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58
src/maths/analysis.tex → src/analysis.tex
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@ -3,7 +3,6 @@
\ger{Analysis}
]{cal}
\Subsection[
\eng{Convolution}
\ger{Faltung / Konvolution}
@ -64,7 +63,7 @@
\end{formula}
\Eng[real]{real}
\Ger[real]{reellwertig}
\begin{formula}{coefficient-complex}
\begin{formula}{coefficient}
\desc{Fourier coefficients}{Complex representation}{}
\desc[german]{Fourierkoeffizienten}{Komplexe Darstellung}{}
\eq{
@ -113,69 +112,14 @@
\end{enumerate}
\Subsection[
\eng{Misc}
\ger{Verschiedenes}
]{misc}
\begin{formula}{stirling-approx}
\desc{Stirling approximation}{}{}
\desc[german]{Stirlingformel}{}{}
\eq{\ln (N!) \approx N \ln(N) - N + \Order(\ln(N))}
\end{formula}
\begin{formula}{error-function}
\desc{Error function}{\erf: \C \to \C}{}
\desc[german]{Fehlerfunktion}{Error function: \erf: \C \to \C}{}
\eq{
\erf(x) &= \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \d t \\
\erfc(x) &= 1 - \erf(x)\\
&= \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \d t
}
\end{formula}
\Section[
\eng{Logarithm}
\ger{Logarithmus}
]{log}
\begin{formula}{identities}
\desc{Logarithm identities}{}{}
\desc[german]{Logarithmus Identitäten}{Logarithmus Rechenregeln}{}
\eq{
\log(xy) &= \log(x) + \log(y) \\
\log \left(\frac{x}{y}\right) &= \log(x) - \log(y) \\
\log \left(x^d\right) &= d\log(x) \\
\log \left(\sqrt[y]{x}\right) &= \frac{\log(x)}{y} \\
x^{\log(y)} &= y^{\log(x)}
}
\end{formula}
\Section[
\eng{List of common integrals}
\ger{Liste nützlicher Integrale}
]{integrals}
\begin{formula}{spherical-coordinates}
\desc{Spherical coordinates}{}{}
\desc[german]{Kugelkoordinaten}{}{}
\eq{
x &= r \sin\phi,\cos\theta \\
y &= r \cos\phi,\cos\theta \\
z &= r \sin\theta
}
\end{formula}
\begin{formula}{spheical-coordinates-int}
\desc{Integration in spherical coordinates}{}{}
\desc[german]{Integration in Kugelkoordinaten}{}{}
\eq{\iiint\d x \d y \d z= \int_0^{\infty} \!\! \int_0^{2\pi} \!\! \int_0^\pi \d r \d\phi\d\theta \, r^2\sin\theta}
\end{formula}
\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}

2
src/atom.tex
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@ -28,7 +28,7 @@
\end{formula}
\begin{formula}{wave_function}
\desc{Wave function}{}{$R_{nl}(r)$ \fqEqRef{qm:h:radial}, $Y_{lm}$ \fqEqRef{qm:spherical_harmonics}}
\desc{Wave function}{}{}
\desc[german]{Wellenfunktion}{}{}
\eq{\psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_{lm}(\theta,\phi)}
\end{formula}

