update
This commit is contained in:
Matthias@Dell
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106
src/atom.tex
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106
src/atom.tex
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\def\masse{m_\textrm{e}}
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\def\grad{\vec{\nabla}}
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\def\vecr{\vec{r}}
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\def\abohr{a_\textrm{B}}
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\Section[
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\eng{Hydrogen Atom}
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\ger{Wasserstoffatom}
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]{h}
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\begin{formula}{reduced_mass}
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\desc{Reduced mass}{}{}
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\desc[german]{Reduzierte Masse}{}{}
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\eq{\mu = \frac{\masse m_\textrm{K}}{\masse + m_\textrm{K}} \explOverEq[\approx]{$\masse \ll m_\textrm{K}$} \masse}
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\end{formula}
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\begin{formula}{potential}
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\desc{Coulumb potential}{For a single electron atom}{$Z$ atomic number}
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\desc[german]{Coulumb potential}{Für ein Einelektronenatom}{$Z$ Ordnungszahl/Kernladungszahl}
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\eq{V(\vecr) = \frac{Z\,e^2}{4\pi\epsilon_0 r}}
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\end{formula}
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\begin{formula}{hamiltonian}
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\desc{Hamiltonian}{}{}
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\desc[german]{Hamiltonian}{}{}
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\eq{\hat{H} &= -\frac{\hbar^2}{2\mu} {\grad_\vec{r}}^2 - V(\vecr) \\
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&= \frac{\hat{p}_r^2}{2\mu} + \frac{\hat{L}^2}{2\mu r} + V(r)}
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\end{formula}
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\begin{formula}{wave_function}
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\desc{Wave function}{}{}
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\desc[german]{Wellenfunktion}{}{}
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\eq{\psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_{lm}(\theta,\phi)}
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\end{formula}
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\begin{formula}{radial}
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\desc{Radial part}{}{$L_r^s(x)$ Laguerre-polynomials}
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\desc[german]{Radialanteil}{}{$L_r^s(x)$ Laguerre-Polynome}
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\eq{
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R_{nl} &= - \sqrt{\frac{(n-l-1)!(2\kappa)^3}{2n[(n+l)!]^3}} (2\kappa r)^l \e^{-\kappa r} L_{n+1}^{2l+1}(2\kappa r)
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\shortintertext{\GT{with}}
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\kappa &= \frac{\sqrt{2\mu\abs{E}}}{\hbar} = \frac{Z}{n \abohr}
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}
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\end{formula}
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\begin{formula}{energy}
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\desc{Energy eigenvalues}{}{}
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\desc[german]{Energieeigenwerte}{}{}
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\eq{E_n &= \frac{Z^2\mu e^4}{n^2(4\pi\epsilon_0)^2 2\hbar^2} = -E_\textrm{H}\frac{Z^2}{n^2}}
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\end{formula}
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\begin{formula}{rydberg_energy}
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\desc{Rydberg energy}{}{}
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\desc[german]{Rydberg-Energy}{}{}
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\eq{E_\textrm{H} = h\,c\,R_\textrm{H} = \frac{\mu e^4}{(4\pi\epsilon_0)^2 2\hbar^2}}
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\end{formula}
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\Subsection[
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\eng{Corrections}
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\ger{Korrekturen}
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]{corrections}
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\Subsubsection[
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\eng{Darwin term}
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\ger{Darwin-Term}
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]{darwin}
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\begin{ttext}{desc}
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\eng{Relativisitc correction: Because of the electrons zitterbewegung, it is not entirely localised. \TODO{fact check}}
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\ger{Relativistische Korrektur: Elektronen führen eine Zitterbewegung aus und sind nicht vollständig lokalisiert.}
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\end{ttext}
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\begin{formula}{energy_shift}
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\desc{Energy shift}{}{}
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\desc[german]{Energieverschiebung}{}{}
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\eq{\Delta E_\textrm{rel} = -E_n \frac{Z^2\alpha^2}{n} \Big(\frac{3}{4n} - \frac{1}{l+ \frac{1}{2}}\Big)}
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\end{formula}
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\begin{formula}{fine_structure_constant}
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\desc{Fine-structure constant}{Sommerfeld constant}{}
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\desc[german]{Feinstrukturkonstante}{Sommerfeldsche Feinstrukturkonstante}{}
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\eq{\alpha = \frac{e^2}{4\pi\epsilon_0\hbar c} \approx \frac{1}{137}}
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\end{formula}
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\Subsubsection[
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\eng{Spin-orbit coupling (LS-coupling)}
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\ger{Spin-Bahn-Kopplung (LS-Kopplung)}
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]{ls_coupling}
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\begin{ttext}{desc}
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\eng{The interaction of the electron spin with the electrostatic field of the nuclei lead to energy shifts.}
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\ger{The Wechselwirkung zwischen dem Elektronenspin und dem elektrostatischen Feld des Kerns führt zu Energieverschiebungen.}
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\end{ttext}
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\begin{formula}{energy_shift}
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\desc{Energy shift}{}{}
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\desc[german]{Energieverschiebung}{}{}
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\eq{\Delta E_\text{LS} = \frac{\mu_0 Z e^2}{8\pi m^2 e\,r^3} \braket{\vec{S} \cdot \vec{L}}}
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\end{formula}
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\begin{formula}{sl}
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\desc{\TODO{name}}{}{}
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\desc[german]{??}{}{}
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\eq{\braket{\vec{S} \cdot \vec{L}} &= \frac{1}{2} \braket{[J^2-L^2-S^2]} \nonumber \\
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&= \frac{\hbar^2}{2}[j(j+1) -l(l+1) -s(s+1)]}
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\end{formula}
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\Subsubsection[
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\eng{Fine-structure}
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\ger{Feinstruktur}
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]{fine_structure}
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\begin{ttext}{desc}
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\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}.
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\ger{Die Feinstruktur vereint relativistische Korrekturen \ref{sec:qm:h:corrections:darwin} und die Spin-Orbit-Kupplung \ref{sec:qm:h:corrections:ls_coupling}.
