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Winkel nach oben oder unten abgelenkt. + \item Beide Ergebnisse sind gleich wahrscheinlich. + \item Wird der Magnet in der $(z, x)$-Ebene gedreht bleiben 1. und 2. erhalten. +\end{enumerate} + +\subsection*{klassische Analyse} +Hamilton Funktion +\begin{equation} + H = \frac{p^2}{2m} - \overrightarrow{\mu} \overrightarrow{B} +\end{equation} +mit $\overrightarrow{\mu}$ mang. Moment.\\ +Kraft +\begin{equation} + F = \nabla ( \overrightarrow{\mu} \cdot \overrightarrow{B} ) +\end{equation} +dominiert +\begin{equation} + F_2 = \mu_z \partial_z B_z \simeq \mu_z \partial_z B_z |_{z = 0} \simeq konst. +\end{equation} +Wir erwarten, dass $\overrightarrow{\mu}$ unpolarisiert ist mit $\mu_z = abs(\mu) \cos \theta$ mit $\theta$ zufällig $p(\theta) = \frac{2\pi}{4\pi} \sin \theta$ und damit auf dem Schirm: + +\begin{figure}[h] +\includegraphics{1-002.pdf} +\caption{klassisches Histogramm} +\end{figure} +Das Ergebnis, insbesondere 3. ist klassisch nicht zu verstehen! + +\section{Schlüsselexperimente} +Kurzdarstellung: + +\begin{figure}[h] +\includegraphics{1-003.pdf} + +bzw. + +\includegraphics{1-004.pdf} +\end{figure} + +$SG, n$ sei ein in $\overrightarrow{n}$ Richtung orientierter Magnet. + +Physikalische Eigenschaft: Spin ($\cequiv$ Auslenkung) in $+\overrightarrow{n}$ Richtung +\begin{equation} + \sigma_n = \underbrace{\pm 1}_\text{mögliche Messwerte} +\end{equation} +\pagebreak %pfusch!! + +\subsection*{Ex. 1} +\begin{figure}[h] +\includegraphics{1-005.pdf} +\end{figure} +Fazit: Wiederholung der gleichen Messung führt auf das identische Ergebnis. + +\subsection*{Ex. 2} +\subsubsection*{a} +\begin{figure}[h] +\includegraphics{1-006.pdf} +\end{figure} + +Fazit: Die $x$-Messung hat den $z$-Spin beeinflusst. + +\subsubsection*{b} +\begin{figure}[h] +\includegraphics{1-007.pdf} +\end{figure} + +\pagebreak %pfusch! + +\subsection*{Ex. 3} +\begin{figure}[h] +\includegraphics{1-008.pdf} +\end{figure} + + +\section*{Superposition VS Messung} +\begin{figure}[h] +\includegraphics{1-009.pdf} +\end{figure} + +\subsection*{Ex. 4} +\begin{figure}[h] +\includegraphics{1-010.pdf} +\end{figure} +Fazit: Wird $\sigma_x$ nicht gemessen bleibt $\sigma_z$ erhalten. + +\pagebreak %pfusch! + +\subsection*{Ex. 5 (Peres)} + +\subsubsection*{a} +\begin{figure}[h] +\includegraphics{1-011.pdf} +\end{figure} + +\subsubsection*{b} +\begin{figure}[h] +\includegraphics{1-012.pdf} +\end{figure} + +\subsubsection*{c} +\begin{figure}[h] +\includegraphics{1-013.pdf} +\end{figure} + +\pagebreak %pfusch! + +\subsubsection*{d} +Wenn der mittlere $SG, x$ immer schwächer wird ($B_x \rightarrow 0$), muss sich das Muster auf dem so verändern +\begin{figure}[h] +\includegraphics{1-014.pdf} +\end{figure} +$\Rightarrow$ Intereferenz diff --git a/kapI-2.tex b/kapI-2.tex new file mode 100644 index 0000000..584e9c6 --- /dev/null +++ b/kapI-2.tex @@ -0,0 +1,330 @@ +\chapter{Mathematische Sprache/Bühne: Hilbert-Raum (endl. Dim.)} +\section{Definition} +\begin{itemize} + \item linerarer Vektorraum $\hilbert$ mit Vektoren $\set{\ket{\phi}, \ket{\psi}}$ + \begin{equation} + c_1 \ket{\phi} + c_2 \ket{\psi} = \ket{c_1 \phi + c_2 \psi} \in \hilbert + \end{equation} + \item Inneres (hermitesches) Produkt: $\hilbert \times \hilbert \rightarrow \setC$ + \begin{equation} + \ket{\phi}, \ket{\psi} \rightarrow \braket{\phi}{\psi} = \braket{\psi}{\phi}^* + \end{equation} + \begin{enumerate} + \item linear im 2. Argument: + $\braket{\phi}{c_1 \psi_1 + c_2 \psi_2} = c_1 \braket{\phi}{\psi_1} + c_2 \braket{\phi}{\psi_2}$ + \item $\underbrace{\braket{\phi}{\phi}}_{\in \setR} > 0$ falls $\ket{\phi} \neq \ket{0}$ + \item antilinear im 1. Argument: + $\braket{c \phi}{\psi} = c^* \braket{\phi}{\psi}$ + \end{enumerate} + \item \underline{ortho}normierte Basis $\set{\ket{n}} = \set{\ket{1}, \ket{2}, ... , \ket{N}}$ + \begin{equation} + \braket{n}{m} = \delta_{n,m} + \end{equation} + Jeder Vektor kann entwickelt werden + \begin{align} + \ket{\psi} &= \sum^{N}_{n=1}c_n \ket{n} &\left| \bra{m} \right.\\ + \braket{m}{\psi} &= \sum^N_{n=1}c_n \braket{m}{n} = c_m\\ + \ket{\psi} &= \sum^N_{n=1} \braket{n}{\psi} \ket{n}\\ + \text{mit} \ket{\phi} &= \sum^{N}_{n=1}d_n \ket{n}\\ + \braket{\phi}{\psi} &= \sum^{N}_{n=1}d_n^* \sum^{N}_{m=1}c_m \braket{n}{m} = \sum^{N}_{n=1}d_n^* c_n + \end{align} + \item Norm + \begin{equation} + \norm{\ket{psi}} \equiv \norm{\phi} \equiv \left( \braket{\phi}{\phi} \right)^\frac{1}{2} = \left( \sum_n c_n^* c_n \right)^\frac{1}{2} + \end{equation} + \item Schwarz'sche Ungleichung + \begin{equation} + \abs{\braket{\chi}{\phi}}^2 \leq \braket{\chi}{\chi}\braket{\phi}{\phi} = \norm{\chi}^2\norm{\phi}^2 + \end{equation} + Gleichheit falls $\ket{\chi} = c \ket{\phi}$ +\end{itemize} + +\section{Lineare Operatoren} +\subsection*{linearer Operator A} +\begin{align} + A \ket{\phi} &= \ket{A \phi}\\ + \text{mit} A\left(c_1\ket{\phi_1}+c_2\ket{phi2}\right) &= c_1 \ket{A\phi_1} + c_2 \ket{A\phi_2} +\end{align} + +\subsection*{``Darstellung'' in Basis} +\begin{align} + \ket{A\psi} &= \sum_n c_n \ket{An} &\left| \bra{m} \right.\\ + \braket{m}{A\psi} &= \sum_n c_m \braket{m}{An} \equiv \sum_n c_n A_{m,n}c_n +\end{align} +$A_{m,n}$ ... ``Matrixelemente'' + +\subsection*{Adjungierter Operator $A^\dagger$} +definiert durch +\begin{align} + \braket{\phi}{A^\dagger \psi} &\equiv \braket{A \phi}{\psi} = \braket{\psi}{A\phi}*\\ + A^\dagger &\equiv \braket{m}{A^\dagger n} = \braket{Am}{n} = \braket{n}{Am}^* = A_{n,m}^*\\ + \left(A^\dagger \right)^\dagger &= A +\end{align} +Produkt $(AB)^\dagger$ +\begin{align} + \braket{\phi}{(AB)^\dagger \psi} = \braket{(AB)\phi}{\psi} &= \braket{B\phi}{A^\dagger \psi}\\ + &= \braket{\phi}{B^\dagger A^\dagger \psi}\\ + \Rightarrow (AB)^\dagger = B^\dagger A^\dagger +\end{align} +Definition: +\begin{equation} + \text{$A$ ist hermitesch} \gdw A^\dagger = A \gdw \underbrace{A_{n,m} = A_{m,n}^*}_\text{Diagonale reel!} +\end{equation} + +\section{Dirac-Notation} +\begin{align} + \ket{A\phi} &= A \ket{\phi}\\ + \bra{A^\dagger \psi} &\stackrel{\text{DN}}{\equiv} \bra{\psi A}\\ + \braket{\psi}{A \phi} &= \braket{A^\dagger \psi}{\phi} \stackrel{\text{DN}}{=} \braket{\psi A}{\phi} \equiv \dirac{\psi}{A}{\phi} +\end{align} +Merke: Operatoren wirken entweder normal nach rechts oder adjungiert nach links.\\[10pt] +Jedem Vektor $\ket{\phi}$ (``ket'') wird ein dualer Vektor $\bra{\phi}$ (``bra'') zugeordnet.\\ +\begin{tabular}[2]{c|c} + ket & bra \\ \hline + $\ket{\phi}$ & $\bra{\phi}$ \\ + $\ket{c_1 \phi}$ & $c_1^* \bra{\phi}$ +\end{tabular} + +\subsection*{Darstellung als Spalten- und Zeilenvektoren} +\begin{align} + \set{\ket{1}, ..., \ket{N}} &\cequiv \set{\inlinematrix{1 \\ 0 \\ : \\ 0}, \inlinematrix{0 \\ 1 \\ : \\ 0}, ..., \inlinematrix{0 \\ : \\ 0 \\ 1}}\\ + \ket{\phi} &\cequiv \inlinematrix{c_1 \\ : \\ c_N}\\ + \bra{\phi} = \sum_n d_n^* \bra{n} &\cequiv (d_1^*, ..., d_N^*) +\end{align} +\begin{equation} + \braket{\phi}{\psi} = \sum_n d_n^* c_n +\end{equation} + +\subsection*{Operator $A$} +\begin{align} + A \ket{\phi} &= \sum_n c_n A \ket{n}\\ + \braket{\phi}{A \psi} &= \sum_{n, m} d_m A_{m,n} c_n\\ + &= (d_1^*, ..., d_N^*) \inlinematrix{A_{1,1} & \cdots & A_{1,N} \\ \vdots & \ddots & \vdots \\ A_{N,1} & \cdots & A_{N,N}} \inlinematrix{c_1 \\ : \\ c_N} +\end{align} +Natürlich ist auch +\begin{equation} + AB \neq BA +\end{equation} +. + +\section{Projektionsoperatoren} +Seien $\hilbert_1$ und $\hilbert_2$ orthogonale Unterräume von $\hilbert$. +Jeder ket $\ket{\phi}$ kann ein geuteig zerlegt werden: +\begin{equation} + \ket{\psi} = \ket{\psi_1} + \ket{\psi_2} +\end{equation} + +\subsection*{Definition} +\begin{equation} + P_1\ket{\psi} \equiv \ket{\psi_1} +\end{equation} +\begin{itemize} + \item $P_1$ linear $\checkmark$ + \item $P_1$ hermitesch + \begin{align} + \braket{\phi}{P_1^\dagger \psi} &= \braket{P_1 \phi}{ \psi}\\ + &= \braket{\psi}{P_1 \phi}^* = \braket{\psi}{\phi_1}^*\\ + &= \braket{\phi_1}{\psi} = \braket{\phi}{\psi_1}\\ + &= \braket{\phi}{P_1 \psi} \checkmark + \end{align} +\end{itemize} + +\subsection*{Projektion auf einen Basisvektor $\ket{n}$} +\begin{equation} + P_n = \ket{n}\bra{n} +\end{equation} +denn +\begin{align} + P_n \ket{\psi} &= \ket{n}\braket{n}{\psi} + &= c_n \ket{n} +\end{align} + +\subsection*{Zerlegung der $\one$} +\begin{equation} + \one = \sum_n \ket{n}\bra{n} +\end{equation} +denn +\begin{align} + \one \ket{\psi} = \ket{\psi} &= \sum_n \ket{n} \braket{n}{\psi}\\ + &= \sum_n \ket{n} c_n +\end{align} + +Als Matrix: +\begin{align} + P_n &= \ket{n}\bra{n} \cequiv \inlinematrix{0\\ :\\ 1\\ :\\ 0} (0, .., 1, .., 0)\\ + \one &= \sum_n P_n = \inlinematrix{1 & & 0 \\ & \ddots & \\ 0 & & 1} +\end{align} + +\section{Unitäre Operatoren} +\subsection*{Definition} +\begin{equation} + \text{$U$ unitär} \gdw UU^\dagger = U^\dagger U = \one \gdw U^\dagger = U^{-1} +\end{equation} + +\subsection*{Satz} +\begin{equation} + \text{$U$ unitär} \gdw \norm{U \phi} = \norm{\phi} +\end{equation} +\paragraph{Beweis:} +\begin{align} + \norm{\phi + c \chi}^2 &= \braket{\phi + c \chi}{\phi + c \chi} = \braket{\phi}{\phi} + 2 \re{c \braket{\phi}{\chi}} + cc^* \braket{\chi}{\chi} \\ + \norm{U (\phi + c \chi)}^2 &= \braket{U \phi}{U \phi} + 2 \re{c \braket{U \phi}{U \chi}} + cc^* \braket{U \chi}{U \chi} +\end{align} + +\paragraph{``$\Leftarrow$'':} +\begin{align} + \re{c \braket{\phi}{\chi}} &= \re{c \braket{U \phi}{U \chi}}\\ + c = 1 \re{\braket{\phi}{\chi}} &= \re{\braket{U \phi}{U \chi}}\\ + c = i \im{\braket{\phi}{\chi}} &= \im{\braket{U \phi}{U \chi}}\\ + \rightarrow \braket{\phi}{\chi} &= \braket{U \phi}{U \chi}\\ + &= \dirac{\phi}{U^\dagger U}{\chi} +\end{align} +mit $U^\dagger U = \one$ + +\paragraph{``$\Rightarrow$'':} +\begin{equation} + \norm{U \phi}^2 = \braket{U \phi}{U \phi} = \dirac{\phi}{U^\dagger U}{\phi} = \braket{\phi}{\phi} = \norm{\phi}^2 +\end{equation} + +\paragraph{Bemerkung:} +Unitäre Operatoren vermitteln Basiswechsel:\\ +gegeben $\set{\ket{n}}$, definiere +\begin{equation} + \set{\ket{n'}} \equiv \set{U \ket{n}} +\end{equation} +denn: +\begin{align} + \braket{m'}{n'} &= \braket{Um}{Un} = \dirac{m}{U^\dagger U}{n} = \braket{m}{n}\\ + &= \delta_{m,n} +\end{align} +\begin{enumerate} + \item $\ket{\psi} = \sum_n c_n \ket{n} = \sum_{n'} c_{n'} \ket{n'}$\\ + mit $c_{n'} = \braket{n'}{\psi} = \braket{Un}{\psi}$ + \item Matrixelemente $A'_{m,n} \equiv \dirac{m'}{A}{n'} = \sum_{k,l} U_{m,k}^\dagger A_{k,l} U_{l,m}$ +\end{enumerate} + +\section{Sektralzerlegung von hermitschen Operatoren} +\subsection*{Satz} +\begin{align} + \text{A hermitesch} \Rightarrow &\text{(1) Eigenwerte $a_n$ sind reell}\\ + &\text{(2) Eigenvektoren zu unterschiedlichen Eigenwerten sind orthognal} +\end{align} +\paragraph{Beweis} +\begin{equation} + A \ket{a_n} = a_n \ket{a_n} +\end{equation} +\subparagraph{zu (1)} +\begin{align} + A \ket{a_n} &= a_n \ket{a_n} &\left| \bra{a_n} \right.\\ + \braket{a_n A}{a_n} &= a_n \braket{a_n}{a_n} +\end{align} +\begin{align} + \braket{a_n A}{a_n} &= \braket{a_n (a_n)}{a_n} = a_n^* \braket{a_n}{a_n} +\end{align} +\begin{math} + \Rightarrow a_n = a_n^* \text{ \QED} +\end{math} +\subparagraph{zu (2)} +\begin{align} + A \ket{a_m} &= a_m \ket{a_n}\\ + A \ket{a_m} &= a_m \ket{a_m} +\end{align} +\begin{align} + \dirac{a_n}{A}{a_m} &= a_m \braket{a_n}{a_m}\\ + &= a_n \braket{a_n}{a_m} +\end{align} +\begin{math} + a_m \neq a_n \Rightarrow \braket{a_n}{a_m} = 0 \text{ \QED} +\end{math} + +\subsection*{Zwei Fälle} +\begin{enumerate} + \item alle $a_n$ unterschiedlich + $\rightarrow \set{\ket{a_n}}$ bildet Basis + \item nicht alle $a_n$ unterschiedlich + Dann gibt es immer eine untäre Transformation $U$ mit + \begin{equation} + U^{-1}AU = \inlinematrix{a_1 & & 0 \\ & \ddots & \\ 0 & & a_N} + \end{equation} + $\rightarrow$ orthogonle Basis konstruiert. +\end{enumerate} + +Sei $g(n)$ die Entartung von Eigenwert $a_n$. Im Unterraum gibt es also $g(n)$ Eigenvektoren: + +$\ket{n,r}$ mit $r=1, ..., g(n)$ +\begin{equation} + A \ket{n,r} = a_n \ket{n,r} +\end{equation} + +\paragraph{Projektion auf diesen Unterraum} +\begin{equation} + P_n = \sum_{r=1}^{g(n)}\ket{n,r}\bra{n,r} +\end{equation} +\begin{equation} + \sum_n P_n = \sum_n \sum_{r=1}^{g(n)}\ket{n,r}\bra{n,r} = \one +\end{equation} + +\begin{align} + A \ket{\psi} = A \one \ket{\psi} &= A \sum_n \sum_{r=1}^{g(n)}\ket{n,r}\braket{n,r}{\psi}\\ + &= \sum_n a_n \sum_{r=1}^{g(n)}\ket{n,r}\braket{n,r}{\psi} +\end{align} +\begin{align} + A &= \sum_n a_n \sum_{r=1}^{g(n)}\ket{n,r}\bra{n,r}\\ + &= \sum_n a_n \ket{n}\bra{n} = \sum_n a_n \ket{a_n}\bra{a_n} & \text{(nicht entartet)} +\end{align} + +\section{Vollständiger Satz kommutierender Operatoren} +\subsection*{Definition} +\begin{equation} + \text{$A,B$ kommutieren} \gdw AB -BA = [A,B] = 0 +\end{equation} + +\subsection*{Satz} +\begin{equation} + \text{$A$, $B$ hermitesch und $[A,B] = 0$} \Rightarrow \text{es existiert eine gemeinsame Eigenbasis} +\end{equation} + +\paragraph{Beweis} +\begin{align} + A \ket{n,r} &= a_n \ket{n,r} & B_\rightarrow\\ + BA \ket{n,r} &= a_n B \ket{n,r}\\ + A (B \ket{n,r}) &= a_n (B \ket{n,r}) +\end{align} +$\rightarrow$ $B \ket{n,r}$ liegt im Untrraum zu $a_n$ +\begin{description} + \item[Fall (1)] $a_n$ nicht entartet ($\ket{n,r} \equiv \ket{n}$) + \begin{equation} + B \ket{n} = b_n \ket{n} + \end{equation} + \item[Fall (2)] $a_n$ entartet + \begin{equation} + \bra{m,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} + \end{equation} + Falls $B^{(n)}$ entartet, gibt es einen dritten Opertor $C$ mit $[A,C] = [B,C] = 0$. +\end{description} + +\subsection*{Definition} +Das Ensemble $set{A^1, ..., A^M}$ wechselseitig kommutierender Operatoren, deren Eigenwerte $(a_{n_1}^1, ..., a_{n_N}^M)$ mit zugehörigen Eigenvektoren $\set{\ket{a_{n_1}^1, ..., a_{n_N}^M}}$ mit +\begin{equation} + A^k \ket{a_{n_1}^1, ..., a_{n_N}^M} = a_{n_k}^k \ket{a_{n_1}^1, ..., a_{n_N}^M} +\end{equation} +eine eindeutige Basis definieren einen vollständigen Satz kommutierender Operatoren (VSKO, CSCO). + +\section{Operatorfunktionen} +Sei $A$ Operator, definiere $f(A)$ +\begin{enumerate} + \item über die Potenzreihe + \begin{equation} + f(A) = \sum_{n=0}^{\inf} \frac{1}{n!} f^{(n)}(n) A^n + \end{equation} + $\lboxed{\text{Bsp: } e^A = \sum_{n=0}^{\inf} \frac{1}{n!} A^n}$ + \item für hermitesche + \begin{align} + f(A) &= \sum_{n=1}^N f(a_n) \ket{a_n}\bra{a_n} & \text{(nicht entartet)}\\ + &= \sum_n \sum_{r=1}^{g(n)} f(a_n) \ket{n,r}\bra{n,r} + \end{align} + beachte: $e^A \cdot e^B \neq e^{A+B}$ +\end{enumerate} diff --git a/kapI-3.tex b/kapI-3.tex new file mode 100644 index 0000000..c4de4dd --- /dev/null +++ b/kapI-3.tex @@ -0,0 +1,134 @@ +\chapter{Prinzipien der Quantenmechanik Teil 1: Kinematik} +\section{Zustands- und Superpositionspostulat} +\paragraph{(P1)} Bei \underline{vollständiger Kenntinis} eines Quantalen Systems wird dieses durch einen (normalisierten) Vektor $\psi$ im zugehörigen Hilbertraum $\hilbert$ repräsentiert.\\ +Mit $\ket{\psi_1}$ und $\ket{\psi_2}$ ist auch die Superposition $\ket{\psi} = c_1 \ket{\psi_1} + c_2 \ket{\psi_2}$ ($c_{1,2} \in \setC$ geeignet normiert) ein möglicher Zustand. +\paragraph{Bemerkung} klassisch: (1) $(q^i, p^i)$ im Phasenraum; (2) $(q^1, p^1) \stackrel{\text{??}}{+} (q^2, p^2)$ ist Unsinn! + +\section{Operator-, Mess- und Präparationspostulat} +\paragraph{(P2a)} +Jeder physikalischen Größe $A$ entspricht ein hermitescher Operator $A$. +\paragraph{(P2b)} +Eine Messung der Größe $A$ im Zustand $\ket{\psi_0}$ ergibt \underline{mit Sicherheit} eine der reellen Eigenwerte $a_j$ ($A \ket{a_j} = a_j \ket{a_j}$). Die Wahrscheinlichkeit genau ein $a_n$ +zu messen beträgt: +\begin{align} + \prob{\left. A \widehat{=} a_n \right| \ket{\psi}} &= \abs{\braket{a_n}{\psi_0}}^2 & \text{nicht entartet}\\ + &= \sum_{r=1}^{g(n)} \abs{\braket{n,r}{\psi_0}}^2 & \text{entartet}\\ + &\lboxed{\stackrel{\text{Not.}}{=} p_n} +\end{align} +\begin{align} + \sum_{n=1}^{N} \prob{} &= \sum_n \abs{\braket{a_n}{\psi_0}}^2\\ + &= \sum_n \braket{\psi_0}{a_n} \braket{a_n}{\psi_0}\\ + &= \sum_n \dirac{\psi_0}{\one}{\psi_0}\\ + &= 1 +\end{align} + +\paragraph{(P2c)} +Unmittelbar nach der Messung mit Messwert $a_n$ ist das System im Zustand +\begin{align} + \ket{\psi} &= P_n \ket{\psi_0}\\ + &= \ket{a_n} & \text{(bei Nichtentartung)}\\ + &= \sum_{r=1}^{g(n)} \ket{n,r} \frac{\braket{n,r}{\psi_0}}{\sum_n \abs{\braket{n,r}{\psi_n}}^2} +\end{align} + +\paragraph{Bemerkung} vergleiche klassisch: (1) Physikalische Größen sind Phasenraumfunktionen ($\vec{r}, \vec{p}, \vec{l}, H, V$), die immer eindeutige Werte haben. (2) Messung ändert Zustand nicht. + +\section{Wichtiger Spezialfall (``Bom'sche Regel'')} +Sei $A$ die Eigenschaft im Zustand $\ket{\chi}$ zu sein, zugehöriger Operator $P_\chi = \ket{\chi}\bra{\chi}$ (Eigenwerte: 1 ... $P_\chi \ket{\chi} = 1 \ket{\chi}$ und 0 ... alle orthogonal).