progress

.gitignore

@@ -0,0 +1,6 @@
+.aux
+out
+*.virtual_documents*
+*ipynb_checkpoints*
+*__pycache__*
+**.pdf

src/analysis.tex

@@ -0,0 +1,125 @@
+\Part[
+    \eng{Calculus}
+    \ger{Analysis}
+    ]{cal}
+
+    \Subsection[
+        \eng{Convolution}
+        \ger{Faltung / Konvolution}
+        ]{conv}
+        \begin{ttext}
+            \eng{Convolution is \textbf{commutative}, \textbf{associative} and \textbf{distributive}.}
+            \ger{Die Faltung ist \textbf{kommutativ}, \textbf{assoziativ} und \textbf{distributiv}}
+        \end{ttext}
+        \begin{formula}{def}
+            \desc{Definition}{}{}
+            \desc[german]{Definition}{}{}
+            \eq{(f*g)(t) = f(t) * g(t) = int_{-\infty}^\infty f(\tau) g(t-\tau) \d \tau}
+        \end{formula}
+        \begin{formula}{notation}
+            \desc{Notation}{}{}
+            \desc[german]{Notation}{}{}
+            \eq{
+                f(t) * g(t-t_0) &= (f*g)(t-t_0) \\
+                f(t-t_0) * g(t-t_0) &= (f*g)(t-2t_0)
+            }
+        \end{formula}
+        \begin{formula}{commutativity}
+            \desc{Commutativity}{}{}
+            \desc[german]{Kommutativität}{}{}
+            \eq{f * g = g * f}
+        \end{formula}
+
+        \begin{formula}{associativity}
+            \desc{Associativity}{}{}
+            \desc[german]{Assoziativität]}{}{}
+            \eq{(f*g)*h = f*(g*h)}
+        \end{formula}
+
+        \begin{formula}{distributivity}
+            \desc{Distributivity}{}{}
+            \desc[german]{Distributivität}{}{}
+            \eq{f * (g + h) = f*g + f*h}
+        \end{formula}
+
+        \begin{formula}{complex_conjugate}
+            \desc{Complex conjugate}{}{}
+            \desc[german]{Komplexe konjugation}{}{}
+            \eq{(f*g)^* = f^* * g^*}
+        \end{formula}
+        
+    \Subsection[
+        \eng{Fourier analysis}
+        \ger{Fourieranalyse}
+        ]{fourier}
+        \Subsubsection[
+            \eng{Fourier series}
+            \ger{Fourierreihe}
+            ]{series}
+            \begin{formula}{series}
+                \desc{Fourier series}{Complex representation}{$f\in \Lebesgue^2(\R,\C)$ $T$-\GT{periodic}}
+                \desc[german]{Fourierreihe}{Komplexe Darstellung}{}
+                \eq{f(t) = \sum_{k=-\infty}^{\infty} c_k \Exp{\frac{2\pi \I kt}{T}}}
+            \end{formula}
+            \Eng[real]{real}
+            \Ger[real]{reellwertig}
+            \begin{formula}{coefficient}
+                \desc{Fourier coefficients}{Complex representation}{}
+                \desc[german]{Fourierkoeffizienten}{Komplexe Darstellung}{}
+                \eq{
+                    c_k &= \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Exp{-\frac{2\pi \I}{T}kt}\d t \quad\text{\GT{for}}\,k\ge0\\
+                    c_{-k} &= \overline{c_k} \quad \text{\GT{if} $f$ \GT{real}}
+                }
+            \end{formula}
+
+            \begin{formula}{series_sincos}
+                \desc{Fourier series}{Sine and cosine representation}{$f\in \Lebesgue^2(\R,\C)$ $T$-\GT{periodic}}
+                \desc[german]{Fourierreihe}{Sinus und Kosinus Darstellung}{}
+                \eq{f(t) = \frac{a_0}{2} + \sum_{k=1}^{\infty} \left(a_k \Cos{\frac{2\pi}{T}kt} + b_k\Sin{\frac{2\pi}{T}kt}\right)}
+            \end{formula}
+            \begin{formula}{coefficient}
+                \desc{Fourier coefficients}{Sine and cosine representation\\If $f$ has point symmetry: $a_{k>0}=0$, if $f$ has axial symmetry: $b_k=0$}{}
+                \desc[german]{Fourierkoeffizienten}{Sinus und Kosinus Darstellung\\Wenn $f$ punktsymmetrisch: $a_{k>0}=0$, wenn $f$ achsensymmetrisch: $b_k=0$}{}
+                \eq{
+                    a_k &= \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Cos{-\frac{2\pi}{T}kt}\d t \quad\text{\GT{for}}\,k\ge0\\
+                    b_k &= \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Sin{-\frac{2\pi}{T}kt}\d t \quad\text{\GT{for}}\,k\ge1\\
+                    a_k &= c_k + c_{-k}      \quad\text{\GT{for}}\,k\ge0\\
+                    b_k &= \I(c_k - c_{-k})  \quad\text{\GT{for}}\,k\ge1
+                }
+            \end{formula}
+        \TODO{cleanup}
+        
+
+
+        \Subsubsection[
+            \eng{Fourier transformation}
+            \ger{Fouriertransformation}
+            ]{trafo}
+            \begin{formula}{transform}
+                \desc{Fourier transform}{}{$\hat{f}:\R^n \mapsto \C$, $\forall f\in L^1(\R^n)$}
+                \desc[german]{Fouriertransformierte}{}{}
+                \eq{\hat{f}(k) \coloneq \frac{1}{\sqrt{2\pi}^n} \int_{\R^n} \e^{-\I kx}f(x)\d x}
+            \end{formula}
+
+            \Eng[linear_in]{linear in}
+            \Ger[linear_in]{linear in}
+            \GT{for} $f\in L^1(\R^n)$:
+            \begin{enumerate}[i)]
+                \item $f \mapsto \hat{f}$ \GT{linear_in} $f$
+                \item $g(x) = f(x-h) \qRarrow \hat{g}(k) = \e^{-\I kn}\hat{f}(k)$
+                \item $g(x) = \e^{ih\cdot x}f(x) \qRarrow \hat{g}(k) = \hat{f}(k-h)$
+                \item $g(\lambda) = f\left(\frac{x}{\lambda}\right) \qRarrow \hat{g}(k)\lambda^n \hat{f}(\lambda k)$
+            \end{enumerate}
+
+
+\Section[
+    \eng{List of common integrals}
+    \ger{Liste nützlicher Integrale}
+    ]{integrals}
+
+    \begin{formula}{riemann_zeta}
+        \desc{Riemann Zeta Function}{}{}
+        \desc[german]{Riemannsche Zeta-Funktion}{}{}
+        \eq{\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \frac{1}{(1-2^{(1-s)})\Gamma(s)} \int_0^\infty \d\eta \frac{\eta^{(s-1)}}{\e^\eta + 1}}
+    \end{formula}
+

src/atom.tex

@@ -1,7 +1,6 @@
-\def\masse{m_\textrm{e}}
+\def\vecr{{\vec{r}}}
-\def\grad{\vec{\nabla}}
+\kef\abohr{a_\textrm{B}}
-\def\vecr{\vec{r}}
+
-\def\abohr{a_\textrm{B}}
 \Section[
     \eng{Hydrogen Atom}
     \ger{Wasserstoffatom}
@@ -23,9 +22,8 @@
         \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) \\
-            % \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)
-            %         &= \frac{\hat{p}_r^2}{2\mu} + \frac{\hat{L}^2}{2\mu r} + V(r)
         }
     \end{formula}
 
@@ -67,7 +65,7 @@
         \eng{Darwin term}
         \ger{Darwin-Term}
         ]{darwin}
-        \begin{ttext}{desc}
+        \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}
@@ -87,7 +85,7 @@
         \eng{Spin-orbit coupling (LS-coupling)}
         \ger{Spin-Bahn-Kopplung (LS-Kopplung)}
         ]{ls_coupling}
-        \begin{ttext}{desc}
+        \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}
@@ -95,7 +93,7 @@
         \begin{formula}{energy_shift}
             \desc{Energy shift}{}{}
             \desc[german]{Energieverschiebung}{}{}
-            \eq{\Delta E_\text{LS} = \frac{\mu_0 Z e^2}{8\pi m^2 e\,r^3} \braket{\vec{S} \cdot \vec{L}}}
+            \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}}{}{}
@@ -108,7 +106,7 @@
         \eng{Fine-structure}
         \ger{Feinstruktur}
         ]{fine_structure}
-        \begin{ttext}{desc}
+        \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}
@@ -122,8 +120,8 @@
     \Subsubsection[
         \eng{Lamb-shift}
         \ger{Lamb-Shift}
-    ]{lamb_shift}
+        ]{lamb_shift}
-        \begin{ttext}{desc}
+        \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}
@@ -136,11 +134,10 @@
     \Subsubsection[
         \eng{Hyperfine structure}
         \ger{Hyperfeinstruktur}
-    ]{hyperfine_structure}
+        ]{hyperfine_structure}
-        \begin{ttext}{desc}
+        \begin{ttext}[desc]
-            \eng{Interaction of the nucleus spin with the magnetic field created by the electron leads to energy shifts. (Lifts degenaracy) }
+            \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}{}{}

src/calculus.tex

@@ -0,0 +1,55 @@
+\Part[
+    \eng{Analysis}
+    \ger{Analysis}
+    ]{ana}
+
+    \Subsection[
+        \eng{Convolution}
+        \ger{Faltung / Konvolution}
+        ]{conv}
+        \begin{ttext}
+            \eng{Convolution is \textbf{commutative}, \textbf{associative} and \textbf{distributive}.}
+            \ger{Die Faltung ist \textbf{kommutativ}, \textbf{assoziativ} und \textbf{distributiv}}
+        \end{ttext}
+        \begin{formula}{def}
+            \desc{Definition}{}{}
+            \desc[german]{Definition}{}{}
+            \eq{(f*g)(t) = f(t) * g(t) = int_{-\infty}^\infty f(\tau) g(t-\tau) \d \tau}
+        \end{formula}
+        \begin{formula}{notation}
+            \desc{Notation}{}{}
+            \desc[german]{Notation}{}{}
+            \eq{
+                f(t) * g(t-t_0) &= (f*g)(t-t_0) \\
+                f(t-t_0) * g(t-t_0) &= (f*g)(t-2t_0)
+            }
+        \end{formula}
+        \begin{formula}{commutativity}
+            \desc{Commutativity}{}{}
+            \desc[german]{Kommutativität}{}{}
+            \eq{f * g = g * f}
+        \end{formula}
+
+        \begin{formula}{associativity}
+            \desc{Associativity}{}{}
+            \desc[german]{Assoziativität]}{}{}
+            \eq{(f*g)*h = f*(g*h)}
+        \end{formula}
+
+        \begin{formula}{distributivity}
+            \desc{Distributivity}{}{}
+            \desc[german]{Distributivität}{}{}
+            \eq{f * (g + h) = f*g + f*h}
+        \end{formula}
+
+        \begin{formula}{complex_conjugate}
+            \desc{Complex conjugate}{}{}
+            \desc[german]{Komplexe konjugation}{}{}
+            \eq{(f*g)^* = f^* * g^*}
+        \end{formula}
+        
+    \Subsection[
+        \eng{Fourier analysis}
+        \ger{Fourieranalyse}
+        ]{fourier}
+

src/circuit.tex

@@ -0,0 +1,79 @@
+
+%  Josephson junction
+\makeatletter
+\pgfcircdeclarebipolescaled{instruments}
+{
+    % put the node text above and centered
+    \anchor{text}{\pgfextracty{\pgf@circ@res@up}{\northeast}
+        \pgfpoint{-.5\wd\pgfnodeparttextbox}{
+            \dimexpr.5\dp\pgfnodeparttextbox+.5\ht\pgfnodeparttextbox+\pgf@circ@res@up\relax
+        }
+    }
+}
+{\ctikzvalof{bipoles/oscope/height}}
+{josephson}
+{\ctikzvalof{bipoles/oscope/height}}
+{\ctikzvalof{bipoles/oscope/width}}
+{
+    % ?
+    \pgf@circ@setlinewidth{bipoles}{\pgfstartlinewidth}
+    \pgfextracty{\pgf@circ@res@up}{\northeast}
+    \pgfextractx{\pgf@circ@res@right}{\northeast}
+    \pgfextractx{\pgf@circ@res@left}{\southwest}
+    \pgfextracty{\pgf@circ@res@down}{\southwest}
+    \pgfmathsetlength{\pgf@circ@res@step}{0.25*\pgf@circ@res@up}
+
+    % \pgfscope % box
+    %     \pgfpathrectanglecorners{\pgfpoint{\pgf@circ@res@left}{\pgf@circ@res@down}}{\pgfpoint{\pgf@circ@res@right}{\pgf@circ@res@up}}
+    %     \pgf@circ@draworfill
+    % \endpgfscope
+    \pgfscope % cross
+      \pgfpathmoveto{\pgfpoint{\pgf@circ@res@left}{\pgf@circ@res@up}}%
+      \pgfpathlineto{\pgfpoint{\pgf@circ@res@right}{\pgf@circ@res@down}}%
+      \pgfpathmoveto{\pgfpoint{\pgf@circ@res@right}{\pgf@circ@res@up}}%
+      \pgfpathlineto{\pgfpoint{\pgf@circ@res@left}{\pgf@circ@res@down}}%
+      \pgfusepath{draw}
+    \endpgfscope
+}
+\def\pgf@circ@josephson@path#1{\pgf@circ@bipole@path{josephson}{#1}}
+\tikzset{josephson/.style = {\circuitikzbasekey, /tikz/to path=\pgf@circ@josephson@path, l=#1}}
+
+
+\pgfcircdeclarebipolescaled{instruments}
+{
+    % put the node text above and centered
+    \anchor{text}{\pgfextracty{\pgf@circ@res@up}{\northeast}
+        \pgfpoint{-.5\wd\pgfnodeparttextbox}{
+            \dimexpr.5\dp\pgfnodeparttextbox+.5\ht\pgfnodeparttextbox+\pgf@circ@res@up\relax
+        }
+    }
+}
+{\ctikzvalof{bipoles/oscope/height}}
+{josephsoncap}
+{\ctikzvalof{bipoles/oscope/height}}
+{\ctikzvalof{bipoles/oscope/width}}
+{
+    % ?
+    \pgf@circ@setlinewidth{bipoles}{\pgfstartlinewidth}
+    \pgfextracty{\pgf@circ@res@up}{\northeast}
+    \pgfextractx{\pgf@circ@res@right}{\northeast}
+    \pgfextractx{\pgf@circ@res@left}{\southwest}
+    \pgfextracty{\pgf@circ@res@down}{\southwest}
+    \pgfmathsetlength{\pgf@circ@res@step}{0.25*\pgf@circ@res@up}
+
+    \pgfscope % box
+        \pgfpathrectanglecorners{\pgfpoint{\pgf@circ@res@left}{\pgf@circ@res@down}}{\pgfpoint{\pgf@circ@res@right}{\pgf@circ@res@up}}
+        \pgf@circ@draworfill
+    \endpgfscope
+    \pgfscope % cross
+      \pgfpathmoveto{\pgfpoint{\pgf@circ@res@left}{\pgf@circ@res@up}}%
+      \pgfpathlineto{\pgfpoint{\pgf@circ@res@right}{\pgf@circ@res@down}}%
+      \pgfpathmoveto{\pgfpoint{\pgf@circ@res@right}{\pgf@circ@res@up}}%
+      \pgfpathlineto{\pgfpoint{\pgf@circ@res@left}{\pgf@circ@res@down}}%
+      \pgfusepath{draw}
+    \endpgfscope
+}
+\def\pgf@circ@josephsoncap@path#1{\pgf@circ@bipole@path{josephsoncap}{#1}}
+\tikzset{josephsoncap/.style = {\circuitikzbasekey, /tikz/to path=\pgf@circ@josephsoncap@path, l=#1}}
+
+

