\def\sigmaxmatrix{\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}}  
\def\sigmaymatrix{\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}} 
\def\sigmazmatrix{\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}} 
\def\sigmaxbraket{\ket{0}\bra{1} + \ket{1}\bra{0}}
\def\sigmaybraket{-i \ket{0}\bra{1} + i \ket{1}\bra{0}}
\def\sigmazbraket{\ket{0}\bra{0} - \ket{1}\bra{1}}

\Part[
    \eng{Quantum Mechanics}
    \ger{Quantenmechanik}
]{qm}
    \Section[
        \eng{Basics}
        \ger{Basics}
    ]{basics}
        \Subsection[
            \eng{Operators}
            \ger{Operatoren}
        ]{op}
            \Ger[row_vector]{Zeilenvektor}
            \Ger[column_vector]{Spaltenvektor}
            \Eng[column_vector]{Column vector}
            \Eng[row_vector]{Row vector}
            \begin{formula}{dirac_notation}
                \desc{Dirac notation}{}{}
                \desc[german]{Dirac-Notation}{}{}
                \eq{
                    \bra{x} \vspace*{1cm} \text{"Bra" \GT{row_vector}} \\
                    \ket{x} \vspace*{1cm} \text{"Ket" \GT{column_vector}} \\
                    \hat{A}\ket{\beta} = \ket{\alpha} \Rightarrow \bra{\alpha} = \bra{\beta} \hat{A}^\dagger
                }
            \end{formula}

            \begin{formula}{dagger}
                \desc{Dagger}{}{}
                \desc[german]{Dagger}{}{}
                \eq{
                    \hat{A}^\dagger &= (\hat{A}^*)^\mathrm{T} \\
                    (c \hat{A})^\dagger &= c^* \hat{A}^\dagger \\
                    (\hat{A}\hat{B})^\dagger &= \hat{B}^\dagger \hat{A}^\dagger \\
                }
            \end{formula}

            \begin{formula}{adjoint_op}
                \desc{Adjoint operator}{}{}
                \desc[german]{Adjungierter operator}{}{}
                \eq{\braket{\alpha|\hat{A}^\dagger|\beta} = \braket{\beta|\hat{A}|\alpha}^*}
            \end{formula}

            \begin{formula}{hermitian_op}
                \desc{Hermitian operator}{}{}
                \desc[german]{Hermitescher operator}{}{}
                \eq{\hat{A} = \hat{A}^\dagger}
            \end{formula}

            \Subsubsection[
                \eng{Measurement}
                \ger{Messung}
            ]{measurement}  
                \begin{ttext}
                    \eng{An observable is a hermition operator acting on $\hat{H}$. The measurement randomly yields one of the eigenvalues of $\hat{O}$ (all real).}
                    \ger{Eine Observable ist ein hermitscher Operator, der auf $\hat{H}$ wirkt. Die Messung ergibt zufällig einen der Eigenwerte von $\hat{O}$, welche alle reell sind.}
                \end{ttext}
                \begin{formula}{name}
                    \desc{Measurement probability}{Probability to measure $\psi$ in state $\lambda$}{}
                    \desc[german]{Messwahrscheinlichkeit}{Wahrscheinlichkeit, $\psi$ im Zustand $\lambda$ zu messen}{}
                    \eq{p(\lambda) = \braket{\psi|\hat{P}_\lambda|\psi}}
                \end{formula}
                \begin{formula}{state_after}
                    \desc{State after measurement}{}{}
                    \desc[german]{Zustand nach der Messung}{}{}
                    \eq{\ket{\psi}_\text{post} = \frac{1}{\sqrt{p(\lambda)}}\hat{P}_\lambda \ket{\psi}}
                \end{formula}
                
            
            \Subsubsection[
                \eng{Pauli matrices}
                \ger{Pauli-Matrizen}
            ]{pauli_matrices}
                \begin{formula}{pauli_matrices}
                    \desc{Pauli matrices}{}{}
                    \desc[german]{Pauli Matrizen}{}{}
                    \newFormulaEntry
                    \begin{alignat}{2}
                        \sigma_x &= \sigmaxmatrix &&= \sigmaxbraket \label{eq:pauli_x} \\
                        \sigma_y &= \sigmaymatrix &&= \sigmaybraket \label{eq:pauli_y} \\
                        \sigma_z &= \sigmazmatrix &&= \sigmazbraket \label{eq:pauli_z}
                    \end{alignat}
                \end{formula}
            % $\sigma_x$ NOT
            % $\sigma_y$ PHASE
            % $\sigma_z$ Sign