\\ +Die Wahrscheinlichkeit ``1'' zu messen im Zustand $\ket{\chi}$: +\begin{equation} + \prob{\left. P_\chi = 1 \right| \ket{\chi}} = \abs{\braket{\chi}{\psi_0}}^2 +\end{equation} + +\section{Erwartungswert und Messwert} +\begin{itemize} + \item Einzelmessung $a_n$ mit Wahrscheinlichkeit $p_n = \abs{\braket{a_n}{\psi}}^2$ + \item Mittelwert vieler Messungen an identisch präparierten System(en) + \begin{align} + < A >_\psi &= \sum_{n=1}^N a_n p_n = \sum_{n=1}^N a_n \braket{a_n}{\psi}\braket{\psi}{a_n} = \sum_n \braket{\psi}{a_n} a_n \braket{a_n}{\psi}\\ + &= \dirac{\psi}{A}{\psi} + \end{align} + ($< A >_\psi$ heißt leider ``Erwartungswert'')\\ + Beispiel: $\ket{\psi} = \ket{x+}$, $\sigma_z$ Messung + \begin{align} + < \sigma_z >_\ket{x+} &= (+1) \prob{\left. \sigma_z \cequiv +1 \right| \ket{x+}} + (-1) \prob{\left. \sigma_z \cequiv -1 \right| \ket{x+}}\\ + &= 1 \frac{1}{2} - 1 \frac{1}{2}\\ + &= 0 + \end{align} +\end{itemize} + +\section{Heisenberg'sche Unschärferelation} +\paragraph{Definition} Sei $(\Delta A)_\psi^2$ ``Dispersion'' oder Streuung mit +\begin{equation} + (\Delta A)_\psi^2 \equiv \sum_n p_n (a_n - < A >_\psi)^2 +\end{equation} +. + +\begin{align} + (\Delta A)_\psi^2 &= \sum_n p_n (a_n - < A >_\psi)^2\\ + &= \sum_n \braket{\psi}{n} \braket{n}{\psi} (\underbrace{a_n^2}_{\boxed{1}} - \underbrace{2 a_n < A >_\psi}_{\boxed{2}} + < A >_\psi^2)\\ + \boxed{1} &= \sum_n \braket{\psi}{n} a_n^2 \braket{n}{\psi}\\ + &= \dirac{\psi}{A^2}{\psi} &\text{denn: } A^2 = \sum_n \ket{n} a_n^2 \bra{n}\\ + \boxed{2} &= -2 < A >_\psi < A >_\psi\\[10pt] + \rightarrow (\Delta A)_\psi^2 &= < A^2 >_\psi - < A >_\psi^2 +\end{align} + +\begin{equation} + (\Delta A)_{\psi_0}^2 = 0 \gdw \ket{\psi} \text{ ist Eigenzustand (-vektor) von A.} +\end{equation} + +\paragraph{Satz} Seien $A$, $B$ physikalische Größen, d.h. hermitesche Operatoren mit $[A,B] = iC$, dann gilt für die Streuung d.h. ``Unschärfe'' der $A$ und $B$ messwerte: +\begin{equation} + (\Delta A)_{\psi_0} \cdot (\Delta B)_{\psi_0} \leq \frac{1}{2} \abs{< C >_{\psi_0}} +\end{equation} + +\subparagraph*{Beweis} +\begin{itemize} + \item $C$ hermitesch: + \begin{align} + [A,B]^\dagger &= (AB)^\dagger - (BA)^\dagger\\ + &= B^\dagger A^\dagger - A^\dagger B^\dagger\\ + &\stackrel{A, B\text{ herm.}}{=} BA - AB\\ + &= [B,A] = -[A,B] + \end{align} + + \begin{equation} + \rightarrow C^\dagger = (-i [A,B])^\dagger = -i [A,B] = C + \end{equation} + \item Seien $A_0 \equiv A - < A >_{\psi_0} \one$ und $B_0 \equiv B - < B >_{\psi_0} \one$. + \begin{align} + \rightarrow [A_0, B_0] &= iC\\ + \text{und } (\Delta A)_{\psi_0}^2 &= < A_0^2 >\\ + (\Delta B)_{\psi_0}^2 &= < B_0^2 > + \end{align} + + \begin{align} + 0 \leq \norm{(A_0 + i \lambda B_0) \psi_0}^2 &=_{\forall \lambda \in \setR} \braket{\psi_0}{(A_0 - i \lambda B_0)(A_0 + i \lambda B_0) \psi_0} & \\ + &= \norm{A_0 \psi_0}^2 + \lambda^2 \norm{B_0 \psi_0}^2 + i \lambda \left( \dirac{\psi_0}{A_0 B_0}{\psi_0} - \dirac{\psi_0}{B_0 A_0}{\psi_0} \right) + \end{align} + + \begin{align} + \rightarrow 0 &\leq \norm{A_0 \psi_0}^2 \lambda^2 \norm{B_0 \psi_0}^2 - \lambda < C >_{\psi_0}\\ + \rightarrow < C >_{\psi_0}^2 &\leq 4 < A_0^2 >_{\psi_0} < B_0^2 >_{\psi_0} + \end{align} +\end{itemize} +\begin{flushright} + $\square$ +\end{flushright} + +\paragraph{Bemerkung} +\begin{itemize} + \item Messung nicht kommutierender Größen können nicht beide beliebig scharf sein. + \item Falls $\ket{\psi_0}$ Eigenzustand von $A$ ($\rightarrow (\Delta A)_{\psi_0}^2$) + \begin{equation} + \rightarrow < C >_{\psi_0} \stackrel{!}{=} 0 + \end{equation} + + Beispiel (Spin $\frac{1}{2}$): + \begin{align} + &A \cequiv \sigma_z, B \cequiv \sigma_x\\ + \rightarrow &C = 2\sigma_y \text{ mit } \ket{\psi_0} = \ket{z+}\\ + \rightarrow &< \sigma_y >_\ket{z+} = 0 + \end{align} + +\end{itemize} + diff --git a/kapI-4.tex b/kapI-4.tex new file mode 100644 index 0000000..01f04fb --- /dev/null +++ b/kapI-4.tex @@ -0,0 +1,207 @@ +\chapter{Quanten-Kinematik des Spin 1/2 Systems} + +\paragraph{Idee} Kombination der Prinzipien und Experimente führt auf $\hilbert$ und die konkreten Formen der Zustände und Operatoren. + +\section{Hilbertraum und $\sigma_z$-Darstellung} +\begin{itemize} + \item immer nur $\pm 1$ als Messwert %TODO $\rightarrow$ $\hilbert = 2$ + \item wähle als Basis $\set{\ket{1}, \ket{2}} = \set{\ket{z+}, \ket{z-}} = \set{\inlinematrix{1\\ 0}, \inlinematrix{0\\ 1}}$ die Eigenvektoren des zu ``Spin in z-Richtung'', $\sigma_z$, gehörenden Operatoren: + \begin{align} + \sigma_z \ket{z+} &= (+1) \ket{z+}\\ + \sigma_z \ket{z-} &= (-1) \ket{z-} + \end{align} + zugehörige Bra: $\bra{z+} \cequiv (1, 0)$ und $\bra{z-} \cequiv (0, 1)$ + \begin{equation} + \sigma_z \cequiv \inlinematrix{1 & 0 \\ 0 & -1} + \end{equation} + \item beliebiger Zustand $\psi = c_1 \ket{z+} + c_2 \ket{z-}$ mit normierung $\braket{\psi}{\psi} = 1 \Rightarrow c_1 c_1^* + c_2 c_2^* = 1$ +\end{itemize} + +\subsection*{Ex. 1} +\begin{figure}[h] + \includegraphics{4-001.pdf} +\end{figure} +\begin{align} + \prob{\left. \sigma_z \cequiv +1 \right| \ket{\psi_0} = \ket{z+}} &= \abs{\braket{z+}{\psi_0}}^2\\ + &= \abs{\braket{z+}{z+}}^2 = 1\\ + \prob{\left. \sigma_z \cequiv -1 \right| \ket{\psi_0} = \ket{z-}} &= \abs{\braket{z-}{\psi_0}}^2\\ + &= \abs{\braket{z-}{z+}}^2 = 0 +\end{align} + +\section{Spin-Operatoren} +\paragraph{Frage} +Welcher Operator $\sigma_z$ entspricht der physikalischen Größe Spin in n-Richtung? + +\paragraph{Wir wissen} +(1) $\sigma_n \cequiv 2 \times 2$ Matrix, hermitesch, $\inlinematrix{a & b \\ b^* & d}$ (2) mit Eigenwetren $\pm 1$: $\tr \sigma_n = 0$, $\det \sigma_n = 1$ +\begin{equation} + \rightarrow \sigma_n = \inlinematrix{a & b \\ b^* & -a} +\end{equation} +\begin{align} + \det \sigma_n = -a^2-bb^* &= -1\\ + \rightarrow a^2+bb^* &= 1 +\end{align} + +\paragraph{Ansatz} +\begin{equation} + \sigma_n = \inlinematrix{\cos \beta & e^{-i \alpha} \sin \beta \\ e^{i \alpha} \sin \beta & -\cos \beta} \text{ mit } \alpha = \alpha (n), \beta = \beta (n) +\end{equation} +\[ + (\alpha (n=z) = 0; \beta (n=z) = 0) +\] + +\paragraph{Eigenvektoren} +\begin{equation} + \sigma_n \underbrace{{\inlinematrix{\cos \frac{\beta}{2} \\ e^{i \alpha} \sin \frac{\beta}{2}}}}_{\equiv \ket{n+}} = +1 \inlinematrix{\cos \frac{\beta}{2} \\ e^{i \alpha} \sin \frac{\beta}{2}} \text{ (Phase ist Konvention)} +\end{equation} + +\begin{equation} + \rightarrow \sigma_n \ket{n+} = \ket{n+} +\end{equation} + +\begin{equation} + \sigma \inlinematrix{\cos \frac{\beta}{2} \\ e^{-i \alpha} \sin \frac{\beta}{2}} = (-1) \inlinematrix{\cos \frac{\beta}{2} \\ e^{-i \alpha} \sin \frac{\beta}{2}} \equiv (-1) \ket{n-} +\end{equation} + +\paragraph{Form von $\sigma_x$} +\begin{equation} + \ket{x+} = \inlinematrix{\cos \frac{\beta_x}{2} \\ e^{i \alpha_x} \sin \frac{\beta_x}{2}} +\end{equation} + +\subsection*{Ex. 2a} +\begin{figure}[h] + \includegraphics{4-002.pdf} +\end{figure} +\begin{align} + \prob{\left. \sigma_x \cequiv +1 \right| \ket{\psi_0} = \ket{z+}} \stackrel{Ex}{=} \frac{1}{2} &=\stackrel{P2b}{=} \abs{\braket{x+}{z+}}^2\\ + &= \abs{(\cos \frac{\beta_x}{2}, e^{-i \alpha_x} \sin \frac{\beta_x}{2}) \inlinematrix{1 \\ 0}}^2\\ + &= \cos^2 \frac{\beta_x}{2} +\end{align} + +\begin{equation} + \rightarrow \beta_x = \frac{\pi}{2} +\end{equation} + +\begin{equation} + \Rightarrow \sigma_x = \inlinematrix{0 & e^{-i \alpha_x} \\ e^{i \alpha_x} & 0} +\end{equation} + +\subsection*{Ex. 2b} +analog zu Ex. 2a: +\begin{equation} + \Rightarrow \sigma_y = \inlinematrix{0 & e^{-i \alpha_y} \\ e^{i \alpha_y} & 0} +\end{equation} + +\subsection*{Ex. 2c} +\begin{figure}[h] + \includegraphics{4-003.pdf} +\end{figure} +\begin{align} + \prob{\left. \sigma \cequiv +1 \right| \ket{x+}} \stackrel{Ex}{=} \frac{1}{2} &\stackrel{P2b}{=} \abs{\braket{y+}{x+}}^2\\ + &= \abs{(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} e^{-i \alpha_y}) \inlinematrix{\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} e^{i \alpha_x}}}^2\\ + &= \abs{\frac{1}{2} (1 + e^{i(\alpha_x - \alpha_y)})}^2\\ + &= \frac{1}{4} \abs{1 + \underbrace{e^{i(\alpha_x - \alpha_y)}}_{\stackrel{!