src/condensed_matter.tex

@@ -0,0 +1,231 @@
+\Part[
+    \eng{Condensed matter physics}
+    \ger{Festkörperphysik}
+    ]{cm}
+\Section[
+    \eng{Bravais lattice}
+    \ger{Bravais-Gitter}
+    ]{bravais}
+
+    % \begin{ttext}
+    %     \eng{
+
+    %     }
+    %     \ger{
+
+    %     }
+    % \end{ttext}
+
+    \eng[bravais_table2]{In 2D, there are 5 different Bravais lattices}
+    \ger[bravais_table2]{In 2D gibt es 5 verschiedene Bravais-Gitter}
+
+    \eng[bravais_table3]{In 3D, there are 14 different Bravais lattices}
+    \ger[bravais_table3]{In 3D gibt es 14 verschiedene Bravais-Gitter}
+
+    \Eng[lattice_system]{Lattice system}
+    \Ger[lattice_system]{Gittersystem}
+    \Eng[crystal_family]{Crystal system}
+    \Ger[crystal_family]{Kristall-system}
+    \Eng[point_group]{Point group}
+    \Ger[point_group]{Punktgruppe}
+    \eng[bravais_lattices]{Bravais lattices}
+    \ger[bravais_lattices]{Bravais Gitter}
+
+    \newcommand\bvimg[1]{\begin{center}\includegraphics[width=0.1\textwidth]{img/bravais/#1.pdf}\end{center}}
+    \renewcommand\tabularxcolumn[1]{m{#1}}
+    \newcolumntype{Z}{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}X}
+    \begin{table}[H]
+        \centering
+        \caption{\gt{bravais_table2}}
+        \label{tab:bravais2}
+    
+        \begin{adjustbox}{width=\textwidth}
+        \begin{tabularx}{\textwidth}{||Z|c|Z|Z||}
+            \hline
+            \multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{2}{c||}{5 \gt{bravais_lattices}} \\ \cline{3-4}
+                                                 & & \GT{primitive} (p) & \GT{centered} (c) \\ \hline
+            \GT{monoclinic} (m) & $\text{C}_\text{2}$   & \bvimg{mp} & \\ \hline
+            \GT{orthorhombic} (o) & $\text{D}_\text{2}$ & \bvimg{op} & \bvimg{oc} \\ \hline
+            \GT{tetragonal} (t) & $\text{D}_\text{4}$   & \bvimg{tp} & \\ \hline
+            \GT{hexagonal} (h) & $\text{D}_\text{6}$    & \bvimg{hp} & \\ \hline
+        \end{tabularx}
+        \end{adjustbox}
+    \end{table}
+    
+
+
+    \begin{table}[H]
+        \centering
+        \caption{\gt{bravais_table3}}
+        \label{tab:bravais3}
+    
+        % \newcolumntype{g}{>{\columncolor[]{0.8}}}
+        \begin{adjustbox}{width=\textwidth}
+            % \begin{tabularx}{\textwidth}{|c|}
+            %     asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
+            %     asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
+            % \end{tabularx}
+            % \begin{tabular}{|c|}
+            %     asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
+            %     asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
+            % \end{tabular}
+            % \\
+        \begin{tabularx}{\textwidth}{||Z|Z|c|Z|Z|Z|Z||}
+            \hline
+            \multirow{2}{*}{\GT{crystal_family}} & \multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{4}{c||}{14 \gt{bravais_lattices}} \\ \cline{4-7}
+                                                 & & & \GT{primitive} (P) & \GT{base_centered} (S) & \GT{body_centered} (I) & \GT{face_centered} (F) \\ \hline
+            \multicolumn{2}{||c|}{\GT{triclinic} (a)} & $\text{C}_\text{i}$ & \bvimg{tP}  &  &  &  \\  \hline
+            \multicolumn{2}{||c|}{\GT{monoclinic} (m)} & $\text{C}_\text{2h}$ & \bvimg{mP} & \bvimg{mS}  &  &  \\ \hline
+            \multicolumn{2}{||c|}{\GT{orthorhombic} (o)} & $\text{D}_\text{2h}$ & \bvimg{oP} & \bvimg{oS} & \bvimg{oI} & \bvimg{oF} \\ \hline
+            \multicolumn{2}{||c|}{\GT{tetragonal} (t)} & $\text{D}_\text{4h}$ & \bvimg{tP} &  & \bvimg{tI} & \\ \hline
+            \multirow{2}{*}{\GT{hexagonal} (h)} & \GT{rhombohedral} & $\text{D}_\text{3d}$ & \bvimg{hR} &  &  & \\ \cline{2-7}
+             & \GT{hexagonal} & $\text{D}_\text{6h}$ & \bvimg{hP} &  &  & \\ \hline
+            \multicolumn{2}{||c|}{\GT{cubic} (c)} & $\text{O}_\text{h}$ & \bvimg{cP} &  & \bvimg{cI} & \bvimg{cF} \\ \hline
+        \end{tabularx}
+        \end{adjustbox}
+    \end{table}
+    
+
+\Section[
+    \eng{Reciprocal lattice}
+    \ger{Reziprokes Gitter}
+    ]{reci}
+    \begin{ttext}
+        \eng{The reciprokal lattice is made up of all the wave vectors $\vec{k}$ that ressemble standing waves with the periodicity of the Bravais lattice.}
+        \ger{Das rezioproke Gitter besteht aus dem dem Satz aller Wellenvektoren $\vec{k}$, die ebene Wellen mit der Periodizität des Bravais-Gitters ergeben.}
+    \end{ttext}
+
+    \begin{formula}{vectors}
+        \desc{Reciprocal lattice vectors}{}{$a_i$ real-space lattice vectors, $V_c$ volume of the primitive lattice cell}
+        \desc[german]{Reziproke Gittervektoren}{}{$a_i$ Bravais-Gitter Vektoren, $V_c$ Volumen der primitiven Gitterzelle}
+        \eq{
+            \vec{b_1} &= \frac{2\pi}{V_c} \vec{a_2} \times \vec{a_3} \\
+            \vec{b_2} &= \frac{2\pi}{V_c} \vec{a_3} \times \vec{a_1} \\
+            \vec{b_3} &= \frac{2\pi}{V_c} \vec{a_1} \times \vec{a_2}
+        }
+    \end{formula}
+    
+
+\Section[
+    \eng{Free electron gas}
+    \ger{Freies Elektronengase}
+]{free_e_gas}
+    \begin{ttext}
+        \eng{Assumptions: electrons can move freely and independent of each other.}
+        \ger{Annahmen: Elektronen bewegen sich frei und unabhänig voneinander.}
+    \end{ttext}
+
+    \begin{formula}{drift_velocity}
+        \desc{Drift velocity}{Velocity component induced by an external force (eg. electric field)}{$v_\text{th}$ thermal velocity}
+        \desc[german]{Driftgeschwindgkeit}{Geschwindigkeitskomponente durch eine externe Kraft (z.B. ein elektrisches Feld)}{$v_\text{th}$ thermische Geschwindigkeit}
+        \eq{\vec{v}_\text{D} = \vec{v} - \vec{v}_\text{th}}
+    \end{formula}
+
+    \begin{formula}{mean_free_time}
+        \desc{Mean free time}{}{}
+        \desc[german]{Streuzeit}{}{}
+        \eq{\tau}
+    \end{formula}
+
+    \begin{formula}{mean_free_path}
+        \desc{Mean free path}{}{}
+        \desc[german]{Mittlere freie Weglänge}{}{}
+        \eq{\ell = \braket{v} \tau}
+    \end{formula}
+
+    \begin{formula}{mobility}
+        \desc{Electrical mobility}{}{$q$ charge, $m$ mass}
+        \desc[german]{Beweglichkeit}{}{$q$ Ladung, $m$ Masse}
+        \eq{\mu = \frac{q \tau}{m}}
+    \end{formula}
+
+    \Subsection[
+        \eng{Drude model}
+        \ger{Drude-Modell}
+        ]{drude}
+        \begin{ttext}
+            \eng{Classical model describing the transport properties of electrons in materials (metals):
+                The material is assumed to be an ion lattice and with freely moving electrons (electron gas). The electrons are 
+                accelerated by an electric field and decelerated through collisions with the lattice ions.
+                The model disregards the Fermi-Dirac partition of the conducting electrons.
+            }
+            \ger{Ein klassisches Model zur Beschreibung der Transporteigenschaften von Elektronen in (v.a.) Metallen:
+                Der Festkörper wird als Ionenkristall mit frei beweglichen Elektronen (Elektronengas). 
+                Die Elektronen werden durch ein Elektrisches Feld $E$ beschleunigt und durch Stöße mit den Gitterionen gebremst.
+                Das Modell vernachlässigt die Fermi-Dirac Verteilung der Leitungselektronen.
+            }
+        \end{ttext}
+        \begin{formula}{motion}
+            \desc{Equation of motion}{}{$v$ electron speed, $\vec{v}_\text{D}$ drift velocity, $\tau$ mean free time between collisions}
+            \desc[german]{Bewegungsgleichung}{}{$v$ Elektronengeschwindigkeit, $\vec{v}_\text{D}$ Driftgeschwindigkeit, $\tau$ Stoßzeit}
+            \eq{\masse \odv{\vec{v}}{t} + \frac{\masse}{\tau} \vec{v}_\text{D} = -e \vec{E}}
+        \end{formula}
+        \begin{formula}{current_density}
+            \desc{Current density}{Ohm's law}{$n$ charge particle density}
+            \desc[german]{Stromdichte}{Ohmsches Gesetz}{$n$ Ladungsträgerdichte}
+            \eq{\vec{j} = -ne\vec{v}_\text{D} = ne\mu \vec{E}}
+        \end{formula}
+        \begin{formula}{conductivity}
+            \desc{Drude-conductivity}{}{}
+            \desc[german]{Drude-Leitfähigkeit}{}{}
+            \eq{\sigma = \frac{\vec{j}}{\vec{E}} = \frac{e^2 \tau n}{\masse} = n e \mu}
+        \end{formula}
+
+    \Subsection[
+        \eng{Sommerfeld model}
+        \ger{Sommerfeld-Modell}
+        ]{sommerfeld}
+        \begin{ttext}
+            \eng{Assumes a gas of free fermions underlying the pauli-exclusion principle. Only electrons in an energy range of $\kB T$ around the Fermi energy $\EFermi$ participate in scattering processes.}
+            \ger{Annahme eines freien Fermionengases, welches dem Pauli-Prinzip unterliegt. Nur Elektronen in einem Energiebereich von $\kB T$ um die Fermi Energe $\EFermi$ nehmen an Streuprozessen teil.}
+        \end{ttext}
+        \begin{formula}{current_density}
+            \desc{Current density}{}{}
+            \desc[german]{Stromdichte}{}{}
+            \eq{\vec{j} = -en\braket{v} = -e n \frac{\hbar}{\masse}\braket{\vec{k}} = -e \frac{1}{V} \sum_{\vec{k},\sigma} \frac{\hbar \vec{k}}{\masse}}
+        \end{formula}
+        \TODO{The formula for the conductivity is the same as in the drude model?}
+
+    \Subsection[
+        \eng{2D electron gas}
+        \ger{2D Elektronengas}
+        ]{2deg}
+
+        \begin{ttext}
+            \eng{Lower dimension gases can be obtained by restricting a 3D gas with infinetly high potential walls on a narrow area with the width $L$.}
+            \ger{
+                Niederdimensionale Elektronengase erhält man, wenn ein 3D Gas durch unendlich hohe Potentialwände auf einem schmalen Bereich mit Breite $L$ eingeschränkt wird.
+            }
+        \end{ttext}
+        \begin{formula}{confinement_energy}
+            \desc{Confinement energy}{Raises ground state energy}{}
+            \desc[german]{Confinement Energie}{Erhöht die Grundzustandsenergie}{}
+            \eq{\Delta E = \frac{\hbar^2 \pi^2}{2\masse L^2}}
+        \end{formula}
+
+        \Eng[plain_wave]{plain wave}
+        \Ger[plain_wave]{ebene Welle}
+        \begin{formula}{energy}
+            \desc{Energy}{}{}
+            \desc[german]{Energie}{}{}
+            \eq{E_n = \underbrace{\frac{\hbar^2 k_\parallel^2}{2\masse}}_\text{$x$-$y$: \GT{plain_wave}} + \underbrace{\frac{\hbar^2 \pi^2}{2\masse L^2} n^2}_\text{$z$}}
+        \end{formula}
+
+    \Subsection[
+        \eng{1D electron gas / quantum wire}
+        \ger{1D Eleltronengas / Quantendraht}
+        ]{1deg}
+
+        \begin{formula}{energy}
+            \desc{Energy}{}{}
+            \desc[german]{Energie}{}{}
+            \eq{E_n = \frac{\hbar^2 k_x^2}{2\masse} + \frac{\hbar^2 \pi^2}{2\masse L_z^2} n_1^2 + \frac{\hbar^2 \pi^2}{2\masse L_y^2} n_2^2}
+        \end{formula}
+        
+    \Subsection[
+        \eng{0D electron gas / quantum dot}
+        \ger{0D Elektronengase / Quantenpunkt}
+        ]{0deg}
+
+        \TODO{TODO}
+        

src/electrodynamics.tex

@@ -0,0 +1,222 @@
+\def\PhiB{\Phi_\text{B}}
+\def\PhiE{\Phi_\text{E}}
+
+\Part[
+    \eng{Electrodynamics}
+    \ger{Elektrodynamik}
+]{ed}
+
+
+\Section[
+    \eng{Maxwell-Equations}
+    \ger{Maxwell-Gleichungen}
+    ]{Maxwell}
+    \begin{formula}{vacuum}
+        \desc{Vacuum}{microscopic formulation}{}
+        \desc[german]{Vakuum}{Mikroskopische Formulierung}{}
+        \eq{
+            \Div \vec{E} &= \frac{\rho_\text{el}}{\epsilon_0} \\
+            \Div \vec{B} &= 0 \\
+            \Rot \vec{E} &= - \odv{\vec{B}}{t} \\
+            \Rot \vec{B} &= \mu_0 \vec{j} + \frac{1}{c^2} \odv{\vec{E}}{t}
+        }
+    \end{formula}
+
+    \begin{formula}{material}
+        \desc{Matter}{Macroscopic formulation}{}
+        \desc[german]{Materie}{Makroskopische Formulierung}{}
+        \eq{
+            \Div \vec{D} &= \rho_\text{el} \\
+            \Div \vec{B} &= 0 \\
+            \Rot \vec{E} &= - \odv{\vec{B}}{t} \\
+            \Rot \vec{H} &= \vec{j} + \odv{\vec{D}}{t}
+        }
+    \end{formula}
+
+\Section[
+    \eng{Fields}
+    \ger{Felder}
+    ]{fields}
+
+    \Subsection[
+        \eng{Electric field}
+        \ger{Elektrisches Feld}
+        ]{mag}
+        \begin{formula}{gauss_law}
+            \desc{Gauss's law for electric fields}{Electric flux through a closed surface is proportional to the electric charge}{$S$ closed surface}
+            \desc[german]{Gaußsches Gesetz für elektrische Felder}{Der magnetische Fluss durch eine geschlossene Fläche ist proportional zur elektrischen Ladung}{$S$ geschlossene Fläche}
+            \eq{\PhiE = \iint_S \vec{E}\cdot\d\vec{S} = \frac{Q}{\varepsilon_0}}
+        \end{formula}
+
+    \Subsection[
+        \eng{Magnetic field}
+        \ger{Magnetfeld}
+        ]{mag}
+
+        \Eng[magnetic_flux]{Magnetix flux density}
+        \Ger[magnetic_flux]{Magnetische Flussdichte}
+
+        % \begin{quantity}{mag_flux}{\Phi}{\Wb}{\kg\m^2\per\s^2\A^1}{scalar}
+        %     \sign{}
+        %     \desc{Magnetic flux density}{}
+        %     \desc[german]{Magnetische Feldstärke}{}
+        % \end{quantity}
+
+        \begin{formula}{magnetic_flux}
+            \desc{Magnetic flux}{}{}
+            \desc[german]{Magnetischer Fluss}{}{}
+            \eq{\PhiB = \iint_A \vec{B}\cdot\d\vec{A}}
+        \end{formula}
+
+        \begin{formula}{gauss_law}
+            \desc{Gauss's law for magnetism}{Magnetic flux through a closed surface is $0$ \Rightarrow there are no magnetic monopoles}{$S$ closed surface}
+            \desc[german]{Gaußsches Gesetz für Magnetismus}{Der magnetische Fluss durch eine geschlossene Fläche ist $0$ \Rightarrow es gibt keine magnetischen Monopole}{$S$ geschlossene Fläche}
+            \eq{\PhiB = \iint_S \vec{B}\cdot\d\vec{S} = 0}
+        \end{formula}
+
+        \begin{formula}{name}
+            \desc{}{}{}
+            \desc[german]{}{}{}
+            \eq{}
+        \end{formula}
+        \begin{formula}{magnetization}
+            \desc{Magnetization}{}{$m$ mag. moment, $V$ volume}
+            \desc[german]{Magnetisierung}{}{$m$ mag. Moment, $V$ Volumen}
+            \eq{\vec{M} = \odv{\vec{m}}{V} = \chi_\text{m} \cdot \vec{H}}
+        \end{formula}
+
+        \begin{formula}{angular_torque}
+            \desc{Torque}{}{$m$ mag. moment}
+            \desc[german]{Drehmoment}{}{$m$ mag. Moment}
+            \eq{\vec{\tau} = \vec{m} \times \vec{B}}
+        \end{formula}
+
+        \begin{formula}{suceptibility}
+            \desc{Susceptibility}{}{}
+            \desc[german]{Suszeptibilität}{}{}
+            \eq{\chi_\text{m} = \pdv{M}{B} = \frac{\mu}{\mu_0} - 1 }
+        \end{formula}
+
+
+        \begin{formula}{poynting}
+            \desc{Poynting vector}{Directional energy flux or power flow of an electromagnetic field [$\si{\W\per\m^2}$]}{}
+            \desc[german]{Poynting-Vektor}{Gerichteter Energiefluss oder Leistungsfluss eines elektromgnetischen Feldes  [$\si{\W\per\m^2}$]}{}
+            \eq{\vec{S} = \vec{E} \times \vec{H}}
+        \end{formula}
+
+
+
+        \Subsection[
+            \eng{Induction}
+            \ger{Unduktion}
+            ]{induction}
+            \begin{formula}{farady_law}
+                \desc{Faraday's law of induction}{}{}
+                \desc[german]{Faradaysche Induktionsgesetz}{}{}
+                \eq{U_\text{ind} = -\odv{}{t} \PhiB = - \odv{}{t} \iint_A\vec{B} \cdot \d\vec{A}}
+            \end{formula}
+
+\Section[
+    \eng{Hall-Effect}
+    \ger{Hall-Effekt}
+    ]{hall}
+
+    \begin{formula}{cyclotron}
+        \desc{Cyclontron frequency}{}{}
+        \desc[german]{Zyklotronfrequenz}{}{}
+        \eq{\omega_\text{c} = \frac{e B}{\masse}}
+    \end{formula}
+    \TODO{Move}
+
+
+    \Subsection[
+        \eng{Classical Hall-Effect}
+        \ger{Klassischer Hall-Effekt}
+        ]{classic}
+        \begin{ttext}
+            \eng{Current flowing in $x$ direction in a conductor ($l \times b \times d$) with a magnetic field $B$ in $z$ direction leads to a hall voltage $U_\text{H}$ in $y$ direction.}
+            \ger{Fließt in einem Leiter ($l \times b \times d$) ein Strom in $x$ Richtung, während der Leiter von einem Magnetfeld $B$ in $z$-Richtung durchdrungen, wird eine Hallspannung $U_\text{H}$ in $y$-Richtung induziert.}
+        \end{ttext}
+        \begin{formula}{voltage}
+            \desc{Hall voltage}{}{$n$ charge carrier density}
+            \desc[german]{Hallspannung}{}{$n$ Ladungsträgerdichte}
+            \eq{U_\text{H} = \frac{I B}{ne d}}
+        \end{formula}
+
+        \begin{formula}{coefficient}
+            \desc{Hall coefficient}{}{}
+            \desc[german]{Hall-Koeffizient}{}{}
+            \eq{R_\text{H} = -\frac{Eg}{j_x Bg} = \frac{1}{ne} = \frac{\rho_{xy}}{B_z}}
+        \end{formula}
+
+        \begin{formula}{resistivity}
+            \desc{Resistivity}{}{}
+            \desc[german]{Spezifischer Widerstand}{}{}
+            \eq{\rho_{xx} &= \frac{\masse}{ne^2\tau} \\ \rho_{xy} &= \frac{B}{ne}}
+        \end{formula}
+
+
+    \Subsection[
+        \eng{Integer quantum hall effect}
+        \ger{Ganzahliger Quantenhalleffekt}
+        ]{quantum}  
+
+        \begin{formula}{conductivity}
+            \desc{Conductivity tensor}{}{}
+            \desc[german]{Leitfähigkeitstensor}{}{}
+            \eq{\sigma = \begin{pmatrix} \sigma_{xy} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix} }
+        \end{formula}
+
+        \begin{formula}{resistivity}
+            \desc{Resistivity tensor}{}{}
+            \desc[german]{Spezifischer Widerstands-tensor}{}{}
+            \eq{
+                \rho = \sigma^{-1}
+                % \sigma = \begin{pmatrix} \sigma_{xy} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix} }
+            }
+        \end{formula}
+
+        \begin{formula}{resistivity}
+            \desc{Resistivity}{}{$\nu \in \mathbb{Z}$}
+            \desc[german]{Spezifischer Hallwiderstand}{}{$\nu \in \mathbb{Z}$}
+            \eq{\rho_{xy} = \frac{2\pi\hbar}{e^2} \frac{1}{\nu}}
+        \end{formula}
+
+        % \begin{formula}{qhe}
+        %     \desc{Integer quantum hall effect}{}{}
+        %     \desc[german]{Ganzahliger Quanten-Hall-Effekt}{}{}
+        %     \fig{img/qhe-klitzing.jpeg}
+        % \end{formula}
+    
+
+    \TODO{sort}
+    \begin{formula}{impedance_c}
+        \desc{Impedance of a capacitor}{}{}
+        \desc[german]{Impedanz eines Kondesnators}{}{}
+        \eq{Z_{C} = \frac{1}{i\omega C}}
+    \end{formula}
+
+    \begin{formula}{impedance_l}
+        \desc{Impedance of an inductor}{}{}
+        \desc[german]{Impedanz eines Induktors}{}{}
+        \eq{Z_{L} = i\omega L}
+    \end{formula}
+
+    \TODO{impedance addition for parallel / linear}
+
+\Section[
+    \eng{Dipole-stuff}
+    \ger{Dipol-zeug}
+]{dipole}
+
+    \begin{formula}{poynting}
+        \desc{Dipole radiation Poynting vector}{}{}
+        \desc[german]{Dipolsrahlung Poynting-Vektor}{}{}
+        \eq{\vec{S} = \left(\frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\right)\frac{\sin^2\theta}{r^2} \vec{r}}
+    \end{formula}
+
+    \begin{formula}{power}
+        \desc{Time-average power}{}{}
+        \desc[german]{Zeitlich mittlere Leistung}{}{}
+        \eq{P = \frac{\mu_0\omega^4 p_0^2}{12\pi c}}
+    \end{formula}