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\end{ttext}
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290
src/main.tex
290
src/main.tex
@ -3,6 +3,7 @@
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\usepackage[english]{babel}
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\usepackage[english]{babel}
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\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
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\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
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\usepackage{mathtools}
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\usepackage{mathtools}
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\usepackage{esdiff} % derivatives
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\usepackage{braket}
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\usepackage{braket}
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\usepackage{graphicx}
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\usepackage{graphicx}
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\usepackage{etoolbox}
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\usepackage{etoolbox}
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@ -24,9 +25,217 @@
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\sisetup{exponent-product=\ensuremath{\cdot}}
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\sisetup{exponent-product=\ensuremath{\cdot}}
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\DeclarePairedDelimiter\abs{\lvert}{\rvert}
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\DeclarePairedDelimiter\abs{\lvert}{\rvert}
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\DeclareMathOperator{\e}{e}
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\usepackage{translations}
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\usepackage{translations}
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\newcommand{\TODO}[1]{{\color{red}#1}}
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% put an explanation above an equal sign
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% [1]: equality sign (or anything else)
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% 2: text (not in math mode!)
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\newcommand{\explUnderEq}[2][=]{%
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\underset{\substack{\uparrow\\\mathrlap{\text{\hspace{-1em}#2}}}}{#1}}
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\newcommand{\explOverEq}[2][=]{%
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\overset{\substack{\mathrlap{\text{\hspace{-1em}#2}}\\\downarrow}}{#1}}
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%
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% TRANSLATION COMMANDS
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%
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% The lower case commands use \fqname based keys, the upper case absolute keys.
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% Example:
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% \dt[example]{german}{Beispiel} % defines the key \fqname:example
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% \ger[example]{Beispiel} % defines the key \fqname:example
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% \DT[example]{german}{Beispiel} % defines the key example
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% \Ger[example]{Beispiel} % defines the key example
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%
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% For ease of use in the ttext environment and the optional argument of the \Part, \Section, ... commands,
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% all "define translation" commands use \fqname as default key
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% Get a translation
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% expandafter required because the translation commands dont expand anything
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% shortcuts for translations
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% 1: key
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\newcommand{\gt}[1]{\expandafter\GetTranslation\expandafter{\fqname:#1}}
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\newcommand{\GT}[1]{\expandafter\GetTranslation\expandafter{#1}}
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\newcommand{\IfTranslationExists}{
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\IfTranslation{\languagename}
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}
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\newcommand{\iftranslation}[1]{\expandafter\IfTranslationExists\expandafter{\fqname:#1}}
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% Define a new translation
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% [1]: key, 2: lang, 3: translation
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\newcommand{\dt}[3][\fqname]{
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% hack because using expandafter on the second arg didnt work
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\def\tempaddtranslation{\addtranslation{#2}}
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\ifstrequal{#1}{\fqname}{
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\expandafter\tempaddtranslation\expandafter{\fqname}{#3}
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}{
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\expandafter\tempaddtranslation\expandafter{\fqname:#1}{#3}
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}
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}
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\newcommand{\DT}[3][\fqname]{
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% hack because using expandafter on the second arg didnt work
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\def\tempaddtranslation{\addtranslation{#2}}
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\ifstrequal{#1}{\fqname}{
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\expandafter\tempaddtranslation\expandafter{\fqname}{#3}
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}{
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\expandafter\tempaddtranslation\expandafter{#1}{#3}
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}
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}
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% [1]: key, 2: translation
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\newcommand{\ger}[2][\fqname]{\dt[#1]{german}{#2}}
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\newcommand{\eng}[2][\fqname]{\dt[#1]{english}{#2}}
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\newcommand{\GER}[2][\fqname]{\DT[#1]{german}{#2}}
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\newcommand{\ENG}[2][\fqname]{\DT[#1]{english}{#2}}
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% use this to define text in different languages for the key <env arg>
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% the translation for <env arg> when the environment ends.
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% (temporarily change fqname to the \fqname:<env arg> to allow
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% the use of \eng and \ger without the key parameter)
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\newenvironment{ttext}[1]{
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\edef\realfqname{\fqname}
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\edef\fqname{\fqname:#1}
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}{
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\expandafter\GT\expandafter{\fqname} \\
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\edef\fqname{\realfqname}
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}
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% "automate" sectioning
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% start <section>, get heading from translation, set label
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% fqname is the fully qualified name: the keys of all previous sections joined with a ':'
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% [1]: code to run after setting \fqname, but before the \part, \section etc
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% 2: key
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\newcommand{\Part}[2][desc]{
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\newpage
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\def\partname{#2}
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\def\sectionname{}
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\def\subsectionname{}
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\def\subsubsectionname{}
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\edef\fqname{\partname}
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#1
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\part{\GT{\fqname}}
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\label{sec:\fqname}
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}
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\newcommand{\Section}[2][]{
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\def\sectionname{#2}
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\def\subsectionname{}
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\def\subsubsectionname{}
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\edef\fqname{\partname:\sectionname}
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#1
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\section{\GT{\fqname}}
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\label{sec:\fqname}
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}
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% \newcommand{\Subsection}[1]{\Subsection{#1}{}}
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\newcommand{\Subsection}[2][]{
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\def\subsectionname{#2}
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\def\subsubsectionname{}
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\edef\fqname{\partname:\sectionname:\subsectionname}
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#1
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\subsection{\GT{\fqname}}
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\label{sec:\fqname}
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}
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\newcommand{\Subsubsection}[2][]{
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\def\subsubsectionname{#2}