}{=}\pm i}}^2 +\end{align} + +\begin{equation} + \rightarrow \alpha_x - \alpha_y = \pm \frac{\pi}{2} +\end{equation} +Konvention: $\alpha_x = 0;$ $\alpha_x = \frac{\pi}{2}$ + +\section*{Zusammenfassung} +% \begin{displaymath} +% \begin{array}{lllr} +% \sigma_z = \inlinematrix{1 & 0 \\ 0 & -1} & \ket{z+} = \inlinematrix{1 \\ 0} & \ket{z-} = \inlinematrix{0 \\ 1} & \refstepcounter{equation}(\theequation)\\ +% \sigma_x = \inlinematrix{0 & 1 \\ 1 & 0} & \ket{x+} = \frac{1}{\sqrt{2}}\inlinematrix{1 \\ 1} & \ket{x-} = \frac{1}{\sqrt{2}}\inlinematrix{1 \\ -1}\\ +% \sigma_y = \inlinematrix{0 & -i \\ i & 0} & \ket{y+} = \frac{1}{\sqrt{2}}\inlinematrix{1 \\ i} & \ket{y-}= \frac{1}{\sqrt{2}}\inlinematrix{1 \\ -i} +% \end{array} +% \end{displaymath} +\begin{align} + \sigma_z &= \inlinematrix{1 & 0 \\ 0 & -1} & \ket{z+} &= \inlinematrix{1 \\ 0} & \ket{z-} &= \inlinematrix{0 \\ 1}\\ + \sigma_x &= \inlinematrix{0 & 1 \\ 1 & 0} & \ket{x+} &= \frac{1}{\sqrt{2}}\inlinematrix{1 \\ 1} & \ket{x-} &= \frac{1}{\sqrt{2}}\inlinematrix{1 \\ -1}\\ + \sigma_y &= \inlinematrix{0 & -i \\ i & 0} & \ket{y+} &= \frac{1}{\sqrt{2}}\inlinematrix{1 \\ i} & \ket{y-} &= \frac{1}{\sqrt{2}}\inlinematrix{1 \\ -i} +\end{align} + +\section*{Paulimatrizen} +\begin{equation} + \sigma_i \sigma_j = \delta_{i,j} + i \sum_k \epsilon_{i,j,k} \sigma_k +\end{equation} + +\begin{align} + \rightarrow \sigma_\alpha^2 &= \one\\ + \sigma_x \sigma_y &= i \sigma_z \text{ etc. cyclisch}\\ + \left[ \sigma_i, \sigma_j \right] &= 2i \epsilon_{i,j,k} \sigma_k +\end{align} + +\section*{Allgemeine Form von $\sigma_n$} +\begin{figure}[h] + \includegraphics{4-004.pdf} +\end{figure} +\begin{equation} + \vec{n} = \inlinematrix{n_x\\ n_y\\ n_z} = \inlinematrix{\sin \theta \cos \phi\\ \sin \theta \sin \phi\\ \cos \theta} +\end{equation}\\ +Die Wahrscheinlichkeiten +\begin{equation} + \prob{\left. \sigma_z \cequiv +1 \right| \ket{n+}} = p_{z+}(n) +\end{equation} +bilden nun die Erwartungswerte: +\begin{align} + < \sigma_x >_\ket{n+} &= \dirac{n+}{\sigma_x}{n+} \left(= p_{x+}(n) - p_{x-}(n) \right)\\ + < \sigma_y >_\ket{n+} &= \dirac{n+}{\sigma_y}{n+}\\ + < \sigma_z >_\ket{n+} &= \dirac{n+}{\sigma_z}{n+} +\end{align}\\ +Fasse die Erwartungswerte zu einem Vektor zusammen: +\begin{equation} + < \vec(\sigma) >_\ket{n+} = \inlinematrix{< \sigma_x >\\ < \sigma_y >\\ < \sigma_z >} +\end{equation}\\ +Es gilt (experimentell oder aus Rotationsinvarianz): +\begin{equation} + < \vec{\sigma} >_\ket{n+} = \vec{n} +\end{equation}\\ +Konkret: +\begin{align} + \dirac{n+}{\sigma_x}{n+} &= \left( \cos \frac{\beta}{2}, e^{-i \alpha} \sin \frac{\beta}{2} \right) \inlinematrix{0 & 1 \\ 1 & 0} \inlinematrix{\cos \frac{\beta}{2} \\ e^{i \alpha} \sin \frac{\beta}{2}}\\ + &= \left( e^{i \alpha} + e^{-i \alpha} \right) \sin \frac{\beta}{2} \cos \frac{\beta}{2}\\ + &= 2 \cos \alpha \frac{1}{2} \sin \beta\\ + &= \cos \alpha \sin \beta\\ + &\stackrel{!}{=} \sin \theta \cos \phi\\[15pt] + \dirac{n+}{\sigma_y}{n+} &= ... = \sin \alpha \sin \beta\\ + &\stackrel{!}{=} \sin \theta \sin \phi\\[15pt] + \dirac{n+}{\sigma_z}{n+} &= ... = \cos \beta\\ + &\stackrel{!}{=} \cos \theta\\[15pt] + \rightarrow \beta &= 0\\ + \alpha &= \phi +\end{align} + +\subsection*{zu Ex 4} +\begin{figure}[h] + \includegraphics{4-005.pdf} +\end{figure} +In den 2. SG,z: +\begin{align} + \ket{\psi} &= \left( \ket{x+} \braket{x+}{z+} + \ket{x-} \braket{x-}{z+} \right)\\ + &= \frac{1}{\sqrt{2}} \inlinematrix{1\\ 1} \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \inlinematrix{1\\ -1} \frac{1}{\sqrt{2}}\\ + &= \inlinematrix{1\\ 0}\\ + &= \ket{z+} +\end{align} +d.h. bei Nichtmessung (von $\sigma_z$) gilt die Superposition der Zustände mit den Wahrscheinlichkeitsamplituden. + +\section{Quantenkryptographie} +\paragraph{Ziel} Abhörsichere Versendung von Codes +\paragraph{Situation} Alice (A) sendet Bob (B) eine Nachricht. Eve (E) versucht heimlich abzuhören ohne dass A und B das merken. +\paragraph{Vorgehensweise} Bennett-Bassard Protokoll (binär $1,0 \cequiv \ket{+}, \ket{-}$) +%TODO: darstellung der basiswahl etc. +\begin{itemize} + \item A wählt für sich zufällig für jedes Bit eine $x$ oder $z$ Basis. + \item B wählt eine zufällige Basis und misst. + \item B sendet A seine Basenwahlsequenz öffentlich. + \item A identifiziert die Bits (genauer: deren Nummer) bei denen sie beide zufällig dieselbe Basis hatten und sendet B öffentlich die Positionen. Beide streichen die anderen Bits. + \item B sendet A öffentlich Position und Ergebnis einer Untermenge der verbl. Bits.: $(2 \rightarrow 0)$, $(4 \rightarrow 1)$, $(8 \rightarrow 0)$ + \item Alice überprüft die Resultate + \begin{itemize} + \item[Fall 1] $100\%$ übereinstimmung $\Rightarrow$ E war nicht dazwischen + \item[Fall 2] $25\%$ Fehler: E hat abgehört + \end{itemize} + Falls E abhört muss sie für jedes Bit eine Basis wählen, zu $50\%$ wählt sie eine andere Basis als A, zu weiteren $50\%$ misst B dann das falsche Bit. + \item Die restlichen Bits stehen nun als Verschlüsselung zur Verfügung. +\end{itemize} diff --git a/kapI-5.tex b/kapI-5.tex new file mode 100644 index 0000000..670bd7a --- /dev/null +++ b/kapI-5.tex @@ -0,0 +1,195 @@ +\chapter{Prinzipien der Quantenmechanik Teil 2: Dynamik} +\section{Dynamisches Prinzip: Schrödinger-Gleichung} +\paragraph{(P3, SG)} Nach einer Präparation und vor der nächsten Messung entwickelt sich ein quantaler Zustand gemäß +\begin{equation} + i\hbar \partial_t \ket{\psi} = H \ket{\psi(t)} +\end{equation} +mit +\begin{itemize} + \item $H$ (eventuell $H(t)$) dem hermiteschen Hamilton-Operator + \item $\hbar \approx 1,05 \cdot 10^{-34}$Js der (neuen) Fundamentalkonstanten +\end{itemize} + +\paragraph{Bemerkung} +\begin{itemize} + \item das klassische analogon zu $H$ ist die Hamilton-Funktion + \item vgl. mit $\dot{q} = \frac{\partial H}{\partial p}$, $\dot{p} = - \frac{\partial H}{\partial q}$ in der klassischen mechanik. +\end{itemize} + +\section{Beispiel: Spin 1/2 im konstanten Magnetfeld} +\begin{itemize} + \item[klassisch:] magn. Moment $\vec{\mu}$ im Magnetfeld $\vec{B}$\\ + Energie: + \begin{equation} + E = - \vec{\mu} \vec{B} + \end{equation} + \item[quantal:] mit $\mu_B \equiv \frac{\abs{e}}{2 m_e c}$ + \begin{equation} + \vec{\mu} = - \underbrace{g}_{text{g-Faktor}} \mu_B \frac{1}{2} \vec{\sigma} + \end{equation} + $\rightarrow$ Hamilton-Operator + \begin{equation} + H = g \mu_B \frac{1}{2} \vec{\sigma} \cdot \vec{B} + \end{equation} + o.B.d.A. mit $\omega \equiv \frac{g \mu_B B}{\hbar}$ und $\vec{B} = B \vec{e_z}$ + \begin{equation} + H = \frac{\hbar}{2} \omega \sigma_z + \end{equation}\\ + $\rightarrow$ SG: + \begin{equation} + i \hbar \partial_t \ket{\psi(t)} = \frac{\hbar}{2} \omega \sigma_z \ket{\psi(t)} + \end{equation}\\ + beliebiger Zustand + \begin{align} + \ket{\psi} &= c_+(t) \ket{z+} + c_-(t) \ket{z-}\\ + &= \inlinematrix{c_+(t) \\ c_-(t)} + \end{align}\\ + $\rightarrow$ SG: + \begin{align} + i\hbar \partial_t \inlinematrix{c_+(t) \\ c_-(t)} &= \frac{\hbar \omega}{2} \inlinematrix{1 & 0 \\ 0 & -1} \inlinematrix{c_+(t) \\ c_-(t)}\\[15pt] + i\hbar \dot{c}_+(t) &= \frac{\hbar \omega}{2} c_+(t)\\ + i\hbar \dot{c}_-(t) &= \frac{- \hbar \omega}{2} c_-(t) + \end{align} + \begin{align} + \rightarrow c_+(t) &= e^{- \frac{i \omega}{2} (t-t_0} c_+(t_0)\\ + c_-(t) &= e^{\frac{i \omega}{2} (t-t_0} c_-(t_0) + \end{align} + + \begin{align} + \rightarrow \ket{\psi(t)} &= e^{-\frac{i \omega}{2} (t-t_0)} c_+(t_0) \ket{z+} + e^{\frac{i \omega}{2} (t-t_0)} c_-(t_0) \ket{z-}\\ + \ket{\psi(t_0)} &= c_+(t_0) \ket{z+} + c_-(t_0) \ket{z-} + \end{align} +\end{itemize} + +\subsection*{Messung zur Zeit t} +\begin{align} + \prob{\left. \sigma_z \cequiv +1 \right| \ket{\psi(t)}} &= \abs{\braket{z+}{\psi(t)}}^2\\ + &= \abs{\bra{z+}\left( e^{-\frac{i \omega}{2} (t-t_0)} c_+(t_0) \ket{z+} + e^{-\frac{i \omega}{2} (t-t_0)} c_-(t_0) \ket{z-} \right)}^2\\ + &= \abs{e^{-\frac{i \omega}{2} (t-t_0)} c_+(t_0)}^2\\ + &= \abs{c_+(t_0)}^2 +\end{align} +$\rightarrow$ unabhängig von $t$! + +\begin{align} + \prob{\left. \sigma_x \cequiv +1\right| \ket{\psi(t)}} &= \abs{\braket{x+}{\psi(t)}}^2\\ + &= \frac{1}{2} \abs{\left( \bra{z+} + \bra{z-} \right) \ket{\psi(t)}}^2\\ + &= \frac{1}{2} \abs{e^{-\frac{i \omega}{2} (t-t_0)} c_+(t_0) + e^{-\frac{i \omega}{2} (t-t_0)} c_-(t_0)}^2 +\end{align} + +\paragraph{Beispiel} +\begin{equation} + \ket{\psi(t_0)} = \ket{x+} = \frac{1}{\sqrt{2}} \inlinematrix{1 \\ 1} +\end{equation} +\begin{figure}[h] + \includegraphics{5-001.pdf} +\end{figure} +\begin{equation} + \prob{\left. \sigma_x \cequiv +1 \right| \ket{\psi(t)}} = \frac{1}{4} 4 \cos^2 \frac{\omega}{2} (t-t_0) +\end{equation} +Mittelwert: +\begin{equation} + < \sigma_x >_\ket{\psi(t)} \equiv < \sigma_x >(t) = \dirac{\psi(t)}{\sigma_x}{\psi(t)} = \cos(\omega(t-t_o)) +\end{equation} +d.h. der Mittelwert präzediert um die z-Achse. + +\section{Spin 1/2 im rotierenden Magnetfeld: Rabi-Oszillationen} +\begin{equation} + \vec{B}(t) = B_z \vec{e}_z + B_1 \left( \cos(\omega t) \vec{e}_x \sin(\omega t) \vec{e}_y \right) +\end{equation} +\begin{figure}[h] + \includegraphics{5-002.pdf} +\end{figure} +\begin{equation} + H(t) = \frac{\hbar \omega_0}{2} \sigma_z + \frac{\hbar \omega_1}{2} \left( \cos(\omega t) \sigma_x \sin(\omega t) \sigma_y \right) +\end{equation} +mit $\omega_{0,1} \equiv \frac{g \mu_B B_{z,1}}{\hbar}$\\[15pt] +in SG: +\begin{align} + i \hbar \inlinematrix{\dot{c}_+(t) \\ \dot{c}_-(t)} &= \hbar \inlinematrix{\frac{\omega_0}{2} & \frac{\omega_1}{2} \cos(\omega t) - i\sin(\omega t) \\ \frac{\omega_1}{2} \cos(\omega t) + i\sin(\omega t) & \frac{\omega_0}{2}} \inlinematrix{c_+(t) \\ c_-(t)}\\ + i \inlinematrix{\dot{c}_+ \\ \dot{c}_-} &= \inlinematrix{\frac{\omega_0}{2} & \frac{\omega_1}{2} e^{-i \omega t} \\ \frac{\omega_1}{2} e^{i \omega t} & \frac{\omega_0}{2}} \inlinematrix{c_+ \\ c_-} +\end{align} +mit $b_\pm \equiv e^{\pm i \frac{\omega}{2}t} c_\pm(t)$ +\begin{align} + i\dot{b}_+ &= -\frac{\omega - \omega_0}{2} b_+ + \frac{\omega_1}{2} b_-\\ + i\dot{b}_- &= \frac{\omega_1}{2} b_+ + \frac{\omega - \omega_0}{2} b_- +\end{align} +\begin{equation} + \ddot{b}_\pm = - \left( \frac{\Omega}{2} \right) b_\pm +\end{equation} +mit $\Omega^2 = (\omega - \omega_0)^2 + \omega_1^2$ vollständig gelöst (bis auf Trivialitäten) + +\paragraph{Konkretes Bsp. ($t_0 = 0$)} +\begin{align} + \ket{\psi(t_0)} &= \ket{z+} \rightarrow \left\lbrace \inlinematrixu{c_+(0) = 1, c_-(0) = 0 \\ b_+(0) = 1, b_-(0) = 0} \right.\\[15pt] + b_-(t) &= -\frac{i \omega}{\Omega} \sin\left( \frac{\Omega}{2} t \right)\\ + \rightarrow c_-(t) &= e^{\frac{i\omega}{2}t} \left( \frac{-i \omega_1}{\Omega} \right) \sin\left( \frac{\Omega}{2}t \right) +\end{align} +\begin{align} + \prob{\left. \sigma_z \cequiv -1 \right| \ket{\psi(t)}} &= \abs{\braket{z-}{\psi(t)}}^2\\ + &= \abs{c_-(t)}^2\\ + &= \left( \frac{\omega_1}{\Omega} \right)^2 \sin^2\left(\frac{\Omega}{2}t\right) +\end{align} +\begin{figure}[h] + \includegraphics{5-003.pdf} +\end{figure} + +\subparagraph{Resonanzfall} +$\omega_1 = \Omega$ d.h. $\omega = \omega_0$, d.h. $B_1$-Feld zirkuliert mit der in 5.2 berechneten Präzessionsfrequenz. +\begin{itemize} + \item Im Resonenzfall flippt der Spin mit Sicherheit auch für kleine $B_1$ + \item Rabi (1939, Nobel '44) misst meangetisches Moment des Protons durch\\ +% \begin{figure}[h] + \includegraphics{5-004.pdf}\\ + \includegraphics{5-005.pdf} +% \end{figure} + \item wichtige Anwendung: \underline{\underline{NMR}} (Idee: Magnetfeld hängt von der lokalen Umgebung ab.) +\end{itemize} + +\section{Transformationsverhalten unter Rotationen} +\subsection*{klassisch} +Vektor $\vec{a}$ wird gedreht mit Matrix $R_z(\varepsilon)$ um den Winkel $\varepsilon$ um die $z$-Achse +\begin{align} + R_z(\varepsilon) &= \inlinematrix{\cos \varepsilon & -\sin \varepsilon & 0 \\ \sin \varepsilon & \cos \varepsilon & 0 \\ 0 & 0 & 1} \stackrel{\text{kleine } \varepsilon}{\approx} \inlinematrix{1-\frac{\varepsilon^2}{2} & -\varepsilon & 0 \\ \varepsilon & 1-\frac{\varepsilon^2}{2} & 0 \\ 0 & 0 & 1}\\ + R_x(\varepsilon) &\approx \inlinematrix{1 & 0 & 0 \\ 0 & 1-\frac{\varepsilon^2}{2} & -\varepsilon \\ 0 & \varepsilon & 1-\frac{\varepsilon^2}{2}}\\ + R_y(\varepsilon) &\approx \inlinematrix{1-\frac{\varepsilon^2}{2} & 0 & \varepsilon \\ 0 & 1 & 0 \\ -\varepsilon & 0 & 1-\frac{\varepsilon^2}{2}} +\end{align} +\begin{equation} + R_x(\varepsilon) R_y(\varepsilon) - R_y(\varepsilon) R_x(\varepsilon) = \inlinematrix{0 & \varepsilon^2 & 0\\ \varepsilon^2 & 0 & 0 \\ 0 & 0 & 0} = R_z(\varepsilon^2) - \one +\end{equation} + +\subsection*{quantal} +Zustand $\ket{\psi}$ rotierten: +\begin{equation} + \ket{\psi} \rightarrow \ket{\tilde{\psi}} = D_{xyz}(\varepsilon)\ket{\psi} +\end{equation} +mit $D_{xyz} = \one - \frac{i \varepsilon}{\hbar} J_{xyz}$.\\[15pt] +endliche Rotationen: +\begin{equation} + D_\alpha(\phi) = \left( 1 - \frac{i}{\hbar} \frac{\phi}{N} J_\alpha \right)^N = e^{-\frac{i}{\hbar} \phi J_\alpha} +\end{equation} +Folgerung: $D_x(\varepsilon)$ erfüllen die Relation (5.42). +\begin{align} + D_x(\varepsilon)D_y(\varepsilon) - D_y(\varepsilon)D_x(\varepsilon) &= \left( 1 - \frac{i \varepsilon}{\hbar} J_x \right) \left( 1 - \frac{i \varepsilon}{\hbar} J_y \right) - \left( 1 - \frac{i \varepsilon}{\hbar} J_y \right) \left( 1 - \frac{i \varepsilon}{\hbar} J_x \right)\\ + &= -\frac{\varepsilon^2}{\hbar^2} \left[J_x, J_y\right] \stackrel{\text{(5.42) !}}{=} \left( 1 - \frac{i \varepsilon^2}{\hbar} J_z \right) +\end{align} +\begin{equation} + \rightarrow \left[ J_x, J_y \right] = i \hbar J_z +\end{equation} +. +\begin{equation} + J_\alpha = \frac{\hbar}{2} \sigma_\alpha \equiv S_\alpha +\end{equation} +$\rightarrow$ endl. Rotation um $n$-Achse +\begin{equation} + D_{\vec{n}} = e^{-\frac{i}{\hbar} \phi \vec{n} \vec{J}} = e^{-i \frac{\phi}{2} \vec{\sigma} \vec{n}} +\end{equation} +Bsp.: $\ket{x+}$ um $\phi$ um die $z$-Achse +\begin{align} + D_z(\phi) \ket{x+} &= e^{-\frac{i}{2} \phi \sigma_z} \left( \frac{1}{\sqrt{2}} \right) \left( \ket{z+} + \ket{z+} \right)\\ + &= \frac{1}{\sqrt{2}} \left( e^{-\frac{i}{2} \phi} \ket{z+} + e^{\frac{i}{2} \phi} \ket{z-} \right)\\ + &= \frac{e^{-\frac{i}{2} \phi}}{\sqrt{2}} \left( \ket{z+} + e^{i \phi} \ket{z-} \right)\\[15pt] + D_z\left(\frac{\pi}{2}\right) \ket{x+} &= \frac{e^{-\frac{i}{4} \pi}}{\sqrt{2}} \left( \ket{z+} + i \ket{z-} \right)\\ + &= \frac{e^{-\frac{i}{4} \pi}}{\sqrt{2}} \ket{x+}\\[15pt] + D_z(2\pi) \ket{x+} &= (-1) \ket{x+}\\[15pt] + D_z(4\pi) \ket{x+} &= \ket{x+} +\end{align} diff --git a/kapI-6.tex b/kapI-6.tex new file mode 100644 index 0000000..1f0c58c --- /dev/null +++ b/kapI-6.tex @@ -0,0 +1,156 @@ +\chapter{Allgemeine Konsequenzen aus der SG für N-Niveau Systeme} +\section{Zeitentwicklungsoperator} +SG: +\begin{equation} + i \hbar \sigma_t \ket{\psi} = H(t) \ket{\psi(t)} +\end{equation} +Lösung durch +\begin{align} + \ket{\psi(t)} &= U(t, t_0) \ket{\psi(t)} &\left| i \hbar \sigma_t \right. +\end{align} + +\begin{align} + \rightarrow i \hbar \sigma_t U(t, t_0) \ket{\psi(t_0)} &= H(t) \ket{\psi(t)}\\ + &= H(t) U(t, t_0) \ket{\psi(t_0)} +\end{align} +\begin{equation} + \rightarrow \boxed{ i \hbar \sigma_t = H(t) U(t, t_0)} +\end{equation} +mit $U(t_0, t_0) = \one$ +\begin{align} + i \hbar \sigma_t U^\dagger(t, t_0) &= U^\dagger(t, t_0) H(t)\\ + \sigma_t \left( U^\dagger U \right) &= \frac{i}{\hbar} \left( U^\dagger H U - U^\dagger H U \right)\\ + &= 0 +\end{align} + +\begin{equation} + U^\dagger U = \one \Rightarrow U \text{ unitär} \Rightarrow \text{Norm bleibt erhalten} +\end{equation} + +\begin{equation} + \norm{\psi(t)} = \norm{\psi(t_0)} +\end{equation} + +Man unterscheidet 3 Fälle für den Hamiltonoperator: +\begin{enumerate} + \item $H$ zeitunabhängig $H(t) = H$:\\ + $\Box$ gelöst durch + \begin{equation} + U(t, t_0) = e^{-\frac{i}{\hbar} (t-t_0) H} + \end{equation} + Beispiel: konst. Magnetfeld + \begin{equation} + H = \hbar \omega \vec{\sigma} \cdot \vec{n} + \end{equation} + bzw. + \begin{align} + H &= \hbar \frac{\omega}{2} \sigma_z\\ + U(t, t_0) &= e^{-\frac{i \omega}{2} (t - t_0) \sigma_z} + \end{align} + \item $H$ zeitabhängig aber $[H(t_1), H(t_2)] = 0$:\\ + Beispiel: + \begin{align} + H(t) &= -\mu B_z(t) + U(t, t_0) &= e^{-\frac{i}{\hbar} \int_{t_0}^t H(t') dt'} + \end{align} + \item $H$ zeitabhängig mit $[H(t_1), H(t_2)] \neq 0$\\ + keine explizite Lösung (siehe QM II) +\end{enumerate} + +\section{Stationäre Zustände} +$H$ sei zeitabhängig.\\ +Eigenvektoren (mit $n = 0, ..., N-1$) +\begin{equation} + H \ket{n} = E_n \ket{n} +\end{equation} +$\ket{0}$ sei Grundzustand (o.B.d.A. $E_0 = 0$) + +Sei $\ket{\psi(t_0)} = \ket{n}$ +\begin{align} + \ket{\psi(t)} &= U(t, t_0) \ket{n}\\ + &= e^{-\frac{i}{\hbar} (t - t_0) H} \ket{n}\\ + &= e^{-\frac{i}{\hbar} (t - t_0) E_n} \ket{n}\\ + &= e^{-i \omega_n (t - t_0)} \text{ mit } \omega_n \equiv \frac{E_n}{\hbar} +\end{align} + +\begin{itemize} + \item Diese Zustände nehmen unter der Dynamik eine Zeitabhängige Phase an. + \item Die Wahrscheinlichkeit des Systems zur Zeit $t$ im Zustand $\ket{\chi}$ zu mussen, ist + \begin{align} + \probb{P_\chi = 1}{\ket{\psi(t)}} &= \abs{\braket{\chi}{\psi(t)}}^2\\ + &= \abs{\dirac{\chi}{e^{-i \omega_n (t - t_0)}}{n}}^2\\ + &= \abs{\braket{\chi}{n}} + \end{align} +\end{itemize} + +Beispiel: +\begin{align} + H &= \frac{\hbar \omega_z}{2} \sigma_z\\ + \ket{\psi(t_0)} &= \ket{z+}\\ + \ket{\chi} &= \ket{x+} +\end{align} + +Zeitentwicklung eines allgemeinen Zustands: +\begin{align} + \ket{\psi(t_0)} &= \sum_{n=0}^{N-1} c_n(t_0) \ket{n}\\[10pt] + \rightarrow \ket{\psi(t)} &= U(t, t_0) \ket{\psi(t_0)}\\ + &= e^{-\frac{i}{\hbar} (t - t_0) H} \sum_{n=0}^{N-1} c_n(t_0) \ket{n}\\ + &= \sum_{n=0}^{N-1} e^{-\frac{i}{\hbar} (t - t_0) E_n} c_n(t_0) \ket{n}\\ + &= \sum_{n=0}^{N-1} c_n(t) \ket{n} +\end{align} + +\section{Zeitentwicklung des Erwartungswerts: Ehrenfest-Theorem} +Sei $A$ eine physikalische Größe (evtl. $A(t)$) +\begin{equation} + _{\psi(t)} = \dirac{\psi(t)}{A(t)}{\psi(t)} +\end{equation} +Es ist +\begin{equation} + \diffT{t} \ket{\psi(t)} = \frac{1}{i \hbar} H(t) \ket{\psi(t)} +\end{equation} +und +\begin{equation} + \diffT{t} \bra{\psi(t)} = -\frac{1}{i \hbar} \bra{\psi(t)} H(t) +\end{equation} +. +\begin{align} + \diffT{t} _{\psi(t)} &= \diffT{t} \dirac{\psi(t)}{A(t)}{\psi(t)}\\ + &= \dirac{\psi(t)}{A(t) \left| \frac{1}{i \hbar} H(t) \right.