%\includegraphics{excs/qm1_blatt03_SS08.pdf}
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\chapter{Quantenmechanik I - Übungsblatt 3}
Zugehörige Voraussetzungen:
Operatoren
Wahrscheinlichkeiten

\section{Aufgabe 7: Eigenzustände und Erwartungswerte von Spin-1/2-Teilchen}
$\vec{n}(\theta,\Phi) = \inlinematrix{\sin(\theta) \cos(\Phi) \\ \sin(\theta) \sin(\Phi) \\ \cos(\Phi)}$ \\
Präparation in $\ket{n_1+} \vec{n_1} = \vec{n}\sbk{\frac{\pi}{4},\frac{\pi}{4}}$ \\
$\Rightarrow$ \\
$\vec{n_1} = \inlinematrix{\cosb{\frac{\pi}{4}} \\ 1 \\ \cosb{\frac{\pi}{4}}}$
$\ket{n_1} = \inlinematrix{\cosb{\frac{\Theta}{2}} \\ e^{\i \phi} \sinb{\frac{\Theta}{2}}}$

\subsection*{a)}
$\vec{n_2} = \vec{n}\sbk{\frac{3\pi}{4},\phi}$
\begin{align}
	p_+\sbk{\phi}	&= \probb{\sigma_{n_2} \cequiv +1}{\ket{n_1+}} \\
					&= \abs{\braket{n_2+}{n_1+}}^2 \\
					&= \frac{1}{4} \sbk{1 + \cosb{\phi+\frac{\pi}{4}}} \\
	p_-\sbk{\phi}	&= 1 - p_+ \\
					&= 1 - \frac{1}{4} \sbk{1 + \cosb{\phi+\frac{\pi}{4}}} \\
					&= \frac{3}{4} - \frac{1}{4} \cosb{\phi+\frac{\pi}{4}} \\
					&= \frac{1}{4} \sbk{3 - \cosb{\phi+\frac{\pi}{4}}} \\
	\ket{n_1+}		&= c_1 \ket{n_2+} + c_2 \ket{n_2-}
\end{align}

\subsection*{b)}
	\begin{align}
		\expval{sigma_z}_{\ket{n_1+}}	&= \probb{\sigma_z \cequiv +1}{\ket{n_1+}} +
											(-1) \probb{\sigma_z \cequiv -1}{\ket{n_1+}} \\
										&= \abs{\inlinematrix{1 & 0} \inlinematrix{\cos(\frac{\pi}{8} \\ e^{\i \frac{\pi}{4}}}}^2 -
											\abs{\inlinematrix{0 & -1} \inlinematrix{\cos(\frac{\pi}{8} \\ e^{\i \frac{\pi}{4}}}}^2 \\
										&= \cos^2(\frac{\pi}{8}) - \sin^2(\frac{\pi}{8}) \\
										&= \cos(\frac{\pi}{4} \\
										&= \frac{\sqrt{2}}{2} \\
		\sbk{\Delta \sigma_x}^2			&= \prob{\sigma_x^2} - \prob{\sigma_x}^2 \\
		\dirac{n_1+}{\sigma_x}{n_1+}	&= \sin(\frac{\pi}{4} \cos(\frac{\pi}{4}	&= \frac{1}{2} \\
		\dirac{n_1+}{\sigma_x^2}{n_1+}	&= \inlinematrix{\cos(\frac{\pi}{8}) & e^{-\frac{\pi}{4}} \sin(\frac{\pi}{8})} \cdot
											\inlinematrix{1 & 0 \\ 1 & 0} \cdot
											\inlinematrix{\cos(\frac{\pi}{8}) \\ e^{\frac{\pi}{4}} \sin(\frac{\pi}{8})} \\
										&\Rightarrow
		\sbk{\Delta \sigma_x}^2			&= 1 - \frac{1}{2}^2 \\
										&= \frac{3}{4} \\
										&\stackrel{\text{analog}}{=} \sbk{\Delta \sigma_y}^2 \\
		\sbk{\Delta \sigma_x} \cdot \sbk{\Delta \sigma_y}	&= \frac{3}{4} \\
		\sbk{\Delta A} \cdot \sbk{\Delta B}	&\geq \frac{1}{2} \abs{\expval{\frac{1}{\i} [A,B]}} \\
		\frac{1}{2} \abs{\expval{\sigma_z}}	&= \frac{1}{\sqrt{2}} &< \frac{3}{4}
	\end{align}
	
