\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}

        \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}{variance}
                \desc{Variance}{}{}
                \desc[german]{Varianz}{}{}
                \eq{\sigma^2 = \braket{(\Delta \hat{A})^2} = \braket{\hat{A}^2} - \braket{\hat{A}}^2}
            \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}
            
            \Subsubsection[
                \eng{Pauli matrices}
                \ger{Pauli-Matrizen}
            ]{pauli_matrices}
                \begin{formula}{pauli_matrices}
                    \desc{Pauli matrices}{}{}
                    \desc[german]{Pauli Matrizen}{}{}
                    \eqAlignedAt{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{formula}
            % $\sigma_x$ NOT
            % $\sigma_y$ PHASE
            % $\sigma_z$ Sign


        \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}

            \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}{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}

        \Subsection[
            \eng{Schrödinger equation}
            \ger{Schrödingergleichung}
        ]{schroedinger_equation}
            \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}{}{}
                \eq{i\hbar\frac{\partial}{\partial t}\psi(x, t) = (- \frac{\hbar^2}{2m} \vec{\nabla}^2 + \vec{V}(x)) \psi(x)}
            \end{formula}
            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}

            \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) \\
                        \diff{\hat{A}_\textrm{H}}{t} = \frac{1}{i\hbar}[\hat{A}_\textrm{H}, \hat{H}_\textrm{H}] + \Big(\diffp{\hat{A}_\textrm{S}}{t}\Big)_\textrm{H}
                    }
                \end{formula}

            \Subsubsection[
                \ger{Korrespondenzprinzip}
                \eng{Correspondence principle}
            ]{correspondence_principle}
                \begin{ttext}{desc}
                    \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{ttext}
                

            \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{Ehrenfesttheorem}{applies to both pictures}{}
                    \desc[german]{Ehrenfest-Theorem}{gilt für beide Bilder}{}
                    \eq{
                        \diff{}{t} \braket{\hat{A}} = \frac{1}{i\hbar}\braket{[\hat{A},\hat{H}]} + \Braket{\diffp{\hat{A}}{t}}
                    }
                \end{formula}
                \begin{formula}{ehrenfest_theorem_x}
                    \desc{}{Example for $x$}{}
                    \desc[german]{}{Beispiel für $x$}{}
                    \eq{m\diff[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}

    \Section[
        \eng{Pertubation theory}
        \ger{Störungstheorie}
    ]{qm_pertubation}
        \eng[desc]{The following holds true if the pertubation $\hat{H_1}$ is sufficently small and the $E^{(0)}_n$ levels are not degenerate.}
        \ger[desc]{Die folgenden Gleichungen gelten wenn $\hat{H_1}$ ausreichend klein ist und die $E_n^{(0)}$ Niveaus nicht entartet sind.}
        \gt{desc}
        \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}


    \Section[
        \eng{Harmonic oscillator}
        \ger{Harmonischer Oszillator}
    ]{qm_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}
            \eng{Creation and Annihilation 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}

            \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{angular_momentum}
    \times 


        \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}