        \Subsection[
            \ger{Wahrscheinlichkeitstheorie}
            \eng{Probability theory}
        ]{probability}
            \begin{formula}{conservation_of_probability}
                \desc{Continuity equation}{}{$\rho$ density of a conserved quantity $q$, $j$ flux density of $q$}
                \desc[german]{Kontinuitätsgleichung}{}{$\rho$ Dichte einer Erhaltungsgröße $q$, $j$ Fluß von $q$}
                \eq{\frac{\partial\rho(\vec{x}, t)}{\partial t} + \nabla \cdot \vec{j}(\vec{x},t) = 0}
            \end{formula}

            \begin{formula}{state_probability}
                \desc{State probability}{}{}
                \desc[german]{Zustandswahrscheinlichkeit}{}{}
                \eq{TODO}
            \end{formula}

            \begin{formula}{dispersion}
                \desc{Dispersion}{}{}
                \desc[german]{Dispersion}{}{}
                \eq{\Delta \hat{A} = \hat{A} - \braket{\hat{A}}}
            \end{formula}

            \begin{formula}{generalized_uncertainty}
                \desc{Generalized uncertainty principle}{}{}
                \desc[german]{Allgemeine Unschärferelation}{}{}
                % \eq{\braket{(\Delta \hat{A})^2} \braket{(\Delta \hat{B})^2} \ge \frac{1}{4} \braket{[\hat{A},\hat{B}]}^2}
                \eq{
                    \sigma_A \sigma_B &\ge \frac{1}{4} \braket{[\hat{A},\hat{B}]}^2 \\
                    \sigma_A \sigma_B &\ge \frac{1}{2} \abs{\braket{[\hat{A},\hat{B}]}}
                }
            \end{formula}


        \Subsection[
            \eng{Commutator}
            \ger{Kommutator}
        ]{commutator}
            \begin{formula}{commutator}
                \desc{Commutator}{}{}
                \desc[german]{Kommutator}{}{}
                \eq{[A,B] = AB - BA}
            \end{formula}

            \begin{formula}{anticommutator}
                \desc{Anticommutator}{}{}
                \desc[german]{Antikommmutator}{}{}
                \eq{\{A,B\} = AB + BA}
            \end{formula}

            \begin{formula}{commutation_relations}\
                \desc{Commutation relations}{}{}
                \desc[german]{Kommutatorrelationen}{}{}
                \eq{[A, BC] = [A, B]C - B[A,C]}
            \end{formula}
            \TODO{add some more?}

            \begin{formula}{function}
                \desc{Commutator involving a function}{}{given $[A,[A,B]] = 0$}
                \desc[german]{Kommutator mit einer Funktion}{}{falls $[A,[A,B]] = 0$} 
                \eq{[f(A) , B] = [A,B]\,\pdv{f}{A}}
            \end{formula}

            \begin{formula}{jacobi_identity}
                \desc{Jacobi identity}{}{}
                \desc[german]{Jakobi-Identität}{}{}
                \eq{[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0}
            \end{formula}

            \begin{formula}{hadamard_lemma}
                \desc{Hadamard's Lemma}{}{}
                \desc[german]{Lemma von Hadamard}{}{}
                \eq{\e^A B \e^{-A} = B + [A,B] + \frac{1}{2!} [A, [A,B]] + \frac{1}{3!} [A, [A, [A, B]]] + \dots}
            \end{formula}   

            \begin{formula}{canon_comm_relation}
                \desc{Canonical commutation relation}{}{$x$, $p$ canonical conjugates}
                \desc[german]{Kanonische Vertauschungsrelationen}{}{$x$, $p$ kanonische konjugierte}
                \eq{
                    [x_i, x_j] &= 0 \\
                    [p_i, p_j] &= 0 \\
                    [x_i, p_j] &= i \hbar \delta_{ij}
                }
            \end{formula}

