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This commit is contained in:
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6
.gitignore
vendored
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6
.gitignore
vendored
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@ -0,0 +1,6 @@
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.aux
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out
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*.virtual_documents*
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*ipynb_checkpoints*
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*__pycache__*
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**.pdf
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125
src/analysis.tex
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125
src/analysis.tex
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@ -0,0 +1,125 @@
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\Part[
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\eng{Calculus}
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\ger{Analysis}
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]{cal}
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\Subsection[
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\eng{Convolution}
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\ger{Faltung / Konvolution}
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]{conv}
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\begin{ttext}
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\eng{Convolution is \textbf{commutative}, \textbf{associative} and \textbf{distributive}.}
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\ger{Die Faltung ist \textbf{kommutativ}, \textbf{assoziativ} und \textbf{distributiv}}
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\end{ttext}
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\begin{formula}{def}
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\desc{Definition}{}{}
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\desc[german]{Definition}{}{}
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\eq{(f*g)(t) = f(t) * g(t) = int_{-\infty}^\infty f(\tau) g(t-\tau) \d \tau}
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\end{formula}
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\begin{formula}{notation}
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\desc{Notation}{}{}
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\desc[german]{Notation}{}{}
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\eq{
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f(t) * g(t-t_0) &= (f*g)(t-t_0) \\
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f(t-t_0) * g(t-t_0) &= (f*g)(t-2t_0)
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}
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\end{formula}
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\begin{formula}{commutativity}
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\desc{Commutativity}{}{}
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\desc[german]{Kommutativität}{}{}
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\eq{f * g = g * f}
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\end{formula}
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\begin{formula}{associativity}
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\desc{Associativity}{}{}
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\desc[german]{Assoziativität]}{}{}
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\eq{(f*g)*h = f*(g*h)}
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\end{formula}
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\begin{formula}{distributivity}
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\desc{Distributivity}{}{}
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\desc[german]{Distributivität}{}{}
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\eq{f * (g + h) = f*g + f*h}
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\end{formula}
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\begin{formula}{complex_conjugate}
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\desc{Complex conjugate}{}{}
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\desc[german]{Komplexe konjugation}{}{}
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\eq{(f*g)^* = f^* * g^*}
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\end{formula}
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\Subsection[
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\eng{Fourier analysis}
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\ger{Fourieranalyse}
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]{fourier}
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\Subsubsection[
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\eng{Fourier series}
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\ger{Fourierreihe}
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]{series}
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\begin{formula}{series}
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\desc{Fourier series}{Complex representation}{$f\in \Lebesgue^2(\R,\C)$ $T$-\GT{periodic}}
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\desc[german]{Fourierreihe}{Komplexe Darstellung}{}
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\eq{f(t) = \sum_{k=-\infty}^{\infty} c_k \Exp{\frac{2\pi \I kt}{T}}}
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\end{formula}
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\Eng[real]{real}
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\Ger[real]{reellwertig}
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\begin{formula}{coefficient}
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\desc{Fourier coefficients}{Complex representation}{}
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\desc[german]{Fourierkoeffizienten}{Komplexe Darstellung}{}
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\eq{
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c_k &= \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Exp{-\frac{2\pi \I}{T}kt}\d t \quad\text{\GT{for}}\,k\ge0\\
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c_{-k} &= \overline{c_k} \quad \text{\GT{if} $f$ \GT{real}}
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}
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\end{formula}
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\begin{formula}{series_sincos}
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\desc{Fourier series}{Sine and cosine representation}{$f\in \Lebesgue^2(\R,\C)$ $T$-\GT{periodic}}
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\desc[german]{Fourierreihe}{Sinus und Kosinus Darstellung}{}
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\eq{f(t) = \frac{a_0}{2} + \sum_{k=1}^{\infty} \left(a_k \Cos{\frac{2\pi}{T}kt} + b_k\Sin{\frac{2\pi}{T}kt}\right)}
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\end{formula}
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\begin{formula}{coefficient}
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\desc{Fourier coefficients}{Sine and cosine representation\\If $f$ has point symmetry: $a_{k>0}=0$, if $f$ has axial symmetry: $b_k=0$}{}
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\desc[german]{Fourierkoeffizienten}{Sinus und Kosinus Darstellung\\Wenn $f$ punktsymmetrisch: $a_{k>0}=0$, wenn $f$ achsensymmetrisch: $b_k=0$}{}
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\eq{
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a_k &= \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Cos{-\frac{2\pi}{T}kt}\d t \quad\text{\GT{for}}\,k\ge0\\
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b_k &= \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Sin{-\frac{2\pi}{T}kt}\d t \quad\text{\GT{for}}\,k\ge1\\
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a_k &= c_k + c_{-k} \quad\text{\GT{for}}\,k\ge0\\
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b_k &= \I(c_k - c_{-k}) \quad\text{\GT{for}}\,k\ge1
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}
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\end{formula}
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\TODO{cleanup}
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\Subsubsection[
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\eng{Fourier transformation}
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\ger{Fouriertransformation}
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]{trafo}
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\begin{formula}{transform}
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\desc{Fourier transform}{}{$\hat{f}:\R^n \mapsto \C$, $\forall f\in L^1(\R^n)$}
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\desc[german]{Fouriertransformierte}{}{}
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\eq{\hat{f}(k) \coloneq \frac{1}{\sqrt{2\pi}^n} \int_{\R^n} \e^{-\I kx}f(x)\d x}
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\end{formula}
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\Eng[linear_in]{linear in}
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\Ger[linear_in]{linear in}
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\GT{for} $f\in L^1(\R^n)$:
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\begin{enumerate}[i)]
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\item $f \mapsto \hat{f}$ \GT{linear_in} $f$
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\item $g(x) = f(x-h) \qRarrow \hat{g}(k) = \e^{-\I kn}\hat{f}(k)$
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\item $g(x) = \e^{ih\cdot x}f(x) \qRarrow \hat{g}(k) = \hat{f}(k-h)$
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\item $g(\lambda) = f\left(\frac{x}{\lambda}\right) \qRarrow \hat{g}(k)\lambda^n \hat{f}(\lambda k)$
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\end{enumerate}
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\Section[
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\eng{List of common integrals}
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\ger{Liste nützlicher Integrale}
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]{integrals}
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\begin{formula}{riemann_zeta}
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\desc{Riemann Zeta Function}{}{}
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\desc[german]{Riemannsche Zeta-Funktion}{}{}
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\eq{\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \frac{1}{(1-2^{(1-s)})\Gamma(s)} \int_0^\infty \d\eta \frac{\eta^{(s-1)}}{\e^\eta + 1}}
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\end{formula}
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31
src/atom.tex
31
src/atom.tex
@ -1,7 +1,6 @@
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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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\def\vecr{{\vec{r}}}
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\kef\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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@ -23,9 +22,8 @@
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\desc[german]{Hamiltonian}{}{}
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% \eq{V(\vecr) = \frac{Z\,e^2}{4\pi\epsilon_0 r}}
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\eq{
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% \hat{H} &= -\frac{\hbar^2}{2\mu} {\grad_\vecr}^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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\hat{H} &= -\frac{\hbar^2}{2\mu} {\Grad_\vecr}^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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}
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\end{formula}
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@ -67,7 +65,7 @@
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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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\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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@ -87,7 +85,7 @@
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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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\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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@ -95,7 +93,7 @@
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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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\eq{\Delta E_\text{LS} = \frac{\mu_0 Z e^2}{8\pi \masse^2\,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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@ -108,7 +106,7 @@
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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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\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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@ -122,8 +120,8 @@
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\Subsubsection[
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\eng{Lamb-shift}
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\ger{Lamb-Shift}
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]{lamb_shift}
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\begin{ttext}{desc}
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]{lamb_shift}
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\begin{ttext}[desc]
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\eng{The interaction of the electron with virtual photons emitted/absorbed by the nucleus leads to a (very small) shift in the energy level.}
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\ger{The Wechselwirkung zwischen dem Elektron und vom Kern absorbierten/emittierten virtuellen Photonen führt zu einer (sehr kleinen) Energieverschiebung.}
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\end{ttext}
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@ -136,11 +134,10 @@
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\Subsubsection[
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\eng{Hyperfine structure}
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\ger{Hyperfeinstruktur}
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]{hyperfine_structure}
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\begin{ttext}{desc}
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\eng{Interaction of the nucleus spin with the magnetic field created by the electron leads to energy shifts. (Lifts degenaracy) }
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]{hyperfine_structure}
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\begin{ttext}[desc]
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\eng{Interaction of the nucleus spin with the magnetic field created by the electron leads to energy shifts. (Lifts degeneracy) }
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\ger{Wechselwirkung von Kernspin mit dem vom Elektron erzeugten Magnetfeld spaltet Energieniveaus}
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\end{ttext}
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\begin{formula}{nuclear_spin}
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\desc{Nuclear spin}{}{}
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55
src/calculus.tex
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55
src/calculus.tex
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\Part[
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\eng{Analysis}
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\ger{Analysis}
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]{ana}
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\Subsection[
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\eng{Convolution}
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\ger{Faltung / Konvolution}
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]{conv}
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\begin{ttext}
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\eng{Convolution is \textbf{commutative}, \textbf{associative} and \textbf{distributive}.}
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\ger{Die Faltung ist \textbf{kommutativ}, \textbf{assoziativ} und \textbf{distributiv}}
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\end{ttext}
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\begin{formula}{def}
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\desc{Definition}{}{}
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\desc[german]{Definition}{}{}
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\eq{(f*g)(t) = f(t) * g(t) = int_{-\infty}^\infty f(\tau) g(t-\tau) \d \tau}
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\end{formula}
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\begin{formula}{notation}
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\desc{Notation}{}{}
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\desc[german]{Notation}{}{}
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\eq{
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f(t) * g(t-t_0) &= (f*g)(t-t_0) \\
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f(t-t_0) * g(t-t_0) &= (f*g)(t-2t_0)
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}
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\end{formula}
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\begin{formula}{commutativity}
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\desc{Commutativity}{}{}
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\desc[german]{Kommutativität}{}{}
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\eq{f * g = g * f}
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\end{formula}
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\begin{formula}{associativity}
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\desc{Associativity}{}{}
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\desc[german]{Assoziativität]}{}{}
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\eq{(f*g)*h = f*(g*h)}
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\end{formula}
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\begin{formula}{distributivity}
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\desc{Distributivity}{}{}
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\desc[german]{Distributivität}{}{}
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\eq{f * (g + h) = f*g + f*h}
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\end{formula}
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\begin{formula}{complex_conjugate}
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\desc{Complex conjugate}{}{}
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\desc[german]{Komplexe konjugation}{}{}
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\eq{(f*g)^* = f^* * g^*}
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\end{formula}
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\Subsection[
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\eng{Fourier analysis}
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\ger{Fourieranalyse}
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]{fourier}
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79
src/circuit.tex
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79
src/circuit.tex
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% Josephson junction
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\makeatletter
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\pgfcircdeclarebipolescaled{instruments}
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{
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% put the node text above and centered
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\anchor{text}{\pgfextracty{\pgf@circ@res@up}{\northeast}
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\pgfpoint{-.5\wd\pgfnodeparttextbox}{
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\dimexpr.5\dp\pgfnodeparttextbox+.5\ht\pgfnodeparttextbox+\pgf@circ@res@up\relax
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}
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}
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}
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{\ctikzvalof{bipoles/oscope/height}}
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{josephson}
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{\ctikzvalof{bipoles/oscope/height}}
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{\ctikzvalof{bipoles/oscope/width}}
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{
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% ?
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\pgf@circ@setlinewidth{bipoles}{\pgfstartlinewidth}
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\pgfextracty{\pgf@circ@res@up}{\northeast}
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\pgfextractx{\pgf@circ@res@right}{\northeast}
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\pgfextractx{\pgf@circ@res@left}{\southwest}
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\pgfextracty{\pgf@circ@res@down}{\southwest}
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\pgfmathsetlength{\pgf@circ@res@step}{0.25*\pgf@circ@res@up}
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% \pgfscope % box
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% \pgfpathrectanglecorners{\pgfpoint{\pgf@circ@res@left}{\pgf@circ@res@down}}{\pgfpoint{\pgf@circ@res@right}{\pgf@circ@res@up}}
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% \pgf@circ@draworfill
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% \endpgfscope
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\pgfscope % cross
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\pgfpathmoveto{\pgfpoint{\pgf@circ@res@left}{\pgf@circ@res@up}}%
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\pgfpathlineto{\pgfpoint{\pgf@circ@res@right}{\pgf@circ@res@down}}%
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\pgfpathmoveto{\pgfpoint{\pgf@circ@res@right}{\pgf@circ@res@up}}%
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\pgfpathlineto{\pgfpoint{\pgf@circ@res@left}{\pgf@circ@res@down}}%
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\pgfusepath{draw}
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\endpgfscope
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}
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\def\pgf@circ@josephson@path#1{\pgf@circ@bipole@path{josephson}{#1}}
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\tikzset{josephson/.style = {\circuitikzbasekey, /tikz/to path=\pgf@circ@josephson@path, l=#1}}
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\pgfcircdeclarebipolescaled{instruments}
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{
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% put the node text above and centered
|
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\anchor{text}{\pgfextracty{\pgf@circ@res@up}{\northeast}
|
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\pgfpoint{-.5\wd\pgfnodeparttextbox}{
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\dimexpr.5\dp\pgfnodeparttextbox+.5\ht\pgfnodeparttextbox+\pgf@circ@res@up\relax
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||||
}
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||||
}
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||||
}
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||||
{\ctikzvalof{bipoles/oscope/height}}
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{josephsoncap}
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||||
{\ctikzvalof{bipoles/oscope/height}}
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{\ctikzvalof{bipoles/oscope/width}}
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{
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||||
% ?
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||||
\pgf@circ@setlinewidth{bipoles}{\pgfstartlinewidth}
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||||
\pgfextracty{\pgf@circ@res@up}{\northeast}
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||||
\pgfextractx{\pgf@circ@res@right}{\northeast}
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||||
\pgfextractx{\pgf@circ@res@left}{\southwest}
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\pgfextracty{\pgf@circ@res@down}{\southwest}
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\pgfmathsetlength{\pgf@circ@res@step}{0.25*\pgf@circ@res@up}
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\pgfscope % box
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||||
\pgfpathrectanglecorners{\pgfpoint{\pgf@circ@res@left}{\pgf@circ@res@down}}{\pgfpoint{\pgf@circ@res@right}{\pgf@circ@res@up}}
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\pgf@circ@draworfill
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\endpgfscope
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\pgfscope % cross
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\pgfpathmoveto{\pgfpoint{\pgf@circ@res@left}{\pgf@circ@res@up}}%
|
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\pgfpathlineto{\pgfpoint{\pgf@circ@res@right}{\pgf@circ@res@down}}%
|
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\pgfpathmoveto{\pgfpoint{\pgf@circ@res@right}{\pgf@circ@res@up}}%
|
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\pgfpathlineto{\pgfpoint{\pgf@circ@res@left}{\pgf@circ@res@down}}%
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\pgfusepath{draw}
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||||
\endpgfscope
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||||
}
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\def\pgf@circ@josephsoncap@path#1{\pgf@circ@bipole@path{josephsoncap}{#1}}
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\tikzset{josephsoncap/.style = {\circuitikzbasekey, /tikz/to path=\pgf@circ@josephsoncap@path, l=#1}}
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||||
|
||||
|
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231
src/condensed_matter.tex
Normal file
231
src/condensed_matter.tex
Normal file
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\Part[
|
||||
\eng{Condensed matter physics}
|
||||
\ger{Festkörperphysik}
|
||||
]{cm}
|
||||
\Section[
|
||||
\eng{Bravais lattice}
|
||||
\ger{Bravais-Gitter}
|
||||
]{bravais}
|
||||
|
||||
% \begin{ttext}
|
||||
% \eng{
|
||||
|
||||
% }
|
||||
% \ger{
|
||||
|
||||
% }
|
||||
% \end{ttext}
|
||||
|
||||
\eng[bravais_table2]{In 2D, there are 5 different Bravais lattices}
|
||||
\ger[bravais_table2]{In 2D gibt es 5 verschiedene Bravais-Gitter}
|
||||
|
||||
\eng[bravais_table3]{In 3D, there are 14 different Bravais lattices}
|
||||
\ger[bravais_table3]{In 3D gibt es 14 verschiedene Bravais-Gitter}
|
||||
|
||||
\Eng[lattice_system]{Lattice system}
|
||||
\Ger[lattice_system]{Gittersystem}
|
||||
\Eng[crystal_family]{Crystal system}
|
||||
\Ger[crystal_family]{Kristall-system}
|
||||
\Eng[point_group]{Point group}
|
||||
\Ger[point_group]{Punktgruppe}
|
||||
\eng[bravais_lattices]{Bravais lattices}
|
||||
\ger[bravais_lattices]{Bravais Gitter}
|
||||
|
||||
\newcommand\bvimg[1]{\begin{center}\includegraphics[width=0.1\textwidth]{img/bravais/#1.pdf}\end{center}}
|
||||
\renewcommand\tabularxcolumn[1]{m{#1}}
|
||||
\newcolumntype{Z}{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}X}
|
||||
\begin{table}[H]
|
||||
\centering
|
||||
\caption{\gt{bravais_table2}}
|
||||
\label{tab:bravais2}
|
||||
|
||||
\begin{adjustbox}{width=\textwidth}
|
||||
\begin{tabularx}{\textwidth}{||Z|c|Z|Z||}
|
||||
\hline
|
||||
\multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{2}{c||}{5 \gt{bravais_lattices}} \\ \cline{3-4}
|
||||
& & \GT{primitive} (p) & \GT{centered} (c) \\ \hline
|
||||
\GT{monoclinic} (m) & $\text{C}_\text{2}$ & \bvimg{mp} & \\ \hline
|
||||
\GT{orthorhombic} (o) & $\text{D}_\text{2}$ & \bvimg{op} & \bvimg{oc} \\ \hline
|
||||
\GT{tetragonal} (t) & $\text{D}_\text{4}$ & \bvimg{tp} & \\ \hline
|
||||
\GT{hexagonal} (h) & $\text{D}_\text{6}$ & \bvimg{hp} & \\ \hline
|
||||
\end{tabularx}
|
||||
\end{adjustbox}
|
||||
\end{table}
|
||||
|
||||
|
||||
|
||||
\begin{table}[H]
|
||||
\centering
|
||||
\caption{\gt{bravais_table3}}
|
||||
\label{tab:bravais3}
|
||||
|
||||
% \newcolumntype{g}{>{\columncolor[]{0.8}}}
|
||||
\begin{adjustbox}{width=\textwidth}
|
||||
% \begin{tabularx}{\textwidth}{|c|}
|
||||
% asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
|
||||
% asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
|
||||
% \end{tabularx}
|
||||
% \begin{tabular}{|c|}
|
||||
% asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
|
||||
% asdfasdfadslfasdfaasdofiuapsdoifuapodisufpaoidsufpaoidsufpaoisdfaoisdfpaosidfupaoidsufpaoidsufpaoidsufpaoisdufpaoidsufpoaiudsfpioaspdoifuaposidufpaoisudpfoiaupsdoifupasodf \\
|
||||
% \end{tabular}
|
||||
% \\
|
||||
\begin{tabularx}{\textwidth}{||Z|Z|c|Z|Z|Z|Z||}
|
||||
\hline
|
||||
\multirow{2}{*}{\GT{crystal_family}} & \multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{4}{c||}{14 \gt{bravais_lattices}} \\ \cline{4-7}
|
||||
& & & \GT{primitive} (P) & \GT{base_centered} (S) & \GT{body_centered} (I) & \GT{face_centered} (F) \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{triclinic} (a)} & $\text{C}_\text{i}$ & \bvimg{tP} & & & \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{monoclinic} (m)} & $\text{C}_\text{2h}$ & \bvimg{mP} & \bvimg{mS} & & \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{orthorhombic} (o)} & $\text{D}_\text{2h}$ & \bvimg{oP} & \bvimg{oS} & \bvimg{oI} & \bvimg{oF} \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{tetragonal} (t)} & $\text{D}_\text{4h}$ & \bvimg{tP} & & \bvimg{tI} & \\ \hline
|
||||
\multirow{2}{*}{\GT{hexagonal} (h)} & \GT{rhombohedral} & $\text{D}_\text{3d}$ & \bvimg{hR} & & & \\ \cline{2-7}
|
||||
& \GT{hexagonal} & $\text{D}_\text{6h}$ & \bvimg{hP} & & & \\ \hline
|
||||
\multicolumn{2}{||c|}{\GT{cubic} (c)} & $\text{O}_\text{h}$ & \bvimg{cP} & & \bvimg{cI} & \bvimg{cF} \\ \hline
|
||||
\end{tabularx}
|
||||
\end{adjustbox}
|
||||
\end{table}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Reciprocal lattice}
|
||||
\ger{Reziprokes Gitter}
|
||||
]{reci}
|
||||
\begin{ttext}
|
||||
\eng{The reciprokal lattice is made up of all the wave vectors $\vec{k}$ that ressemble standing waves with the periodicity of the Bravais lattice.}
|
||||
\ger{Das rezioproke Gitter besteht aus dem dem Satz aller Wellenvektoren $\vec{k}$, die ebene Wellen mit der Periodizität des Bravais-Gitters ergeben.}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{vectors}
|
||||
\desc{Reciprocal lattice vectors}{}{$a_i$ real-space lattice vectors, $V_c$ volume of the primitive lattice cell}
|
||||
\desc[german]{Reziproke Gittervektoren}{}{$a_i$ Bravais-Gitter Vektoren, $V_c$ Volumen der primitiven Gitterzelle}
|
||||
\eq{
|
||||
\vec{b_1} &= \frac{2\pi}{V_c} \vec{a_2} \times \vec{a_3} \\
|
||||
\vec{b_2} &= \frac{2\pi}{V_c} \vec{a_3} \times \vec{a_1} \\
|
||||
\vec{b_3} &= \frac{2\pi}{V_c} \vec{a_1} \times \vec{a_2}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Free electron gas}
|
||||
\ger{Freies Elektronengase}
|
||||
]{free_e_gas}
|
||||
\begin{ttext}
|
||||
\eng{Assumptions: electrons can move freely and independent of each other.}
|
||||
\ger{Annahmen: Elektronen bewegen sich frei und unabhänig voneinander.}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{drift_velocity}
|
||||
\desc{Drift velocity}{Velocity component induced by an external force (eg. electric field)}{$v_\text{th}$ thermal velocity}
|
||||
\desc[german]{Driftgeschwindgkeit}{Geschwindigkeitskomponente durch eine externe Kraft (z.B. ein elektrisches Feld)}{$v_\text{th}$ thermische Geschwindigkeit}
|
||||
\eq{\vec{v}_\text{D} = \vec{v} - \vec{v}_\text{th}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{mean_free_time}
|
||||
\desc{Mean free time}{}{}
|
||||
\desc[german]{Streuzeit}{}{}
|
||||
\eq{\tau}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{mean_free_path}
|
||||
\desc{Mean free path}{}{}
|
||||
\desc[german]{Mittlere freie Weglänge}{}{}
|
||||
\eq{\ell = \braket{v} \tau}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{mobility}
|
||||
\desc{Electrical mobility}{}{$q$ charge, $m$ mass}
|
||||
\desc[german]{Beweglichkeit}{}{$q$ Ladung, $m$ Masse}
|
||||
\eq{\mu = \frac{q \tau}{m}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Drude model}
|
||||
\ger{Drude-Modell}
|
||||
]{drude}
|
||||
\begin{ttext}
|
||||
\eng{Classical model describing the transport properties of electrons in materials (metals):
|
||||
The material is assumed to be an ion lattice and with freely moving electrons (electron gas). The electrons are
|
||||
accelerated by an electric field and decelerated through collisions with the lattice ions.