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\edef\fqname{\partname:\sectionname:\subsectionname:\subsubsectionname}
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#1
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\subsubsection{\GT{\fqname}}
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\label{sec:\fqname}
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}
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\usepackage{xstring}
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\newcommand{\insertEquationLine}[2]{
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\par\noindent\ignorespaces
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% \textcolor{gray}{\hrule}
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\vspace{0.5\baselineskip}
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% \fbox{
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\begin{minipage}{0.3\textwidth}
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\iftranslation{#1}{
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\raggedright
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\gt{#1}
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}{}
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\iftranslation{#1_desc}{
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\\ {\color{darkgray} \gt{#1_desc}}
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}{}
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\end{minipage}
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% }
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\hfill
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\fbox{
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\begin{minipage}{0.6\textwidth}
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% \vspace{-\baselineskip} % remove the space that comes from starting a new paragraph
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#2 %
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\noindent\iftranslation{#1_defs}{
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\begingroup
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\color{darkgray}
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\gt{#1_defs}
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% \edef\temp{\GT{#1_defs}}
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% \expandafter\StrSubstitute\expandafter{\temp}{:}{\\}
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\endgroup
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}{}
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% \vspace{-\baselineskip} % remove the space that comes from starting a new paragraph
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\end{minipage}
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}
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\textcolor{lightgray}{\hrule}
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\vspace{0.5\baselineskip}
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% \par
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% \hrule
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}
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\newcommand{\insertEquation}[2]{
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\insertEquationLine{#1}{
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\begin{align}
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\label{eq:\fqname:#1}
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#2
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\end{align}
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}
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}
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\newcommand{\insertFLAlign}[2]{ % eq name, #cols, eq
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\insertEquationLine{#1}{%
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\begin{flalign}%
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% dont place label when one is provided
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% \IfSubStringInString{label}\unexpanded{#3}{}{
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% \label{eq:#1}
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% }
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#2%
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\end{flalign}
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}
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}
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\newcommand{\insertAlignedAt}[3]{ % eq name, #cols, eq
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\insertEquationLine{#1}{%
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\begin{alignat}{#2}%
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% dont place label when one is provided
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% \IfSubStringInString{label}\unexpanded{#3}{}{
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% \label{eq:#1}
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% }
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#3%
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\end{alignat}
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}
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}
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\newenvironment{formula}[1]{
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% key
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\newcommand{\desc}[4][english]{
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% language, name, description, definitions
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\dt[#1]{##1}{##2}
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\ifblank{##3}{}{\dt[#1_desc]{##1}{##3}}
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\ifblank{##4}{}{\dt[#1_defs]{##1}{##4}}
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}
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\newcommand{\eq}[1]{
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\insertEquation{#1}{##1}
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}
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\newcommand{\eqAlignedAt}[2]{
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\insertAlignedAt{#1}{##1}{##2}
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}
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\newcommand{\eqFLAlign}[1]{
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\insertFLAlign{#1}{##1}
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}
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}{\ignorespacesafterend}
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\title{Formelsammlung}
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\title{Formelsammlung}
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\author{Matthias Quintern}
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\author{Matthias Quintern}
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\date{\today}
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\date{\today}
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@ -34,10 +243,9 @@
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\begin{document}
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\begin{document}
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\maketitle
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\maketitle
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% \thispagestyle{empty}
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\tableofcontents
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% \tableofcontents
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\newpage
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% \newpage
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\setcounter{page}{1}
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% \setcounter{page}{1}
|
|
||||||
|
|
||||||
% \nuwcommand{\eq}[4][desc]{
|
% \nuwcommand{\eq}[4][desc]{
|
||||||
% \vspace*{0.1cm}
|
% \vspace*{0.1cm}
|
||||||
@ -56,75 +264,8 @@
|
|||||||
% \end{minipage}
|
% \end{minipage}
|
||||||
% \newline
|
% \newline
|
||||||
% }
|
% }
|
||||||
\newcommand{\insertEquation}[2]{
|
|
||||||
\vspace*{0.1cm}
|
|
||||||
\begin{minipage}{0.3\textwidth}
|
|
||||||
\IfTranslation{\languagename}{#1}{
|
|
||||||
\raggedright
|
|
||||||
\GetTranslation{#1}
|
|
||||||
}{}
|
|
||||||
\IfTranslation{\languagename}{#1_desc}{
|
|
||||||
\\ {\color{gray} \GetTranslation{#1_desc}}
|
|
||||||
}{}
|
|
||||||
\end{minipage}
|
|
||||||
\begin{minipage}{0.7\textwidth}
|
|
||||||
\begin{align}
|
|
||||||
\label{eq:#1}
|
|
||||||
#2
|
|
||||||
\end{align}
|
|
||||||
\IfTranslation{\languagename}{#1_defs}{
|
|
||||||
{\color{gray} \GetTranslation{#1_defs}}
|
|
||||||
}{}
|
|
||||||
\end{minipage}
|
|
||||||
\newline
|
|
||||||
}
|
|
||||||
|
|
||||||
\newcommand{\insertAlignedAt}[3]{ % eq name, #cols, eq
|
|
||||||
\vspace*{0.1cm}
|
|
||||||
\begin{minipage}{0.3\textwidth}
|
|
||||||
\IfTranslation{\languagename}{#1}{
|
|
||||||
\raggedright
|
|
||||||
\GetTranslation{#1}
|
|
||||||
}{}
|
|
||||||
\IfTranslation{\languagename}{#1_desc}{
|
|
||||||
\\ {\color{gray} \GetTranslation{#1_desc}}
|
|
||||||
}{}
|
|
||||||
\end{minipage}
|
|
||||||
\begin{minipage}{0.7\textwidth}
|
|
||||||
\begin{alignat}{#2}
|
|
||||||
% dont place label when one is provided
|
|
||||||
\IfSubStringinString{label}{#3}{}{
|
|
||||||
\label{eq:#1}
|
|
||||||
}
|
|
||||||
#3
|
|
||||||
\end{alignat}
|
|
||||||
\IfTranslation{\languagename}{#1_defs}{
|
|
||||||
{\color{gray} \GetTranslation{#1_defs}}
|
|
||||||
}{}
|
|
||||||
\end{minipage}
|
|
||||||
\newline
|
|
||||||
}
|
|
||||||
|
|
||||||
\newenvironment{formula}[1]{
|
|
||||||
\newcommand{\desc}[4][english]{
|
|
||||||
% language, name, description, definitions
|
|
||||||
\definetranslation{##1}{#1}{##2}
|
|
||||||
\ifblank{##3}{}{\definetranslation{##1}{#1_desc}{##3}}
|
|
||||||
\ifblank{##4}{}{\definetranslation{##1}{#1_defs}{##4}}
|
|
||||||
}
|
|
||||||
\newcommand{\eq}[1]{
|
|
||||||
\insertEquation{#1}{##1}
|
|
||||||
}
|
|
||||||
\newcommand{\eqAlignedAt}[2]{
|
|
||||||
\insertAlignedAt{#1}{##1}{##2}
|
|
||||||
}
|
|
||||||
}{}
|
|
||||||
|
|
||||||
|
|
||||||
\newcommand{\GT}{\GetTranslation}
|
|
||||||
\newcommand{\dt}{\definetranslation}
|
|
||||||
\newcommand{\ger}{\definetranslation{german}}
|
|
||||||
\newcommand{\eng}{\definetranslation{english}}
|
|
||||||
% \newcommand{\eqd}[5][desc]{
|
% \newcommand{\eqd}[5][desc]{
|
||||||
% \vspace*{0.1cm}
|
% \vspace*{0.1cm}
|
||||||
% \begin{minipage}{0.3\textwidth}
|
% \begin{minipage}{0.3\textwidth}
|
||||||
@ -143,10 +284,17 @@
|
|||||||
% \end{minipage}
|
% \end{minipage}
|
||||||
% \newline
|
% \newline
|
||||||
% }
|
% }
|
||||||
|
\IfSubStringInString{lol}{lol\frac{asdsd}{lol} & l}{YES!}{
|
||||||
|
NO!
