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.aux
out
*.virtual_documents*
*ipynb_checkpoints*
*__pycache__*
**.pdf

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\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}

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src/atom.tex
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\def\masse{m_\textrm{e}}
\def\grad{\vec{\nabla}}
\def\vecr{\vec{r}}
\def\abohr{a_\textrm{B}}
\def\vecr{{\vec{r}}}
\kef\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)
% &= \frac{\hat{p}_r^2}{2\mu} + \frac{\hat{L}^2}{2\mu r} + V(r)
\hat{H} &= -\frac{\hbar^2}{2\mu} {\Grad_\vecr}^2 - V(\vecr) \\
&= \frac{\hat{p}_r^2}{2\mu} + \frac{\hat{L}^2}{2\mu r} + V(r)
}
\end{formula}
@ -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}
\begin{ttext}{desc}
]{lamb_shift}
\begin{ttext}[desc]
\eng{The interaction of the electron with virtual photons emitted/absorbed by the nucleus leads to a (very small) shift in the energy level.}
\ger{The Wechselwirkung zwischen dem Elektron und vom Kern absorbierten/emittierten virtuellen Photonen führt zu einer (sehr kleinen) Energieverschiebung.}
\end{ttext}
@ -136,11 +134,10 @@
\Subsubsection[
\eng{Hyperfine structure}
\ger{Hyperfeinstruktur}
]{hyperfine_structure}
\begin{ttext}{desc}
\eng{Interaction of the nucleus spin with the magnetic field created by the electron leads to energy shifts. (Lifts degenaracy) }
]{hyperfine_structure}
\begin{ttext}[desc]
\eng{Interaction of the nucleus spin with the magnetic field created by the electron leads to energy shifts. (Lifts degeneracy) }
\ger{Wechselwirkung von Kernspin mit dem vom Elektron erzeugten Magnetfeld spaltet Energieniveaus}
\end{ttext}
\begin{formula}{nuclear_spin}
\desc{Nuclear spin}{}{}

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\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}

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% 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}}

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\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}

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\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}

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\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}
}

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\Part[
\eng{Analysis}
\ger{Analysis}
]{ana}
\eng{Geometry}
\ger{Geometrie}
]{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}

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\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}

47
src/macros.tex Normal file
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@ -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}}

178
src/main.tex
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@ -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)
@ -41,6 +52,7 @@
\underset{\substack{\uparrow\\\mathrlap{\text{\hspace{-1em}#2}}}}{#1}}
\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{\ENG}[2][\fqname]{\DT[#1]{english}{#2}}
\newcommand{\Ger}[2][\fqname]{\DT[#1]{german}{#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]{
\insertEquationLine{#1}{
\begin{align}
\label{eq:\fqname:#1}
#2
\end{align}
}
}
\input{circuit.tex}
\newcommand{\insertFLAlign}[2]{ % eq name, #cols, eq
\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}
% some translations
\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}

4
src/mechanics.tex
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@ -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 $\vec{x} = \begin{pmatrix} \cos\varphi \\ \sin\varphi \end{pmatrix}$.

198
src/probability_theory.tex Normal file
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@ -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}

418
src/quantum_computing.tex
View File

@ -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}
\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}
\ger{Wenn zwei Supraleiter durch einen dünnen Isolator getrennt sind, können Cooper-Paare durch den Isolator tunneln.}
\end{ttext}
\eng{Superconducting qubits}
\ger{Supraleitende qubits}
]{scq}
\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}
\Subsection[
\eng{Building blocks}
\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}{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}{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}{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}

220
src/quantum_mechanics.tex
View File

@ -17,10 +17,10 @@
\eng{Operators}
\ger{Operatoren}
]{op}
\GER[row_vector]{Zeilenvektor}
\GER[column_vector]{Spaltenvektor}
\ENG[column_vector]{Column vector}
\ENG[row_vector]{Row vector}
\Ger[row_vector]{Zeilenvektor}
\Ger[column_vector]{Spaltenvektor}
\Eng[column_vector]{Column vector}
\Eng[row_vector]{Row vector}
\begin{formula}{dirac_notation}
\desc{Dirac notation}{}{}
\desc[german]{Dirac-Notation}{}{}
@ -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}{}{}
@ -116,27 +110,40 @@
\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}
\desc{Canonical commutation relation}{}{$x$, $p$ canonical conjugates}
\desc[german]{Kanonische Vertauschungsrelationen}{}{$x$, $p$ kanonische konjugierte}
@ -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}
\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}
\eng{Time evolution}
\ger{Zeitentwicklug}
]{time}
The time evolution of the Hamiltonian is given by:
\begin{formula}{time_evolution_op}
\desc{Time evolution operator}{}{$U$ unitary}
@ -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}
\eng[desc]{The following holds true if the pertubation $\hat{H_1}$ is sufficently small and the $E^{(0)}_n$ levels are not degenerate.}
\ger[desc]{Die folgenden Gleichungen gelten wenn $\hat{H_1}$ ausreichend klein ist und die $E_n^{(0)}$ Niveaus nicht entartet sind.}
\gt{desc}
]{qm_pertubation}
\begin{ttext}
\eng[desc]{The following holds true if the pertubation $\hat{H_1}$ is sufficently small and the $E^{(0)}_n$ levels are not degenerate.}
\ger[desc]{Die folgenden Gleichungen gelten wenn $\hat{H_1}$ ausreichend klein ist und die $E_n^{(0)}$ Niveaus nicht entartet sind.}
\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}

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import qutip as qt

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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

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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)

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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")

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numpy
matplotlib
scqubits
qutip

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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")

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\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 Normal file
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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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After

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

208
src/svgs/td_gay_lussac.svg Normal file
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@ -0,0 +1,208 @@
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src/topo.tex
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\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}

31
src/translations.tex
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@ -1,8 +1,27 @@
\ENG[angle_deg]{Degree}
\GER[angle_deg]{Grad}
\Eng[angle_deg]{Degree}
\Ger[angle_deg]{Grad}
\ENG[angle_rad]{Radian}
\GER[angle_rad]{Rad}
\Eng[angle_rad]{Radian}
\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}