|
||||
The model disregards the Fermi-Dirac partition of the conducting electrons.
|
||||
}
|
||||
\ger{Ein klassisches Model zur Beschreibung der Transporteigenschaften von Elektronen in (v.a.) Metallen:
|
||||
Der Festkörper wird als Ionenkristall mit frei beweglichen Elektronen (Elektronengas).
|
||||
Die Elektronen werden durch ein Elektrisches Feld $E$ beschleunigt und durch Stöße mit den Gitterionen gebremst.
|
||||
Das Modell vernachlässigt die Fermi-Dirac Verteilung der Leitungselektronen.
|
||||
}
|
||||
\end{ttext}
|
||||
\begin{formula}{motion}
|
||||
\desc{Equation of motion}{}{$v$ electron speed, $\vec{v}_\text{D}$ drift velocity, $\tau$ mean free time between collisions}
|
||||
\desc[german]{Bewegungsgleichung}{}{$v$ Elektronengeschwindigkeit, $\vec{v}_\text{D}$ Driftgeschwindigkeit, $\tau$ Stoßzeit}
|
||||
\eq{\masse \odv{\vec{v}}{t} + \frac{\masse}{\tau} \vec{v}_\text{D} = -e \vec{E}}
|
||||
\end{formula}
|
||||
\begin{formula}{current_density}
|
||||
\desc{Current density}{Ohm's law}{$n$ charge particle density}
|
||||
\desc[german]{Stromdichte}{Ohmsches Gesetz}{$n$ Ladungsträgerdichte}
|
||||
\eq{\vec{j} = -ne\vec{v}_\text{D} = ne\mu \vec{E}}
|
||||
\end{formula}
|
||||
\begin{formula}{conductivity}
|
||||
\desc{Drude-conductivity}{}{}
|
||||
\desc[german]{Drude-Leitfähigkeit}{}{}
|
||||
\eq{\sigma = \frac{\vec{j}}{\vec{E}} = \frac{e^2 \tau n}{\masse} = n e \mu}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Sommerfeld model}
|
||||
\ger{Sommerfeld-Modell}
|
||||
]{sommerfeld}
|
||||
\begin{ttext}
|
||||
\eng{Assumes a gas of free fermions underlying the pauli-exclusion principle. Only electrons in an energy range of $\kB T$ around the Fermi energy $\EFermi$ participate in scattering processes.}
|
||||
\ger{Annahme eines freien Fermionengases, welches dem Pauli-Prinzip unterliegt. Nur Elektronen in einem Energiebereich von $\kB T$ um die Fermi Energe $\EFermi$ nehmen an Streuprozessen teil.}
|
||||
\end{ttext}
|
||||
\begin{formula}{current_density}
|
||||
\desc{Current density}{}{}
|
||||
\desc[german]{Stromdichte}{}{}
|
||||
\eq{\vec{j} = -en\braket{v} = -e n \frac{\hbar}{\masse}\braket{\vec{k}} = -e \frac{1}{V} \sum_{\vec{k},\sigma} \frac{\hbar \vec{k}}{\masse}}
|
||||
\end{formula}
|
||||
\TODO{The formula for the conductivity is the same as in the drude model?}
|
||||
|
||||
\Subsection[
|
||||
\eng{2D electron gas}
|
||||
\ger{2D Elektronengas}
|
||||
]{2deg}
|
||||
|
||||
\begin{ttext}
|
||||
\eng{Lower dimension gases can be obtained by restricting a 3D gas with infinetly high potential walls on a narrow area with the width $L$.}
|
||||
\ger{
|
||||
Niederdimensionale Elektronengase erhält man, wenn ein 3D Gas durch unendlich hohe Potentialwände auf einem schmalen Bereich mit Breite $L$ eingeschränkt wird.
|
||||
}
|
||||
\end{ttext}
|
||||
\begin{formula}{confinement_energy}
|
||||
\desc{Confinement energy}{Raises ground state energy}{}
|
||||
\desc[german]{Confinement Energie}{Erhöht die Grundzustandsenergie}{}
|
||||
\eq{\Delta E = \frac{\hbar^2 \pi^2}{2\masse L^2}}
|
||||
\end{formula}
|
||||
|
||||
\Eng[plain_wave]{plain wave}
|
||||
\Ger[plain_wave]{ebene Welle}
|
||||
\begin{formula}{energy}
|
||||
\desc{Energy}{}{}
|
||||
\desc[german]{Energie}{}{}
|
||||
\eq{E_n = \underbrace{\frac{\hbar^2 k_\parallel^2}{2\masse}}_\text{$x$-$y$: \GT{plain_wave}} + \underbrace{\frac{\hbar^2 \pi^2}{2\masse L^2} n^2}_\text{$z$}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{1D electron gas / quantum wire}
|
||||
\ger{1D Eleltronengas / Quantendraht}
|
||||
]{1deg}
|
||||
|
||||
\begin{formula}{energy}
|
||||
\desc{Energy}{}{}
|
||||
\desc[german]{Energie}{}{}
|
||||
\eq{E_n = \frac{\hbar^2 k_x^2}{2\masse} + \frac{\hbar^2 \pi^2}{2\masse L_z^2} n_1^2 + \frac{\hbar^2 \pi^2}{2\masse L_y^2} n_2^2}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{0D electron gas / quantum dot}
|
||||
\ger{0D Elektronengase / Quantenpunkt}
|
||||
]{0deg}
|
||||
|
||||
\TODO{TODO}
|
||||
|
||||
222
src/electrodynamics.tex
Normal file
222
src/electrodynamics.tex
Normal file
@ -0,0 +1,222 @@
|
||||
\def\PhiB{\Phi_\text{B}}
|
||||
\def\PhiE{\Phi_\text{E}}
|
||||
|
||||
\Part[
|
||||
\eng{Electrodynamics}
|
||||
\ger{Elektrodynamik}
|
||||
]{ed}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Maxwell-Equations}
|
||||
\ger{Maxwell-Gleichungen}
|
||||
]{Maxwell}
|
||||
\begin{formula}{vacuum}
|
||||
\desc{Vacuum}{microscopic formulation}{}
|
||||
\desc[german]{Vakuum}{Mikroskopische Formulierung}{}
|
||||
\eq{
|
||||
\Div \vec{E} &= \frac{\rho_\text{el}}{\epsilon_0} \\
|
||||
\Div \vec{B} &= 0 \\
|
||||
\Rot \vec{E} &= - \odv{\vec{B}}{t} \\
|
||||
\Rot \vec{B} &= \mu_0 \vec{j} + \frac{1}{c^2} \odv{\vec{E}}{t}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{material}
|
||||
\desc{Matter}{Macroscopic formulation}{}
|
||||
\desc[german]{Materie}{Makroskopische Formulierung}{}
|
||||
\eq{
|
||||
\Div \vec{D} &= \rho_\text{el} \\
|
||||
\Div \vec{B} &= 0 \\
|
||||
\Rot \vec{E} &= - \odv{\vec{B}}{t} \\
|
||||
\Rot \vec{H} &= \vec{j} + \odv{\vec{D}}{t}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Fields}
|
||||
\ger{Felder}
|
||||
]{fields}
|
||||
|
||||
\Subsection[
|
||||
\eng{Electric field}
|
||||
\ger{Elektrisches Feld}
|
||||
]{mag}
|
||||
\begin{formula}{gauss_law}
|
||||
\desc{Gauss's law for electric fields}{Electric flux through a closed surface is proportional to the electric charge}{$S$ closed surface}
|
||||
\desc[german]{Gaußsches Gesetz für elektrische Felder}{Der magnetische Fluss durch eine geschlossene Fläche ist proportional zur elektrischen Ladung}{$S$ geschlossene Fläche}
|
||||
\eq{\PhiE = \iint_S \vec{E}\cdot\d\vec{S} = \frac{Q}{\varepsilon_0}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Magnetic field}
|
||||
\ger{Magnetfeld}
|
||||
]{mag}
|
||||
|
||||
\Eng[magnetic_flux]{Magnetix flux density}
|
||||
\Ger[magnetic_flux]{Magnetische Flussdichte}
|
||||
|
||||
% \begin{quantity}{mag_flux}{\Phi}{\Wb}{\kg\m^2\per\s^2\A^1}{scalar}
|
||||
% \sign{}
|
||||
% \desc{Magnetic flux density}{}
|
||||
% \desc[german]{Magnetische Feldstärke}{}
|
||||
% \end{quantity}
|
||||
|
||||
\begin{formula}{magnetic_flux}
|
||||
\desc{Magnetic flux}{}{}
|
||||
\desc[german]{Magnetischer Fluss}{}{}
|
||||
\eq{\PhiB = \iint_A \vec{B}\cdot\d\vec{A}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gauss_law}
|
||||
\desc{Gauss's law for magnetism}{Magnetic flux through a closed surface is $0$ \Rightarrow there are no magnetic monopoles}{$S$ closed surface}
|
||||
\desc[german]{Gaußsches Gesetz für Magnetismus}{Der magnetische Fluss durch eine geschlossene Fläche ist $0$ \Rightarrow es gibt keine magnetischen Monopole}{$S$ geschlossene Fläche}
|
||||
\eq{\PhiB = \iint_S \vec{B}\cdot\d\vec{S} = 0}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{name}
|
||||
\desc{}{}{}
|
||||
\desc[german]{}{}{}
|
||||
\eq{}
|
||||
\end{formula}
|
||||
\begin{formula}{magnetization}
|
||||
\desc{Magnetization}{}{$m$ mag. moment, $V$ volume}
|
||||
\desc[german]{Magnetisierung}{}{$m$ mag. Moment, $V$ Volumen}
|
||||
\eq{\vec{M} = \odv{\vec{m}}{V} = \chi_\text{m} \cdot \vec{H}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{angular_torque}
|
||||
\desc{Torque}{}{$m$ mag. moment}
|
||||
\desc[german]{Drehmoment}{}{$m$ mag. Moment}
|
||||
\eq{\vec{\tau} = \vec{m} \times \vec{B}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{suceptibility}
|
||||
\desc{Susceptibility}{}{}
|
||||
\desc[german]{Suszeptibilität}{}{}
|
||||
\eq{\chi_\text{m} = \pdv{M}{B} = \frac{\mu}{\mu_0} - 1 }
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{poynting}
|
||||
\desc{Poynting vector}{Directional energy flux or power flow of an electromagnetic field [$\si{\W\per\m^2}$]}{}
|
||||
\desc[german]{Poynting-Vektor}{Gerichteter Energiefluss oder Leistungsfluss eines elektromgnetischen Feldes [$\si{\W\per\m^2}$]}{}
|
||||
\eq{\vec{S} = \vec{E} \times \vec{H}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Induction}
|
||||
\ger{Unduktion}
|
||||
]{induction}
|
||||
\begin{formula}{farady_law}
|
||||
\desc{Faraday's law of induction}{}{}
|
||||
\desc[german]{Faradaysche Induktionsgesetz}{}{}
|
||||
\eq{U_\text{ind} = -\odv{}{t} \PhiB = - \odv{}{t} \iint_A\vec{B} \cdot \d\vec{A}}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Hall-Effect}
|
||||
\ger{Hall-Effekt}
|
||||
]{hall}
|
||||
|
||||
\begin{formula}{cyclotron}
|
||||
\desc{Cyclontron frequency}{}{}
|
||||
\desc[german]{Zyklotronfrequenz}{}{}
|
||||
\eq{\omega_\text{c} = \frac{e B}{\masse}}
|
||||
\end{formula}
|
||||
\TODO{Move}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Classical Hall-Effect}
|
||||
\ger{Klassischer Hall-Effekt}
|
||||
]{classic}
|
||||
\begin{ttext}
|
||||
\eng{Current flowing in $x$ direction in a conductor ($l \times b \times d$) with a magnetic field $B$ in $z$ direction leads to a hall voltage $U_\text{H}$ in $y$ direction.}
|
||||
\ger{Fließt in einem Leiter ($l \times b \times d$) ein Strom in $x$ Richtung, während der Leiter von einem Magnetfeld $B$ in $z$-Richtung durchdrungen, wird eine Hallspannung $U_\text{H}$ in $y$-Richtung induziert.}
|
||||
\end{ttext}
|
||||
\begin{formula}{voltage}
|
||||
\desc{Hall voltage}{}{$n$ charge carrier density}
|
||||
\desc[german]{Hallspannung}{}{$n$ Ladungsträgerdichte}
|
||||
\eq{U_\text{H} = \frac{I B}{ne d}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{coefficient}
|
||||
\desc{Hall coefficient}{}{}
|
||||
\desc[german]{Hall-Koeffizient}{}{}
|
||||
\eq{R_\text{H} = -\frac{Eg}{j_x Bg} = \frac{1}{ne} = \frac{\rho_{xy}}{B_z}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{resistivity}
|
||||
\desc{Resistivity}{}{}
|
||||
\desc[german]{Spezifischer Widerstand}{}{}
|
||||
\eq{\rho_{xx} &= \frac{\masse}{ne^2\tau} \\ \rho_{xy} &= \frac{B}{ne}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Integer quantum hall effect}
|
||||
\ger{Ganzahliger Quantenhalleffekt}
|
||||
]{quantum}
|
||||
|
||||
\begin{formula}{conductivity}
|
||||
\desc{Conductivity tensor}{}{}
|
||||
\desc[german]{Leitfähigkeitstensor}{}{}
|
||||
\eq{\sigma = \begin{pmatrix} \sigma_{xy} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix} }
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{resistivity}
|
||||
\desc{Resistivity tensor}{}{}
|
||||
\desc[german]{Spezifischer Widerstands-tensor}{}{}
|
||||
\eq{
|
||||
\rho = \sigma^{-1}
|
||||
% \sigma = \begin{pmatrix} \sigma_{xy} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix} }
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{resistivity}
|
||||
\desc{Resistivity}{}{$\nu \in \mathbb{Z}$}
|
||||
\desc[german]{Spezifischer Hallwiderstand}{}{$\nu \in \mathbb{Z}$}
|
||||
\eq{\rho_{xy} = \frac{2\pi\hbar}{e^2} \frac{1}{\nu}}
|
||||
\end{formula}
|
||||
|
||||
% \begin{formula}{qhe}
|
||||
% \desc{Integer quantum hall effect}{}{}
|
||||
% \desc[german]{Ganzahliger Quanten-Hall-Effekt}{}{}
|
||||
% \fig{img/qhe-klitzing.jpeg}
|
||||
% \end{formula}
|
||||
|
||||
|
||||
\TODO{sort}
|
||||
\begin{formula}{impedance_c}
|
||||
\desc{Impedance of a capacitor}{}{}
|
||||
\desc[german]{Impedanz eines Kondesnators}{}{}
|
||||
\eq{Z_{C} = \frac{1}{i\omega C}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{impedance_l}
|
||||
\desc{Impedance of an inductor}{}{}
|
||||
\desc[german]{Impedanz eines Induktors}{}{}
|
||||
\eq{Z_{L} = i\omega L}
|
||||
\end{formula}
|
||||
|
||||
\TODO{impedance addition for parallel / linear}
|
||||
|
||||
\Section[
|
||||
\eng{Dipole-stuff}
|
||||
\ger{Dipol-zeug}
|
||||
]{dipole}
|
||||
|
||||
\begin{formula}{poynting}
|
||||
\desc{Dipole radiation Poynting vector}{}{}
|
||||
\desc[german]{Dipolsrahlung Poynting-Vektor}{}{}
|
||||
\eq{\vec{S} = \left(\frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\right)\frac{\sin^2\theta}{r^2} \vec{r}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{power}
|
||||
\desc{Time-average power}{}{}
|
||||
\desc[german]{Zeitlich mittlere Leistung}{}{}
|
||||
\eq{P = \frac{\mu_0\omega^4 p_0^2}{12\pi c}}
|
||||
\end{formula}
|
||||
213
src/environments.tex
Normal file
213
src/environments.tex
Normal file
@ -0,0 +1,213 @@
|
||||
\def\descwidth{0.3\textwidth}
|
||||
\def\eqwidth{0.6\textwidth}
|
||||
|
||||
% [1]: minipage width
|
||||
% 2: fqname of name
|
||||
% 3: fqname of a translation that holds the explanation
|
||||
\newcommand{\NameWithExplanation}[3][\descwidth]{
|
||||
\begin{minipage}{#1}
|
||||
\iftranslation{#2}{
|
||||
\raggedright
|
||||
\gt{#2}
|
||||
}{}
|
||||
\iftranslation{#3}{
|
||||
\\ {\color{darkgray} \gt{#3}}
|
||||
}{}
|
||||
\end{minipage}
|
||||
}
|
||||
|
||||
% [1]: minipage width
|
||||
% 2: content
|
||||
% 3: fqname of a translation that holds the explanation
|
||||
\newcommand{\ContentBoxWithExplanation}[3][\eqwidth]{
|
||||
\fbox{
|
||||
\begin{minipage}{#1}
|
||||
% \vspace{-\baselineskip} % remove the space that comes from starting a new paragraph
|
||||
#2
|
||||
\noindent\iftranslation{#3}{
|
||||
\begingroup
|
||||
\color{darkgray}
|
||||
\gt{#3}
|
||||
% \edef\temp{\GT{#1_defs}}
|
||||
% \expandafter\StrSubstitute\expandafter{\temp}{:}{\\}
|
||||
\endgroup
|
||||
}{}
|
||||
% \vspace{-\baselineskip} % remove the space that comes from starting a new paragraph
|
||||
\end{minipage}
|
||||
}
|
||||
|
||||
}
|
||||
|
||||
% 1: fqname, optional with #1_defs and #1_desc defined
|
||||
% 2: content
|
||||
\newcommand{\NameLeftContentRight}[2]{
|
||||
\par\noindent\ignorespaces
|
||||
% \textcolor{gray}{\hrule}
|
||||
\vspace{0.5\baselineskip}
|
||||
\NameWithExplanation[\descwidth]{#1}{#1_desc}
|
||||
\hfill
|
||||
\ContentBoxWithExplanation[\eqwidth]{#2}{#1_defs}
|
||||
\textcolor{lightgray}{\hrule}
|
||||
\vspace{0.5\baselineskip}
|
||||
% \par
|
||||
% \hrule
|
||||
}
|
||||
|
||||
\newcommand{\insertEquation}[2]{
|
||||
\NameLeftContentRight{#1}{
|
||||
\begin{align}
|
||||
\label{eq:\fqname:#1}
|
||||
#2
|
||||
\end{align}
|
||||
}
|
||||
}
|
||||
|
||||
\newcommand{\insertFLAlign}[2]{ % eq name, #cols, eq
|
||||
\NameLeftContentRight{#1}{%
|
||||
\begin{flalign}%
|
||||
% dont place label when one is provided
|
||||
% \IfSubStringInString{label}\unexpanded{#3}{}{
|
||||
% \label{eq:#1}
|
||||
% }
|
||||
#2%
|
||||
\end{flalign}
|
||||
}
|
||||
}
|
||||
|
||||
\newcommand{\insertAlignedAt}[3]{ % eq name, #cols, eq
|
||||
\NameLeftContentRight{#1}{%
|
||||
\begin{alignat}{#2}%
|
||||
% dont place label when one is provided
|
||||
% \IfSubStringInString{label}\unexpanded{#3}{}{
|
||||
% \label{eq:#1}
|
||||
% }
|
||||
#3%
|
||||
\end{alignat}
|
||||
}
|
||||
}
|
||||
|
||||
\newcommand\luaexpr[1]{\directlua{tex.sprint(#1)}}
|
||||
% 1: fqname
|
||||
% 2: file path
|
||||
% 3: equation
|
||||
\newcommand{\insertEquationWithFigure}[4][0.55]{
|
||||
\par\noindent\ignorespaces
|
||||
% \textcolor{gray}{\hrule}
|
||||
\vspace{0.5\baselineskip}
|
||||
\begin{minipage}{#1\textwidth}
|
||||
\NameWithExplanation[\textwidth]{#2}{#2_desc}
|
||||
% TODO: why is this ignored
|
||||
\vspace{1.0cm}
|
||||
% TODO: fix box is too large without 0.9
|
||||
\ContentBoxWithExplanation[0.90\textwidth]{
|
||||
\begin{align}
|
||||
\label{eq:\fqname:#2}
|
||||
#4
|
||||
\end{align}
|
||||
}{#2_defs}
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{\luaexpr{1.0-#1}\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{#3}
|
||||
\label{fig:\fqname:#2}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\textcolor{lightgray}{\hrule}
|
||||
\vspace{0.5\baselineskip}
|
||||
|
||||
}
|
||||
|
||||
\newenvironment{formula}[1]{
|
||||
% key
|
||||
\newcommand{\desc}[4][english]{
|
||||
% language, name, description, definitions
|
||||
\dt[#1]{##1}{##2}
|
||||
\ifblank{##3}{}{\dt[#1_desc]{##1}{##3}}
|
||||
\ifblank{##4}{}{\dt[#1_defs]{##1}{##4}}
|
||||
}
|
||||
\newcommand{\eq}[1]{
|
||||
\insertEquation{#1}{##1}
|
||||
}
|
||||
\newcommand{\eqAlignedAt}[2]{
|
||||
\insertAlignedAt{#1}{##1}{##2}
|
||||
}
|
||||
\newcommand{\eqFLAlign}[1]{
|
||||
\insertFLAlign{#1}{##1}
|
||||
}
|
||||
\newcommand{\figeq}[2]{
|
||||
\insertEquationWithFigure{#1}{##1}{##2}
|
||||
}
|
||||
\newcommand{\content}[1]{
|
||||
\NameLeftContentRight{#1}{##1}
|
||||
}
|
||||
}{\ignorespacesafterend}
|
||||
|
||||
|
||||
\newenvironment{quantity}[5]{
|
||||
% key, symbol, si unit, si base units, comment (key to translation)
|
||||
\newcommand{\desc}[3][english]{
|
||||
% language, name, description
|
||||
\DT[qty:#1]{}{##1}{##2}
|
||||
\ifblank{##3}{}{\DT[qty:#1_desc]{##1}{##3}}
|
||||
}
|
||||
\newcommand{\eq}[1]{
|
||||
\insertEquation{#1}{##1}
|
||||
}
|
||||
\newcommand{\eqAlignedAt}[2]{
|
||||
\insertAlignedAt{#1}{##1}{##2}
|
||||
}
|
||||
\newcommand{\eqFLAlign}[1]{
|
||||
\insertFLAlign{#1}{##1}
|
||||
}
|
||||
|
||||
\edef\qtyname{#1}
|
||||
\edef\qtysign{#2}
|
||||
\edef\qtyunit{#3}
|
||||
\edef\qtybaseunits{#4}
|
||||
\edef\qtycomment{#5}
|
||||
}
|
||||
{
|
||||