|
||||||
|
}
|
||||||
|
\input{translations.tex}
|
||||||
|
|
||||||
\input{trigonometry.tex}
|
\input{trigonometry.tex}
|
||||||
|
|
||||||
\input{quantum_mechanics.tex}
|
\input{quantum_mechanics.tex}
|
||||||
|
\input{atom.tex}
|
||||||
|
|
||||||
|
\input{quantum_computing.tex}
|
||||||
|
|
||||||
%\newpage
|
%\newpage
|
||||||
% \bibliographystyle{plain}
|
% \bibliographystyle{plain}
|
||||||
|
|||||||
40
src/quantum_computing.tex
Normal file
40
src/quantum_computing.tex
Normal file
@ -0,0 +1,40 @@
|
|||||||
|
\Part[
|
||||||
|
\eng{Quantum Computing}
|
||||||
|
\ger{Quantencomputing}
|
||||||
|
]{qubit}
|
||||||
|
|
||||||
|
\Section[
|
||||||
|
\eng{Qubits}
|
||||||
|
\ger{Qubits}
|
||||||
|
]{qubit}
|
||||||
|
\begin{formula}{bloch_sphere}
|
||||||
|
\desc{Bloch sphere}{}{}
|
||||||
|
\desc[german]{Bloch-Sphäre}{}{}
|
||||||
|
\eq{
|
||||||
|
\ket{\psi} &= \alpha \ket{0} + \beta \ket{1} \\
|
||||||
|
&= \cos \frac{\theta}{2} \e^{i\phi_\alpha} \ket{0} + \sin{\frac{\theta}{2} \e^{i\phi_\beta}} \ket{1} \\
|
||||||
|
&= \e^{i\phi_\alpha} \cos\frac{\theta}{2} \ket{0} + \sin\frac{\theta}{2} \e^{i\phi} \ket{1}
|
||||||
|
}
|
||||||
|
\end{formula}
|
||||||
|
|
||||||
|
\Section[
|
||||||
|
\eng{Gates}
|
||||||
|
\ger{Gates}
|
||||||
|
]{gates}
|
||||||
|
\begin{formula}{gates}
|
||||||
|
\desc{}{}{}
|
||||||
|
\desc[german]{}{}{}
|
||||||
|
\eqAlignedAt{2}{
|
||||||
|
& \text{\gt{bitflip}:} & \hat{X} &= \sigma_x = \sigmaxmatrix \\
|
||||||
|
& \text{\gt{bitphaseflip}:} & \hat{Y} &= \sigma_y = \sigmaymatrix \\
|
||||||
|
& \text{\gt{phaseflip}:} & \hat{Z} &= \sigma_z = \sigmazmatrix \\
|
||||||
|
& \text{\gt{hadamard}:} & \hat{H} &= \frac{1}{\sqrt{2}}(\hat{X}-\hat{Z}) = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}
|
||||||
|
}
|
||||||
|
\end{formula}
|
||||||
|
% \begin{itemize}
|
||||||
|
% \item \gt{bitflip}: $\hat{X} = \sigma_x = \sigmaxmatrix$
|
||||||
|
% \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}
|
||||||
|
|
||||||
|
|
||||||
@ -1,15 +1,26 @@
|
|||||||
\eng{quantum_mechanics}{Quantum Mechanics}
|
\def\sigmaxmatrix{\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}}
|
||||||
\ger{quantum_mechanics}{Quantenmechanik}
|
\def\sigmaymatrix{\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}}
|
||||||
|
\def\sigmazmatrix{\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}}
|
||||||
|
\def\sigmaxbraket{\ket{0}\bra{1} + \ket{1}\bra{0}}
|
||||||
|
\def\sigmaybraket{-i \ket{0}\bra{1} + i \ket{1}\bra{0}}
|
||||||
|
\def\sigmazbraket{\ket{0}\bra{0} - \ket{1}\bra{1}}
|
||||||
|
|
||||||
\eng{operators}{Operators}
|
\Part[
|
||||||
\ger{operators}{Operatoren}
|
\eng{Quantum Mechanics}
|
||||||
|
\ger{Quantenmechanik}
|
||||||
\eng{hosc}{Harmonic oscillator}
|
]{qm}
|
||||||
\ger{hosc}{Harmonischer Oszillator}
|
\Section[
|
||||||
|
\eng{Basics}
|
||||||
\part{\GT{quantum_mechanics}}
|
\ger{Basics}
|
||||||
\section{Basics}
|
]{basics}
|
||||||
\subsection{\GT{operators}}
|
\Subsection[
|
||||||
|
\eng{Operators}
|
||||||
|
\ger{Operatoren}
|
||||||
|
]{op}
|
||||||
|
\GER[row_vector]{Zeilenvektor}
|
||||||
|
\GER[column_vector]{Spaltenvektor}
|
||||||
|
\ENG[column_vector]{Column vector}
|
||||||
|
\ENG[row_vector]{Row vector}
|
||||||
\begin{formula}{dirac_notation}
|
\begin{formula}{dirac_notation}
|
||||||
\desc{Dirac notation}{}{}
|
\desc{Dirac notation}{}{}
|
||||||
\desc[german]{Dirac-Notation}{}{}
|
\desc[german]{Dirac-Notation}{}{}
|
||||||
@ -42,7 +53,10 @@
|
|||||||
\eq{\hat{A} = \hat{A}^\dagger}
|
\eq{\hat{A} = \hat{A}^\dagger}
|
||||||
\end{formula}
|
\end{formula}
|
||||||
|
|
||||||
\subsection{\GT{qm_probability}}
|
\Subsection[
|
||||||
|
\ger{Wahrscheinlichkeitstheorie}
|
||||||
|
\eng{Probability theory}
|
||||||
|
]{probability}
|
||||||
\begin{formula}{conservation_of_probability}
|
\begin{formula}{conservation_of_probability}
|
||||||
\desc{Continuity equation}{}{$\rho$ density of a conserved quantity $q$, $j$ flux density of $q$}
|
\desc{Continuity equation}{}{$\rho$ density of a conserved quantity $q$, $j$ flux density of $q$}
|
||||||