}{\psi(t)} - \frac{1}{i \hbar} \dirac{\psi(t)}{H(t)|A(t)}{\psi(t)}\\ &~~+ \dirac{\psi(t)}{\diffP{t}A(t)}{\psi(t)}\\ + &= \frac{1}{\hbar} \dirac{\psi(t)}{[H, A]}{\psi(t)} + \dirac{\psi(t)}{\diffPs{t} A}{\psi(t)}\\ + &= \frac{i}{\hbar} <[H, A]>_{\psi(t)} + <\diffPs{t}A(t)>_{\psi(t)}\\ + &~~\text{(Ehrenfest-Theorem)} +\end{align} + +\paragraph*{Konsequenz} +\begin{itemize} + \item $[H,H] = 0 \rightarrow \diffT{t} = <\diffPs{t} H>$\\ + falls $H$ zeitunabhängig $\rightarrow \diffT{t} = 0$\\[15pt] + $\rightarrow$ $H$ entspricht Energie. + \item falls $A$ zeitunabhängig und $[A, H] = 0$\\[15pt] + $\rightarrow$ Erwartungswert von $A$ zeitunabhängig\\ + $\rightarrow$ ``A ist eine Konstante der Bewegung''\\[15pt] + dann auch $\diffT{t} = 0$\\ + $\rightarrow \probb{A \cequiv a_n}{\psi(t)} = \text{konst.}$! + \item vgl. K.M. + \begin{equation} + \diffT{t}f(p, q, t) = \lbrace f,H \rbrace + \diffPfrac{f}{t} + \end{equation} +\end{itemize} + +\section{Schrödinger- VS Heisenberg-Bild} +bisher: Operatoren und Zustände zeitabhängig $\rightarrow$ ``S.-Bild'' + +\paragraph*{Heisenberg-Bild} +\begin{align} + _\ket{\psi}(t) &= \dirac{\psi(t)}{A}{\psi(t)}\\ + &= \dirac{\psi(0)}{e^{\frac{i}{\hbar}H(t)} A e^{\frac{-i}{\hbar}H(t)}}{\psi(0)}\\ + &= \dirac{\psi_H}{A_H(t)}{\psi_H} +\end{align} +$\rightarrow$ Zustand zeitunabhängig und Opeator zeitabhängig\\ +Ãœbergang von S.-Bild zu H.-Bild durch unitäre Transformation: +\begin{align} + \ket{\psi_H} &= \underbrace{e^{\frac{i}{\hbar}H(t)}}_{U^\dagger(t,0)} \ket{\psi(t)}\\ + A_H(t) &= U^\dagger(t,0) A(t) U(t,0)\\ + i\hbar \diffT{t} A_H(t) &= [A_H, H] + i \hbar \left( \diffPfrac{A}{t} \right)_H +\end{align} + diff --git a/kapII-0.tex b/kapII-0.tex new file mode 100644 index 0000000..5347781 --- /dev/null +++ b/kapII-0.tex @@ -0,0 +1,24 @@ +\chapter{Wellenmechanik in Ortsdarstellung: Grundregeln und Beispiel} +\section{Konkrete Form der Postulate} +\paragraph*{(P1)} Bei vollst. Kenntnis kann ein 1-dimensionales quantales System zu jedem Zeitpunkt durch eine komplexe Wellenfunktion $\psi(x,t_0)$ (mit $\intgr{-\infty}{+\infty}{\abs{\psi(x,t_0)}^2}{x} = 1$) repräsentiert werden. +\paragraph{(P2)} Die Warscheinlichkeit das System zur Zeit $t_0$ +\begin{itemize} + \item am Ort $x$ zu messen ist + \begin{equation} + \rho(x) = \abs{\psi(x,t_0)}^2 + \end{equation} + \item mit dem Impuls $p$ zu messen ist + \begin{equation} + \rho(p) = \abs{\psi(p,t_0)}^2 + \end{equation} + mit + \begin{equation} + \psi(p,t_0) = \frac{1}{\sqrt{2 \pi \hbar}} \intgr{-\infty}{+\infty}{e^{-\frac{i p x}{\hbar}} \psi(x,t_0)}{x} + \end{equation} +\end{itemize} +Konsequenz für die Erwartungswerte +\begin{align} + (t_0) &\equiv \intgr{-\infty}{+\infty}{x~\rho(x,t)}{x} = \intgr{-\infty}{+\infty}{\psi^*(x,t_0) \cdot x \psi(x,t_0)}{x}\\[15pt] +

(t_0) &= \intgr{-\infty}{+\infty}{p~\rho(p,t_0)}{p} = \intgr{-\infty}{+\infty}{\intgr{-\infty}{+\infty}{\frac{1}{\sqrt{2 \pi \hbar}} e^\frac{i p x}{\hbar} \psi^*(x,t_0)}{x}~p~\intgr{-\infty}{+\infty}{e^{-\frac{i x' p}{\hbar}} \psi(x',t_0)}{x'}}{p}\\ + &= \intgru{\frac{1}{\sqrt{2 \pi \hbar}} \intgru{\psi^*(x,t_0) e^\frac{i p x}{\hbar}}{p} \intgru{\frac{1}{\sqrt{2 \pi \hbar} \psi(x,t) (i \hbar \partial_{x'})e^\frac{-i x' p}{\hbar}}}{x'}}{x} +\end{align} diff --git a/kapII-1.tex b/kapII-1.tex new file mode 100644 index 0000000..626246e --- /dev/null +++ b/kapII-1.tex @@ -0,0 +1,210 @@ +\chapter{Grundlagen} +\section{Kommutatorrelation $[\hat{x}, \hat{p}] = i \hbar$} +\paragraph*{klassisch} $H = \frac{p^2}{2m} + V(x)$ +\paragraph*{quantal} $\hat{x} = ?; \hat{p} = ?;$ +\begin{equation} + H = \frac{\hat{p}^2}{2m} + V(\hat{x}) +\end{equation} + +sinnvolle Forderung (Motivation) +\begin{equation} + \diffT{t} \dirac{\psi}{\hat{x}}{\psi} \stackrel{!}{=} \frac{\dirac{\psi}{p}{\psi}}{m} +\end{equation} +(Mittelwerte für alle $\ket{\psi}$ verhalten sich klassisch: $p = m \dot{x}$) + +\begin{align} + \diffT{t} \dirac{\psi}{\hat{x}}{\psi} &= \frac{1}{i \hbar} \dirac{\psi}{\hat{x}}{H \psi} - \dirac{\psi H}{\hat{x}}{\psi}\\ + &= -\frac{i}{\hbar} \dirac{\psi}{[\hat{x}, H]}{\psi} +\end{align} + +\begin{align} + [\hat{x}, H] &= \left[\hat{x}, \frac{\hat{p}^2}{2m} + V(\hat{x}) \right]\\ + &= \left[ \hat{x}, \frac{\hat{p}^2}{2m} \right]\\ + &= \frac{1}{2m} \left( \hat{p} [\hat{x}, \hat{p}] + [\hat{x}, \hat{p}]\hat{p} \right)\\ + \rightarrow [\hat{x}, \hat{p}] &= i \hbar +\end{align} + +vergleiche klassisch $\left\lbrace q,p \right\rbrace = 1$ + +\paragraph*{Lemma} +\begin{equation} + [f(\hat{x}), \hat{p}] = i \hbar f'(\hat{x}) +\end{equation} +\subparagraph*{Beweis} +für +\begin{equation} + f(\hat{x}) = \sum_{n=0}^{\infty}a_n \hat{x}^n +\end{equation} +äquivalent zu +\begin{equation} + [\hat{x}, \hat{p}] = n \hat{x}^{n-1} \cdot i\hbar +\end{equation} +vollständige Induktion:\\ +$(n := 1)$ +\begin{equation} + [\hat{x}, \hat{p}] = i \hbar +\end{equation} +$(n := n+1)$ +\begin{align} + [\hat{x}^{n+1}, \hat{p}] &= \hat{x} \underbrace{[\hat{x}^n, \hat{p}]}_{n(\hat{x}^{n+1}) i\hbar} + \underbrace{[\hat{x}, \hat{p}]}_{i\hbar} \hat{x}^n\\ + &= (n+1)\hat{x}^n i\hbar +\end{align} +\begin{flushright} + $\Box$ +\end{flushright} +Wie verhält sich $\diffT{t} \dirac{\psi}{\hat{p}}{\psi}$ ? +\begin{align} + \rightarrow \diffT{t} \dirac{\psi}{\hat{p}}{\psi} &= -\frac{i}{\hbar} \dirac{\psi}{[\hat{p}, H]}{\psi}\\ + &= -\frac{i}{\hbar} \dirac{\psi}{[\hat{x}, V(\hat{x})]}{\psi}\\ + &= -\dirac{\psi}{V'(\hat{x})}{\psi}\\ + &\neq -V'(_\psi) \text{ falls $V(x)$ nicht quadratisch} +\end{align} + +\section{Ortsdatsrellung (Analogie zu Spin $\frac{1}{2}$)} +\begin{tabular}{l|c|c} + & Spin & Teilchen \\ \hline\hline + Basis & $\sigma_z \ket{z\pm} = \pm 1 \ket{z\pm}$ & $\hat{x} \ket{x} = x \ket{x}$ \\ \hline + Orthogonalität & + $\begin{array}[t]{rl}\braket{z+}{z+} &= 1\\ &= \braket{z-}{z-}\\ \braket{z+}{z-} &= \braket{z-}{z+}\\ &= 0\end{array}$ & + $\braket{x'}{x} = \delta(x'-x)$ \\ \hline + Zustände in Basis entwickelt & $\begin{array}[t]{rl} \ket{\psi} &= \one \ket{\psi}\\ &= \ket{z+}\braket{z+}{\psi} + \ket{z-}\braket{z-}{\psi} \end{array}$ & $\begin{array}[t]{rl} \ket{\psi} &= \one \ket{\psi}\\ &= \intgr{-\infty}{+\infty}{\ket{x}\underbrace{\braket{x}{\psi}}_{\psi(x)}}{x}\\ &= \intgr{-\infty}{+\infty}{\psi(x) \ket{x}}{x} \end{array}$ +\end{tabular} + +\paragraph*{Normierung} +\begin{align} + 1 \stackrel{!}{=} \braket{\psi}{\psi} &= \intgr{-\infty}{+\infty}{\psi^* \bra{x}}{x} \intgr{-\infty}{+\infty}{\psi(x') \ket{x'}}{x'}\\ + \intgru{\intgru{\psi(x)\psi(x')\underbrace{\braket{x}{x'}}_{\delta(x-x')}}{x'}}{x}\\ + &= \intgru{\psi^*(x)\psi(x)}{x}\\ + &= \intgru{(\psi(x))^2}{x} +\end{align} +$\rightarrow$ Zulässige Zustände haben eine ``Wellenfunktion'', die quadrat-integrabel ist. + +\paragraph*{Erwartungswert einer Ortsmessung} +\begin{align} + <\hat{x}>_\psi &= \dirac{\psi}{\hat{x}}{\psi}\\ + &= \intgru{\dirac{\psi}{\hat{x}}{x}\braket{x}{\psi}}{x}\\ + &= \intgru{\bra{\psi} \cdot x \ket{x} \braket{x}{\psi}}{x}\\ + &= \intgru{x\psi^*(x)\psi(x)}{x}\\ + &= \intgru{x\abs{\psi(x)}^2}{x}\\ + &\equiv \intgru{x \rho(x)}{x} +\end{align} +mit $\rho(x) \equiv \abs{\psi(x)}^2 = \psi^*(x) \psi(x)$ der Wahrscheinlichkeitsdichte das Teilchen im Zustand $\ket{\psi}$ am Ort $x$ zu messen! +\subparagraph*{Hinweis:} +\begin{equation} + \intgr{-\infty}{+\infty}{\rho(x)}{x} = 1 +\end{equation} + +\paragraph*{``Matrixelemente'' des Impulsoperators in Ortsdarstellung} +(vgl. Spins $\sigma_y = \inlinematrix{0& -i\\ i& 0}$) +\begin{equation} + \dirac{x'}{\hat{p}}{x} = ? +\end{equation} + +\begin{align} + \hat{x}\hat{p} - \hat{p}\hat{x} &= i \hbar & \left| \ket{x} \right.\\ + \hat{x}\hat{p} \ket{x}- \hat{p}\hat{x} \ket{x} &= i \hbar \ket{x} & \left| \bra{x'} \right.