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87
src/atom.tex
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@ -21,14 +21,20 @@
\begin{formula}{hamiltonian}
\desc{Hamiltonian}{}{}
\desc[german]{Hamiltonian}{}{}
\eq{\hat{H} &= -\frac{\hbar^2}{2\mu} {\grad_\vec{r}}^2 - V(\vecr) \\
&= \frac{\hat{p}_r^2}{2\mu} + \frac{\hat{L}^2}{2\mu r} + V(r)}
% \eq{V(\vecr) = \frac{Z\,e^2}{4\pi\epsilon_0 r}}
\eq{
% \hat{H} &= -\frac{\hbar^2}{2\mu} {\grad_\vecr}^2 - V(\vecr)
% &= \frac{\hat{p}_r^2}{2\mu} + \frac{\hat{L}^2}{2\mu r} + V(r)
}
\end{formula}
\begin{formula}{wave_function}
\desc{Wave function}{}{}
\desc[german]{Wellenfunktion}{}{}
\eq{\psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_{lm}(\theta,\phi)}
\end{formula}
\begin{formula}{radial}
\desc{Radial part}{}{$L_r^s(x)$ Laguerre-polynomials}
\desc[german]{Radialanteil}{}{$L_r^s(x)$ Laguerre-Polynome}
@ -38,11 +44,13 @@
\kappa &= \frac{\sqrt{2\mu\abs{E}}}{\hbar} = \frac{Z}{n \abohr}
}
\end{formula}
\begin{formula}{energy}
\desc{Energy eigenvalues}{}{}
\desc[german]{Energieeigenwerte}{}{}
\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}}
\end{formula}
\begin{formula}{rydberg_energy}
\desc{Rydberg energy}{}{}
\desc[german]{Rydberg-Energy}{}{}
@ -99,8 +107,77 @@
\Subsubsection[
\eng{Fine-structure}
\ger{Feinstruktur}
]{fine_structure}
]{fine_structure}
\begin{ttext}{desc}
\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}.
\ger{Die Feinstruktur vereint relativistische Korrekturen \ref{sec:qm:h:corrections:darwin} und die Spin-Orbit-Kupplung \ref{sec:qm:h:corrections:ls_coupling}.
\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}.}
\ger{Die Feinstruktur vereint relativistische Korrekturen \ref{sec:qm:h:corrections:darwin} und die Spin-Orbit-Kupplung \ref{sec:qm:h:corrections:ls_coupling}.}
\end{ttext}
\begin{formula}{energy_shift}
\desc{Energy shift}{}{}
\desc[german]{Energieverschiebung}{}{}
\eq{\Delta E_\textrm{FS} = \frac{Z^2\alpha^2}{n}\Big(\frac{1}{j+\frac{1}{2}} - \frac{3}{4n}\Big)}
\end{formula}
\Subsubsection[
\eng{Lamb-shift}
\ger{Lamb-Shift}
]{lamb_shift}
\begin{ttext}{desc}
\eng{The interaction of the electron with virtual photons emitted/absorbed by the nucleus leads to a (very small) shift in the energy level.}
\ger{The Wechselwirkung zwischen dem Elektron und vom Kern absorbierten/emittierten virtuellen Photonen führt zu einer (sehr kleinen) Energieverschiebung.}
\end{ttext}
\begin{formula}{energy}
\desc{Potential energy}{}{$\delta r$ pertubation of $r$}
\desc[german]{Potentielle Energy}{}{$\delta r$ Schwankung von $r$}
\eq{\braket{E_\textrm{pot}} = -\frac{Z e^2}{4\pi\epsilon_0} \Braket{\frac{1}{r+\delta r}}}
\end{formula}
\Subsubsection[
\eng{Hyperfine structure}
\ger{Hyperfeinstruktur}
]{hyperfine_structure}
\begin{ttext}{desc}
\eng{Interaction of the nucleus spin with the magnetic field created by the electron leads to energy shifts. (Lifts degenaracy) }
\ger{Wechselwirkung von Kernspin mit dem vom Elektron erzeugten Magnetfeld spaltet Energieniveaus}
\end{ttext}
\begin{formula}{nuclear_spin}
\desc{Nuclear spin}{}{}
\desc[german]{Kernspin}{}{}
\eq{\vec{F} &= \vec{J} + \vec{I} \\
\abs{\vec{I}} &= \sqrt{i(i+1)}\hbar \\
I_z &= m_i\hbar \\
