qm1-script/ueb1.tex

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\chapter{Quantenmechanik I - Übungsblatt 1}
\section{Aufgabe 1: Stern-Gerlach Experimente}

\section{Aufgabe 2: Pauli Matrizen}
\subsection*{a)}

	\begin{math}
	\textbf{\sigma} 	= (\sigma_x, \sigma_y, \sigma_z) \\
	\textbf{\a},\textbf{\b} \in \setR
	\end{math}
	
	\begin{align}
	
	(\textbf{a \cdot \sigma})(\textbf{b \cdot \sigma})	&= \one (\textbf{a \cdot b} + \i \textbf{\sigma} \cdot (\textbf{\a} \times \textbf{b}) \\
	\sum a_\alpha b_\beta \sigma_\alpha \sigma_\beta	&=  \\
	\sum a_\alpha b_\beta ( \krondelta{\alpha \beta} \one	+ \i \levicivita{\alpha,\beta,\gamma} \sigma_\gamma )	&= \\
	\sum a_\alpha b_\beta \krondelta{\alpha \beta} \cdot \one + \i a_\alpha b_\beta \levicivita{\alpha,\beta,\gamma} \sigma_\gamma &= \\
	\one (\textbf{a \cdot b} + \i \sigma \cdot (a \times b)
	\end{align}
\subsection*{b)}
\subsection*{c)}
\subsection*{d)}

\section{Aufgabe 3: Operator-Identitäten}
\subsection*{a)}
\subsection*{b)}
\subsection*{c)}