\subsection*{c)}
\begin{align}
	\ket{\Psi}							&= N (\ket{Z+} + e^{\i \alpha} \ket{Z-}) = N \inlinematrix{1 \\ e^{\i \alpha}} \\
	\braket{\Psi}{\Psi}					&= 1 \\
	N^2 \sbk{1 + e^{\i \alpha - \i \alpha)}}	&\deq 1 \\
										&\Rightarrow \\
	N									&= \frac{1}{\sqrt{2}} \\ \\
	P_+									&= \abs{\braket{x+}{\Psi}}^2 \\
										&= \abs{\frac{1}{\sqrt{2}} \inlinematrix{1 \\ 1} \cdot \frac{1}{\sqrt{2}} \inlinematrix{1 \\ e^{\i \alpha}}}^2 \\
										&= \frac{1}{4} \abs{1 + e^{\i \alpha}}^2 \\
										&= \frac{1}{2} \sbk{1 + \cos(\alpha)}
\end{align}


\section{Aufgabe 8: Teilchen mit Spin 1}
\subsection*{a)}
\includegraphics{grafiken/U_A6_a.pdf}
\subsection*{b)}
$[\Sigma_\alpha, \Sigma_\beta] = \i \Sigma_{\alpha, \beta, \gamma} \Sigma_\gamma$
mit allen $\Sigma_{x,y,z}$ durch testen.
\subsection*{c)}
\begin{align}
	\Sigma^2	&= \Sigma_x^2 + \Sigma_y^2 + \Sigma_z^2 \\
				&= 2 \one
\end{align}

\subsection*{d)}
\begin{math}
	\ket{x+}, \ket{x-}, \ket{x0}, \ket{\Psi} = \ket{n_0}, n(\Theta,\Phi)
\end{math}
\begin{align}
	p_0					&\Rightarrow	&\ket{x0}				&= \frac{1}{\sqrt{2}} \cdot \inlinematrix{1 \\ 0 \\ -1} \\
	p_+					&\Rightarrow	&\ket{x+}				&= \frac{1}{2} \cdot \inlinematrix{1 \\ \frac{1}{\sqrt{2}} \\ 1} \\
	p_-					&\Rightarrow	&\ket{x-}				&= \frac{1}{2} \cdot \inlinematrix{-1 \\ \frac{1}{\sqrt{2}} \\ -1}
	\vec{\Sigma_n}		&= 				&\Sigma \cdot \vec{n}	&= \inlinematrix{\cos(\Theta) & \frac{1}{\sqrt{2}} \sin(\Theta) e^{-\ \phi} & 0 \\
																					\frac{1}{\sqrt{2}} \sin(\Phi) e^{\i \phi} & 0 & \frac{1}{\sqrt{2}}	\sin(\Theta) e^{-\i \phi} \\
																					0	& \frac{1}{\sqrt{2}} \sin(\Theta) e^{\i \phi} & \cos(\Theta)}
						&\Rightarrow
	\ket{n_0}			&				&						&= \frac{1}{\sqrt{2}} \inlinematrix{-\sin(\Theta) e^{\i \Phi} \\ \sqrt{2} \cos(\Theta) \\ \sin(\Theta) e^{\i \Phi}}
\end{align}
\begin{align}
	p_+		&= \abs{\braket{x_1}{n_0}}^2		&= \ldots \frac{1}{2} \cos^2(\Theta) + \frac{1}{2} \sin^2(\Theta) \sin^2(\Phi) = p_- \\
	p_0		&= 1- 2 p_+							&= 1 - 2 p_-
\end{align}

\subsection*{e)}
\begin{align}
	\expval{\Sigma_x}_{\ket{\Psi}}	&= \dirac{n_0}{\Sigma_x}{n_0} \\
									&= +1 + 1 p_+ + p_0 + (-1) p_- \\
									&= 0 \\
	\ket{\Psi}						&= c_1 \ket{x+} + c_2 \ket{x-} + c_3 \ket{x0} \\
	\sbk{\Delta \Sigma_x}^2		&= \langle \Sigma_x^2 \rangle - {\langle \Sigma_x \rangle}^2 \\
									&= c_2 \inlinematrix{1 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 0 & 1} %ist das richtig?
	\dirac{\Psi}{\Sigma_x^2}{\Psi}	&= c_1^2 \cdot 1^2 + c_2^2 \cdot (-1)^2 + c_3^2 \cdot 0^2 \\
									&\stackrel{c_1 = c_2}{=} 2 c_1^2 \\
									&= \cos^2(\Theta) + \sin^2(\Theta) \cdot \sin^2(\phi) %Ist das richtig?
\end{align}
