übungsblatt 9

ueb9.tex

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+\chapter{Quantenmechanik I - Übungsblatt 9}
+\section{Aufgabe 22: Elektron im ortsunabhängigen Magnetfeld:}
+\subsection*{a)}
+\begin{align}
+	H	&= H_0 \otimes \one - \sum_{n=1}^2 \ket{n} \bra{n} \otimes \frac{\hbar \omega_n}{2} \sigma_z	&\left| \omega_n = \gamma B_n \right. \\
+		&= \inlinematrix{0 & -A \one \\ -A \one & 0} - \inlinematrix{\hbar \gamma B_1 \frac{\sigma_z}{2} & 0 \\ 0 & -\hbar \gamma B_2 \frac{\sigma_2}{2}} \\
+		&= \inlinearray{cc|cc}{-\frac{\hbar \omega_1}{2} & -A & 0 & 0 \\ -A  -\frac{\hbar \omega_2}{2} & 0 & 0 \\ \hline
+									0 & 0 & \frac{\hbar \omega_1}{2} & -A \\ 0 & 0 & -A & \frac{\hbar \omega_2}{2}} \\ \\
+		&= \inlinematrix{H\uparrow & 0 \\ 0 & H\downarrow}
+\end{align}
+
+\subsection*{b)}
+\equationblock{\lambda_{1 \ldots 4} \frac{\pm \sqrt{\hbar^2 \sbk{\omega_1-\omega_2}^2 + 16 A^2} \pm \hbar \sbk{\omega_1 + \omega_2}}{4}}
+\subsection*{c)}
+$\pm \sqrt{x^2 + 1} \pm 2 x$
+%da kommt nen bild rein
+
+\section{Aufgabe 23: Kohärente Zustände (Glauber-Zustände):}
+\subsection*{a)}
+%alles richtig hier?
+$a \ket{\alpha} = \alpha \ket{\alpha}$
+$\hat{n} = a^\dagger a$
+$\ket{n} = \frac{1}{\sqrt{n!}} \sbk{a^dagger}^n \ket{0}$
+$\braket{\alpha}{\alpha} = 1$
+\begin{align}
+	a \ket{\alpha}	&= \sum_{n=0} c_n(x) a \ket{n}		&\left| a \ket{n} = \sqrt{n} \ket{n-1} \right. \\
+					&= \sum_{n \geq 0} c_n(x) \sqrt{n} \ket{n-1} \\
+					&= \alpha \ket{\alpha} \\
+					&= \alpha \sum_{n \geq 0} c_n(\alpha) \ket{n} \\
+					&= \sum_{n \geq 0} \alpha \braket{n(\alpha)}{n} \\
+					&= \sum_{n \geq 0} \alpha \braket{n-\i \alpha}{n - 1} \\ %HÄ?
+\end{align}
+$\Rightarrow$
+\begin{align}
+	c_n(\alpha) \sqrt{n}	&= \alpha c_{n-1}(\alpha) \\
+	c_n(\alpha)				&= \frac{\alpha}{\sqrt{n}} c_{n-1}(\alpha) \\
+							&= \frac{\alpha^2}{\sqrt{n!}} c_0(alpha) \\
+	\ket{\alpha}			&= \sbk{\sum_{n \geq 0} \frac{\alpha^n}{\sqrt{n!}} \ket{n}} \cdot c_0(\alpha)	&\left| \bra{\alpha} \right. \\
+	c_0(\alpha)				&= e^{-\abs{\alpha}^2 \frac{1}{2}} \\
+	\sum_{n \geq 0} \frac{\alpha^n}{\sqrt{n!}} \ket{n}	&= \sum_{n \geq 0} \frac{\alpha^n}{\sqrt{n!}} \frac{1}{\sqrt{n!}} \sbk{a^\dagger}^n \ket{0} \\
+							&= e^{\alpha a^\dagger} \ket{0} \\
+	\ket{\alpha}			&= \underbrace{e^{-\frac{1}{2} \abs{\alpha}^2} e^{\alpha a^\dagger}}_{c(\alpha)} \ket{0}
+\end{align}
+
+\subsection*{b)}
+\begin{align}
+	\dirac{\alpha}{\hat{n}}{\alpha}	&= \dirac{\alpha}{a^\dagger a}{\alpha} \\
+									&= \braket{a \alpha}{a x} \\
+									&= \braket{\alpha}{\alpha} \alpha^\ast \alpha \\
+									&= \abs{\alpha}^2 \\
+	\sbk{\Delta \hat{n}}^2			&= \dirac{\alpha}{\hat{n}^2}{\alpha} - \abs{\alpha}^4 \\
+									&= \dirac{\alpha}{a^\dagger a a^\dagger a}{x} - \abs{\alpha}^4 \\
+									&= \abs{\alpha}^2 \dirac{\alpha}{a a^\dagger}{\alpha} - \abs{\alpha}^4 \\
+									&= \abs{\alpha}^2 \sbk{\abs{\alpha}^2 + 1} - \abs{\alpha}^4 \\
+									&= \abs{\alpha}^2 \\
+									&= \ssbk{\hat{n}}
+\end{align}
+
+%fehlt da was?
