\newcommand\abs[1]{\left\lvert #1 \right\rvert}
\newcommand\avg[1]{\left\langle #1 \right\rangle}
\newcommand\diff{\text{d}}
\newcommand\Diff{\text{D}}
\newcommand\degree{^{\circ}}
\newcommand\ihat{\hat{\imath}}
\newcommand\jhat{\hat{\jmath}}
\newcommand\Jhat{\hat{J}}
\newcommand\khat{\hat{k}}
\newcommand\ham{\hat{H}}
\newcommand\Lhat{\hat{L}}
\newcommand\ehat{\hat{e}}
\newcommand\ex{\hat{e}_x}
\newcommand\ey{\hat{e}_y}
\newcommand\ez{\hat{e}_z}
\newcommand\et{\hat{e}_t}
\newcommand\matx[1]{\underline{\underline{\mathbf{#1}}}}
\newcommand\bra[1]{\langle #1\lvert}
\newcommand\ket[1]{\lvert #1\rangle}
\newcommand\braket[2]{\langle #1 \lvert #2 \rangle}
\newcommand\Braket[3]{\langle #1 \lvert #2 \lvert #3 \rangle}
\newcommand\stf[1]{\times 10^{#1}}
\newcommand\sol{_{\odot}}
\newcommand\const{\text{const.}}
\newcommand\tot{\text{tot}}
\newcommand\curl[1]{\mathcal{#1}}
\newcommand\zed{\mathcal{Z}}
\let\oldphi\phi
\renewcommand\phi\varphi
\let\oldepsilon\epsilon
\renewcommand\epsilon\varepsilon
\newcommand\diffr[3]{\dfrac{\diff^{#3} #1}{\diff #2^{#3}}}
\newcommand\diffrat[4]{\left.\dfrac{\diff^{#3} #1}{\diff #2^{#3}}\right\lvert_{#4}\!\!\!\!}
\newcommand\partr[3]{\dfrac{\partial^{#3} #1}{\partial #2^{#3}}}
\newcommand\partrat[4]{\left.\dfrac{\partial^{#3} #1}{\partial #2^{#3}}\right\lvert_{#4}\!\!\!\!}
\newcommand\at[1]{\left. #1 \right\lvert}
\newcommand\elmnt[3]{\phantom{}_{#1}^{#2}\text{#3}}
\newcommand\AA{\mathring{A}}