1\rangle}

\begin{document}
\[
\begin{tikzcd}
\ket{\vec{\mu}-\vec{\alpha}}
  \arrow[r,bend left,"E^{+}",
    start anchor={[yshift=1.5pt]real east},end anchor={[yshift=1.5pt]real west}] &
\ket{\vec{\mu}}
  \arrow[l,bend left,"E^{-}",
    start anchor={[yshift=-1.5pt]real west},end anchor={[yshift=-1.5pt]real east}]
  \arrow[r,bend left,"E^{+}",
    start anchor={[yshift=1.5pt]real east},end anchor={[yshift=1.5pt]real west}] &
\ket{\vec{\mu}+\vec{\alpha}}
  \arrow[l,bend left,"E^{-}",
    start anchor={[yshift=-1.5pt]real west},end anchor={[yshift=-1.5pt]real east}]
\end{tikzcd}
\]
\end{document}

Some good soul might help pushing the arrow tip at the center.

Answer 3

If you want to consider using pstricks, it can be done with the psmatrix environment. I add the nccmath package, to have medium-sized formulae, esvectfor better looking vector arrows, and mathtools to define the pair of | … > delimiters. Alternatively, you can use the braket package, but you then lose the possibility (not used here) of fine-tuning the size of delimiters according to context.

\documentclass[pdf]{article}
\usepackage{pstricks-add}
\usepackage{esvect}
\usepackage{mathtools, nccmath}
\DeclarePairedDelimiter{\ket}\lvert\rangle

\begin{document}

\[ %
\psset{ArrowInside=->, ArrowInsidePos=0.57, arrowinset=0.2 , linewidth=0.5pt, arrows=c-c, offset=1ex, nodesep=3pt, arcangle=20, labelsep =2pt}
\begin{psmatrix}
%%% nodes
  \ket{\vv{\mu} - \vv{\alpha}}\rnode{l1}{}\hspace{3em}\rnode{l2}{}\ket{\vv{\mu}}\rnode{r1}{}\hspace{3em} \rnode{r2}{} \ket{\vv{\mu} + \vv{\alpha}}
%%%arrows
\ncarc{l1}{l2}\naput{\medmath{E_+}}\ncarc{r1}{r2}\naput{\medmath{E_+}}
\psset{offset=0ex, nodesep=1.5pt}
\ncarc{l2}{l1}\naput{\medmath{E_-}} \ncarc{r2}{r1}\naput{\medmath{E_-}}
\end{psmatrix}
\]%

\end{document}