Merge branch 'master' of git+ssh://stbuehler.de:522/qm1-script
This commit is contained in:
Oliver Groß
commit
cc42cb882c
2
.gitignore
vendored
2
.gitignore
vendored
@ -4,4 +4,4 @@
|
||||
*.toc
|
||||
*.out
|
||||
theo2.pdf
|
||||
|
||||
theo2.kilepr
|
||||
|
||||
@ -6,6 +6,7 @@
|
||||
% undendlich dim. raum
|
||||
% was zum henker ist ein operator
|
||||
% impulsoperator
|
||||
%definition bedingte wahrscheinlichkeiten
|
||||
|
||||
\chapter{Notationen}
|
||||
\section{Dirac-Notation}
|
||||
|
||||
@ -298,10 +298,10 @@ $\rightarrow$ $B \ket{n,r}$ liegt im Untrraum zu $a_n$
|
||||
\end{equation}
|
||||
\item[Fall (2)] $a_n$ entartet
|
||||
\begin{equation}
|
||||
\bra{m,s} \cdot B \cdot \ket{n,r} = B_{s,r}^{(n)}
|
||||
\bra{n,s} \cdot B \cdot \ket{n,r} = B_{s,r}^{(n)}
|
||||
\end{equation}
|
||||
\begin{equation}
|
||||
B \cequiv \inlinematrix{\boxed{B^{(1)}} & & 0 \\ & \boxed{B^{(2)}} & & \\ & & \boxed{B} & \\ 0 & & & \boxed{B^{(4)}}} \rightarrow \text{kann diagonalisiert werden in jedem Kästchen}
|
||||
B \cequiv \inlinematrix{\boxed{B^{(1)}} & & 0 \\ & \boxed{B^{(2)}} & & \\ & & \boxed{B^{(3)}} & \\ 0 & & & \boxed{B^{(4)}}} \rightarrow \text{kann diagonalisiert werden in jedem Kästchen}
|
||||
\end{equation}
|
||||
Falls $B^{(n)}$ entartet, gibt es einen dritten Opertor $C$ mit $[A,C] = [B,C] = 0$.
|
||||
\end{description}
|
||||
|
||||
32
kapIV-2.tex
32
kapIV-2.tex
@ -60,20 +60,20 @@ entsprechend
|
||||
\begin{equation}
|
||||
E_0 = \hbar \omega \left( n + \frac{1}{2} \right) + \lambda \dirac{n}{x^4}{n}
|
||||
\end{equation}
|
||||
%\begin{figure}[H] \centering
|
||||
%\includegraphics{pdf/III/02-01-00.pdf}
|
||||
%\end{figure}
|
||||
\begin{figure}[H] \centering
|
||||
\includegraphics{pdf/IV/02-01-00.pdf}
|
||||
\end{figure}
|
||||
Konsequenz?
|
||||
\begin{itemize}
|
||||
\item für $\lambda$ negativ $\abs{\lambda} \ll 1$
|
||||
%\begin{figure}[H] \centering
|
||||
%\includegraphics{pdf/III/02-01-01.pdf}
|
||||
%\caption{gestrichelte Kurve entspricht $\frac{m}{2} \omega x^2 \abs{\lambda} x^4$}
|
||||
%\end{figure}
|
||||
\begin{figure}[H] \centering
|
||||
\includegraphics{pdf/IV/02-01-01.pdf}
|
||||
\caption{gestrichelte Kurve entspricht $\frac{m}{2} \omega x^2 \abs{\lambda} x^4$}
|
||||
\end{figure}
|
||||
\item für $\lambda$ positiv
|
||||
%\begin{figure}[H] \centering
|
||||
%\includegraphics{pdf/III/02-01-02.pdf}
|
||||
%\end{figure}
|
||||
\begin{figure}[H] \centering
|
||||
\includegraphics{pdf/IV/02-01-02.pdf}
|
||||
\end{figure}
|
||||
\end{itemize}
|
||||
volle Rechnung zeigt:
|
||||
\begin{equation}
|
||||
@ -89,9 +89,9 @@ mit
|
||||
\item Obige Formel wegen Energienenner nicht anwendbar bei Entartung.
|
||||
\item sehr relevant: Aufhebeung der Entartung durch Störung
|
||||
\end{itemize}
|
||||
%\begin{figure}[H] \centering
|
||||
%\includegraphics{pdf/III/02-02-00.pdf}
|
||||
%\end{figure}
|
||||
\begin{figure}[H] \centering
|
||||
\includegraphics{pdf/IV/02-02-00.pdf}
|
||||
\end{figure}
|
||||
\paragraph{Ansatz}
|
||||
\begin{align}
|
||||
\ket{a} &= \sum_{\alpha \in D} c_\alpha \ket{\alpha} + \sum_{\mu \notin D} d_\mu \ket{\mu}\\[10pt]
|
||||
@ -140,7 +140,7 @@ Störung: $H_1$ sei E'feld
|
||||
\begin{itemize}
|
||||
\item in Ordnung $\lambda$ d.h. in $O(\abs{E})$:
|
||||
\begin{equation}
|
||||
\dirac{1,0,0}{H_1}{1,0,0} = \intgru{\Phi^*_{1,0,0} (\vec{r}) \cdor e \abs{E} z \Phi_{1,0,0}(\vec{r})}{r^3} = 0
|
||||
\dirac{1,0,0}{H_1}{1,0,0} = \intgru{\Phi^*_{1,0,0} (\vec{r}) \cdot e \abs{E} \hat{z} \Phi_{1,0,0}(\vec{r})}{r^3} = 0
|
||||
\end{equation}
|
||||
\item in Ordnung $\abs{E}^2$:
|
||||
\begin{align}
|
||||
@ -179,11 +179,11 @@ diagonalisieren: $\dirac{\beta}{H_1}{\alpha}$
|
||||
\begin{enumerate}
|
||||
\item Diagonalelemente
|
||||
\begin{equation}
|
||||
\dirac{\alpha}{H_1}{\alpha} \tilde \intgru{\cos\theta\abs{\Phi_{n,l,m}}^2}{(\cos\theta)} \equiv 0
|
||||
\dirac{\alpha}{H_1}{\alpha} \sim \intgru{\cos\theta\abs{\Phi_{n,l,m}}^2}{(\cos\theta)} \equiv 0
|
||||
\end{equation}
|
||||
\item Nichtdiagonalelemente
|
||||
\begin{equation}
|
||||
\dirac{n',l',m'}{\hat{z}}{n,l,m} \tilde \intgru{e^{i(m-m')\phi}}{\phi} \tilde \krondelta{m,m'}
|
||||
\dirac{n',l',m'}{\hat{z}}{n,l,m} \sim \intgru{e^{i(m-m')\phi}}{\phi} \sim \krondelta{m,m'}
|
||||
\end{equation}
|
||||
mit
|
||||
\begin{equation}
|
||||
|
||||
192
kapIV-2.tex.backup
Normal file
192
kapIV-2.tex.backup
Normal file
@ -0,0 +1,192 @@
|
||||
\chapter{Zeitabhängige Störungstheorie}
|
||||
Für $H = H_0 + \lambda_1$ mit $H_0$ exakt lösbar und $\lambda \ll 1$, lässt sich das Problem perturativ angeben.
