Kapitel IV 2 WIP
This commit is contained in:
Oliver Groß
parent
156958c602
commit
2bf6c32cb0
@ -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}
|
||||
|
||||
209
kapIV-2.tex
Normal file
209
kapIV-2.tex
Normal file
@ -0,0 +1,209 @@
|
||||
\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}{2,1,0} \equiv V_0
|
||||
\end{equation}
|
||||
\end{enumerate}
|
||||
d.h. das Eigenwertproblem lautet
|
||||
\begin{equation}
|
||||
\inlinematrix{0& V_0& 0& 0\\ V_0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0} \inlinematrix{c_1\\ c_2\\ c_3\\ c_4} = \left( E_a - E_\alpha \right) \inlinematrix{c_1\\ c_2\\ c_3\\ c_4}
|
||||
\end{equation}
|
||||
triviale Eigenvektoren $\inlinematrix{0\\ 0\\ 1\\ 0}$ und $\inlinematrix{0\\ 0\\ 0\\ 1}$ zum Eigenwert $E_a = E_\alpha$ reduziert auf
|
||||
\begin{equation}
|
||||
\inlinematrix{0& V_0\\ V_0& 0} \inlinematrix{c_1\\ c_2} = \left( E_a - E_\alpha \right) \inlinematrix{c_1\\ c_2}
|
||||
\end{equation}
|
||||
also
|
||||
\begin{align}
|
||||
\frac{1}{\sqrt{2}} \inlinematrix{1\\ 1\\ 0\\ 0} &\text{ zu } E_a - E_\alpha = +V_0\\
|
||||
\frac{1}{\sqrt{2}} \inlinematrix{1\\ -1\\ 0\\ 0} &\text{ zu } E_a - E_\alpha = -V_0\\
|
||||
\end{align}
|
||||
mit
|
||||
\begin{equation}
|
||||
V_0 = -3 e \abs{E} a_0
|
||||
\end{equation}
|
||||
101
theo2.kilepr
101
theo2.kilepr
@ -1,13 +1,14 @@
|
||||
[General]
|
||||
def_graphic_ext=
|
||||
img_extIsRegExp=false
|
||||
img_extensions=.eps .jpg .jpeg .png .pdf .ps .fig .gif
|
||||
kileprversion=2
|
||||
kileversion=2.0
|
||||
lastDocument=kapIV-1.tex
|
||||
kileversion=2.0.85
|
||||
lastDocument=theo2.tex
|
||||
masterDocument=
|
||||
name=Theo2
|
||||
pkg_extIsRegExp=false
|
||||
pkg_extensions=.cls .sty
|
||||
pkg_extensions=.cls .sty .bbx .cbx .lbx
|
||||
src_extIsRegExp=false
|
||||
src_extensions=.tex .ltx .latex .dtx .ins
|
||||
|
||||
@ -15,12 +16,37 @@ src_extensions=.tex .ltx .latex .dtx .ins
|
||||
MakeIndex=
|
||||
QuickBuild=PDFLaTeX+ViewPDF
|
||||
|
||||
[document-settings,item:kapIV-1.tex]
|
||||
Bookmarks=
|
||||
Encoding=UTF-8
|
||||
Highlighting=LaTeX
|
||||
Indentation Mode=normal
|
||||
Mode=LaTeX
|
||||
ReadWrite=true
|
||||
|
||||
[document-settings,item:kapIV-2.tex]
|
||||
Bookmarks=
|
||||
Encoding=UTF-8
|
||||
Highlighting=LaTeX
|
||||
Indentation Mode=normal
|
||||
Mode=LaTeX
|
||||
ReadWrite=true
|
||||
|
||||
[document-settings,item:theo2.tex]
|
||||
Bookmarks=
|
||||
Encoding=UTF-8
|
||||
Highlighting=LaTeX
|
||||
Indentation Mode=normal
|
||||
Mode=LaTeX
|
||||
ReadWrite=true
|
||||
|
||||
[item:formelsammlung.tex]
|
||||
archive=true
|
||||
column=115
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=42
|
||||
mode=
|
||||
open=false
|
||||
order=6
|
||||
|
||||
@ -30,6 +56,7 @@ column=17
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=103
|
||||
mode=
|
||||
open=false
|
||||
order=3
|
||||
|
||||
@ -39,6 +66,7 @@ column=13
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=163
|
||||
mode=
|
||||
open=false
|
||||
order=4
|
||||
|
||||
@ -48,6 +76,7 @@ column=0
|
||||
encoding=
|
||||
highlight=LaTeX
|
||||
line=0
|
||||
mode=
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
@ -57,6 +86,7 @@ column=6
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=192
|
||||
mode=
|
||||
open=false
|
||||
order=5
|
||||
|
||||
@ -66,6 +96,7 @@ column=7
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=148
|
||||
mode=
|
||||
open=false
|
||||
order=6
|
||||
|
||||
@ -75,6 +106,7 @@ column=4
|
||||
encoding=
|
||||
highlight=LaTeX
|
||||
line=2
|
||||
mode=
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
@ -84,6 +116,7 @@ column=12
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=122
|
||||
mode=
|
||||
open=false
|
||||
order=1
|
||||
|
||||
@ -93,6 +126,7 @@ column=113
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=67
|
||||
mode=
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
@ -102,6 +136,7 @@ column=9
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=91
|
||||
mode=
|
||||