\desc[german]{Kontinuitätsgleichung}{}{$\rho$ Dichte einer Erhaltungsgröße $q$, $j$ Fluß von $q$}
|
\desc[german]{Kontinuitätsgleichung}{}{$\rho$ Dichte einer Erhaltungsgröße $q$, $j$ Fluß von $q$}
|
||||||
@ -77,14 +91,17 @@
|
|||||||
}
|
}
|
||||||
\end{formula}
|
\end{formula}
|
||||||
|
|
||||||
\subsubsection{\GT{pauli_matrices}}
|
\Subsubsection[
|
||||||
|
\eng{Pauli matrices}
|
||||||
|
\ger{Pauli-Matrizen}
|
||||||
|
]{pauli_matrices}
|
||||||
\begin{formula}{pauli_matrices}
|
\begin{formula}{pauli_matrices}
|
||||||
\desc{Pauli matrices}{}{}
|
\desc{Pauli matrices}{}{}
|
||||||
\desc[german]{Pauli Matrizen}{}{}
|
\desc[german]{Pauli Matrizen}{}{}
|
||||||
\eqAlignedAt{2}{
|
\eqAlignedAt{2}{
|
||||||
\sigma_x &= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} &&= \ket{0}\bra{1} + \ket{1}\bra{0} \label{eq:pauli_x} \\
|
\sigma_x &= \sigmaxmatrix &&= \sigmaxbraket \label{eq:pauli_x} \\
|
||||||
\sigma_y &= \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} &&= -i \ket{0}\bra{1} + i \ket{1}\bra{0} \label{eq:pauli_y} \\
|
\sigma_y &= \sigmaymatrix &&= \sigmaybraket \label{eq:pauli_y} \\
|
||||||
\sigma_z &= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} &&= \ket{0}\bra{0} - \ket{1}\bra{1} \label{eq:pauli_z}
|
\sigma_z &= \sigmazmatrix &&= \sigmazbraket \label{eq:pauli_z}
|
||||||
}
|
}
|
||||||
\end{formula}
|
\end{formula}
|
||||||
% $\sigma_x$ NOT
|
% $\sigma_x$ NOT
|
||||||
@ -92,7 +109,10 @@
|
|||||||
% $\sigma_z$ Sign
|
% $\sigma_z$ Sign
|
||||||
|
|
||||||
|
|
||||||
\subsection{Kommutator}
|
\Subsection[
|
||||||
|
\eng{Commutator}
|
||||||
|
\ger{Kommutator}
|
||||||
|
]{commutator}
|
||||||
\begin{formula}{commutator}
|
\begin{formula}{commutator}
|
||||||
\desc{Commutator}{}{}
|
\desc{Commutator}{}{}
|
||||||
\desc[german]{Kommutator}{}{}
|
\desc[german]{Kommutator}{}{}
|
||||||
@ -127,7 +147,10 @@
|
|||||||
}
|
}
|
||||||
\end{formula}
|
\end{formula}
|
||||||
|
|
||||||
\subsection{Schrödinger Gleichungen}
|
\Subsection[
|
||||||
|
\eng{Schrödinger equation}
|
||||||
|
\ger{Schrödingergleichung}
|
||||||
|
]{schroedinger_equation}
|
||||||
\begin{formula}{energy_operator}
|
\begin{formula}{energy_operator}
|
||||||
\desc{Energy operator}{}{}
|
\desc{Energy operator}{}{}
|
||||||
\desc[german]{Energieoperator}{}{}
|
\desc[german]{Energieoperator}{}{}
|
||||||
@ -157,21 +180,121 @@
|
|||||||
\desc[german]{Schrödingergleichung}{}{}
|
\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)}
|
\eq{i\hbar\frac{\partial}{\partial t}\psi(x, t) = (- \frac{\hbar^2}{2m} \vec{\nabla}^2 + \vec{V}(x)) \psi(x)}
|
||||||
\end{formula}
|
\end{formula}
|
||||||
The time evolution of the Hamiltonian is given by:\\
|
The time evolution of the Hamiltonian is given by:
|
||||||
% \eq{Time evolution}{\hat{H}\ket{\psi} = E\ket{\psi}}{sg_time}
|
\begin{formula}{time_evolution_op}
|
||||||
% \subsection{Creation and Annihilation operators}
|
\desc{Time evolution operator}{}{$U$ unitary}
|
||||||
% \eq{Annihilation operator}{\hat{a} = }{c\hat{a}_op_annihilation}
|
\desc[german]{Zeitentwicklungsoperator}{}{$U$ unitär}
|
||||||
% \eq{Creation operator}{\hat{a}^\dagger = }{c\hat{a}_op_creation}
|
\eq{\ket{\psi(t)} = \hat{U}(t, t_0) \ket{\psi(t_0)}}
|
||||||
% \eq{Commutator}{[\hat{a},\hat{a}^\dagger] = 1}{c\hat{a}_op_commutator}
|
\end{formula}
|
||||||
% \eq{}{
|
|
||||||
% \hat{a} \ket{n} &= \sqrt{n}\ket{n-1} \\
|
|
||||||
% \hat{a}^\dagger \ket{n} &= \sqrt{n+1}\ket{n+1} \\
|
|
||||||
% \ket{n} &= \frac{1}{\sqrt{n!}} (\hat{a}^\dagger)^n \ket{0}
|
|
||||||
% }{ca_op_on_state}
|
|
||||||
% \eq{Heisenberg}{\frac{dA}{dt}=\frac{\partial A}{\partial t}+\frac{[A,H]}{i\hbar}}{heisenberg}
|
|
||||||
|
|
||||||
\section{\GT{hosc}}
|
\Subsubsection[
|
||||||
\begin{formula}{hosc_hamiltonian}
|
\eng{Schrödinger- and Heisenberg-pictures}
|
||||||
|
\ger{Schrödinger- und Heisenberg-Bild}
|
||||||
|
]{s_h_pictures}
|
||||||
|
\eng[s_h_pictures_desc]{
|
||||||
|
In the \textbf{Schrödinger picture}, the time dependecy is in the states
|
||||||
|
while in the \textbf{Heisenberg picture} the observables (operators) are time dependent.