|
||||
\section{Nichtentarteter Fall}
|
||||
gegeben:
|
||||
\begin{equation}
|
||||
\left(H_0 - E_\alpha \right) \ket{\alpha} = 0 ~ \left[ H_0 \ket{\alpha} = E_\alpha \ket{\alpha} \right]
|
||||
\end{equation}
|
||||
suche:
|
||||
\begin{equation}
|
||||
\left( H - E_a \right) \ket{a} = 0 \label{stern00}
|
||||
\end{equation}
|
||||
mit $\ket{a} \rightarrow \ket{\alpha}$ (für $x \rightarrow 0$) eindeutig, da nicht entartet.
|
||||
\paragraph{Strategie} Wir entwickeln nach $\lambda$
|
||||
\begin{equation}
|
||||
\ket{a} = c_\alpha \ket{\alpha} + \sum_{\beta \neq \alpha} d_\beta \ket{\beta} \text{ und } d_\beta = O(\lambda)
|
||||
\end{equation}
|
||||
Norm:
|
||||
\begin{align}
|
||||
\abs{c_\alpha}^2 + \sum_{\beta \neq \alpha} \abs{d_\beta}^2 &= 1\\
|
||||
\rightarrow c_\alpha &= 1 - O(\lambda^2)
|
||||
\end{align}
|
||||
einsetzen in (\ref{stern00}):
|
||||
\begin{align}
|
||||
0 &= \left( H - E_\alpha \right) \ket{\alpha}\\
|
||||
0 &= \left( H_0 - \lambda H_1 - E_\alpha \right) \left( c_\alpha \ket{\alpha} + \sum_{\beta \neq \alpha} d_\beta \ket{\beta} \right) &\left| \bra{\gamma} ~ \gamma \neq \alpha \right.\\
|
||||
&= c_\alpha \lambda \dirac{\gamma}{H_1}{\alpha} + \sum_{\beta \neq \alpha} d_\beta \left( E_\beta \krondelta{\beta,\gamma} + \lambda \dirac{\gamma}{H_1}{\beta} - E_a \krondelta{\beta,\gamma} \right)\\
|
||||
0 &= \underbrace{c_\alpha}_{1-O(\lambda^2)} \lambda \dirac{\gamma}{H_1}{\alpha} + d_\gamma \left( E_\gamma - E_a \right) + \underbrace{\lambda \sum_{\beta \neq \alpha} d_\beta \dirac{\gamma}{H_1}{\beta}}_{O(\lambda^2)}
|
||||
\end{align}
|
||||
in niedrigster Ordnung erhält man:
|
||||
\begin{align}
|
||||
\rightarrow d_\gamma &= \lambda \frac{\dirac{\gamma}{H_1}{\alpha}}{\underbrace{E_a}_{E_\alpha - O(\lambda)} - E_\gamma} + O(\lambda^2)\\
|
||||
\rightarrow d_\beta &= \lambda \frac{\dirac{\beta}{H_1}{\alpha}}{E_\alpha - E_\beta}\\
|
||||
\ket{a} = \ket{\alpha} + \lambda \sum_{\beta \neq \alpha} \frac{\dirac{\beta}{H_1}{\alpha}}{E_\alpha - E_\beta} \ket{\beta} + O(\lambda^2)
|
||||
\end{align}
|
||||
gut falls:
|
||||
\begin{equation}
|
||||
\lambda \frac{\dirac{\beta}{H_1}{\alpha}}{E_\alpha - E_\beta} \ll 1 ~ \forall \beta \neq \alpha
|
||||
\end{equation}
|
||||
Energieverschiebung:
|
||||
\begin{align}
|
||||
0 &= \left( H - E_\alpha \right) \ket{a} &\left| \ket{\alpha} \right.\\
|
||||
0 &= \bra{\alpha} \left( H_0 + \lambda_1 H_1 - E_a \right) \left( \ket{\alpha} + \lambda \sum_{\beta \neq \alpha} \frac{\dirac{\beta}{H_1}{\alpha}}{E_\alpha - E_\beta} \ket{\beta} + O(\lambda^2) \right)\\
|
||||
E_a &= E_ \alpha + \lambda \dirac{\alpha}{H_1}{\alpha} + \lambda^2 \sum_{\beta \neq \alpha} \frac{\dirac{\alpha}{H_1}{\beta} \dirac{\beta}{H_1}{\alpha}}{E_\alpha - E_\beta} + O(\lambda^3)
|
||||
\end{align}
|
||||
\paragraph{Fazit}
|
||||
\begin{enumerate}
|
||||
\item In der führenden Ordnung ist die Energieverschiebung das Matrixelement der Störung im ungestörten Zustand.
|
||||
\item Falls $\dirac{\alpha}{H_1}{\alpha}$ aus Symmetriegründen verschwindet, dann tragen alle Zustände zu nichtverschwindenden Korrekturen bei!
|
||||
\end{enumerate}
|
||||
\paragraph{Beispiel}
|
||||
\begin{align}
|
||||
H &= \frac{p^2}{2m} + \frac{m}{2} \omega^2 x^2 + \lambda \underbrace{x^4}_{H_1}\\
|
||||
H_0 \ket{n} &= \hbar \omega \left( n + \frac{1}{2} \right) \ket{n}
|
||||
\end{align}
|
||||
Grundzustandsverschiebung
|
||||
\begin{equation}
|
||||
E_0 = \hbar \omega \frac{1}{2} + \lambda \dirac{0}{x^4}{0}
|
||||
\end{equation}
|
||||
entsprechend
|
||||
\begin{equation}
|
||||
E_0 = \hbar \omega \left( n + \frac{1}{2} \right) + \lambda \dirac{n}{x^4}{n}
|
||||
\end{equation}
|
||||
%\begin{figure}[H] \centering
|
||||
%\includegraphics{pdf/III/02-01-00.pdf}
|
||||
%\end{figure}
|
||||
Konsequenz?