    \Section[
        \eng{Schrödinger equation}
        \ger{Schrödingergleichung}
    ]{se}
        \begin{formula}{energy_operator}
            \desc{Energy operator}{}{}
            \desc[german]{Energieoperator}{}{}
            \eq{E = i\hbar \frac{\partial}{\partial t}}
        \end{formula}

        \begin{formula}{momentum_operator}
            \desc{Momentum operator}{}{}
            \desc[german]{Impulsoperator}{}{}
            \eq{\vec{p} = -i\hbar \vec{\nabla_x}}
        \end{formula}

        \begin{formula}{space_operator}
            \desc{Space operator}{}{}
            \desc[german]{Ortsoperator}{}{}
            \eq{\vec{x} = i\hbar \vec{\nabla_p}}
        \end{formula}

        \begin{formula}{stationary_schroedinger_equation}
            \desc{Stationary Schrödingerequation}{}{}
            \desc[german]{Stationäre Schrödingergleichung}{}{}
            \eq{\hat{H}\ket{\psi} = E\ket{\psi}}
        \end{formula}

        \begin{formula}{schroedinger_equation}
            \desc{Schrödinger equation}{}{}
            \desc[german]{Schrödingergleichung}{}{}
            \abbrLabel{SE}
            \eq{i\hbar\frac{\partial}{\partial t}\psi(x, t) = (- \frac{\hbar^2}{2m} \vec{\nabla}^2 + \vec{V}(x)) \psi(x)}
        \end{formula}

        \begin{formula}{hellmann_feynmann} \absLabel
            \desc{Hellmann-Feynman-Theorem}{Derivative of the energy to a parameter}{}
            \desc[german]{Hellmann-Feynman-Theorem}{Abletiung der Energie nach einem Parameter}{}
            \eq{
                \odv{E_\lambda}{\lambda} = \int \d^3r \psi^*_\lambda \odv{\hat{H}_\lambda}{\lambda} \psi_\lambda = \Braket{\psi(\lambda)|\odv{\hat{H}_{\lambda}}{\lambda}|\psi(\lambda)}
            }
        \end{formula}

        \begin{formula}{variational_principle}
            \desc{Variational principle}{}{}
            \desc[german]{Variationsprinzip}{}{}
            \ttxt{\eng{
                    If $\hat{H}\psi = E\psi$, then $E_0 \le E = \Braket{\psi|\hat{H}|\psi}$. The ground state can thus be found by varying $\psi$ until $E$ is minimized.
                }\ger{
                    Wenn $\hat{H}\psi = E\psi$, dann ist $E_0 \le E = \Braket{\psi|\hat{H}|\psi}$. Der Grundzustand kann daher gefunden werden, indem $\psi$ variiert wird bis die Energie minimiert ist.
            }}
        \end{formula}

        \Subsection[
            \eng{Time evolution}
            \ger{Zeitentwicklug}
            ]{time}
            The time evolution of the Hamiltonian is given by:
            \begin{formula}{time_evolution_op}
                \desc{Time evolution operator}{}{$U$ unitary}
                \desc[german]{Zeitentwicklungsoperator}{}{$U$ unitär}
                \eq{\ket{\psi(t)} = \hat{U}(t, t_0) \ket{\psi(t_0)}}
            \end{formula}

            \begin{formula}{von_neumann}
                \desc{Von-Neumann Equation}{Time evolution of the density operator in the Schrödinger picture. Qm analog to the Liouville equation \ref{eq:mech:liouville:todo}}{}
                \desc[german]{Von-Neumann Gleichung}{Zeitentwicklung des Dichteoperators im Schödingerbild. Qm. Analogon zur Liouville-Gleichung \ref{eq:mech:liouville:todo}}{}
                \eq{\pdv{\hat{\rho}}{t} = - \frac{i}{\hbar}[\hat{H}, \hat{\rho}]}
            \end{formula}

            \begin{formula}{lindblad}
                \desc{Lindblad master equation}{Generalization of von-Neummann equation for open quantum systems}{$h$ positive semidifnite matrix, $\hat{A}$ arbitrary operator}
                \desc[german]{Lindblad-Mastergleichung}{Verallgemeinerung der von-Neumman Gleichung für offene Quantensysteme}{$h$ positiv-semifinite Matrix, $\hat{A}$ beliebiger Operator}
                \eq{\dot{\rho} = \underbrace{-\frac{i}{\hbar} [\hat{H}, \rho]}_\text{reversible} + \underbrace{\sum_{n.m} h_{nm} \left(\hat{A}_n\rho \hat{A}_{m^\dagger} - \frac{1}{2}\left\{\hat{A}_m^\dagger \hat{A}_n,\rho \right\}\right)}_\text{irreversible}}
            \end{formula}