src/environments.tex

@@ -0,0 +1,213 @@
+\def\descwidth{0.3\textwidth}
+\def\eqwidth{0.6\textwidth}
+
+% [1]: minipage width
+% 2: fqname of name
+% 3: fqname of a translation that holds the explanation
+\newcommand{\NameWithExplanation}[3][\descwidth]{
+    \begin{minipage}{#1}
+        \iftranslation{#2}{
+            \raggedright
+            \gt{#2}
+        }{} 
+        \iftranslation{#3}{
+           \\ {\color{darkgray} \gt{#3}}
+        }{}
+    \end{minipage}
+}
+
+% [1]: minipage width
+% 2: content
+% 3: fqname of a translation that holds the explanation
+\newcommand{\ContentBoxWithExplanation}[3][\eqwidth]{
+    \fbox{
+    \begin{minipage}{#1}
+        % \vspace{-\baselineskip}  % remove the space that comes from starting a new paragraph
+        #2
+        \noindent\iftranslation{#3}{
+            \begingroup
+                \color{darkgray} 
+                \gt{#3}
+                % \edef\temp{\GT{#1_defs}}
+                % \expandafter\StrSubstitute\expandafter{\temp}{:}{\\}
+            \endgroup
+        }{}
+        % \vspace{-\baselineskip}  % remove the space that comes from starting a new paragraph
+    \end{minipage}
+    }
+
+}
+
+% 1: fqname, optional with #1_defs and #1_desc defined
+% 2: content
+\newcommand{\NameLeftContentRight}[2]{
+    \par\noindent\ignorespaces
+    % \textcolor{gray}{\hrule}
+    \vspace{0.5\baselineskip}
+    \NameWithExplanation[\descwidth]{#1}{#1_desc}
+    \hfill
+    \ContentBoxWithExplanation[\eqwidth]{#2}{#1_defs}
+    \textcolor{lightgray}{\hrule}
+    \vspace{0.5\baselineskip}
+    % \par
+    % \hrule
+}
+
+\newcommand{\insertEquation}[2]{
+    \NameLeftContentRight{#1}{
+        \begin{align}
+            \label{eq:\fqname:#1}
+            #2
+        \end{align}
+    }
+}
+
+\newcommand{\insertFLAlign}[2]{  % eq name, #cols, eq
+    \NameLeftContentRight{#1}{%
+        \begin{flalign}%
+            % dont place label when one is provided
+            % \IfSubStringInString{label}\unexpanded{#3}{}{
+            %     \label{eq:#1}
+            % }
+            #2%
+        \end{flalign}
+    }
+}
+
+\newcommand{\insertAlignedAt}[3]{  % eq name, #cols, eq
+    \NameLeftContentRight{#1}{%
+        \begin{alignat}{#2}%
+            % dont place label when one is provided
+            % \IfSubStringInString{label}\unexpanded{#3}{}{
+            %     \label{eq:#1}
+            % }
+            #3%
+        \end{alignat}
+    }
+}
+
+\newcommand\luaexpr[1]{\directlua{tex.sprint(#1)}}
+% 1: fqname
+% 2: file path
+% 3: equation
+\newcommand{\insertEquationWithFigure}[4][0.55]{
+    \par\noindent\ignorespaces
+    % \textcolor{gray}{\hrule}
+    \vspace{0.5\baselineskip}
+    \begin{minipage}{#1\textwidth}
+        \NameWithExplanation[\textwidth]{#2}{#2_desc}
+        % TODO: why is this ignored
+        \vspace{1.0cm}
+        % TODO: fix box is too large without 0.9
+        \ContentBoxWithExplanation[0.90\textwidth]{
+            \begin{align}
+                \label{eq:\fqname:#2}
+                #4
+            \end{align}
+        }{#2_defs}
+    \end{minipage}
+    \hfill
+    \begin{minipage}{\luaexpr{1.0-#1}\textwidth}
+        \begin{figure}[H]
+            \centering
+            \includegraphics[width=\textwidth]{#3}
+            \label{fig:\fqname:#2}
+        \end{figure}
+    \end{minipage}
+    \textcolor{lightgray}{\hrule}
+    \vspace{0.5\baselineskip}
+
+}
+
+\newenvironment{formula}[1]{
+    % key
+    \newcommand{\desc}[4][english]{
+        % language, name, description, definitions
+        \dt[#1]{##1}{##2}
+        \ifblank{##3}{}{\dt[#1_desc]{##1}{##3}}
+        \ifblank{##4}{}{\dt[#1_defs]{##1}{##4}}
+    }
+    \newcommand{\eq}[1]{
+        \insertEquation{#1}{##1}
+    }
+    \newcommand{\eqAlignedAt}[2]{
+        \insertAlignedAt{#1}{##1}{##2}
+    }
+    \newcommand{\eqFLAlign}[1]{
+        \insertFLAlign{#1}{##1}
+    }
+    \newcommand{\figeq}[2]{
+        \insertEquationWithFigure{#1}{##1}{##2}
+    }
+    \newcommand{\content}[1]{
+        \NameLeftContentRight{#1}{##1}
+    }
+}{\ignorespacesafterend}
+
+
+\newenvironment{quantity}[5]{
+    % key, symbol, si unit, si base units, comment (key to translation)
+    \newcommand{\desc}[3][english]{
+        % language, name, description
+        \DT[qty:#1]{}{##1}{##2}
+        \ifblank{##3}{}{\DT[qty:#1_desc]{##1}{##3}}
+    }
+    \newcommand{\eq}[1]{
+        \insertEquation{#1}{##1}
+    }
+    \newcommand{\eqAlignedAt}[2]{
+        \insertAlignedAt{#1}{##1}{##2}
+    }
+    \newcommand{\eqFLAlign}[1]{
+        \insertFLAlign{#1}{##1}
+    }
+
+    \edef\qtyname{#1}
+    \edef\qtysign{#2}
+    \edef\qtyunit{#3}
+    \edef\qtybaseunits{#4}
+    \edef\qtycomment{#5}
+}
+{
+    Quantity: \expandafter\GT\expandafter{qty:\qtyname}: \GT{qty:\qtyname_desc} \\
+    $\qtysign$ $[\SI{\qtyunit}] = [\SI{\qtybaseunits}]$ - \qtycomment \\
+    \ignorespacesafterend
+}
+    \def\distrightwidth{0.45\textwidth}
+    \def\distleftwidth{0.45\textwidth}
+    
+    % Table for distributions
+    % create entries for parameters using \disteq
+    \newenvironment{distribution}[0]{
+        % 1: param name (translation key)
+        % 2: math
+        \newcommand{\disteq}[2]{
+            % add links to some names
+            \directlua{
+                local cases = {
+                    pdf         = "eq:pt:distributions:pdf",
+                    pmf         = "eq:pt:distributions:pdf",
+                    cdf         = "eq:pt:distributions:cdf",
+                    mean        = "eq:pt:mean",
+                    variance    = "eq:pt:variance"
+                }
+                if cases["\luaescapestring{#1}"] \string~= nil then
+                    tex.sprint("\\hyperref["..cases["\luaescapestring{#1}"].."]{\\GT{#1}}")
+                else
+                    tex.sprint("\\GT{#1}")
+                end
+            }
+            & #2 \\ \hline
+        }
+        \hfill
+        \begin{minipage}{\distrightwidth}
+            \begingroup
+                \setlength{\tabcolsep}{0.9em}  % horizontal
+                \renewcommand{\arraystretch}{2}  % vertical
+                \begin{tabular}{|l|>{$\displaystyle}c<{$}|}
+                \hline
+    }{
+                \end{tabular}
+            \endgroup
+        \end{minipage}
+    }

src/geometry.tex

@@ -1,8 +1,7 @@
-
 \Part[
-    \eng{Analysis}
+    \eng{Geometry}
-    \ger{Analysis}
+    \ger{Geometrie}
-    ]{ana}
+    ]{geo}
 
 \Section[
     \eng{Trigonometry}
@@ -18,14 +17,14 @@
 \begin{formula}{sine}
     \desc{Sine}{}{}
     \desc[german]{Sinus}{}{}
-    \eq{\sin(x) &= \sum_{n=0}^{\infty} \frac{x^{(2n+1)}}{(2n+1)!} \\
+    \eq{\sin(x) &= \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{(2n+1)}}{(2n+1)!} \\
                 &= \frac{e^{ix}-e^{-ix}}{2i}}
 \end{formula}
 
 \begin{formula}{cosine}
     \desc{Cosine}{}{}
     \desc[german]{Kosinus}{}{}
-    \eq{\cos(x) &= \sum_{n=0}^{\infty} \frac{x^{(2n)}}{(2n)!} \\
+    \eq{\cos(x) &= \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{(2n)}}{(2n)!} \\
                 &= \frac{e^{ix}+e^{-ix}}{2}}
 \end{formula}
 
@@ -72,6 +71,12 @@
         }
     \end{formula}
 
+    \begin{formula}{name}
+        \desc{}{}{$\tan\theta = b$}
+        \desc[german]{}{}{$\tan\theta = b$}
+        \eq{\cos x + b\sin x = \sqrt{1 + b^2}\cos(x-\theta)}
+    \end{formula}
+
 
     \Subsection[
         \eng{Table of values}

src/linalg.tex

@@ -0,0 +1,118 @@
+\def\id{\mathbb{1}}
+
+\Part[
+    \eng{Linear algebra}
+    \ger{Lineare Algebra}
+    ]{linalg}
+
+    \Section[
+        \eng{Determinant}
+        \ger{Determinante}
+        ]{determinant}
+        \begin{formula}{2x2}
+            \desc{2x2 matrix}{}{}
+            \desc[german]{2x2 Matrix}{}{}
+            \eq{\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = a\,d -c\,b}
+        \end{formula}   
+        \begin{formula}{3x3}
+            \desc{3x3 matrix (Rule of Sarrus)}{}{}
+            \desc[german]{3x3 Matrix (Regel von Sarrus)}{}{}
+            \eq{\det \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = 
+                a\,e\,i + b\,f\,g + c\,d\,h - g\,e\,c - h\,f\,a-i\,d\,b}
+        \end{formula}   
+        \begin{formula}{leibniz}
+            \desc{Leibniz formla}{}{}
+            \desc[german]{Leibniz-Formel}{}{}
+            \eq{\det(A) = \sum_{\sigma \in S_n}\Big(\sgn(\sigma) \prod_{i=1}^n a_{i,\sigma(i)}\Big)}
+        \end{formula}
+        
+        \begin{formula}{product}
+            \desc{Product}{}{}
+            \desc[german]{Produkt}{}{}
+            \eq{\det(A \, B) = \det(A) \det(B)}
+        \end{formula}
+        \begin{formula}{inverse}
+            \desc{Inverse}{}{}
+            \desc[german]{Inverse}{}{}
+            \eq{\det(A^{-1}) = \det(A)^{-1}}
+        \end{formula}
+        \begin{formula}{transposed}
+            \desc{Transposed}{}{}
+            \desc[german]{Transponiert}{}{}
+            \eq{\det(A^{\T}) = \det(A)}
+        \end{formula}
+
+
+    \Section[
+        
+    ]{zeug}
+
+        \begin{formula}{unitary}
+            \desc{Unitary matrix}{}{}
+            \desc[german]{Unitäre Matrix}{}{}
+            \eq{U ^\dagger U = \id}
+        \end{formula}
+
+        \begin{formula}{svd}
+            \desc{Singular value decomposition}{Factorization of complex matrices through rotating \rightarrow rescaling \rightarrow rotation.}{$A$: $m\times n$ matrix, $U$: $m\times m$ unitary matrix, $\Lambda$: $m\times n$ rectangular diagonal matrix with non-negative numbers on the diagonal, $V$: $n\times n$ unitary matrix}
+            \desc[german]{Singulärwertzerlegung}{Faktorisierung einer reellen oder komplexen Matrix durch Rotation \rightarrow Skalierung \rightarrow Rotation.}{}
+            \eq{A = U \Lambda V}
+        \end{formula}
+
+        \begin{formula}{rotation2}
+            \desc{2D rotation matrix}{}{}
+            \desc[german]{2D Rotationsmatrix}{}{}
+            \eq{R = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} }
+        \end{formula}
+
+        \begin{formula}{rotation3}
+            \desc{3D rotation matrices}{}{}
+            \desc[german]{3D Rotationsmatrizen}{}{}
+            \eq{
+                R_x &= \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix}  \\
+                R_y &= \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\  -\sin\theta & 0 & \cos\theta \end{pmatrix}  \\
+                R_z &= \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} 
+            }
+        \end{formula}
+
+        \begin{formula}{rotation_properties}
+            \desc{Properites of rotation matrices}{}{$n$ dimension, SO($n$) special othogonal group}
+            \desc[german]{Eigenschaften von Rotationsmatrizen}{}{$n$ Dimension, SO($n$) spezielle orthognale Gruppe}
+            \eq{
+                R^{\T} = R^{-1} \\
+                \det R = 1 \\
+                R \in \text{SO($n$)}
+            }
+        \end{formula}
+
+
+    \Section[
+        \eng{Eigenvalues}
+        \ger{Eigenwerte}
+    ]{eigen}
+        \begin{formula}{values}
+            \desc{Eigenvalue equation}{}{$\lambda$ eigenvalue, $v$ eigenvector}
+            \desc[german]{Eigenwert-Gleichung}{}{$\lambda$ Eigenwert, $v$ Eigenvektor}
+            \eq{A v = \lambda v}
+        \end{formula}
+
+        \begin{formula}{charac_poly}
+            \desc{Characteristic polynomial}{Zeros are the eigenvalues of $A$}{}
+            \desc[german]{Charakteristisches Polynom}{Nullstellen sind die Eigenwerte von $A$}{}
+            \eq{\chi_A = \det(A - \lambda \id) \overset{!}{=} 0}
+        \end{formula}
+
+        \begin{formula}{kramer}
+            \desc{Kramer's theorem}{If $H$ is invariant under $T$ and $\ket{\psi}$ is an eigenstate of $H$, then $T\ket{\psi}$ is also am eigenstate of $H$}{}
+            \desc[german]{Kramers-Theorem}{Wenn $H$ invariant unter $T$ ist und $\ket{\psi}$ ein Eigenzustand von $H$ ist, dann ist $T \ket{\psi}$ auch ein Eigenzustand von $H$}{}
+            \eq{T H T^\dagger = H \quad\wedge\quad H \ket{\psi} = E \ket{\psi} \quad\Rightarrow\quad H T \ket{\psi} = E T \ket{\psi}}
+        \end{formula}
+
+        \begin{formula}{eigendecomp}
+            \desc{Eigendecomposition}{}{$A$ diagonalizable, columns of $V$ are eigenvectors $v_i$, $\Lambda$ diagonal matrix with eigenvalues $\lambda_i$ on the diagonal}
+            \desc[german]{Eigenwertzerlegung}{}{$A$ diagonalisierbar, Spalten von $V$ sind die Eigenvektoren $v_i$, $\Lambda$ Diagonalmatrix mit Eigenwerten $\lambda_$ auf der Diagonalen}
+            \eq{A = V \Lambda V^{-1}}
+        \end{formula}
+
+        \TODO{Jordan stuff, blockdiagonal matrices, permutations, skalar product lapacescher entwicklungssatz maybe, cramers rule}
+

src/macros.tex

@@ -0,0 +1,47 @@
+\def\Grad{\vec{\nabla}}
+\def\Div{\vec{\nabla} \cdot}
+\def\Rot{\vec{\nabla} \times}
+\def\vecr{\vec{r}}
+
+\def\kB{k_\text{B}}
+\def\EFermi{E_\text{F}}
+
+\def\masse{m_\textrm{e}}
+
+\def\R{\mathbb{R}}
+\def\C{\mathbb{C}}
+\def\Z{\mathbb{Z}}
+\def\N{\mathbb{N}}
+
+\def\Four{\mathcal{F}}  % Fourier transform
+\def\Lebesgue{\mathcal{L}}  % Lebesgue
+\def\Order{\mathcal{O}} 
+
+% complex, may be changed later to idot or upright...
+\def\I{i}
+
+\def\sdots{\,\dots\,}
+\def\qdots{\quad\dots\quad}
+\def\qRarrow{\quad\Rightarrow\quad}
+
+\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
+\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor}
+\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil}
+
+\DeclareMathOperator{\e}{e}
+\DeclareMathOperator{\T}{T}  % transposed
+\DeclareMathOperator{\sgn}{sgn}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\const}{const}
+\DeclareMathOperator{\erf}{erf}
+% diff, for integrals and stuff
+% \DeclareMathOperator{\dd}{d}
+\renewcommand*\d{\mathop{}\!\mathrm{d}}
+
+% functions with paranthesis
+\newcommand\CmdWithParenthesis[2]{
+    #1\left(#2\right)
+}
+\newcommand\Exp[1]{\CmdWithParenthesis{\exp}{#1}}
+\newcommand\Sin[1]{\CmdWithParenthesis{\sin}{#1}}
+\newcommand\Cos[1]{\CmdWithParenthesis{\cos}{#1}}

src/main.tex

@@ -3,17 +3,30 @@
 \usepackage[german]{babel}
 \usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
 \usepackage{mathtools}
+\usepackage{MnSymbol}  % for >>> \ggg sign
 % \usepackage{esdiff} % derivatives
 % esdiff breaks when taking \dot{q} has argument
 \usepackage{derivative}
+\usepackage{bbold}  % \mathbb font
 \usepackage{braket}
 \usepackage{graphicx}
 \usepackage{etoolbox}
+\usepackage{expl3} % switch case and other stuff
 \usepackage{substr}
 \usepackage{xcolor}
 \usepackage{float}
+\usepackage{tikz} % drawings
+\usetikzlibrary{decorations.pathmorphing}
+\usetikzlibrary{calc}
+\usepackage{circuitikz}
 \usepackage[hidelinks]{hyperref}
 \usepackage{subcaption}
+\usepackage[shortlabels]{enumitem} % easily change enum symbols to i), a. etc
+\usepackage{colortbl}  % color table
+\usepackage{tabularx}  % bravais table
+\usepackage{adjustbox}
+\usepackage{multirow}  % for superconducting qubit table
+\usepackage{hhline}    % for superconducting qubit table
 \hypersetup{colorlinks   = true,    % Colours links instead of ugly boxes
             urlcolor     = blue,    % Colour for external hyperlinks
             linkcolor    = cyan,    % Colour of internal links
@@ -26,13 +39,11 @@
 \sisetup{per-mode = power}
 \sisetup{exponent-product=\ensuremath{\cdot}}
 
-\DeclarePairedDelimiter\abs{\lvert}{\rvert}
-\DeclareMathOperator{\e}{e}
-\DeclareMathOperator{\d}{d}
 
 \usepackage{translations}
 
 \newcommand{\TODO}[1]{{\color{red}TODO:#1}}
+\newcommand{\ts}{\textsuperscript}
 
 % put an explanation above an equal sign
 % [1]: equality sign (or anything else)
@@ -42,6 +53,7 @@
 \newcommand{\explOverEq}[2][=]{%
     \overset{\substack{\mathrlap{\text{\hspace{-1em}#2}}\\\downarrow}}{#1}}
     
+
 %
 % TRANSLATION COMMANDS
 %
@@ -91,14 +103,14 @@
 \newcommand{\ger}[2][\fqname]{\dt[#1]{german}{#2}}
 \newcommand{\eng}[2][\fqname]{\dt[#1]{english}{#2}}
 
-\newcommand{\GER}[2][\fqname]{\DT[#1]{german}{#2}}
+\newcommand{\Ger}[2][\fqname]{\DT[#1]{german}{#2}}
-\newcommand{\ENG}[2][\fqname]{\DT[#1]{english}{#2}}
+\newcommand{\Eng}[2][\fqname]{\DT[#1]{english}{#2}}
 
 % use this to define text in different languages for the key <env arg>
 % the translation for <env arg> when the environment ends.
 % (temporarily change fqname to the \fqname:<env arg> to allow
 % the use of \eng and \ger without the key parameter)
-\newenvironment{ttext}[1]{
+\newenvironment{ttext}[1][desc]{
     \edef\realfqname{\fqname}
     \edef\fqname{\fqname:#1}
 }{
@@ -148,101 +160,27 @@
     \label{sec:\fqname}
 }
 
+\newcommand{\fqEqRef}[1]{
+    \hyperref[eq:#1]{\GT{#1}}
+}
+\newcommand{\fqSecRef}[1]{
+    \hyperref[sec:#1]{\GT{#1}}
+}
+
 \usepackage{xstring}
 
-\newcommand{\insertEquationLine}[2]{
-    \par\noindent\ignorespaces
-    % \textcolor{gray}{\hrule}
-    \vspace{0.5\baselineskip}
-    % \fbox{
-    \begin{minipage}{0.3\textwidth}
-        \iftranslation{#1}{
-            \raggedright
-            \gt{#1}
-        }{} 
-        \iftranslation{#1_desc}{
-           \\ {\color{darkgray} \gt{#1_desc}}
-        }{}
-    \end{minipage}
-    % }
-    \hfill
-    \fbox{
-    \begin{minipage}{0.6\textwidth}
-        % \vspace{-\baselineskip}  % remove the space that comes from starting a new paragraph
-        #2 %
-        \noindent\iftranslation{#1_defs}{
-            \begingroup
-                \color{darkgray} 
-                \gt{#1_defs}
-                % \edef\temp{\GT{#1_defs}}
-                % \expandafter\StrSubstitute\expandafter{\temp}{:}{\\}
-            \endgroup
-        }{}
-        % \vspace{-\baselineskip}  % remove the space that comes from starting a new paragraph
-    \end{minipage}
-    }
-    \textcolor{lightgray}{\hrule}
-    \vspace{0.5\baselineskip}
-    % \par
-    % \hrule
-}
 
-\newcommand{\insertEquation}[2]{
+\input{circuit.tex}
-    \insertEquationLine{#1}{
-        \begin{align}
-            \label{eq:\fqname:#1}
-            #2
-        \end{align}
-    }
-}
 
-\newcommand{\insertFLAlign}[2]{  % eq name, #cols, eq
+% some translations
-    \insertEquationLine{#1}{%
-        \begin{flalign}%
-            % dont place label when one is provided
-            % \IfSubStringInString{label}\unexpanded{#3}{}{
-            %     \label{eq:#1}
-            % }
-            #2%
-        \end{flalign}
-    }
-}
-
-\newcommand{\insertAlignedAt}[3]{  % eq name, #cols, eq
-    \insertEquationLine{#1}{%
-        \begin{alignat}{#2}%
-            % dont place label when one is provided
-            % \IfSubStringInString{label}\unexpanded{#3}{}{
-            %     \label{eq:#1}
-            % }
-            #3%
-        \end{alignat}
-    }
-}
-
-\newenvironment{formula}[1]{
-    % key
-    \newcommand{\desc}[4][english]{
-        % language, name, description, definitions
-        \dt[#1]{##1}{##2}
-        \ifblank{##3}{}{\dt[#1_desc]{##1}{##3}}
-        \ifblank{##4}{}{\dt[#1_defs]{##1}{##4}}
-    }
-    \newcommand{\eq}[1]{
-        \insertEquation{#1}{##1}
-    }
-    \newcommand{\eqAlignedAt}[2]{
-        \insertAlignedAt{#1}{##1}{##2}
-    }
-    \newcommand{\eqFLAlign}[1]{
-        \insertFLAlign{#1}{##1}
-    }
-}{\ignorespacesafterend}
 