Quantity: \expandafter\GT\expandafter{qty:\qtyname}: \GT{qty:\qtyname_desc} \\
|
||||
$\qtysign$ $[\SI{\qtyunit}] = [\SI{\qtybaseunits}]$ - \qtycomment \\
|
||||
\ignorespacesafterend
|
||||
}
|
||||
\def\distrightwidth{0.45\textwidth}
|
||||
\def\distleftwidth{0.45\textwidth}
|
||||
|
||||
% Table for distributions
|
||||
% create entries for parameters using \disteq
|
||||
\newenvironment{distribution}[0]{
|
||||
% 1: param name (translation key)
|
||||
% 2: math
|
||||
\newcommand{\disteq}[2]{
|
||||
% add links to some names
|
||||
\directlua{
|
||||
local cases = {
|
||||
pdf = "eq:pt:distributions:pdf",
|
||||
pmf = "eq:pt:distributions:pdf",
|
||||
cdf = "eq:pt:distributions:cdf",
|
||||
mean = "eq:pt:mean",
|
||||
variance = "eq:pt:variance"
|
||||
}
|
||||
if cases["\luaescapestring{#1}"] \string~= nil then
|
||||
tex.sprint("\\hyperref["..cases["\luaescapestring{#1}"].."]{\\GT{#1}}")
|
||||
else
|
||||
tex.sprint("\\GT{#1}")
|
||||
end
|
||||
}
|
||||
& #2 \\ \hline
|
||||
}
|
||||
\hfill
|
||||
\begin{minipage}{\distrightwidth}
|
||||
\begingroup
|
||||
\setlength{\tabcolsep}{0.9em} % horizontal
|
||||
\renewcommand{\arraystretch}{2} % vertical
|
||||
\begin{tabular}{|l|>{$\displaystyle}c<{$}|}
|
||||
\hline
|
||||
}{
|
||||
\end{tabular}
|
||||
\endgroup
|
||||
\end{minipage}
|
||||
}
|
||||
@ -1,8 +1,7 @@
|
||||
|
||||
\Part[
|
||||
\eng{Analysis}
|
||||
\ger{Analysis}
|
||||
]{ana}
|
||||
\eng{Geometry}
|
||||
\ger{Geometrie}
|
||||
]{geo}
|
||||
|
||||
\Section[
|
||||
\eng{Trigonometry}
|
||||
@ -18,14 +17,14 @@
|
||||
\begin{formula}{sine}
|
||||
\desc{Sine}{}{}
|
||||
\desc[german]{Sinus}{}{}
|
||||
\eq{\sin(x) &= \sum_{n=0}^{\infty} \frac{x^{(2n+1)}}{(2n+1)!} \\
|
||||
\eq{\sin(x) &= \sum_{n=0}^{\infty} (-1)^{n} \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)!} \\
|
||||
\eq{\cos(x) &= \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{(2n)}}{(2n)!} \\
|
||||
&= \frac{e^{ix}+e^{-ix}}{2}}
|
||||
\end{formula}
|
||||
|
||||
@ -72,6 +71,12 @@
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{name}
|
||||
\desc{}{}{$\tan\theta = b$}
|
||||
\desc[german]{}{}{$\tan\theta = b$}
|
||||
\eq{\cos x + b\sin x = \sqrt{1 + b^2}\cos(x-\theta)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Table of values}
|
||||
760
src/img/bravais2/aP.svg
Normal file
760
src/img/bravais2/aP.svg
Normal file
File diff suppressed because one or more lines are too long
801
src/img/bravais2/cF.svg
Normal file
801
src/img/bravais2/cF.svg
Normal file
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801
src/img/bravais2/cI.svg
Normal file
801
src/img/bravais2/cI.svg
Normal file
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772
src/img/bravais2/cP.svg
Normal file
772
src/img/bravais2/cP.svg
Normal file
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743
src/img/bravais2/hP.svg
Normal file
743
src/img/bravais2/hP.svg
Normal file
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768
src/img/bravais2/hR.svg
Normal file
768
src/img/bravais2/hR.svg
Normal file
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772
src/img/bravais2/mP.svg
Normal file
772
src/img/bravais2/mP.svg
Normal file
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762
src/img/bravais2/mS.svg
Normal file
762
src/img/bravais2/mS.svg
Normal file
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760
src/img/bravais2/oF.svg
Normal file
760
src/img/bravais2/oF.svg
Normal file
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759
src/img/bravais2/oI.svg
Normal file
759
src/img/bravais2/oI.svg
Normal file
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769
src/img/bravais2/oP.svg
Normal file
769
src/img/bravais2/oP.svg
Normal file
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766
src/img/bravais2/oS.svg
Normal file
766
src/img/bravais2/oS.svg
Normal file
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760
src/img/bravais2/tI.svg
Normal file
760
src/img/bravais2/tI.svg
Normal file
File diff suppressed because one or more lines are too long
750
src/img/bravais2/tP.svg
Normal file
750
src/img/bravais2/tP.svg
Normal file
File diff suppressed because one or more lines are too long
BIN
src/img/qhe-klitzing.jpeg
Normal file
BIN
src/img/qhe-klitzing.jpeg
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 260 KiB |
118
src/linalg.tex
Normal file
118
src/linalg.tex
Normal file
@ -0,0 +1,118 @@
|
||||
\def\id{\mathbb{1}}
|
||||
|
||||
\Part[
|
||||
\eng{Linear algebra}
|
||||
\ger{Lineare Algebra}
|
||||
]{linalg}
|
||||
|
||||
\Section[
|
||||
\eng{Determinant}
|
||||
\ger{Determinante}
|
||||
]{determinant}
|
||||
\begin{formula}{2x2}
|
||||
\desc{2x2 matrix}{}{}
|
||||
\desc[german]{2x2 Matrix}{}{}
|
||||
\eq{\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = a\,d -c\,b}
|
||||
\end{formula}
|
||||
\begin{formula}{3x3}
|
||||
\desc{3x3 matrix (Rule of Sarrus)}{}{}
|
||||
\desc[german]{3x3 Matrix (Regel von Sarrus)}{}{}
|
||||
\eq{\det \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} =
|
||||
a\,e\,i + b\,f\,g + c\,d\,h - g\,e\,c - h\,f\,a-i\,d\,b}
|
||||
\end{formula}
|
||||
\begin{formula}{leibniz}
|
||||
\desc{Leibniz formla}{}{}
|
||||
\desc[german]{Leibniz-Formel}{}{}
|
||||
\eq{\det(A) = \sum_{\sigma \in S_n}\Big(\sgn(\sigma) \prod_{i=1}^n a_{i,\sigma(i)}\Big)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{product}
|
||||
\desc{Product}{}{}
|
||||
\desc[german]{Produkt}{}{}
|
||||
\eq{\det(A \, B) = \det(A) \det(B)}
|
||||
\end{formula}
|
||||
\begin{formula}{inverse}
|
||||
\desc{Inverse}{}{}
|
||||
\desc[german]{Inverse}{}{}
|
||||
\eq{\det(A^{-1}) = \det(A)^{-1}}
|
||||
\end{formula}
|
||||
\begin{formula}{transposed}
|
||||
\desc{Transposed}{}{}
|
||||
\desc[german]{Transponiert}{}{}
|
||||
\eq{\det(A^{\T}) = \det(A)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Section[
|
||||
|
||||
]{zeug}
|
||||
|
||||
\begin{formula}{unitary}
|
||||
\desc{Unitary matrix}{}{}
|
||||
\desc[german]{Unitäre Matrix}{}{}
|
||||
\eq{U ^\dagger U = \id}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{svd}
|
||||
\desc{Singular value decomposition}{Factorization of complex matrices through rotating \rightarrow rescaling \rightarrow rotation.}{$A$: $m\times n$ matrix, $U$: $m\times m$ unitary matrix, $\Lambda$: $m\times n$ rectangular diagonal matrix with non-negative numbers on the diagonal, $V$: $n\times n$ unitary matrix}
|
||||
\desc[german]{Singulärwertzerlegung}{Faktorisierung einer reellen oder komplexen Matrix durch Rotation \rightarrow Skalierung \rightarrow Rotation.}{}
|
||||
\eq{A = U \Lambda V}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{rotation2}
|
||||
\desc{2D rotation matrix}{}{}
|
||||
\desc[german]{2D Rotationsmatrix}{}{}
|
||||
\eq{R = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} }
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{rotation3}
|
||||
\desc{3D rotation matrices}{}{}
|
||||
\desc[german]{3D Rotationsmatrizen}{}{}
|
||||
\eq{
|
||||
R_x &= \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix} \\
|
||||
R_y &= \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix} \\
|
||||
R_z &= \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{rotation_properties}
|
||||
\desc{Properites of rotation matrices}{}{$n$ dimension, SO($n$) special othogonal group}
|
||||
\desc[german]{Eigenschaften von Rotationsmatrizen}{}{$n$ Dimension, SO($n$) spezielle orthognale Gruppe}
|
||||
\eq{
|
||||
R^{\T} = R^{-1} \\
|
||||
\det R = 1 \\
|
||||
R \in \text{SO($n$)}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Eigenvalues}
|
||||
\ger{Eigenwerte}
|
||||
]{eigen}
|
||||
\begin{formula}{values}
|
||||
\desc{Eigenvalue equation}{}{$\lambda$ eigenvalue, $v$ eigenvector}
|
||||
\desc[german]{Eigenwert-Gleichung}{}{$\lambda$ Eigenwert, $v$ Eigenvektor}
|
||||
\eq{A v = \lambda v}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{charac_poly}
|
||||
\desc{Characteristic polynomial}{Zeros are the eigenvalues of $A$}{}
|
||||
\desc[german]{Charakteristisches Polynom}{Nullstellen sind die Eigenwerte von $A$}{}
|
||||
\eq{\chi_A = \det(A - \lambda \id) \overset{!}{=} 0}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{kramer}
|
||||
\desc{Kramer's theorem}{If $H$ is invariant under $T$ and $\ket{\psi}$ is an eigenstate of $H$, then $T\ket{\psi}$ is also am eigenstate of $H$}{}
|
||||
\desc[german]{Kramers-Theorem}{Wenn $H$ invariant unter $T$ ist und $\ket{\psi}$ ein Eigenzustand von $H$ ist, dann ist $T \ket{\psi}$ auch ein Eigenzustand von $H$}{}
|
||||
\eq{T H T^\dagger = H \quad\wedge\quad H \ket{\psi} = E \ket{\psi} \quad\Rightarrow\quad H T \ket{\psi} = E T \ket{\psi}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{eigendecomp}
|
||||
\desc{Eigendecomposition}{}{$A$ diagonalizable, columns of $V$ are eigenvectors $v_i$, $\Lambda$ diagonal matrix with eigenvalues $\lambda_i$ on the diagonal}
|
||||
\desc[german]{Eigenwertzerlegung}{}{$A$ diagonalisierbar, Spalten von $V$ sind die Eigenvektoren $v_i$, $\Lambda$ Diagonalmatrix mit Eigenwerten $\lambda_$ auf der Diagonalen}
|
||||
\eq{A = V \Lambda V^{-1}}
|
||||
\end{formula}
|
||||
|
||||
\TODO{Jordan stuff, blockdiagonal matrices, permutations, skalar product lapacescher entwicklungssatz maybe, cramers rule}
|
||||
|
||||
47
src/macros.tex
Normal file
47
src/macros.tex
Normal file
@ -0,0 +1,47 @@
|
||||
\def\Grad{\vec{\nabla}}
|
||||
\def\Div{\vec{\nabla} \cdot}
|
||||
\def\Rot{\vec{\nabla} \times}
|
||||
\def\vecr{\vec{r}}
|
||||
|
||||
\def\kB{k_\text{B}}
|
||||
\def\EFermi{E_\text{F}}
|
||||
|
||||
\def\masse{m_\textrm{e}}
|
||||
|
||||
\def\R{\mathbb{R}}
|
||||
\def\C{\mathbb{C}}
|
||||
\def\Z{\mathbb{Z}}
|
||||
\def\N{\mathbb{N}}
|
||||
|
||||
\def\Four{\mathcal{F}} % Fourier transform
|
||||
\def\Lebesgue{\mathcal{L}} % Lebesgue
|
||||
\def\Order{\mathcal{O}}
|
||||
|
||||
% complex, may be changed later to idot or upright...
|
||||
\def\I{i}
|
||||
|
||||
\def\sdots{\,\dots\,}
|
||||
\def\qdots{\quad\dots\quad}
|
||||
\def\qRarrow{\quad\Rightarrow\quad}
|
||||
|
||||
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
|
||||
\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor}
|
||||
\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil}
|
||||
|
||||
\DeclareMathOperator{\e}{e}
|
||||
\DeclareMathOperator{\T}{T} % transposed
|
||||
\DeclareMathOperator{\sgn}{sgn}
|
||||
\DeclareMathOperator{\tr}{tr}
|
||||
\DeclareMathOperator{\const}{const}
|
||||
\DeclareMathOperator{\erf}{erf}
|
||||
% diff, for integrals and stuff
|
||||
% \DeclareMathOperator{\dd}{d}
|
||||
\renewcommand*\d{\mathop{}\!\mathrm{d}}
|
||||
|
||||
% functions with paranthesis
|
||||
\newcommand\CmdWithParenthesis[2]{
|
||||
#1\left(#2\right)
|
||||
}
|
||||
\newcommand\Exp[1]{\CmdWithParenthesis{\exp}{#1}}
|
||||
\newcommand\Sin[1]{\CmdWithParenthesis{\sin}{#1}}
|
||||
\newcommand\Cos[1]{\CmdWithParenthesis{\cos}{#1}}
|
||||
178
src/main.tex
178
src/main.tex
@ -3,17 +3,30 @@
|
||||
\usepackage[german]{babel}
|
||||
\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
|
||||
\usepackage{mathtools}
|
||||
\usepackage{MnSymbol} % for >>> \ggg sign
|
||||
% \usepackage{esdiff} % derivatives
|
||||
% esdiff breaks when taking \dot{q} has argument
|
||||
\usepackage{derivative}
|
||||
\usepackage{bbold} % \mathbb font
|
||||
\usepackage{braket}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{etoolbox}
|
||||
\usepackage{expl3} % switch case and other stuff
|
||||
\usepackage{substr}
|
||||
\usepackage{xcolor}
|
||||
\usepackage{float}
|
||||
\usepackage{tikz} % drawings
|
||||
\usetikzlibrary{decorations.pathmorphing}
|
||||
\usetikzlibrary{calc}
|
||||
\usepackage{circuitikz}
|
||||
\usepackage[hidelinks]{hyperref}
|
||||
\usepackage{subcaption}
|
||||
\usepackage[shortlabels]{enumitem} % easily change enum symbols to i), a. etc
|
||||
\usepackage{colortbl} % color table
|
||||
\usepackage{tabularx} % bravais table
|
||||
\usepackage{adjustbox}
|
||||
\usepackage{multirow} % for superconducting qubit table
|
||||
\usepackage{hhline} % for superconducting qubit table
|
||||
\hypersetup{colorlinks = true, % Colours links instead of ugly boxes
|
||||
urlcolor = blue, % Colour for external hyperlinks
|
||||
linkcolor = cyan, % Colour of internal links
|
||||
@ -26,13 +39,11 @@
|
||||
\sisetup{per-mode = power}
|
||||
\sisetup{exponent-product=\ensuremath{\cdot}}
|
||||
|
||||
\DeclarePairedDelimiter\abs{\lvert}{\rvert}
|
||||
\DeclareMathOperator{\e}{e}
|
||||
\DeclareMathOperator{\d}{d}
|
||||
|
||||
\usepackage{translations}
|
||||
|
||||
\newcommand{\TODO}[1]{{\color{red}TODO:#1}}
|
||||
\newcommand{\ts}{\textsuperscript}
|
||||
|
||||
% put an explanation above an equal sign
|
||||
% [1]: equality sign (or anything else)
|
||||
@ -42,6 +53,7 @@
|
||||
\newcommand{\explOverEq}[2][=]{%
|
||||
\overset{\substack{\mathrlap{\text{\hspace{-1em}#2}}\\\downarrow}}{#1}}
|
||||
|
||||
|
||||
%
|
||||
% TRANSLATION COMMANDS
|
||||
%
|
||||
@ -91,14 +103,14 @@
|
||||
\newcommand{\ger}[2][\fqname]{\dt[#1]{german}{#2}}
|
||||
\newcommand{\eng}[2][\fqname]{\dt[#1]{english}{#2}}
|
||||
|
||||
\newcommand{\GER}[2][\fqname]{\DT[#1]{german}{#2}}
|
||||
\newcommand{\ENG}[2][\fqname]{\DT[#1]{english}{#2}}
|
||||
\newcommand{\Ger}[2][\fqname]{\DT[#1]{german}{#2}}
|
||||
\newcommand{\Eng}[2][\fqname]{\DT[#1]{english}{#2}}
|
||||
|
||||
% use this to define text in different languages for the key <env arg>
|
||||
% the translation for <env arg> when the environment ends.
|
||||
% (temporarily change fqname to the \fqname:<env arg> to allow
|
||||
% the use of \eng and \ger without the key parameter)
|
||||
\newenvironment{ttext}[1]{
|
||||
\newenvironment{ttext}[1][desc]{
|
||||
\edef\realfqname{\fqname}
|
||||
\edef\fqname{\fqname:#1}
|
||||
}{
|
||||
@ -148,101 +160,27 @@
|
||||
\label{sec:\fqname}
|
||||
}
|
||||
|
||||
\newcommand{\fqEqRef}[1]{
|
||||
\hyperref[eq:#1]{\GT{#1}}
|
||||
}
|
||||
\newcommand{\fqSecRef}[1]{
|
||||
\hyperref[sec:#1]{\GT{#1}}
|
||||
}
|
||||
|
||||
\usepackage{xstring}
|
||||
|
||||
\newcommand{\insertEquationLine}[2]{
|
||||
\par\noindent\ignorespaces
|
||||
% \textcolor{gray}{\hrule}
|
||||
\vspace{0.5\baselineskip}
|
||||
% \fbox{
|
||||
\begin{minipage}{0.3\textwidth}
|
||||
\iftranslation{#1}{
|
||||
\raggedright
|
||||
\gt{#1}
|
||||
}{}
|
||||
\iftranslation{#1_desc}{
|
||||
\\ {\color{darkgray} \gt{#1_desc}}
|
||||
}{}
|
||||
\end{minipage}
|
||||
% }
|
||||
\hfill
|
||||
\fbox{
|
||||
\begin{minipage}{0.6\textwidth}
|
||||
% \vspace{-\baselineskip} % remove the space that comes from starting a new paragraph
|
||||
#2 %
|
||||
\noindent\iftranslation{#1_defs}{
|
||||
\begingroup
|
||||
\color{darkgray}
|
||||
\gt{#1_defs}
|
||||
% \edef\temp{\GT{#1_defs}}
|
||||
% \expandafter\StrSubstitute\expandafter{\temp}{:}{\\}
|
||||
\endgroup
|
||||
}{}
|
||||
% \vspace{-\baselineskip} % remove the space that comes from starting a new paragraph
|
||||
\end{minipage}
|
||||
}
|
||||
\textcolor{lightgray}{\hrule}
|
||||
\vspace{0.5\baselineskip}
|
||||
% \par
|
||||
% \hrule
|
||||
}
|
||||
|
||||
\newcommand{\insertEquation}[2]{
|
||||
\insertEquationLine{#1}{
|
||||
\begin{align}
|
||||
\label{eq:\fqname:#1}
|
||||
#2
|
||||
\end{align}
|
||||
}
|
||||
}
|
||||
\input{circuit.tex}
|
||||
|
||||
\newcommand{\insertFLAlign}[2]{ % eq name, #cols, eq
|
||||
\insertEquationLine{#1}{%
|
||||
\begin{flalign}%
|
||||
% dont place label when one is provided
|
||||
% \IfSubStringInString{label}\unexpanded{#3}{}{
|
||||
% \label{eq:#1}
|
||||
% }
|
||||
#2%
|
||||
\end{flalign}
|
||||
}
|
||||
}
|
||||
|
||||
\newcommand{\insertAlignedAt}[3]{ % eq name, #cols, eq
|
||||
\insertEquationLine{#1}{%
|
||||
\begin{alignat}{#2}%
|
||||
% dont place label when one is provided
|
||||
% \IfSubStringInString{label}\unexpanded{#3}{}{
|
||||
% \label{eq:#1}
|
||||
% }
|
||||
#3%
|
||||
\end{alignat}
|
||||
}
|
||||
}
|
||||
|
||||
\newenvironment{formula}[1]{
|
||||
% key
|
||||
\newcommand{\desc}[4][english]{
|
||||
% language, name, description, definitions
|
||||
\dt[#1]{##1}{##2}
|
||||
\ifblank{##3}{}{\dt[#1_desc]{##1}{##3}}
|
||||
\ifblank{##4}{}{\dt[#1_defs]{##1}{##4}}
|
||||
}
|
||||
\newcommand{\eq}[1]{
|
||||
\insertEquation{#1}{##1}
|
||||
}
|
||||
\newcommand{\eqAlignedAt}[2]{
|
||||
\insertAlignedAt{#1}{##1}{##2}
|
||||
}
|
||||
\newcommand{\eqFLAlign}[1]{
|
||||
\insertFLAlign{#1}{##1}
|
||||
}
|
||||
}{\ignorespacesafterend}
|
||||
% some translations
|
||||
|
||||
\title{Formelsammlung}
|
||||
\author{Matthias Quintern}
|
||||
\date{\today}
|
||||
|
||||
\input{macros.tex}
|
||||
\input{environments.tex}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
@ -250,55 +188,29 @@
|
||||
\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{\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
|
||||
% }
|
||||
\IfSubStringInString{lol}{lol\frac{asdsd}{lol} & l}{YES!}{
|
||||
NO!
