\def\vecr{{\vec{r}}}
\def\abohr{a_\textrm{B}}

\Section[
    \eng{Hydrogen Atom}
    \ger{Wasserstoffatom}
    ]{h}

    \begin{formula}{reduced_mass}
        \desc{Reduced mass}{}{}
        \desc[german]{Reduzierte Masse}{}{}
        \eq{\mu = \frac{\masse m_\textrm{K}}{\masse + m_\textrm{K}} \explOverEq[\approx]{$\masse \ll m_\textrm{K}$} \masse}
    \end{formula}   

    \begin{formula}{potential}
        \desc{Coulumb potential}{For a single electron atom}{$Z$ atomic number}
        \desc[german]{Coulumb potential}{Für ein Einelektronenatom}{$Z$ Ordnungszahl/Kernladungszahl}
        \eq{V(\vecr) = \frac{Z\,e^2}{4\pi\epsilon_0 r}}
    \end{formula}
    \begin{formula}{hamiltonian}
        \desc{Hamiltonian}{}{}
        \desc[german]{Hamiltonian}{}{}
        % \eq{V(\vecr) = \frac{Z\,e^2}{4\pi\epsilon_0 r}}
        \eq{
            \hat{H} &= -\frac{\hbar^2}{2\mu} {\Grad_\vecr}^2 - V(\vecr) \\
                    &= \frac{\hat{p}_r^2}{2\mu} + \frac{\hat{L}^2}{2\mu r} + V(r)
        }
    \end{formula}

    \begin{formula}{wave_function}
        \desc{Wave function}{}{$R_{nl}(r)$ \fqEqRef{qm:h:radial}, $Y_{lm}$ \fqEqRef{qm:spherical_harmonics}}
        \desc[german]{Wellenfunktion}{}{}
        \eq{\psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_{lm}(\theta,\phi)}
    \end{formula}

    \begin{formula}{radial}
        \desc{Radial part}{}{$L_r^s(x)$ Laguerre-polynomials}
        \desc[german]{Radialanteil}{}{$L_r^s(x)$ Laguerre-Polynome}
        \eq{
            R_{nl} &= - \sqrt{\frac{(n-l-1)!(2\kappa)^3}{2n[(n+l)!]^3}} (2\kappa r)^l \e^{-\kappa r} L_{n+1}^{2l+1}(2\kappa r)
            \shortintertext{\GT{with}} 
            \kappa &= \frac{\sqrt{2\mu\abs{E}}}{\hbar} = \frac{Z}{n \abohr}
        }
    \end{formula}

    \begin{formula}{energy}
        \desc{Energy eigenvalues}{}{}
        \desc[german]{Energieeigenwerte}{}{}
        \eq{E_n &= \frac{Z^2\mu e^4}{n^2(4\pi\epsilon_0)^2 2\hbar^2} = -E_\textrm{H}\frac{Z^2}{n^2}}
    \end{formula}

    \begin{formula}{rydberg_constant_heavy}
        \desc{Rydberg constant}{for heavy atoms}{\ConstRef{electron_mass}, \ConstRef{elementary_charge}, \QtyRef{vacuum_permittivity}, \ConstRef{planck}, \ConstRef{vacuum_speed_of_light}}
        \desc[german]{Rydberg-Konstante}{für schwere Atome}{}
        \constant{R_\infty}{exp}{
            \val{10973731.568157(12)}{\per\m}
        }
        \eq{
            R_\infty = \frac{m_e e^4}{8\epsilon_0^2 h^3 c}
        }
    \end{formula}

    \begin{formula}{rydberg_constant_corrected}
        \desc{Rydberg constant}{corrected for nucleus mass $M$}{\QtyRef{rydberg_constant_heavy}, $\mu = \left(\frac{1}{m_\txe} + \frac{1}{M}\right)^{-1}$ \GT{reduced_mass}, \ConstRef{electron_mass}}
        \desc[german]{Rydberg Konstante}{korrigiert für Kernmasse $M$}{}
        \eq{R_\txM = \frac{\mu}{m_\txe} R_\infty}
    \end{formula}