|
||||||
|
}
|
||||||
|
\ger[s_h_pictures_desc]{Im Schrödinger-Bild sind die Zustände zeitabhänig, im Heisenberg-Bild
|
||||||
|
sind die Observablen (Operatoren) zeitabhänig
|
||||||
|
}
|
||||||
|
\gt{s_h_pictures_desc}\\
|
||||||
|
\begin{formula}{schroediner_time_evolution}
|
||||||
|
\desc{Schrödinger time evolution}{}{}
|
||||||
|
\desc[german]{Schrödinger Zeitentwicklug}{}{}
|
||||||
|
\eq{
|
||||||
|
\ket{\psi(t)_\textrm{S}} = \hat{U}(t,t_0)\ket{\psi(t_0)}
|
||||||
|
}
|
||||||
|
\end{formula}
|
||||||
|
|
||||||
|
\begin{formula}{heisenberg_time_evolution}
|
||||||
|
\desc{Heisenberg time evolution}{}{\textrm{H} and \textrm{S} being the Heisenberg and Schrödinger picture, respectively}
|
||||||
|
\desc[german]{Heisenberg Zeitentwicklung}{}{mit \textrm{H} und \textrm{S} dem Heisenberg- und Schrödinger-Bild}
|
||||||
|
\eq{
|
||||||
|
\ket{\psi_\mathrm{H}} = \ket{\psi_\mathrm{S}(t_0)} \\
|
||||||
|
A_\textrm{H} = U^\dagger(t,t_0)A_\textrm{S}U(t,t_0) \\
|
||||||
|
\diff{\hat{A}_\textrm{H}}{t} = \frac{1}{i\hbar}[\hat{A}_\textrm{H}, \hat{H}_\textrm{H}] + \Big(\diffp{\hat{A}_\textrm{S}}{t}\Big)_\textrm{H}
|
||||||
|
}
|
||||||
|
\end{formula}
|
||||||
|
|
||||||
|
\Subsubsection[
|
||||||
|
\ger{Korrespondenzprinzip}
|
||||||
|
\eng{Correspondence principle}
|
||||||
|
]{correspondence_principle}
|
||||||
|
\begin{ttext}{desc}
|
||||||
|
\ger{Die klassischen Bewegungsgleichungen lassen sich als Grenzfall (große Quantenzahlen) aus der Quantenmechanik ableiten.}
|
||||||
|
\eng{The classical mechanics can be derived from quantum mechanics in the limit of large quantum numbers.}
|
||||||
|
\end{ttext}
|
||||||
|
|
||||||
|
|
||||||
|
\Subsubsection[
|
||||||
|
\eng{Ehrenfest theorem}
|
||||||
|
\ger{Ehrenfest-Theorem}
|
||||||
|
]{ehrenfest_theorem}
|
||||||
|
\GT{see_also} \ref{sec:qm:basics:schroedinger_equation:correspondence_principle}
|
||||||
|
\begin{formula}{ehrenfest_theorem}
|
||||||
|
\desc{Ehrenfesttheorem}{applies to both pictures}{}
|
||||||
|
\desc[german]{Ehrenfest-Theorem}{gilt für beide Bilder}{}
|
||||||
|
\eq{
|
||||||
|
\diff{}{t} \braket{\hat{A}} = \frac{1}{i\hbar}\braket{[\hat{A},\hat{H}]} + \Braket{\diffp{\hat{A}}{t}}
|
||||||
|
}
|
||||||
|
\end{formula}
|
||||||
|
\begin{formula}{ehrenfest_theorem_x}
|
||||||
|
\desc{}{Example for $x$}{}
|
||||||
|
\desc[german]{}{Beispiel für $x$}{}
|
||||||
|
\eq{m\diff[2]{}{t}\braket{x} = -\braket{\nabla V(x)} = \braket{F(x)}}
|
||||||
|
\end{formula}
|
||||||
|
% \eq{Time evolution}{\hat{H}\ket{\psi} = E\ket{\psi}}{sg_time}
|
||||||
|
|
||||||
|
\Section[
|
||||||
|
\eng{Pertubation theory}
|
||||||
|
\ger{Störungstheorie}
|
||||||
|
]{qm_pertubation}
|
||||||
|
\eng[desc]{The following holds true if the pertubation $\hat{H_1}$ is sufficently small and the $E^{(0)}_n$ levels are not degenerate.}
|
||||||
|
\ger[desc]{Die folgenden Gleichungen gelten wenn $\hat{H_1}$ ausreichend klein ist und die $E_n^{(0)}$ Niveaus nicht entartet sind.}
|
||||||
|
\gt{desc}
|
||||||
|
\begin{formula}{pertubation_hamiltonian}
|
||||||
|
\desc{Hamiltonian}{}{}
|
||||||
|
\desc[german]{Hamiltonian}{}{}
|
||||||
|
\eq{\hat{H} = \hat{H_0} + \lambda \hat{H_1}}
|
||||||
|
\end{formula}
|
||||||
|
|
||||||
|
\begin{formula}{pertubation_series}
|
||||||
|
\desc{Power series}{}{}
|
||||||
|
\desc[german]{Potenzreihe}{}{}
|
||||||
|
\eq{
|
||||||
|
E_n &= E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + ... \\
|
||||||
|
\ket{\psi_n} &= \ket{\psi_n^{(0)}} + \lambda \ket{\psi_n^{(1)}} + \lambda^2 \ket{\psi_n^{(2)}} + ...
