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\title{Zusammenfassung Quantenmechanik}
\author{}
\date{}

\begin{document}
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\maketitle

\section{Operatoren}
\begin{tabular}{p{.03\textwidth}p{.42\textwidth}p{.42\textwidth}}
& Ortsdarstellung:
\begin{eqnarray*}
\hat x &=& x \\
\hat p &=& -i\hbar\stackrel{x}{\nabla}
\end{eqnarray*}
& Impulsdarstellung:
\begin{eqnarray*}
\hat x &=& i\hbar\stackrel{p}{\nabla}  \\
\hat p &=& p
\end{eqnarray*}
\end{tabular}

\subsection{Hamiltonoperatoren}
\begin{labeling}{harmonischer Oszillator:}
\item[freies Teilchen:] $\hat H = \frac{\hat p^2}{2m}$
\item[stationär:] $\hat H = \frac{\hat p^2}{2m} + V(\hat x)$
\item[harmonischer Oszillator:] $\hat H = \frac{\hat p^2}{2m} + \frac{1}{2}m\omega^2\hat x^2$
\item[starrer Rotator:] $\hat H = \frac{1}{2\Theta} \hat L^2$
\end{labeling}

\subsection{weitere Operatoren}
\begin{tabular}{p{.03\textwidth}p{.28\textwidth}p{.28\textwidth}p{.28\textwidth}}
& $\hat E = i\hbar\frac{\partial}{\partial t}$ & $\hat U(t) = \e^{-\frac{i}{\hbar} \hat H t}$ \medskip\\
& $\hat a = \sqrt{\frac{m\omega}{2\hbar}}\left(\hat x + \frac{i}{m\omega}\hat p\right)$ & $\hat a^\dagger = \sqrt{\frac{m\omega}{2\hbar}}\left(\hat x - \frac{i}{m\omega}\hat p\right)$ \medskip\\
& $\hat S_i = \frac{\hbar}{2} \sigma_i$ & $\hat L = \hat x \times \hat p$ & $\hat L_\pm =  \hat L_x \pm i \hat L_y$

\end{tabular}

\subsubsection{Anwendung der Operatoren auf jeweilige Eigenzustände}
\begin{tabular}{p{.03\textwidth}p{.50\textwidth}p{.34\textwidth}}
& $\hat a \ket{n}=\sqrt{n}\cdot \ket{n-1}$ & $\hat a^\dagger \ket{n} = \sqrt{n+1}\cdot \ket{n+1}$\medskip\\
& ${\hat L}^2 \ket{lm}= \hbar^2 l(l+)\ket{lm}$ & $\hat L_3\ket{lm}= \hbar m \ket{lm}$\smallskip\\
& $\hat L_\pm\ket{lm}= \hbar \sqrt{l(l+1)-m(m\pm 1)}\ket{l\,m\pm 1}$
\end{tabular}


\section{Kontinuitätsgleichung}
\[ \frac{\partial}{\partial t}\,\rho(\vec x, t) + \vec \nabla \cdot \vec j(\vec x, t) = 0 \]
\begin{eqnarray*}
\text{mit:} \quad \rho(\vec x, t) &\!\!\!=\!\!\!& \Psi^*\Psi \\
\vec j(\vec x,t) &\!\!\!=\!\!\!& -\frac{\hbar i}{2m}\left[ \Psi^*\left(\vec \nabla \Psi\right) -\left(\vec \nabla \Psi^* \right)\Psi\right]
\end{eqnarray*}


\section{Schrödingergleichung}
\[\hat H \Psi = \hat E \Psi \quad \Longleftrightarrow \quad \left( \frac{\hat p^2}{2m} + V(\hat x)\right) \Psi = i\hbar\frac{\partial}{\partial t}\Psi\]
\subsection{stationäre Schrödingergleichung}
\[ \hat H\,\Phi(\vec x) = E\,\Phi(\vec x) \qquad\quad \text{mit } \Psi(\vec x,t) = \Phi(\vec x)\ \e^{-\frac{i}{\hbar}E\cdot t} \text{ stationäre Zustände} \]