|
||||
}
|
||||
\input{translations.tex}
|
||||
|
||||
\input{trigonometry.tex}
|
||||
\input{linalg.tex}
|
||||
|
||||
\input{geometry.tex}
|
||||
|
||||
\input{analysis.tex}
|
||||
|
||||
\input{probability_theory.tex}
|
||||
|
||||
\input{mechanics.tex}
|
||||
|
||||
\input{statistical_mechanics.tex}
|
||||
|
||||
\input{electrodynamics.tex}
|
||||
|
||||
\input{quantum_mechanics.tex}
|
||||
\input{atom.tex}
|
||||
|
||||
\input{condensed_matter.tex}
|
||||
|
||||
\input{topo.tex}
|
||||
|
||||
\input{quantum_computing.tex}
|
||||
|
||||
\input{many-body-simulations.tex}
|
||||
|
||||
@ -8,7 +8,7 @@
|
||||
\eng{Lagrange formalism}
|
||||
\ger{Lagrange Formalismus}
|
||||
]{lagrange}
|
||||
\begin{ttext}{desc}
|
||||
\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.
|
||||
}
|
||||
@ -16,7 +16,7 @@
|
||||
da das Aufstellen der Lagrange-Funktion mit geeigneten generalisierten Koordinaten oft relativ einfach ist.
|
||||
}
|
||||
\end{ttext}
|
||||
\begin{ttext}{generalized_coords}
|
||||
\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 $\vec{x} = \begin{pmatrix} \cos\varphi \\ \sin\varphi \end{pmatrix}$.
|
||||
|
||||
198
src/probability_theory.tex
Normal file
198
src/probability_theory.tex
Normal file
@ -0,0 +1,198 @@
|
||||
\Part[
|
||||
\eng{Probability theory}
|
||||
\ger{Wahrscheinlichkeitstheorie}
|
||||
]{pt}
|
||||
|
||||
\begin{formula}{mean}
|
||||
\desc{Mean}{}{}
|
||||
\desc[german]{Mittelwert}{}{}
|
||||
\eq{\braket{x} = \int w(x)\, x\, \d x}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{variance}
|
||||
\desc{Variance}{}{}
|
||||
\desc[german]{Varianz}{}{}
|
||||
\eq{\sigma^2 = (\Delta \hat{x})^2 = \braket{\hat{x}^2} - \braket{\hat{x}}^2 = \braket{(x - \braket{x})^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{std_deviation}
|
||||
\desc{Standard deviation}{}{}
|
||||
\desc[german]{Standardabweichung}{}{}
|
||||
\eq{\sigma = \sqrt{(\Delta x)^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{median}
|
||||
\desc{Median}{Value separating lower half from top half}{$x$ dataset with $n$ elements}
|
||||
\desc[german]{Median}{Teilt die untere von der oberen Hälfte}{$x$ Reihe mit $n$ Elementen}
|
||||
\eq{
|
||||
\textrm{med}(x) = \left\{ \begin{array}{ll} x_{(n+1)/2} & \text{$n$ \GT{odd}} \\ \frac{x_{(n/2)}+x_{((n/2)+1)}}{2} & \text{$n$ \GT{even}} \end{array} \right
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{pdf}
|
||||
\desc{Probability density function}{Random variable has density $f$. The integral gives the probability of $X$ taking a value $x\in[a,b]$.}{$f$ normalized: $\int_{-\infty}^\infty f(x) \d x= 1$}
|
||||
\desc[german]{Wahrscheinlichkeitsdichtefunktion}{Zufallsvariable hat Dichte $f$. Das Integral gibt Wahrscheinlichkeit an, dass $X$ einen Wert $x\in[a,b]$ annimmt}{$f$ normalisiert $\int_{-\infty}^\infty f(x) \d x= 1$}
|
||||
\eq{P([a,b]) := \int_a^b f(x) \d x}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{cdf}
|
||||
\desc{Cumulative distribution function}{}{$f$ probability density function}
|
||||
\desc[german]{Kumulative Verteilungsfunktion}{}{$f$ Wahrscheinlichkeitsdichtefunktion}
|
||||
\eq{F(x) = \int_{-\infty}^x f(t) \d t}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Distributions}
|
||||
\ger{Verteilungen}
|
||||
]{distributions}
|
||||
|
||||
|
||||
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Gauß/Normal distribution}
|
||||
\ger{Gauß/Normal-Verteilung}
|
||||
]{normal}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_gauss.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{\mu \in \R,\quad \sigma^2 \in \R}
|
||||
\disteq{support}{x \in \R}
|
||||
\disteq{pdf}{\frac{1}{\sqrt{2\pi\sigma^2}}\exp \left(-\frac{(x-\mu)^2}{2\sigma^2}\right)}
|
||||
\disteq{cdf}{\frac{1}{2}\left[1 + \erf \left(\frac{x-\mu}{\sqrt{2}\sigma}\right)\right]}
|
||||
\disteq{mean}{\mu}
|
||||
\disteq{median}{\mu}
|
||||
\disteq{variance}{\sigma^2}
|
||||
\end{distribution}
|
||||
|
||||
\begin{formula}{standard_normal_distribution}
|
||||
\desc{Density function of the standard normal distribution}{$\mu = 0$, $\sigma = 1$}{}
|
||||
\desc[german]{Dichtefunktion der Standard-Normalverteilung}{$\mu = 0$, $\sigma = 1$}{}
|
||||
\eq{\varphi(x) = \frac{1}{\sqrt{2\pi}} \e^{-\frac{1}{2}x^2}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Cauchys / Lorentz distribution}
|
||||
\ger{Cauchy / Lorentz-Verteilung}
|
||||
]{cauchy}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_cauchy.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{x_0 \in \R,\quad \gamma \in \R}
|
||||
\disteq{support}{x \in \R}
|
||||
\disteq{pdf}{\frac{1}{\pi\gamma\left[1+\left(\frac{x-x_0}{\gamma}\right)^2\right]}}
|
||||
\disteq{cdf}{\frac{1}{\pi}\arctan\left(\frac{x-x_0}{\gamma}\right) + \frac{1}{2}}
|
||||
\disteq{mean}{\text{\GT{undefined}}}
|
||||
\disteq{median}{x_0}
|
||||
\disteq{variance}{\text{\GT{undefined}}}
|
||||
\end{distribution}
|
||||
|
||||
\begin{ttext}
|
||||
\eng{Also known as \textbf{Cauchy-Lorentz distribution}, \textbf{Lorentz(ian) function}, \textbf{Breit-Wigner distribution}.}
|
||||
\ger{Auch bekannt als \textbf{Cauchy-Lorentz Verteilung}, \textbf{Lorentz Funktion}, \textbf{Breit-Wigner Verteilung}.}
|
||||
\end{ttext}
|
||||
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Binomial distribution}
|
||||
\ger{Binomialverteilung}
|
||||
]{binomial}
|
||||
\begin{ttext}
|
||||
\eng{For the number of trials going to infinity ($n\to\infty$), the binomial distribution converges to the \hyperref[sec:pb:distributions::poisson]{poisson distribution}}
|
||||
\ger{Geht die Zahl der Versuche gegen unendlich ($n\to\infty$), konvergiert die Binomualverteilung gegen die \hyperref[sec:pb:distributions::poisson]{Poissonverteilung}}
|
||||
\end{ttext}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_binomial.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{n \in \Z, \quad p \in [0,1],\quad q = 1 - p}
|
||||
\disteq{support}{k \in \{0,\,1,\,\dots,\,n\}}
|
||||
\disteq{pmf}{\binom{n}{k} p^k q^{n-k}}
|
||||
% \disteq{cdf}{\text{regularized incomplete beta function}}
|
||||
\disteq{mean}{np}
|
||||
\disteq{median}{\floor{np} \text{ or } \ceil{np}}
|
||||
\disteq{variance}{npq = np(1-p)}
|
||||
\end{distribution}
|
||||
\Subsubsection[
|
||||
\eng{Poisson distribution}
|
||||
\ger{Poissonverteilung}
|
||||
]{poisson}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_poisson.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{\lambda \in (0,\infty)}
|
||||
\disteq{support}{k \in \N}
|
||||
\disteq{pmf}{\frac{\lambda^k \e^{-\lambda}}{k!}}
|
||||
\disteq{cdf}{\e^{-\lambda} \sum_{j=0}^{\floor{k}} \frac{\lambda^j}{j!}}
|
||||
\disteq{mean}{\lambda}
|
||||
\disteq{median}{\approx\floor*{\lambda + \frac{1}{3} - \frac{1}{50\lambda}}}
|
||||
\disteq{variance}{\lambda}
|
||||
\end{distribution}
|
||||
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Maxwell-Boltzmann distribution}
|
||||
\ger{Maxwell-Boltzmann Verteilung}
|
||||
]{maxwell-boltzmann}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/distribution_maxwell-boltzmann.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{a > 0}
|
||||
\disteq{support}{x \in (0, \infty)}
|
||||
\disteq{pdf}{\sqrt{\frac{2}{\pi}} \frac{x^2}{a^3} \exp\left(-\frac{x^2}{2a^2}\right)}
|
||||
\disteq{cdf}{\erf \left(\frac{x}{\sqrt{2} a}\right) - \sqrt{\frac{2}{\pi}} \frac{x}{a} \exp\left({\frac{-x^2}{2a^2}}\right)}
|
||||
\disteq{mean}{2a \frac{2}{\pi}}
|
||||
\disteq{median}{}
|
||||
\disteq{variance}{\frac{a^2(3\pi-8)}{\pi}}
|
||||
\end{distribution}
|
||||
|
||||
|
||||
|
||||
% \begin{distribution}{maxwell-boltzmann}
|
||||
% \distdesc{Maxwell-Boltzmann distribution}{}
|
||||
% \distdesc[german]{Maxwell-Boltzmann Verteilung}{}
|
||||
% \disteq{parameters}{}
|
||||
% \disteq{pdf}{}
|
||||
% \disteq{cdf}{}
|
||||
% \disteq{mean}{}
|
||||
% \disteq{median}{}
|
||||
% \disteq{variance}{}
|
||||
% \end{distribution}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Central limit theorem}
|
||||
\ger{Zentraler Grenzwertsatz}
|
||||
]{cls}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
Suppose $X_1, X_2, \dots$ is a sequence of independent and identically distributed random variables with $\braket{X_i} = \mu$ and $(\Delta X_i)^2 = \sigma^2 < \infty$.
|
||||
As $N$ approaches infinity, the random variables $\sqrt{N}(\bar{X}_N - \mu)$ converge to a normal distribution $\mathcal{N}(0, \sigma^2)$.
|
||||
\\ That means that the variance scales with $\frac{1}{\sqrt{N}}$ and statements become accurate for large $N$.
|
||||
}
|
||||
\ger{
|
||||
Sei $X_1, X_2, \dots$ eine Reihe unabhängiger und gleichverteilter Zufallsvariablen mit $\braket{X_i} = \mu$ und $(\Delta X_i)^2 = \sigma^2 < \infty$.
|
||||
Für $N$ gegen unendlich konvergieren die Zufallsvariablen $\sqrt{N}(\bar{X}_N - \mu)$ zu einer Normalverteilung $\mathcal{N}(0, \sigma^2)$.
|
||||
\\ Das bedeutet, dass die Schwankung mit $\frac{1}{\sqrt{N}}$ wächst und Aussagen für große $N$ scharf werden.
|
||||
}
|
||||
\end{ttext}
|
||||
|
||||
@ -1,7 +1,7 @@
|
||||
\Part[
|
||||
\eng{Quantum Computing}
|
||||
\ger{Quantencomputing}
|
||||
]{qubit}
|
||||
]{qc}
|
||||
|
||||
\Section[
|
||||
\eng{Qubits}
|
||||
@ -36,38 +36,396 @@
|
||||
% \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}
|
||||
|
||||
\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}
|
||||
\eng{Superconducting qubits}
|
||||
\ger{Supraleitende qubits}
|
||||
]{scq}
|
||||
|
||||
\Subsection[
|
||||
\eng{Building blocks}
|
||||
\ger{Bauelemente}
|
||||
]{elements}
|
||||
\Subsubsection[
|
||||
\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. The Josephson junction is a non-linear inductor.}
|
||||
\ger{Wenn zwei Supraleiter durch einen dünnen Isolator getrennt sind, können Cooper-Paare durch den Isolator tunneln. Der Josephson-Kontakt ist ein nicht-linearer Induktor.}
|
||||
\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}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{SQUID}
|
||||
\ger{SQUID}
|
||||
]{squid}
|
||||
\ctikzsubcircuitdef{squidloop}{n, s, nw, ne, se, sw}{
|
||||
% start at top
|
||||
coordinate(#1-n)
|
||||
(#1-n)
|
||||
to ++(-1, 0) coordinate(#1-nw)
|
||||
to[josephsoncap=$\phi_1$] ++(0,-2) coordinate(#1-sw)
|
||||
to ++(1,0) coordinate(#1-s) to ++(1,0) coordinate(#1-se)
|
||||
to[josephsoncap=$\phi_2$] ++(0,2) coordinate(#1-ne)
|
||||
to ++(-1,0)
|
||||
(#1-s) % leave at bottom
|
||||
}
|
||||
\begin{formula}{circuit}
|
||||
\desc{SQUID}{Superconducting quantum interference device, consists of parallel \hyperref{sec:qc:scq:josephson_junction}{josephson junctions}, can be used to measure extremely weak magnetic fields}{}
|
||||
\desc[german]{SQUID}{Superconducting quantum interference device, besteht aus parralelen \hyperref{sec:qc:scq:josephson_junction}{Josephson Junctions} und kann zur Messung extrem schwacher Magnetfelder genutzt werden}{}
|
||||
\content{
|
||||
\centering
|
||||
\begin{circuitikz}
|
||||
\draw (0, 0) \squidloop{loop}{};
|
||||
\end{circuitikz}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{$\hat{\phi}$ phase difference across the junction}
|
||||
\desc[german]{Hamiltonian}{}{$\hat{\phi}$ Phasendifferenz an einer Junction}
|
||||
\eq{\hat{H} &= -E_{\text{J}1} \cos\hat{\phi}_{1} - E_{\text{J}2} \cos\hat{\phi}_{2}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Josephson Qubit??}
|
||||
\ger{TODO}
|
||||
]{josephson_qubit}
|
||||
\begin{circuitikz}
|
||||
\draw (0,0) to[capacitor] (0,2);
|
||||
\draw (0,0) to (2,0);
|
||||
\draw (0,2) to (2,2);
|
||||
\draw (2,0) to[josephson] (2,2);
|
||||
|
||||
\draw[->] (3,1) -- (4,1);
|
||||
\draw (5,0) to[josephsoncap=$C_\text{J}$] (5,2);
|
||||
\end{circuitikz}
|
||||
\TODO{Include schaltplan}
|
||||
|
||||
\begin{circuitikz}
|
||||
\draw (0,0) to[sV=$V_\text{g}$] (0,2);
|
||||
\draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
|
||||
\draw (2,2) to (4,2);
|
||||
\draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
|
||||
\draw (4,0) to[capacitor=$C_C$] (4,2);
|
||||
\draw (0,0) to (2,0);
|
||||
\draw (2,0) to (4,0);
|
||||
\end{circuitikz}
|
||||
|
||||
\begin{formula}{charging_energy}
|
||||
\desc{Charging energy / electrostatic energy}{}{}
|
||||
\desc[german]{Ladeenergie?}{}{}
|
||||
\eq{E_\text{C} = \frac{(2e)^2}{C}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{josephson_energy}
|
||||
\desc{Josephson energy}{}{}
|
||||
\desc[german]{Josephson-Energie?}{}{}
|
||||
\eq{E_\text{J} = \frac{I_0 \phi_0}{2\pi}}
|
||||
\end{formula}
|
||||
\TODO{Was ist I0}
|
||||
|
||||
\begin{formula}{inductive_energy}
|
||||
\desc{Inductive energy}{}{}
|
||||
\desc[german]{Induktive Energie}{}{}
|
||||
\eq{E_\text{L} = \frac{\varphi_0^2}{L}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gate_charge}
|
||||
\desc{Gate charge}{or offset charge}{}
|
||||
\desc[german]{Gate Ladung}{auch Offset charge}{}
|
||||
\eq{n_\text{g}=\frac{C_g V_\text{g}}{2e}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{anharmonicity}
|
||||
\desc{Anharmonicity}{}{}
|
||||
\desc[german]{Anharmonizität}{}{}
|
||||
\eq{\alpha \coloneq \omega_{1\leftrightarrow 2} - \omega_{0\leftrightarrow 1}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{table}[h!]
|
||||
\centering
|
||||
\begin{tabular}{ p{1cm} |p{1cm}||p{2.8cm}|p{2cm}|p{2cm}|p{2cm}|}
|
||||
\multicolumn{1}{c}{}& \multicolumn{1}{c}{} &\multicolumn{4}{c}{$E_L/(E_J-E_L)$} \\
|
||||
\cline{3-6}
|
||||
\multicolumn{1}{c}{} & & $0$ & $\ll$ 1 & $\sim 1$ & $\gg 1$\\
|
||||
\hhline{~|=====|}
|
||||
\multirow{4}{*}{$E_J/E_C$} & $\ll 1$ & cooper-pair box & & & \\
|
||||
\cline{2-6}
|
||||
& $\sim 1$ & quantronium & fluxonium & &\\
|
||||
\cline{2-6}
|
||||
& $\gg 1$ &transmon & & & flux qubit\\
|
||||
\cline{2-6}
|
||||
& $\ggg 1$ & & & phase qubit & \\
|
||||
\cline{2-6}
|
||||
\end{tabular}
|
||||
\caption{``periodic table'' of superconducting quantum circuits}
|
||||
\label{Juncatalog}
|
||||
|
||||
\end{table}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Cooper Pair Box (CPB) qubit}
|
||||
\ger{Cooper Paar Box (QPB) Qubit}
|
||||
]{cpb}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
= voltage bias junction\\= charge qubit?
|
||||
\begin{itemize}
|
||||
\item large anharmonicity
|
||||
\item sensitive to charge noise
|
||||
\end{itemize}
|
||||
}
|
||||
\ger{}
|
||||
\end{ttext}
|
||||
\begin{circuitikz}
|
||||
\draw (0,0) to[sV=$V_\text{g}$] (0,2);
|
||||
% \draw (0,0) to (2,0);
|
||||
\draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
|
||||
\draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
|
||||
\draw (0,0) to (2,0);
|
||||
\end{circuitikz}
|
||||
|
||||
|
||||
\TODO{maybe include graphic from page 48, intro to qed circuits}
|
||||
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{}
|
||||
\desc[german]{Hamiltonian}{}{}
|
||||
\eq{\hat{H} &= 4 E_C(\hat{n} - n_\text{g})^2 - E_\text{J} \cos\hat{\phi} \\
|
||||
&=\sum_n \left[4 E_C (n-n_\text{g})^2 \ket{n}\bra{n} - \frac{E_\text{J}}{2}\ket{n}\bra{n+1}+\ket{n+1}\bra{n}\right] }
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Transmon qubit}
|
||||
\ger{Transmon Qubit}
|
||||
]{transmon}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Transmon qubit}{
|
||||
Josephson junction with a shunt \textbf{capacitance}.
|
||||
\begin{itemize}
|
||||
\item charge noise insensitive
|
||||
\item small anharmonicity
|
||||
\end{itemize}
|
||||
}{}
|
||||
\desc[ger]{Transmon Qubit}{
|
||||
Josephson-Kontakt mit einem parallelen \textbf{kapzitiven Element}.
