Kapitel III.1 fertig

kapIII-1.tex

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