 \title{Formelsammlung}
 \author{Matthias Quintern}
 \date{\today}
 
+\input{macros.tex}
+\input{environments.tex}
+
 \begin{document}
 
 \maketitle
@@ -250,55 +188,29 @@
 \newpage
 \setcounter{page}{1}
 
-% \nuwcommand{\eq}[4][desc]{
-%     \vspace*{0.1cm}
-%     \begin{minipage}{0.3\textwidth}
-%         \raggedright
-%         .#2 \\
-%         \ifstrequal{#1}{desc}{}{
-%             {\color{gray}#1}
-%         }
-%     \end{minipage}
-%     \begin{minipage}{0.7\textwidth}
-%         \begin{align}
-%             \label{eq:#4}
-%             #3
-%         \end{align}
-%     \end{minipage}
-%     \newline
-% }
-
-
-% \newcommand{\eqd}[5][desc]{
-%     \vspace*{0.1cm}
-%     \begin{minipage}{0.3\textwidth}
-%         \raggedright
-%         .#2 \\
-%         \ifstrequal{#1}{desc}{}{
-%             {\color{gray}#1}
-%         }
-%     \end{minipage}
-%     \begin{minipage}{0.7\textwidth}
-%         \begin{align}
-%             \label{eq:#5}
-%             #3
-%         \end{align}
-%         {\color{gray}with: #4}
-%     \end{minipage}
-%     \newline
-% }
-\IfSubStringInString{lol}{lol\frac{asdsd}{lol} & l}{YES!}{
-    NO!
-}
 \input{translations.tex}
 
-\input{trigonometry.tex}
+\input{linalg.tex}
+
+\input{geometry.tex}
+
+\input{analysis.tex}
+
+\input{probability_theory.tex}
 
 \input{mechanics.tex}
 
+\input{statistical_mechanics.tex}
+
+\input{electrodynamics.tex}
+    
 \input{quantum_mechanics.tex}
 \input{atom.tex}
 
+\input{condensed_matter.tex}
+
+\input{topo.tex}
+
 \input{quantum_computing.tex}
 
 \input{many-body-simulations.tex}

src/mechanics.tex

@@ -8,7 +8,7 @@
     \eng{Lagrange formalism}
     \ger{Lagrange Formalismus}
 ]{lagrange}
-    \begin{ttext}{desc}
+    \begin{ttext}[desc]
         \eng{The Lagrange formalism is often the most simple approach the get the equations of motion,
             because with suitable generalied coordinates obtaining the Lagrange function is often relatively easy.
         }
@@ -16,7 +16,7 @@
             da das Aufstellen der Lagrange-Funktion mit geeigneten generalisierten Koordinaten oft relativ einfach ist.
         }
     \end{ttext}
-    \begin{ttext}{generalized_coords}
+    \begin{ttext}[generalized_coords]
         \eng{
             The generalized coordinates are choosen so that the cronstraints are automatically fullfilled.
             For example, the generalized coordinate for a 2D pendelum is $q=\varphi$, with $\vec{x} = \begin{pmatrix} \cos\varphi \\ \sin\varphi \end{pmatrix}$.

src/probability_theory.tex

@@ -0,0 +1,198 @@
+\Part[
+    \eng{Probability theory}
+    \ger{Wahrscheinlichkeitstheorie}
+    ]{pt}
+
+    \begin{formula}{mean}
+        \desc{Mean}{}{}
+        \desc[german]{Mittelwert}{}{}
+        \eq{\braket{x} = \int w(x)\, x\, \d x}
+    \end{formula}
+
+    \begin{formula}{variance}
+        \desc{Variance}{}{}
+        \desc[german]{Varianz}{}{}
+        \eq{\sigma^2 = (\Delta \hat{x})^2 = \braket{\hat{x}^2} - \braket{\hat{x}}^2 = \braket{(x - \braket{x})^2}}
+    \end{formula}
+    
+    \begin{formula}{std_deviation}
+        \desc{Standard deviation}{}{}
+        \desc[german]{Standardabweichung}{}{}
+        \eq{\sigma = \sqrt{(\Delta x)^2}}
+    \end{formula}
+
+    \begin{formula}{median}
+        \desc{Median}{Value separating lower half from top half}{$x$ dataset with $n$ elements}
+        \desc[german]{Median}{Teilt die untere von der oberen Hälfte}{$x$ Reihe mit $n$ Elementen}
+        \eq{
+            \textrm{med}(x) = \left\{ \begin{array}{ll} x_{(n+1)/2} & \text{$n$ \GT{odd}} \\ \frac{x_{(n/2)}+x_{((n/2)+1)}}{2} & \text{$n$ \GT{even}} \end{array} \right
+        }
+    \end{formula}
+
+    \begin{formula}{pdf}
+        \desc{Probability density function}{Random variable has density $f$. The integral gives the probability of $X$ taking a value $x\in[a,b]$.}{$f$ normalized: $\int_{-\infty}^\infty f(x) \d x= 1$}
+        \desc[german]{Wahrscheinlichkeitsdichtefunktion}{Zufallsvariable hat Dichte $f$. Das Integral gibt Wahrscheinlichkeit an, dass $X$ einen Wert $x\in[a,b]$ annimmt}{$f$ normalisiert $\int_{-\infty}^\infty f(x) \d x= 1$}
+        \eq{P([a,b]) := \int_a^b f(x) \d x}
+    \end{formula}
+
+    \begin{formula}{cdf}
+        \desc{Cumulative distribution function}{}{$f$ probability density function}
+        \desc[german]{Kumulative Verteilungsfunktion}{}{$f$ Wahrscheinlichkeitsdichtefunktion}
+        \eq{F(x) = \int_{-\infty}^x f(t) \d t}
+    \end{formula}
+
+\Section[
+    \eng{Distributions}
+    \ger{Verteilungen}
+    ]{distributions}
+
+
+
+
+    \Subsubsection[
+        \eng{Gauß/Normal distribution}
+        \ger{Gauß/Normal-Verteilung}
+        ]{normal}
+        \begin{minipage}{\distleftwidth}
+            \begin{figure}[H]
+                \centering
+                \includegraphics[width=\textwidth]{img/distribution_gauss.pdf}
+            \end{figure}
+        \end{minipage}
+        \begin{distribution}
+            \disteq{parameters}{\mu \in \R,\quad \sigma^2 \in \R}
+            \disteq{support}{x \in \R}
+            \disteq{pdf}{\frac{1}{\sqrt{2\pi\sigma^2}}\exp \left(-\frac{(x-\mu)^2}{2\sigma^2}\right)}
+            \disteq{cdf}{\frac{1}{2}\left[1 + \erf \left(\frac{x-\mu}{\sqrt{2}\sigma}\right)\right]}
+            \disteq{mean}{\mu}
+            \disteq{median}{\mu}
+            \disteq{variance}{\sigma^2}
+        \end{distribution}
+
+        \begin{formula}{standard_normal_distribution}
+            \desc{Density function of the standard normal distribution}{$\mu = 0$, $\sigma = 1$}{}
+            \desc[german]{Dichtefunktion der Standard-Normalverteilung}{$\mu = 0$, $\sigma = 1$}{}
+            \eq{\varphi(x) = \frac{1}{\sqrt{2\pi}} \e^{-\frac{1}{2}x^2}}
+        \end{formula}
+
+    \Subsubsection[
+        \eng{Cauchys / Lorentz distribution}
+        \ger{Cauchy / Lorentz-Verteilung}
+        ]{cauchy}
+        \begin{minipage}{\distleftwidth}
+            \begin{figure}[H]
+                \centering
+                \includegraphics[width=\textwidth]{img/distribution_cauchy.pdf}
+            \end{figure}
+        \end{minipage}
+        \begin{distribution}
+            \disteq{parameters}{x_0 \in \R,\quad \gamma \in \R}
+            \disteq{support}{x \in \R}
+            \disteq{pdf}{\frac{1}{\pi\gamma\left[1+\left(\frac{x-x_0}{\gamma}\right)^2\right]}}
+            \disteq{cdf}{\frac{1}{\pi}\arctan\left(\frac{x-x_0}{\gamma}\right) + \frac{1}{2}}
+            \disteq{mean}{\text{\GT{undefined}}}
+            \disteq{median}{x_0}
+            \disteq{variance}{\text{\GT{undefined}}}
+        \end{distribution}
+
+        \begin{ttext}
+            \eng{Also known as \textbf{Cauchy-Lorentz distribution}, \textbf{Lorentz(ian) function}, \textbf{Breit-Wigner distribution}.}
+            \ger{Auch bekannt als \textbf{Cauchy-Lorentz Verteilung}, \textbf{Lorentz Funktion}, \textbf{Breit-Wigner Verteilung}.}
+        \end{ttext}
+
+
+    \Subsubsection[
+        \eng{Binomial distribution}
+        \ger{Binomialverteilung}
+        ]{binomial}
+        \begin{ttext}
+            \eng{For the number of trials going to infinity ($n\to\infty$), the binomial distribution converges to the   \hyperref[sec:pb:distributions::poisson]{poisson distribution}}
+            \ger{Geht die Zahl der Versuche gegen unendlich ($n\to\infty$), konvergiert die Binomualverteilung gegen die \hyperref[sec:pb:distributions::poisson]{Poissonverteilung}}
+        \end{ttext}
+        \begin{minipage}{\distleftwidth}
+            \begin{figure}[H]
+                \centering
+                \includegraphics[width=\textwidth]{img/distribution_binomial.pdf}
+            \end{figure}
+        \end{minipage}
+        \begin{distribution}
+            \disteq{parameters}{n \in \Z, \quad p \in [0,1],\quad q = 1 - p}
+            \disteq{support}{k \in \{0,\,1,\,\dots,\,n\}}
+            \disteq{pmf}{\binom{n}{k} p^k q^{n-k}}
+            % \disteq{cdf}{\text{regularized incomplete beta function}}
+            \disteq{mean}{np}
+            \disteq{median}{\floor{np} \text{ or } \ceil{np}}
+            \disteq{variance}{npq = np(1-p)}
+        \end{distribution}
+    \Subsubsection[
+        \eng{Poisson distribution}
+        \ger{Poissonverteilung}
+        ]{poisson}
+        \begin{minipage}{\distleftwidth}
+            \begin{figure}[H]
+                \centering
+                \includegraphics[width=\textwidth]{img/distribution_poisson.pdf}
+            \end{figure}
+        \end{minipage}
+        \begin{distribution}
+            \disteq{parameters}{\lambda \in (0,\infty)}
+            \disteq{support}{k \in \N}
+            \disteq{pmf}{\frac{\lambda^k \e^{-\lambda}}{k!}}
+            \disteq{cdf}{\e^{-\lambda} \sum_{j=0}^{\floor{k}} \frac{\lambda^j}{j!}}
+            \disteq{mean}{\lambda}
+            \disteq{median}{\approx\floor*{\lambda + \frac{1}{3} - \frac{1}{50\lambda}}}
+            \disteq{variance}{\lambda}
+        \end{distribution}
+                    
+
+    \Subsubsection[
+        \eng{Maxwell-Boltzmann distribution}
+        \ger{Maxwell-Boltzmann Verteilung}
+        ]{maxwell-boltzmann}
+        \begin{minipage}{\distleftwidth}
+            \begin{figure}[H]
+                \centering
+                \includegraphics[width=\textwidth]{img/distribution_maxwell-boltzmann.pdf}
+            \end{figure}
+        \end{minipage}
+        \begin{distribution}
+            \disteq{parameters}{a > 0}
+            \disteq{support}{x \in (0, \infty)}
+            \disteq{pdf}{\sqrt{\frac{2}{\pi}} \frac{x^2}{a^3} \exp\left(-\frac{x^2}{2a^2}\right)}
+            \disteq{cdf}{\erf \left(\frac{x}{\sqrt{2} a}\right) - \sqrt{\frac{2}{\pi}} \frac{x}{a} \exp\left({\frac{-x^2}{2a^2}}\right)}
+            \disteq{mean}{2a \frac{2}{\pi}}
+            \disteq{median}{}
+            \disteq{variance}{\frac{a^2(3\pi-8)}{\pi}}
+        \end{distribution}
+
+
+
+    % \begin{distribution}{maxwell-boltzmann}
+    %     \distdesc{Maxwell-Boltzmann distribution}{}
+    %     \distdesc[german]{Maxwell-Boltzmann Verteilung}{}
+    %     \disteq{parameters}{}
+    %     \disteq{pdf}{}
+    %     \disteq{cdf}{}
+    %     \disteq{mean}{}
+    %     \disteq{median}{}
+    %     \disteq{variance}{}
+    % \end{distribution}
+
+
+    \Subsection[
+        \eng{Central limit theorem}
+        \ger{Zentraler Grenzwertsatz}
+        ]{cls}  
+        \begin{ttext}
+            \eng{
+                Suppose $X_1, X_2, \dots$ is a sequence of independent and identically distributed random variables with $\braket{X_i} = \mu$ and $(\Delta X_i)^2 = \sigma^2 < \infty$.
+                As $N$ approaches infinity, the random variables $\sqrt{N}(\bar{X}_N - \mu)$ converge to a normal distribution $\mathcal{N}(0, \sigma^2)$.
+                \\ That means that the variance scales with $\frac{1}{\sqrt{N}}$ and statements become accurate for large $N$.
+            }
+            \ger{
+                Sei $X_1, X_2, \dots$ eine Reihe unabhängiger und gleichverteilter Zufallsvariablen mit $\braket{X_i} = \mu$ und $(\Delta X_i)^2 = \sigma^2 < \infty$.
+                Für $N$ gegen unendlich konvergieren die Zufallsvariablen $\sqrt{N}(\bar{X}_N - \mu)$ zu einer Normalverteilung $\mathcal{N}(0, \sigma^2)$.
+                \\ Das bedeutet, dass die Schwankung mit $\frac{1}{\sqrt{N}}$ wächst und Aussagen für große $N$ scharf werden.
+            }
+        \end{ttext}
+

src/quantum_computing.tex

@@ -1,7 +1,7 @@
 \Part[
     \eng{Quantum Computing}
     \ger{Quantencomputing}
-]{qubit}
+]{qc}
 
