commit ac406b43a6a598a39a88f358c87f15d93993a672 Author: Matthias@Dell Date: Sat May 18 20:32:21 2024 +0200 initial commit diff --git a/src/main.tex b/src/main.tex new file mode 100755 index 0000000..c589425 --- /dev/null +++ b/src/main.tex @@ -0,0 +1,154 @@ +\documentclass[11pt, a4paper]{article} +% \usepackage[utf8]{inputenc} +\usepackage[english]{babel} +\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry} +\usepackage{mathtools} +\usepackage{braket} +\usepackage{graphicx} +\usepackage{etoolbox} +\usepackage{substr} +\usepackage{xcolor} +\usepackage{float} +\usepackage[hidelinks]{hyperref} +\usepackage{subcaption} +\hypersetup{colorlinks = true, % Colours links instead of ugly boxes + urlcolor = blue, % Colour for external hyperlinks + linkcolor = cyan, % Colour of internal links + citecolor = red % Colour of citations + } +% \usepackage[version=4,arrows=pgf-filled]{mhchem} +\usepackage{siunitx} +\sisetup{output-decimal-marker = {,}} +\sisetup{separate-uncertainty} +\sisetup{per-mode = power} +\sisetup{exponent-product=\ensuremath{\cdot}} + +\DeclarePairedDelimiter\abs{\lvert}{\rvert} + +\usepackage{translations} + +\title{Formelsammlung} +\author{Matthias Quintern} +\date{\today} + +\begin{document} + +\maketitle +% \thispagestyle{empty} +% \tableofcontents +% \newpage +% \setcounter{page}{1} + +% \nuwcommand{\eq}[4][desc]{ +% \vspace*{0.1cm} +% \begin{minipage}{0.3\textwidth} +% \raggedright +% .#2 \\ +% \ifstrequal{#1}{desc}{}{ +% {\color{gray}#1} +% } +% \end{minipage} +% \begin{minipage}{0.7\textwidth} +% \begin{align} +% \label{eq:#4} +% #3 +% \end{align} +% \end{minipage} +% \newline +% } +\newcommand{\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]{ +% \vspace*{0.1cm} +% \begin{minipage}{0.3\textwidth} +% \raggedright +% .#2 \\ +% \ifstrequal{#1}{desc}{}{ +% {\color{gray}#1} +% } +% \end{minipage} +% \begin{minipage}{0.7\textwidth} +% \begin{align} +% \label{eq:#5} +% #3 +% \end{align} +% {\color{gray}with: #4} +% \end{minipage} +% \newline +% } + +\input{trigonometry.tex} + +\input{quantum_mechanics.tex} + +%\newpage +% \bibliographystyle{plain} +% \bibliography{ref} +\end{document} diff --git a/src/quantum_mechanics.tex b/src/quantum_mechanics.tex new file mode 100644 index 0000000..c2f50d4 --- /dev/null +++ b/src/quantum_mechanics.tex @@ -0,0 +1,211 @@ +\eng{quantum_mechanics}{Quantum Mechanics} +\ger{quantum_mechanics}{Quantenmechanik} + +\eng{operators}{Operators} +\ger{operators}{Operatoren} + +\eng{hosc}{Harmonic oscillator} +\ger{hosc}{Harmonischer Oszillator} + +\part{\GT{quantum_mechanics}} +\section{Basics} + \subsection{\GT{operators}} + \begin{formula}{dirac_notation} + \desc{Dirac notation}{}{} + \desc[german]{Dirac-Notation}{}{} + \eq{ + \bra{x} \vspace*{1cm} \text{"Bra" \GT{row_vector}} \\ + \ket{x} \vspace*{1cm} \text{"Ket" \GT{column_vector}} \\ + \hat{A}\ket{\beta} = \ket{\alpha} \Rightarrow \bra{\alpha} = \bra{\beta} \hat{A}^\dagger + } + \end{formula} + + \begin{formula}{dagger} + \desc{Dagger}{}{} + \desc[german]{Dagger}{}{} + \eq{ + \hat{A}^\dagger &= (\hat{A}^*)^\mathrm{T} \\ + (c \hat{A})^\dagger &= c^* \hat{A}^\dagger \\ + (\hat{A}\hat{B})^\dagger &= \hat{B}^\dagger \hat{A}^\dagger \\ + } + \end{formula} + + \begin{formula}{adjoint_op} + \desc{Adjoint operator}{}{} + \desc[german]{Adjungierter operator}{}{} + \eq{\braket{\alpha|\hat{A}^\dagger|\beta} = \braket{\beta|\hat{A}|\alpha}^*} + \end{formula} + + \begin{formula}{hermitian_op} + \desc{Hermitian operator}{}{} + \desc[german]{Hermitescher operator}{}{} + \eq{\hat{A} = \hat{A}^\dagger} + \end{formula} + + \subsection{\GT{qm_probability}} + \begin{formula}{conservation_of_probability} + \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 $q$} + \eq{\frac{\partial\rho(\vec{x}, t)}{\partial t} + \nabla \cdot \vec{j}(\vec{x},t) = 0} + \end{formula} + + \begin{formula}{state_probability} + \desc{State probability}{}{} + \desc[german]{Zustandswahrscheinlichkeit}{}{} + \eq{TODO} + \end{formula} + + \begin{formula}{dispersion} + \desc{Dispersion}{}{} + \desc[german]{Dispersion}{}{} + \eq{\Delta \hat{A} = \hat{A} - \braket{\hat{A}}} + \end{formula} + + \begin{formula}{variance} + \desc{Variance}{}{} + \desc[german]{Varianz}{}{} + \eq{\sigma^2 = \braket{(\Delta \hat{A})^2} = \braket{\hat{A}^2} - \braket{\hat{A}}^2} + \end{formula} + + \begin{formula}{generalized_uncertainty} + \desc{Generalized uncertainty principle}{}{} + \desc[german]{Allgemeine Unschärferelation}{}{} + % \eq{\braket{(\Delta \hat{A})^2} \braket{(\Delta \hat{B})^2} \ge \frac{1}{4} \braket{[\hat{A},\hat{B}]}^2} + \eq{ + \sigma_A \sigma_B &\ge \frac{1}{4} \braket{[\hat{A},\hat{B}]}^2 \\ + \sigma_A \sigma_B &\ge \frac{1}{2} \abs{\braket{[\hat{A},\hat{B}]}} + } + \end{formula} + + \subsubsection{\GT{pauli_matrices}} + \begin{formula}{pauli_matrices} + \desc{Pauli matrices}{}{} + \desc[german]{Pauli Matrizen}{}{} + \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_y &= \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} &&= -i \ket{0}\bra{1} + i \ket{1}\bra{0} \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} + } + \end{formula} + % $\sigma_x$ NOT + % $\sigma_y$ PHASE + % $\sigma_z$ Sign + + + \subsection{Kommutator} + \begin{formula}{commutator} + \desc{Commutator}{}{} + \desc[german]{Kommutator}{}{} + \eq{[a,b] = ab - ba} + \end{formula} + + \begin{formula}{anticommutator} + \desc{Anticommutator}{}{} + \desc[german]{Antikommmutator}{}{} + \eq{\{a,b\} = ab + ba} + \end{formula} + + \begin{formula}{commutation_relations}\ + \desc{Commutation relations}{}{} + \desc[german]{Kommutatorrelationen}{}{} + \eq{[a, bc] = \{a, b\}c - b\{a,c\}} + \end{formula} + + \begin{formula}{jacobi_identity} + \desc{Jacobi identity}{}{} + \desc[german]{Jakobi-Identität}{}{} + \eq{[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0} + \end{formula} + + \begin{formula}{canon_comm_relation} + \desc{Canonical commutation relation}{}{$x$, $p$ canonical conjugates} + \desc[german]{Kanonische Vertauschungsrelationen}{}{$x$, $p$ kanonische konjugierte} + \eq{ + [x_i, x_j] &= 0 \\ + [p_i, p_j] &= 0 \\ + [x_i, p_j] &= i \hbar \delta_{ij} + } + \end{formula} + + \subsection{Schrödinger Gleichungen} + \begin{formula}{energy_operator} + \desc{Energy operator}{}{} + \desc[german]{Energieoperator}{}{} + \eq{E = i\hbar \frac{\partial}{\partial t}} + \end{formula} + + \begin{formula}{momentum_operator} + \desc{Momentum operator}{}{} + \desc[german]{Impulsoperator}{}{} + \eq{\vec{p} = -i\hbar \vec{\nabla_x}} + \end{formula} + + \begin{formula}{space_operator} + \desc{Space operator}{}{} + \desc[german]{Ortsoperator}{}{} + \eq{\vec{x} = i\hbar \vec{\nabla_p}} + \end{formula} + + \begin{formula}{stationary_schroedinger_equation} + \desc{Stationary Schrödingerequation}{}{} + \desc[german]{Stationäre Schrödingergleichung}{}{} + \eq{\hat{H}\ket{\psi} = E\ket{\psi}} + \end{formula} + + \begin{formula}{schroedinger_equation} + \desc{Schrödinger equation}{}{} + \desc[german]{Schrödingergleichung}{}{} + \eq{i\hbar\frac{\partial}{\partial t}\psi(x, t) = (- \frac{\hbar^2}{2m} \vec{\nabla}^2 + \vec{V}(x)) \psi(x)} + \end{formula} + The time evolution of the Hamiltonian is given by:\\ + % \eq{Time evolution}{\hat{H}\ket{\psi} = E\ket{\psi}}{sg_time} + % \subsection{Creation and Annihilation operators} + % \eq{Annihilation operator}{\hat{a} = }{c\hat{a}_op_annihilation} + % \eq{Creation operator}{\hat{a}^\dagger = }{c\hat{a}_op_creation} + % \eq{Commutator}{[\hat{a},\hat{a}^\dagger] = 1}{c\hat{a}_op_commutator} + % \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}} + \begin{formula}{hosc_hamiltonian} + \desc{Hamiltonian}{}{} + \desc[german]{Hamiltonian}{}{} + \eq{ + H&=\frac{p^2}{2m}+\frac{1}{2} m\omega^2 x^2\\ + &=\frac{1}{2} \hbar\omega+\omega a^\dagger a + } + \end{formula} + + % \begin{align} + % \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}} \\ + % HAu_E=(E-\hbar\omega)Au_E \\ + % u_n=\frac{1}{\sqrt{n!}}\left(\frac{A^\dagger}{\sqrt{\hbar}}\right)^nu_0 \\ + % u_0=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right) \\ + % E_n=( \frac{1}{2} +n)\hbar\omega + % \end{equation} + + + % \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} + diff --git a/src/trigonometry.tex b/src/trigonometry.tex new file mode 100644 index 0000000..9993887 --- /dev/null +++ b/src/trigonometry.tex @@ -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}