|
||||
\begin{itemize}
|
||||
\item Charge noise resilient
|
||||
\item Geringe Anharmonizität $\alpha$
|
||||
\end{itemize}
|
||||
}{}
|
||||
\content{
|
||||
\centering
|
||||
\begin{circuitikz}
|
||||
\draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
|
||||
to[capacitor=$C_\text{g}$] ++(2,0)
|
||||
to[capacitor=$C_C$] ++(0,-3)
|
||||
to ++(-2,0);
|
||||
\draw (2,3) to ++(2,0) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to ++(0,-0.5) to ++(-2,0);
|
||||
\end{circuitikz}
|
||||
\\\TODO{Ist beim Transmon noch die Voltage source dran?}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{}
|
||||
\desc[german]{Hamiltonian}{}{}
|
||||
\eq{\hat{H} &= 4 E_C\hat{n}^2 - E_\text{J} \cos\hat{\phi}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Tunable Transmon qubit}
|
||||
\ger{Tunable Transmon Qubit}
|
||||
]{tunable}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Frequency tunable transmon}{By using a \fqSecRef{qc:scq:elements:squid} instead of a \fqSecRef{qc:scq:elements:josephson_junction}, the qubit is frequency tunable through an external field}{}
|
||||
\desc[german]{}{Durch Nutzung eines \fqSecRef{qc:scq:elements:squid} anstatt eines \fqSecRef{qc:scq:elements:josephson_junction}s, ist die Frequenz des Qubits durch ein externes Magnetfeld einstellbar}{}
|
||||
\content{
|
||||
\centering
|
||||
\begin{circuitikz}
|
||||
\draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
|
||||
to[capacitor=$C_\text{g}$] ++(2,0)
|
||||
to[capacitor=$C_C$] ++(0,-3)
|
||||
to ++(-2,0);
|
||||
\draw (2,3) to ++(3,0) to ++(0,-0.5) \squidloop{loop}{SQUID} to ++(0,-0.5) to ++(-3,0);
|
||||
\end{circuitikz}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{energy}
|
||||
\desc{Josephson energy}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ asymmetry}
|
||||
\desc[german]{Josephson Energie}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ Asymmetrie}
|
||||
\eq{E_\text{J,eff}(\Phi_\text{ext}) = (E_\text{J1}+E_\text{J2}) \sqrt{\cos^2\left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right) + d^2 \sin \left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right)}}
|
||||
\end{formula}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{}
|
||||
\desc[german]{Hamiltonian}{}{}
|
||||
\eq{\hat{H} = 4E_C \hat{n}^2 - \frac{1}{2} E_\text{J,eff}(\Phi_\text{ext}) \sum_{n}\left[\ket{n}\bra{n+1} + \ket{n+1}\bra{n}\right]}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Flux qubit}
|
||||
\ger{Flux Qubit}
|
||||
]{flux}
|
||||
\TODO{TODO}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Flux qubit / Persistent current qubit}{}{}
|
||||
\desc[german]{Flux Qubit / Persistent current qubit}{}{}
|
||||
\content{
|
||||
\centering
|
||||
\begin{circuitikz}
|
||||
\draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
|
||||
to ++(2,0) coordinate(top1);
|
||||
\draw[color=gray] (top1) to[capacitor=$C_C$] ++(0,-3);
|
||||
\draw (top1) to ++(2,0) coordinate(top2) to ++(0,-0.5)
|
||||
to[josephsoncap=$C_\text{J}$] ++(-0,-2) to ++(0,-0.5) to ++(-2,0) to ++(-2,0);
|
||||
\draw (top2) to ++(2,0) to[cute inductor=$E_L$] ++(0,-3) to ++(-2,0);
|
||||
\node at (5,0.5) {$\Phi_\text{ext}$};
|
||||
\end{circuitikz}
|
||||
\\\TODO{Ist beim Fluxonium noch die Voltage source dran?}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Fluxonium qubit}
|
||||
\ger{Fluxonium Qubit}
|
||||
]{fluxonium}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Fluxonium qubit}{
|
||||
Josephson junction with a shunt \textbf{inductance}. Instead of having to tunnel, cooper pairs can move to the island via the inductance.
|
||||
The inductance consists of many parallel Josephson Junctions to avoid parasitic capacitances.
|
||||
}{}
|
||||
\desc[german]{Fluxonium Qubit}{
|
||||
Josephson-Kontakt mit einem parallelen \textbf{induktiven Element}.
|
||||
Anstatt zu tunneln, können die Cooper-Paare über das induktive Element auf die Insel gelangen.
|
||||
Das induktive Element besteht aus sehr vielen parallelen Josephson-Kontakten um parisitische Kapazitäten zu vermeiden.
|
||||
}{}
|
||||
\content{
|
||||
\centering
|
||||
\begin{circuitikz}
|
||||
\draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
|
||||
to ++(2,0) coordinate(top1);
|
||||
\draw[color=gray] (top1) to[capacitor=$C_C$] ++(0,-3);
|
||||
\draw (top1) to ++(2,0) coordinate(top2) to ++(0,-0.5)
|
||||
to[josephsoncap=$C_\text{J}$] ++(-0,-2) to ++(0,-0.5) to ++(-2,0) to ++(-2,0);
|
||||
\draw (top2) to ++(2,0) to[cute inductor=$E_L$] ++(0,-3) to ++(-2,0);
|
||||
\node at (5,0.5) {$\Phi_\text{ext}$};
|
||||
\end{circuitikz}
|
||||
\\\TODO{Ist beim Fluxonium noch die Voltage source dran?}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\def\temp{$E_\text{C} = \frac{(2e)^2}{2C}, E_\text{L} = \frac{\varphi_0^2}{2L}, \delta_\text{s} = \frac{\varphi_\text{s}}{\varphi_0}$}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{\temp}
|
||||
\desc[german]{Hamiltonian}{}{\temp}
|
||||
\eq{\hat{H} = 4E_\text{C} \hat{n}^2 - E_\text{J} \cos \hat{\delta} + E_\text{L}(\hat{\delta} - \delta_\text{s})^2}
|
||||
\end{formula}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{img/qubit_flux_onium.pdf}
|
||||
\caption{img/}
|
||||
\label{fig:img-}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Two-level system}
|
||||
\ger{Zwei-Niveau System}
|
||||
]{stuff}
|
||||
|
||||
\begin{formula}{resonance_frequency}
|
||||
\desc{Resonance frequency}{}{}
|
||||
\desc[german]{Ressonanzfrequenz}{}{}
|
||||
\eq{\omega_{21} = \frac{E_2 - E_1}{\hbar}}
|
||||
\end{formula}
|
||||
\TODO{sollte das nicht 10 sein?}
|
||||
|
||||
\begin{formula}{rabi_oscillation}
|
||||
\desc{Rabi oscillations}{}{$\omega_{21}$ resonance frequency of the energy transition, $\Omega$ Rabi frequency}
|
||||
\desc[german]{Rabi-Oszillationen}{}{$\omega_{21}$ Resonanzfrequenz des Energieübergangs, $\Omega$ Rabi-Frequenz}
|
||||
\eq{\Omega_ TODO}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Ramsey interferometry}
|
||||
\ger{Ramsey Interferometrie}
|
||||
]{ramsey}
|
||||
|
||||
\begin{ttext}
|
||||
\eng{$\ket{0} \xrightarrow{\frac{\pi}{2}\,\text{pulse}}$ precession in $xy$ plane for time $\tau$ $\xrightarrow{\frac{\pi}{2}\,\text{pulse}}$ measurement}
|
||||
\ger{q}
|
||||
\end{ttext}
|
||||
|
||||
\Section[
|
||||
\eng{Noise and decoherence}
|
||||
\ger{Noise und Dekohärenz}
|
||||
]{noise}
|
||||
\begin{formula}{long}
|
||||
\desc{Longitudinal relaxation rate}{$\Gamma_{1\downarrow}$: $\ket{1}\rightarrow \ket{0}$ \\ $\Gamma_{1\uparrow}$: $\ket{0}\rightarrow \ket{1}$}{}
|
||||
\desc[german]{Longitudinale Relaxationsrate}{$\Gamma_{1\downarrow}$: $\ket{1}\rightarrow \ket{0}$ \\ $\Gamma_{1\uparrow}$: $\ket{0}\rightarrow \ket{1}$}{}
|
||||
\eq{\Gamma_1 = \frac{1}{T_1} = \Gamma_{1\uparrow} + \Gamma_{1\downarrow}}
|
||||
\end{formula}
|
||||
|
||||
\begin{ttext}[long]
|
||||
\eng{$\Gamma_{1\uparrow}$ is supressed because of detailed balance}
|
||||
\ger{$\Gamma_{1\uparrow}$ ist unterdrückt wegen detailed balance}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{dephasing}
|
||||
\desc{Pure dephasing rate}{}{}
|
||||
\desc[german]{Reine Phasenverschiebung}{}{}
|
||||
\eq{\Gamma_\phi}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{trans}
|
||||
\desc{Transversal relaxation rate}{}{}
|
||||
\desc[german]{Transversale Relaxationsrate}{}{}
|
||||
\eq{\Gamma_2 = \frac{1}{T_2} = \frac{\Gamma_1}{2} + \Gamma_\phi}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bloch_redfield}
|
||||
\desc{Bloch-Redfield density matrix}{2-level System weakly coupled to noise sources with short correlation time}{}
|
||||
\desc[german]{Bloch-Redfield Dichtematrix}{2-Niveau System schwach an Noise Quellen mit kurzer Korrelationszeit gekoppelt}{}
|
||||
\eq{\rho_\text{BR} = \begin{pmatrix} 1+(\abs{\alpha}^2-1)\e^{-\Gamma_1 t} & \alpha \beta^* \e^{-\Gamma_2 t} \\
|
||||
\alpha^*\beta \e^{-\Gamma_2 t} & \abs{\beta}^2 \e^{-\Gamma_1 t} \end{pmatrix} }
|
||||
\end{formula}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{img/qubit_transmon.pdf}
|
||||
\caption{Transmon and so TODO}
|
||||
\label{fig:img-qubit_transmon-pdf}
|
||||
\end{figure}
|
||||
|
||||
|
||||
|
||||
|
||||
\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}
|
||||
|
||||
|
||||
@ -17,10 +17,10 @@
|
||||
\eng{Operators}
|
||||
\ger{Operatoren}
|
||||
]{op}
|
||||
\GER[row_vector]{Zeilenvektor}
|
||||
\GER[column_vector]{Spaltenvektor}
|
||||
\ENG[column_vector]{Column vector}
|
||||
\ENG[row_vector]{Row vector}
|
||||
\Ger[row_vector]{Zeilenvektor}
|
||||
\Ger[column_vector]{Spaltenvektor}
|
||||
\Eng[column_vector]{Column vector}
|
||||
\Eng[row_vector]{Row vector}
|
||||
\begin{formula}{dirac_notation}
|
||||
\desc{Dirac notation}{}{}
|
||||
\desc[german]{Dirac-Notation}{}{}
|
||||
@ -75,12 +75,6 @@
|
||||
\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}{}{}
|
||||
@ -116,25 +110,38 @@
|
||||
\begin{formula}{commutator}
|
||||
\desc{Commutator}{}{}
|
||||
\desc[german]{Kommutator}{}{}
|
||||
\eq{[a,b] = ab - ba}
|
||||
\eq{[A,B] = AB - BA}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{anticommutator}
|
||||
\desc{Anticommutator}{}{}
|
||||
\desc[german]{Antikommmutator}{}{}
|
||||
\eq{\{a,b\} = ab + ba}
|
||||
\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\}}
|
||||
\eq{[A, BC] = [A, B]C - B[A,C]}
|
||||
\end{formula}
|
||||
\TODO{add some more?}
|
||||
|
||||
\begin{formula}{function}
|
||||
\desc{Commutator involving a function}{}{given $[A,[A,B]] = 0$}
|
||||
\desc[german]{Kommutator mit einer Funktion}{}{falls $[A,[A,B]] = 0$}
|
||||
\eq{[f(A) , B] = [A,B]\,\pdv{f}{A}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{jacobi_identity}
|
||||
\desc{Jacobi identity}{}{}
|
||||
\desc[german]{Jakobi-Identität}{}{}
|
||||
\eq{[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0}
|
||||
\eq{[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{hadamard_lemma}
|
||||
\desc{Hadamard's Lemma}{}{}
|
||||
\desc[german]{Lemma von Hadamard}{}{}
|
||||
\eq{\e^A B \e^{-A} = B + [A,B] + \frac{1}{2!} [A, [A,B]] + \frac{1}{3!} [A, [A, [A, B]]] + \dots}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{canon_comm_relation}
|
||||
@ -147,39 +154,44 @@
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Schrödinger equation}
|
||||
\ger{Schrödingergleichung}
|
||||
]{schroedinger_equation}
|
||||
\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}
|
||||
|
||||
\Subsection[
|
||||
\eng{Schrödinger equation}
|
||||
\ger{Schrödingergleichung}
|
||||
]{schroedinger_equation}
|
||||
\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}
|
||||
\eng{Time evolution}
|
||||
\ger{Zeitentwicklug}
|
||||
]{time}
|
||||
The time evolution of the Hamiltonian is given by:
|
||||
\begin{formula}{time_evolution_op}
|
||||
\desc{Time evolution operator}{}{$U$ unitary}
|
||||
@ -187,6 +199,21 @@
|
||||
\eq{\ket{\psi(t)} = \hat{U}(t, t_0) \ket{\psi(t_0)}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{von_neumann}
|
||||
\desc{Von-Neumann Equation}{Time evolution of the density operator in the Schrödinger picture. Qm analog to the Liouville equation \ref{eq:mech:liouville:todo}}{}
|
||||
\desc[german]{Von-Neumann Gleichung}{Zeitentwicklung des Dichteoperators im Schödingerbild. Qm. Analogon zur Liouville-Gleichung \ref{eq:mech:liouville:todo}}{}
|
||||
\eq{\pdv{\hat{\rho}}{t} = - \frac{i}{\hbar}[\hat{H}, \hat{\rho}]}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{lindblad}
|
||||
\desc{Lindblad master equation}{Generalization of von-Neummann equation for open quantum systems}{$h$ positive semidifnite matrix, $\hat{A}$ arbitrary operator}
|
||||
\desc[german]{Lindblad-Mastergleichung}{Verallgemeinerung der von-Neumman Gleichung für offene Quantensysteme}{$h$ positiv-semifinite Matrix, $\hat{A}$ beliebiger Operator}
|
||||
\eq{\dot{\rho} = \underbrace{-\frac{i}{\hbar} [\hat{H}, \rho]}_\text{reversible} + \underbrace{\sum_{n.m} h_{nm} \left(\hat{A}_n\rho \hat{A}_{m^\dagger} - \frac{1}{2}\left\{\hat{A}_m^\dagger \hat{A}_n,\rho \right\}\right)}_\text{irreversible}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\TODO{unitary transformation of time dependent H}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Schrödinger- and Heisenberg-pictures}
|
||||
\ger{Schrödinger- und Heisenberg-Bild}
|
||||
@ -217,20 +244,10 @@
|
||||
}
|
||||
\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}
|
||||
]{ehrenfest_theorem}
|
||||
\GT{see_also} \ref{sec:qm:basics:schroedinger_equation:correspondence_principle}
|
||||
\begin{formula}{ehrenfest_theorem}
|
||||
\desc{Ehrenfesttheorem}{applies to both pictures}{}
|
||||
@ -244,15 +261,27 @@
|
||||
\desc[german]{}{Beispiel für $x$}{}
|
||||
\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}
|
||||
% \eq{Time evolution}{\hat{H}\ket{\psi} = E\ket{\psi}}{sg_time}
|
||||
|
||||
\Subsection[
|
||||
\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}
|
||||
|
||||
|
||||
|
||||
\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}
|
||||
]{qm_pertubation}
|
||||
\begin{ttext}
|
||||
\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.}
|
||||
\end{ttext}
|
||||
\begin{formula}{pertubation_hamiltonian}
|
||||
\desc{Hamiltonian}{}{}
|
||||
\desc[german]{Hamiltonian}{}{}
|
||||
@ -289,6 +318,12 @@
|
||||
% \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}{golden_rule}
|
||||
\desc{Fermi\'s golden rule}{Transition rate from initial state $\ket{i}$ under a pertubation $H^1$ to final state $\ket{f}$}{}
|
||||
\desc[german]{Fermis goldene Regel}{Übergangsrate des initial Zustandes $\ket{i}$ unter einer Störung $H^1$ zum Endzustand $\ket{f}$}{}
|
||||
\eq{\Gamma_{i\to f} = \frac{2\pi}{\hbar} \abs*{\braket{f | H^1 | i}}^2\,\rho(E_f)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Harmonic oscillator}
|
||||
@ -404,3 +439,64 @@
|
||||
\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}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Symmetries}
|
||||
\ger{Symmetrien}
|
||||
]{symmetry}
|
||||
\begin{ttext}[desc]
|
||||
\eng{Most symmetry operators are unitary \ref{sec:linalg:unitary} because the norm of a state must be invariant under transformations of space, time and spin.}
|
||||
\ger{Die meisten Symmetrieoperatoren sind unitär \ref{sec:linalg:unitary}, da die Norm eines Zustands invariant unter Raum-, Zeit- und Spin-Transformationen sein muss.}
|
||||
\end{ttext}
|
||||
\begin{formula}{invariance}
|
||||
\desc{Invariance}{$\hat{H}$ is invariant under a symmetrie described by $\hat{U}$ if this holds}{}
|
||||
\desc[german]{Invarianz}{$\hat{H}$ is invariant unter der von $\hat{U}$ beschriebenen Symmetrie wenn gilt:}{}
|
||||
\eq{\hat{U}\hat{H}\hat{U}^\dagger = \hat{H} \Leftrightarrow [\hat{U}, \hat{H}] = 0}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Time-reversal symmetry}
|
||||
\ger{Zeitumkehrungssymmetrie}
|
||||
]{time_reversal}
|
||||
|
||||
\begin{formula}{time}
|
||||
\desc{Time-reversal symmetry}{}{}
|
||||
\desc[german]{Zeitumkehrungssymmetrie}{}{}
|
||||
\eq{T: t \to -t}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{antiunitary}
|
||||
\desc{Anti-unitary}{}{}
|
||||
\desc[german]{Antiunitär}{}{}
|
||||
\eq{T^2 = -1}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Two-level systems (TLS)}
|
||||
\ger{Zwei-Niveau System (TLS)}
|
||||
]{tls}
|
||||
\begin{formula}{james_cummings}
|
||||
\desc{James-Cummings Hamiltonian}{TLS interacting with optical cavity}{$\hat{E} = E_\text{ZPF}(\hat{a} + \hat{a}^\dagger)$ field operator with bosonic ladder operators, $\hat{S} = \hat{\sigma}^\dagger + \hat{\sigma}$ polarization operator with ladder operators of the TLS}
|
||||
\desc[german]{James-Cummings Hamiltonian}{TLS interagiert mit resonantem Lichtfeld}{$\hat{E} = E_\text{ZPF}(\hat{a} + \hat{a}^\dagger)$ Feldoperator mit bosonischen Leiteroperatoren, $\hat{S} = \hat{\sigma}^\dagger + \hat{\sigma}$ Polarisationsoperator mit Leiteroperatoren des TLS}
|
||||
\eq{H &= \underbrace{\hbar\omega_c \hat{a}^\dagger \hat{a}}_\text{\GT{field}}
|
||||
+ \underbrace{\hbar\omega_\text{a} \frac{\hat{\sigma}_z}{2}}_\text{\GT{atom}}
|
||||
+ \underbrace{\frac{\hbar\Omega}{2} \hat{E} \hat{S}}_\text{int} \\
|
||||
\shortintertext{\GT{after} \hyperref[eq:qm:other:RWS]{RWA}:} \\
|
||||
&= \hbar\omega_c \hat{a}^\dagger \hat{a}
|
||||
+ \hbar\omega_\text{a} \hat{\sigma}^\dagger \hat{\sigma}
|
||||
+ \frac{\hbar\Omega}{2} (\hat{a}\hat{\sigma^\dagger} + \hat{a}^\dagger \hat{\sigma})
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Other}
|
||||
\ger{Sonstiges}
|
||||
]{other}
|
||||
\begin{formula}{RWS}
|
||||
\desc{Rotating Wave Approximation (RWS)}{Rapidly oscilating terms are neglected}{$\omega_\text{L}$ light frequency, $\omega_0$ transition frequency}
|
||||
\desc[german]{Rotating Wave Approximation / Drehwellennäherung (RWS)}{Schnell oscillierende Terme werden vernachlässigt}{$\omega_\text{L}$ Frequenz des Lichtes, $\omega_0$ Übergangsfrequenz}
|
||||
\eq{\Delta\omega \coloneq \abs{\omega_0 - \omega_\text{L}} \ll \abs{\omega_0 + \omega_\text{L}} \approx 2\omega_0}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
2
src/scripts/bloch_sphere.py
Normal file
2
src/scripts/bloch_sphere.py
Normal file
@ -0,0 +1,2 @@
|
||||
import qutip as qt
|
||||
|
||||
189
src/scripts/crystal_lattices-Copy1.ipynb
Normal file
189
src/scripts/crystal_lattices-Copy1.ipynb
Normal file
File diff suppressed because one or more lines are too long
441
src/scripts/crystal_lattices.ipynb
Normal file
441
src/scripts/crystal_lattices.ipynb
Normal file
File diff suppressed because one or more lines are too long
106
src/scripts/distributions.py
Normal file
106
src/scripts/distributions.py
Normal file
@ -0,0 +1,106 @@
|
||||
from numpy import fmax
|
||||
from plot import *
|
||||
|
||||
|
||||
def get_fig():
|
||||
fig, ax = plt.subplots(figsize=size_half_half)
|
||||
ax.grid()
|
||||
ax.set_xlabel(f"$x$")
|
||||
ax.set_ylabel("PDF")
|
||||
return fig, ax
|
||||
|
||||
# GAUSS / NORMAL
|
||||
def fgauss(x, mu, sigma_sqr):
|
||||
return 1 / (np.sqrt(2 * np.pi * sigma_sqr)) * np.exp(-(x - mu)**2 / (2 * sigma_sqr))
|
||||
|
||||
def gauss():
|
||||
fig, ax = get_fig()
|
||||
x = np.linspace(-5, 5, 300)
|
||||
for mu, sigma_sqr in [(0, 1), (0, 0.2), (0, 5), (-2, 2)]:
|
||||
y = fgauss(x, mu, sigma_sqr)
|
||||
label = texvar("mu", mu) + ", " + texvar("sigma^2", sigma_sqr)
|
||||
ax.plot(x, y, label=label)
|
||||
ax.legend()
|
||||
return fig
|
||||
export(gauss(), "distribution_gauss")
|
||||
|
||||
# CAUCHY / LORENTZ
|
||||
def fcauchy(x, x_0, gamma):
|
||||
return 1 / (np.pi * gamma * (1 + ((x - x_0)/gamma)**2))
|
||||
|
||||
def cauchy():
|
||||
fig, ax = get_fig()
|
||||
x = np.linspace(-5, 5, 300)
|
||||
for x_0, gamma in [(0, 1), (0, 0.5), (0, 2), (-2, 1)]:
|
||||
y = fcauchy(x, x_0, gamma)
|
||||
label = f"$x_0 = {x_0}$ , {texvar('gamma', gamma)}"
|
||||
ax.plot(x, y, label=label)
|
||||
ax.legend()
|
||||
return fig
|
||||
export(cauchy(), "distribution_cauchy")
|
||||
|
||||
# MAXWELL-BOLTZMANN
|
||||
def fmaxwell(x, a):
|
||||
return np.sqrt(2/np.pi) * x**2 / a**3 * np.exp(-x**2 /(2*a**2))
|
||||
|
||||
def maxwell():
|
||||
fig, ax = get_fig()
|
||||
x = np.linspace(0, 20, 300)
|
||||
for a in [1, 2, 5]:
|
||||
y = fmaxwell(x, a)
|
||||
label = f"$a = {a}$"
|
||||
ax.plot(x, y, label=label)
|
||||
ax.legend()
|
||||
return fig
|
||||
|
||||
export(maxwell(), "distribution_maxwell-boltzmann")
|
||||
|
||||
|
||||
# POISSON
|
||||
def fpoisson(k, l):
|
||||
return l**k * np.exp(-l) / scp.special.factorial(k)
|
||||
|
||||
def poisson():
|
||||
fig, ax = get_fig()
|
||||
k = np.arange(0, 21, dtype=int)
|
||||
for l in [1, 4, 10]:
|
||||
y = fpoisson(k, l)
|
||||
label = texvar("lambda", l)
|
||||
ax.plot(k, y, color="#555")
|
||||
ax.scatter(k, y, label=label)
|
||||
ax.set_xlabel(f"$k$")
|
||||
ax.set_ylabel(f"PMF")
|
||||
ax.legend()
|
||||
return fig
|
||||
|
||||
export(poisson(), "distribution_poisson")
|
||||
|
||||
# BINOMIAL
|
||||
def binom(n, k):
|
||||
return scp.special.factorial(n) / (
|
||||
scp.special.factorial(k) *
|
||||
scp.special.factorial((n-k))
|
||||
)
|
||||
def fbinomial(k, n, p):
|
||||
return binom(n, k) * p**k * (1-p)**(n-k)
|
||||
|
||||
def binomial():
|
||||
fig, ax = get_fig()
|
||||
n = 20
|
||||
k = np.arange(0, n+1, dtype=int)
|
||||
for p in [0.3, 0.5, 0.7]:
|
||||
y = fbinomial(k, n, p)
|
||||
label = f"$n={n}$, $p={p}$"
|
||||
ax.plot(k, y, color="#555")
|
||||
ax.scatter(k, y, label=label)
|
||||
ax.set_xlabel(f"$k$")
|
||||
ax.set_ylabel(f"PMF")
|
||||
ax.legend()
|
||||
return fig
|
||||
|
||||
export(binomial(), "distribution_binomial")
|
||||
|
||||
# FERMI-DIRAC
|
||||
|
||||
# BOSE-EINSTEIN
|
||||
|
||||
34
src/scripts/plot.py
Normal file
34
src/scripts/plot.py
Normal file
@ -0,0 +1,34 @@
|
||||
import os
|
||||
import matplotlib.pyplot as plt
|
||||
import numpy as np
|
||||
import math
|
||||
import scipy as scp
|
||||
|
||||
outdir = "../img/"
|
||||
filetype = ".pdf"
|
||||
skipasserts = False
|
||||
|
||||
full = 8
|
||||
size_half_half = (full/2, full/2)
|
||||
size_third_half = (full/3, full/2)
|
||||
size_half_third = (full/2, full/3)
|
||||
|
||||
def texvar(var, val, math=True):
|
||||
s = "$" if math else ""
|
||||
s += f"\\{var} = {val}"
|
||||
if math: s += "$"
|
||||
return s
|
||||
|
||||
def export(fig, name):
|
||||
if not skipasserts:
|
||||
assert os.path.abspath(".").endswith("scripts"), "Please run from the `scripts` directory"
|
||||
filename = os.path.join(outdir, name + filetype)
|
||||
fig.savefig(filename, bbox_inches="tight")
|
||||
|
||||
|
||||
@np.vectorize
|
||||
def smooth_step(x: float, left_edge: float, right_edge: float):
|
||||
x = (x - left_edge) / (right_edge - left_edge)
|
||||
if x <= 0: return 0.
