initial commit
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Matthias@Dell
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src/main.tex
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154
src/main.tex
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\documentclass[11pt, a4paper]{article}
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% \usepackage[utf8]{inputenc}
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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{mathtools}
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\usepackage{braket}
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\usepackage{graphicx}
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\usepackage{etoolbox}
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\usepackage{substr}
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\usepackage{xcolor}
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\usepackage{float}
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\usepackage[hidelinks]{hyperref}
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\usepackage{subcaption}
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\hypersetup{colorlinks = true, % Colours links instead of ugly boxes
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urlcolor = blue, % Colour for external hyperlinks
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linkcolor = cyan, % Colour of internal links
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citecolor = red % Colour of citations
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}
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% \usepackage[version=4,arrows=pgf-filled]{mhchem}
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\usepackage{siunitx}
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\sisetup{output-decimal-marker = {,}}
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\sisetup{separate-uncertainty}
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\sisetup{per-mode = power}
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\sisetup{exponent-product=\ensuremath{\cdot}}
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\DeclarePairedDelimiter\abs{\lvert}{\rvert}
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\usepackage{translations}
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\title{Formelsammlung}
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\author{Matthias Quintern}
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\date{\today}
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\begin{document}
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\maketitle
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% \thispagestyle{empty}
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% \tableofcontents
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% \newpage
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% \setcounter{page}{1}
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% \nuwcommand{\eq}[4][desc]{
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% \vspace*{0.1cm}
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% \begin{minipage}{0.3\textwidth}
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% \raggedright
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% .#2 \\
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% \ifstrequal{#1}{desc}{}{
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% {\color{gray}#1}
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% }
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% \end{minipage}
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% \begin{minipage}{0.7\textwidth}
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% \begin{align}
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% \label{eq:#4}
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% #3
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% \end{align}
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% \end{minipage}
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% \newline
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% }
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\newcommand{\insertEquation}[2]{
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\vspace*{0.1cm}
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\begin{minipage}{0.3\textwidth}
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\IfTranslation{\languagename}{#1}{
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\raggedright
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\GetTranslation{#1}
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}{}
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\IfTranslation{\languagename}{#1_desc}{
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\\ {\color{gray} \GetTranslation{#1_desc}}
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}{}
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\end{minipage}
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\begin{minipage}{0.7\textwidth}
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\begin{align}
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\label{eq:#1}
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#2
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\end{align}
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\IfTranslation{\languagename}{#1_defs}{
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{\color{gray} \GetTranslation{#1_defs}}
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}{}
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\end{minipage}
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\newline
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}
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\newcommand{\insertAlignedAt}[3]{ % eq name, #cols, eq
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\vspace*{0.1cm}
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\begin{minipage}{0.3\textwidth}
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\IfTranslation{\languagename}{#1}{
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\raggedright
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\GetTranslation{#1}
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}{}
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\IfTranslation{\languagename}{#1_desc}{
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\\ {\color{gray} \GetTranslation{#1_desc}}
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}{}
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\end{minipage}
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\begin{minipage}{0.7\textwidth}
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\begin{alignat}{#2}
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% dont place label when one is provided
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\IfSubStringinString{label}{#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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\IfTranslation{\languagename}{#1_defs}{