|
||||
\begin{itemize}
|
||||
\item für $\lambda$ negativ $\abs{\lambda} \ll 1$
|
||||
%\begin{figure}[H] \centering
|
||||
%\includegraphics{pdf/III/02-01-01.pdf}
|
||||
%\caption{gestrichelte Kurve entspricht $\frac{m}{2} \omega x^2 \abs{\lambda} x^4$}
|
||||
%\end{figure}
|
||||
\item für $\lambda$ positiv
|
||||
%\begin{figure}[H] \centering
|
||||
%\includegraphics{pdf/III/02-01-02.pdf}
|
||||
%\end{figure}
|
||||
\end{itemize}
|
||||
volle Rechnung zeigt:
|
||||
\begin{equation}
|
||||
E_n(\lambda) = \hbar \omega \left( n + \frac{1}{2} \right) + A^{(1)}_n (\lambda) + A^{(2)}_n (\lambda)
|
||||
\end{equation}
|
||||
mit
|
||||
\begin{equation}
|
||||
A^{(1)}_n (\lambda) = O(\lambda) ~ ; ~ A^{(2)}_n (\lambda) = O \left(e^{-\frac{1}{\lambda}} \right)
|
||||
\end{equation}
|
||||
|
||||
\section{Entarteter Fall}
|
||||
\begin{itemize}
|
||||
\item Obige Formel wegen Energienenner nicht anwendbar bei Entartung.
|
||||
\item sehr relevant: Aufhebeung der Entartung durch Störung
|
||||
\end{itemize}
|
||||
%\begin{figure}[H] \centering
|
||||
%\includegraphics{pdf/III/02-02-00.pdf}
|
||||
%\end{figure}
|
||||
\paragraph{Ansatz}
|
||||
\begin{align}
|
||||
\ket{a} &= \sum_{\alpha \in D} c_\alpha \ket{\alpha} + \sum_{\mu \notin D} d_\mu \ket{\mu}\\[10pt]
|
||||
0 &= \left( H - E_a \right) \ket{a}\\
|
||||
\rightarrow 0 &= \left( H_0 + \lambda H_1 - E_a \right) \left( \sum_{\mu \notin D} d_\mu \ket{\mu} \right)
|
||||
\end{align}
|
||||
\paragraph*{Projektion auf $\bra{\nu} \notin D$}
|
||||
\begin{align}
|
||||
0 &= \lambda \sum_\alpha c_\alpha \dirac{\nu}{H_1}{\alpha} d_\nu \left( E_\nu - E_a \right) + \lambda \sum_{\mu \notin D} d_\mu \dirac{\nu}{H_1}{\mu}\\
|
||||
\rightarrow d_\nu &= \lambda \frac{\sum_\alpha c_a \dirac{\nu}{H_1}{\alpha}}{\underbrace{E_a}_{E_\alpha} - E_\nu} + o(\lambda^2)
|
||||
\end{align}
|
||||
\paragraph*{Projektion auf $\bra{\beta} \in D$}
|
||||
\begin{align}
|
||||
0 &= \sum_\alpha c_\alpha \left( E_\alpha - E_a \right) \krondelta{\alpha,\beta} + \lambda \sum_\alpha c_\alpha \dirac{\beta}{H_1}{\alpha} + \lambda \sum_\mu d_\mu \dirac{\beta}{H_1}{\alpha}\\
|
||||
0 &= \sum_\alpha c_\alpha \left( \left( E_a - E_\alpha \right) \krondelta{\alpha,\beta} + \underbrace{\lambda \dirac{\beta}{H_1}{\alpha} + \lambda^2 \sum_{\mu \notin D} \frac{\dirac{\beta}{H_1}{\mu} \dirac{\mu}{H_1}{\alpha}}{E_\alpha E_\mu}}_{\equiv \dirac{\beta}{H_\text{eff}}{\alpha}} \right) + O(\lambda^3)
|
||||
\end{align}
|
||||
$\forall \beta = 1, ..., N$ mit $N$ die Dimension von $D$\\
|
||||
d.h. LGS für die $c_\alpha$ hat nichttriviale Lösung falls
|
||||
\begin{equation}
|
||||
\detb{E_\alpha \krondelta{\alpha,\beta} + \dirac{\beta}{H_\text{eff}}{\alpha} - E_\alpha \krondelta{\alpha,\beta}} = 0
|
||||
\end{equation}
|
||||
d.h. wir müssen $H_0 + H_\text{eff}$ im Unterraum $D$ diagonalisieren
|
||||
\paragraph{Fazit}
|
||||
\begin{enumerate}
|
||||
\item in Ordnung $\lambda$ reicht es $H_1$ im entarteten Unterraum zu diagonalisieren
|
||||
\item falls $H_1$ die Entartung nicht aufhebt, muss der $\lambda^2$-Term mitgenommen werden
|
||||
\end{enumerate}
|
||||
|
||||
\section{Beispiel: Stark-Effekt}
|
||||
H-Atom:
|
||||
\begin{equation}
|
||||
H_0 \ket{n,l,m} = E_n \ket{n,l,m}
|
||||
\end{equation}
|
||||
mit
|
||||
\begin{equation}
|
||||
E_n = \frac{l^2}{2 a_0} \frac{1}{n^2} \text{; Entartung } g(n) = n^2
|
||||
\end{equation}
|
||||
Störung: $H_1$ sei E'feld
|
||||
\begin{equation}
|
||||
H_1 = e \abs{E} \hat{z} \text{ für } \lambda = 1
|
||||
\end{equation}
|
||||
\paragraph{Erster Grundzustand}
|
||||
\begin{equation}
|
||||
\ket{1,0,0} \rightarrow \braket{r,\theta,\varphi}{1,0,0} = \Phi_{1,0,0}(\vec{r}) = \frac{U_{1,0}}{r}Y_{0,0} = \frac{1}{\sqrt{\pi} a_0^\frac{3}{2}}e^{-\frac{r}{a_0}}
|
||||
\end{equation}
|
||||
\begin{itemize}
|
||||
\item in Ordnung $\lambda$ d.h. in $O(\abs{E})$:
|
||||
\begin{equation}
|
||||
\dirac{1,0,0}{H_1}{1,0,0} = \intgru{\Phi^*_{1,0,0} (\vec{r}) \cdor e \abs{E} z \Phi_{1,0,0}(\vec{r})}{r^3} = 0
|
||||
\end{equation}
|
||||
\item in Ordnung $\abs{E}^2$:
|
||||
\begin{align}
|
||||
E_a = E_\alpha + ... &= E_1 + \sum_{\beta,\alpha} \frac{\abs{\dirac{\beta}{H_1}{\alpha}}^2}{E_\beta - E_\alpha}\\