|
||||
elif x >= 1: return 1.
|
||||
else: return 3*(x**2) - 2*(x**3)
|
||||
90
src/scripts/qubits.py
Normal file
90
src/scripts/qubits.py
Normal file
@ -0,0 +1,90 @@
|
||||
from plot import *
|
||||
import scqubits as scq
|
||||
import qutip as qt
|
||||
|
||||
# flux = scq.FluxQubit()
|
||||
fluxonium = scq.Fluxonium(EJ=9, EC=3, EL=0.5, flux=1, cutoff=100)
|
||||
|
||||
|
||||
|
||||
def _plot_transmon_n_wavefunctions(qubit: scq.Transmon, fig_ax, which=[0,1]):
|
||||
fig, ax = fig_ax
|
||||
ax.set_ylabel(r"$|\psi_i(n)|^2$")
|
||||
ax.set_xlabel(r"$n$")
|
||||
colors = "brgy"
|
||||
for i in which:
|
||||
wf = qubit.numberbasis_wavefunction(which=i)
|
||||
x = wf.basis_labels
|
||||
y = wf.amplitudes
|
||||
offset = (len(which)/2 - i) * 0.2
|
||||
ax.bar(x-offset, y, width=0.2, align='center', label=f"$i={i}$")
|
||||
xlim = (-4, 4)
|
||||
ax.set_xlim(*xlim)
|
||||
ax.set_xticks(np.arange(xlim[0], xlim[1]+1))
|
||||
|
||||
def _plot_transmon(qubit: scq.Transmon, ngs, fig, axs):
|
||||
_,_ = qubit.plot_evals_vs_paramvals("ng", ngs, fig_ax=(fig, axs[0]), evals_count=5, subtract_ground=False)
|
||||
_,_ = qubit.plot_wavefunction(fig_ax=(fig, axs[1]), which=[0, 1, 2], mode="abs_sqr")
|
||||
_plot_transmon_n_wavefunctions(qubit, (fig, axs[2]), which=[0, 1, 2])
|
||||
qubit.ng = 0.5
|
||||
_plot_transmon_n_wavefunctions(qubit, (fig, axs[3]), which=[0, 1, 2])
|
||||
qubit.ng = 0
|
||||
|
||||
|
||||
def transmon_cpb():
|
||||
EC = 1
|
||||
qubit = scq.Transmon(EJ=30, EC=EC, ng=0, ncut=30)
|
||||
ngs = np.linspace(-2, 2, 200)
|
||||
fig, axs = plt.subplots(4, 3, squeeze=True, figsize=(full,full))
|
||||
axs = axs.T
|
||||
qubit.ng = 0
|
||||
qubit.EJ = 0.1 * EC
|
||||
title = lambda x: f"$E_J/E_C = {x}$"
|
||||
_plot_transmon(qubit, ngs, fig, axs[0])
|
||||
axs[0][0].set_title("Cooper-Pair-Box\n"+title(qubit.EJ))
|
||||
|
||||
qubit.EJ = EC
|
||||
_plot_transmon(qubit, ngs, fig, axs[1])
|
||||
axs[1][0].set_title("Quantronium\n"+title(qubit.EJ))
|
||||
|
||||
qubit.EJ = 20 * EC
|
||||
_plot_transmon(qubit, ngs, fig, axs[2])
|
||||
axs[2][0].set_title("Transmon\n"+title(qubit.EJ))
|
||||
|
||||
for ax in axs[1:,:].flatten(): ax.set_ylabel("")
|
||||
for ax in axs[:,0].flatten():
|
||||
ax.set_xticks([-2, -1, -0.5, 0, 0.5, 1, 2])
|
||||
ax.set_xticklabels(["-2", "-1", "", "0", "", "1", "2"])
|
||||
ylim = ax.get_ylim()
|
||||
ax.vlines([-1, -0.5], ymin=ylim[0], ymax=ylim[1], color="#aaa", linestyle="dotted")
|
||||
axs[0][2].legend()
|
||||
fig.tight_layout()
|
||||
return fig
|
||||
|
||||
export(transmon_cpb(), "qubit_transmon")
|
||||
|
||||
|
||||
def flux_onium():
|
||||
fig, axs = plt.subplots(1, 2, squeeze=True, figsize=(full,full))
|
||||
fluxs = np.linspace(-2, 2, 101)
|
||||
EJ = 35.0
|
||||
alpha = 0.6
|
||||
fluxqubit = scq.FluxQubit(EJ1 = EJ,
|
||||
EJ2 = EJ,
|
||||
EJ3 = alpha*EJ,
|
||||
ECJ1 = 1.0,
|
||||
ECJ2 = 1.0,
|
||||
ECJ3 = 1.0/alpha,
|
||||
ECg1 = 50.0,
|
||||
ECg2 = 50.0,
|
||||
ng1 = 0.0,
|
||||
ng2 = 0.0,
|
||||
flux = 0.5,
|
||||
ncut = 10)
|
||||
fluxqubit.plot_evals_vs_paramvals("flux", fluxs, evals_count=5, subtract_ground=True, fig_ax=(fig, axs[0]))
|
||||
|
||||
fluxonium = scq.Fluxonium(EJ=9, EC=3, EL=0.5, flux=1, cutoff=100)
|
||||
fluxonium.plot_evals_vs_paramvals("flux", fluxs, evals_count=5, subtract_ground=True, fig_ax=(fig, axs[1]))
|
||||
return fig
|
||||
|
||||
export(flux_onium(), "qubit_flux_onium")
|
||||
5
src/scripts/requirements.txt
Normal file
5
src/scripts/requirements.txt
Normal file
@ -0,0 +1,5 @@
|
||||
numpy
|
||||
matplotlib
|
||||
scqubits
|
||||
qutip
|
||||
|
||||
120
src/scripts/stat-mech.py
Normal file
120
src/scripts/stat-mech.py
Normal file
@ -0,0 +1,120 @@
|
||||
from plot import *
|
||||
|
||||
def flennard_jones(r, epsilon, sigma):
|
||||
return 4 * epsilon * ((sigma/r)**12 - (sigma/r)**6)
|
||||
|
||||
def lennard_jones():
|
||||
fig, ax = plt.subplots(figsize=size_half_half)
|
||||
ax.grid()
|
||||
ax.set_xlabel(r"$r$")
|
||||
ax.set_ylabel(r"$V(r)$")
|
||||
|
||||
r = np.linspace(0.5, 4, 300)
|
||||
for epsilon, sigma in [(1, 1), (1, 2)]:
|
||||
y = flennard_jones(r, epsilon, sigma)
|
||||
label = texvar("epsilon", epsilon) + ", " + texvar("sigma", sigma)
|
||||
ax.plot(r, y, label=label)
|
||||
ax.legend()
|
||||
ax.set_ylim(-1.1, 1.1)
|
||||
return fig
|
||||
export(lennard_jones(), "potential_lennard_jones")
|
||||
|
||||
# BOLTZMANN / FERMI-DIRAC / BOSE-EINSTEN DISTRIBUTIONS
|
||||
def fboltzmann(x):
|
||||
return 1 / np.exp(x)
|
||||
def fbose_einstein(x):
|
||||
return 1 / (np.exp(x) -1)
|
||||
def ffermi_dirac(x):
|
||||
return 1 / (np.exp(x) + 1)
|
||||
|
||||
|
||||
def id_qgas():
|
||||
fig, ax = plt.subplots(figsize=size_half_half)
|
||||
ax.grid()
|
||||
ax.set_xlabel(r"$\beta(\epsilon-\mu)$")
|
||||
ax.set_ylabel(r"$\langle n(\epsilon)\rangle$")
|
||||
x = np.linspace(-4, 4, 300)
|
||||
xbe = np.linspace(0.01, 4, 300)
|
||||
yb = fboltzmann(x)
|
||||
ybe = fbose_einstein(xbe)
|
||||
yfd = ffermi_dirac(x)
|
||||
ax.vlines([0], ymin=-10, ymax=10, color="black", linestyle="dashed")
|
||||
ax.plot(xbe, ybe, label="Bose-Einstein")
|
||||
ax.plot(x, yb, label="Boltzmann")
|
||||
ax.plot(x, yfd, label="Fermi-Dirac")
|
||||
ax.legend()
|
||||
ax.set_ylim(-0.1, 4)
|
||||
return fig
|
||||
export(id_qgas(), "td_id_qgas_distributions")
|
||||
|
||||
@np.vectorize
|
||||
def fstep(x):
|
||||
return 1 if x >= 0 else 0
|
||||
|
||||
def fermi_occupation():
|
||||
fig, ax = plt.subplots(figsize=size_half_third)
|
||||
# ax.grid()
|
||||
# ax.set_xlabel(r"$\beta(\epsilon-\mu)$")
|
||||
ax.set_xticks([0])
|
||||
ax.set_xticklabels([r"$\epsilon=\mu$"])
|
||||
ax.set_ylabel(r"$\langle n(\epsilon)\rangle$")
|
||||
x = np.linspace(-4, 4, 300)
|
||||
ystep = fstep(-x)
|
||||
yfd = ffermi_dirac(x)
|
||||
ax.vlines([0], ymin=-10, ymax=10, color="black", linestyle="dashed")
|
||||
ax.plot(x, ystep, label="$T = 0$")
|
||||
ax.plot(x, yfd, label=r"$T > 0$")
|
||||
ax.legend()
|
||||
ax.set_ylim(-0.1, 1.1)
|
||||
return fig
|
||||
export(fermi_occupation(), "td_fermi_occupation")
|
||||
|
||||
def fermi_heat_capacity():
|
||||
fig, ax = plt.subplots(figsize=size_half_third)
|
||||
# ax.grid()
|
||||
# ax.set_xlabel(r"$\beta(\epsilon-\mu)$")
|
||||
x = np.linspace(0, 4, 100)
|
||||
T_F = 1
|
||||
a = np.pi**2 / (2 * T_F)
|
||||
|
||||
def linear(x):
|
||||
return x * a
|
||||
X_32 = 3/2 /(np.pi**2 / (2 * T_F))
|
||||
|
||||
|
||||
low_temp_Cv = linear(x)
|
||||
ax.plot(x, low_temp_Cv, color="orange", linestyle="dashed", label=r"${\pi^2}/{2}\,{T}/{T_\text{F}}$")
|
||||
ax.hlines([3/2], xmin=0, xmax=10, color="blue", linestyle="dashed", label="Petit-Dulong")
|
||||
@np.vectorize
|
||||
def unphysical_f(x):
|
||||
# exponential
|
||||
# find ṕoint where unshifted exponential has slope of the linear
|
||||
# f = 3/2 - exp(-A*x)
|
||||
# f' = A exp(-A*x) = a
|
||||
# x = -log(a/A) / A
|
||||
A = 5
|
||||
x_unshifted = -np.log(a/A) / A
|
||||
# shift so that this the exponential intersects the linear at this point
|
||||
y_intersect = 3/2 - np.exp(-A*x_unshifted)
|
||||
# find intersect x value of linear
|
||||
# x = y/a
|
||||
x_intersect = y_intersect / a
|
||||
# shift exp so that x_intersect becomes x_unshifted
|
||||
if x > x_intersect: return 3/2 - np.exp(-A * (x-(x_intersect - x_unshifted)))
|
||||
else: return a * x
|
||||
# ax.plot(x, smoothing, label="smooth")
|
||||
y = unphysical_f(x)
|
||||
ax.plot(x, y, color="black")
|
||||
ax.legend(loc="lower right")
|
||||
|
||||
|
||||
ax.set_xticks([0, T_F])
|
||||
ax.set_xticklabels(["0", r"$T_F$"])
|
||||
ax.set_yticks([0, 3/2])
|
||||
ax.set_yticklabels(["0", r"$3/2$"])
|
||||
ax.set_ylabel(r"${C_V}/{N k_\text{B}}$")
|
||||
ax.set_xlim(0, 1.4 * T_F)
|
||||
ax.set_ylim(0, 2)
|
||||
return fig
|
||||
export(fermi_heat_capacity(), "td_fermi_heat_capacity")
|
||||
|
||||
757
src/statistical_mechanics.tex
Normal file
757
src/statistical_mechanics.tex
Normal file
@ -0,0 +1,757 @@
|
||||
\Part[
|
||||
\eng{Statistichal Mechanics}
|
||||
\ger{Statistische Mechanik}
|
||||
]{stat}
|
||||
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
\textbf{Intensive quantities:} Additive for subsystems (system size dependent): $S(\lambda E, \lambda V, \lambda N) = \lambda S(E, V, N)$\\
|
||||
\textbf{Extensive quantities:} Independent of system size, ratio of two intensive quantities
|
||||
}
|
||||
\ger{
|
||||
\textbf{Intensive Größen:} Additiv für Subsysteme (Systemgrößenabhänig): $S(\lambda E, \lambda V, \lambda N) = \lambda S(E, V, N)$\\
|
||||
\textbf{Extensive Größen:} Unabhängig der Systemgröße, Verhältnis zweier intensiver Größen
|
||||
}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{liouville}
|
||||
\desc{Liouville equation}{}{$\{\}$ poisson bracket}
|
||||
\desc[german]{Liouville-Gleichung}{}{$\{\}$ Poisson-Klammer}
|
||||
\eq{\pdv{\rho}{t} = - \sum_{i=1}^{N} \left(\pdv{\rho}{q_i} \pdv{H}{p_i} - \pdv{\rho}{p_i} \pdv{H}{q_i} \right) = \{H, \rho\}}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Entropy}
|
||||
\ger{Entropie}
|
||||
]{entropy}
|
||||
|
||||
\begin{formula}{properties}
|
||||
\desc{Positive-definite and additive}{}{}
|
||||
\desc[german]{Positiv Definit und Additiv}{}{}
|
||||
\eq{
|
||||
S &\ge 0 \\
|
||||
S(E_1, E_2) &= S_1 + S_2
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{von_neumann}
|
||||
\desc{Von-Neumann}{}{$\rho$ density matrix}
|
||||
\desc[german]{Von-Neumann}{}{$\rho$ Dichtematrix}
|
||||
\eq{S = - \kB \braket{\log \rho} = - \kB \tr(\rho \log\rho)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gibbs}
|
||||
\desc{Gibbs}{}{$p_n$ probability for micro state $n$}
|
||||
\desc[german]{Gibbs}{}{$p_n$ Wahrscheinlichkeit für Mikrozustand $n$}
|
||||
\eq{S = - \kB \sum_n p_n \log p_n}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{boltzmann}
|
||||
\desc{Boltzmann}{}{$\Omega$ \#micro states}
|
||||
\desc[german]{Boltzmann}{}{$\Omega$ \#Mikrozustände}
|
||||
\eq{S = \kB \log\Omega}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{temp}
|
||||
\desc{Temperature}{}{}
|
||||
\desc[german]{Temperatur}{}{}
|
||||
\eq{\frac{1}{T} \coloneq \pdv{S}{E}_V}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{pressure}
|
||||
\desc{Pressure}{}{}
|
||||
\desc[german]{Druck}{}{}
|
||||
\eq{p = T \pdv{S}{V}_E}
|
||||
\end{formula}
|
||||
|
||||
\Part[
|
||||
\eng{Thermodynamics}
|
||||
\ger{Thermodynamik}
|
||||
]{td}
|
||||
|
||||
\begin{formula}{therm_wavelength}
|
||||
\desc{Thermal wavelength}{}{}
|
||||
\desc[german]{Thermische Wellenlänge}{}{}
|
||||
\eq{\lambda = \frac{\hbar}{\sqrt{2\pi m \kB T}}}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Processes}
|
||||
\ger{Prozesse}
|
||||
]{process}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
\begin{itemize}
|
||||
\item \textbf{isobaric}: constant pressure $p = \const$
|
||||
\item \textbf{isochoric}: constant volume $V = \const$
|
||||
\item \textbf{isothermal}: constant temperature $T = \const$
|
||||
\item \textbf{isentropic}: constant entropy $S = \const$
|
||||
\item \textbf{isenthalpic}: constant entalphy $H = \const$
|
||||
\item \textbf{adiabatic}: no heat transfer $\Delta Q=0$
|
||||
\item \textbf{quasistatic}: happens so slow, the system always stays in td. equilibrium
|
||||
\item \textbf{reversivle}: reversible processes are always quasistatic and no entropie is created $\Delta S = 0$
|
||||
\end{itemize}
|
||||
}
|
||||
\ger{
|
||||
\begin{itemize}
|
||||
\item \textbf{isobar}: konstanter Druck $p = \const$
|
||||
\item \textbf{isochor}: konstantes Volumen $V = \const$
|
||||
\item \textbf{isotherm}: konstante Temperatur $T = \const$
|
||||
\item \textbf{isentrop}: konstante Entropie $S = \const$
|
||||
\item \textbf{isenthalp}: konstante Entalphie $H = \const$
|
||||
\item \textbf{adiabatisch}: kein Wärmeübertrag $\Delta Q=0$
|
||||
\item \textbf{quasistatsch}: läuft so langsam ab, dass das System durchgehend im t.d Equilibrium bleibt
|
||||
\item \textbf{reversibel}: reversible Prozesse sind immer quasistatisch und es wird keine Entropie erzeugt $Delta S = 0$
|
||||
\end{itemize}
|
||||
}
|
||||
\end{ttext}
|
||||
|
||||
\Subsection[
|
||||
\eng{Irreversible gas expansion (Gay-Lussac experiment)}
|
||||
\ger{Irreversible Gasexpansion (Gay-Lussac-Versuch)}
|
||||
]{gay}
|
||||
|
||||
\begin{minipage}{0.6\textwidth}
|
||||
\vfill
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
A classical gas in a system with volume $V_1$ is separated from another system with volume $V_2$.
|
||||
In the Gay-Lussac experiment, the separation is removed and the gas flows into $V_2$.