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{\color{gray} \GetTranslation{#1_defs}}
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}{}
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\end{minipage}
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\newline
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}
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\newenvironment{formula}[1]{
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\newcommand{\desc}[4][english]{
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% language, name, description, definitions
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\definetranslation{##1}{#1}{##2}
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\ifblank{##3}{}{\definetranslation{##1}{#1_desc}{##3}}
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\ifblank{##4}{}{\definetranslation{##1}{#1_defs}{##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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}{}
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\newcommand{\GT}{\GetTranslation}
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\newcommand{\dt}{\definetranslation}
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\newcommand{\ger}{\definetranslation{german}}
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\newcommand{\eng}{\definetranslation{english}}
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% \newcommand{\eqd}[5][desc]{
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% \vspace*{0.1cm}
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% \begin{minipage}{0.3\textwidth}
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% \raggedright
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% .#2 \\
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% \ifstrequal{#1}{desc}{}{
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% {\color{gray}#1}
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% }
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% \end{minipage}
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% \begin{minipage}{0.7\textwidth}
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% \begin{align}
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% \label{eq:#5}
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% #3
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% \end{align}
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% {\color{gray}with: #4}
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% \end{minipage}
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% \newline
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% }
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\input{trigonometry.tex}
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\input{quantum_mechanics.tex}
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%\newpage
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% \bibliographystyle{plain}
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% \bibliography{ref}
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\end{document}
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211
src/quantum_mechanics.tex
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211
src/quantum_mechanics.tex
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\eng{quantum_mechanics}{Quantum Mechanics}
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\ger{quantum_mechanics}{Quantenmechanik}
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\eng{operators}{Operators}
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\ger{operators}{Operatoren}
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\eng{hosc}{Harmonic oscillator}
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\ger{hosc}{Harmonischer Oszillator}
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\part{\GT{quantum_mechanics}}
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\section{Basics}
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\subsection{\GT{operators}}
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\begin{formula}{dirac_notation}
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\desc{Dirac notation}{}{}
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\desc[german]{Dirac-Notation}{}{}
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\eq{
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\bra{x} \vspace*{1cm} \text{"Bra" \GT{row_vector}} \\
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\ket{x} \vspace*{1cm} \text{"Ket" \GT{column_vector}} \\
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\hat{A}\ket{\beta} = \ket{\alpha} \Rightarrow \bra{\alpha} = \bra{\beta} \hat{A}^\dagger
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}
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\end{formula}
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\begin{formula}{dagger}
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\desc{Dagger}{}{}
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\desc[german]{Dagger}{}{}
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\eq{
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\hat{A}^\dagger &= (\hat{A}^*)^\mathrm{T} \\
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(c \hat{A})^\dagger &= c^* \hat{A}^\dagger \\
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(\hat{A}\hat{B})^\dagger &= \hat{B}^\dagger \hat{A}^\dagger \\
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}
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\end{formula}
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\begin{formula}{adjoint_op}
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\desc{Adjoint operator}{}{}
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\desc[german]{Adjungierter operator}{}{}
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\eq{\braket{\alpha|\hat{A}^\dagger|\beta} = \braket{\beta|\hat{A}|\alpha}^*}
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\end{formula}
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\begin{formula}{hermitian_op}
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\desc{Hermitian operator}{}{}
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\desc[german]{Hermitescher operator}{}{}
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\eq{\hat{A} = \hat{A}^\dagger}
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\end{formula}
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\subsection{\GT{qm_probability}}
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\begin{formula}{conservation_of_probability}
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\desc{Continuity equation}{}{$\rho$ density of a conserved quantity $q$, $j$ flux density of $q$}
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\desc[german]{Kontinuitätsgleichung}{}{$\rho$ Dichte einer Erhaltungsgröße $q$, $j$ Fluß von $q$}