|
||||
&= \left( E_1 \sum_{n=2}^\infty \sum_{l=0}^{n-1} \sum_{m=-l}^{+l} \frac{\abs{\dirac{n,l,m}{H_1}{1,0,0}}^2}{E_1 - E_n} + \int \text{Kontinuum} \right) + O(E^3)
|
||||
\end{align}
|
||||
mit
|
||||
\begin{align}
|
||||
\dirac{n,l,m}{\hat{z}}{1,0,0} &= \intgr{0}{-\infty}{r \intgr{-1}{+1}{\intgr{0}{2\pi}{\frac{U_{n,r}(r)}{r} Y^*_{l,m}(\theta,\varphi) r \underbrace{\cos\theta}_{Y_{1,0} \frac{1}{\sqrt{\pi} a_0^\frac{3}{2}}e^{-\frac{r}{a_0}}}}{\varphi}}{(\cos\theta)}}{r}\\
|
||||
&= \krondelta{m,0} \krondelta{l,1} \frac{1}{\sqrt{3}} \int U_{n,1}(r) r U_{1,0}(r)
|
||||
\end{align}
|
||||
\end{itemize}
|
||||
am meisten trägt $n=2$ bei
|
||||
\begin{align}
|
||||
\dirac{2,1,0}{\hat{z}}{1,0,0} &= \frac{a_0}{\sqrt{3}} \intgr{0}{\infty}{\frac{r^2 e^{-\frac{r}{2}}}{2 \sqrt{6}} r \left( 2r e^{-r} \right)}{r}\\
|
||||
&= \frac{2^7 \sqrt{2}}{3^5} a_0
|
||||
\end{align}
|
||||
und damit ist
|
||||
\begin{equation}
|
||||
E_{1,0,0} = E_1 - 1,48 a_0^3 \abs{E}^2
|
||||
\end{equation}
|
||||
und für alle weiteren $n$
|
||||
\begin{equation}
|
||||
E_{1,0,0} \rightarrow E_1 - \frac{9}{4} a_0^3 \abs{E}^2
|
||||
\end{equation}
|
||||
\paragraph{linerarer Stark-Effekt für n=2}
|
||||
\begin{align}
|
||||
\braket{r,\theta,\varphi}{2,0,0} &= \Phi_{2,0,0}(\vec{r}) = \frac{1}{\sqrt{2 a_0^3}} \left(1 - \frac{r}{2a_0} \right) e^{-\frac{r}{2a_0}} Y_{0,0}\\
|
||||
\braket{r,\theta,\varphi}{2,1,m} &= \Phi_{2,1,m} (\vec{r}) = \frac{1}{24 a_0^3} \frac{r}{a_0} e^{-\frac{r}{2a_0}} Y_{l,m}
|
||||
\end{align}
|
||||
im entartetetn Unterraum $\setCond{\bra{\alpha}}{\alpha = 1,2,3,4}$ mit
|
||||
\begin{equation}
|
||||
\bra{1} = \bra{1,0,0} ;~ \bra{2} = \bra{2,1,0} ;~ \bra{3} = \bra{2,1,1} ;~ \bra{4} = \bra{2,1,-1}
|
||||
\end{equation}
|
||||
diagonalisieren: $\dirac{\beta}{H_1}{\alpha}$
|
||||
\begin{enumerate}
|
||||
\item Diagonalelemente
|
||||
\begin{equation}
|
||||
\dirac{\alpha}{H_1}{\alpha} \tilde \intgru{\cos\theta\abs{\Phi_{n,l,m}}^2}{(\cos\theta)} \equiv 0
|
||||
\end{equation}
|
||||
\item Nichtdiagonalelemente
|
||||
\begin{equation}
|
||||
\dirac{n',l',m'}{\hat{z}}{n,l,m} \tilde \intgru{e^{i(m-m')\phi}}{\phi} \tilde \krondelta{m,m'}
|
||||
\end{equation}
|
||||
mit
|
||||
\begin{equation}
|
||||
\dirac{1}{H_1}{2} = \dirac{2}{H_1}{1} = \dirac{2,0,0}{e\abs{E}z}{×}
|
||||
\end{equation}
|
||||
\end{enumerate}
|
||||
69
kapIV-3.tex
Normal file
69
kapIV-3.tex
Normal file
@ -0,0 +1,69 @@
|
||||
\chapter{Bohr'sche Näherung für Streutheorie}
|
||||
% \begin{figure}[H] \centering
|
||||
% \includegraphics{pdf/IV/03-00-00.pdf}
|
||||
% \end{figure}
|
||||
\section{Geometrie}
|
||||
% \begin{figure}[H] \centering
|
||||
% \includegraphics{pdf/IV/03-01-00.pdf}
|
||||
% \end{figure}
|
||||
\section{Stat SG}
|
||||
\equationblock{\Phi\sbk{\vec{r}} = \Phi^\text{in}\sbk{\vec{n}} + \Phi^\text{ex}\sbk{\vec{r}}}
|
||||
mit
|
||||
\equationblock{\Phi^\text{ex}\sbk{\vec{r}} \longrightarrow^{\vec{r}\rightarrow\infty} f\sbk{\theta,\Phi} \frac{e^{\i k \vec{r}}}{r}}
|
||||
und
|
||||
\equationblock{\Phi^\text{in}\sbk{\vec{r}} = e^{i \vec{k} \vec{r}}}
|
||||
SG:
|
||||
\begin{align}
|
||||
\sbk{-\frac{\hbar^2}{2 \mu} \vec{\nabla}^2 + V\sbk{\vec{r}}} \Phi\sbk{\vec{r}} &= E \Phi\sbk{\vec{r}} &\left| -\frac{2 \mu}{\hbar^2} \right. \\
|
||||
\sbk{\vec{\nabla}^2 - V\sbk{\vec{r}}} \Phi\sbk{\vec{r}} &= -\frac{2 \mu}{\hbar^2} E \Phi\sbk{\vec{r}} &\left; E = \frac{\hbar^2}{2 \mu} k^2 \right. \\
|
||||
\rightarrow \sbk{\vec{\nabla}^2 + k^2} \Phi\sbk{\vec{r}} &= V\sbk{\vec{r}} \Phi\sbk{r}
|
||||
\end{align}
|
||||
|
||||
Green's Funktion Ansatz
|
||||
\begin{align}
|
||||
&\sbk{\vec{\nabla}^2 + k^2} G^0\sbk{\vec{r}-\vec{r'}} = \delta\sbk{\vec{r}-\vec{r'}} \\
|
||||
&\Phi\sbk{\vec{r}} = \intgru{}{r'} G^\sbk{0}\sbk{\vec{r}-\vec{r'}} V\sbk{\vec{r'}} \Phi\sbk{\vec{r'}} + \underbrace{\Phi^\sbk{0}\sbk{\vec{r}}}_{\text{beliebige Lösung der homogenen Gl.}}
|
||||
\end{align}
|
||||
\begin{enumerate}
|
||||
\item für $V = 0$: \equationblock{\Phi\sbk{\vec{r}} = \Phi^\sbk{0}\sbk{\vec{r}} = e^{\i \vec{k} \vec{r}}}
|
||||
\item in $\bigOb{V}$: \equationblock{\Phi{\vec{r}} \approx \sbk{\intgru{}{\vec{r}} G^\sbk{0}\sbk{\vec{r}-\vec{r'}} v\sbk{\vec{r}} e^{\i \vec{k} \vec{r}}} + e^{\i \vec{r} \vec{r}}}