            \TODO{unitary transformation of time dependent H}

            \Subsubsection[
                \eng{Schrödinger- and Heisenberg-pictures}
                \ger{Schrödinger- und Heisenberg-Bild}
            ]{s_h_pictures}
                \eng[s_h_pictures_desc]{
                    In the \textbf{Schrödinger picture}, the time dependecy is in the states
                    while in the \textbf{Heisenberg picture} the observables (operators) are time dependent.
                }
                \ger[s_h_pictures_desc]{Im Schrödinger-Bild sind die Zustände zeitabhänig, im Heisenberg-Bild
                    sind die Observablen (Operatoren) zeitabhänig
                }
                \gt{s_h_pictures_desc}\\
                \begin{formula}{schroediner_time_evolution}
                    \desc{Schrödinger time evolution}{}{}
                    \desc[german]{Schrödinger Zeitentwicklug}{}{}
                    \eq{
                        \ket{\psi(t)_\textrm{S}} = \hat{U}(t,t_0)\ket{\psi(t_0)}
                    }
                \end{formula}

                \begin{formula}{heisenberg_time_evolution}
                    \desc{Heisenberg time evolution}{}{\textrm{H} and \textrm{S} being the Heisenberg and Schrödinger picture, respectively}
                    \desc[german]{Heisenberg Zeitentwicklung}{}{mit \textrm{H} und \textrm{S} dem Heisenberg- und Schrödinger-Bild}
                    \eq{
                        \ket{\psi_\mathrm{H}} = \ket{\psi_\mathrm{S}(t_0)} \\
                        A_\textrm{H} = U^\dagger(t,t_0)A_\textrm{S}U(t,t_0) \\
                        \odv{\hat{A}_\textrm{H}}{t} = \frac{1}{i\hbar}[\hat{A}_\textrm{H}, \hat{H}_\textrm{H}] + \Big(\pdv{\hat{A}_\textrm{S}}{t}\Big)_\textrm{H}
                    }
                \end{formula}

            \Subsubsection[
                \eng{Ehrenfest theorem}
                \ger{Ehrenfest-Theorem}
                ]{ehrenfest_theorem}
                \GT{see_also} \ref{sec:qm:basics:schroedinger_equation:correspondence_principle}
                \begin{formula}{ehrenfest_theorem}
                    \desc{Ehrenfest theorem}{applies to both pictures}{}
                    \desc[german]{Ehrenfest-Theorem}{gilt für beide Bilder}{}
                    \eq{
                        \odv{}{t} \braket{\hat{A}} = \frac{1}{i\hbar}\braket{[\hat{A},\hat{H}]} + \Braket{\pdv{\hat{A}}{t}}
                    }
                \end{formula}
                \begin{formula}{ehrenfest_theorem_x}
                    \desc{Ehrenfest theorem example}{Example for $x$}{}
                    \desc[german]{Ehrenfest-Theorem Beispiel}{Beispiel für $x$}{}
                    \eq{m\odv[2]{}{t}\braket{x} = -\braket{\nabla V(x)} = \braket{F(x)}}
                \end{formula}
                % \eq{Time evolution}{\hat{H}\ket{\psi} = E\ket{\psi}}{sg_time}

            % TODO: wo gehört das hin?
            \begin{formula}{correspondence_principle}
                \desc{Correspondence principle}{}{}
                \desc[german]{Korrespondenzprinzip}{}{}
                \ttxt{
                    \ger{Die klassischen Bewegungsgleichungen lassen sich als Grenzfall (große Quantenzahlen) aus der Quantenmechanik ableiten.}
                    \eng{The classical mechanics can be derived from quantum mechanics in the limit of large quantum numbers.}
                }
            \end{formula}
                