 \Section[
     \eng{Qubits}
@@ -36,38 +36,396 @@
     %     \item \gt{bitphaseflip}: $\hat{Y} = \sigma_y = \sigmaymatrix$
     %     \item \gt{phaseflip}: $\hat{Z} = \sigma_z = \sigmazmatrix$ \item \gt{hadamard}: $\hat{H} = \frac{1}{\sqrt{2}}(\hat{X}-\hat{Z}) = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$
     % \end{itemize}
-
 \Section[
-    \eng{Josephson Junction}
+    \eng{Superconducting qubits}
-    \ger{Josephson-Kontakt}
+    \ger{Supraleitende qubits}
-    ]{josephson_junction}
+    ]{scq}
-    \begin{ttext}{desc}
+
-        \eng{When two superconductors are separated by a thin isolator, Cooper pairs can tunnel through the insulator}
+    \Subsection[
-        \ger{Wenn zwei Supraleiter durch einen dünnen Isolator getrennt sind, können Cooper-Paare durch den Isolator tunneln.} 
+        \eng{Building blocks}
-    \end{ttext}
+        \ger{Bauelemente}
+        ]{elements}
+        \Subsubsection[
+            \eng{Josephson Junction}
+            \ger{Josephson-Kontakt}
+            ]{josephson_junction}
+            \begin{ttext}[desc]
+                \eng{When two superconductors are separated by a thin isolator, Cooper pairs can tunnel through the insulator. The Josephson junction is a non-linear inductor.}
+                \ger{Wenn zwei Supraleiter durch einen dünnen Isolator getrennt sind, können Cooper-Paare durch den Isolator tunneln. Der Josephson-Kontakt ist ein nicht-linearer Induktor.} 
+            \end{ttext}
+
+            \begin{formula}{hamiltonian}
+                \desc{Josephson-Hamiltonian}{}{}
+                \desc[german]{Josephson-Hamiltonian}{}{}
+                \eq{
+                    \hat{H}_\text{J} &= - \frac{E_\text{J}}{2} \sum_n [\ket{n}\bra{n+1} + \ket{n+1}\bra{n}]
+                }
+            \end{formula}
+
+            \begin{formula}{1st_josephson_relation}
+                \desc{1. Josephson relation}{Dissipationless supercurrent accros junction at zero applied voltage}{$I_\text{C}=\frac{2e}{\hbar}E_\text{J}$ critical current, $\delta$ phase difference accross junction}
+                \desc[german]{1. Josephson Gleichung}{Dissipationsloser Suprastrom durch die Kreuzung ohne angelegte Spannung}{$I_\text{C}=\frac{2e}{\hbar}E_\text{J}$ kritischer Strom, $\delta$ Phasendifferenz zwischen den Supraleitern}
+                \eq{\hat{I}\ket{\delta} = I_\text{C}\sin\delta \ket{\delta}}
+            \end{formula}
+
+            \begin{formula}{2nd_josephson_relation}
+                \desc{2. Josephson relation}{superconducting phase change is proportional to applied voltage}{$\varphi_0=\frac{\hbar}{2e}$ reduced flux quantum}
+                \desc[german]{2. Josephson Gleichung}{Supraleitende Phasendifferenz is proportional zur angelegten Spannung}{$\varphi_0=\frac{\hbar}{2e}$ reduziertes Flussquantum}
+                \eq{\odv{\hat{\delta}}{t}=\frac{1}{i\hbar}[\hat{H},\hat{\delta}] = -\frac{2eU}{i\hbar}[\hat{n},\hat{\delta}] = \frac{1}{\varphi_0} U}
+            \end{formula}
+
+        \Subsubsection[
+            \eng{SQUID}
+            \ger{SQUID}
+            ]{squid}
+            \ctikzsubcircuitdef{squidloop}{n, s, nw, ne, se, sw}{
+                % start at top
+                coordinate(#1-n)
+                (#1-n)
+                to ++(-1, 0) coordinate(#1-nw)
+                to[josephsoncap=$\phi_1$] ++(0,-2) coordinate(#1-sw)
+                to ++(1,0) coordinate(#1-s) to ++(1,0) coordinate(#1-se)
+                to[josephsoncap=$\phi_2$] ++(0,2) coordinate(#1-ne)
+                to ++(-1,0)
+                (#1-s)  % leave at bottom
+            }
+            \begin{formula}{circuit}
+                \desc{SQUID}{Superconducting quantum interference device, consists of parallel \hyperref{sec:qc:scq:josephson_junction}{josephson junctions}, can be used to measure extremely weak magnetic fields}{}
+                \desc[german]{SQUID}{Superconducting quantum interference device, besteht aus parralelen \hyperref{sec:qc:scq:josephson_junction}{Josephson Junctions} und kann zur Messung extrem schwacher Magnetfelder genutzt werden}{}
+                \content{
+                    \centering
+                    \begin{circuitikz}
+                        \draw (0, 0) \squidloop{loop}{};
+                    \end{circuitikz}
+                }
+            \end{formula}
+            \begin{formula}{hamiltonian}
+                \desc{Hamiltonian}{}{$\hat{\phi}$ phase difference across the junction}
+                \desc[german]{Hamiltonian}{}{$\hat{\phi}$ Phasendifferenz an einer Junction}
+                \eq{\hat{H} &= -E_{\text{J}1} \cos\hat{\phi}_{1} - E_{\text{J}2} \cos\hat{\phi}_{2}}
+            \end{formula}
+
+        \Subsection[
+            \eng{Josephson Qubit??}
+            \ger{TODO}
+            ]{josephson_qubit}
+            \begin{circuitikz}
+              \draw (0,0) to[capacitor] (0,2);
+              \draw (0,0) to (2,0);
+              \draw (0,2) to (2,2);
+              \draw (2,0) to[josephson] (2,2);
+
+              \draw[->] (3,1) -- (4,1);
+              \draw (5,0) to[josephsoncap=$C_\text{J}$] (5,2);
+            \end{circuitikz}
+            \TODO{Include schaltplan}
+
+            \begin{circuitikz}
+              \draw (0,0) to[sV=$V_\text{g}$] (0,2);
+              \draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
+              \draw (2,2) to (4,2);
+              \draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
+              \draw (4,0) to[capacitor=$C_C$] (4,2);
+              \draw (0,0) to (2,0);
+              \draw (2,0) to (4,0);
+            \end{circuitikz}
+
+            \begin{formula}{charging_energy}
+                \desc{Charging energy / electrostatic energy}{}{}
+                \desc[german]{Ladeenergie?}{}{}
+                \eq{E_\text{C} = \frac{(2e)^2}{C}}
+            \end{formula}
+
+            \begin{formula}{josephson_energy}
+                \desc{Josephson energy}{}{}
+                \desc[german]{Josephson-Energie?}{}{}
+                \eq{E_\text{J} = \frac{I_0 \phi_0}{2\pi}}
+            \end{formula}
+            \TODO{Was ist I0}
+
+            \begin{formula}{inductive_energy}
+                \desc{Inductive energy}{}{}
+                \desc[german]{Induktive Energie}{}{}
+                \eq{E_\text{L} = \frac{\varphi_0^2}{L}}
+            \end{formula}
+
+            \begin{formula}{gate_charge}
+                \desc{Gate charge}{or offset charge}{}
+                \desc[german]{Gate Ladung}{auch Offset charge}{}
+                \eq{n_\text{g}=\frac{C_g V_\text{g}}{2e}}
+            \end{formula}
+
+            \begin{formula}{anharmonicity}
+                \desc{Anharmonicity}{}{}
+                \desc[german]{Anharmonizität}{}{}
+                \eq{\alpha \coloneq \omega_{1\leftrightarrow 2} - \omega_{0\leftrightarrow 1}}
+            \end{formula}
+
+
+            \begin{table}[h!]
+            \centering
+                \begin{tabular}{  p{1cm} |p{1cm}||p{2.8cm}|p{2cm}|p{2cm}|p{2cm}|}
+                    \multicolumn{1}{c}{}& \multicolumn{1}{c}{} &\multicolumn{4}{c}{$E_L/(E_J-E_L)$} \\
+                    \cline{3-6}
+                    \multicolumn{1}{c}{} & & $0$ & $\ll$ 1 & $\sim 1$ & $\gg 1$\\
+                    \hhline{~|=====|}
+                    \multirow{4}{*}{$E_J/E_C$} & $\ll 1$ & cooper-pair box  &     &  & \\
+                    \cline{2-6}
+                    & $\sim 1$ & quantronium &   fluxonium  &    &\\
+                    \cline{2-6}
+                    & $\gg 1$ &transmon & & &  flux qubit\\
+                    \cline{2-6}
+                    & $\ggg 1$ & & & phase qubit & \\
+                    \cline{2-6}
+                    \end{tabular}
+                    \caption{``periodic table'' of superconducting quantum circuits}
+                    \label{Juncatalog}
+
+                \end{table}
+
+
+        \Subsection[
+            \eng{Cooper Pair Box (CPB) qubit}
+            \ger{Cooper Paar Box (QPB) Qubit}
+            ]{cpb}
+            \begin{ttext}
+                \eng{
+                    = voltage bias junction\\= charge qubit?
+                    \begin{itemize}
+                        \item large anharmonicity
+                        \item sensitive to charge noise
+                    \end{itemize}
+                }
+                \ger{}
+            \end{ttext}
+            \begin{circuitikz}
+              \draw (0,0) to[sV=$V_\text{g}$] (0,2);
+              % \draw (0,0) to (2,0);
+              \draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
+              \draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
+              \draw (0,0) to (2,0);
+            \end{circuitikz}
+
+
+            \TODO{maybe include graphic from page 48, intro to qed circuits}
+
+            \begin{formula}{hamiltonian}
+                \desc{Hamiltonian}{}{}
+                \desc[german]{Hamiltonian}{}{}
+                \eq{\hat{H} &= 4 E_C(\hat{n} - n_\text{g})^2 - E_\text{J} \cos\hat{\phi} \\
+                            &=\sum_n \left[4 E_C  (n-n_\text{g})^2 \ket{n}\bra{n} - \frac{E_\text{J}}{2}\ket{n}\bra{n+1}+\ket{n+1}\bra{n}\right] }
+            \end{formula}
+
+        \Subsection[
+            \eng{Transmon qubit}
+            \ger{Transmon Qubit}
+            ]{transmon}
+            \begin{formula}{circuit}
+                \desc{Transmon qubit}{
+                    Josephson junction with a shunt \textbf{capacitance}.
+                    \begin{itemize}
+                        \item charge noise insensitive
+                        \item small anharmonicity
+                    \end{itemize}
+            }{}
+            \desc[ger]{Transmon Qubit}{
+                    Josephson-Kontakt mit einem parallelen \textbf{kapzitiven Element}.
+                    \begin{itemize}
+                        \item Charge noise resilient
+                        \item Geringe Anharmonizität $\alpha$
+                    \end{itemize}
+                }{}
+                \content{
+                    \centering
+                    \begin{circuitikz}
+                        \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
+                        to[capacitor=$C_\text{g}$] ++(2,0)
+                        to[capacitor=$C_C$] ++(0,-3)
+                        to ++(-2,0);
+                        \draw (2,3) to ++(2,0) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to ++(0,-0.5) to ++(-2,0);
+                    \end{circuitikz}
+                    \\\TODO{Ist beim Transmon noch die Voltage source dran?}
+                }
+            \end{formula}
+
+            \begin{formula}{hamiltonian}
+                \desc{Hamiltonian}{}{}
+                \desc[german]{Hamiltonian}{}{}
+                \eq{\hat{H} &= 4 E_C\hat{n}^2 - E_\text{J} \cos\hat{\phi}}
+            \end{formula}
+
+            \Subsubsection[
+                \eng{Tunable Transmon qubit}
+                \ger{Tunable Transmon Qubit}
+                ]{tunable}
+                \begin{formula}{circuit}
+                    \desc{Frequency tunable transmon}{By using a \fqSecRef{qc:scq:elements:squid} instead of a \fqSecRef{qc:scq:elements:josephson_junction}, the qubit is frequency tunable through an external field}{}
+                    \desc[german]{}{Durch Nutzung eines \fqSecRef{qc:scq:elements:squid} anstatt eines \fqSecRef{qc:scq:elements:josephson_junction}s, ist die Frequenz des Qubits durch ein externes Magnetfeld einstellbar}{}
+                    \content{
+                        \centering
+                        \begin{circuitikz}
+                            \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
+                            to[capacitor=$C_\text{g}$] ++(2,0)
+                            to[capacitor=$C_C$] ++(0,-3)
+                            to ++(-2,0);
+                            \draw (2,3) to ++(3,0) to ++(0,-0.5) \squidloop{loop}{SQUID} to ++(0,-0.5) to ++(-3,0);
+                        \end{circuitikz}
+                    }
+                \end{formula}
+
+                \begin{formula}{energy}
+                    \desc{Josephson energy}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ asymmetry}
+                    \desc[german]{Josephson Energie}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ Asymmetrie}
+                    \eq{E_\text{J,eff}(\Phi_\text{ext}) = (E_\text{J1}+E_\text{J2}) \sqrt{\cos^2\left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right) + d^2 \sin \left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right)}}
+                \end{formula}
+                \begin{formula}{hamiltonian}
+                    \desc{Hamiltonian}{}{}
+                    \desc[german]{Hamiltonian}{}{}
+                    \eq{\hat{H} = 4E_C \hat{n}^2 - \frac{1}{2} E_\text{J,eff}(\Phi_\text{ext}) \sum_{n}\left[\ket{n}\bra{n+1} + \ket{n+1}\bra{n}\right]}
+                \end{formula}
+
+
+        \Subsection[
+            \eng{Flux qubit}
+            \ger{Flux Qubit}
+            ]{flux}
+            \TODO{TODO}
+            \begin{formula}{circuit}
+                \desc{Flux qubit / Persistent current qubit}{}{}
+                \desc[german]{Flux Qubit / Persistent current qubit}{}{}
+                \content{
+                    \centering
+                    \begin{circuitikz}
+                        \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
+                        to ++(2,0) coordinate(top1);
+                        \draw[color=gray] (top1) to[capacitor=$C_C$] ++(0,-3);
+                        \draw (top1) to ++(2,0) coordinate(top2) to ++(0,-0.5)
+                        to[josephsoncap=$C_\text{J}$] ++(-0,-2) to ++(0,-0.5) to ++(-2,0) to ++(-2,0);
+                        \draw (top2) to ++(2,0) to[cute inductor=$E_L$] ++(0,-3) to ++(-2,0);
+                        \node at (5,0.5) {$\Phi_\text{ext}$};
+                    \end{circuitikz}
+                    \\\TODO{Ist beim Fluxonium noch die Voltage source dran?}
+                }
+            \end{formula}
+
+
+        \Subsection[
+            \eng{Fluxonium qubit}
+            \ger{Fluxonium Qubit}
+            ]{fluxonium}
+            \begin{formula}{circuit}
+                \desc{Fluxonium qubit}{
+                    Josephson junction with a shunt \textbf{inductance}. Instead of having to tunnel, cooper pairs can move to the island via the inductance.
+                    The inductance consists of many parallel Josephson Junctions to avoid parasitic capacitances.
+                }{}
+                \desc[german]{Fluxonium Qubit}{
+                    Josephson-Kontakt mit einem parallelen \textbf{induktiven Element}.
+                    Anstatt zu tunneln, können die Cooper-Paare über das induktive Element auf die Insel gelangen.
+                    Das induktive Element besteht aus sehr vielen parallelen Josephson-Kontakten um parisitische Kapazitäten zu vermeiden.
+                }{}
+                \content{
+                    \centering
+                    \begin{circuitikz}
+                        \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
+                        to ++(2,0) coordinate(top1);
+                        \draw[color=gray] (top1) to[capacitor=$C_C$] ++(0,-3);
+                        \draw (top1) to ++(2,0) coordinate(top2) to ++(0,-0.5)
+                        to[josephsoncap=$C_\text{J}$] ++(-0,-2) to ++(0,-0.5) to ++(-2,0) to ++(-2,0);
+                        \draw (top2) to ++(2,0) to[cute inductor=$E_L$] ++(0,-3) to ++(-2,0);
+                        \node at (5,0.5) {$\Phi_\text{ext}$};
+                    \end{circuitikz}
+                    \\\TODO{Ist beim Fluxonium noch die Voltage source dran?}
+                }
+            \end{formula}
+
+            \def\temp{$E_\text{C} = \frac{(2e)^2}{2C}, E_\text{L} = \frac{\varphi_0^2}{2L}, \delta_\text{s} = \frac{\varphi_\text{s}}{\varphi_0}$}
+            \begin{formula}{hamiltonian}
+                \desc{Hamiltonian}{}{\temp}
+                \desc[german]{Hamiltonian}{}{\temp}
+                \eq{\hat{H} = 4E_\text{C} \hat{n}^2 - E_\text{J} \cos \hat{\delta} + E_\text{L}(\hat{\delta} - \delta_\text{s})^2}
+            \end{formula}
+
+            \begin{figure}[h]
+                \centering
+                \includegraphics[width=0.8\textwidth]{img/qubit_flux_onium.pdf}
+                \caption{img/}
+                \label{fig:img-}
+            \end{figure}
+
+
+    \Section[
+        \eng{Two-level system}
+        \ger{Zwei-Niveau System}
+        ]{stuff}
+
+        \begin{formula}{resonance_frequency}
+            \desc{Resonance frequency}{}{}
+            \desc[german]{Ressonanzfrequenz}{}{}
+            \eq{\omega_{21} = \frac{E_2 - E_1}{\hbar}}
+        \end{formula}
+        \TODO{sollte das nicht 10 sein?}
+
+        \begin{formula}{rabi_oscillation}
+            \desc{Rabi oscillations}{}{$\omega_{21}$ resonance frequency of the energy transition, $\Omega$ Rabi frequency}
+            \desc[german]{Rabi-Oszillationen}{}{$\omega_{21}$ Resonanzfrequenz des Energieübergangs, $\Omega$ Rabi-Frequenz}
+            \eq{\Omega_ TODO}
+        \end{formula}
+
+        \Subsection[
+            \eng{Ramsey interferometry}
+            \ger{Ramsey Interferometrie}
+            ]{ramsey}
+
+            \begin{ttext}
+                \eng{$\ket{0} \xrightarrow{\frac{\pi}{2}\,\text{pulse}}$ precession in $xy$ plane for time $\tau$ $\xrightarrow{\frac{\pi}{2}\,\text{pulse}}$ measurement}
+                \ger{q}
+            \end{ttext}
+
+    \Section[
+        \eng{Noise and decoherence}
+        \ger{Noise und Dekohärenz}
+        ]{noise}
+        \begin{formula}{long}
+            \desc{Longitudinal relaxation rate}{$\Gamma_{1\downarrow}$: $\ket{1}\rightarrow \ket{0}$ \\ $\Gamma_{1\uparrow}$: $\ket{0}\rightarrow \ket{1}$}{}
+            \desc[german]{Longitudinale Relaxationsrate}{$\Gamma_{1\downarrow}$: $\ket{1}\rightarrow \ket{0}$ \\ $\Gamma_{1\uparrow}$: $\ket{0}\rightarrow \ket{1}$}{}
+            \eq{\Gamma_1 = \frac{1}{T_1} = \Gamma_{1\uparrow} + \Gamma_{1\downarrow}}
+        \end{formula}
+
+        \begin{ttext}[long]
+            \eng{$\Gamma_{1\uparrow}$ is supressed because of detailed balance}
+            \ger{$\Gamma_{1\uparrow}$ ist unterdrückt wegen detailed balance}
+        \end{ttext}
+
+        \begin{formula}{dephasing}
+            \desc{Pure dephasing rate}{}{}
+            \desc[german]{Reine Phasenverschiebung}{}{}
+            \eq{\Gamma_\phi}
+        \end{formula}
+
+        \begin{formula}{trans}
+            \desc{Transversal relaxation rate}{}{}
+            \desc[german]{Transversale Relaxationsrate}{}{}
+            \eq{\Gamma_2 = \frac{1}{T_2} = \frac{\Gamma_1}{2} + \Gamma_\phi}
+        \end{formula}
+
+        \begin{formula}{bloch_redfield}
+            \desc{Bloch-Redfield density matrix}{2-level System weakly coupled to noise sources with short correlation time}{}
+            \desc[german]{Bloch-Redfield Dichtematrix}{2-Niveau System schwach an Noise Quellen mit kurzer Korrelationszeit gekoppelt}{}
+            \eq{\rho_\text{BR} = \begin{pmatrix} 1+(\abs{\alpha}^2-1)\e^{-\Gamma_1 t} & \alpha \beta^* \e^{-\Gamma_2 t} \\ 
+            \alpha^*\beta \e^{-\Gamma_2 t} & \abs{\beta}^2 \e^{-\Gamma_1 t} \end{pmatrix} }
+        \end{formula}
+
+        \begin{figure}[h]
+            \centering
+            \includegraphics[width=0.8\textwidth]{img/qubit_transmon.pdf}
+            \caption{Transmon and so TODO}
+            \label{fig:img-qubit_transmon-pdf}
+        \end{figure}
+        
+
+
         
-    \begin{formula}{hamiltonian}
-        \desc{Josephson-Hamiltonian}{}{}
-        \desc[german]{Josephson-Hamiltonian}{}{}
-        \eq{
-            \hat{H}_\text{J} &= - \frac{E_\text{J}}{2} \sum_n [\ket{n}\bra{n+1} + \ket{n+1}\bra{n}]
-        }
-    \end{formula}
     
-    \begin{formula}{1st_josephson_relation}
-        \desc{1. Josephson relation}{Dissipationless supercurrent accros junction at zero applied voltage}{$I_\text{C}=\frac{2e}{\hbar}E_\text{J}$ critical current, $\delta$ phase difference accross junction}
-        \desc[german]{1. Josephson Gleichung}{Dissipationsloser Suprastrom durch die Kreuzung ohne angelegte Spannung}{$I_\text{C}=\frac{2e}{\hbar}E_\text{J}$ kritischer Strom, $\delta$ Phasendifferenz zwischen den Supraleitern}
-        \eq{\hat{I}\ket{\delta} = I_\text{C}\sin\delta \ket{\delta}}
-    \end{formula}
 
-    \begin{formula}{2nd_josephson_relation}
-        \desc{2. Josephson relation}{superconducting phase change is proportional to applied voltage}{$\varphi_0=\frac{\hbar}{2e}$ reduced flux quantum}
-        \desc[german]{2. Josephson Gleichung}{Supraleitende Phasendifferenz is proportional zur angelegten Spannung}{$\varphi_0=\frac{\hbar}{2e}$ reduziertes Flussquantum}
-        \eq{\odv{\hat{\delta}}{t}=\frac{1}{i\hbar}[\hat{H},\hat{\delta}] = -\frac{2eU}{i\hbar}[\hat{n},\hat{\delta}] = \frac{1}{\varphi_0} U}
-    \end{formula}
 
-\Section[
-    \eng{Cooper Pair Box (CPB) qubit}
-    \ger{Cooper Paar Box (QPB) Qubit}
-    ]{cpb}
 

src/quantum_mechanics.tex

@@ -17,10 +17,10 @@
             \eng{Operators}
             \ger{Operatoren}
         ]{op}
-            \GER[row_vector]{Zeilenvektor}
+            \Ger[row_vector]{Zeilenvektor}
-            \GER[column_vector]{Spaltenvektor}
+            \Ger[column_vector]{Spaltenvektor}
-            \ENG[column_vector]{Column vector}
+            \Eng[column_vector]{Column vector}
-            \ENG[row_vector]{Row vector}
+            \Eng[row_vector]{Row vector}
             \begin{formula}{dirac_notation}
                 \desc{Dirac notation}{}{}
                 \desc[german]{Dirac-Notation}{}{}
@@ -75,12 +75,6 @@
                 \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}{}{}
@@ -116,25 +110,38 @@
             \begin{formula}{commutator}
                 \desc{Commutator}{}{}
                 \desc[german]{Kommutator}{}{}
-                \eq{[a,b] = ab - ba}
+                \eq{[A,B] = AB - BA}
             \end{formula}
 
             \begin{formula}{anticommutator}
                 \desc{Anticommutator}{}{}
                 \desc[german]{Antikommmutator}{}{}
-                \eq{\{a,b\} = ab + ba}
+                \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\}}
+                \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}
+                \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}
@@ -147,39 +154,44 @@
                 }
             \end{formula}
 
+    \Section[
+        \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}
+
         \Subsection[
-            \eng{Schrödinger equation}
+            \eng{Time evolution}
-            \ger{Schrödingergleichung}
+            \ger{Zeitentwicklug}
-        ]{schroedinger_equation}
+            ]{time}
-            \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}
@@ -187,6 +199,21 @@
                 \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}
@@ -217,20 +244,10 @@
                     }
                 \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}
+                ]{ehrenfest_theorem}
                 \GT{see_also} \ref{sec:qm:basics:schroedinger_equation:correspondence_principle}
                 \begin{formula}{ehrenfest_theorem}
                     \desc{Ehrenfesttheorem}{applies to both pictures}{}
@@ -244,15 +261,27 @@
                     \desc[german]{}{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}
+                % \eq{Time evolution}{\hat{H}\ket{\psi} = E\ket{\psi}}{sg_time}
+
+            \Subsection[
+                \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}
+                
+
 
