19 lines
620 B
TeX
19 lines
620 B
TeX
%--------------------------------------------------------------
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\begin{frame}
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\frametitle{Radial Wave Equation}
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\begin{block}{}
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By induction,
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$$\color{ec}
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\Psi_{\ell} = -\Big(\frac{\partial}{\partial r} - \frac{\ell}{r}\Big)\Psi_{\ell -1}, \hspace{2em} D^+_\ell = -\Big(\frac{\partial}{\partial r} - \frac{\ell}{r}\Big)
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$$
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If
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$$\color{ec}
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-\partial^2_t\Psi_{\ell-1} + \partial^2_r\Psi_{\ell-1} - \frac{\ell(\ell-1)}{r^2}\Psi_{\ell-1} = 0,
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$$
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Then
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$$\color{ec}
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-\partial^2_t(D^+_\ell \Psi_{\ell -1}) +\partial^2_r(D^+_\ell \Psi_{\ell-1}) - \frac{\ell(\ell +1)}{r^2}D^+_\ell \Psi_{\ell-1} = 0.
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$$
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\end{block}
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\end{frame}
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