    \Section[
        \eng{Pertubation theory}
        \ger{Störungstheorie}
        ]{qm_pertubation}
        \begin{ttext}
            \eng{The following holds true if the pertubation $\hat{H_1}$ is sufficently small and the $E^{(0)}_n$ levels are not degenerate.}
            \ger{Die folgenden Gleichungen gelten wenn $\hat{H_1}$ ausreichend klein ist und die $E_n^{(0)}$ Niveaus nicht entartet sind.}
        \end{ttext}
        \begin{formula}{pertubation_hamiltonian}
            \desc{Hamiltonian}{}{}
            \desc[german]{Hamiltonian}{}{}
            \eq{\hat{H} = \hat{H_0} + \lambda \hat{H_1}}
        \end{formula}

        \begin{formula}{pertubation_series}
            \desc{Power series}{}{}
            \desc[german]{Potenzreihe}{}{}
            \eq{
                E_n &= E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + ... \\
                \ket{\psi_n} &= \ket{\psi_n^{(0)}} + \lambda \ket{\psi_n^{(1)}} + \lambda^2 \ket{\psi_n^{(2)}} + ...
            }
        \end{formula}
        \begin{formula}{1o_energy}
            \desc{1. order energy shift}{}{}
            \desc[german]{Energieverschiebung 1. Ordnung}{}{}
            \eq{E_n^{(1)} = \Braket{\psi_n^{(0)}|\hat{H_1}|\psi_n^{(0)}}}
        \end{formula}
        \begin{formula}{1o_state}
            \desc{1. order states}{}{}
            \desc[german]{Zustände}{}{}
            \eq{\ket{\psi_n^{(1)}} = \sum_{k\neq n}\frac{\Braket{\psi_k^{(0)}|\hat{H_1}|\psi_n^{(0)}}}{E_n^{(0)} - E_k^{(0)}}\ket{\psi_k^{(0)}}}
        \end{formula}
        \begin{formula}{2o_energy}
            \desc{2. order energy shift}{}{}
            \desc[german]{Energieverschiebung 2. Ordnung}{}{}
            % \eq{E_n^{(1)} = \Braket{\psi_n^{(0)}|\hat{H_1}|\psi_n^{(0)}}}
            \eq{E_n^{(2)} = \sum_{k\neq n}\frac{\abs{\Braket{\psi_k^{(0)}|\hat{H_1}|\psi_n^{(0)}}}^2}{E_n^{(0)} - E_k^{(0)}}}
        \end{formula}
        % \begin{formula}{qm:pertubation:}
        %     \desc{1. order states}{}{}
        %     \desc[german]{Zustände}{}{}
        %     \eq{\ket{\psi_n^{(1)}} = \sum_{k\neq n}\frac{\Braket{\psi_k^{(0)}|\hat{H_1}|\psi_n^{(0)}}}{E_n^{(0)} - E_k^{(0)}}\ket{\psi_k^{(0)}}}
        % \end{formula}

        \begin{formula}{golden_rule}
            \desc{Fermi's golden rule}{Transition rate from initial state $\ket{i}$ under a pertubation $H^1$ to final state $\ket{f}$}{}
            \desc[german]{Fermis goldene Regel}{Übergangsrate des initial Zustandes $\ket{i}$ unter einer Störung $H^1$ zum Endzustand $\ket{f}$}{}
            \eq{\Gamma_{i\to f} = \frac{2\pi}{\hbar} \abs{\braket{f | H^1 | i}}^2\,\rho(E_f)}
        \end{formula}


    \Section[
        \eng{Harmonic oscillator}
        \ger{Harmonischer Oszillator}
    ]{hosc}
        \begin{formula}{hamiltonian}
            \desc{Hamiltonian}{}{}
            \desc[german]{Hamiltonian}{}{}
            \eq{
                H&=\frac{p^2}{2m}+\frac{1}{2} m\omega^2 x^2\\
                 &=\frac{1}{2} \hbar\omega+\omega a^\dagger a
             }
        \end{formula}

        \begin{formula}{hosc_spectrum}
            \desc{Energy spectrum}{}{}
            \desc[german]{Energiespektrum}{}{}
            \eq{E_n = \hbar\omega \Big(\frac{1}{2} + n\Big)}
        \end{formula}