|
||||
\item im Prinzip iterieren:
|
||||
\begin{align}
|
||||
\Phi &= \Phi^0 + \G^\sbk{0}v \Phi^0 + G^\sbk{0} v G^\sbk{0} \Phi^\sbk{0} + \ldots
|
||||
&= \frac{1}{1 - G^\sbk{0} v} \Phi^\sbk{0} &\left( geometrische Reihe \right)
|
||||
\end{align}
|
||||
formal exakt, praktische ziemlich nutzlos
|
||||
\end{enumerate}
|
||||
|
||||
\section{Berechnung der Green'schen Funktion}
|
||||
\begin{align}
|
||||
\sbk{\vec{\nabla}^2 + k^2} G\sbk{\vec{u}} &= \delta\sbk{\vec{u}} &\left| \intgru{e^{-\i \vec{q} \vec{u}}}{\vec{u}} \right. \\
|
||||
\sbk{-\vec{q}^2 + k^2} G\sbk{q} &= 1 \\
|
||||
G\sbk{q} &= \frac{1}{k^2 - q^2} \\
|
||||
G\sbk{\vec{ u}} &= \intgru{\frac{1}{\sbk{2 \pi}^2} \frac{1}{k^2 - q^2} e^{\i \vec{q} \vec{u}}}{q} \\
|
||||
&= \frac{1}{4 \pi^2} \intgr{-1}{+1}{\intgr{0}{\infty}{q^2 \frac{1}{k^2 - q^2} e^{\i q u \cosb{\theta}}}{q}}{\sbk{\cosb{\theta}}} \\
|
||||
&= \frac{1}{4 \pi^2} \intgr{0}{\infty}{\frac{q^2}{\i q u} \sbk{e^{\i q u} - e^{-\i q u}} \frac{1}{k^2 - q^2}}{q} \\
|
||||
&= \frac{1}{4 \pi^2} \frac{1}{\i n} \intgrinf{\frac{q e^{\i q n}}{k^2 -q^2}}{q}
|
||||
&= \frac{1}{4 \pi^2} \frac{1}{\i n} \intgrinf{\frac{q e^{\i q u}}{k^2 - q^2 + \i \epsilon}}{q} \\
|
||||
\text{Residuensatz} \Rightarrow &= \frac{1}{4 \pi^2} \frac{1}{\i u} 2 \pi \i \underbrace{Res\sbk{q=k}_{\epsilon \rightarrow 0}}_{-\frac{k}{2 k} e^{\i k u} \\
|
||||
G\sbk{u} &= -\frac{1}{4 \pi u} e^{\i k u}
|
||||
\end{align}
|
||||
% \begin{figure}[H] \centering
|
||||
% \includegraphics{pdf/IV/03-03-00.pdf}
|
||||
% \end{figure}
|
||||
|
||||
\section{Bohr'sche Näherung}
|
||||
\equationblock{}
|
||||
|
||||
\section{Streuamplitude und differentieller Wirkungsquerschnitt}
|
||||
|
||||
Bsp.: abgeschirmtes Coulomb-Potential
|
||||
Yukawa Potential
|
||||
|
||||
\begin{align}
|
||||
V\sbk{r} &= \frac{l^2}{r} l^{-\frac{r}{r_0}} \\
|
||||
V\sbk{q} &= \frac{4 \pi l^2}{q}\intgr{0}{\infty}{\sinb{q r'} l^{-\frac{r}{r_0}}}{r'} \\
|
||||
&= 4 \pi l^2 \frac{1}{q^2 + \frac{1}{r_0^2}} \\
|
||||
\diffTfrac{r}{\Omega} &= \frac{l^2}{4 \mu^2 \tilde{V}^4 \sin^4\sbk{\frac{\theta}{2}}} &\left \tilde{V} = \frac{\hbar k}{\mu} \left.
|
||||
\end{align}
|
||||
Rutherford'sche Streuquerschnitt für das Coulomb-Problem
|
||||
163
kapV-1.tex
Normal file
163
kapV-1.tex
Normal file
@ -0,0 +1,163 @@
|
||||
\chapter{Reine Zustände}
|
||||
\section{Postulate}
|
||||
\begin{itemize}
|
||||
\item P1: Bei vollständiger Kenntnis (Präparation) wird ein System durch einen normierten Vektor
|
||||
\equationblock{\ket{\Psi} \in \hilbert} beschrieben
|
||||
\item P2a: Jeder physikalischen Größe entspricht ein hermitescher Operator
|
||||
\equationblock{A = \sum a_n \ket{n} \bra{n} \text{(Spektraldarstellung)}}
|
||||
mit Eigenzuständen $\ket{n}$ und reellen Eigenwerten $a_n$
|
||||
\equationblock{A = \sum a_n \ket{n} \bra{n} = \sum_\nu a_\nu P_\nu}
|
||||
mit $P_\nu = \sum_{n : a_n = a_\nu} \ket{n} \bra{n}$
|
||||
\item P2b: Eine Messung von A im Zustand $\ket{\Psi}$ gibt Sicherheit einen der Eigenzustände $a_\nu$
|
||||
Die Wahrscheinlichkeit, $a_\mu$ zu messen ist:
|
||||
\begin{align}
|
||||
\probb{A \cequiv a_\mu}{\ket{\Psi_0}} &= \dirac{\Psi_0}{P_\mu}{\Psi_0} \\
|
||||
&= \braket{\Psi}{n} \braket{m}{\Psi} \\
|
||||
&= \spbk{\braket{m}{\Psi}}^2 \\
|
||||
&= \sum_k \dirac{\Psi_0}{P_\mu}{k} \braket{k}{\Psi_0} \\
|
||||
&= \sum_k \braket{k}{\Psi_0} \dirac{\Psi_0}{P_\mu}{k} \\
|
||||
&= \tr\sbk{\ket{\Psi_0} \bra{\Psi_0} P_\mu} \\
|
||||
&= \tr\sbk{P_{\Psi_0} P_\mu}
|
||||
\end{align}
|
||||
mit $P_{\Psi_{0}} = \ket{\Psi_0} \bra{\Psi_0}$ \\
|
||||
Konsequenz: ``Erwarutngswert'' oder Mittelwert über viele Messungen in identisch präparierten Zustand $\ket{\Psi_0}$
|
||||
\begin{align}
|
||||
\expval{A_{\Psi_0}} &= \sum_\nu a_\nu \probb{A \cequiv a_\nu}{\ket{\Psi_0}} \\
|
||||
&= \sum_\nu a_\nu \dirac{\Psi_0}{P_\nu}{\Psi_0} \\
|
||||
&= \dirac{\Psi_0}{A}{\Psi_0}
|
||||
\end{align}
|
||||
\item P2c: Unmittelbar nach der Messung des Messwertes $a_\mu$ ist das System im Zustand
|
||||
\equationblock{\ket{\Psi} = \frac{O_\mu \ket{\Psi_0}}{\norm{P_\mu} \ket{\Psi_0}} \stackrel{a_\mu nicht entartet}{} \ket{m}}
|
||||
