Übungsblatt 7
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ueb7.tex
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ueb7.tex
@ -73,6 +73,7 @@ mit den \hyperlink{fs_mtrx_inv_2d}{Inversen} von: (1) und (2) % geschweifte klam
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\inlinematrix{A & B} = \inlinematrix{\frac{1}{2} & -\frac{\i}{2k}\\ \frac{1}{2} & \frac{\i}{2k}} \inlinematrix{1 & 1 \\ q & -q} \inlinematrix{C & D}
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\end{math}
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\subsection*{c)}
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$\inlinematrix{C & D}$ eingesetzt ergibt:
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\begin{align}
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\tilde{E} &= \frac{2 k a}{e^{ika} \sbk{(q^2 - k^2) \cdot \i sinh(q \cdot a) - 2 k q cosh(qa)}} \\ % ist das richtig?
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@ -82,10 +83,49 @@ $\inlinematrix{C & D}$ eingesetzt ergibt:
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T &= \frac{1}{\sbk{\frac{a^2 - k^2}{2 q k}} \cdot 2 sinh^2(q a) + cosh^2(q a)}
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\end{align}
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\subsection*{c)}
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\section{Aufgabe 19: Doppeltes Delta-Potential}
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$V(x) = - \frac{\hbar^2}{m} \kappa_0 \sbk{\delta\sbk{x -a} + \delta\sbk{x + a}}$
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$E<0$
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$\sbk{-\frac{\hbar^2}{2 m} \diffPs{x}^2 - \frac{\hbar^2}{m} \kappa_0 \sbk{\delta\sbk{x - a} + \delta\sbk{x + a}}} \Phi(x) = E \Phi(x)$
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$\Phi'_I(-a) - \Phi'_{II}(-a) = - 2 \kappa_0 \Phi_I(-a)$
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Mit $k^2 = \frac{-2 m E}{\hbar^2}$
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Für I:
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\begin{align}
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\psi_I &= A e^{k x} \\
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\psi_{II} &= B e^{-k x} + c e^{k x} \\
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\psi_{III} &= D e^{-k x} \\
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\abs{A} &= \abs{D} \\
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A &= \pm D \\
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B &= \pm C \\
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0 &= B e^{2 k a} - C + A \sbk{1 - 2 \frac{k_0}{k}} \\
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0 &= C e^{2 k a} - B + D \sbk{1 - 2 \frac{k_0}{k}} \\
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\frac{k_0}{k} \sbk{e^{- 2 k a} \pm 1} &= \pm 1 \\
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e^{-2 k a} &= \frac{k}{k_0} - 1 \\
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e^{-2 k a} &= -\frac{k}{k_0} + 1 \\
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\end{align}
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%grafik einfügen
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\begin{align}
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\kappa_0 &= 1 \\
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\kappa_1 &= 1,11 \\
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\kappa_2 &= 0,80 \\
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E &= -\frac{k^2 \hbar^2}{2 m}
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\end{align}
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\subsection*{a)}
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\subsection*{b)}
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$k_0 a \gg 1$ $\Rightarrow$ $\kappa \approx \kappa_0$
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$k_\pm = k_0 \sbk{1 \pm e^{-2 \kappa_0 a}}$
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\subsection*{c)}
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\subsection*{d)}
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$a \rightarrow 0$
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\begin{align}
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\kappa_+ &\approx \kappa_0 \sbk{1 + 1 - 2k + a} \\
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&\approx 2 \kappa_0
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\end{align}
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