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

59
src/condensed_matter.tex
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@ -2,7 +2,6 @@
\eng{Condensed matter physics}
\ger{Festkörperphysik}
]{cm}
\TODO{Bonds, hybridized orbitals, tight binding}
\Section[
\eng{Bravais lattice}
\ger{Bravais-Gitter}
@ -85,17 +84,6 @@
\end{tabularx}
\end{adjustbox}
\end{table}
family of plane that are equivalent due to crystal symmetry
\begin{formula}{miller}
\desc{Miller index}{}{}
\desc[german]{Millersche Indizes}{}{}
\eq{
(hkl) & \text{\GT{plane}}\\
[hkl] & \text{\GT{direction}}\\
\{hkl\} & \text{\GT{millerFamily}}
}
\end{formula}
\Section[
@ -407,50 +395,3 @@ family of plane that are equivalent due to crystal symmetry
\centering
\includegraphics[width=\textwidth]{img/cm_mbe_english.pdf}
\end{minipage}
\Section[
\eng{Superconductivity}
\ger{Supraleitung}
]{sc}
\begin{ttext}
\eng{
Materials for which the electric resistance jumps to 0 under a critical temperature.
\\\textbf{Type I}: Has a single critical magnetic field at which the superconuctor becomes a normal conductor.
\\\textbf{Type II}: Has two critical
}
\ger{Materialien, bei denen der elektrische Widerstand beim unterschreiten einer kritischen Temperatur auf 0 springt.}
\end{ttext}
\begin{formula}{meissner_effect}
\desc{Meißner-Ochsenfeld effect}{Perfect diamagnetism}{}
\desc[german]{Meißner-Ochsenfeld Effekt}{Perfekter Diamagnetismus}{}
\ttxt{
\eng{Blabla }
\ger{Blubb blubb }
}
\end{formula}
\Subsection[
\eng{London equation}
\ger{London-Gleichungen}
]{london}
\begin{formula}{first}
% \vec{j} = \frac{nq\hbar}{m}\Grad S - \frac{nq^2}{m}\vec{A}
\desc{First London Equation}{}{$\vec{j}$ current density, $n$, $m$, $q$ density, mass and charge of superconduticng particles}
\desc[german]{Erste London-Gleichung}{}{$\vec{j}$ Stromdichte, $n$, $m$, $q$ Dichte, Masse und Ladung der supraleitenden Teilchen}
\eq{
\partical_t \vec{j} = \frac{nq^2}{m}\vec{E}
}
\end{formula}
\begin{formula}{second}
\desc{Second London Equation}{}{$\vec{j}$ current density, $n$, $m$, $q$ density, mass and charge of superconduticng particles}
\desc[german]{Zweite London-Gleichung}{}{$\vec{j}$ Stromdichte, $n$, $m$, $q$ Dichte, Masse und Ladung der supraleitenden Teilchen}
\eq{
\Rot \vec{j} = -\frac{nq^2}{m} \vec{B}
}
\end{formula}
\begin{formula}{penetration_depth}
\desc{London penetration depth}{}{}
\desc[german]{London Eindringtiefe}{}{}
\eq{\lambda_\textrm{L} = \sqrt{\frac{m}{\mu_0 nq^2}}}
\end{formula}

1
src/electrodynamics.tex
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@ -32,7 +32,6 @@
\Rot \vec{H} &= \vec{j} + \odv{\vec{D}}{t}
}
\end{formula}
\TODO{Polarization, Magnetisation}
\Section[
\eng{Fields}

0
src/maths/geometry.tex → src/geometry.tex
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0
src/maths/linalg.tex → src/linalg.tex
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15
src/main.tex Executable file → Normal file
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@ -145,10 +145,13 @@
\input{util/translations.tex}
\input{maths/linalg.tex}
\input{maths/geometry.tex}
\input{maths/analysis.tex}
\input{maths/probability_theory.tex}
\input{linalg.tex}
\input{geometry.tex}
\input{analysis.tex}
\input{probability_theory.tex}
\input{mechanics.tex}
@ -161,9 +164,9 @@
\input{condensed_matter.tex}
\input{topo.tex}
% \input{topo.tex}
\input{quantum_computing.tex}
% \input{quantum_computing.tex}
% \input{many-body-simulations.tex}