|
||||
}
|
||||
\ger{
|
||||
Ein klassisches Gas in einem System mit Volumen $V_1$ ist getrennt von einem zweiten System mit Volumen $V_2$.
|
||||
Beim Gay-Lussac Versuch wird die Trennwand entfern und das Gas fließt in das Volumen $V_2$.
|
||||
}
|
||||
\end{ttext}
|
||||
\vfill
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{0.3\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/td_gay_lussac.pdf}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
|
||||
\begin{formula}{entropy}
|
||||
\desc{Entropy change}{}{}
|
||||
\desc[german]{Entropieänderung}{}{}
|
||||
\eq{\Delta S=N\kB \ln \left(\frac{V_1 + V_2}{V_1}\right) > 0}
|
||||
\end{formula}
|
||||
|
||||
\TODO{Reversible}
|
||||
|
||||
\TODO{Quasistatischer T-Ausgleich}
|
||||
|
||||
\TODO{Joule-Thompson Prozess}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Phase transitions}
|
||||
\ger{Phasenübergänge}
|
||||
]{phases}
|
||||
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
A phase transition is a discontinuity in the free energy $F$ or Gibbs energy $G$ or in one of their derivatives.
|
||||
The degree of the phase transition is the degree of the derivative which exhibits the discontinuity.
|
||||
}
|
||||
\ger{
|
||||
Ein Phasenübergang ist eine Unstetigkeit in the Freien Energie $F$ oder in der Gibbs-Energie $G$ oder in ihrer Ableitungen.
|
||||
Die Ordnung des Phasenübergangs ist die Ordnung der Ableitung, in welcher die Unstetigkeit auftritt.
|
||||
}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{latent_heat}
|
||||
\desc{Latent heat}{Heat required to bring substance from phase 1 to phase 2}{$\Delta S$ entropy change of the phase transition}
|
||||
\desc[german]{Latente Wärme}{Für den Phasenübergang von Phase 1 nach Phase 2 benötigte Wärme}{$\Delta S$ Entropieänderung des Phasenübergangs}
|
||||
\eq{Q_\text{L} = T \Delta S}
|
||||
\end{formula}
|
||||
\begin{formula}{clausis_clapeyron}
|
||||
\desc{Clausius-Clapyeron equation}{Slope of the coexistence curve}{$\Delta V$ Volume change of the phase transition}
|
||||
\desc[german]{Clausius-Clapeyron Gleichung}{Steigung der Phasengrenzlinie}{$\Delta V$ Volumenänderung des Phasenübergangs}
|
||||
\eq{\odv{p}{T} = \frac{Q_\text{L}}{T\Delta V}}
|
||||
\end{formula}
|
||||
\begin{formula}{coexistence}
|
||||
\desc{Phase transition}{At the coexistence curve}{}
|
||||
\desc[german]{Phasenübergang}{Im Koexistenzbereich}{}
|
||||
\eq{G_1 = G_2 \\ \shortintertext{\GT{and_therefore}} \mu_1 = \mu_2}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gibbs_phase_rule}
|
||||
\desc{Gibbs rule / Phase rule}{}{$c$ \#components, $f$ \#degrees of freedom, $p$ \#phases}
|
||||
\desc[german]{Gibbsche Phasenregel}{}{$c$ \#Komponenten, $f$ \#Freiheitsgrade, $p$ \#Phasen}
|
||||
\eq{f = c - p + 2}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Osmosis}
|
||||
\ger{Osmose}
|
||||
]{osmosis}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
Osmosis is the spontaneous net movement or diffusion of solvent molecules
|
||||
through a selectively-permeable membrane, which allows through the solvent molecules, but not the solute molecules.
|
||||
The direction of the diffusion is from a region of high water potential
|
||||
(region of lower solute concentration) to a region of low water potential
|
||||
(region of higher solute concentration), in the direction that tends to equalize the solute concentrations on the two sides.
|
||||
}
|
||||
\ger{
|
||||
Osmosis ist die spontane Passage oder Diffusion Lösungsmittelmolekülen durch eine semi-permeable Membran
|
||||
die für das Lösungsmittel, jedoch nicht die darin gelösten Stoffe durchlässig ist.
|
||||
Die Richtung der Diffusion ist vom Gebiet mit hohem chemischen Potential (niedrigere Konzentration des gelösten Stoffes)
|
||||
in das mit niedrigem chemischem Potential (höherere Konzentraion des gelösten Stoffes), sodass die Konzentration des gelösten Stoffes ausgeglichen wird.
|
||||
}
|
||||
\end{ttext}
|
||||
\begin{formula}{osmosis}
|
||||
\desc{Osmotic pressure}{}{$N_c$ \#dissolved particles}
|
||||
\desc[german]{Osmotischer Druck / Van-\'t-hoffsches Gesetz}{}{$N_c$ \#gelöster Teilchen}
|
||||
\eq{p_\text{osm} = \kB T \frac{N_c}{V}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Material properties}
|
||||
\ger{Materialeigenschaften}
|
||||
]{}
|
||||
\begin{formula}{heat_cap}
|
||||
\desc{Heat capacity}{}{$Q$ heat}
|
||||
\desc[german]{Wärmekapazität}{}{$Q$ Wärme}
|
||||
\eq{c = \frac{Q}{\Delta T}}
|
||||
\end{formula}
|
||||
\begin{formula}{heat_cap_V}
|
||||
\desc{Isochoric heat capacity}{}{$U$ internal energy}
|
||||
\desc[german]{Isochore Wärmekapazität}{}{$U$ innere Energie}
|
||||
\eq{c_v = \pdv{Q}{T}_V = \pdv{U}{T}_V}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{heat_cap_p}
|
||||
\desc{Isobaric heat capacity}{}{$H$ enthalpy}
|
||||
\desc[german]{Isobare Wärmekapazität}{}{$H$ Enthalpie}
|
||||
\eq{c_p = \pdv{Q}{T}_P = \pdv{H}{T}_P}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bulk_modules}
|
||||
\desc{Bulk modules}{}{$p$ pressure, $V$ initial volume}
|
||||
\desc[german]{Kompressionsmodul}{}{$p$ Druck, $V$ Anfangsvolumen}
|
||||
\eq{K = -V \odv{p}{V} }
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{compressibility}
|
||||
\desc{Compressibility}{}{}
|
||||
\desc[german]{Kompressibilität}{}{}
|
||||
\eq{\kappa = -\frac{1}{V} \pdv{V}{p} }
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{compressibility_T}
|
||||
\desc{Isothermal compressibility}{}{}
|
||||
\desc[german]{Isotherme Kompressibilität}{}{}
|
||||
\eq{\kappa_T = -\frac{1}{V} \pdv{V}{p}_{T} = \frac{1}{K}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{compressibility_S}
|
||||
\desc{Adiabatic compressibility}{}{}
|
||||
\desc[german]{Adiabatische Kompressibilität}{}{}
|
||||
\eq{\kappa_S = -\frac{1}{V} \pdv{V}{p}_{S}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{therm_expansion}
|
||||
\desc{Thermal expansion coefficient}{}{}
|
||||
\desc[german]{Thermaler Ausdehnungskoeffizient}{}{}
|
||||
\eq{\alpha = \frac{1}{V} \pdv{V}{T}_{p,N}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Laws of thermodynamics}
|
||||
\ger{Hauptsätze der Thermodynamik}
|
||||
]{laws}
|
||||
|
||||
\Subsection[
|
||||
\eng{Zeroeth law}
|
||||
\ger{Nullter Hauptsatz}
|
||||
]{law0}
|
||||
\begin{ttext}
|
||||
\eng{If two systems are each in thermal equilibrium with a third, they are also in thermal equilibrium with each other.}
|
||||
\ger{Wenn sich zwei Siesteme jeweils im thermischen Gleichgewicht mit einem dritten befinden, befinden sie sich auch untereinander im thermischen Gleichgewicht.}
|
||||
\end{ttext}
|
||||
|
||||
\Eng[teq]{th. eq.}
|
||||
\Ger[teq]{th. GGW.}
|
||||
\def\ggwarrow{\overset{\GT{teq}}{\leftrightarrow}}
|
||||
\begin{equation}
|
||||
\label{eq:\fqname}
|
||||
A \ggwarrow C \quad\wedge\quad B \ggwarrow C \quad\Rightarrow\quad A \ggwarrow B
|
||||
\end{equation}
|
||||
|
||||
\Subsection[
|
||||
\eng{First law}
|
||||
\ger{Erster Hauptsatz}
|
||||
]{law1}
|
||||
\begin{ttext}
|
||||
\eng{In a process without transfer of matter, the change in internal energy, $\Delta U$, of a thermodynamic system is equal to the energy gained as heat, $Q$, less the thermodynamic work, W, done by the system on its surroundings.}
|
||||
\ger{In einem abgeschlossenem System ist die Änderung der inneren Energie $U$ gleich der gewonnenen Wärme $Q$ minus der vom System an der Umgebung verrichteten Arbeit $W$.}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{internal_energy}
|
||||
\desc{Internal energy change}{}{}
|
||||
\desc[german]{Änderung der inneren Energie}{}{}
|
||||
\eq{
|
||||
\Delta U &= \delta Q - \delta W \\
|
||||
\d U &= T \d S - p \d V
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Second law}
|
||||
\ger{Zweiter Hauptsatz}
|
||||
]{law2}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
\textbf{Clausius}: Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.\\
|
||||
\textbf{Kelvin}: It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.
|
||||
}
|
||||
\ger{
|
||||
\textbf{Clausius}: Es gibt keine Zustansänderung, deren einziges Ergebnis die Übertragung von Wärme von einem Körper niederer Temperatur auf einen Körper höherer Temperatur ist.\\
|
||||
\textbf{Kelvin}: Es ist unmöglich, eine periodisch arbeitende Maschine zu konstruieren, die weiter nichts bewirkt als Hebung einer Last und Abkühlung eines Wärmereservoirs.
|
||||
}
|
||||
\end{ttext}
|
||||
\Subsection[
|
||||
\eng{Third law}
|
||||
\ger{Dritter Hauptsatz}
|
||||
]{law3}
|
||||
\begin{ttext}
|
||||
\eng{It is impussible to cool a system to absolute zero.}
|
||||
\ger{Es ist unmöglich, ein System bis zum absoluten Nullpunkt abzukühlen.}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{3d_law}
|
||||
\desc{Entropy density}{}{$s = \frac{S}{N}$}
|
||||
\desc[german]{Entropiedichte}{}{$s = \frac{S}{N}$}
|
||||
\eq{
|
||||
\lim_{T\to 0} s(T) &= 0 \\
|
||||
\shortintertext{\GT{and_therefore_also}} \\
|
||||
\lim_{T\to 0} c_V &= 0
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Ensembles}
|
||||
\ger{Ensembles}
|
||||
]{ensembles}
|
||||
|
||||
|
||||
|
||||
\begin{table}[H]
|
||||
\centering
|
||||
\caption{caption}
|
||||
\label{tab:\fqname}
|
||||
|
||||
\begin{tabular}{l|c|c|c}
|
||||
& \gt{mk} & \gt{k} & \gt{gk} \\ \hline
|
||||
\GT{variables} & $E$, $V,$ $N$ & $T$, $V$, $N$ & $T$, $V$, $\mu$ \\ \hline
|
||||
\GT{partition_sum} & $\Omega = \sum_n 1$ & $Z = \sum_n \e^{-\beta E_n}$ & $Z_\text{g} = \sum_{n} \e^{-\beta(E_n - \mu N_n)}$ \\ \hline
|
||||
\GT{probability} & $p_n = \frac{1}{\Omega}$ & $p_n = \frac{\e^{-\beta E_n}}{Z}$ & $p_n = \frac{\e^{-\beta (E_n - \mu N_n}}{Z_\text{g}}$ \\ \hline
|
||||
\GT{td_pot} & $S = \kB\ln\Omega$ & $F = - \kB T \ln Z$ & $ \Phi = - \kB T \ln Z$ \\ \hline
|
||||
\GT{pressure} & $p = T \pdv{S}{V}_{E,N}$ &$p = -\pdv{F}{V}_{T,N}$ & $p = -\pdv{\Phi}{V}_{T,\mu} = -\frac{\Phi}{V}$ \\ \hline
|
||||
\GT{entropy} & $S = \kB = \ln\Omega$ & $S = -\pdv{F}{T}_{V,N}$ & $S = -\pdv{\Phi}{T}_{V,\mu}$ \\ \hline
|
||||
\end{tabular}
|
||||
\end{table}
|
||||
|
||||
\begin{formula}{ergodic_hypo}
|
||||
\desc{Ergodic hypothesis}{Over a long periode of time, all accessible microstates in the phase space are equiprobable}{$A$ Observable}
|
||||
\desc[german]{Ergodenhypothese}{Innerhalb einer langen Zeitspanne sind alle energetisch erreichbaren Mikrozustände im Phasenraum gleich wahrscheinlich}{$A$ Messgröße}
|
||||
\eq{\braket{A}_\text{\GT{time}} = \braket{A}_\text{\GT{ensemble}}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Potentials}
|
||||
\ger{Potentiale}
|
||||
]{pots}
|
||||
\begin{formula}{internal_energy}
|
||||
\desc{Internal energy}{}{}
|
||||
\desc[german]{Innere Energie}{}{}
|
||||
\eq{\d U(S,V,N) = T\d S -p\d V + \mu\d N}
|
||||
\end{formula}
|
||||
\begin{formula}{enthalpy}
|
||||
\desc{Enthalpy}{}{}
|
||||
\desc[german]{Enthalpie}{}{}
|
||||
\eq{\d H(S,p,N) = T\d S +V\d p + \mu\d N}
|
||||
\end{formula}
|
||||
\begin{formula}{gibbs_energy}
|
||||
\desc{Gibbs energy}{}{}
|
||||
\desc[german]{Gibbsche Energie}{}{}
|
||||
\eq{\d G(T,p,N) = -S\d T + V\d p + \mu\d N}
|
||||
\end{formula}
|
||||
\begin{formula}{free_energy}
|
||||
\desc{Free energy / Helmholtz energy }{}{}
|
||||
\desc[german]{Freie Energie / Helmholtz Energie}{}{}
|
||||
\eq{\d F(T,V,N) = -S\d T -p\d V + \mu\d N}
|
||||
\end{formula}
|
||||
\begin{formula}{grand_canon_pot}
|
||||
\desc{Grand canonical potential}{}{}
|
||||
\desc[german]{Großkanonisches Potential}{}{}
|
||||
\eq{\d \Phi(T,V,\mu) = -S\d T -p\d V - N\d\mu}
|
||||
\end{formula}
|
||||
|
||||
\TODO{Maxwell Relationen, TD Quadrat}
|
||||
|
||||
\Section[
|
||||
\eng{Ideal gas}
|
||||
\ger{Ideales Gas}
|
||||
]{id_gas}
|
||||
\begin{ttext}
|
||||
\eng{The ideal gas consists of non-interacting, undifferentiable particles.}
|
||||
\ger{Das ideale Gas besteht aus nicht-wechselwirkenden, ununterscheidbaren Teilchen.}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{phase_space_vol}
|
||||
\desc{Phase space volume}{$3N$ sphere}{$N$ \#particles, $h^{3N}$ volume of a microstate, $N!$ particles are undifferentiable}
|
||||
\desc[german]{}{$3N$ Kugel}{$N$ \#Teilchen, $h^{3N}$ Volumen eines Mikrozustandes, $N!$ Teilchen sind ununterscheidbar}
|
||||
\eq{
|
||||
\Omega(E) &= \int_V\d^3q_1 \sdots \int_V\d^3q_N \int \d^3p_1 \sdots \int\d^3p_N \frac{1}{N!\,h^{3N}} \Theta\left(E - \sum_{i} \frac{\vec{p_i}^2}{2m}\right) \\
|
||||
&= \left(\frac{V}{N}\right)^N \left(\frac{4\pi m E}{3 h^2 N}\right)^{\frac{3N}{2}} \e^\frac{5N}{2}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{entropy}
|
||||
\desc{Entropy}{}{}
|
||||
\desc[german]{Entropie}{}{}
|
||||
\eq{
|
||||
S = \frac{5}{2} N\kB + N\kB\ln\left(\frac{V}{N}\left(\frac{2\pi m E}{3 h^2 N}\right)^{\frac{3}{2}}\right)
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{id_gas_eq}
|
||||
\desc{Ideal gas equation}{}{}
|
||||
\desc[german]{Ideale Gasgleichung}{Thermische Zustandsgleichung idealer Gase}{}
|
||||
\eq{pV &= nRT \\ &= N\kB T}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{equation_of_state}
|
||||
\desc{Equation of state}{}{}
|
||||
\desc[german]{Kalorische Zustangsgleichung}{}{}
|
||||
\eq{U = \frac{3}{2} N\kB T}
|
||||
\end{formula}
|
||||
|
||||
% \Subsubsection[
|
||||
% \eng{Equipartitiontheorem}
|
||||
% \ger{Äquipartitionstheorem}
|
||||
% ]{equipart}
|
||||
\begin{formula}{equipart}
|
||||
\desc{Equipartitiontheorem}{Each degree of freedom contributes $U_\text{D}$ (for classical particle systems)}{}
|
||||
\desc[german]{Äquipartitionstheorem}{Jedem Freiheitsgrad steht die Energie $U_\text{D}$ zur Verfügung}{}
|
||||
\eq{U_\text{D} = \frac{1}{2} \kB T}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{maxwell_velocity}
|
||||
\desc{Maxwell velocity distribution}{See also \ref{sec:pt:distributions::maxwell-boltzmann}}{}
|
||||
\desc[german]{Maxwellsche Geschwindigkeitsverteilung}{Siehe auch \ref{sec:pt:distributions::maxwell-boltzmann}}{}
|
||||
\eq{w(v) \d v = 4\pi \left(\frac{\beta m}{2\pi}\right)^\frac{3}{2} v^2 \e^{-\frac{\beta m v^2}{2}} \d v}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{avg_velocity}
|
||||
\desc{Average quadratic velocity}{per particle in a 3D gas}{}
|
||||
\desc[german]{Mittlere quadratosche Geschwindigkeit}{pro Teilchen im 3D-Gas}{}
|
||||
\eq{\braket{v^2} = \int_0^\infty \d v\,v^2 w(v) = \frac{3\kB T}{m}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Molecule gas}
|
||||
\ger{Molekülgas}
|
||||
]{molecule_gas}
|
||||
|
||||
\begin{formula}{desc}
|
||||
\desc{Molecule gas}{2 particles of mass $M$ connected by a ``spring'' with distance $L$}{}
|
||||
\desc[german]{Molekülgas}{2 Teilchen der Masse $M$ sind verbunden durch eine ``Feder'' mit Länge $L$}{}
|
||||
\content{
|
||||
% \begin{figure}[h]
|
||||
\centering
|
||||
\tikzstyle{spring}=[thick,decorate,decoration={coil,aspect=0.8,amplitude=5,pre length=0.1cm,post length=0.1cm,segment length=10}]
|
||||
\begin{tikzpicture}
|
||||
\def\radius{0.5}
|
||||
\coordinate (left) at (-3, 0);
|
||||
\coordinate (right) at (3, 0);
|
||||
\draw (left) circle (\radius);
|
||||
\draw[spring] ($(left) + (\radius,0)$) -- ($(right) - (\radius,0)$);
|
||||
\draw (right) circle (\radius);
|
||||
\end{tikzpicture}
|
||||
% \end{figure}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{translation}
|
||||
\desc{Translation}{}{$n_i \in \N_0$, $i=x,\,y,\,z$}
|
||||
\desc[german]{Translation}{}{$n_i \in \N_0$, $i=x,\,y,\,z$}
|
||||
\eq{p_i &= \frac{2\pi\hbar}{L}n_i \\
|
||||
E_\text{kin} &= \frac{\vec{p}_r^2}{2M}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{vibration}
|
||||
\desc{Vibration}{}{$n \in \N_0$}
|
||||
\desc[german]{Schwingungen}{}{$n \in \N_0$}
|
||||
\eq{E_\text{vib} = \hbar \omega \left(n+\frac{1}{2}\right)}
|
||||
\end{formula}
|
||||
\begin{formula}{rotation}
|
||||
\desc{Rotation}{}{$j\in \N_0$}
|
||||
\desc[german]{Rotation}{}{$j\in \N_0$}
|
||||
\eq{E_\text{rot} = \frac{\hbar^2}{2I}j(j+1)}
|
||||
\end{formula}
|
||||
\TODO{Diagram für verschiedene Temperaturen, Weiler Skript p.83}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Real gas}
|
||||
\ger{Reales Gas}
|
||||
]{real_gas}
|
||||
|
||||
\Subsection[
|
||||
\eng{Virial expansion}
|
||||
\ger{Virialentwicklung}
|
||||
]{virial}
|
||||
\begin{ttext}
|
||||
\eng{Expansion of the pressure $p$ in a power series of the density $\rho$.}
|
||||
\ger{Entwicklung desw Drucks $p$ in eine Potenzreihe der Dichte $\rho$.}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{series}
|
||||
\desc{Virial expansion}{The 2\ts{nd} and 3\ts{d} virial coefficient are tabelated for many substances}{$B$ and $C$ 2\ts{nd} and 3\ts{d} virial coefficient, $\rho = \frac{N}{V}$}
|
||||
\desc[german]{Virialentwicklung}{Der zweite und dritte Virialkoeffizient ist für viele Substanzen tabelliert}{$B$ und $C$ 2. und 3. Virialkoeffizient, $\rho = \frac{N}{V}$}
|
||||
\eq{p = \kB T \rho\,[1 + B(T) \rho + C(T) \rho^2 + \dots]}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{mayer_function}
|
||||
\desc{Mayer function}{}{$V(i,j)$ pair potential}
|
||||
\desc[german]{Mayer-Funktion}{}{$V(i,j)$ Paarpotential}
|
||||
\eq{f(\vec{r}) = \e^{-\beta V(i,j)} - 1}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{second_coefficient}
|
||||
\desc{Second virial coefficient}{Depends on pair potential between two molecules}{}
|
||||
\desc[german]{Zweiter Virialkoeffizient}{Hängt vom Paarpotential zweier Moleküle ab}{}
|
||||
\eq{B = -\frac{1}{2} \int_V \d^3 \vec{r} f(\vec{r})}
|
||||
% b - \frac{a}{\kB T}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{lennard_jones}
|
||||
