qm1-script/ueb4.tex

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\chapter{Quantenmechanik I - Übungsblatt 4}
\section{Aufgabe 9: Zeitentwicklung eines allgemeinen Zweizustandssystems}
\begin{math}
	H = \hbar \inlinematrix{A & B \\ B & -A} \\
	A = \Omega \cosb{2 \theta} \\
	B = \Omega \sinb{2 \theta} \\
	E_\pm = \pm \hbar \Omega
\end{math}

\subsection*{a)}
\begin{align}
	\ket{\chi_+}	&= \inlinematrix{\sinb{2 \theta} \\ 1 - \cosb{2 \theta}}	&= \inlinematrix{\cosb{\theta} \\ \sinb{\theta}} \\
	\ket{\chi_-}	&= 															&= \inlinematrix{\sinb{\theta} \\ -\cosb{\theta}}
\end{align}

\subsection*{b)}
\begin{align}
	\i \hbar \diffPs{t} \ket{\psi}	&= H \ket{\psi} \\
	\ket{\dot{\psi}}				&= -\i \Omega \inlinematrix{\cosb{2 \theta} & \sinb{2 \theta} \\ \sinb{2 \theta} & -\cosb{2 \theta}} \ket{\psi} \\
	\ket{\psi(t)}					&= c_1 \cdot e^{\i \Omega t} \ket{\chi+} + c_2 \cdot e^{-\i \Omega t} \ket{\chi-} \\
	\ket{\psi(0)}					&= \inlinematrix{\lambda \\ \mu} \\
	\inlinematrix{c_+(t) \\ c_-(t)}	&= \sbk{\lambda \cosb{\theta} + \mu \sinb{\theta}} \cdot \inlinematrix{\cosb{\theta} \\ \sinb{\theta}} \cdot e^{-\i t} +
										\sbk{\lambda \sinb{\theta} - \mu \cosb{\theta}} \cdot \inlinematrix{\sinb{\theta} \\ -\cosb{\theta}} \cdot e^{\i \Omega t}
\end{align}

\subsection*{c)}
%hier stimmt evtl. was nicht
\begin{align}
	\ket{\psi(0)}					&= \inlinematrix{0 \\ 1} \\
	\abs{\braket{+}{\psi(t)}}^2		&= \sin^2\sbk{\Omega t} \cdot \sin^2\sbk{2 \theta} \\
	\abs{\braket{-}{\psi(t)}}^2		&= \cos^2\sbk{\Omega t} + \sin^2\sbk{\Omega t} \cdot \cos^2\sbk{2 \Omega}
\end{align}


\section{Aufgabe 10: Zerfall eines instabilen Zustandes}
\subsection*{a)}
\subsection*{b)}
\subsection*{c)}
\subsection*{d)}
\subsection*{e)}
\subsection*{f)}