    \begin{formula}{rydberg_energy}
        \desc{Rydberg energy}{Energy unit}{\ConstRef{rydberg_constant_heavy}, \ConstRef{planck}, \ConstRef{vacuum_speed_of_light}}
        \desc[german]{Rydberg-Energy}{Energie Einheit}{}
        \eq{1\,\text{Ry} = hc\,R_\infty}
    \end{formula}

    \begin{formula}{bohr_radius}
        \desc{Bohr radius}{}{\ConstRef{vacuum_permittivity}, \ConstRef{electron_mass}}
        \desc[german]{Bohrscher Radius}{}{}
        \constant{a_0}{exp}{
            \val{5.29177210544(82) \xE{-11}}{\m}
        }
        \eq{a_0 = \frac{4\pi \epsilon_0 \hbar^2}{e^2 m_\txe}}
    \end{formula}


\Subsection[
    \eng{Corrections}
    \ger{Korrekturen}
    ]{corrections}

    \Subsubsection[
        \eng{Darwin term}
        \ger{Darwin-Term}
        ]{darwin}
        \begin{ttext}[desc]
            \eng{Relativisitc correction: Because of the electrons zitterbewegung, it is not entirely localised. \TODO{fact check}}
            \ger{Relativistische Korrektur: Elektronen führen eine Zitterbewegung aus und sind nicht vollständig lokalisiert.}
        \end{ttext}
        \begin{formula}{energy_shift}
            \desc{Energy shift}{}{}
            \desc[german]{Energieverschiebung}{}{}
            \eq{\Delta E_\textrm{rel} = -E_n \frac{Z^2\alpha^2}{n} \Big(\frac{3}{4n} - \frac{1}{l+ \frac{1}{2}}\Big)}
        \end{formula}

        \begin{formula}{fine_structure_constant}
            \desc{Fine-structure constant}{Sommerfeld constant}{}
            \desc[german]{Feinstrukturkonstante}{Sommerfeldsche Feinstrukturkonstante}{}
            \eq{\alpha = \frac{e^2}{4\pi\epsilon_0\hbar c} \approx \frac{1}{137}}
        \end{formula}

    \Subsubsection[
        \eng{Spin-orbit coupling (LS-coupling)}
        \ger{Spin-Bahn-Kopplung (LS-Kopplung)}
        ]{ls_coupling}
        \begin{ttext}[desc]
            \eng{The interaction of the electron spin with the electrostatic field of the nuclei lead to energy shifts.}
            \ger{The Wechselwirkung zwischen dem Elektronenspin und dem elektrostatischen Feld des Kerns führt zu Energieverschiebungen.}
        \end{ttext}

        \begin{formula}{energy_shift}
            \desc{Energy shift}{}{}
            \desc[german]{Energieverschiebung}{}{}
            \eq{\Delta E_\text{LS} = \frac{\mu_0 Z e^2}{8\pi \masse^2\,r^3} \braket{\vec{S} \cdot \vec{L}}}
        \end{formula}
        \begin{formula}{sl}
            \desc{\TODO{name}}{}{}
            \desc[german]{??}{}{}
            \eq{\braket{\vec{S} \cdot \vec{L}} &= \frac{1}{2} \braket{[J^2-L^2-S^2]} \nonumber \\
                                               &= \frac{\hbar^2}{2}[j(j+1) -l(l+1) -s(s+1)]}
        \end{formula}

    \Subsubsection[
        \eng{Fine-structure}
        \ger{Feinstruktur}
        ]{fine_structure}
        \begin{ttext}[desc]
            \eng{The fine-structure combines relativistic corrections \ref{sec:qm:h:corrections:darwin} and the spin-orbit coupling \ref{sec:qm:h:corrections:ls_coupling}.}
            \ger{Die Feinstruktur vereint relativistische Korrekturen \ref{sec:qm:h:corrections:darwin} und die Spin-Orbit-Kupplung \ref{sec:qm:h:corrections:ls_coupling}.}
        \end{ttext}
        \begin{formula}{energy_shift}
            \desc{Energy shift}{}{}
            \desc[german]{Energieverschiebung}{}{}
            \eq{\Delta E_\textrm{FS} = \frac{Z^2\alpha^2}{n}\Big(\frac{1}{j+\frac{1}{2}} - \frac{3}{4n}\Big)}
        \end{formula}