\item P3: Nach einer Messung oder Präparation entwickelt sich der Zustand nach der Schrödingergleichung:
|
||||
\equationblock{\i \hbar \diffPs{t} \ket{\Psi_0\sbk{t}} = H\sbk{t} \ket{\Psi\sbk{t}}}
|
||||
mit dem (hermiteschen) Hamiltonoperator $H\sbk{t}$
|
||||
\end{itemize}
|
||||
|
||||
\section{Einfaches Beispiel mit Spin $\frac{1}{2}$}
|
||||
% \begin{figure}[H] \centering
|
||||
% \includegraphics{pdf/V/01-02-00.pdf}
|
||||
% \end{figure}
|
||||
\begin{itemize}
|
||||
\item P1: $\hilbert = \setC^2$ \\
|
||||
Basis: $\ket{z+}$, $\ket{z-}$ \\
|
||||
allgemeiner Zustand: $\ket{\Psi} = c_1 \ket{z+} + c_2 \ket{z-}$ mit $\spbk{c_1}^2 + \spbk{^2} = 1$
|
||||
\item P2a: Mögliche physikalische Größen: Messung durch SG in $\vec{n}$ Richtung:
|
||||
% \begin{figure}[H] \centering
|
||||
% \includegraphics{pdf/V/01-02-01.pdf}
|
||||
% \end{figure}
|
||||
mögliche Messwerte: Eigenwerte von $V_n = \pm 1$ \\
|
||||
Eigenvektoren $\ket{n+} = \inlinematrixu{\cosb{\frac{\Theta}{2}} \\ e^{\i \Phi} \sinb{\frac{\Theta}{2}}}$
|
||||
$n = \inlinematrixu{\sinb{\Theta}\cosb{\Phi} \\ \sinb{\Theta}\sinb{\Phi} \\ \cosb{\Theta}}$
|
||||
\item P2b: $\probb{\Sigma_n \cequiv +1}{\ket{\Psi_0}} = \spbk{\braket{n+}{\Psi_0}}^2$
|
||||
\item P2c: Nach der Messung von +1 mit Sicherheit im Zustand $\ket{n+}$ \\
|
||||
Besipiel für den Erwartungswert:
|
||||
\begin{align}
|
||||
\expval{\Sigma_n}_\ket{\Psi_0} &= \\
|
||||
\dirac{\Psi_0}{\Sigma_n}{\Psi_0} &=
|
||||
\inlinematrixu{c_1^\ast & c_2} \inlinematrixu{n_z & n_x - \i n_y \\ n_x + \i n_y & -n_z} \inlinematrixu{c_1 \\ c_2}
|
||||
\end{align}
|
||||
\item P3: Dynamik im Magnetfeld:
|
||||
\equationblock{H\sbk{t} = - \vec{mu} \vec{B}\sbk{t} = g \mu_b \frac{1}{2} \vec{\Sigma} \cdot \vec{B}\sbk{t}}
|
||||
Beispiel: $\vec{B}\sbk{t} ) B_z \vec{e_z} \rightarrow H = \frac{\hbar \omega}{2} \Sigma_z$
|
||||
mit $\omega = g \mu_b \frac{B}{\hbar}$
|
||||
$\text{SG(P3)} = \i \hbar \diffPs{t} \ket{\Psi} = \hbar \frac{\hbar}{2} \Sigma_z \ket{\Psi}$
|
||||
$\Psi\sbk{t} = e^{-\frac{\i}{\hbar} H t} \ket{\Psi_0} =$ \\
|
||||
$c_1\sbk{0} e^{-\frac{\i \omega t}{2}} \ket{z+} + c_2\sbk{0} e^{+\frac{\i \omega t}{2}} \ket{z-}$
|
||||
\end{itemize}
|
||||
|
||||
\chapter{Gemische: Statistischer Operator}
|
||||
\section{Motivation: Ein Spiel}
|
||||
% \begin{figure}[H] \centering
|
||||
% \includegraphics{pdf/V/02-01-00.pdf}
|
||||
% \end{figure}
|
||||
|
||||
Alice sendet Bob mit Wahrscheinlichkeit $p_+ = \frac{1}{2}$ den Zustand $\ket{z+}$ und mit $p_- = \frac{1}{2}$ den Zustand $\ket{z-}$
|
||||
|
||||
Bob weiss nicht ob Alice $\ket{z+}$ oder $\ket{z-}$ geschicket hat. Bob darf aber ein beliebiges Stern-Gerlach-Experiment durchführen.
|
||||
|
||||
Frage: Wie soll Bob seinen Einganszustand beschreiben?
|
||||
\begin{enumerate}
|
||||
\item Bobs Experiment(e) zeigen: \\
|
||||
$\expval{\Sigma_z} = \expval{\Sigma_x} = \expval{\Sigma_x} = \expval{\Sigma_n}$ \\
|
||||
Es gibt kein $\ket{\Psi_0}$ mit $\expval{\Sigma_n}_\ket{\Psi} = 0 \forall n$ \\
|
||||
Bobs Kenntnis ist unvollständig (Eingangspräparation)
|
||||
\item Bobs Input besteht aus einem \textbf{klassischen} Ensemble (Gesamtheit), in dem mit \textbf{klassischer} Wahrscheinlichkeit $p_+ = \frac{1}{2}$ der Zustand $\ket{z+}$ und mit \textbf{klassischer} Wahrscheinlichkeit $p_- = \frac{1}{2}$ der Zustand $\ket{z-}$ enthalten ist.
|
||||
\end{enumerate}
|
||||
|
||||
\section{Definition des statistischen Operators (Dichtematrix; engl. density matrix)}
|
||||
Sei $\rho$ ein Operator $\hilbert \rightarrow \hilbert$:
|
||||
\begin{enumerate}
|
||||
\item $\rho = \rho^\dagger$
|
||||
\item $\tr\sbk{\rho} = 1$
|
||||
\item $\dirac{\psi}{\rho}{\psi} \geq 0 \forall \ket{\psi} \in \hilbert$
|
||||
\end{enumerate}
|
||||
bzw. in irgend einer Basis $\sgbk{\ket{n}}$
|
||||
\begin{enumerate}
|
||||
\item $\rho_{nn} = \dirac{n}{\rho}{m} \rho^\ast_{mn}$
|
||||
\item $\sum_n \rho_{nn} = 1$
|
||||
\item $\sum_{n,m} c_n^\ast \rho_{nm} c_m \geq \forall c_n$ mit $\sum_n \spbk{c_n}^2$
|
||||
\end{enumerate}
|
||||
|
||||
\section{Gemisch}
|
||||
Definition: Ein quantales enthält mit Wahrscheinlichkeit $p_i$ den reinen Zustand $\ket{\psi_i}$ $\sbk{i = 1\ldots M}$ $M$ beliebig, im Allgemeinen ist $M \neq \dim \hilbert$.