m_i &= -i, -i+1, \dots, i-1, i
}
\end{formula}
\begin{formula}{angular_momentum}
\desc{Combined angular momentum}{}{}
\desc[german]{Gesamtdrehimpuls}{}{}
\eq{\vec{F} &= \vec{J} + \vec{I} \\
\abs{\vec{F}} &= \sqrt{f(f+1)}\hbar \\
F_z &= m_f\hbar
}
\end{formula}
\begin{formula}{selection_rule}
\desc{Selection rule}{}{}
\desc[german]{Auswahlregel}{}{}
\eq{f &= j \pm i \\ m_f &= -f,-f+1,\dots,f-1,f}
\end{formula}
\begin{formula}{constant}
\desc{Hyperfine structure constant}{}{$B_\textrm{HFS}$ hyperfine field, $\mu_\textrm{K}$ nuclear magneton, $g_i$ nuclear g-factor \ref{qm:h:lande}}
\desc[german]{Hyperfeinstrukturkonstante}{}{$B_\textrm{HFS}$ Hyperfeinfeld, $\mu_\textrm{K}$ Kernmagneton, $g_i$ Kern-g-Faktor \ref{qm:h:lande}}
\eq{A = \frac{g_i \mu_\textrm{K} B_\textrm{HFS}}{\sqrt{j(j+1)}}}
\end{formula}
\begin{formula}{energy_shift}
\desc{Energy shift}{}{}
\desc[german]{Energieverschiebung}{}{}
\eq{\Delta H_\textrm{HFS} = \frac{A}{2}[f(f+1) - j(j+1) -i(i+1)]}
\end{formula}
\TODO{landé factor}
\Subsection[
\eng{Effects in magnetic field}
\ger{Effekte im Magnetfeld}
]{mag_effects}
\TODO{all}

13
src/main.tex Executable file → Normal file
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@ -1,9 +1,11 @@
\documentclass[11pt, a4paper]{article}
% \usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[german]{babel}
\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
\usepackage{mathtools}
\usepackage{esdiff} % derivatives
% \usepackage{esdiff} % derivatives
% esdiff breaks when taking \dot{q} has argument
\usepackage{derivative}
\usepackage{braket}
\usepackage{graphicx}
\usepackage{etoolbox}
@ -26,10 +28,11 @@
\DeclarePairedDelimiter\abs{\lvert}{\rvert}
\DeclareMathOperator{\e}{e}
\DeclareMathOperator{\d}{d}
\usepackage{translations}
\newcommand{\TODO}[1]{{\color{red}#1}}
\newcommand{\TODO}[1]{{\color{red}TODO:#1}}
% put an explanation above an equal sign
% [1]: equality sign (or anything else)
@ -290,12 +293,16 @@
\input{translations.tex}
\input{trigonometry.tex}
\input{mechanics.tex}
\input{quantum_mechanics.tex}
\input{atom.tex}
\input{quantum_computing.tex}
\input{many-body-simulations.tex}
%\newpage
% \bibliographystyle{plain}
% \bibliography{ref}

10
src/many-body-simulations.tex Normal file
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@ -0,0 +1,10 @@
\Part[
\eng{Many-body simulations}
\ger{Vielteilchen Simulationen}
]{mbsim}
\Section[
\eng{Importance sampling}
\ger{Importance sampling / Stichprobenentnahme nach Wichtigkeit}
]{importance_sampling}

56
src/mechanics.tex Normal file
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@ -0,0 +1,56 @@
\Part[
\eng{Mechanics}
\ger{Mechanik}
]{mech}
\def\lagrange{\mathcal{L}}
\Section[
\eng{Lagrange formalism}
\ger{Lagrange Formalismus}
]{lagrange}
\begin{ttext}{desc}
\eng{The Lagrange formalism is often the most simple approach the get the equations of motion,
because with suitable generalied coordinates obtaining the Lagrange function is often relatively easy.
}
\ger{Der Lagrange-Formalsismus ist oft der einfachste Weg die Bewegungsgleichungen zu erhalten,
da das Aufstellen der Lagrange-Funktion mit geeigneten generalisierten Koordinaten oft relativ einfach ist.