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\eq{\frac{\partial\rho(\vec{x}, t)}{\partial t} + \nabla \cdot \vec{j}(\vec{x},t) = 0}
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\end{formula}
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\begin{formula}{state_probability}
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\desc{State probability}{}{}
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\desc[german]{Zustandswahrscheinlichkeit}{}{}
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\eq{TODO}
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\end{formula}
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\begin{formula}{dispersion}
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\desc{Dispersion}{}{}
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\desc[german]{Dispersion}{}{}
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\eq{\Delta \hat{A} = \hat{A} - \braket{\hat{A}}}
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\end{formula}
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\begin{formula}{variance}
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\desc{Variance}{}{}
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\desc[german]{Varianz}{}{}
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\eq{\sigma^2 = \braket{(\Delta \hat{A})^2} = \braket{\hat{A}^2} - \braket{\hat{A}}^2}
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\end{formula}
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\begin{formula}{generalized_uncertainty}
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\desc{Generalized uncertainty principle}{}{}
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\desc[german]{Allgemeine Unschärferelation}{}{}
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% \eq{\braket{(\Delta \hat{A})^2} \braket{(\Delta \hat{B})^2} \ge \frac{1}{4} \braket{[\hat{A},\hat{B}]}^2}
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\eq{
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\sigma_A \sigma_B &\ge \frac{1}{4} \braket{[\hat{A},\hat{B}]}^2 \\
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\sigma_A \sigma_B &\ge \frac{1}{2} \abs{\braket{[\hat{A},\hat{B}]}}
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}
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\end{formula}
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\subsubsection{\GT{pauli_matrices}}
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\begin{formula}{pauli_matrices}
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\desc{Pauli matrices}{}{}
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\desc[german]{Pauli Matrizen}{}{}
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\eqAlignedAt{2}{
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\sigma_x &= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} &&= \ket{0}\bra{1} + \ket{1}\bra{0} \label{eq:pauli_x} \\
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\sigma_y &= \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} &&= -i \ket{0}\bra{1} + i \ket{1}\bra{0} \label{eq:pauli_y} \\
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\sigma_z &= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} &&= \ket{0}\bra{0} - \ket{1}\bra{1} \label{eq:pauli_z}
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}
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\end{formula}
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% $\sigma_x$ NOT
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% $\sigma_y$ PHASE
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% $\sigma_z$ Sign
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\subsection{Kommutator}
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\begin{formula}{commutator}
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\desc{Commutator}{}{}
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\desc[german]{Kommutator}{}{}
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\eq{[a,b] = ab - ba}
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\end{formula}
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\begin{formula}{anticommutator}
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\desc{Anticommutator}{}{}
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\desc[german]{Antikommmutator}{}{}
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\eq{\{a,b\} = ab + ba}
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\end{formula}
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\begin{formula}{commutation_relations}\
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\desc{Commutation relations}{}{}
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\desc[german]{Kommutatorrelationen}{}{}
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\eq{[a, bc] = \{a, b\}c - b\{a,c\}}
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\end{formula}
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\begin{formula}{jacobi_identity}
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\desc{Jacobi identity}{}{}
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\desc[german]{Jakobi-Identität}{}{}
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\eq{[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0}
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\end{formula}
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\begin{formula}{canon_comm_relation}
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\desc{Canonical commutation relation}{}{$x$, $p$ canonical conjugates}
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\desc[german]{Kanonische Vertauschungsrelationen}{}{$x$, $p$ kanonische konjugierte}
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\eq{
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[x_i, x_j] &= 0 \\
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[p_i, p_j] &= 0 \\
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[x_i, p_j] &= i \hbar \delta_{ij}
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}
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\end{formula}
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\subsection{Schrödinger Gleichungen}
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\begin{formula}{energy_operator}
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\desc{Energy operator}{}{}
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\desc[german]{Energieoperator}{}{}
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\eq{E = i\hbar \frac{\partial}{\partial t}}
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\end{formula}
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\begin{formula}{momentum_operator}
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\desc{Momentum operator}{}{}
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\desc[german]{Impulsoperator}{}{}
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\eq{\vec{p} = -i\hbar \vec{\nabla_x}}
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\end{formula}
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\begin{formula}{space_operator}