     \Section[
         \eng{Pertubation theory}
         \ger{Störungstheorie}
-    ]{qm_pertubation}
+        ]{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.}
+        \begin{ttext}
-        \ger[desc]{Die folgenden Gleichungen gelten wenn $\hat{H_1}$ ausreichend klein ist und die $E_n^{(0)}$ Niveaus nicht entartet sind.}
+            \eng[desc]{The following holds true if the pertubation $\hat{H_1}$ is sufficently small and the $E^{(0)}_n$ levels are not degenerate.}
-        \gt{desc}
+            \ger[desc]{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}{}{}
@@ -289,6 +318,12 @@
         %     \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}
@@ -404,3 +439,64 @@
                 \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{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} \hyperref[eq:qm:other:RWS]{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}{RWS}
+            \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}
+
+

src/scripts/bloch_sphere.py

@@ -0,0 +1,2 @@
+import qutip as qt
+

src/scripts/distributions.py

@@ -0,0 +1,106 @@
+from numpy import fmax
+from plot import *
+
+
+def get_fig():
+    fig, ax = plt.subplots(figsize=size_half_half)
+    ax.grid()
+    ax.set_xlabel(f"$x$")
+    ax.set_ylabel("PDF")
+    return fig, ax
+
+# GAUSS / NORMAL
+def fgauss(x, mu, sigma_sqr):
+    return 1 / (np.sqrt(2 * np.pi * sigma_sqr)) * np.exp(-(x - mu)**2 / (2 * sigma_sqr))
+
+def gauss():
+    fig, ax = get_fig()
+    x = np.linspace(-5, 5, 300)
+    for mu, sigma_sqr in [(0, 1), (0, 0.2), (0, 5), (-2, 2)]:
+        y = fgauss(x, mu, sigma_sqr)
+        label = texvar("mu", mu) + ", " + texvar("sigma^2", sigma_sqr)
+        ax.plot(x, y, label=label)
+    ax.legend()
+    return fig
+export(gauss(), "distribution_gauss")
+
+# CAUCHY / LORENTZ
+def fcauchy(x, x_0, gamma):
+    return 1 / (np.pi * gamma * (1 + ((x - x_0)/gamma)**2))
+
+def cauchy():
+    fig, ax = get_fig()
+    x = np.linspace(-5, 5, 300)
+    for x_0, gamma in [(0, 1), (0, 0.5), (0, 2), (-2, 1)]:
+        y = fcauchy(x, x_0, gamma)
+        label = f"$x_0 = {x_0}$ , {texvar('gamma', gamma)}"
+        ax.plot(x, y, label=label)
+    ax.legend()
+    return fig
+export(cauchy(), "distribution_cauchy")
+
+# MAXWELL-BOLTZMANN
+def fmaxwell(x, a):
+    return np.sqrt(2/np.pi) * x**2 / a**3 * np.exp(-x**2 /(2*a**2))
+
+def maxwell():
+    fig, ax = get_fig()
+    x = np.linspace(0, 20, 300)
+    for a in [1, 2, 5]:
+        y = fmaxwell(x, a)
+        label = f"$a = {a}$"
+        ax.plot(x, y, label=label)
+    ax.legend()
+    return fig
+
+export(maxwell(), "distribution_maxwell-boltzmann")
+
+
+# POISSON
+def fpoisson(k, l):
+    return l**k * np.exp(-l) / scp.special.factorial(k)
+
+def poisson():
+    fig, ax = get_fig()
+    k = np.arange(0, 21, dtype=int)
+    for l in [1, 4, 10]:
+        y = fpoisson(k, l)
+        label = texvar("lambda", l)
+        ax.plot(k, y, color="#555")
+        ax.scatter(k, y, label=label)
+    ax.set_xlabel(f"$k$")
+    ax.set_ylabel(f"PMF")
+    ax.legend()
+    return fig
+
+export(poisson(), "distribution_poisson")
+
+# BINOMIAL
+def binom(n, k):
+    return scp.special.factorial(n) / (
+        scp.special.factorial(k) *
+        scp.special.factorial((n-k))
+    )
+def fbinomial(k, n, p):
+    return binom(n, k) * p**k * (1-p)**(n-k)
+
+def binomial():
+    fig, ax = get_fig()
+    n = 20
+    k = np.arange(0, n+1, dtype=int)
+    for p in [0.3, 0.5, 0.7]:
+        y = fbinomial(k, n, p)
+        label = f"$n={n}$, $p={p}$"
+        ax.plot(k, y, color="#555")
+        ax.scatter(k, y, label=label)
+    ax.set_xlabel(f"$k$")
+    ax.set_ylabel(f"PMF")
+    ax.legend()
+    return fig
+
+export(binomial(), "distribution_binomial")
+
+# FERMI-DIRAC
+
+# BOSE-EINSTEIN
+

src/scripts/plot.py

@@ -0,0 +1,34 @@
+import os
+import matplotlib.pyplot as plt
+import numpy as np
+import math
+import scipy as scp
+
+outdir = "../img/"
+filetype = ".pdf"
+skipasserts = False
+
+full = 8
+size_half_half = (full/2, full/2)
+size_third_half = (full/3, full/2)
+size_half_third = (full/2, full/3)
+
+def texvar(var, val, math=True):
+    s = "$" if math else ""
+    s += f"\\{var} = {val}"
+    if math: s += "$"
+    return s
+
+def export(fig, name):
+    if not skipasserts:
+        assert os.path.abspath(".").endswith("scripts"), "Please run from the `scripts` directory"
+    filename = os.path.join(outdir, name + filetype)
+    fig.savefig(filename, bbox_inches="tight")
+
+
+@np.vectorize
+def smooth_step(x: float, left_edge: float, right_edge: float):
+    x = (x - left_edge) / (right_edge - left_edge)
+    if x <= 0: return 0.
+    elif x >= 1: return 1.
+    else: return 3*(x**2) - 2*(x**3)

src/scripts/qubits.py

@@ -0,0 +1,90 @@
+from plot import *
+import scqubits as scq
+import qutip as qt
+
+# flux = scq.FluxQubit()
+fluxonium = scq.Fluxonium(EJ=9, EC=3, EL=0.5, flux=1, cutoff=100)
+
+
+
+def _plot_transmon_n_wavefunctions(qubit: scq.Transmon, fig_ax, which=[0,1]):
+    fig, ax = fig_ax
+    ax.set_ylabel(r"$|\psi_i(n)|^2$")
+    ax.set_xlabel(r"$n$")
+    colors = "brgy"
+    for i in which:
+        wf = qubit.numberbasis_wavefunction(which=i)
+        x = wf.basis_labels
+        y = wf.amplitudes
+        offset = (len(which)/2 - i) * 0.2
+        ax.bar(x-offset, y, width=0.2, align='center', label=f"$i={i}$")
+    xlim = (-4, 4)
+    ax.set_xlim(*xlim)
+    ax.set_xticks(np.arange(xlim[0], xlim[1]+1))
+
+def _plot_transmon(qubit: scq.Transmon, ngs, fig, axs):
+    _,_ = qubit.plot_evals_vs_paramvals("ng", ngs, fig_ax=(fig, axs[0]), evals_count=5, subtract_ground=False)
+    _,_ = qubit.plot_wavefunction(fig_ax=(fig, axs[1]), which=[0, 1, 2], mode="abs_sqr")
+    _plot_transmon_n_wavefunctions(qubit, (fig, axs[2]), which=[0, 1, 2])
+    qubit.ng = 0.5
+    _plot_transmon_n_wavefunctions(qubit, (fig, axs[3]), which=[0, 1, 2])
+    qubit.ng = 0
+
+
+def transmon_cpb():
+    EC = 1
+    qubit = scq.Transmon(EJ=30, EC=EC, ng=0, ncut=30)
+    ngs = np.linspace(-2, 2, 200)
+    fig, axs = plt.subplots(4, 3, squeeze=True, figsize=(full,full))
+    axs = axs.T
+    qubit.ng = 0
+    qubit.EJ = 0.1 * EC
+    title = lambda x: f"$E_J/E_C = {x}$"
+    _plot_transmon(qubit, ngs, fig, axs[0])
+    axs[0][0].set_title("Cooper-Pair-Box\n"+title(qubit.EJ))
+
+    qubit.EJ = EC
+    _plot_transmon(qubit, ngs,  fig, axs[1])
+    axs[1][0].set_title("Quantronium\n"+title(qubit.EJ))
+
+    qubit.EJ = 20 * EC
+    _plot_transmon(qubit, ngs, fig, axs[2])
+    axs[2][0].set_title("Transmon\n"+title(qubit.EJ))
+
+    for ax in axs[1:,:].flatten(): ax.set_ylabel("")
+    for ax in axs[:,0].flatten():
+        ax.set_xticks([-2, -1, -0.5, 0, 0.5, 1, 2])
+        ax.set_xticklabels(["-2", "-1", "", "0", "", "1", "2"])
+        ylim = ax.get_ylim()
+        ax.vlines([-1, -0.5], ymin=ylim[0], ymax=ylim[1], color="#aaa", linestyle="dotted")
+    axs[0][2].legend()
+    fig.tight_layout()
+    return fig
+
+export(transmon_cpb(), "qubit_transmon")
+
+
+def flux_onium():
+    fig, axs = plt.subplots(1, 2, squeeze=True, figsize=(full,full))
+    fluxs = np.linspace(-2, 2, 101)
+    EJ = 35.0
+    alpha = 0.6
+    fluxqubit = scq.FluxQubit(EJ1 = EJ,
+                                  EJ2 = EJ,
+                                  EJ3 = alpha*EJ,
+                                  ECJ1 = 1.0,
+                                  ECJ2 = 1.0,
+                                  ECJ3 = 1.0/alpha,
+                                  ECg1 = 50.0,
+                                  ECg2 = 50.0,
+                                  ng1 = 0.0,
+                                  ng2 = 0.0,
+                                  flux = 0.5,
+                                  ncut = 10)
+    fluxqubit.plot_evals_vs_paramvals("flux", fluxs, evals_count=5, subtract_ground=True, fig_ax=(fig, axs[0]))
+
+    fluxonium = scq.Fluxonium(EJ=9, EC=3, EL=0.5, flux=1, cutoff=100)
+    fluxonium.plot_evals_vs_paramvals("flux", fluxs, evals_count=5, subtract_ground=True, fig_ax=(fig, axs[1]))
+    return fig
+
+export(flux_onium(), "qubit_flux_onium")

src/scripts/requirements.txt

@@ -0,0 +1,5 @@
+numpy
+matplotlib
+scqubits
+qutip
+

src/scripts/stat-mech.py

@@ -0,0 +1,120 @@
+from plot import *
+
+def flennard_jones(r, epsilon, sigma):
+    return 4 * epsilon * ((sigma/r)**12 - (sigma/r)**6)
+
+def lennard_jones():
+    fig, ax = plt.subplots(figsize=size_half_half)
+    ax.grid()
+    ax.set_xlabel(r"$r$")
+    ax.set_ylabel(r"$V(r)$")
+
+    r = np.linspace(0.5, 4, 300)
+    for epsilon, sigma in [(1, 1), (1, 2)]:
+        y = flennard_jones(r, epsilon, sigma)
+        label = texvar("epsilon", epsilon) + ", " + texvar("sigma", sigma)
+        ax.plot(r, y, label=label)
+    ax.legend()
+    ax.set_ylim(-1.1, 1.1)
+    return fig
+export(lennard_jones(), "potential_lennard_jones")
+
+# BOLTZMANN / FERMI-DIRAC / BOSE-EINSTEN DISTRIBUTIONS
+def fboltzmann(x):
+    return  1 / np.exp(x)
+def fbose_einstein(x):
+    return  1 / (np.exp(x) -1)
+def ffermi_dirac(x):
+    return  1 / (np.exp(x) + 1)
+
+
+def id_qgas():
+    fig, ax = plt.subplots(figsize=size_half_half)
+    ax.grid()
+    ax.set_xlabel(r"$\beta(\epsilon-\mu)$")
+    ax.set_ylabel(r"$\langle n(\epsilon)\rangle$")
+    x = np.linspace(-4, 4, 300)
+    xbe = np.linspace(0.01, 4, 300)
+    yb = fboltzmann(x)
+    ybe = fbose_einstein(xbe)
+    yfd = ffermi_dirac(x)
+    ax.vlines([0], ymin=-10, ymax=10, color="black", linestyle="dashed")
+    ax.plot(xbe, ybe, label="Bose-Einstein")
+    ax.plot(x, yb, label="Boltzmann")
+    ax.plot(x, yfd, label="Fermi-Dirac")
+    ax.legend()
+    ax.set_ylim(-0.1, 4)
+    return fig
+export(id_qgas(), "td_id_qgas_distributions")
+
+@np.vectorize
+def fstep(x):
+    return 1 if x >= 0 else 0
+
+def fermi_occupation():
+    fig, ax = plt.subplots(figsize=size_half_third)
+    # ax.grid()
+    # ax.set_xlabel(r"$\beta(\epsilon-\mu)$")
+    ax.set_xticks([0])
+    ax.set_xticklabels([r"$\epsilon=\mu$"])
+    ax.set_ylabel(r"$\langle n(\epsilon)\rangle$")
+    x = np.linspace(-4, 4, 300)
+    ystep = fstep(-x)
+    yfd = ffermi_dirac(x)
+    ax.vlines([0], ymin=-10, ymax=10, color="black", linestyle="dashed")
+    ax.plot(x, ystep, label="$T = 0$")
+    ax.plot(x, yfd, label=r"$T > 0$")
+    ax.legend()
+    ax.set_ylim(-0.1, 1.1)
+    return fig
+export(fermi_occupation(), "td_fermi_occupation")
+
+def fermi_heat_capacity():
+    fig, ax = plt.subplots(figsize=size_half_third)
+    # ax.grid()
+    # ax.set_xlabel(r"$\beta(\epsilon-\mu)$")
+    x = np.linspace(0, 4, 100)
+    T_F = 1
+    a = np.pi**2 / (2 * T_F)
+
+    def linear(x):
+        return x * a
+    X_32 = 3/2 /(np.pi**2 / (2 * T_F))
+
+
+    low_temp_Cv = linear(x)
+    ax.plot(x, low_temp_Cv, color="orange", linestyle="dashed", label=r"${\pi^2}/{2}\,{T}/{T_\text{F}}$")
+    ax.hlines([3/2], xmin=0, xmax=10, color="blue", linestyle="dashed", label="Petit-Dulong")
+    @np.vectorize
+    def unphysical_f(x):
+        # exponential 
+        # find ṕoint where unshifted exponential has slope of the linear
+        # f = 3/2 - exp(-A*x)
+        # f' = A exp(-A*x) = a
+        # x = -log(a/A) / A
+        A = 5
+        x_unshifted = -np.log(a/A) / A
+        # shift so that this the exponential intersects the linear at this point
+        y_intersect = 3/2 - np.exp(-A*x_unshifted)
+        # find intersect x value of linear
+        # x = y/a
+        x_intersect = y_intersect / a
+        # shift exp so that x_intersect becomes x_unshifted
+        if x > x_intersect: return 3/2 - np.exp(-A * (x-(x_intersect - x_unshifted)))
+        else: return a * x
+    # ax.plot(x, smoothing, label="smooth")
+    y = unphysical_f(x)
+    ax.plot(x, y, color="black")
+    ax.legend(loc="lower right")
+
+
+    ax.set_xticks([0, T_F])
+    ax.set_xticklabels(["0", r"$T_F$"])
+    ax.set_yticks([0, 3/2])
+    ax.set_yticklabels(["0", r"$3/2$"])
+    ax.set_ylabel(r"${C_V}/{N k_\text{B}}$")
+    ax.set_xlim(0, 1.4 * T_F)
+    ax.set_ylim(0, 2)
+    return fig
+export(fermi_heat_capacity(), "td_fermi_heat_capacity")
+