|
||||
\begin{enumerate}
|
||||
\item $\sum_i^M p_i = 1$
|
||||
\item $\braket{\psi_i}{\psi_j} \neq \delta_{ij}$ erlaubt
|
||||
\end{enumerate}
|
||||
Dieses quantale Gemisch wird durch den statistischen Operator $\rho = \sum_{i=1}^M p_i \ket{\psi_i}\bra{\psi_i} = \sum_{i=1}^M p_i P_{\psi_i}$.
|
||||
|
||||
Alice präpariert $\ket{z+}$ und $\ket{z-}$ Zustand, sie würfelt und wählt dann mit Wahrscheinlichkeit $\frac{1}{2}$ $\ket{z+}$ und mit Wahrscheinlichkeit $\frac{1}{2}$ $\ket{z-}$, die sie zu Bob schickt.
|
||||
Frage: Wie soll Bob den Eingangszustand beschreiben?
|
||||
|
||||
Bobs mögliche Messwerte $\sigma_n$ sind immer noch $\pm1$.
|
||||
\begin{align}
|
||||
\prob{\sigma_z \cequiv +1} &= p_{z+} &= \frac{1}{2} \\
|
||||
\prob{\sigma_z \cequiv -1} &= p_{z-} &= \frac{1}{2} \\
|
||||
\prob{\sigma_n \cequiv +1} &= p_{z+} &= p_{z-} \probb{\sigma_z \cequiv +1}{\ket{z+}} + p_{z-} \probb{\sigma_z \cequiv -1}{\ket{z+}}
|
||||
\end{align}
|
||||
|
||||
Bsp: $\vec{n} = \vec{e_x}$
|
||||
\begin{align}
|
||||
\prob{\sigma_x \cequiv 1} &= \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{1}{2} &= \frac{1}{2} \\
|
||||
\prob{\sigma_x \cequiv -1} &= &= \frac{1}{2}
|
||||
\end{align}
|
||||
$\Rightarrow \ssbk{\sigma_x} = 0$
|
||||
Check:
|
||||
\begin{enumerate}
|
||||
\item $\rho^\dagger = \sum_i p_i \ket{\psi_1} \bra{\psi_1} = \rho$
|
||||
\item \begin{align}
|
||||
\tr\sbk{\rho} &= \sum_i \dirac{n}{\sum_{i=1}^M P_i}{\psi_i} \braket{\psi_i}{n} \\
|
||||
&= \sum_{i=1}^M P_i \spbk{\underbrace{\braket{n}{\psi_i}}_{=1}}^2 = 1
|
||||
\end{align}
|
||||
\item \begin{align}
|
||||
\dirac{\psi}{\rho}{\psi} &= \dirac{\psi}{\sum_{i=1}^M P_i}{\psi_i} \braket{\psi_i}{\psi} \\
|
||||
&= \sum_{i=1}^M P_i \ssbk{\underbrace{\braket{\psi_i}{\psi}}_{\geq 0}}^2
|
||||
\end{align}
|
||||
\end{enumerate}
|
||||
Bemerkung:
|
||||
\begin{enumerate}
|
||||
\item Als Spezialfall enthält der Begriff Gemisch auch den reinen Zustand. $M=1$ gibt $\rho= \ket{\psi_1} \bra{\psi_1} = P_{\ket{\psi_1}}$
|
||||
\item für einen reinen Zustand gilt: $\rho^2 = \rho$
|
||||
\end{enumerate}
|
||||
Beweis: $\rho^2 = \rho \cdot \rho = \ket{\psi_1} \braket{\psi_1}{\psi_1} \bra{\psi_1} = \ket{\psi_1} \bra{\psi_1} = \rho$
|
||||
|
||||
Beispiel:
|
||||
Alice präpariert mit Wahrscheinlichkeit $p_1$ den Zustand $\ket{z+}$ und mit $p_2$ den Zustand $\ket{x+}$ $\sbk{p_1 + p_2 = 1}$
|
||||
\begin{align}
|
||||
\rho &= p_1 \ket{z+} \bra{z+} + p_2 \ket{x+} \bra{x+} \\
|
||||
&= p_1 \inlinematrixu{1 \\ 0} \inlinematrixu{1 & 0} + p_2 \inlinematrixu{\frac{\sqrt{2 }}{2} \\ \frac{\sqrt{2}}{2}} \inlinematrixu{\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}} \\
|
||||
&= p_1 \inlinematrixu{1 & 0 \\ 0 & 0} + p_2 \inlinematrixu{\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2}} \\
|
||||
&= \inlinematrixu{p_1 + \frac{p_2}{2} & \frac{p_2}{2} \\ \frac{p_2}{2} & \frac{p_2}{2}}
|
||||
\end{align}
|
||||
$\rho^2 = \rho \gdw p_1=1 \text{xor} p_1=0$
|
||||
|
||||
Bemerkung:
|
||||
Die Darstellungen eines Gemisches eines
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
5
math.tex
5
math.tex
@ -26,6 +26,7 @@
|
||||
\newcommand{\im}[1]{{\text{Im}\left( #1 \right)}}
|
||||
\newcommand{\tr}{{\text{tr}}}
|
||||
\newcommand{\sign}{{\text{sign}}}
|
||||
% \newcommand{\dim}{\text{dim}}
|
||||
|
||||
\newcommand{\QED}{\begin{large}\textbf{\checkmark}\end{large}}
|
||||
\newcommand{\cequiv}{\stackrel{\scriptscriptstyle\wedge}{=}} %ContextEQUIvalent
|
||||
@ -44,6 +45,8 @@
|
||||
\newcommand{\sbk}[1]{\left( #1 \right)}
|
||||
\newcommand{\sqbk}[1]{\left[ #1 \right]}
|
||||
\newcommand{\ssbk}[1]{\left< #1 \right>}
|
||||
\newcommand{\spbk}[1]{{\left| #1 \right|}}
|
||||
\newcommand{\sgbk}[1]{\left\{ #1 \right\}}
|
||||
\newcommand{\detb}[1]{\det\sqbk{#1}}
|
||||
\newcommand{\cosb}[1]{\cos\sbk{#1}}
|
||||
\newcommand{\sinb}[1]{\sin\sbk{#1}}
|
||||
@ -60,5 +63,7 @@
|
||||
|
||||
\newcommand{\levicivita}[1]{\varepsilon_{#1}}
|
||||
\newcommand{\krondelta}[1]{\delta_{#1}}
|
||||
% \newcommand{\sum}[3]{\Sigma_{#1}^{#2} #3}
|
||||
% \newcommand{\prod}[3]{\Pi_{#1}^{#2} #3}
|
||||
|
||||
\newcommand{\equationblock}[1]{\begin{equation} #1 \end{equation}}
|
||||
|
||||
295
theo2.kilepr
295
theo2.kilepr
@ -1,295 +0,0 @@
|
||||
[General]
|
||||
img_extIsRegExp=false
|
||||
img_extensions=.eps .jpg .jpeg .png .pdf .ps .fig .gif
|
||||
kileprversion=2
|
||||
kileversion=2.0
|
||||
lastDocument=kapIV-2.tex
|
||||
masterDocument=
|
||||
name=Theo2
|
||||
pkg_extIsRegExp=false
|
||||
pkg_extensions=.cls .sty
|
||||
src_extIsRegExp=false
|
||||
src_extensions=.tex .ltx .latex .dtx .ins
|
||||
|
||||
[Tools]
|
||||
MakeIndex=
|
||||
QuickBuild=PDFLaTeX+ViewPDF
|
||||
|
||||
[item:formelsammlung.tex]
|
||||
archive=true
|
||||
column=115
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=42
|
||||
open=false
|
||||
order=6
|
||||
|
||||
[item:kapI-1.tex]
|
||||
archive=true
|
||||
column=17
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=103
|
||||
open=false
|
||||
order=3
|
||||
|
||||
[item:kapI-2.tex]
|
||||
archive=true
|
||||
column=13
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=163
|
||||
open=false
|
||||
order=4
|
||||
|
||||
[item:kapI-3.tex]
|
||||
archive=true
|
||||
column=0
|
||||
encoding=
|
||||
highlight=LaTeX
|
||||
line=0
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