\desc{Lennard-Jones potential}{Potential between two molecules. Attractive for $r > \sigma$, repulsive for $r < \sigma$}{}
|
||||
\desc[german]{Lennard-Jones-Potential}{Potential zwischen zwei Molekülen. Attraktiv für $r > \sigma$, repulsiv für $r < \sigma$}{}
|
||||
\figeq{img/potential_lennard_jones.pdf}{V(r) = 4\epsilon \left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right]}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Van der Waals equation}
|
||||
\ger{Van der Waals Gleichung}
|
||||
]{vdw}
|
||||
\begin{ttext}
|
||||
\eng{Assumes a hard-core potential with a weak attraction.}
|
||||
\ger{Annahme eines Harte-Kugeln Potentials mit einer schwachen Anziehung}
|
||||
\end{ttext}
|
||||
\begin{formula}{partition_sum}
|
||||
\desc{Partition sum}{}{$a$ internal pressure}
|
||||
\desc[german]{Zustandssumme}{}{$a$ Kohäsionsdruck}
|
||||
\eq{Z_N = \frac{(V-V_0)^N}{\lambda^{3N}N!} \e^{\frac{\beta N^2 a}{V}}}
|
||||
\end{formula}
|
||||
\begin{formula}{equation}
|
||||
\desc{Van der Waals equation}{}{$b$ co-volume?}
|
||||
\desc[german]{Van der Waals-Gleichung}{}{$b$ Kovolumen}
|
||||
\eq{p = \frac{N \kB T}{V-b} - \frac{N^2 a}{V^2}}
|
||||
\end{formula}
|
||||
\TODO{sometimes N is included in a, b}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Ideal quantum gas}
|
||||
\ger{Ideales Quantengas}
|
||||
]{id_qgas}
|
||||
\def\bosfer{$\pm$: {$\text{bos} \atop \text{fer}$}}
|
||||
|
||||
\begin{formula}{fugacity}
|
||||
\desc{Fugacity}{}{}
|
||||
\desc[german]{Fugazität}{}{}
|
||||
\eq{z = \e^{\mu\beta} = \e^\frac{\mu}{\kB T}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{occupation}
|
||||
\desc{Occupation number}{}{$r$ states}
|
||||
\desc[german]{Besetzungszahl}{}{$r$ Zustände}
|
||||
\eq{\sum_{r} n_r = N}
|
||||
\end{formula}
|
||||
\begin{formula}{undiff_particles}
|
||||
\desc{Undifferentiable particles}{}{$p_i$ state}
|
||||
\desc[german]{Ununterscheidbare Teilchen}{}{$p_i$ Zustand}
|
||||
\eq{\ket{p_1,p_2,\dots,p_N} = \ket{p_1}\ket{p_2}\dots \ket{p_N}}
|
||||
\end{formula}
|
||||
\begin{formula}{parity}
|
||||
\desc{Applying the parity operator}{yields a \textit{symmetric} (Bosons) and a \textit{antisymmetic} (Fermions) solution}{$\hat{P}_{12}$ parity operator swaps $1$ and $2$, \bosfer}
|
||||
\desc[german]{Anwenden des Paritätsoperators}{gibt eine \textit{symmetrische} (Bosonen) und eine \textit{antisymmetrische} (Fermionen) Lösung}{$\hat{O}_{12}$ Paritätsoperator tauscht $1$ und $2$, \bosfer}
|
||||
\eq{\hat{P}_{12} \psi(p_i(\vec{r}_1),\,p_j(\vec{r}_2)) = \pm \psi(p_i(\vec{r}_1),\,p_j(\vec{r}_2))}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{spin_degeneracy_factor}
|
||||
\desc{Spin degeneracy factor}{}{$s$ spin}
|
||||
\desc[german]{Spinentartungsfaktor}{}{$s$ Spin}
|
||||
\eq{g_s = 2s+1}
|
||||
\end{formula}
|
||||
\begin{formula}{dos}
|
||||
\desc{Density of states}{}{$g_s$ \fqEqRef{td:id_qgas:spin_degeneracy_factor}}
|
||||
\desc[german]{Zustandsdichte}{}{$g_s$ \fqEqRef{td:id_qgas:spin_degeneracy_factor}}
|
||||
\eq{g(\epsilon) = g_s \frac{V}{4\pi} \left(\frac{2m}{\hbar^2}\right)^\frac{3}{2} \sqrt{\epsilon}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{occupation_number_per_e}
|
||||
\desc{Occupation number per energy}{}{\fqEqRef{td:id_qgas:dos}, \bosfer}
|
||||
\desc[german]{Besetzungszahl pro Energie}{}{\fqEqRef{td:id_qgas:dos}, \bosfer}
|
||||
\eq{n(\epsilon)\, \d\epsilon &= \frac{g(\epsilon)}{\e^{\beta(\epsilon - \mu)} \mp 1}\,\d\epsilon}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{occupation_number}
|
||||
\desc{Occupation number}{}{\bosfer}
|
||||
\desc[german]{Besetzungszahl}{}{\bosfer}
|
||||
\figeq{img/td_id_qgas_distributions.pdf}{%
|
||||
\braket{n(\epsilon)} &= \frac{1}{\e^{\beta(\epsilon - \mu)} \mp 1} \\
|
||||
\shortintertext{\GT{for} $\epsilon - \mu \gg \kB T$}
|
||||
&= \frac{1}{\e^{\beta(\epsilon - \mu)}}
|
||||
}
|
||||
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{particle_number}
|
||||
\desc{Number of particles}{}{}
|
||||
\desc[german]{Teilchenzahl}{}{}
|
||||
\eq{\braket{N} = \int_0^\infty n(\epsilon) \d\epsilon}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{energy}
|
||||
\desc{Energy}{Equal to the classical ideal gas}{}
|
||||
\desc[german]{Energie}{Gleich wie beim klassischen idealen Gas}{}
|
||||
\eq{\braket{E} = \int_0^\infty \epsilon n(\epsilon)\,\d\epsilon = \frac{3}{2} pV}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{equation_of_state}
|
||||
\desc{Equation of state}{Bosons: decreased pressure, they like to cluster\\Fermions: increased pressure because of the Pauli principle}{\bosfer, $v = \frac{V}{N}$ specific volume}
|
||||
\desc[german]{Zustandsgleichung}{Bosonen: verringerter Druck da sie clustern\\Fermionen: erhöhter Druck durch das Pauli-Prinzip}{\bosfer, $v = \frac{V}{N}$ spezifisches Volumen}
|
||||
\eq{
|
||||
pV &= \kB T \ln Z_g \\
|
||||
\shortintertext{\GT{after} \GT{td:real_gas:virial}}
|
||||
&= N \kB T \left[1 \mp \frac{\lambda^3}{2^{5/2} g v} + \Order\left(\left(\frac{\lambda^3}{v}\right)^2\right)\right]
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{relevance}
|
||||
\desc{Relevance of qm. corrections}{Corrections become relevant when the particle distance is in the order of the thermal wavelength}{}
|
||||
\desc[german]{Relevanz der qm. Korrekturen}{Korrekturen werden relevant, wenn der Teilchenabstand in der Größenordnung der thermischen Wellenlänge ist}{}
|
||||
\eq{\left(\frac{V}{N}\right)^\frac{1}{3} \sim \frac{\lambda}{g_s^\frac{1}{3}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{generalized_zeta}
|
||||
\desc{Generalized zeta function}{}{}
|
||||
\desc[german]{Verallgemeinerte Zeta-Funktion}{}{}
|
||||
\eq{\left \begin{array}{l}g_\nu(z)\\f_\nu(z)\end{array}\right\} \coloneq \frac{1}{\Gamma(\nu)} \int_0^\infty \d x\, \frac{x^{\nu-1}}{\e^x z^{-1} \mp 1}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Bosons}
|
||||
\ger{Bosonen}
|
||||
]{bos}
|
||||
\begin{formula}{partition_sum}
|
||||
\desc{Partition sum}{}{$p \in\N_0$}
|
||||
\desc[german]{Zustandssumme}{}{$p \in\N_0$}
|
||||
\eq{Z_\text{g} = \prod_{p} \frac{1}{1-\e^{-\beta(\epsilon_p - \mu)}}}
|
||||
\end{formula}
|
||||
\begin{formula}{occupation}
|
||||
\desc{Occupation number}{Bose-Einstein distribution}{}
|
||||
\desc[german]{Besetzungszahl}{Bose-Einstein Verteilung}{}
|
||||
\eq{\braket{n_p} = \frac{1}{\e^{\beta(\epsilon-\mu)}-1}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Fermions}
|
||||
\ger{Fermionen}
|
||||
]{fer}
|
||||
\begin{formula}{partition_sum}
|
||||
\desc{Partition sum}{}{$p = 0,\,1$}
|
||||
\desc[german]{Zustandssumme}{}{$p = 0,\,1$}
|
||||
\eq{Z_\text{g} = \prod_{p} \left(1+\e^{-\beta(\epsilon_p - \mu)\right)}}
|
||||
\end{formula}
|
||||
\begin{formula}{occupation}
|
||||
\desc{Occupation number}{Fermi-Dirac distribution. At $T=0$ \textit{Fermi edge} at $\epsilon=\mu$}{}
|
||||
\desc[german]{Besetzungszahl}{Fermi-Dirac Verteilung}{Bei $T=0$ \textit{Fermi-Kante} bei $\epsilon=\mu$}
|
||||
\figeq{img/td_fermi_occupation.pdf}{\braket{n_p} = \frac{1}{\e^{\beta(\epsilon-\mu)}+1}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{slater_determinant}
|
||||
\desc{Slater determinant}{}{}
|
||||
\desc[german]{Slater-Determinante}{}{}
|
||||
\eq{
|
||||
\psi(\vecr_1,\vecr_2,\dots,\vecr_N) = \frac{1}{\sqrt{N!}}
|
||||
\begin{vmatrix}
|
||||
p_1(\vecr_1) & p_2(\vecr_1) & \dots & p_N(\vecr_1) \\
|
||||
p_1(\vecr_2) & p_2(\vecr_2) & \dots & p_N(\vecr_2) \\
|
||||
\vdots & \vdots & \ddots & \vdots \\
|
||||
p_1(\vecr_N) & p_2(\vecr_N) & \dots & p_N(\vecr_N) \\
|
||||
\end{vmatrix}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{fermi_energy}
|
||||
\desc{Fermi energy}{}{}
|
||||
\desc[german]{Fermienergie}{}{}
|
||||
\eq{\epsilon_\text{F} \coloneq \mu(T = 0)}
|
||||
\end{formula}
|
||||
\begin{formula}{fermi_temperature}
|
||||
\desc{Fermi temperature}{}{}
|
||||
\desc[german]{Fermi Temperatur}{}{}
|
||||
\eq{T_\text{F} \coloneq \frac{\epsilon_\text{F}}{\kB}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{fermi_impulse}
|
||||
\desc{Fermi impulse}{Radius of the \textit{Fermi sphere} in impulse space. States with $p_\text{F}$ are in the \textit{Fermi surface}}{}
|
||||
\desc[german]{Fermi-Impuls}{Radius der \textit{Fermi-Kugel} im Impulsraum. Zustände mit $P_\text{F}$ sind auf der \textit{Fermi-Fläche}}{}
|
||||
\eq{p_\text{F} = \hbar k_\text{F} = (2mE_\text{F})^\frac{1}{2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{specific_density}
|
||||
\desc{Specific density}{}{$f$ \fqEqRef{td:id_qgas:generalized_zeta}, $g$ degeneracy factor, $z$ \fqEqRef{td:id_qgas:fugacity}}
|
||||
\desc[german]{Spezifische Dichte}{}{$f$ \fqEqRef{td:id_qgas:generalized_zeta}, $g$ Entartungsfaktor, $z$ \fqEqRef{td:id_qgas:fugacity}}
|
||||
\eq{v = \frac{N}{V} = \frac{g}{\lambda^3}f_{3/2}(z)}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Strong degeneracy}
|
||||
\ger{Starke Entartung}
|
||||
]{degenerate}
|
||||
\eng[low_temps]{for low temperatures $T \ll T_\text{F}$}
|
||||
\ger[low_temps]{für geringe Temperaturen $T\ll T_\text{F}$}
|
||||
|
||||
\begin{formula}{sommerfeld}
|
||||
\desc{Sommerfeld expansion}{\gt{low_temps}}{}
|
||||
\desc[german]{Sommerfeld-Entwicklung}{\gt{low_temps}}{}
|
||||
\eq{f_\nu(z) = \frac{(\ln z)^\nu}{\Gamma(\nu+1)} \left(1+\frac{\pi^6}{6}\frac{\nu(\nu-1)}{(\ln z)^2} + \dots\right)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{energy_density}
|
||||
\desc{Energy density}{}{}
|
||||
\desc[german]{Energiedichte}{}{}
|
||||
\eq{
|
||||
\frac{E}{V} &= \frac{3}{2}\frac{g}{\lambda^3} \kB T f_{5/2}(z) \\
|
||||
\shortintertext{\GT{td:id_qgas:fer:degenerate:sommerfeld}}
|
||||
&\approx \frac{3}{5} \frac{N}{V} E_\text{F} \left(1+\frac{5\pi^2}{12}\left(\frac{\kB T}{E_\text{F}}\right)^2 \right)
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{heat_cap}
|
||||
\desc{Heat capacity}{\gt{low_temps}}{differs from \fqEqRef{td:TODO:petit_dulong}}
|
||||
\desc[german]{Wärmecapacity}{\gt{low_temps}}{weicht ab vom \fqEqRef{td:TODO:petit_dulong}}
|
||||
\figeq{img/td_fermi_heat_capacity.pdf}{C_V = \pdv{E}{T}_V = N\kB \frac{\pi}{2} \left(\frac{T}{T_\text{F}}\right)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\TODO{Entartung und Sommerfeld}
|
||||
\TODO{DULONG-PETIT Gesetz}
|
||||
|
||||
118
src/svgs/bravais/aP.svg
Normal file
118
src/svgs/bravais/aP.svg
Normal file
@ -0,0 +1,118 @@
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After Width: | Height: | Size: 14 KiB |
6
src/svgs/bravais/fix.sh
Normal file
6
src/svgs/bravais/fix.sh
Normal file
@ -0,0 +1,6 @@
|
||||
mkdir -p out
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||||
for file in bravais/*; do
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||||
link=https:$(grep "Original file" "$file" | grep -Po '(?<=href=")[^"]+')
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wget "$link" -O "$file"
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sleep 1
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done
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344
src/svgs/bravais/hP.svg
Normal file
344
src/svgs/bravais/hP.svg
Normal file
@ -0,0 +1,344 @@
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7
src/svgs/convertToPdf.sh
Executable file
7
src/svgs/convertToPdf.sh
Executable file
@ -0,0 +1,7 @@
|
||||
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for d in $(find . -type d); do
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|
||||
for file in $(find . -type f -name '*.svg'); do
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208
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208
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84
src/topo.tex
84
src/topo.tex
@ -1,3 +1,85 @@
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\Part{Topo}
|
||||
\Section{berry_phase}
|
||||
\Section[
|
||||
\eng{Berry phase / Geometric phase}
|
||||
\ger{Berry-Phase / Geometrische Phase}
|
||||
]{berry_phase}
|
||||
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\begin{ttext}[desc]
|
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\eng{
|
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While adiabatically traversing a closed through the parameter space $R(t)$, the wave function of a systems
|
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may pick up an additional phase $\gamma$.\\
|
||||
If $\vec{R}(t)$ varies adiabatically (slowly) and the system is initially in eigenstate $\ket{n}$,
|
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it will stay in an Eigenstate throughout the process (quantum adiabtic theorem).
|
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}
|
||||
\ger{
|
||||
Beim adiabatischem Durchlauf eines geschlossenen Weges durch den Parameterraum $R(t)$ kann die Wellenfunktion eines Systems
|
||||
eine zusätzliche Phase $\gamma$ erhalten.\\
|
||||
Wenn $\vec{R}(t)$ adiabatisch (langsam) variiert und das System anfangs im Eigenzustand $\ket{n}$ ist,
|
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bleibt das System während dem Prozess in einem Eigenzustand (Adiabatisches Theorem der Quantenmechanik).
|
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}
|
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\end{ttext}
|
||||
\Eng[dynamic_phase]{Dynamical Phase}
|
||||
\Eng[berry_phase]{Berry Phase}
|
||||
\Ger[dynamic_phase]{Dynamische Phase}
|
||||
\Ger[berry_phase]{Berry Phase}
|
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\begin{formula}{schroedinger_equation}
|
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\desc{Schrödinger equation}{}{}
|
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\desc[german]{Schrödinger Gleichung}{}{}
|
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\eq{H(\vec{R}(t)) \ket{n(\vec{R}(t))} = \epsilon(\vec{R}(t)) \ket{n(\vec{R}(t))}}
|
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\end{formula}
|
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\begin{formula}{wavefunction}
|
||||
\desc{Wave function}{After full adiabtic loop in $\vec{R}$}{}
|
||||
\desc[german]{Wellenfunktion}{Nach vollem adiabtischem Umlauf in $\vec{R}$}{}
|
||||
\eq{\ket{\psi_n(t)} = \underbrace{\e^{i\gamma_n(t)}}_\text{\GT{berry_phase}}
|
||||
\underbrace{\e^{\frac{-i}{\hbar} \int^r \epsilon_n(\vec{R}(t`))\d t}}_\text{\GT{dynamic_phase}} \ket{n(\vec{R}(t))}
|
||||
}
|
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\end{formula}
|
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\begin{formula}{berry_connection}
|
||||
\desc{Berry connection}{}{}
|
||||
\desc[german]{Berry connection}{}{}
|
||||
\eq{A_n(\vec{R}) = i\braket{\psi | \nabla_R | \psi}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{berry_curve}
|
||||
\desc{Berry curvature}{Gauge invariant}{}
|
||||
\desc[german]{Berry-Krümmung}{Eichinvariant}{}
|
||||
\eq{\vec{\Omega}_n = \Grad_R \times A_n(\vec{R})}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{berry_phase}
|
||||
\desc{Berry phase}{Gauge invariant up to $2\pi$}{}
|
||||
\desc[german]{Berry-Phase}{Eichinvariant bis auf $2\pi$}{}
|
||||
\eq{\gamma_n = \oint_C \d \vec{R} \cdot A_n(\vec{R}) = \int_S \d\vec{S} \cdot \vec{\Omega}_n(\vec{R})}
|
||||
\end{formula}
|
||||
|
||||
\begin{ttext}[chern_number_desc]
|
||||
\eng{The Berry flux through any 2D closed surface is quantized by the \textbf{Chern number}.
|
||||
If there is time-reversal symmetry, the Chern-number is 0.
|
||||
}
|
||||
\ger{Der Berry-Fluß durch eine geschlossene 2D Fl[cher is quantisiert durch die \textbf{Chernzahl}
|
||||
Bei erhaltener Zeitumkehrungssymmetrie ist die Chernzahl 0.
|
||||
}
|
||||
\end{ttext}
|
||||
\begin{formula}{chern_number}
|
||||
\desc{Chern number}{Eg. number of Berry curvature monopoles in the Brillouin zone (then $\vec{R} = \vec{k}$)}{$\vec{S}$ closed surface in $\vec{R}$-space. A \textit{Chern insulator} is a 2D insulator with $C_n \neq 0$}
|
||||
\desc[german]{Chernuzahl}{Z.B. Anzahl der Berry-Krümmungs-Monopole in der Brilouinzone (dann ist $\vec{R} = \vec{k}$). Ein \textit{Chern-Isolator} ist ein 2D Isolator mit $C_n\neq0$}{$\vec{S}$ geschlossene Fläche im $\vec{R}$-Raum}
|
||||
\eq{C_n = \frac{1}{2\pi} \oint \d \vec{S}\ \cdot \vec{\Omega}_n(\vec{R})}
|
||||
\end{formula}
|
||||
|
||||
\TODO{Hall conductance of 2D band insulator (lecture 4 revision)}
|
||||
\begin{formula}{hall_conductance}
|
||||
\desc{Hall conductance of a 2D band insulator}{}{}
|
||||
\desc[german]{Hall-Leitfähigkeit eines 2D Band-Isolators}{}{}
|
||||
\eq{\vec{\sigma}_{xy} = \sum_n \frac{e^2}{h} \int_\text{\GT{occupied}} \d^2k\, \frac{\Omega_{xy}^n}{2\pi} = \sum_n C_n \frac{e^2}{h}}
|
||||
\end{formula}
|
||||
|
||||
\begin{ttext}
|
||||
\eng{A 2D insulator with a non-zero }
|
||||
|
||||
\end{ttext}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
@ -1,8 +1,27 @@
|
||||
\ENG[angle_deg]{Degree}
|
||||
\GER[angle_deg]{Grad}
|
||||
\Eng[angle_deg]{Degree}
|
||||
\Ger[angle_deg]{Grad}
|
||||
|
||||
\ENG[angle_rad]{Radian}
|
||||
\GER[angle_rad]{Rad}
|
||||
\Eng[angle_rad]{Radian}
|
||||
\Ger[angle_rad]{Rad}
|
||||
|
||||
\Eng[see_also]{See also}
|
||||
\Ger[see_also]{Siehe auch}
|
||||
|
||||
\Eng[and_therefore]{and therefore}
|
||||
\Ger[and_therefore]{und damit}
|
||||
|
||||
\Eng[and_therefore_also]{and therefore also}
|
||||
\Ger[and_therefore_also]{und damit auch}
|
||||
|
||||
\Eng[time]{Time}
|
||||
\Ger[time]{Zeit}
|
||||
|
||||
\Eng[ensemble]{Ensemble}
|
||||
\Ger[ensemble]{Ensemble}
|
||||
|
||||
\Eng[even]{even}
|
||||
\Ger[even]{gerade}
|
||||
|
||||
\Eng[odd]{odd}
|
||||
\Ger[odd]{ungerade}
|
||||
|
||||
\ENG[see_also]{See also}
|
||||
\GER[see_also]{Siehe auch}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user