|
||||||
|
}
|
||||||
|
\end{formula}
|
||||||
|
\begin{formula}{1o_energy}
|
||||||
|
\desc{1. order energy shift}{}{}
|
||||||
|
\desc[german]{Energieverschiebung 1. Ordnung}{}{}
|
||||||
|
\eq{E_n^{(1)} = \Braket{\psi_n^{(0)}|\hat{H_1}|\psi_n^{(0)}}}
|
||||||
|
\end{formula}
|
||||||
|
\begin{formula}{1o_state}
|
||||||
|
\desc{1. order states}{}{}
|
||||||
|
\desc[german]{Zustände}{}{}
|
||||||
|
\eq{\ket{\psi_n^{(1)}} = \sum_{k\neq n}\frac{\Braket{\psi_k^{(0)}|\hat{H_1}|\psi_n^{(0)}}}{E_n^{(0)} - E_k^{(0)}}\ket{\psi_k^{(0)}}}
|
||||||
|
\end{formula}
|
||||||
|
\begin{formula}{2o_energy}
|
||||||
|
\desc{2. order energy shift}{}{}
|
||||||
|
\desc[german]{Energieverschiebung 2. Ordnung}{}{}
|
||||||
|
% \eq{E_n^{(1)} = \Braket{\psi_n^{(0)}|\hat{H_1}|\psi_n^{(0)}}}
|
||||||
|
\eq{E_n^{(2)} = \sum_{k\neq n}\frac{\abs*{\Braket{\psi_k^{(0)}|\hat{H_1}|\psi_n^{(0)}}}^2}{E_n^{(0)} - E_k^{(0)}}}
|
||||||
|
\end{formula}
|
||||||
|
% \begin{formula}{qm:pertubation:}
|
||||||
|
% \desc{1. order states}{}{}
|
||||||
|
% \desc[german]{Zustände}{}{}
|
||||||
|
% \eq{\ket{\psi_n^{(1)}} = \sum_{k\neq n}\frac{\Braket{\psi_k^{(0)}|\hat{H_1}|\psi_n^{(0)}}}{E_n^{(0)} - E_k^{(0)}}\ket{\psi_k^{(0)}}}
|
||||||
|
% \end{formula}
|
||||||
|
|
||||||
|
|
||||||
|
\Section[
|
||||||
|
\eng{Harmonic oscillator}
|
||||||
|
\ger{Harmonischer Oszillator}
|
||||||
|
]{qm_hosc}
|
||||||
|
\begin{formula}{hamiltonian}
|
||||||
\desc{Hamiltonian}{}{}
|
\desc{Hamiltonian}{}{}
|
||||||
\desc[german]{Hamiltonian}{}{}
|
\desc[german]{Hamiltonian}{}{}
|
||||||
\eq{
|
\eq{
|
||||||
@ -180,6 +303,67 @@
|
|||||||
}
|
}
|
||||||
\end{formula}
|
\end{formula}
|
||||||
|
|
||||||
|
\begin{formula}{hosc_spectrum}
|
||||||
|
\desc{Energy spectrum}{}{}
|
||||||
|
\desc[german]{Energiespektrum}{}{}
|
||||||
|
\eq{E_n = \hbar\omega \Big(\frac{1}{2} + n\Big)}
|
||||||
|
\end{formula}
|
||||||
|
|
||||||
|
\GT{see_also} \ref{sec:qm:hosc:c_a_ops}
|
||||||
|
|
||||||
|
\Subsection[
|
||||||
|
\ger{Erzeugungs und Vernichtungsoperatoren}
|
||||||
|
\eng{Creation and Annihilation operators}
|
||||||
|
]{c_a_ops}
|
||||||
|
\begin{formula}{c_a_ops_def}
|
||||||
|
\desc{Particle number operator/occupation number operator}{}{$\ket{n}$ = Fock states, $\hat{a}$ = Annihilation operator, $\hat{a}^\dagger$ = Creation operator}
|
||||||
|
\desc[german]{Teilchenzahloperator/Besetzungszahloperator}{}{$\ket{n}$ = Fock-Zustände, $\hat{a}$ = Vernichtungsoperator, $\hat{a}^\dagger$ = Erzeugungsoperator}
|
||||||
|
\eq{
|
||||||
|
\hat{N} &:= a^\dagger a \\
|
||||||
|
\hat{N}\ket{n} &= n \ket{N}
|
||||||
|
}
|
||||||
|
\end{formula}
|
||||||
|
|
||||||
|
\begin{formula}{c_a_commutator}
|
||||||
|
\desc{Commutator}{}{}
|
||||||
|
\desc[german]{Kommutator}{}{}
|
||||||
|
\eq{
|
||||||
|
[\hat{a},\hat{a}^\dagger] &= 1 \\
|
||||||
|
[N, \hat{a}] &= -\hat{a} \\
|
||||||
|
[N, \hat{a}^\dagger] &= \hat{a}^\dagger
|
||||||
|
}
|
||||||
|
\end{formula}
|
||||||
|
|
||||||
|
\begin{formula}{c_a_on_state}
|
||||||
|
\desc{Application on states}{}{}
|
||||||
|
\desc[german]{Anwendung auf Zustände}{}{}
|
||||||
|
\eq{
|
||||||
|
\hat{a} \ket{n} &= \sqrt{n}\ket{n-1} \\
|
||||||
|
\hat{a}^\dagger \ket{n} &= \sqrt{n+1}\ket{n+1} \\
|
||||||
|
\ket{n} &= \frac{1}{\sqrt{n!}} (\hat{a}^\dagger)^n \ket{0}
|
||||||
|
}
|
||||||
|
\end{formula}
|
||||||
|
|
||||||
|
\Subsubsection[
|
||||||
|
\eng{Harmonischer Oszillator}
|
||||||
|
\ger{Harmonic Oscillator}
|
||||||
|
]{hosc}
|
||||||
|
\begin{formula}{c_a_ops}
|
||||||
|
\desc{Harmonic oscillator}{}{}
|
||||||
|
\desc[german]{Harmonischer Oszillator}{}{}
|
||||||
|
\eq{
|
||||||
|
% \tilde{X} &= \sqrt{\frac{m\omega}{\hbar}} \hat{x} &= \frac{1}{\sqrt{2}} (\hat{a} + \hat{a}^\dagger) \\
|
||||||
|
% \tilde{P} &= \frac{1}{\sqrt{m\omega\hbar}} \hat{p} &= \frac{-i}{\sqrt{2}} (\hat{a} - \hat{a}^\dagger) \\
|
||||||
|
\hat{x} &= \sqrt{\frac{\hbar}{2m\omega}} (\hat{a} + \hat{a}^\dagger) \\
|
||||||
|
\hat{p} &= -i\sqrt{\frac{m\omega\hbar}{2}} (\hat{a} - \hat{a}^\dagger) \\
|
||||||
|
