\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}