\\ + \dirac{x'}{\hat{x}\hat{p}}{x} - \dirac{x'}{\hat{p}\hat{x}}{x} &= i\hbar \braket{x'}{x}\\ + (x' - x) \underbrace{\dirac{x'}{p}{x}}_{\equiv i\hbar g(x' - x)} &= i\hbar \delta(x' - x) +\end{align} +mit $U g(u) = \delta(u)$ + +%TODO: folgendes abgleichen +% \subparagraph*{Behauptung} +% \begin{equation} +% g(u) = \delta'(u) +% \end{equation} +% +% \subparagraph*{Beweis} +% \begin{align} +% -u \delta'(u) &= \delta(u)\\ +% -\intgru{u \delta'(u) f(u-u_0)}{u} &= \intgru{\delta(u) f(u-u_0)}{u}\\ +% \intgru{\delta(u)}\left(u f(u - u_0)\right)'{u} &= +% \end{align} + +also: +\begin{equation} + \dirac{x'}{\hat{p}}{x} = i\hbar \delta'(x'-x) +\end{equation} +Wirkung $\hat{p}$ auf $\ket{\psi}$: +\begin{align} + \dirac{x'}{\hat{p}}{\psi} &= \intgru{\dirac{x}{\hat{p}}{x}\braket{x}{\psi}}{x}\\ + &= \intgru{(-i\hbar) \delta'(x'-x) \psi(x)}{x}\\ + &= (-i\hbar) \intgru{\delta(x'-x) \diffP{x} \psi(x)}{x}\\ + &= (-i\hbar) \psi'(x') +\end{align} +daher auch +\begin{equation} + \hat{p} \cequiv -i\hbar \diffP{x} +\end{equation} + +\section{Schrödingergleichung in Ortsdarstellung} +SG: +\begin{align} + i\hbar \partial_t \ket{\psi} &= H \ket{\psi}\\ + &= \left( \frac{\hat{p}^2}{2m} + V(\hat{x}) \right) \ket{\psi} & \left| \bra{x} \right.\\ + i\hbar \partial_t \braket{x}{\psi} &= \dirac{x}{\frac{\hat{p}^2}{2m}}{\psi}\\ + i\hbar \partial_t \psi(x,t) &= \left\lbrace \frac{1}{2m} \left( -i\hbar \diffP{x} \right)^2 + V(x) \right\rbrace \psi(x) +\end{align} +Falls $V(x)$ zeitabhängig ist gibt es stationäre Zustände: +\begin{equation} + \psi(x,t) = e^{-\frac{i}{\hbar} E_n t} \psi_n(x) +\end{equation} +\begin{equation} + E_n \psi_n(x) = \left( -\frac{i\hbar}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right) \psi_n(x) +\end{equation} +Normierbare Lösung $\intgr{-\infty}{+\infty}{\abs{\psi(x)}^2}{x} = 1$ nur für bestimmte $E_n$! + +\section{Impuls-Operator} +Zu $\hat{x} \ket{x} = x\ket{x}$ bilder wir die Analogie $\hat{p}\ket{p} = p \ket{p}$. + +\paragraph*{Was ist $\braket{x}{p}$?} +\begin{align} + \hat{p}\ket{p} &= p \ket{p} &\left| \bra{x} \right.\\ + \dirac{x}{\hat{p}}{p} &= p \braket{x}{p}\\ + \intgr{-\infty}{+\infty}{\dirac{x}{\hat{p}}{x'}\braket{x'}{p}}{x'} &= p \braket{x}{p}\\ + \intgr{-\infty}{+\infty}{-i\hbar \delta'(x - x') \braket{x'}{p}}{x'} &= p \braket{x}{p}\\ + \rightarrow -i\hbar \diffP{x} \braket{x}{p} &= p \braket{x}{p}\\ + \braket{x}{p} &= c e^{\frac{ip}{\hbar}x} +\end{align} +d.h.: Die Eigenzustände des Impulsoperators in Ortsdarstellung ist eine ebene Welle. + +\paragraph*{Normierung} +\begin{align} + \braket{p'}{p} &\stackrel{!}{=} \delta(p'-p)\\[15pt] + \intgr{-\infty}{+\infty}{\braket{p'}{x}\braket{x}{p}}{x} &= \intgr{-\infty}{+\infty}{c^* e^{\frac{i p' x}{\hbar}} c e^{\frac{i p x}{\hbar}}}{x}\\ + &= 2 \pi c c^* \delta\left(\frac{p - p'}{\hbar}\right)\\ + &= 2 \pi \hbar c c^* \delta(p - p') +\end{align} +$\rightarrow$ $c = \frac{1}{\sqrt{2 \pi \hbar}}$ ist korrekte Normierung! +\begin{equation} + \braket{x}{p} = \frac{1}{\sqrt{2 \pi \hbar}} e^{\frac{i p x}{\hbar}} +\end{equation} + +\paragraph*{Erwartungswert des Impulses} +\begin{align} +

_\psi &= \dirac{\psi}{\hat{p}}{\psi}\\ + &= \intgru{\dirac{\psi}{\hat{p}}{p}\braket{p}{\psi}}{p}\\ + &= \intgru{p\abs{\braket{p}{\psi}}^2}{p} +\end{align} +d.h. $\rho(p) \equiv \abs{\braket{p}{\psi}}^2$ ist die Wahrscheinlichkeitsdichte im Zustand $\ket{\psi}$ den Impuls $p$ zu messen.\\[15pt] +Für gegebenes $\braket{p}{\psi} = \psi(x)$ gilt +\begin{align} + \braket{p}{\psi} &= \intgr{-\infty}{+\infty}{\braket{p}{x}\braket{x}{\psi}}{x}\\ + &= \intgru{\frac{1}{\sqrt{2 \pi \hbar}}e^{\frac{-i p x}{\hbar}}\psi(x)}{x} \equiv \psi(p) +\end{align} +$\psi(p)$ ist (leicht anders normierte) Furiertransformierte von $\psi(x)$. + +\begin{align} + \dirac{\psi}{p}{\psi} &= \intgru{\intgru{\braket{\psi}{x}\underbrace{\dirac{x}{\hat{p}}{x'}}_{-i \hbar \delta'(x - x')}\braket{x'}{\psi}}{x'}}{x}\\ + &= \intgru{\intgru{(-i \hbar) \delta'(x - x') \psi^*(x) \psi(x')}{x'}}{x}\\ + &= i \hbar \intgru{\intgru{\delta(x - x') {\psi^*}'(x) \psi(x')}{x'}}{x}\\ + &= i \hbar \intgru{{\psi^*}'(x') \psi(x')}{x'}\\ +

_\psi&= i \hbar \intgru{{\psi^*}'(x) \psi(x)}{x} +\end{align} + +\paragraph*{Schrödingergleichung in Impulsdarstellung} +\begin{align} + i \hbar \partial_t\ket{\psi} &= H \ket{\psi} & \left| \bra{p} \right.\\ + i \hbar \partial_t\braket{p}{\psi} &= \dirac{p}{\frac{\hat{p}^2}{2m} + V(x)}{\psi}\\ + &= \left\lbrace \frac{p^2}{2m} \psi(p) + \dirac{p}{V(\hat{x})}{\psi} \right\rbrace +\end{align} diff --git a/kapII-2.tex b/kapII-2.tex new file mode 100644 index 0000000..657ffc0 --- /dev/null +++ b/kapII-2.tex @@ -0,0 +1,162 @@ +\chapter{Freie quantale Teilchen} +\section{Ebene Wellen} +$V(x) \equiv 0$: +\begin{equation} + i \hbar \ket{\psi} = \frac{p^2}{2m} \ket{\psi} +\end{equation} + +\paragraph*{stationäre Lösung} +\begin{equation} + E \ket{\psi} = \frac{\hat{p}^2}{2m} \ket{\psi} +\end{equation} +Lösungen sind Eigenzustände zu $\hat{p}$: +$\hat{p} \ket{p} = p \ket{p}$: +\begin{align} + H \ket{p} &= \frac{p^2}{2m} \ket{p}\\ + &\equiv E \ket{p} +\end{align} +alle $E > 0$ sind möglich.\\[15pt] +Zustände sind zweifach entartet. Zu $E > 0$ gibt es $\ket{\sqrt{2m E}}$ und $\ket{-\sqrt{2m E}}$ als mögliche Eigenzustände. (Vergleiche klassiche Mechanik: $E$ vorgegeben, dann auch $p = \pm \sqrt{2m E}$ als mögliche Bahnen) + +\paragraph*{Lösung in Ortsdarstellung} +\begin{align} + \braket{x}{\pm \sqrt{2m E}} &= \braket{x}{\pm p}\\ + &= \frac{1}{\sqrt{2 \pi \hbar}} e^{\pm \frac{i p x}{\hbar}}\\ + &= \frac{1}{\sqrt{2 \pi \hbar}} e^{\pm \frac{i}{\hbar}\sqrt{2m E} \cdot x}\\ + &= \psi_\pm(x) +\end{align} + +Sei (für $p > 0$): +\begin{equation} + \psi(x, t = 0) = \frac{1}{\sqrt{2 \pi \hbar}} e^{\frac{i p x}{\hbar}} +\end{equation} +dann: +\begin{align} + \psi(x, t) &= e^{-\frac{i}{\hbar} E \cdot t} \psi(x, 0)\\ + &= \frac{1}{\sqrt{2 \pi \hbar}} e^{i \left( \frac{p}{\hbar} x - \frac{E}{\hbar} t \right) }\\ + &= \frac{1}{\sqrt{2 \pi \hbar}} e^{-i(k x - \omega t)} +\end{align} +$\rightarrow$ rechtslaufende Welle\\[15pt] +Entsprechend für $p = -\sqrt{2m E}$ gilt: +\begin{equation} + \psi(x,t) = \frac{1}{\sqrt{2 \pi \hbar}} e^{i(-k x - \omega t)} +\end{equation} +$\rightarrow$ linkslaufende Welle + +\section{Propagator} +$\ket{\psi(t_0)}$ gegeben als $\psi(x,t_0) = \braket{x}{\psi(t_0)}$. Gesucht: $\ket{\psi(t)}$: +\begin{align} + \psi(x,t) = \braket{x}{\psi(t)} &= \dirac{x}{e^{-\frac{i}{\hbar} (t - t_0) H}}{\psi(t_0)}\\ + &= \intgr{-\infty}{+\infty}{\dirac{x}{e^{-\frac{i}{\hbar} \frac{\hat{p}^2}{2m} (t - t_0)}}{p}\bra{p}\psi(t_0)}{p}\\ + &= \intgru{e^{\frac{-i p^2}{\hbar 2m} (t - t_0)} \frac{1}{\sqrt{2 \pi \hbar}} e^{\frac{i p x}{\hbar}} \psi(p, t_0)}{p}\\ + &= \intgru{\intgru{e^{-\frac{i p^2}{2m \hbar} (t - t_0)} \frac{1}{2 \pi \hbar} e^{\frac{i p}{\hbar} (x - x')} \psi(x', t_0)}{x'}}{x}\\ + &= \intgru{\sqrt{\frac{m}{2 \pi \hbar i (t - t_0))}} e^{\frac{i m (x - x')}{2 \hbar (t - t_0)}} \psi(x', 0)}{x'}\\ + &= \intgru{\dirac{x}{U(t, t_0)}{x'}\braket{x'}{\psi(t_0)}}{x'} +\end{align} +$\rightarrow$ Matrixelemente des Zeitentwicklungsoperators in Ortsdarstellung + +\subsection*{Analogie: Diffusion} +Sei $c(x, t)$ die Dichte der blauen Tinte. +%\begin{figure}[h] +%\includegraphics{8-001.pdf} +%\caption{$c(x,t_0)$} +%\end{figure} +Diffusionsgleichung: +\begin{equation} + \partial_t c(x, t) = D \partial_x^2 c(x, t) +\end{equation} +Lösung: +\begin{equation} + c(x, t) = \intgru{\frac{1}{\sqrt{D t 4 \pi}} c(x',t_0)}{x'} +\end{equation} +QM: +\begin{align} + i\hbar \partial_t \psi(x,t) &= -\frac{\hbar^2}{2m} \partial_x^2 \psi(x,t)\\ + \partial_t \psi(x,t) &= i \frac{\hbar}{2m} \partial_x^2 \psi(x,t) +\end{align} +\section{Gauss'sches Wellenpacket} +%\begin{figure}[h] +%\includegraphics{8-002.pdf} +%\caption{$\abs{\psi^2(x)}$ lokalisiert um die Null mit Breite $~\Delta$} +%\end{figure} +\begin{equation} + \psi(x',0) = \frac{1}{(\pi \Delta^2)^{\frac{1}{4}}} +\end{equation} +\begin{itemize} + \item Die Wahrscheinlichkeit Teilchen am Ort $x'$ zu messen sei + \begin{equation} + \rho(x') \equiv \abs{\psi(x',0)}^2 + \end{equation} + . + \item Die Wahrscheinlichkeit Teilchen mit Impuls $p$ zu messen sei + \begin{equation} + \rho(p) \equiv \abs{\psi(p,0)}^2 + \end{equation} + . +\end{itemize} +\begin{align} + \psi(p,0) \equiv \braket{p}{\psi(0)} &= \intgr{-\infty}{+\infty}{\braket{p}{x'} \braket{x'}{\psi(0)}}{x'}\\ + &= \intgr{-\infty}{+\infty}{\frac{e^\frac{-i p x'}{\hbar}}{\sqrt{2 \pi \hbar}} \frac{1}{\left(\pi \Delta^2\right)^\frac{1}{4}} e^\frac{i p_0 x'}{\hbar} e^\frac{-{x'}^2}{2 \Delta^2}}{x'}\\ + &= \frac{\Delta^\frac{1}{2}}{\pi^\frac{1}{4} \hbar^\frac{1}{2}} e^\frac{-(p - p_0)^2 \Delta^2}{2 \hbar} +\end{align} +Also ergibt sich für $\rho(p)$ +\begin{equation} + \rho(p) = \frac{\Delta}{\pi^\frac{1}{2} \hbar} e^\frac{-(p - p_0)^2}{\hbar^2 \Delta^{-2}} +\end{equation} +%\begin{figure}[h] +%\includegraphics{8-002.pdf} +%\caption{$\abs{\psi^2(x)}$ lokalisiert um die Null mit Breite $~\Delta$} +%\end{figure} +\paragraph*{Dispersion, Unschärfe} +\begin{align} + (\Delta x)_\ket{\psi(0)} &= \sqrt{ - ^2} = \sqrt{\dirac{\psi(0)}{\hat{x}^2}{\psi(0)}}\\ + &= \frac{\Delta}{\sqrt{2}}\\[15pt] + (\Delta p)_\ket{\psi(0)} &= \sqrt{ -