}
\end{ttext}
\begin{ttext}{generalized_coords}
\eng{
The generalized coordinates are choosen so that the cronstraints are automatically fullfilled.
For example, the generalized coordinate for a 2D pendelum is $q=\varphi$, with $\vec{x} = \begin{pmatrix} \cos\varphi \\ \sin\varphi \end{pmatrix}$.
}
\ger{
Die generalisierten Koordinaten werden so gewählt, dass die Zwangsbedingungen automatisch erfüllt sind.
Zum Beispiel findet man für ein 2D Pendel die generalisierte Koordinate $q=\varphi$, mit $\vec{x} = \begin{pmatrix} \cos\varphi \\ \sin\varphi \end{pmatrix}$.
}
\end{ttext}
\begin{formula}{lagrangian}
\desc{Lagrange function}{}{$T$ kinetic energy, $V$ potential energy }
\desc[german]{Lagrange-Funktion}{}{$T$ kinetische Energie, $V$ potentielle Energie}
\eq{\lagrange = T - V}
\end{formula}
\begin{formula}{equation}
\desc{Lagrange equations (2nd type)}{}{$q$ generalized coordinates}
\desc[german]{Lagrange-Gleichungen (zweiter Art)}{}{$q$ generalisierte Koordinaten}
\eq{\odv{}{t} \pdv{\lagrange}{\dot{q_i}} - \pdv{\lagrange}{q_i} = 0}
\end{formula}
\begin{formula}{momentum}
\desc{Canocial Momentum}{}{}
\desc[german]{Kanonischer Impuls}{}{}
\eq{p = \pdv{\lagrange}{\dot{q}}}
\end{formula}
\begin{formula}{hamiltonian}
\desc{Hamiltonian}{Hamiltonian can be derived from the Lagrangian using a Legendre transformation}{}
\desc[german]{Hamiltonian}{Den Hamiltonian bekommt man aus dem Lagrangian über eine Legendre Transformation}{}
\eq{H(q,p) = p\,\dot{q}-\lagrange\big(q,\dot{q}(q,p)\big)}
\end{formula}
\TODO{Legendre trafo}

33
src/quantum_computing.tex
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@ -37,4 +37,37 @@
% \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}
\Section[
\eng{Josephson Junction}
\ger{Josephson-Kontakt}
]{josephson_junction}
\begin{ttext}{desc}
\eng{When two superconductors are separated by a thin isolator, Cooper pairs can tunnel through the insulator}
\ger{Wenn zwei Supraleiter durch einen dünnen Isolator getrennt sind, können Cooper-Paare durch den Isolator tunneln.}
\end{ttext}
\begin{formula}{hamiltonian}
\desc{Josephson-Hamiltonian}{}{}
\desc[german]{Josephson-Hamiltonian}{}{}
\eq{
\hat{H}_\text{J} &= - \frac{E_\text{J}}{2} \sum_n [\ket{n}\bra{n+1} + \ket{n+1}\bra{n}]
}
\end{formula}
\begin{formula}{1st_josephson_relation}
\desc{1. Josephson relation}{Dissipationless supercurrent accros junction at zero applied voltage}{$I_\text{C}=\frac{2e}{\hbar}E_\text{J}$ critical current, $\delta$ phase difference accross junction}
\desc[german]{1. Josephson Gleichung}{Dissipationsloser Suprastrom durch die Kreuzung ohne angelegte Spannung}{$I_\text{C}=\frac{2e}{\hbar}E_\text{J}$ kritischer Strom, $\delta$ Phasendifferenz zwischen den Supraleitern}
\eq{\hat{I}\ket{\delta} = I_\text{C}\sin\delta \ket{\delta}}
\end{formula}
\begin{formula}{2nd_josephson_relation}
\desc{2. Josephson relation}{superconducting phase change is proportional to applied voltage}{$\varphi_0=\frac{\hbar}{2e}$ reduced flux quantum}
\desc[german]{2. Josephson Gleichung}{Supraleitende Phasendifferenz is proportional zur angelegten Spannung}{$\varphi_0=\frac{\hbar}{2e}$ reduziertes Flussquantum}
\eq{\odv{\hat{\delta}}{t}=\frac{1}{i\hbar}[\hat{H},\hat{\delta}] = -\frac{2eU}{i\hbar}[\hat{n},\hat{\delta}] = \frac{1}{\varphi_0} U}
\end{formula}
\Section[
\eng{Cooper Pair Box (CPB) qubit}
\ger{Cooper Paar Box (QPB) Qubit}
]{cpb}

24