open=false
|
||||
order=7
|
||||
|
||||
@ -111,6 +146,7 @@ column=12
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=123
|
||||
mode=
|
||||
open=false
|
||||
order=2
|
||||
|
||||
@ -120,6 +156,7 @@ column=13
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=39
|
||||
mode=
|
||||
open=false
|
||||
order=2
|
||||
|
||||
@ -129,6 +166,7 @@ column=29
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=92
|
||||
mode=
|
||||
open=false
|
||||
order=3
|
||||
|
||||
@ -138,6 +176,7 @@ column=8
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=0
|
||||
mode=
|
||||
open=false
|
||||
order=4
|
||||
|
||||
@ -147,6 +186,7 @@ column=15
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=39
|
||||
mode=
|
||||
open=false
|
||||
order=5
|
||||
|
||||
@ -156,15 +196,17 @@ column=20
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=1
|
||||
mode=
|
||||
open=false
|
||||
order=6
|
||||
|
||||
[item:kapIII-3.tex]
|
||||
archive=true
|
||||
column=108
|
||||
column=14
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=338
|
||||
line=319
|
||||
mode=
|
||||
open=false
|
||||
order=7
|
||||
|
||||
@ -174,24 +216,37 @@ column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=195
|
||||
mode=
|
||||
open=false
|
||||
order=8
|
||||
|
||||
[item:kapIV-1.tex]
|
||||
archive=true
|
||||
column=104
|
||||
column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=53
|
||||
line=0
|
||||
mode=LaTeX
|
||||
open=true
|
||||
order=1
|
||||
|
||||
[item:math.tex]
|
||||
[item:kapIV-2.tex]
|
||||
archive=true
|
||||
column=34
|
||||
column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=40
|
||||
line=0
|
||||
mode=LaTeX
|
||||
open=true
|
||||
order=2
|
||||
|
||||
[item:math.tex]
|
||||
archive=true
|
||||
column=17
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=46
|
||||
mode=
|
||||
open=false
|
||||
order=2
|
||||
|
||||
@ -201,6 +256,7 @@ column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=18
|
||||
mode=
|
||||
open=false
|
||||
order=2
|
||||
|
||||
@ -210,15 +266,17 @@ column=0
|
||||
encoding=
|
||||
highlight=
|
||||
line=0
|
||||
mode=
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
[item:theo2.tex]
|
||||
archive=true
|
||||
column=24
|
||||
column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=70
|
||||
line=0
|
||||
mode=LaTeX
|
||||
open=true
|
||||
order=0
|
||||
|
||||
@ -228,6 +286,7 @@ column=2147483647
|
||||
encoding=
|
||||
highlight=
|
||||
line=0
|
||||
mode=
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
@ -237,6 +296,7 @@ column=64
|
||||
encoding=
|
||||
highlight=
|
||||
line=0
|
||||
mode=
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
@ -246,6 +306,7 @@ column=2147483647
|
||||
encoding=
|
||||
highlight=
|
||||
line=0
|
||||
mode=
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
@ -255,6 +316,7 @@ column=2147483647
|
||||
encoding=
|
||||
highlight=
|
||||
line=0
|
||||
mode=
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
@ -264,6 +326,7 @@ column=0
|
||||
encoding=
|
||||
highlight=LaTeX
|
||||
line=0
|
||||
mode=
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
@ -273,6 +336,7 @@ column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=119
|
||||
mode=
|
||||
open=false
|
||||
order=-1
|
||||
|
||||
@ -282,5 +346,18 @@ column=0
|
||||
encoding=UTF-8
|
||||
highlight=LaTeX
|
||||
line=0
|
||||
mode=
|
||||
open=false
|
||||
order=1
|
||||
|
||||
[view-settings,view=0,item:kapIV-1.tex]
|
||||
CursorColumn=0
|
||||
CursorLine=0
|
||||
|
||||
[view-settings,view=0,item:kapIV-2.tex]
|
||||
CursorColumn=0
|
||||
CursorLine=0
|
||||
|
||||
[view-settings,view=0,item:theo2.tex]
|
||||
CursorColumn=0
|
||||
CursorLine=0
|
||||
|
||||
28
theo2.tex
28
theo2.tex
@ -51,21 +51,21 @@
|
||||
\part{Näherungsmethoden}
|
||||
\label{IV}
|
||||
\include{kapIV-1}
|
||||
\include{kapIV-2}
|
||||
% \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{Formelsammlung}
|
||||
\label{FS}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user