\hat{H} &= \frac{\hat{p}^2}{2m} + \frac{m\omega^2 \hat{x}^2}{2} &= \hbar\omega\Big(a^\dagger a + \frac{1}{2}\Big) \\
|
||||||
|
a &= \frac{1}{\sqrt{2}} (\tilde{X} + i\tilde{P}) \\
|
||||||
|
a^\dagger &= \frac{1}{\sqrt{2}} (\tilde{X} - i\tilde{P})
|
||||||
|
% \hat{a}^\dagger ? \sqrt{\frac{}{}}
|
||||||
|
}
|
||||||
|
\end{formula}
|
||||||
|
% \eq{Heisenberg}{\frac{dA}{dt}=\frac{\partial A}{\partial t}+\frac{[A,H]}{i\hbar}}{heisenberg}
|
||||||
|
|
||||||
% \begin{align}
|
% \begin{align}
|
||||||
% \label{eq:k}
|
% \label{eq:k}
|
||||||
% A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} \\
|
% A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} \\
|
||||||
@ -190,9 +374,10 @@
|
|||||||
% E_n=( \frac{1}{2} +n)\hbar\omega
|
% E_n=( \frac{1}{2} +n)\hbar\omega
|
||||||
% \end{equation}
|
% \end{equation}
|
||||||
|
|
||||||
|
\Section{angular_momentum}
|
||||||
|
\times
|
||||||
|
|
||||||
|
|
||||||
% \eq[
|
|
||||||
% ]
|
|
||||||
\begin{formula}{bloch_waves}
|
\begin{formula}{bloch_waves}
|
||||||
\desc{Bloch waves}{
|
\desc{Bloch waves}{
|
||||||
Solve the stat. SG in periodic potential with period
|
Solve the stat. SG in periodic potential with period
|
||||||
|
|||||||
3
src/topo.tex
Normal file
3
src/topo.tex
Normal file
@ -0,0 +1,3 @@
|
|||||||
|
\Part{Topo}
|
||||||
|
\Section{berry_phase}
|
||||||
|
|
||||||
8
src/translations.tex
Normal file
8
src/translations.tex
Normal file
@ -0,0 +1,8 @@
|
|||||||
|
\ENG[angle_deg]{Degree}
|
||||||
|
\GER[angle_deg]{Grad}
|
||||||
|
|
||||||
|
\ENG[angle_rad]{Radian}
|
||||||
|
\GER[angle_rad]{Rad}
|
||||||
|
|
||||||
|
\ENG[see_also]{See also}
|
||||||
|
\GER[see_also]{Siehe auch}
|
||||||
@ -1,4 +1,14 @@
|
|||||||
|
|
||||||
|
\Part[
|
||||||
|
\eng{Analysis}
|
||||||
|
\ger{Analysis}
|
||||||
|
]{ana}
|
||||||
|
|
||||||
|
\Section[
|
||||||
|
\eng{Trigonometry}
|
||||||
|
\ger{Trigonometrie}
|
||||||
|
]{trig}
|
||||||
|
|
||||||
\begin{formula}{exponential_function}
|
\begin{formula}{exponential_function}
|
||||||
\desc{Exponential function}{}{}
|
\desc{Exponential function}{}{}
|
||||||
\desc[german]{Exponentialfunktion}{}{}
|
\desc[german]{Exponentialfunktion}{}{}
|
||||||
@ -33,17 +43,13 @@
|
|||||||
\end{formula}
|
\end{formula}
|
||||||
|
|
||||||
|
|
||||||
\definetranslation{german}{angle_deg}{Grad}
|
|
||||||
\definetranslation{english}{angle_deg}{Degree}
|
|
||||||
\definetranslation{german}{angle_rad}{Rad}
|
|
||||||
\definetranslation{english}{angle_rad}{Radian}
|
|
||||||
\begin{table}[h]
|
\begin{table}[h]
|
||||||
\centering
|
\centering
|
||||||
% \caption{caption}
|
% \caption{caption}
|
||||||
\label{tab:sin_cos_table}
|
\label{tab:sin_cos_table}
|
||||||
\begin{tabular}{c|c|c|c|c|c|c|c|c}
|
\begin{tabular}{c|c|c|c|c|c|c|c|c}
|
||||||
\GetTranslation{angle_deg} & 0° & 30° & 45° & 60° & 90° & 120° & 180° & 270° \\ \hline
|
\GT{angle_deg} & 0° & 30° & 45° & 60° & 90° & 120° & 180° & 270° \\ \hline
|
||||||
\GetTranslation{angle_rad} & $0$ & $\frac{\pi}{6}$ & $\frac{\pi}{4}$ & $\frac{\sqrt{\pi}}{3}$ & $\frac{\pi}{2}$ & $\frac{2\pi}{3}$ & $\pi$ & $\frac{3\pi}{2}$ \\ \hline
|
\GT{angle_rad} & $0$ & $\frac{\pi}{6}$ & $\frac{\pi}{4}$ & $\frac{\sqrt{\pi}}{3}$ & $\frac{\pi}{2}$ & $\frac{2\pi}{3}$ & $\pi$ & $\frac{3\pi}{2}$ \\ \hline
|
||||||
$\sin(x)$ & $0$ & $\frac{1}{2} $ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{3}}{2}$ & $1 $ & $\frac{\sqrt{3}}{2}$ & $ 0$ & $-1 $ \\
|
$\sin(x)$ & $0$ & $\frac{1}{2} $ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{3}}{2}$ & $1 $ & $\frac{\sqrt{3}}{2}$ & $ 0$ & $-1 $ \\
|
||||||
$\cos(x)$ & $1$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{1}{2} $ & $0 $ & $\frac{-1}{2} $ & $-1$ & $ 0 $ \\
|
$\cos(x)$ & $1$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{1}{2} $ & $0 $ & $\frac{-1}{2} $ & $-1$ & $ 0 $ \\
|
||||||
$\tan(x)$ & $0$ & $\frac{1}{\sqrt{3}}$ & $\frac{1}{\sqrt{2}}$ & $\frac{1}{2} $ & $\infty$ & $-\sqrt{3} $ & $ 0$ & $\infty$ \\
|
$\tan(x)$ & $0$ & $\frac{1}{\sqrt{3}}$ & $\frac{1}{\sqrt{2}}$ & $\frac{1}{2} $ & $\infty$ & $-\sqrt{3} $ & $ 0$ & $\infty$ \\
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user