70 lines
3.6 KiB
TeX
70 lines
3.6 KiB
TeX
\chapter{Bohr'sche Näherung für Streutheorie}
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% \begin{figure}[H] \centering
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% \includegraphics{pdf/IV/03-00-00.pdf}
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% \end{figure}
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\section{Geometrie}
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% \begin{figure}[H] \centering
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% \includegraphics{pdf/IV/03-01-00.pdf}
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% \end{figure}
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\section{Stat SG}
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\equationblock{\Phi\sbk{\vec{r}} = \Phi^\text{in}\sbk{\vec{n}} + \Phi^\text{ex}\sbk{\vec{r}}}
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mit
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\equationblock{\Phi^\text{ex}\sbk{\vec{r}} \longrightarrow^{\vec{r}\rightarrow\infty} f\sbk{\theta,\Phi} \frac{e^{\i k \vec{r}}}{r}}
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und
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\equationblock{\Phi^\text{in}\sbk{\vec{r}} = e^{i \vec{k} \vec{r}}}
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SG:
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\begin{align}
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\sbk{-\frac{\hbar^2}{2 \mu} \vec{\nabla}^2 + V\sbk{\vec{r}}} \Phi\sbk{\vec{r}} &= E \Phi\sbk{\vec{r}} &\left| -\frac{2 \mu}{\hbar^2} \right. \\
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\sbk{\vec{\nabla}^2 - V\sbk{\vec{r}}} \Phi\sbk{\vec{r}} &= -\frac{2 \mu}{\hbar^2} E \Phi\sbk{\vec{r}} &\left; E = \frac{\hbar^2}{2 \mu} k^2 \right. \\
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\rightarrow \sbk{\vec{\nabla}^2 + k^2} \Phi\sbk{\vec{r}} &= V\sbk{\vec{r}} \Phi\sbk{r}
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\end{align}
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Green's Funktion Ansatz
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\begin{align}
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&\sbk{\vec{\nabla}^2 + k^2} G^0\sbk{\vec{r}-\vec{r'}} = \delta\sbk{\vec{r}-\vec{r'}} \\
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&\Phi\sbk{\vec{r}} = \intgru{}{r'} G^\sbk{0}\sbk{\vec{r}-\vec{r'}} V\sbk{\vec{r'}} \Phi\sbk{\vec{r'}} + \underbrace{\Phi^\sbk{0}\sbk{\vec{r}}}_{\text{beliebige Lösung der homogenen Gl.}}
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\end{align}
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\begin{enumerate}
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\item für $V = 0$: \equationblock{\Phi\sbk{\vec{r}} = \Phi^\sbk{0}\sbk{\vec{r}} = e^{\i \vec{k} \vec{r}}}
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\item in $\bigOb{V}$: \equationblock{\Phi{\vec{r}} \approx \sbk{\intgru{}{\vec{r}} G^\sbk{0}\sbk{\vec{r}-\vec{r'}} v\sbk{\vec{r}} e^{\i \vec{k} \vec{r}}} + e^{\i \vec{r} \vec{r}}}
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\item im Prinzip iterieren:
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\begin{align}
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\Phi &= \Phi^0 + \G^\sbk{0}v \Phi^0 + G^\sbk{0} v G^\sbk{0} \Phi^\sbk{0} + \ldots
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&= \frac{1}{1 - G^\sbk{0} v} \Phi^\sbk{0} &\left( geometrische Reihe \right)
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\end{align}
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formal exakt, praktische ziemlich nutzlos
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\end{enumerate}
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\section{Berechnung der Green'schen Funktion}
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\begin{align}
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\sbk{\vec{\nabla}^2 + k^2} G\sbk{\vec{u}} &= \delta\sbk{\vec{u}} &\left| \intgru{e^{-\i \vec{q} \vec{u}}}{\vec{u}} \right. \\
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\sbk{-\vec{q}^2 + k^2} G\sbk{q} &= 1 \\
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G\sbk{q} &= \frac{1}{k^2 - q^2} \\
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G\sbk{\vec{ u}} &= \intgru{\frac{1}{\sbk{2 \pi}^2} \frac{1}{k^2 - q^2} e^{\i \vec{q} \vec{u}}}{q} \\
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&= \frac{1}{4 \pi^2} \intgr{-1}{+1}{\intgr{0}{\infty}{q^2 \frac{1}{k^2 - q^2} e^{\i q u \cosb{\theta}}}{q}}{\sbk{\cosb{\theta}}} \\
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&= \frac{1}{4 \pi^2} \intgr{0}{\infty}{\frac{q^2}{\i q u} \sbk{e^{\i q u} - e^{-\i q u}} \frac{1}{k^2 - q^2}}{q} \\
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&= \frac{1}{4 \pi^2} \frac{1}{\i n} \intgrinf{\frac{q e^{\i q n}}{k^2 -q^2}}{q}
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&= \frac{1}{4 \pi^2} \frac{1}{\i n} \intgrinf{\frac{q e^{\i q u}}{k^2 - q^2 + \i \epsilon}}{q} \\
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\text{Residuensatz} \Rightarrow &= \frac{1}{4 \pi^2} \frac{1}{\i u} 2 \pi \i \underbrace{Res\sbk{q=k}_{\epsilon \rightarrow 0}}_{-\frac{k}{2 k} e^{\i k u} \\
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G\sbk{u} &= -\frac{1}{4 \pi u} e^{\i k u}
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\end{align}
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% \begin{figure}[H] \centering
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% \includegraphics{pdf/IV/03-03-00.pdf}
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% \end{figure}
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\section{Bohr'sche Näherung}
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\equationblock{}
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\section{Streuamplitude und differentieller Wirkungsquerschnitt}
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Bsp.: abgeschirmtes Coulomb-Potential
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Yukawa Potential
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\begin{align}
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V\sbk{r} &= \frac{l^2}{r} l^{-\frac{r}{r_0}} \\
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V\sbk{q} &= \frac{4 \pi l^2}{q}\intgr{0}{\infty}{\sinb{q r'} l^{-\frac{r}{r_0}}}{r'} \\
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&= 4 \pi l^2 \frac{1}{q^2 + \frac{1}{r_0^2}} \\
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\diffTfrac{r}{\Omega} &= \frac{l^2}{4 \mu^2 \tilde{V}^4 \sin^4\sbk{\frac{\theta}{2}}} &\left \tilde{V} = \frac{\hbar k}{\mu} \left.
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\end{align}
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Rutherford'sche Streuquerschnitt für das Coulomb-Problem
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