55
src/maths/probability_theory.tex → src/probability_theory.tex
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@ -4,27 +4,21 @@
]{pt}
\begin{formula}{mean}
\desc{Mean}{Expectation value}{}
\desc[german]{Mittelwert}{Erwartungswert}{}
\desc{Mean}{}{}
\desc[german]{Mittelwert}{}{}
\eq{\braket{x} = \int w(x)\, x\, \d x}
\end{formula}
\begin{formula}{variance}
\desc{Variance}{Square of the \fqEqRef{pt:std-deviation}}{}
\desc[german]{Varianz}{Quadrat der\fqEqRef{pt:std-deviation}}{}
\eq{\sigma^2 = (\Delta \hat{x})^2 = \Braket{\hat{x}^2} - \braket{\hat{x}}^2 = \braket{(x - \braket{x})^2}}
\end{formula}
\begin{formula}{covariance}
\desc{Covariance}{}{}
\desc[german]{Kovarianz}{}{}
\eq{\cov(x,y) = \sigma(x,y) = \sigma_{XY} = \Braket{(x-\braket{x})\,(y-\braket{y})}}
\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}
\begin{formula}{std_deviation}
\desc{Standard deviation}{}{}
\desc[german]{Standardabweichung}{}{}
\eq{\sigma = \sqrt{\sigma^2} = \sqrt{(\Delta x)^2}}
\eq{\sigma = \sqrt{(\Delta x)^2}}
\end{formula}
\begin{formula}{median}
@ -198,38 +192,3 @@
}
\end{ttext}
\Section[
\eng{Propagation of uncertainty / error}
\ger{Fehlerfortpflanzung}
]{error}
\begin{formula}{generalised}
\desc{Generalized error propagation}{}{$V$ \fqEqRef{pt:covariance} matrix, $J$ \fqEqRef{ana:jacobi-matrix}}
\desc[german]{Generalisiertes Fehlerfortpflanzungsgesetz}{$V$ \fqEqRef{pt:covariance} Matrix, $J$ \fqEqRef{ana:jacobi-matrix}}{}
\eq{V_y = J(x) \cdot V_x \cdot J^{\T} (x)}
\end{formula}
\begin{formula}{uncorrelated}
\desc{Propagation of uncorrelated errors}{Linear approximation}{}
\desc[german]{Fortpflanzung unabhängiger fehlerbehaftete Größen}{Lineare Näherung}{}
\eq{u_y = \sqrt{ \sum_{i} \left(\pdv{y}{x_i}\cdot u_i\right)^2}}
\end{formula}
\begin{formula}{weight}
\desc{Weight}{Variance is a possible choice for a weight}{$\sigma$ \fqEqRef{pt:variance}}
\desc[german]{Gewicht}{Varianz ist eine mögliche Wahl für ein Gewicht}{}
\eq{w_i = \frac{1}{\sigma_i^2}}
\end{formula}
\begin{formula}{weighted-mean}
\desc{Weighted mean}{}{$w_i$ \fqEqRef{pt:error:weight}}
\desc[german]{Gewichteter Mittelwert}{}{}
\eq{\overline{x} = \frac{\sum_{i} (x_i w_i)}{\sum_i w_i}}
\end{formula}
\begin{formula}{weighted-mean-error}
\desc{Variance of weighted mean}{}{$w_i$ \fqEqRef{pt:error:weight}}
\desc[german]{Varianz des gewichteten Mittelwertes}{}{}
\eq{\sigma^2_{\overline{x}} = \frac{1}{\sum_i w_i}}
\end{formula}

6
src/quantum_mechanics.tex
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@ -379,12 +379,6 @@
}
\end{formula}
\begin{formula}{c_a_matrices}
\desc{Matrix forms}{}{}
\desc[german]{Matrix-Form}{}{}
\eq{\TODO{TODO}}
\end{formula}
\Subsubsection[
\eng{Harmonischer Oszillator}
\ger{Harmonic Oscillator}

