$$ \newcommand{\vb}{\mathbf} \newcommand{\wt}{\widetilde} \newcommand{\mc}{\mathcal} \newcommand{\bmc}[1]{\boldsymbol{\mathcal{#1}}} \newcommand{\sup}[1]{^{\text{#1}}} \newcommand{\pard}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\VMV}[3]{ \Big\langle #1 \Big| #2 \Big| #3 \Big\rangle} $$