44 lines
1.7 KiB
TeX
44 lines
1.7 KiB
TeX
%\includegraphics{excs/qm1_blatt04_SS08.pdf}
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%\pagebreak
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\chapter{Quantenmechanik I - Übungsblatt 4}
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\section{Aufgabe 9: Zeitentwicklung eines allgemeinen Zweizustandssystems}
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\begin{math}
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H = \hbar \inlinematrix{A & B \\ B & -A} \\
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A = \Omega \cosb{2 \theta} \\
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B = \Omega \sinb{2 \theta} \\
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E_\pm = \pm \hbar \Omega
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\end{math}
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\subsection*{a)}
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\begin{align}
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\ket{\chi_+} &= \inlinematrix{\sinb{2 \theta} \\ 1 - \cosb{2 \theta}} &= \inlinematrix{\cosb{\theta} \\ \sinb{\theta}} \\
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\ket{\chi_-} &= &= \inlinematrix{\sinb{\theta} \\ -\cosb{\theta}}
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\end{align}
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\subsection*{b)}
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\begin{align}
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\i \hbar \diffPs{t} \ket{\psi} &= H \ket{\psi} \\
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\ket{\dot{\psi}} &= -\i \Omega \inlinematrix{\cosb{2 \theta} & \sinb{2 \theta} \\ \sinb{2 \theta} & -\cosb{2 \theta}} \ket{\psi} \\
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\ket{\psi(t)} &= c_1 \cdot e^{\i \Omega t} \ket{\chi+} + c_2 \cdot e^{-\i \Omega t} \ket{\chi-} \\
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\ket{\psi(0)} &= \inlinematrix{\lambda \\ \mu} \\
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\inlinematrix{c_+(t) \\ c_-(t)} &= \sbk{\lambda \cosb{\theta} + \mu \sinb{\theta}} \cdot \inlinematrix{\cosb{\theta} \\ \sinb{\theta}} \cdot e^{-\i t} +
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\sbk{\lambda \sinb{\theta} - \mu \cosb{\theta}} \cdot \inlinematrix{\sinb{\theta} \\ -\cosb{\theta}} \cdot e^{\i \Omega t}
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\end{align}
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\subsection*{c)}
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%hier stimmt evtl. was nicht
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\begin{align}
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\ket{\psi(0)} &= \inlinematrix{0 \\ 1} \\
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\abs{\braket{+}{\psi(t)}}^2 &= \sin^2\sbk{\Omega t} \cdot \sin^2\sbk{2 \theta} \\
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\abs{\braket{-}{\psi(t)}}^2 &= \cos^2\sbk{\Omega t} + \sin^2\sbk{\Omega t} \cdot \cos^2\sbk{2 \Omega}
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\end{align}
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\section{Aufgabe 10: Zerfall eines instabilen Zustandes}
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\subsection*{a)}
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\subsection*{b)}
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\subsection*{c)}
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\subsection*{d)}
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\subsection*{e)}
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\subsection*{f)} |