^2} = ...\\ + &= \frac{\hbar}{\sqrt{2} \Delta} +\end{align} +Heisenberg: +\begin{equation} + (\Delta x)_\ket{\psi(0)} (\Delta p)_\ket{\psi(0)} \geq \frac{1}{2}\dirac{\psi_0}{[x,p]}{\psi_0} = \frac{\hbar}{2} +\end{equation} +für Gauss'sches Wellenpacket ist Gleichheit erreicht. +\paragraph*{Dynamik} +\begin{align} + \psi(x,t) &= \intgr{-\infty}{+infty}{U(x,t; x',t_0)}{x'} &\left| t_0 = 0; ~ U(x,t; x',t_0) = \dirac{x}{U(t,t_0}{x'} \right.\\ + &= \left( \sqrt{\pi} \left( \Delta + \frac{i \hbar t}{m \Delta} \right) \right)^{-\frac{1}{2}} e^\frac{-\left(x - \frac{p_0 t}{m} \right)^2}{2 \Delta^2 \left( 1 + i \hbar \frac{t}{m \Delta^2} \right)} e^{\frac{i p_0}{\hbar} \left( x - \frac{p_0 t}{m} \right)} +\end{align} +\begin{equation} + \rho(x,t) = \abs{\psi(x,t)}^2 +\end{equation} +%\begin{figure}[h] +%\includegraphics{8-002.pdf} +%\caption{$\abs{\psi^2(x)}$ lokalisiert um die Null mit Breite $~\Delta$} +%\end{figure} +\begin{equation} + \Delta(t) = \Delta^{(0)} \sqrt{1 + \frac{\hbar^2 t^2}{m^2 \Delta^4}} +\end{equation} + +\paragraph{Fazit} +\begin{enumerate} + \item \begin{equation} + (t) = p_0 \frac{t}{m} =

(t) \frac{t}{m} \text{ und }

(t) = p_0 + \end{equation} + Mittelwerte verhalten sich klassisch (Ehrenfest!). + \item \begin{equation} + (\Delta x)(t) = \frac{\Delta}{\sqrt{2}} \left( 1 + \frac{\hbar^2 t^2}{m^2 \Delta^4} \right)^\frac{1}{2} + \end{equation} + Breite im Ort läuft auseinander. Sie ändert sich sinifikant auf der Zeitskala: + \begin{equation} + t^* = \frac{m}{\hbar} \Delta + \end{equation} + Beispiel makroskopisch: + \begin{align} + m &=0,1kg; ~ \Delta = 10^{-3}m; ~ \hbar = 10^{-23}Js\\[15pt] + t^* &= \frac{10^{-1} 10^{-6}}{10^{-34}}s = 10^{27}s + \end{align} + Elektron: + \begin{align} + m_e &= 10^{-30}kg; ~ \Delta = 10^{-10}m; ~ \hbar = 10^{-23}Js\\[15pt] + t^* &= \frac{10^{-30} 10^{-20}}{10^{-34}}s = 10^{-16}s + \end{align} + \item Impulse verbreitern sich analog. +\end{enumerate} diff --git a/kapII-3.tex b/kapII-3.tex new file mode 100644 index 0000000..2d091f0 --- /dev/null +++ b/kapII-3.tex @@ -0,0 +1,27 @@ +\chapter{Potentialstufen und Potentialtöpfe} +\section{Einschub: Wahrscheinlichkeitsstrom} +\begin{equation} + \rho(x,t) = \psi(x,t) \psi^*(x,t) +\end{equation} +ist die Wahrscheinlichkeit das Teilchen am Ort x zu messen. + +\begin{align} + \partial_t \rho(x,t) &= \psi^*(x,t)~\partial_t \psi(x,t) + \psi(x,t)~\partial_t \psi^*(x,t)\\ + &= \psi^*(x,t) \left( \frac{1}{i \hbar} \left( -\frac{\hbar^2}{2m} \partial_x^2 - V(x)\right) \psi(x,t) \right) + + \psi^*(x,t) \left( -\frac{1}{i \hbar} \left( -\frac{\hbar^2}{2m} \partial_x^2 + V(x)\right) \psi(x,t) \right)\\ + &= -\frac{1}{\hbar} \left( -\frac{\hbar^2}{2m} \psi^*(x,t)~\partial_x^2\psi(x,t) + \frac{\hbar^2}{2m} \psi(x,t)~\partial_x^2\psi^*(x,t) \right)\\ + &= \frac{i \hbar}{2m} \partial_x \left( \psi^*~\partial_x \psi - \psi~\partial_x \psi^* \right) \equiv -\partial_x j(x,t) +\end{align} +mit +\begin{equation} + j(x,t) \equiv \frac{\hbar}{m} \im{\psi^*(x,t)~\partial_x \psi(x,t)} +\end{equation} +der Wahscheinlichkeitsstromdichte (``Kontinuitätsgleichung''; gilt für jede Erhaltungsgröße). +\paragraph*{Beispiel: Ebene Welle} +\begin{align} + \psi(x,t) &= \frac{1}{\sqrt{2 \pi \hbar}} e^{\frac{i p_0}{\hbar}x - \omega t}\\ + j(x,t) &= \frac{1}{\sqrt{2 \pi \hbar}} \frac{\hbar}{m} \im{\frac{i p}{\hbar}}%\\ +% &= \frac{1}{\sqrt{2 \pi \hbar}} \frac{p}{} +\end{align} + +\section{Streuung an der Potentialstufe} \ No newline at end of file diff --git a/math.tex b/math.tex new file mode 100644 index 0000000..5d53c6e --- /dev/null +++ b/math.tex @@ -0,0 +1,31 @@ +\usepackage{amsmath} +%\usepackage{mathtools} +\usepackage{bbm} +\newcommand{\set}[1]{\left\lbrace #1 \right\rbrace } +\newcommand{\setZ}{\mathbbm{Z}} +\newcommand{\setN}{\mathbbm{N}} +\newcommand{\setQ}{\mathbbm{Q}} +\newcommand{\setR}{\mathbbm{R}} +\newcommand{\setC}{\mathbbm{C}} +\newcommand{\inlinematrix}[1]{\begin{pmatrix} #1 \end{pmatrix}} +\newcommand{\inlinematrixu}[1]{\begin{matrix} #1 \end{matrix}} +\newcommand{\inlinematrixd}[2]{\left( \begin{matrix} #1\vphantom{#2} \end{matrix} ~\right| \left. \begin{matrix} \vphantom{#1}#2 \end{matrix} \right)} +\newcommand{\inlinematrixdet}[1]{\begin{vmatrix} #1 \end{vmatrix}} + +\newcommand{\norm}[1]{\left\Vert #1 \right\Vert} +\newcommand{\scalar}[2]{\langle #1 , #2 \rangle} +\newcommand{\abs}[1]{{\left| #1 \right|}} +\newcommand{\gdw}{\Leftrightarrow} 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Grundlagen} +\label{I} +\include{kapI-1} +\include{kapI-2} +\include{kapI-3} +\include{kapI-4} +\include{kapI-5} +\include{kapI-6} +\part{Quantale Systeme in einer Raumdimension} +\label{II} +\include{kapII-0} +\include{kapII-1} +\include{kapII-2} +\include{kapII-3} + +% \part{Übungsmitschrieb} +% \label{UE} +% \include{ueb1} +% \include{ueb2} +% \include{ueb3} +% \include{ueb4} +% \include{ueb5} +% \include{ueb6} + + +\end{document} diff --git a/ueb1.tex b/ueb1.tex new file mode 100644 index 0000000..8df4854 --- /dev/null +++ b/ueb1.tex @@ -0,0 +1,16 @@ +%\includegraphics{excs/qm1_blatt01_SS08.pdf} +%\pagebreak + +\chapter{Quantenmechanik I - Übungsblatt 1} +\section{Aufgabe 1: Stern-Gerlach Experimente} + +\section{Aufgabe 2: Pauli Matrizen} +\subsection*{a)} +\subsection*{b)} +\subsection*{c)} +\subsection*{d)} + +\section{Aufgabe 3: Operator-Identitäten} +\subsection*{a)} +\subsection*{b)} +\subsection*{c)} \ No newline at end of file diff --git a/ueb2.tex b/ueb2.tex new file mode 100644 index 0000000..9215eef --- /dev/null +++ b/ueb2.tex @@ -0,0 +1,17 @@ +%\includegraphics{excs/qm1_blatt02_SS08.pdf} +%\pagebreak + +\chapter{Quantenmechanik I - Übungsblatt 2} +\section{Aufgabe 4: Unitäre Operatoren} +\subsection*{a)} +\subsection*{b)} + +\section{Aufgabe 5: Spur und Determinante} +\subsection*{a)} +\subsection*{b)} +\subsection*{c)} +\subsection*{d)} + +\section{Aufgabe 6: Hermitesche Matrizen} +\subsection*{a)} +\subsection*{b)} diff --git a/ueb3.tex b/ueb3.tex new file mode 100644 index 0000000..41efa54 --- /dev/null +++ b/ueb3.tex @@ -0,0 +1,15 @@ +%\includegraphics{excs/qm1_blatt03_SS08.pdf} +%\pagebreak + +\chapter{Quantenmechanik I - Übungsblatt 3} +\section{Aufgabe 7: Eigenzustände und Erwartungswerte von Spin-1/2-Teilchen} +\subsection*{a)} +\subsection*{b)} +\subsection*{c)} + +\section{Aufgabe 8: Teilchen mit Spin 1} +\subsection*{a)} +\subsection*{b)} +\subsection*{c)} +\subsection*{d)} +\subsection*{e)} \ No newline at end of file diff --git a/ueb4.tex b/ueb4.tex new file mode 100644 index 0000000..8dad127 --- /dev/null +++ b/ueb4.tex @@ -0,0 +1,16 @@ +%\includegraphics{excs/qm1_blatt04_SS08.pdf} +%\pagebreak + +\chapter{Quantenmechanik I - Übungsblatt 4} +\section{Aufgabe 9: Zeitentwicklung eines allgemeinen Zweizustandssystems} +\subsection*{a)} +\subsection*{b)} +\subsection*{c)} + +\section{Aufgabe 10: Zerfall eines instabilen Zustandes} +\subsection*{a)} +\subsection*{b)} +\subsection*{c)} +\subsection*{d)} +\subsection*{e)} +\subsection*{f)} \ No newline at end of file diff --git a/ueb5.tex b/ueb5.tex new file mode 100644 index 0000000..36e1e9f --- /dev/null +++ b/ueb5.tex @@ -0,0 +1,16 @@ +%\includegraphics{excs/qm1_blatt05_SS08.pdf} +%\pagebreak + +\chapter{Quantenmechanik I - Übungsblatt 5} +\section{Aufgabe 11: Spin-1-Teilchen im konstanten Magnetfeld} +\subsection*{a)} +\subsection*{b)} +\subsection*{c)} + +\section{Aufgabe 12: Das Ethen-Molekül} + +\section{Aufgabe 13: Das Benzol-Molekül} +\subsection*{a)} +\subsection*{b)} +\subsection*{c)} +\subsection*{d)} \ No newline at end of file diff --git a/ueb6.tex b/ueb6.tex new file mode 100644 index 0000000..a281369 --- /dev/null +++ b/ueb6.tex @@ -0,0 +1,13 @@ +%\includegraphics{excs/qm1_blatt06_SS08.pdf} +%\pagebreak + +\chapter{Quantenmechanik I - Übungsblatt 6} +\section{Aufgabe 14: Spin-1-Teilchen} +\subsection*{a)} +\subsection*{b)} + +\section{Aufgabe 15: Benzol-Molekül (Teil 2)} +\subsection*{a)} +\subsection*{b)} + +\section{Aufgabe 16: Schrödingergleichung in Impulsraumdarstellung} \ No newline at end of file