        \GT{see_also} \ref{sec:qm:hosc:c_a_ops}

        \Subsection[
            \ger{Erzeugungs und Vernichtungsoperatoren / Leiteroperatoren}
            \eng{Creation and Annihilation operators / Ladder operators}
        ]{c_a_ops}
            \begin{formula}{c_a_ops_def}
                \desc{Particle number operator/occupation number operator}{}{$\ket{n}$ = Fock states, $\hat{a}$ = Annihilation operator, $\hat{a}^\dagger$ = Creation operator}
                \desc[german]{Teilchenzahloperator/Besetzungszahloperator}{}{$\ket{n}$ = Fock-Zustände, $\hat{a}$ = Vernichtungsoperator, $\hat{a}^\dagger$ = Erzeugungsoperator}
                \eq{
                    \hat{N} &:= a^\dagger a \\
                    \hat{N}\ket{n} &= n \ket{N}
                }
            \end{formula}

            \begin{formula}{c_a_commutator}
                \desc{Commutator}{}{}
                \desc[german]{Kommutator}{}{}
                \eq{
                    [\hat{a},\hat{a}^\dagger] &= 1 \\
                    [N, \hat{a}] &= -\hat{a} \\
                    [N, \hat{a}^\dagger] &= \hat{a}^\dagger
                }
            \end{formula}

            \begin{formula}{c_a_on_state}
                \desc{Application on states}{}{}
                \desc[german]{Anwendung auf Zustände}{}{}
                \eq{
                    \hat{a} \ket{n} &= \sqrt{n}\ket{n-1} \\
                    \hat{a}^\dagger \ket{n} &= \sqrt{n+1}\ket{n+1} \\
                    \ket{n} &= \frac{1}{\sqrt{n!}} (\hat{a}^\dagger)^n \ket{0}
                }
            \end{formula}

            \begin{formula}{c_a_matrices}
                \desc{Matrix forms}{}{}
                \desc[german]{Matrix-Form}{}{}
                \eq{
                    \hat{n} &= \begin{pmatrix}
                        0 & 0 & 0 & 0 \\
                        0 & 1 & 0 & 0 \\
                        0 & 0 & \ddots & 0 \\
                        0 & 0 & 0 & N
                    \end{pmatrix} \\
                    \hat{a} &= \begin{pmatrix}
                        0 & \sqrt{1} & 0      & 0 \\
                        0 & 0        & \ddots & 0 \\
                        0 & 0        & 0      & \sqrt{N} \\
                        0 & 0        & 0      & 0 
                    \end{pmatrix} \\
                    \hat{a}^\dagger &= \begin{pmatrix}
                        0        & 0      & 0        & 0 \\
                        \sqrt{1} & 0      & 0        & 0 \\
                        0        & \ddots & 0        & 0 \\
                        0        & 0      & \sqrt{N} & 0
                    \end{pmatrix}
                }
            \end{formula}

            \Subsubsection[
                \eng{Harmonischer Oszillator}
                \ger{Harmonic Oscillator}
            ]{hosc}
                \begin{formula}{c_a_ops}
                    \desc{Harmonic oscillator}{}{}
                    \desc[german]{Harmonischer Oszillator}{}{}
                    \eq{
                        % \tilde{X} &= \sqrt{\frac{m\omega}{\hbar}} \hat{x}  &= \frac{1}{\sqrt{2}} (\hat{a} + \hat{a}^\dagger) \\
                        % \tilde{P} &= \frac{1}{\sqrt{m\omega\hbar}} \hat{p} &= \frac{-i}{\sqrt{2}} (\hat{a} - \hat{a}^\dagger) \\
                        \hat{x} &= \sqrt{\frac{\hbar}{2m\omega}} (\hat{a} + \hat{a}^\dagger) \\
                        \hat{p} &= -i\sqrt{\frac{m\omega\hbar}{2}} (\hat{a} - \hat{a}^\dagger) \\
                        \hat{H} &= \frac{\hat{p}^2}{2m} + \frac{m\omega^2 \hat{x}^2}{2} &= \hbar\omega\Big(a^\dagger a + \frac{1}{2}\Big) \\
                        a &= \frac{1}{\sqrt{2}} (\tilde{X} + i\tilde{P}) \\
                        a^\dagger &= \frac{1}{\sqrt{2}} (\tilde{X} - i\tilde{P})
                        % \hat{a}^\dagger ? \sqrt{\frac{}{}}
                    }
                \end{formula}
            % \eq{Heisenberg}{\frac{dA}{dt}=\frac{\partial A}{\partial t}+\frac{[A,H]}{i\hbar}}{heisenberg}