8
src/readme.md
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@ -20,18 +20,10 @@ The `<partname>:...:<lowest section name>` will be defined as `fqname` (fully qu
- figure: `fig`
- parts, (sub)sections: `sec`
### Reference functions
Functions that create a hyperlink (and use the translation of the target element as link name):
- `\fqSecRef{}`
- `\fqEqRef{}`
## Multilanguage
All text should be defined as a translation (`translations` package, see `util/translation.tex`) and then used using the `gt` or `GT` macros.
The english translation of any key must be defined, because it will also be used as fallback.
Lower case macros are relative to the current `fqname`, while upper case macros are absolute.
Never make a macro that would have to be changed if a new language was added, eg dont do
```tex
% 1: key, 2: english version, 3: german version

43
src/statistical_mechanics.tex
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@ -332,7 +332,7 @@
\desc[german]{Entropiedichte}{}{$s = \frac{S}{N}$}
\eq{
\lim_{T\to 0} s(T) &= 0 \\
\shortintertext{\GT{and_therefore_also}}
\shortintertext{\GT{and_therefore_also}} \\
\lim_{T\to 0} c_V &= 0
}
\end{formula}
@ -376,21 +376,21 @@
\desc[german]{Innere Energie}{}{}
\eq{\d U(S,V,N) = T\d S -p\d V + \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}{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{Free enthalpy / Gibbs energy}{}{}
\desc[german]{Freie Entahlpie / Gibbs-Energie}{}{}
\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}{}{}
@ -398,27 +398,6 @@
\end{formula}
\TODO{Maxwell Relationen, TD Quadrat}
\begin{formula}{td-square}
\desc{Thermodynamic squre}{}{}
\desc[german]{Themodynamisches Quadrat}{Guggenheim Quadrat}{}
\content{
\begin{tikzpicture}
\draw[thick] (0,0) grid (3,3);
\node at (0.5, 2.5) {$-S$};
\node at (1.5, 2.5) {\color{blue}$U$};
\node at (2.5, 2.5) {$V$};
\node at (0.5, 1.5) {\color{blue}$H$};
\node at (2.5, 1.5) {\color{blue}$F$};
\node at (0.5, 0.5) {$-p$};
\node at (1.5, 0.5) {\color{blue}$G$};
\node at (2.5, 0.5) {$T$};
\end{tikzpicture}
\begin{ttext}
\eng{The corners opposite from the potential are the coefficients and each coefficients differential is opposite to it.}
\ger{Die Ecken gegenüber des Potentials sind die Koeffizienten, das Differential eines Koeffizienten ist in der Ecke gegenüber.}
\end{ttext}
}
\end{formula}
\Section[
\eng{Ideal gas}
@ -558,8 +537,8 @@
\end{formula}
\begin{formula}{lennard_jones}
\desc{Lennard-Jones potential}{Potential between two molecules. Attractive for $r > \sigma$, repulsive for $r < \sigma$.\\ In condensed matter: Attraction due to Landau Dispersion \TODO{verify} and repulsion due to Pauli exclusion principle.}{}
\desc[german]{Lennard-Jones-Potential}{Potential zwischen zwei Molekülen. Attraktiv für $r > \sigma$, repulsiv für $r < \sigma$.\\ In Festkörpern: Anziehung durch Landau Dispesion und Abstoßung durch Pauli-Prinzip.}{}
\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}
@ -679,7 +658,7 @@
\eng{Bosons}
\ger{Bosonen}
]{bos}
\begin{formula}{partition-sum}
\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)}}}

0
src/svgs/convertToPdf.sh Normal file → Executable file
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43
src/util/environments.tex
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@ -15,13 +15,6 @@
\def\descwidth{0.3\textwidth}
\def\eqwidth{0.6\textwidth}
%
% FORMULA ENVIRONMENT
% The following commands are meant to be used with the formula environment
%
% Name in black and below description in gray
% [1]: minipage width
% 2: fqname of name
% 3: fqname of a translation that holds the explanation
@ -140,54 +133,32 @@
}
% 1: key
\newenvironment{formula}[1]{
% [1]: language
% 2: name
% 3: description
% 4: definitions/links
% 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}}
}
% 1: equation for align environment
\newcommand{\eq}[1]{
\insertEquation{#1}{##1}
}
% 1: equation for alignat environment
\newcommand{\eqAlignedAt}[2]{
\insertAlignedAt{#1}{##1}{##2}
}
% 1: equation for flalign environment
\newcommand{\eqFLAlign}[1]{
\insertFLAlign{#1}{##1}
}
% 1: file path
% 2: equation
\newcommand{\figeq}[2]{
\insertEquationWithFigure{#1}{##1}{##2}
}
% 1: any content
\newcommand{\content}[1]{
\NameLeftContentRight{#1}{##1}
}
% 1: content for the ttext environment
\newcommand{\ttxt}[1]{
\NameLeftContentRight{#1}{
\begin{ttext}[#1:desc]
##1
\end{ttext}
}
}
}{\ignorespacesafterend}
%
% QUANTITY
%
\newenvironment{quantity}[5]{
% key, symbol, si unit, si base units, comment (key to translation)
\newcommand{\desc}[3][english]{
@ -195,6 +166,15 @@
\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}
@ -210,9 +190,6 @@
%
% DISTRIBUTION
%
\def\distrightwidth{0.45\textwidth}
\def\distleftwidth{0.45\textwidth}

2
src/util/macros.tex
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@ -37,8 +37,6 @@
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\const}{const}
\DeclareMathOperator{\erf}{erf}
\DeclareMathOperator{\erfc}{erfc}
\DeclareMathOperator{\cov}{cov}
% diff, for integrals and stuff
% \DeclareMathOperator{\dd}{d}
\renewcommand*\d{\mathop{}\!\mathrm{d}}