src/quantum_mechanics.tex
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@ -213,7 +213,7 @@
\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}
\odv{\hat{A}_\textrm{H}}{t} = \frac{1}{i\hbar}[\hat{A}_\textrm{H}, \hat{H}_\textrm{H}] + \Big(\pdv{\hat{A}_\textrm{S}}{t}\Big)_\textrm{H}
}
\end{formula}
@ -236,13 +236,13 @@
\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}}
\odv{}{t} \braket{\hat{A}} = \frac{1}{i\hbar}\braket{[\hat{A},\hat{H}]} + \Braket{\pdv{\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)}}
\eq{m\odv[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}
@ -374,10 +374,10 @@
% E_n=( \frac{1}{2} +n)\hbar\omega
% \end{equation}
\Section{angular_momentum}
\times
\Section[
\eng{Angular momentum}
\ger{Drehmoment}
]{angular_momentum}
\begin{formula}{bloch_waves}
\desc{Bloch waves}{
Solve the stat. SG in periodic potential with period
@ -394,3 +394,13 @@
\eq{\psi_k(\vec{r}) = e^{i \vec{k}\cdot \vec{r}} \cdot u_{\vec{k}}(\vec{r})}
\end{formula}
\Subsection[
\eng{Aharanov-Bohm effect}
\ger{Aharanov-Bohm Effekt}
]{aharanov_bohm}
\begin{formula}{phase}
\desc{Acquired phase}{Electron along a closed loop aquires a phase proportional to the enclosed magnetic flux}{}
\desc[german]{Erhaltene Phase}{Elektron entlang eines geschlossenes Phase erhält eine Phase die proportional zum eingeschlossenen magnetischem Fluss ist}{}
\eq{\delta = \frac{2 e}{\hbar} \oint \vec{A}\cdot \d\vec{s} = \frac{2 e}{\hbar} \Phi}
\end{formula}
\TODO{replace with loop intergral symbol and add more info}

62
src/trigonometry.tex
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@ -42,16 +42,54 @@
\eq{\cosh(x) &= \cos{ix} \\ &= \frac{e^{x}+e^{-x}}{2}}
\end{formula}
\Subsection[
\eng{Various theorems}
\ger{Verschiedene Theoreme}
]{theorems}
\begin{formula}{sum}
\desc{}{}{}
\desc[german]{}{}{}
\eq{1 &= \sin^2 x + \cos^2 x}
\end{formula}
\begin{table}[h]
\centering
% \caption{caption}
\label{tab:sin_cos_table}
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\GT{angle_deg} & 0° & 30° & 45° & 60° & 90° & 120° & 180° & 270° \\ \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 $ \\
$\cos(x)$ & $1$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{2}}{2}$ & $\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}
\begin{formula}{addition_theorems}
\desc{Addition theorems}{}{}
\desc[german]{Additionstheoreme}{}{}
\eq{
\sin(x\pm y) &= \sin x \cos x \pm \cos x \sin y \\
\cos(x\pm y) &= \cos x \cos x \mp \sin x \sin y \\
\tan(x\pm y) &= \frac{\sin(x \pm y)}{\cos(x \pm y)} = \frac{\tan x\pm \tan y}{1\mp \tan x \tan y}
}
\end{formula}
\begin{formula}{double_angle}
\desc{Double angle}{}{}
\desc[german]{Doppelwinkelfunktionen}{}{}
\eq{
\sin 2x &= 2\sin x \cos x \\
\cos 2x &= \cos^2 x - \sin^2 x = 1 - 2\sin^2 x \\
\tan 2x &= \frac{2\tan x}{1 - \tan^2x}
}
\end{formula}
\Subsection[
\eng{Table of values}
\ger{Wertetabelle}
]{value_table}
\begingroup
\setlength{\tabcolsep}{0.9em} % horizontal
\renewcommand{\arraystretch}{2} % vertical
\begin{table}[h]
\centering
% \caption{caption}
\label{tab:sin_cos_table}
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\GT{angle_deg} & 0° & 30° & 45° & 60° & 90° & 120° & 180° & 270° \\ \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 $ \\
$\cos(x)$ & $1$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{2}}{2}$ & $\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}
\endgroup