work

src/atom.tex

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

src/calculus.tex

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

src/condensed_matter.tex

@@ -2,6 +2,7 @@
     \eng{Condensed matter physics}
     \ger{Festkörperphysik}
     ]{cm}
+    \TODO{Bonds, hybridized orbitals, tight binding}
 \Section[
     \eng{Bravais lattice}
     \ger{Bravais-Gitter}
@@ -85,6 +86,17 @@
         \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[
     \eng{Reciprocal lattice}
@@ -395,3 +407,50 @@
             \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}
+

src/electrodynamics.tex

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

src/main.tex

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

src/maths/analysis.tex

@@ -3,6 +3,7 @@
     \ger{Analysis}
     ]{cal}
 
+
     \Subsection[
         \eng{Convolution}
         \ger{Faltung / Konvolution}
@@ -63,7 +64,7 @@
             \end{formula}
             \Eng[real]{real}
             \Ger[real]{reellwertig}
-            \begin{formula}{coefficient}
+            \begin{formula}{coefficient-complex}
                 \desc{Fourier coefficients}{Complex representation}{}
                 \desc[german]{Fourierkoeffizienten}{Komplexe Darstellung}{}
                 \eq{
@@ -112,14 +113,69 @@
             \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}
 
+
+

src/maths/probability_theory.tex

@@ -4,21 +4,27 @@
     ]{pt}
 
     \begin{formula}{mean}
-        \desc{Mean}{}{}
+        \desc{Mean}{Expectation value}{}
-        \desc[german]{Mittelwert}{}{}
+        \desc[german]{Mittelwert}{Erwartungswert}{}
         \eq{\braket{x} = \int w(x)\, x\, \d x}
     \end{formula}
 
     \begin{formula}{variance}
-        \desc{Variance}{}{}
+        \desc{Variance}{Square of the \fqEqRef{pt:std-deviation}}{}
-        \desc[german]{Varianz}{}{}
+        \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}}
+        \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}{covariance}
+        \desc{Covariance}{}{}
+        \desc[german]{Kovarianz}{}{}
+        \eq{\cov(x,y) = \sigma(x,y) = \sigma_{XY} = \Braket{(x-\braket{x})\,(y-\braket{y})}}
+    \end{formula}
+    
+    \begin{formula}{std-deviation}
         \desc{Standard deviation}{}{}
         \desc[german]{Standardabweichung}{}{}
-        \eq{\sigma = \sqrt{(\Delta x)^2}}
+        \eq{\sigma = \sqrt{\sigma^2} = \sqrt{(\Delta x)^2}}
     \end{formula}
 
     \begin{formula}{median}
@@ -192,3 +198,38 @@
             }
         \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}
+
+

src/quantum_mechanics.tex

@@ -379,6 +379,12 @@
                 }
             \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}

src/readme.md

@@ -20,10 +20,18 @@ 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

src/statistical_mechanics.tex

@@ -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{Gibbs energy}{}{}
+            \desc{Free enthalpy / Gibbs energy}{}{}
-            \desc[german]{Gibbsche Energie}{}{}
+            \desc[german]{Freie Entahlpie / Gibbs-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}{}{}
@@ -398,6 +398,27 @@
         \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}
@@ -537,8 +558,8 @@
         \end{formula}
 
         \begin{formula}{lennard_jones}
-            \desc{Lennard-Jones potential}{Potential between two molecules. Attractive for $r > \sigma$, repulsive for $r < \sigma$}{}
+            \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 $r < \sigma$}{}
+            \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.}{}
             \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}
 
@@ -658,7 +679,7 @@
         \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)}}}

src/util/environments.tex

@@ -15,6 +15,13 @@
 \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
@@ -133,32 +140,54 @@
 
 }
 
+% 1: key
 \newenvironment{formula}[1]{
-    % key
+    % [1]: language
+    % 2: name
+    % 3: description
+    % 4: definitions/links
     \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]{
@@ -166,15 +195,6 @@
         \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}
@@ -190,6 +210,9 @@
 
 
 
+%
+% DISTRIBUTION
+%
 \def\distrightwidth{0.45\textwidth}
 \def\distleftwidth{0.45\textwidth}
 

src/util/macros.tex

@@ -37,6 +37,8 @@
 \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}}