src/statistical_mechanics.tex

@@ -0,0 +1,757 @@
+\Part[
+    \eng{Statistichal Mechanics}
+    \ger{Statistische Mechanik}
+    ]{stat}
+
+    \begin{ttext}
+        \eng{
+            \textbf{Intensive quantities:} Additive for subsystems (system size dependent): $S(\lambda E, \lambda V, \lambda N)  = \lambda S(E, V, N)$\\
+            \textbf{Extensive quantities:} Independent of system size, ratio of two intensive quantities
+        }
+        \ger{
+            \textbf{Intensive Größen:} Additiv für Subsysteme (Systemgrößenabhänig): $S(\lambda E, \lambda V, \lambda N)  = \lambda S(E, V, N)$\\
+            \textbf{Extensive Größen:} Unabhängig der Systemgröße, Verhältnis zweier intensiver Größen
+        }
+    \end{ttext}
+
+    \begin{formula}{liouville}
+        \desc{Liouville equation}{}{$\{\}$ poisson bracket}
+        \desc[german]{Liouville-Gleichung}{}{$\{\}$ Poisson-Klammer}
+        \eq{\pdv{\rho}{t} = - \sum_{i=1}^{N} \left(\pdv{\rho}{q_i} \pdv{H}{p_i} - \pdv{\rho}{p_i} \pdv{H}{q_i} \right) = \{H, \rho\}}
+    \end{formula}
+
+    \Section[
+        \eng{Entropy}
+        \ger{Entropie}
+        ]{entropy}
+
+        \begin{formula}{properties}
+            \desc{Positive-definite and additive}{}{}
+            \desc[german]{Positiv Definit und Additiv}{}{}
+            \eq{
+                S &\ge 0 \\
+                S(E_1, E_2) &= S_1 + S_2
+            }
+        \end{formula}
+
+        \begin{formula}{von_neumann}
+            \desc{Von-Neumann}{}{$\rho$ density matrix}
+            \desc[german]{Von-Neumann}{}{$\rho$ Dichtematrix}
+            \eq{S = - \kB \braket{\log \rho} = - \kB \tr(\rho \log\rho)}
+        \end{formula}
+
+        \begin{formula}{gibbs}
+            \desc{Gibbs}{}{$p_n$ probability for micro state $n$}
+            \desc[german]{Gibbs}{}{$p_n$ Wahrscheinlichkeit für Mikrozustand $n$}
+            \eq{S = - \kB \sum_n p_n \log p_n}
+        \end{formula}
+
+        \begin{formula}{boltzmann}
+            \desc{Boltzmann}{}{$\Omega$ \#micro states}
+            \desc[german]{Boltzmann}{}{$\Omega$ \#Mikrozustände}
+            \eq{S = \kB \log\Omega}
+        \end{formula}
+
+        \begin{formula}{temp}
+            \desc{Temperature}{}{}
+            \desc[german]{Temperatur}{}{}
+            \eq{\frac{1}{T} \coloneq \pdv{S}{E}_V}
+        \end{formula}
+
+        \begin{formula}{pressure}
+            \desc{Pressure}{}{}
+            \desc[german]{Druck}{}{}
+            \eq{p = T \pdv{S}{V}_E}
+        \end{formula}
+
+\Part[
+    \eng{Thermodynamics}
+    \ger{Thermodynamik}
+    ]{td}
+
+    \begin{formula}{therm_wavelength}
+        \desc{Thermal wavelength}{}{}
+        \desc[german]{Thermische Wellenlänge}{}{}
+        \eq{\lambda = \frac{\hbar}{\sqrt{2\pi m \kB T}}}
+    \end{formula}
+
+    \Section[
+        \eng{Processes}
+        \ger{Prozesse}
+    ]{process}
+    \begin{ttext}
+        \eng{
+            \begin{itemize}
+                \item \textbf{isobaric}: constant pressure $p = \const$
+                \item \textbf{isochoric}: constant volume $V = \const$
+                \item \textbf{isothermal}: constant temperature $T = \const$
+                \item \textbf{isentropic}: constant entropy $S = \const$
+                \item \textbf{isenthalpic}: constant entalphy $H = \const$
+                \item \textbf{adiabatic}: no heat transfer $\Delta Q=0$
+                \item \textbf{quasistatic}: happens so slow, the system always stays in td. equilibrium
+                \item \textbf{reversivle}: reversible processes are always quasistatic and no entropie is created $\Delta S = 0$
+            \end{itemize}
+        }
+        \ger{
+            \begin{itemize}
+                \item \textbf{isobar}: konstanter Druck $p = \const$
+                \item \textbf{isochor}: konstantes Volumen $V = \const$
+                \item \textbf{isotherm}: konstante Temperatur $T = \const$
+                \item \textbf{isentrop}: konstante Entropie $S = \const$
+                \item \textbf{isenthalp}: konstante Entalphie $H = \const$
+                \item \textbf{adiabatisch}: kein Wärmeübertrag $\Delta Q=0$
+                \item \textbf{quasistatsch}: läuft so langsam ab, dass das System durchgehend im t.d Equilibrium bleibt
+                \item \textbf{reversibel}: reversible Prozesse sind immer quasistatisch und es wird keine Entropie erzeugt $Delta S = 0$
+            \end{itemize}
+        }
+    \end{ttext}
+
+    \Subsection[
+        \eng{Irreversible gas expansion (Gay-Lussac experiment)}
+        \ger{Irreversible Gasexpansion (Gay-Lussac-Versuch)}
+        ]{gay}
+
+        \begin{minipage}{0.6\textwidth}
+            \vfill
+            \begin{ttext}
+                \eng{
+                    A classical gas in a system with volume $V_1$ is separated from another system with volume $V_2$. 
+                    In the Gay-Lussac experiment, the separation is removed and the gas flows into $V_2$.
+                }
+                \ger{
+                    Ein klassisches Gas in einem System mit Volumen $V_1$ ist getrennt von einem zweiten System mit Volumen $V_2$.
+                    Beim Gay-Lussac Versuch wird die Trennwand entfern und das Gas fließt in das Volumen $V_2$.
+                }
+            \end{ttext}
+            \vfill
+        \end{minipage}
+        \hfill
+        \begin{minipage}{0.3\textwidth}
+            \begin{figure}[H]
+                \centering
+                \includegraphics[width=\textwidth]{img/td_gay_lussac.pdf}
+            \end{figure}
+        \end{minipage}
+
+        \begin{formula}{entropy}
+            \desc{Entropy change}{}{}
+            \desc[german]{Entropieänderung}{}{}
+            \eq{\Delta S=N\kB \ln \left(\frac{V_1 + V_2}{V_1}\right) > 0}
+        \end{formula}
+
+        \TODO{Reversible}
+
+        \TODO{Quasistatischer T-Ausgleich}
+
+        \TODO{Joule-Thompson Prozess}
+
+
+    \Section[
+        \eng{Phase transitions}
+        \ger{Phasenübergänge}
+        ]{phases}
+
+        \begin{ttext}
+            \eng{
+                A phase transition is a discontinuity in the free energy $F$ or Gibbs energy $G$ or in one of their derivatives.
+                The degree of the phase transition is the degree of the derivative which exhibits the discontinuity.
+            }
+            \ger{
+                Ein Phasenübergang ist eine Unstetigkeit in the Freien Energie $F$ oder in der Gibbs-Energie $G$ oder in ihrer Ableitungen.
+                Die Ordnung des Phasenübergangs ist die Ordnung der Ableitung, in welcher die Unstetigkeit auftritt.
+            }
+        \end{ttext}
+
+        \begin{formula}{latent_heat}
+            \desc{Latent heat}{Heat required to bring substance from phase 1 to phase 2}{$\Delta S$ entropy change of the phase transition}
+            \desc[german]{Latente Wärme}{Für den Phasenübergang von Phase 1 nach Phase 2 benötigte Wärme}{$\Delta S$ Entropieänderung des Phasenübergangs}
+            \eq{Q_\text{L} = T \Delta S}
+        \end{formula}
+        \begin{formula}{clausis_clapeyron}
+            \desc{Clausius-Clapyeron equation}{Slope of the coexistence curve}{$\Delta V$ Volume change of the phase transition}
+            \desc[german]{Clausius-Clapeyron Gleichung}{Steigung der Phasengrenzlinie}{$\Delta V$ Volumenänderung des Phasenübergangs}
+            \eq{\odv{p}{T} = \frac{Q_\text{L}}{T\Delta V}}
+        \end{formula}
+        \begin{formula}{coexistence}
+            \desc{Phase transition}{At the coexistence curve}{}
+            \desc[german]{Phasenübergang}{Im Koexistenzbereich}{}
+            \eq{G_1 = G_2 \\ \shortintertext{\GT{and_therefore}} \mu_1 = \mu_2}
+        \end{formula}
+
+        \begin{formula}{gibbs_phase_rule}
+            \desc{Gibbs rule / Phase rule}{}{$c$ \#components, $f$ \#degrees of freedom, $p$ \#phases}
+            \desc[german]{Gibbsche Phasenregel}{}{$c$ \#Komponenten, $f$ \#Freiheitsgrade, $p$ \#Phasen}
+            \eq{f = c - p + 2}
+        \end{formula}
+
+        \Subsubsection[
+            \eng{Osmosis}
+            \ger{Osmose}
+            ]{osmosis}
+            \begin{ttext}
+                \eng{
+                    Osmosis is the spontaneous net movement or diffusion of solvent molecules 
+                    through a selectively-permeable membrane, which allows through the solvent molecules, but not the solute molecules.
+                    The direction of the diffusion is from a region of high water potential 
+                    (region of lower solute concentration) to a region of low water potential 
+                    (region of higher solute concentration), in the direction that tends to equalize the solute concentrations on the two sides.
+                }
+                \ger{
+                    Osmosis ist die spontane Passage oder Diffusion Lösungsmittelmolekülen durch eine semi-permeable Membran
+                    die für das Lösungsmittel, jedoch nicht die darin gelösten Stoffe durchlässig ist.
+                    Die Richtung der Diffusion ist vom Gebiet mit hohem chemischen Potential (niedrigere Konzentration des gelösten Stoffes)
+                    in das mit niedrigem chemischem Potential (höherere Konzentraion des gelösten Stoffes), sodass die Konzentration des gelösten Stoffes ausgeglichen wird.
+                }
+            \end{ttext}
+        \begin{formula}{osmosis}
+            \desc{Osmotic pressure}{}{$N_c$ \#dissolved particles}
+            \desc[german]{Osmotischer Druck / Van-\'t-hoffsches Gesetz}{}{$N_c$ \#gelöster Teilchen}
+            \eq{p_\text{osm} = \kB T \frac{N_c}{V}}
+        \end{formula}   
+
+
+    \Subsection[
+        \eng{Material properties}
+        \ger{Materialeigenschaften}
+        ]{}
+        \begin{formula}{heat_cap}
+            \desc{Heat capacity}{}{$Q$ heat}
+            \desc[german]{Wärmekapazität}{}{$Q$ Wärme}
+            \eq{c = \frac{Q}{\Delta T}}
+        \end{formula}
+        \begin{formula}{heat_cap_V}
+            \desc{Isochoric heat capacity}{}{$U$ internal energy}
+            \desc[german]{Isochore Wärmekapazität}{}{$U$ innere Energie}
+            \eq{c_v = \pdv{Q}{T}_V = \pdv{U}{T}_V}
+        \end{formula}
+
+        \begin{formula}{heat_cap_p}
+            \desc{Isobaric heat capacity}{}{$H$ enthalpy}
+            \desc[german]{Isobare Wärmekapazität}{}{$H$ Enthalpie}
+            \eq{c_p = \pdv{Q}{T}_P = \pdv{H}{T}_P}
+        \end{formula}
+
+        \begin{formula}{bulk_modules}
+            \desc{Bulk modules}{}{$p$ pressure, $V$ initial volume}
+            \desc[german]{Kompressionsmodul}{}{$p$ Druck, $V$ Anfangsvolumen}
+            \eq{K = -V \odv{p}{V} }
+        \end{formula}
+
+        \begin{formula}{compressibility}
+            \desc{Compressibility}{}{}
+            \desc[german]{Kompressibilität}{}{}
+            \eq{\kappa = -\frac{1}{V} \pdv{V}{p} }
+        \end{formula}
+
+        \begin{formula}{compressibility_T}
+            \desc{Isothermal compressibility}{}{}
+            \desc[german]{Isotherme Kompressibilität}{}{}
+            \eq{\kappa_T = -\frac{1}{V} \pdv{V}{p}_{T} = \frac{1}{K}}
+        \end{formula}
+
+        \begin{formula}{compressibility_S}
+            \desc{Adiabatic compressibility}{}{}
+            \desc[german]{Adiabatische Kompressibilität}{}{}
+            \eq{\kappa_S = -\frac{1}{V} \pdv{V}{p}_{S}}
+        \end{formula}
+
+        \begin{formula}{therm_expansion}
+            \desc{Thermal expansion coefficient}{}{}
+            \desc[german]{Thermaler Ausdehnungskoeffizient}{}{}
+            \eq{\alpha = \frac{1}{V} \pdv{V}{T}_{p,N}}
+        \end{formula}
+
+
+
+\Section[
+    \eng{Laws of thermodynamics}
+    \ger{Hauptsätze der Thermodynamik}
+    ]{laws}
+
+    \Subsection[
+        \eng{Zeroeth law}
+        \ger{Nullter Hauptsatz}
+        ]{law0}
+        \begin{ttext}
+            \eng{If two systems are each in thermal equilibrium with a third, they are also in thermal equilibrium with each other.}
+            \ger{Wenn sich zwei Siesteme jeweils im thermischen Gleichgewicht mit einem dritten befinden, befinden sie sich auch untereinander im thermischen Gleichgewicht.}
+        \end{ttext}
+
+        \Eng[teq]{th. eq.}
+        \Ger[teq]{th. GGW.}
+        \def\ggwarrow{\overset{\GT{teq}}{\leftrightarrow}}
+        \begin{equation}
+            \label{eq:\fqname}
+            A \ggwarrow C \quad\wedge\quad B \ggwarrow C \quad\Rightarrow\quad A \ggwarrow B
+        \end{equation}
+
+    \Subsection[
+        \eng{First law}
+        \ger{Erster Hauptsatz}
+        ]{law1}
+        \begin{ttext}
+            \eng{In a process without transfer of matter, the change in internal energy, $\Delta U$, of a thermodynamic system is equal to the energy gained as heat, $Q$, less the thermodynamic work, W, done by the system on its surroundings.}
+            \ger{In einem abgeschlossenem System ist die Änderung der inneren Energie $U$ gleich der gewonnenen Wärme $Q$ minus der vom System an der Umgebung verrichteten Arbeit $W$.}
+        \end{ttext}
+
+        \begin{formula}{internal_energy}
+            \desc{Internal energy change}{}{}
+            \desc[german]{Änderung der inneren Energie}{}{}
+            \eq{
+                \Delta U &= \delta Q - \delta W \\
+                \d U &= T \d S - p \d V
+            }
+        \end{formula}
+        
+
+    \Subsection[
+        \eng{Second law}
+        \ger{Zweiter Hauptsatz}
+        ]{law2}
+        \begin{ttext}
+            \eng{
+                \textbf{Clausius}: Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.\\
+                \textbf{Kelvin}:  It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.
+            }
+            \ger{
+                \textbf{Clausius}: Es gibt keine Zustansänderung, deren einziges Ergebnis die Übertragung von Wärme von einem Körper niederer Temperatur auf einen Körper höherer Temperatur ist.\\
+                \textbf{Kelvin}: Es ist unmöglich, eine periodisch arbeitende Maschine zu konstruieren, die weiter nichts bewirkt als Hebung einer Last und Abkühlung eines Wärmereservoirs.
+            }
+        \end{ttext}
+    \Subsection[
+        \eng{Third law}
+        \ger{Dritter Hauptsatz}
+        ]{law3}
+        \begin{ttext}
+            \eng{It is impussible to cool a system to absolute zero.}
+            \ger{Es ist unmöglich, ein System bis zum absoluten Nullpunkt abzukühlen.}
+        \end{ttext}
+
+        \begin{formula}{3d_law}
+            \desc{Entropy density}{}{$s = \frac{S}{N}$}
+            \desc[german]{Entropiedichte}{}{$s = \frac{S}{N}$}
+            \eq{
+                \lim_{T\to 0} s(T) &= 0 \\
+                \shortintertext{\GT{and_therefore_also}} \\
+                \lim_{T\to 0} c_V &= 0
+            }
+        \end{formula}   
+
+\Section[
+    \eng{Ensembles}
+    \ger{Ensembles}
+    ]{ensembles}
+
+
+
+    \begin{table}[H]
+        \centering
+        \caption{caption}
+        \label{tab:\fqname}
+    
+        \begin{tabular}{l|c|c|c}
+             &  \gt{mk} & \gt{k}  & \gt{gk}  \\ \hline
+            \GT{variables} & $E$, $V,$ $N$ & $T$, $V$, $N$ & $T$, $V$, $\mu$ \\ \hline
+            \GT{partition_sum} & $\Omega = \sum_n 1$ & $Z = \sum_n \e^{-\beta E_n}$ & $Z_\text{g} = \sum_{n} \e^{-\beta(E_n - \mu N_n)}$ \\ \hline
+            \GT{probability} & $p_n = \frac{1}{\Omega}$ & $p_n = \frac{\e^{-\beta E_n}}{Z}$ & $p_n = \frac{\e^{-\beta (E_n - \mu N_n}}{Z_\text{g}}$ \\ \hline
+            \GT{td_pot} & $S = \kB\ln\Omega$ & $F = - \kB T \ln Z$ & $ \Phi = - \kB T \ln Z$ \\ \hline
+            \GT{pressure} & $p = T \pdv{S}{V}_{E,N}$ &$p = -\pdv{F}{V}_{T,N}$ & $p = -\pdv{\Phi}{V}_{T,\mu} = -\frac{\Phi}{V}$ \\ \hline
+            \GT{entropy} & $S = \kB = \ln\Omega$ & $S = -\pdv{F}{T}_{V,N}$ & $S = -\pdv{\Phi}{T}_{V,\mu}$ \\ \hline
+        \end{tabular}
+    \end{table}
+
+    \begin{formula}{ergodic_hypo}
+        \desc{Ergodic hypothesis}{Over a long periode of time, all accessible microstates in the phase space are equiprobable}{$A$ Observable}
+        \desc[german]{Ergodenhypothese}{Innerhalb einer langen Zeitspanne sind alle energetisch erreichbaren Mikrozustände im Phasenraum gleich wahrscheinlich}{$A$ Messgröße}
+        \eq{\braket{A}_\text{\GT{time}} = \braket{A}_\text{\GT{ensemble}}}
+    \end{formula}
+
+
+    \Subsection[
+        \eng{Potentials}
+        \ger{Potentiale}
+        ]{pots}
+        \begin{formula}{internal_energy}
+            \desc{Internal energy}{}{}
+            \desc[german]{Innere Energie}{}{}
+            \eq{\d U(S,V,N) = T\d S -p\d V + \mu\d N}
+        \end{formula}
+        \begin{formula}{enthalpy}
+            \desc{Enthalpy}{}{}
+            \desc[german]{Enthalpie}{}{}
+            \eq{\d H(S,p,N) = T\d S +V\d p + \mu\d N}
+        \end{formula}
+        \begin{formula}{gibbs_energy}
+            \desc{Gibbs energy}{}{}
+            \desc[german]{Gibbsche Energie}{}{}
+            \eq{\d G(T,p,N) = -S\d T + V\d p + \mu\d N}
+        \end{formula}
+        \begin{formula}{free_energy}
+            \desc{Free energy / Helmholtz energy }{}{}
+            \desc[german]{Freie Energie / Helmholtz Energie}{}{}
+            \eq{\d F(T,V,N) = -S\d T -p\d V + \mu\d N}
+        \end{formula}
+        \begin{formula}{grand_canon_pot}
+            \desc{Grand canonical potential}{}{}
+            \desc[german]{Großkanonisches Potential}{}{}
+            \eq{\d \Phi(T,V,\mu) = -S\d T -p\d V - N\d\mu}
+        \end{formula}
+
+        \TODO{Maxwell Relationen, TD Quadrat}
+
+\Section[
+    \eng{Ideal gas}
+    \ger{Ideales Gas}
+    ]{id_gas}
+    \begin{ttext}
+        \eng{The ideal gas consists of non-interacting, undifferentiable particles.}
+        \ger{Das ideale Gas besteht aus nicht-wechselwirkenden, ununterscheidbaren Teilchen.}
+    \end{ttext}
+
+    \begin{formula}{phase_space_vol}
+        \desc{Phase space volume}{$3N$ sphere}{$N$ \#particles, $h^{3N}$ volume of a microstate, $N!$ particles are undifferentiable}
+        \desc[german]{}{$3N$ Kugel}{$N$ \#Teilchen, $h^{3N}$ Volumen eines Mikrozustandes, $N!$ Teilchen sind ununterscheidbar}
+        \eq{
+            \Omega(E) &= \int_V\d^3q_1 \sdots \int_V\d^3q_N \int \d^3p_1 \sdots \int\d^3p_N \frac{1}{N!\,h^{3N}} \Theta\left(E - \sum_{i} \frac{\vec{p_i}^2}{2m}\right) \\
+                      &= \left(\frac{V}{N}\right)^N \left(\frac{4\pi m E}{3 h^2 N}\right)^{\frac{3N}{2}} \e^\frac{5N}{2}
+        }
+    \end{formula}
+    \begin{formula}{entropy}
+        \desc{Entropy}{}{}
+        \desc[german]{Entropie}{}{}
+        \eq{
+            S = \frac{5}{2} N\kB + N\kB\ln\left(\frac{V}{N}\left(\frac{2\pi m E}{3 h^2 N}\right)^{\frac{3}{2}}\right)
+        }
+    \end{formula}
+
+    \begin{formula}{id_gas_eq}
+        \desc{Ideal gas equation}{}{}
+        \desc[german]{Ideale Gasgleichung}{Thermische Zustandsgleichung idealer Gase}{}
+        \eq{pV &= nRT \\ &= N\kB T}
+    \end{formula}
+
+    \begin{formula}{equation_of_state}
+        \desc{Equation of state}{}{}
+        \desc[german]{Kalorische Zustangsgleichung}{}{}
+        \eq{U = \frac{3}{2} N\kB T}
+    \end{formula}
+
+    % \Subsubsection[
+    %     \eng{Equipartitiontheorem}
+    %     \ger{Äquipartitionstheorem}
+    %     ]{equipart}
+    \begin{formula}{equipart}
+        \desc{Equipartitiontheorem}{Each degree of freedom contributes $U_\text{D}$ (for classical particle systems)}{}
+        \desc[german]{Äquipartitionstheorem}{Jedem Freiheitsgrad steht die Energie $U_\text{D}$ zur Verfügung}{}
+        \eq{U_\text{D} = \frac{1}{2} \kB T}
+    \end{formula}
+
+
+    \begin{formula}{maxwell_velocity}
+        \desc{Maxwell velocity distribution}{See also \ref{sec:pt:distributions::maxwell-boltzmann}}{}
+        \desc[german]{Maxwellsche Geschwindigkeitsverteilung}{Siehe auch \ref{sec:pt:distributions::maxwell-boltzmann}}{}
+        \eq{w(v) \d v = 4\pi \left(\frac{\beta m}{2\pi}\right)^\frac{3}{2} v^2 \e^{-\frac{\beta m v^2}{2}} \d v}
+    \end{formula}
+
+    \begin{formula}{avg_velocity}
+        \desc{Average quadratic velocity}{per particle in a 3D gas}{}
+        \desc[german]{Mittlere quadratosche Geschwindigkeit}{pro Teilchen im 3D-Gas}{}
+        \eq{\braket{v^2} = \int_0^\infty \d v\,v^2 w(v) = \frac{3\kB T}{m}}
+    \end{formula}
+
+    \Subsubsection[
+        \eng{Molecule gas}
+        \ger{Molekülgas}
+        ]{molecule_gas}
+
+        \begin{formula}{desc}
+            \desc{Molecule gas}{2 particles of mass $M$ connected by a ``spring'' with distance $L$}{}
+            \desc[german]{Molekülgas}{2 Teilchen der Masse $M$ sind verbunden durch eine ``Feder'' mit Länge $L$}{}
+            \content{
+                % \begin{figure}[h]
+                \centering
+                \tikzstyle{spring}=[thick,decorate,decoration={coil,aspect=0.8,amplitude=5,pre length=0.1cm,post length=0.1cm,segment length=10}]
+                \begin{tikzpicture}
+                    \def\radius{0.5}
+                    \coordinate (left) at (-3, 0);
+                    \coordinate (right) at (3, 0);
+                    \draw (left) circle (\radius);
+                    \draw[spring] ($(left) + (\radius,0)$)  -- ($(right) - (\radius,0)$);
+                    \draw (right) circle (\radius);
+                \end{tikzpicture}
+                % \end{figure}
+            }
+        \end{formula}
+
+        \begin{formula}{translation}
+            \desc{Translation}{}{$n_i \in \N_0$, $i=x,\,y,\,z$}
+            \desc[german]{Translation}{}{$n_i \in \N_0$, $i=x,\,y,\,z$}
+            \eq{p_i &= \frac{2\pi\hbar}{L}n_i \\ 
+                E_\text{kin} &= \frac{\vec{p}_r^2}{2M}
+            }
+        \end{formula}
+        \begin{formula}{vibration}
+            \desc{Vibration}{}{$n \in \N_0$}
+            \desc[german]{Schwingungen}{}{$n \in \N_0$}
+            \eq{E_\text{vib} = \hbar \omega \left(n+\frac{1}{2}\right)}
+        \end{formula}
+        \begin{formula}{rotation}
+            \desc{Rotation}{}{$j\in \N_0$}
+            \desc[german]{Rotation}{}{$j\in \N_0$}
+            \eq{E_\text{rot} = \frac{\hbar^2}{2I}j(j+1)}
+        \end{formula}
+        \TODO{Diagram für verschiedene Temperaturen, Weiler Skript p.83}
+
+
+\Section[
+    \eng{Real gas}
+    \ger{Reales Gas}
+    ]{real_gas}
+
+    \Subsection[
+        \eng{Virial expansion}
+        \ger{Virialentwicklung}
+        ]{virial}
+        \begin{ttext}
+            \eng{Expansion of the pressure $p$ in a power series of the density $\rho$.}
+            \ger{Entwicklung desw Drucks $p$ in eine Potenzreihe der Dichte $\rho$.}
+        \end{ttext}
+
+        \begin{formula}{series}
+            \desc{Virial expansion}{The 2\ts{nd} and 3\ts{d} virial coefficient are tabelated for many substances}{$B$ and $C$ 2\ts{nd} and 3\ts{d} virial coefficient, $\rho = \frac{N}{V}$}
+            \desc[german]{Virialentwicklung}{Der zweite und dritte Virialkoeffizient ist für viele Substanzen tabelliert}{$B$ und $C$ 2. und 3. Virialkoeffizient, $\rho = \frac{N}{V}$}
+            \eq{p = \kB T \rho\,[1 + B(T) \rho + C(T) \rho^2 + \dots]}
+        \end{formula}
+
+        \begin{formula}{mayer_function}
+            \desc{Mayer function}{}{$V(i,j)$ pair potential}
+            \desc[german]{Mayer-Funktion}{}{$V(i,j)$ Paarpotential}
+            \eq{f(\vec{r}) = \e^{-\beta V(i,j)} - 1}
+        \end{formula}
+
+        \begin{formula}{second_coefficient}
+            \desc{Second virial coefficient}{Depends on pair potential between two molecules}{}
+            \desc[german]{Zweiter Virialkoeffizient}{Hängt vom Paarpotential zweier Moleküle ab}{}
+            \eq{B = -\frac{1}{2} \int_V \d^3 \vec{r} f(\vec{r})}
+            % b - \frac{a}{\kB T}}
+        \end{formula}
+
+        \begin{formula}{lennard_jones}
+            \desc{Lennard-Jones potential}{Potential between two molecules. Attractive for $r > \sigma$, repulsive for $r < \sigma$}{}
+            \desc[german]{Lennard-Jones-Potential}{Potential zwischen zwei Molekülen. Attraktiv für $r > \sigma$, repulsiv für $r < \sigma$}{}
+            \figeq{img/potential_lennard_jones.pdf}{V(r) = 4\epsilon \left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right]}
+        \end{formula}
+
+    \Subsection[
+        \eng{Van der Waals equation}
+        \ger{Van der Waals Gleichung}
+        ]{vdw}
+        \begin{ttext}
+            \eng{Assumes a hard-core potential with a weak attraction.}
+            \ger{Annahme eines Harte-Kugeln Potentials mit einer schwachen Anziehung}
+        \end{ttext}
+        \begin{formula}{partition_sum}
+            \desc{Partition sum}{}{$a$ internal pressure}
+            \desc[german]{Zustandssumme}{}{$a$ Kohäsionsdruck}
+            \eq{Z_N = \frac{(V-V_0)^N}{\lambda^{3N}N!} \e^{\frac{\beta N^2 a}{V}}}
+        \end{formula}
+        \begin{formula}{equation}
+            \desc{Van der Waals equation}{}{$b$ co-volume?}
+            \desc[german]{Van der Waals-Gleichung}{}{$b$ Kovolumen}
+            \eq{p = \frac{N \kB T}{V-b} - \frac{N^2 a}{V^2}}
+        \end{formula}
+        \TODO{sometimes N is included in a, b}
+
+
+\Section[
+    \eng{Ideal quantum gas}
+    \ger{Ideales Quantengas}
+    ]{id_qgas}
+    \def\bosfer{$\pm$: {$\text{bos} \atop \text{fer}$}}
+
+    \begin{formula}{fugacity}
+        \desc{Fugacity}{}{}
+        \desc[german]{Fugazität}{}{}
+        \eq{z = \e^{\mu\beta} = \e^\frac{\mu}{\kB T}}
+    \end{formula}
+
+    \begin{formula}{occupation}
+        \desc{Occupation number}{}{$r$ states}
+        \desc[german]{Besetzungszahl}{}{$r$ Zustände}
+        \eq{\sum_{r} n_r = N}
+    \end{formula}   
+    \begin{formula}{undiff_particles}
+        \desc{Undifferentiable particles}{}{$p_i$ state}
+        \desc[german]{Ununterscheidbare Teilchen}{}{$p_i$ Zustand}
+        \eq{\ket{p_1,p_2,\dots,p_N} = \ket{p_1}\ket{p_2}\dots \ket{p_N}}
+    \end{formula}
+    \begin{formula}{parity}
+        \desc{Applying the parity operator}{yields a \textit{symmetric} (Bosons) and a \textit{antisymmetic} (Fermions) solution}{$\hat{P}_{12}$ parity operator swaps $1$ and $2$, \bosfer}
+        \desc[german]{Anwenden des Paritätsoperators}{gibt eine \textit{symmetrische} (Bosonen) und eine \textit{antisymmetrische} (Fermionen) Lösung}{$\hat{O}_{12}$ Paritätsoperator tauscht $1$ und $2$, \bosfer}
+        \eq{\hat{P}_{12} \psi(p_i(\vec{r}_1),\,p_j(\vec{r}_2)) = \pm \psi(p_i(\vec{r}_1),\,p_j(\vec{r}_2))}
+    \end{formula}
+
+    \begin{formula}{spin_degeneracy_factor}
+        \desc{Spin degeneracy factor}{}{$s$ spin}
+        \desc[german]{Spinentartungsfaktor}{}{$s$ Spin}
+        \eq{g_s = 2s+1}
+    \end{formula}
+    \begin{formula}{dos}
+        \desc{Density of states}{}{$g_s$ \fqEqRef{td:id_qgas:spin_degeneracy_factor}}
+        \desc[german]{Zustandsdichte}{}{$g_s$ \fqEqRef{td:id_qgas:spin_degeneracy_factor}}
+        \eq{g(\epsilon) = g_s \frac{V}{4\pi} \left(\frac{2m}{\hbar^2}\right)^\frac{3}{2} \sqrt{\epsilon}}
+    \end{formula}
+
+    \begin{formula}{occupation_number_per_e}
+        \desc{Occupation number per energy}{}{\fqEqRef{td:id_qgas:dos}, \bosfer}
+        \desc[german]{Besetzungszahl pro Energie}{}{\fqEqRef{td:id_qgas:dos}, \bosfer}
+        \eq{n(\epsilon)\, \d\epsilon &= \frac{g(\epsilon)}{\e^{\beta(\epsilon - \mu)} \mp 1}\,\d\epsilon}
+    \end{formula}
+    
+    \begin{formula}{occupation_number}
+        \desc{Occupation number}{}{\bosfer}
+        \desc[german]{Besetzungszahl}{}{\bosfer}
+        \figeq{img/td_id_qgas_distributions.pdf}{%
+            \braket{n(\epsilon)} &= \frac{1}{\e^{\beta(\epsilon - \mu)} \mp 1} \\
+            \shortintertext{\GT{for} $\epsilon - \mu \gg \kB T$}
+                        &= \frac{1}{\e^{\beta(\epsilon - \mu)}}
+        }
+         
+    \end{formula}
+
+
+    \begin{formula}{particle_number}
+        \desc{Number of particles}{}{}
+        \desc[german]{Teilchenzahl}{}{}
+        \eq{\braket{N} = \int_0^\infty n(\epsilon) \d\epsilon}
+    \end{formula}
+
+    \begin{formula}{energy}
+        \desc{Energy}{Equal to the classical ideal gas}{}
+        \desc[german]{Energie}{Gleich wie beim klassischen idealen Gas}{}
+        \eq{\braket{E} = \int_0^\infty \epsilon n(\epsilon)\,\d\epsilon = \frac{3}{2} pV}
+    \end{formula}
+
+
+    \begin{formula}{equation_of_state}
+        \desc{Equation of state}{Bosons: decreased pressure, they like to cluster\\Fermions: increased pressure because of the Pauli principle}{\bosfer, $v = \frac{V}{N}$ specific volume}
+        \desc[german]{Zustandsgleichung}{Bosonen: verringerter Druck da sie clustern\\Fermionen: erhöhter Druck durch das Pauli-Prinzip}{\bosfer, $v = \frac{V}{N}$ spezifisches Volumen}
+        \eq{
+            pV &= \kB T \ln Z_g \\
+            \shortintertext{\GT{after} \GT{td:real_gas:virial}}
+               &= N \kB T \left[1 \mp \frac{\lambda^3}{2^{5/2} g v} + \Order\left(\left(\frac{\lambda^3}{v}\right)^2\right)\right]
+        }
+    \end{formula}
+    \begin{formula}{relevance}
+        \desc{Relevance of qm. corrections}{Corrections become relevant when the particle distance is in the order of the thermal wavelength}{}
+        \desc[german]{Relevanz der qm. Korrekturen}{Korrekturen werden relevant, wenn der Teilchenabstand in der Größenordnung der thermischen Wellenlänge ist}{}
+        \eq{\left(\frac{V}{N}\right)^\frac{1}{3} \sim \frac{\lambda}{g_s^\frac{1}{3}}}
+    \end{formula}
+
+    \begin{formula}{generalized_zeta}
+        \desc{Generalized zeta function}{}{}
+        \desc[german]{Verallgemeinerte Zeta-Funktion}{}{}
+        \eq{\left \begin{array}{l}g_\nu(z)\\f_\nu(z)\end{array}\right\} \coloneq \frac{1}{\Gamma(\nu)} \int_0^\infty \d x\, \frac{x^{\nu-1}}{\e^x z^{-1} \mp 1}}
+    \end{formula}
+
+    \Subsection[
+        \eng{Bosons}
+        \ger{Bosonen}
+        ]{bos}
+        \begin{formula}{partition_sum}
+            \desc{Partition sum}{}{$p \in\N_0$}
+            \desc[german]{Zustandssumme}{}{$p \in\N_0$}
+            \eq{Z_\text{g} = \prod_{p} \frac{1}{1-\e^{-\beta(\epsilon_p - \mu)}}}
+        \end{formula}
+        \begin{formula}{occupation}
+            \desc{Occupation number}{Bose-Einstein distribution}{}
+            \desc[german]{Besetzungszahl}{Bose-Einstein Verteilung}{}
+            \eq{\braket{n_p} = \frac{1}{\e^{\beta(\epsilon-\mu)}-1}}
+        \end{formula}
+
+
+    \Subsection[
+        \eng{Fermions}
+        \ger{Fermionen}
+        ]{fer}
+        \begin{formula}{partition_sum}
+            \desc{Partition sum}{}{$p = 0,\,1$}
+            \desc[german]{Zustandssumme}{}{$p = 0,\,1$}
+            \eq{Z_\text{g} = \prod_{p} \left(1+\e^{-\beta(\epsilon_p - \mu)\right)}}
+        \end{formula}
+        \begin{formula}{occupation}
+            \desc{Occupation number}{Fermi-Dirac distribution. At $T=0$ \textit{Fermi edge} at $\epsilon=\mu$}{}
+            \desc[german]{Besetzungszahl}{Fermi-Dirac Verteilung}{Bei $T=0$ \textit{Fermi-Kante} bei $\epsilon=\mu$}
+            \figeq{img/td_fermi_occupation.pdf}{\braket{n_p} = \frac{1}{\e^{\beta(\epsilon-\mu)}+1}}
+        \end{formula}
+
+        \begin{formula}{slater_determinant}
+            \desc{Slater determinant}{}{}
+            \desc[german]{Slater-Determinante}{}{}
+            \eq{
+                \psi(\vecr_1,\vecr_2,\dots,\vecr_N) = \frac{1}{\sqrt{N!}} 
+                \begin{vmatrix} 
+                    p_1(\vecr_1) & p_2(\vecr_1) & \dots & p_N(\vecr_1) \\
+                    p_1(\vecr_2) & p_2(\vecr_2) & \dots & p_N(\vecr_2) \\
+                    \vdots       & \vdots       & \ddots & \vdots \\
+                    p_1(\vecr_N) & p_2(\vecr_N) & \dots & p_N(\vecr_N) \\
+                \end{vmatrix}
+            }
+        \end{formula}
+
+        \begin{formula}{fermi_energy}
+            \desc{Fermi energy}{}{}
+            \desc[german]{Fermienergie}{}{}
+            \eq{\epsilon_\text{F} \coloneq \mu(T = 0)}
+        \end{formula}
+        \begin{formula}{fermi_temperature}
+            \desc{Fermi temperature}{}{}
+            \desc[german]{Fermi Temperatur}{}{}
+            \eq{T_\text{F} \coloneq \frac{\epsilon_\text{F}}{\kB}}
+        \end{formula}
+
+        \begin{formula}{fermi_impulse}
+            \desc{Fermi impulse}{Radius of the \textit{Fermi sphere} in impulse space. States with $p_\text{F}$ are in the \textit{Fermi surface}}{}
+            \desc[german]{Fermi-Impuls}{Radius der \textit{Fermi-Kugel} im Impulsraum. Zustände mit $P_\text{F}$ sind auf der \textit{Fermi-Fläche}}{}
+            \eq{p_\text{F} = \hbar k_\text{F} = (2mE_\text{F})^\frac{1}{2}}
+        \end{formula}
+
+        \begin{formula}{specific_density}
+            \desc{Specific density}{}{$f$ \fqEqRef{td:id_qgas:generalized_zeta}, $g$ degeneracy factor, $z$ \fqEqRef{td:id_qgas:fugacity}}
+            \desc[german]{Spezifische Dichte}{}{$f$ \fqEqRef{td:id_qgas:generalized_zeta}, $g$ Entartungsfaktor, $z$ \fqEqRef{td:id_qgas:fugacity}}
+            \eq{v = \frac{N}{V} = \frac{g}{\lambda^3}f_{3/2}(z)}
+        \end{formula}
+
+        \Subsubsection[
+            \eng{Strong degeneracy}
+            \ger{Starke Entartung}
+            ]{degenerate}
+            \eng[low_temps]{for low temperatures $T \ll T_\text{F}$}
+            \ger[low_temps]{für geringe Temperaturen $T\ll T_\text{F}$}
+
+            \begin{formula}{sommerfeld}
+                \desc{Sommerfeld expansion}{\gt{low_temps}}{}
+                \desc[german]{Sommerfeld-Entwicklung}{\gt{low_temps}}{}
+                \eq{f_\nu(z) = \frac{(\ln z)^\nu}{\Gamma(\nu+1)} \left(1+\frac{\pi^6}{6}\frac{\nu(\nu-1)}{(\ln z)^2} + \dots\right)}
+            \end{formula}
+
+            \begin{formula}{energy_density}
+                \desc{Energy density}{}{}
+                \desc[german]{Energiedichte}{}{}
+                \eq{
+                    \frac{E}{V} &= \frac{3}{2}\frac{g}{\lambda^3} \kB T f_{5/2}(z) \\
+                    \shortintertext{\GT{td:id_qgas:fer:degenerate:sommerfeld}}
+                                &\approx \frac{3}{5} \frac{N}{V} E_\text{F} \left(1+\frac{5\pi^2}{12}\left(\frac{\kB T}{E_\text{F}}\right)^2 \right)
+                }
+            \end{formula}
+
+            \begin{formula}{heat_cap}
+                \desc{Heat capacity}{\gt{low_temps}}{differs from \fqEqRef{td:TODO:petit_dulong}}
+                \desc[german]{Wärmecapacity}{\gt{low_temps}}{weicht ab vom \fqEqRef{td:TODO:petit_dulong}}
+                \figeq{img/td_fermi_heat_capacity.pdf}{C_V = \pdv{E}{T}_V = N\kB \frac{\pi}{2} \left(\frac{T}{T_\text{F}}\right)}
+            \end{formula}
+
+            
+        \TODO{Entartung und Sommerfeld}
+        \TODO{DULONG-PETIT Gesetz}
+