    \Subsubsection[
        \eng{Lamb-shift}
        \ger{Lamb-Shift}
        ]{lamb_shift}
        \begin{ttext}[desc]
            \eng{The interaction of the electron with virtual photons emitted/absorbed by the nucleus leads to a (very small) shift in the energy level.}
            \ger{The Wechselwirkung zwischen dem Elektron und vom Kern absorbierten/emittierten virtuellen Photonen führt zu einer (sehr kleinen) Energieverschiebung.}
        \end{ttext}
        \begin{formula}{energy}
            \desc{Potential energy}{}{$\delta r$ pertubation of $r$}
            \desc[german]{Potentielle Energy}{}{$\delta r$ Schwankung von $r$}
            \eq{\braket{E_\textrm{pot}} = -\frac{Z e^2}{4\pi\epsilon_0} \Braket{\frac{1}{r+\delta r}}}
        \end{formula}

    \Subsubsection[
        \eng{Hyperfine structure}
        \ger{Hyperfeinstruktur}
        ]{hyperfine_structure}
        \begin{ttext}[desc]
            \eng{Interaction of the nucleus spin with the magnetic field created by the electron leads to energy shifts. (Lifts degeneracy) }
            \ger{Wechselwirkung von Kernspin mit dem vom Elektron erzeugten Magnetfeld spaltet Energieniveaus}
        \end{ttext}
        \begin{formula}{nuclear_spin}
            \desc{Nuclear spin}{}{}
            \desc[german]{Kernspin}{}{}
            \eq{\vec{F} &= \vec{J} + \vec{I} \\
                \abs{\vec{I}} &= \sqrt{i(i+1)}\hbar \\
                I_z &= m_i\hbar \\
                m_i &= -i, -i+1, \dots, i-1, i
            }
        \end{formula}
        \begin{formula}{angular_momentum}
            \desc{Combined angular momentum}{}{}
            \desc[german]{Gesamtdrehimpuls}{}{}
            \eq{\vec{F} &= \vec{J} + \vec{I} \\
                \abs{\vec{F}} &= \sqrt{f(f+1)}\hbar \\
                F_z &= m_f\hbar
            }
        \end{formula}
        \begin{formula}{selection_rule}
            \desc{Selection rule}{}{}
            \desc[german]{Auswahlregel}{}{}
            \eq{f &= j \pm i \\ m_f &= -f,-f+1,\dots,f-1,f}
        \end{formula}
        \begin{formula}{constant}
            \desc{Hyperfine structure constant}{}{$B_\textrm{HFS}$ hyperfine field, $\mu_\textrm{K}$ nuclear magneton, $g_i$ nuclear g-factor \ref{qm:h:lande}}
            \desc[german]{Hyperfeinstrukturkonstante}{}{$B_\textrm{HFS}$ Hyperfeinfeld, $\mu_\textrm{K}$ Kernmagneton, $g_i$ Kern-g-Faktor \ref{qm:h:lande}}
            \eq{A = \frac{g_i \mu_\textrm{K} B_\textrm{HFS}}{\sqrt{j(j+1)}}}
        \end{formula}
        \begin{formula}{energy_shift}
            \desc{Energy shift}{}{}
            \desc[german]{Energieverschiebung}{}{}
            \eq{\Delta H_\textrm{HFS} = \frac{A}{2}[f(f+1) - j(j+1) -i(i+1)]}
        \end{formula}
        \TODO{landé factor}

\Subsection[
    \eng{Effects in magnetic field}
    \ger{Effekte im Magnetfeld}
]{mag_effects}
    \TODO{all} 
    \\\TODO{Hunds rules}

\Subsection[
    \eng{misc}
    \ger{Sonstiges}
]{other}
    \begin{formula}{auger_effect}
        \desc{Auger-Meitner-Effekt}{Auger-Effect}{}
        \desc[german]{Auger-Meitner-Effekt}{Auger-Effekt}{}
        \ttxt{
            \eng{An excited electron relaxes into a lower, unoccupied energy level. The released energy causes the emission of another electron in a higher energy level (Auger-Electron)}
            \ger{Ein angeregtes Elektron fällt in ein unbesetztes, niedrigeres Energieniveau zurück. Durch die frei werdende Energie verlässt ein Elektron aus einer höheren Schale das Atom (Auger-Elektron).}
        }
    \end{formula}