        % \begin{align}
        %     \label{eq:k}
        %     A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} \\
        %     A^\dagger=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x-\frac{ip}{\sqrt{2m\omega}} \\
        %     HAu_E=(E-\hbar\omega)Au_E \\
        %     u_n=\frac{1}{\sqrt{n!}}\left(\frac{A^\dagger}{\sqrt{\hbar}}\right)^nu_0 \\
        %     u_0=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right) \\
        %     E_n=( \frac{1}{2} +n)\hbar\omega
        % \end{equation}

    \Section[
        \eng{Angular momentum}
        \ger{Drehmoment}
        ]{angular_momentum}

        \Subsection[
            \eng{Aharanov-Bohm effect}
            \ger{Aharanov-Bohm Effekt}
            ]{aharanov_bohm}
            \begin{formula}{phase}
                \desc{Acquired phase}{Electron along a closed loop aquires a phase proportional to the enclosed magnetic flux}{}
                \desc[german]{Erhaltene Phase}{Elektron entlang eines geschlossenes Phase erhält eine Phase die proportional zum eingeschlossenen magnetischem Fluss ist}{}
                \eq{\delta = \frac{2 e}{\hbar} \oint \vec{A}\cdot \d\vec{s} = \frac{2 e}{\hbar} \Phi}
            \end{formula}
            \TODO{replace with loop intergral symbol and add more info}
    \Section[
        \eng{Periodic potentials}
        \ger{Periodische Potentiale}
    ]{periodic}
        \begin{formula}{bloch_waves}
            \desc{Bloch waves}{
                Solve the stat. SG in periodic potential with period 
                $\vec{R}$: $V(\vec{r}) = V(\vec{r} + \vec{R})$\\
            }{
                $\vec{k}$ arbitrary, $u$ periodic function
            }
            \desc[german]{Blochwellen}{
                Lösen stat. SG im periodischen Potential mit Periode 
                $\vec{R}$: $V(\vec{r}) = V(\vec{r} + \vec{R})$\\
            }{
                $\vec{k}$ beliebig, $u$ periodische Funktion
            }
            \eq{\psi_k(\vec{r}) = e^{i \vec{k}\cdot \vec{r}} \cdot u_{\vec{k}}(\vec{r})}
        \end{formula}

        \begin{formula}{periodicity}
            \desc{Periodicity}{}{\QtyRef{lattice_vector}, \QtyRef{reciprocal_lattice_vector}}
            \desc[german]{Periodizität}{}{}
            \eq{
                u_\vec{k}(\vec{r} + \vec{R}) = u_\vec{k}(\vec{r}) \\
                \psi_{\vec{k}+\vec{G}}(\vec{r}) = \psi_\vec{k}(\vec{r})
            }
        \end{formula}


    \Section[
        \eng{Symmetries}
        \ger{Symmetrien}
        ]{symmetry}
        \begin{ttext}[desc]
            \eng{Most symmetry operators are unitary \ref{sec:linalg:unitary} because the norm of a state must be invariant under transformations of space, time and spin.}
            \ger{Die meisten Symmetrieoperatoren sind unitär \ref{sec:linalg:unitary}, da die Norm eines Zustands invariant unter Raum-, Zeit- und Spin-Transformationen sein muss.}
        \end{ttext}
        \begin{formula}{invariance}
            \desc{Invariance}{$\hat{H}$ is invariant under a symmetrie described by $\hat{U}$ if this holds}{}
            \desc[german]{Invarianz}{$\hat{H}$ is invariant unter der von $\hat{U}$ beschriebenen Symmetrie wenn gilt:}{}
            \eq{\hat{U}\hat{H}\hat{U}^\dagger = \hat{H} \Leftrightarrow [\hat{U}, \hat{H}] = 0}
        \end{formula}