[item:kapI-4.tex]
|
||||
archive=true
|
||||
column=6
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=192
|
||||
open=false
|
||||
order=5
|
||||
|
||||
[item:kapI-5.tex]
|
||||
archive=true
|
||||
column=7
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=148
|
||||
open=false
|
||||
order=6
|
||||
|
||||
[item:kapI-6.tex]
|
||||
archive=true
|
||||
column=4
|
||||
encoding=
|
||||
highlight=LaTeX
|
||||
line=2
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
[item:kapII-0.tex]
|
||||
archive=true
|
||||
column=12
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=122
|
||||
open=false
|
||||
order=1
|
||||
|
||||
[item:kapII-1.tex]
|
||||
archive=true
|
||||
column=113
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=67
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
[item:kapII-2.tex]
|
||||
archive=true
|
||||
column=9
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=91
|
||||
open=false
|
||||
order=7
|
||||
|
||||
[item:kapII-3.tex]
|
||||
archive=true
|
||||
column=12
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=123
|
||||
open=false
|
||||
order=2
|
||||
|
||||
[item:kapII-4.tex]
|
||||
archive=true
|
||||
column=13
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=39
|
||||
open=false
|
||||
order=2
|
||||
|
||||
[item:kapII-5.tex]
|
||||
archive=true
|
||||
column=29
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=92
|
||||
open=false
|
||||
order=3
|
||||
|
||||
[item:kapIII-0.tex]
|
||||
archive=true
|
||||
column=8
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=0
|
||||
open=false
|
||||
order=4
|
||||
|
||||
[item:kapIII-1.tex]
|
||||
archive=true
|
||||
column=15
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=39
|
||||
open=false
|
||||
order=5
|
||||
|
||||
[item:kapIII-2.tex]
|
||||
archive=true
|
||||
column=20
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=1
|
||||
open=false
|
||||
order=6
|
||||
|
||||
[item:kapIII-3.tex]
|
||||
archive=true
|
||||
column=14
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=319
|
||||
open=false
|
||||
order=7
|
||||
|
||||
[item:kapIII-4.tex]
|
||||
archive=true
|
||||
column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=195
|
||||
open=false
|
||||
order=8
|
||||
|
||||
[item:kapIV-1.tex]
|
||||
archive=true
|
||||
column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=36
|
||||
open=true
|
||||
order=1
|
||||
|
||||
[item:kapIV-2.tex]
|
||||
archive=true
|
||||
column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=209
|
||||
open=true
|
||||
order=2
|
||||
|
||||
[item:math.tex]
|
||||
archive=true
|
||||
column=17
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=46
|
||||
open=false
|
||||
order=2
|
||||
|
||||
[item:physics.tex]
|
||||
archive=true
|
||||
column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=18
|
||||
open=false
|
||||
order=2
|
||||
|
||||
[item:theo2.kilepr]
|
||||
archive=true
|
||||
column=0
|
||||
encoding=
|
||||
highlight=
|
||||
line=0
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
[item:theo2.tex]
|
||||
archive=true
|
||||
column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=47
|
||||
open=true
|
||||
order=0
|
||||
|
||||
[item:ueb1.tex]
|
||||
archive=true
|
||||
column=2147483647
|
||||
encoding=
|
||||
highlight=
|
||||
line=0
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
[item:ueb2.tex]
|
||||
archive=true
|
||||
column=64
|
||||
encoding=
|
||||
highlight=
|
||||
line=0
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
[item:ueb3.tex]
|
||||
archive=true
|
||||
column=2147483647
|
||||
encoding=
|
||||
highlight=
|
||||
line=0
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
[item:ueb4.tex]
|
||||
archive=true
|
||||
column=2147483647
|
||||
encoding=
|
||||
highlight=
|
||||
line=0
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
[item:ueb5.tex]
|
||||
archive=true
|
||||
column=0
|
||||
encoding=
|
||||
highlight=LaTeX
|
||||
line=0
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
[item:ueb6.tex]
|
||||
archive=true
|
||||
column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=119
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
[item:ueb7.tex]
|
||||
archive=true
|
||||
column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=0
|
||||
open=false
|
||||
order=1
|
||||
32
theo2.tex
32
theo2.tex
@ -48,24 +48,30 @@
|
||||
\include{kapIII-3}
|
||||
\include{kapIII-4}
|
||||
|
||||
|
||||
\part{Näherungsmethoden}
|
||||
\label{IV}
|
||||
\include{kapIV-1}
|
||||
\include{kapIV-2}
|
||||
|
||||
% \part{Übungsmitschrieb}
|
||||
% \label{UE}
|
||||
% \include{ueb1}
|
||||
% \include{ueb2}
|
||||
% \include{ueb3}
|
||||
% \include{ueb4}
|
||||
% \include{ueb5}
|
||||
% \include{ueb6}
|
||||
% \include{ueb7}
|
||||
% \include{ueb8}
|
||||
% \include{ueb9}
|
||||
% \include{ueb10}
|
||||
% \include{ueb11}
|
||||
\part{Übungsmitschrieb}
|
||||
\label{UE}
|
||||
\include{ueb1}
|
||||
\include{ueb2}
|
||||
\include{ueb3}
|
||||
\include{ueb4}
|
||||
\include{ueb5}
|
||||
\include{ueb6}
|
||||
\include{ueb7}
|
||||
\include{ueb8}
|
||||
\include{ueb9}
|
||||
\include{ueb10}
|
||||
\include{ueb11}
|
||||
|
||||
\part{Wiederholung}
|
||||
\label{V}
|
||||
\include{kapV-1}
|
||||
|
||||
|
||||
\part{Formelsammlung}
|
||||
\label{FS}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user