\newcommand{\ket}[1]{\ensuremath{\vert#1\rangle}} \newcommand{\bra}[1]{\ensuremath{\langle#1\vert}} \newcommand{\braket}[2]{\ensuremath{\langle#1\vert#2\rangle}} \newcommand{\braketoperator}[3]{\ensuremath{\langle#1\vert#2\vert#3\rangle}} \newcommand{\doublebraketoperator}[3]{\left\langle \!\! \left\langle #1 \left\vert #2 \right\vert #3 \right\rangle \!\! \right\rangle} \newcommand{\pd}[2]{\ensuremath{\frac{\partial#1}{\partial#2}}} \newcommand{\degree}{\ensuremath{^\circ}}