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src/svgs/bravais/fix.sh

@@ -0,0 +1,6 @@
+mkdir -p out
+for file in bravais/*; do
+    link=https:$(grep "Original file" "$file" | grep -Po '(?<=href=")[^"]+')
+    wget "$link" -O "$file"
+    sleep 1
+done

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src/svgs/convertToPdf.sh

@@ -0,0 +1,7 @@
+IFS=$'\n'
+for d in $(find . -type d); do
+    mkdir -p "../img/$d"
+done
+for file in $(find . -type f -name '*.svg'); do
+    inkscape -o "../img/${file%.*}.pdf" "$file"
+done

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src/topo.tex

@@ -1,3 +1,85 @@
 \Part{Topo}
-\Section{berry_phase}
+\Section[
+    \eng{Berry phase / Geometric phase}
+    \ger{Berry-Phase / Geometrische Phase}
+     ]{berry_phase}
+
+    \begin{ttext}[desc]
+        \eng{
+            While adiabatically traversing a closed through the parameter space $R(t)$, the wave function of a systems
+            may pick up an additional phase $\gamma$.\\
+            If $\vec{R}(t)$ varies adiabatically (slowly) and the system is initially in eigenstate $\ket{n}$, 
+            it will stay in an Eigenstate throughout the process (quantum adiabtic theorem).
+        }
+        \ger{
+            Beim adiabatischem Durchlauf eines geschlossenen Weges durch den Parameterraum $R(t)$ kann die Wellenfunktion eines Systems
+            eine zusätzliche Phase $\gamma$ erhalten.\\
+            Wenn $\vec{R}(t)$ adiabatisch (langsam) variiert und das System anfangs im Eigenzustand $\ket{n}$ ist, 
+            bleibt das System während dem Prozess in einem Eigenzustand (Adiabatisches Theorem der Quantenmechanik).
+        }
+    \end{ttext}
+    \Eng[dynamic_phase]{Dynamical Phase}
+    \Eng[berry_phase]{Berry Phase}
+    \Ger[dynamic_phase]{Dynamische Phase}
+    \Ger[berry_phase]{Berry Phase}
+    \begin{formula}{schroedinger_equation}
+        \desc{Schrödinger equation}{}{}
+        \desc[german]{Schrödinger Gleichung}{}{}
+    \eq{H(\vec{R}(t)) \ket{n(\vec{R}(t))} = \epsilon(\vec{R}(t)) \ket{n(\vec{R}(t))}}
+    \end{formula}
+    \begin{formula}{wavefunction}
+        \desc{Wave function}{After full adiabtic loop in $\vec{R}$}{}
+        \desc[german]{Wellenfunktion}{Nach vollem adiabtischem Umlauf in $\vec{R}$}{}
+        \eq{\ket{\psi_n(t)} = \underbrace{\e^{i\gamma_n(t)}}_\text{\GT{berry_phase}} 
+            \underbrace{\e^{\frac{-i}{\hbar} \int^r \epsilon_n(\vec{R}(t`))\d t}}_\text{\GT{dynamic_phase}} \ket{n(\vec{R}(t))}
+        }
+    \end{formula}
+
+    \begin{formula}{berry_connection}
+        \desc{Berry connection}{}{}
+        \desc[german]{Berry connection}{}{}
+        \eq{A_n(\vec{R}) = i\braket{\psi | \nabla_R | \psi}}
+    \end{formula}
+
+    \begin{formula}{berry_curve}
+        \desc{Berry curvature}{Gauge invariant}{}
+        \desc[german]{Berry-Krümmung}{Eichinvariant}{}
+        \eq{\vec{\Omega}_n = \Grad_R \times A_n(\vec{R})}
+    \end{formula}
+
+    \begin{formula}{berry_phase}
+        \desc{Berry phase}{Gauge invariant up to $2\pi$}{}
+        \desc[german]{Berry-Phase}{Eichinvariant bis auf $2\pi$}{}
+        \eq{\gamma_n = \oint_C \d \vec{R} \cdot A_n(\vec{R}) = \int_S \d\vec{S} \cdot \vec{\Omega}_n(\vec{R})}
+    \end{formula}
+
+    \begin{ttext}[chern_number_desc]
+        \eng{The Berry flux through any 2D closed surface is quantized by the \textbf{Chern number}.
+            If there is time-reversal symmetry, the Chern-number is 0.
+        }
+        \ger{Der Berry-Fluß durch eine geschlossene 2D Fl[cher is quantisiert durch die \textbf{Chernzahl}
+            Bei erhaltener Zeitumkehrungssymmetrie ist die Chernzahl 0.
+        }
+    \end{ttext}
+    \begin{formula}{chern_number}
+        \desc{Chern number}{Eg. number of Berry curvature monopoles in the Brillouin zone (then $\vec{R} = \vec{k}$)}{$\vec{S}$ closed surface in $\vec{R}$-space. A \textit{Chern insulator} is a 2D insulator with $C_n \neq 0$}
+        \desc[german]{Chernuzahl}{Z.B. Anzahl der Berry-Krümmungs-Monopole in der Brilouinzone (dann ist $\vec{R} = \vec{k}$). Ein \textit{Chern-Isolator} ist ein 2D Isolator mit $C_n\neq0$}{$\vec{S}$ geschlossene Fläche im $\vec{R}$-Raum}
+        \eq{C_n = \frac{1}{2\pi} \oint \d \vec{S}\ \cdot \vec{\Omega}_n(\vec{R})}
+    \end{formula}
+
+    \TODO{Hall conductance of 2D band insulator (lecture 4 revision)}
+    \begin{formula}{hall_conductance}
+        \desc{Hall conductance of a 2D band insulator}{}{}
+        \desc[german]{Hall-Leitfähigkeit eines 2D Band-Isolators}{}{}
+        \eq{\vec{\sigma}_{xy} = \sum_n \frac{e^2}{h} \int_\text{\GT{occupied}} \d^2k\, \frac{\Omega_{xy}^n}{2\pi} = \sum_n C_n \frac{e^2}{h}}
+    \end{formula}
+
+    \begin{ttext}
+        \eng{A 2D insulator with a non-zero }
+        
+    \end{ttext}
+
+
+
+    
     

src/translations.tex

@@ -1,8 +1,27 @@
-\ENG[angle_deg]{Degree}
+\Eng[angle_deg]{Degree}
-\GER[angle_deg]{Grad}
+\Ger[angle_deg]{Grad}
 
-\ENG[angle_rad]{Radian}
+\Eng[angle_rad]{Radian}
-\GER[angle_rad]{Rad}
+\Ger[angle_rad]{Rad}
+
+\Eng[see_also]{See also}
+\Ger[see_also]{Siehe auch}
+
+\Eng[and_therefore]{and therefore}
+\Ger[and_therefore]{und damit}
+
+\Eng[and_therefore_also]{and therefore also}
+\Ger[and_therefore_also]{und damit auch}
+
+\Eng[time]{Time}
+\Ger[time]{Zeit}
+
+\Eng[ensemble]{Ensemble}
+\Ger[ensemble]{Ensemble}
+
+\Eng[even]{even}
+\Ger[even]{gerade}
+
+\Eng[odd]{odd}
+\Ger[odd]{ungerade}
 
-\ENG[see_also]{See also}
-\GER[see_also]{Siehe auch}