        \Subsection[
            \eng{Time-reversal symmetry}
            \ger{Zeitumkehrungssymmetrie}
            ]{time_reversal}

            \begin{formula}{time}
                \desc{Time-reversal symmetry}{}{}
                \desc[german]{Zeitumkehrungssymmetrie}{}{}
                \eq{T: t \to -t}
            \end{formula}

            \begin{formula}{antiunitary}
                \desc{Anti-unitary}{}{}
                \desc[german]{Antiunitär}{}{}
                \eq{T^2 = -1}
            \end{formula}

    \Section[
        \eng{Two-level systems (TLS)}
        \ger{Zwei-Niveau System (TLS)}
        ]{tls}
        \begin{formula}{james_cummings}
            \desc{James-Cummings Hamiltonian}{TLS interacting with optical cavity}{$\hat{E} = E_\text{ZPF}(\hat{a} + \hat{a}^\dagger)$ field operator with bosonic ladder operators, $\hat{S} = \hat{\sigma}^\dagger + \hat{\sigma}$ polarization operator with ladder operators of the TLS}
            \desc[german]{James-Cummings Hamiltonian}{TLS interagiert mit resonantem Lichtfeld}{$\hat{E} = E_\text{ZPF}(\hat{a} + \hat{a}^\dagger)$ Feldoperator mit bosonischen Leiteroperatoren, $\hat{S} = \hat{\sigma}^\dagger + \hat{\sigma}$ Polarisationsoperator mit Leiteroperatoren des TLS}
            \eq{H &= \underbrace{\hbar\omega_c \hat{a}^\dagger \hat{a}}_\text{\GT{field}} 
                    + \underbrace{\hbar\omega_\text{a} \frac{\hat{\sigma}_z}{2}}_\text{\GT{atom}} 
                    + \underbrace{\frac{\hbar\Omega}{2} \hat{E} \hat{S}}_\text{int} \\
                \shortintertext{\GT{after} \fRef[RWA]{qm:other:RWA}:} \\
                  &= \hbar\omega_c \hat{a}^\dagger \hat{a} 
                    + \hbar\omega_\text{a} \hat{\sigma}^\dagger \hat{\sigma}
                    + \frac{\hbar\Omega}{2} (\hat{a}\hat{\sigma^\dagger} + \hat{a}^\dagger \hat{\sigma})
            }
        \end{formula}

    \Section[
        \eng{Other}
        \ger{Sonstiges}
    ]{other}
        \begin{formula}{RWA}
            \desc{Rotating Wave Approximation (RWS)}{Rapidly oscilating terms are neglected}{$\omega_\text{L}$ light frequency, $\omega_0$ transition frequency}
            \desc[german]{Rotating Wave Approximation / Drehwellennäherung (RWS)}{Schnell oscillierende Terme werden vernachlässigt}{$\omega_\text{L}$ Frequenz des Lichtes, $\omega_0$ Übergangsfrequenz}
            \eq{\Delta\omega \coloneq \abs{\omega_0 - \omega_\text{L}} \ll \abs{\omega_0 + \omega_\text{L}} \approx 2\omega_0}
        \end{formula}

        \begin{formula}{adiabatic_theorem} \absLabel
            \desc{Adiabatic theorem}{}{}
            \desc[german]{Adiabatentheorem}{}{}
            \ttxt{
                \eng{A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.}
                \ger{Ein quantenmechanisches System bleibt in im derzeitigen Eigenzustand falls eine Störung langsam genug wirkt und der Eigenwert durch eine Lücke vom Rest des Spektrums getrennt ist.}
            }
        \end{formula}

        \begin{formula}{slater_det}
            \desc{Slater determinant}{Construction of a fermionic (antisymmetric) many-particle wave function from single-particle wave functions}{}
            \desc[german]{Slater Determinante}{Konstruktion einer fermionischen (antisymmetrischen) Vielteilchen Wellenfunktion aus ein-Teilchen Wellenfunktionen}{}
            \eq{
                \Psi(q_1, \dots, q_N) = \frac{1}{\sqrt{N!}}
                \begin{vmatrix}
                \phi_a(q_1) & \phi_a(q_2) & \cdots & \phi_a(q_N) \\
                \phi_b(q_1) & \phi_b(q_2) & \cdots & \phi_b(q_N) \\
                \vdots & \vdots & \ddots & \vdots \\
                \phi_z(q_1) & \phi_z(q_2) & \cdots & \phi_z(q_N)
                \end{vmatrix}
            }
        \end{formula}