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\desc{Space operator}{}{}
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\desc[german]{Ortsoperator}{}{}
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\eq{\vec{x} = i\hbar \vec{\nabla_p}}
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\end{formula}
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\begin{formula}{stationary_schroedinger_equation}
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\desc{Stationary Schrödingerequation}{}{}
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\desc[german]{Stationäre Schrödingergleichung}{}{}
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\eq{\hat{H}\ket{\psi} = E\ket{\psi}}
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\end{formula}
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\begin{formula}{schroedinger_equation}
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\desc{Schrödinger equation}{}{}
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\desc[german]{Schrödingergleichung}{}{}
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\eq{i\hbar\frac{\partial}{\partial t}\psi(x, t) = (- \frac{\hbar^2}{2m} \vec{\nabla}^2 + \vec{V}(x)) \psi(x)}
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\end{formula}
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The time evolution of the Hamiltonian is given by:\\
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% \eq{Time evolution}{\hat{H}\ket{\psi} = E\ket{\psi}}{sg_time}
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% \subsection{Creation and Annihilation operators}
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% \eq{Annihilation operator}{\hat{a} = }{c\hat{a}_op_annihilation}
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% \eq{Creation operator}{\hat{a}^\dagger = }{c\hat{a}_op_creation}
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% \eq{Commutator}{[\hat{a},\hat{a}^\dagger] = 1}{c\hat{a}_op_commutator}
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% \eq{}{
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% \hat{a} \ket{n} &= \sqrt{n}\ket{n-1} \\
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% \hat{a}^\dagger \ket{n} &= \sqrt{n+1}\ket{n+1} \\
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% \ket{n} &= \frac{1}{\sqrt{n!}} (\hat{a}^\dagger)^n \ket{0}
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% }{ca_op_on_state}
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% \eq{Heisenberg}{\frac{dA}{dt}=\frac{\partial A}{\partial t}+\frac{[A,H]}{i\hbar}}{heisenberg}
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\section{\GT{hosc}}
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|
\begin{formula}{hosc_hamiltonian}
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\desc{Hamiltonian}{}{}
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||||||
|
\desc[german]{Hamiltonian}{}{}
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||||||
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\eq{
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H&=\frac{p^2}{2m}+\frac{1}{2} m\omega^2 x^2\\
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&=\frac{1}{2} \hbar\omega+\omega a^\dagger a
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|
}
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\end{formula}
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||||||
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||||||
|
% \begin{align}
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||||||
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% \label{eq:k}
|
||||||
|
% A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} \\
|
||||||
|
% A^\dagger=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x-\frac{ip}{\sqrt{2m\omega}} \\
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% HAu_E=(E-\hbar\omega)Au_E \\
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||||||
|
% u_n=\frac{1}{\sqrt{n!}}\left(\frac{A^\dagger}{\sqrt{\hbar}}\right)^nu_0 \\
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||||||
|
% u_0=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right) \\
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||||||
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% E_n=( \frac{1}{2} +n)\hbar\omega
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||||||
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% \end{equation}
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||||||
|
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||||||
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||||||
|
% \eq[
|
||||||
|
% ]
|
||||||
|
\begin{formula}{bloch_waves}
|
||||||
|
\desc{Bloch waves}{
|
||||||
|
Solve the stat. SG in periodic potential with period
|
||||||
|
$\vec{R}$: $V(\vec{r}) = V(\vec{r} + \vec{R})$\\
|
||||||
|
}{
|
||||||
|
$\vec{k}$ arbitrary, $u$ periodic function
|
||||||
|
}
|
||||||
|
\desc[german]{Blochwellen}{
|
||||||
|
Lösen stat. SG im periodischen Potential mit Periode
|
||||||
|
$\vec{R}$: $V(\vec{r}) = V(\vec{r} + \vec{R})$\\
|
||||||
|
}{
|
||||||
|
$\vec{k}$ beliebig, $u$ periodische Funktion
|
||||||
|
}
|
||||||
|
\eq{\psi_k(\vec{r}) = e^{i \vec{k}\cdot \vec{r}} \cdot u_{\vec{k}}(\vec{r})}
|
||||||
|
\end{formula}
|
||||||
|
|
||||||
51
src/trigonometry.tex
Normal file
51
src/trigonometry.tex
Normal file
@ -0,0 +1,51 @@
|
|||||||
|
|
||||||
|
\begin{formula}{exponential_function}
|
||||||
|
\desc{Exponential function}{}{}
|
||||||
|
\desc[german]{Exponentialfunktion}{}{}
|
||||||
|
\eq{\exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}}
|
||||||
|
\end{formula}
|
||||||
|
|
||||||
|
\begin{formula}{sine}
|
||||||
|
\desc{Sine}{}{}
|
||||||
|
\desc[german]{Sinus}{}{}
|
||||||
|
\eq{\sin(x) &= \sum_{n=0}^{\infty} \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)!} \\
|
||||||
|
&= \frac{e^{ix}+e^{-ix}}{2}}
|
||||||
|
\end{formula}
|
||||||
|
|
||||||
|
|
||||||
|
\begin{formula}{hyperbolic_sine}
|
||||||
|
\desc{Hyperbolic sine}{}{}
|
||||||
|
\desc[german]{Sinus hyperbolicus}{}{}
|
||||||
|
\eq{\sinh(x) &= -i\sin{ix} \\ &= \frac{e^{x}-e^{-x}}{2}}
|
||||||
|
\end{formula}
|
||||||
|
|
||||||
|
\begin{formula}{hyperbolic_cosine}
|
||||||
|
\desc{Hyperbolic cosine}{}{}
|
||||||
|
\desc[german]{Kosinus hyperbolicus}{}{}
|
||||||
|
\eq{\cosh(x) &= \cos{ix} \\ &= \frac{e^{x}+e^{-x}}{2}}
|
||||||
|
\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]
|
||||||
|
\centering
|
||||||
|
% \caption{caption}
|
||||||
|
\label{tab:sin_cos_table}
|
||||||
|
\begin{tabular}{c|c|c|c|c|c|c|c|c}
|
||||||
|
\GetTranslation{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
|
||||||
|
$\sin(x)$ & $0$ & $\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 $ \\
|
||||||
|
$\tan(x)$ & $0$ & $\frac{1}{\sqrt{3}}$ & $\frac{1}{\sqrt{2}}$ & $\frac{1}{2} $ & $\infty$ & $-\sqrt{3} $ & $ 0$ & $\infty$ \\
|
||||||
|
\end{tabular}
|
||||||
|
\end{table}
|
||||||
Loading…
Reference in New Issue
Block a user