\section{Kommutatoren}
\[ \kom A B := AB - BA \]
\subsection{wichtige Relationen}
\[ \kom A {B+C} = \kom A B + \kom A C \]
\[ \kom A {B\ C} = B \kom A C + \kom A B C \]
\[\bigl[A,\kom B C\bigr] + \bigl[B,\kom C A\bigr] + \bigl[C,\kom A B\bigr] = 0 \]
\subsection{wichtige Kommutatoren}
\begin{tabular}{p{.03\textwidth}p{.28\textwidth}p{.28\textwidth}p{.28\textwidth}}
& $\kom {\hat x}{\hat p} = i\hbar$ & $\kom {f(\hat x)}{\hat p} = i\hbar\,f'(\hat x)$ & $\kom {\hat x}{f(\hat p)} = i\hbar\,f'(\hat p)$ \medskip\\
& $\kom {\hat a^\dagger}{\hat a} = 1$ \medskip\\
& $\kom{\hat L_i}{\hat L_j} = i\hbar\,\varepsilon_{ijk}\,\hat L_k$ & $\kom{\hat L_i}{\hat x_j} = i\hbar\,\varepsilon_{ijk}\,\hat x_k$ & $\kom{\hat L_i}{\hat p_j} = i\hbar\,\varepsilon_{ijk}\,\hat p_k$ \medskip\\
& $\kom{\hat L^2}{\hat L_z} = 0$ & \multicolumn{2}{l}{$\kom{\hat L_+}{\hat L_-} = -2i \kom{\hat L_x}{\hat L_y} = 2 \hbar \hat L_z$} \medskip\\
& $\kom{\hat L^2}{\hat L_\pm} =0$ & \multicolumn{2}{l}{$\kom{\hat L_\pm}{\hat L_z} = -i \hbar\,\hat L_y \pm i\left(i\hbar\,\hat L_x\right) = \pm\hbar\,\hat L_\pm$} \medskip\\
\end{tabular}

\section{Pauli-Matrizen}
\[ \sigma_i = \left( \begin{array}{cc}0&1\\1&0\end{array}\right), \qquad \sigma_2 = \left( \begin{array}{cc}0&-i\\i&0\end{array}\right), \qquad \sigma_3 = \left( \begin{array}{cc}1&0\\0&-1\end{array}\right) \]
\subsection{Eigenschaften}
\begin{itemize}
\item $ \mathrm{Tr}\ \sigma_i = 0$
\item $\sigma_i \cdot \sigma_j = \delta_{ij} 1\!\!1 + i\,\varepsilon_{ijk}\,\sigma_k$
\item $\kom{\sigma_i}{\sigma_j} = 2i\,\varepsilon_{ijk}\,\sigma_k$
\end{itemize}


\section{Dirac-Notation}
\begin{tabular}{p{.03\textwidth}p{.28\textwidth}p{.28\textwidth}p{.28\textwidth}}
& $\Braket{a|b}^* = \Braket{b|a}$ & $\Braket{a|\hat O|b}^* = \Braket{b|\hat O^\dagger|a}$ \medskip\\
& $\Psi(x) = \Braket{x|\Psi}$ & $\widetilde\Psi(p) = \Braket{p|\Psi}$ \medskip\\
& $\hat x\Ket x = x\Ket x$& $\hat p\Ket p = p\Ket p$ \medskip\\
& $\displaystyle \int\limits_{-\infty}^\infty \dd^3 x \ket x \bra x = 1\!\!1$ & $\displaystyle \int\limits_{-\infty}^\infty \frac{\dd^3 p}{\left(2\pi\hbar\right)^3} \ket x \bra x = 1\!\!1$ & (Vollständigkeit) \medskip\\
& $\Braket{x|x'} = \delta(x-x')$ & $\Braket{p|p'} = \left(2\pi\hbar\right)^3 \delta(p-p') $ & (Orthogonalität) \medskip\\
& $\Braket{x|p} = \e^{i\frac{x\cdot p}{\hbar}}$ & $\Braket{p|x} = \e^{-i\frac{p\cdot x}{\hbar}}$
\end{tabular}


\section{Störungstheorie}
\subsection{zeitunabhängig}
Hamiltonoperator $\hat H = \hat H_0 + \hat V \qquad \text{ mit } \hat V \text{: Störpotential}$ \medskip\\
bekannt: $\Hat H_0 \Ket{n^{(0)}} = \Hat E_n^{(0)} \Ket{n^{(0)}}$ \\
Energiekorrektur: \qquad\qquad\qquad $\displaystyle E_n^{(1)} = \braket{n^{(0)} | \hat V | n^{(0)}} \qquad\qquad\qquad E_n^{(2)} = \sum\limits_{m \neq n} \frac{\left|\braket{m^{(0)} | \hat V | n^{(0)}}\right|^2}{E_n^{(0)} - E_m^{(0)}}$ \\
Korrektur zur Wellenfunktion: \qquad $\displaystyle \ket{n^{(1)}} = \sum\limits_{m \neq n} \ket{m^{(0)}} \frac{\braket{m^{(0)} | \hat V | n^{(0)}}}{E_n^{(0)} - E_m^{(0)}}$


\end{document}