41 lines
1.7 KiB
TeX
41 lines
1.7 KiB
TeX
\begin{landscape}
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\chapter{Übersicht und Notation}
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\begin{tabular}{l||l|l|l}
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& $d = 1$ & $d = 2$ & $d = 3$ \\ \hline \hline
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SG: $i\hbar \ket{\psi} = H \ket{\psi}$ &
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$\checkmark$ &
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$\checkmark$ &
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$\checkmark$ \\ \hline
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Hamiltonoperator: $H$ &
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$\frac{\hat{p}^2}{2m} + V(\hat{x})$ &
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$\frac{\hat{p}_x^2}{2m} + \frac{\hat{p}_y^2}{2m} + V(\hat{x}, \hat{y})$ &
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$\frac{\hat{p}_x^2}{2m} + \frac{\hat{p}_y^2}{2m} + \frac{\hat{p}_z^2}{2m} + V(\hat{x}, \hat{y}, \hat{z})$ \\ \hline
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Kommutator &
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$[\hat{x}, \hat{y}] = i\hbar$ &
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$[\hat{x}_k, \hat{y}_l] = i\hbar \krondelta{k,l};~ k, l = 1, 2$ &
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$[\hat{x}_k, \hat{y}_l] = i\hbar \krondelta{k,l};~ k, l = 1, 2, 3$ \\ \hline
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stationäre SG: $H \ket{\phi} = E \ket{\psi}$ &
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gewöhnliche DGL &
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partielle DGL &
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partielle DGL\\ \hline
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Rotationsinvarianz &
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Irrelevant &
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$V(x,y) = V(\rho)$; $\rho \equiv \sqrt{x^2 + y^2}$ &
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$V(x,y,z) = V(r)$; $r \equiv \sqrt{x^2 + y^2 + z^2}$ \\ \hline
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Ortseingen-vektoren &
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$\hat{x}\ket{x} = x \ket{x}$ &
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$\ket{x,y} \cequiv \ket{\rho, \theta}$ &
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$\ket{x,y,z} \cequiv \ket{\rho, \theta, \varphi}$ \\ \hline
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Norm der Wellenfunktion &
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\begin{math}1 = \intgr{-\infty}{+\infty}{\abs{\psi(x)}^2}{x}\end{math} &
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$\begin{array}{l}
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1 = \intgr{-\infty}{+\infty}{\intgr{-\infty}{+\infty}{\abs{\psi(x,y)}^2}{y}}{x} \\
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\hphantom{1} = \intgr{0}{\infty}{\rho \intgr{0}{2\pi}{\abs{\psi(\rho, \varphi)}^2}{\varphi}}{\rho}
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\end{array}$ &
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$\begin{array}{l}
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1 = \intgru{\intgru{\intgru{\abs{\psi(x,y,z)}^2}{x}}{y}}{z} \\
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\hphantom{1} = \intgr{0}{+\infty}{r^2 \intgr{-1}{+1}{\intgr{0}{2\pi}{\abs{\psi}^2}{\varphi}}{(\cos\theta)}}{r}
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\end{array}$ \\ \hline
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\end{tabular}
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\end{landscape}
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