initial commit
This commit is contained in:
Oliver Groß
commit
3d7261e3e1
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\chapter{Stern-Gerlach-Experimente}
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||||
\section{Versuchsaufbau (1921)}
|
||||
\begin{figure}[h]
|
||||
\includegraphics{1-001.pdf}
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||||
\caption{Versuchsskizze}
|
||||
\end{figure}
|
||||
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||||
\paragraph{Ergebnis}
|
||||
\begin{enumerate}
|
||||
\item Jedes einzelne Atom wird entweder einen festen 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
|
||||
330
kapI-2.tex
Normal file
330
kapI-2.tex
Normal file
@ -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}
|
||||
134
kapI-3.tex
Normal file
134
kapI-3.tex
Normal file
@ -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}
|
||||
|
||||
207
kapI-4.tex
Normal file
207
kapI-4.tex
Normal file
@ -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}
|
||||
195
kapI-5.tex
Normal file
195
kapI-5.tex
Normal file
@ -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}
|
||||
156
kapI-6.tex
Normal file
156
kapI-6.tex
Normal file
@ -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}
|
||||
<A(t)>_{\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} <A(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 $[H,H] = 0 \rightarrow \diffT{t} <H> = <\diffPs{t} H>$\\
|
||||
falls $H$ zeitunabhängig $\rightarrow \diffT{t}<H> = 0$\\[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 $\diffT{t} <f(A)> = 0$\\
|
||||
$\rightarrow \probb{A \cequiv a_n}{\psi(t)} = \text{konst.}$!
|
||||
\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}
|
||||
<A>_\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\\
|
||||
Übergang 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}
|
||||
|
||||
24
kapII-0.tex
Normal file
24
kapII-0.tex
Normal file
@ -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}
|
||||
<x>(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]
|
||||
<p>(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}
|
||||
210
kapII-1.tex
Normal file
210
kapII-1.tex
Normal file
@ -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'(<x>_\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}
|
||||
<p>_\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'}\\
|
||||
<p>_\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}
|
||||
162
kapII-2.tex
Normal file
162
kapII-2.tex
Normal file
@ -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{<x^2> - <x>^2} = \sqrt{\dirac{\psi(0)}{\hat{x}^2}{\psi(0)}}\\
|
||||
&= \frac{\Delta}{\sqrt{2}}\\[15pt]
|
||||
(\Delta p)_\ket{\psi(0)} &= \sqrt{<p^2> - <p>^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}
|
||||
<x>(t) = p_0 \frac{t}{m} = <p>(t) \frac{t}{m} \text{ und } <p>(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}
|
||||
27
kapII-3.tex
Normal file
27
kapII-3.tex
Normal file
@ -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}
|
||||
31
math.tex
Normal file
31
math.tex
Normal file
@ -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}
|
||||
%\newcommand{\iff}{\Leftrightarrow}
|
||||
\newcommand{\re}[1]{{\text{Re}\left( #1 \right)}}
|
||||
\newcommand{\im}[1]{{\text{Im}\left( #1 \right)}}
|
||||
\newcommand{\tr}{{\text{tr}}}
|
||||
|
||||
\newcommand{\QED}{\begin{large}\textbf{\checkmark}\end{large}}
|
||||
\newcommand{\cequiv}{\stackrel{\scriptscriptstyle\wedge}{=}}
|
||||
|
||||
\newcommand{\diffP}[1]{\frac{\partial}{\partial #1}}
|
||||
\newcommand{\diffT}[1]{\frac{\text{d}}{\text{d} #1}}
|
||||
\newcommand{\diffPfrac}[2]{\frac{\partial #1}{\partial #2}}
|
||||
\newcommand{\intgr}[4]{\int_{#1}^{#2} #3 ~\text{d}#4}
|
||||
\newcommand{\intgru}[2]{\int #1 ~\text{d}#2}
|
||||
11
physics.tex
Normal file
11
physics.tex
Normal file
@ -0,0 +1,11 @@
|
||||
\newcommand{\bra}[1]{{\left< #1 \right|}}
|
||||
\newcommand{\ket}[1]{{\left| #1 \right>}}
|
||||
\newcommand{\braket}[2]{{\left\langle #1\vphantom{#2}\right.\left|\vphantom{#1}#2\right\rangle}}
|
||||
\newcommand{\dirac}[3]{{\left\langle #1\vphantom{#2}\vphantom{#3}\right| #2 \left|\vphantom{#1}\vphantom{#2}#3\right\rangle}}
|
||||
|
||||
\newcommand{\hilbert}{\mathcal{H}}
|
||||
\newcommand{\one}{\mathbbm{1}}
|
||||
|
||||
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theo2.tex
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theo2.tex
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\documentclass[a4paper,10pt,oneside]{book}
|
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\usepackage[utf8]{inputenc}
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\usepackage{ngerman}
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\usepackage{graphics}
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\newcommand{\lboxed}[1]{\left\lceil #1 \right\rfloor}
|
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|
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\title{Theoretische Physik 2 Vorlesungs- und Übungsmitschrieb}
|
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\author{Daniel Bahrdt, Oliver Groß}
|
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|
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\begin{document}
|
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\maketitle
|
||||
\tableofcontents
|
||||
|
||||
\part{Spin 1/2 Systeme - 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}
|
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\include{kapII-0}
|
||||
\include{kapII-1}
|
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\include{kapII-2}
|
||||
\include{kapII-3}
|
||||
|
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% \part{Übungsmitschrieb}
|
||||
% \label{UE}
|
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% \include{ueb1}
|
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% \include{ueb2}
|
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% \include{ueb3}
|
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% \include{ueb4}
|
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% \include{ueb5}
|
||||
% \include{ueb6}
|
||||
|
||||
|
||||
\end{document}
|
||||
16
ueb1.tex
Normal file
16
ueb1.tex
Normal file
@ -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)}
|
||||
17
ueb2.tex
Normal file
17
ueb2.tex
Normal file
@ -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)}
|
||||
15
ueb3.tex
Normal file
15
ueb3.tex
Normal file
@ -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)}
|
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16
ueb4.tex
Normal file
16
ueb4.tex
Normal file
@ -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)}
|
||||
|
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\section{Aufgabe 10: Zerfall eines instabilen Zustandes}
|
||||
\subsection*{a)}
|
||||
\subsection*{b)}
|
||||
\subsection*{c)}
|
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\subsection*{d)}
|
||||
\subsection*{e)}
|
||||
\subsection*{f)}
|
||||
16
ueb5.tex
Normal file
16
ueb5.tex
Normal file
@ -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)}
|
||||
13
ueb6.tex
Normal file
13
ueb6.tex
Normal file
@ -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}
|
||||
Loading…
Reference in New Issue
Block a user