+
+\subsection*{c)}
+\begin{align} %komisch hier. sicher viel falsch
+	\ket{\psi(0)}	&= \sum_{n=0}^{\infty} c_r \ket{n}
+	\ket{\psi(t)}	&= e^{-\frac{\i}{\hbar} H t} \\
+					&= \sum_{n=0}^{\infty} e^{-\i \sbk{n + \frac{1}{2}} \omega t} \\
+					&= e^{-\frac{\abs{\alpha}^2}{\hbar}} e^{-\frac{\i}{2} \omega t} \sum_{n=0}^{\infty} \frac{\alpha^n e^{-\i \omega t n}}{\sqrt{n!}} \ket{n} \\
+					&= e^{\alpha \i \frac{1}{2} e^{-\i \omega t} I^2} \sum_{n=0}^\infty \frac{\sbk{\alpha e^{-\i \omega t}}^2}{\sqrt{n!}} \ket{n}
+	\ket{\alpha}	&= e^{-\frac{\abs{\alpha}^2}{2}} \sum_n \frac{\alpha^n}{\sqrt{n!}} \ket{n} \\
+	\ket{\alpha(t)}	&= \sum_n e^{-\i \sbk{n + \frac{1}{2}} \omega t} \frac{\alpha^n}{\sqrt{n!}} \ket{n} c_0
+\end{align}
+
+mit $\tilde{\alpha} = \alpha e^{-\i \omega t}$
+
+
+%anderes blatt
+\begin{align}
+	\left<x\right>		&= \dirac{\tilde{\alpha}(t)}{\frac{1}{\sqrt{2}} \sbk{a + a^\dagger}}{\tilde{\alpha}(t)}
+	\ket{\psi(t)}		&= e^{-\frac{\i H t}{\hbar}} \ket{\alpha_0} \\
+						&= \sum_n c_0 \frac{\alpha_0^2}{\sqrt{n!}} e^{-\frac{\i \sbk{n +\frac{1}{2}} \omega t}{\hbar}} \ket{n} \\
+						&= e^{-\i \frac{1}{2} \omega t} \ket{\alpha(t)} \\
+	\left<x\right>		&= \dirac{\alpha(t)}{\frac{1}{\sqrt{2}} \sbk{a + a^\dagger}}{\alpha(t)} \\
+						&= \frac{1}{\sqrt{2}} \sbk{\alpha^\ast(t) + \alpha(t)}
+\end{align}
+mit $\alpha(t) = \alpha e^{-\i \omega t}$
+
+\begin{align}
+	\ssbk{x}			&= \frac{\alpha_0}{\sqrt{2}} \sbk{e^{\i \omega t} + e^{-\i \omega t}} \\
+						&= \frac{a_0}{\sqrt{2}} 2 \cosb{\omega t} \\
+	\ssbk{p}			&= \dirac{\alpha(t)}{\frac{\i}{\sqrt{2}} \sbk{a - a^\dagger}}{\alpha(t)} \\
+						&= - \sqrt{2} a_0 \sinb{\omega t}
+	\ssbk{x^2}			&= \dirac{\alpha(t)}{\frac{1}{2} \sbk{a a + a^\dagger a^\dagger - a a^\dagger + a^\dagger a}}{\alpha(t)} \\
+						&= \frac{1}{2} \sbk{\sbk{\alpha^\ast + \alpha}^2 + 1} \\
+						&= \sbk{1 + \ssbk{\hat{x}}^2} \frac{1}{2}
+	\ssbk{p^2}			&= \sbk{\ssbk{p}^2 + 1} \frac{1}{2}
+	\sbk{\Delta x}^2	&= \left<x^2\right> - \left<x\right>^2	&= \frac{1}{2} \\
+	\sbk{\Delta p}^2	&= \frac{1}{2} \\
+	\Delta p \Delta x	&= \frac{1}{2}
+\end{align}
+
+
+
+\subsection*{d)}
+\begin{align}
+	\braket{\alpha'}{\alpha}	&= \sum_m \sum_n c_m^\ast(\alpha) c_n(\alpha) \braket{m}{n} \\
+								&= c_0^\ast(\alpha) c_0^\ast(\alpha!) \sum_n \frac{\sbk{\alpha^\ast \alpha}^n}{n!} \\
+								&= e^{-\frac{1}{2} \abs{\alpha}^2} - \frac{1}{2} \abs{\alpha}^2 + \alpha^\ast \alpha \\
+	\intgrinf{\ket{\alpha} \braket{\alpha}}{^2\alpha}	&= \sum_m \sum_n \frac{1}{\sqrt{m!} \sqrt{n!}} \ket{n} \bra{m} \intgrinf{\sbk{\alpha^ast}^m \sbk{\alpha^\ast}^n e^{- \abs{\alpha}^2}}{^2\alpha} \\
+								&= \sum_m \sum_n \frac{1}{\sqrt{m!} \sqrt{n!}} \ket{n} \bra{m} \intgrinf{\abs{\alpha}^{m+n} e^{\i \sbk{n - m} \phi} e^{-\abs{\alpha}^2}}{\alpha} \\
+								&= \sum_m \sum_n \frac{1}{\sqrt{m!} \sqrt{n!}} \ket{n} \bra{m}
+								\int_\infty^\infty ~\text{ d}\alpha \int_\infty^\infty ~\text{ d}\phi e^{\i \sbk{n -m} \phi} \abs{\alpha}^{m+n} e^{-\abs{\alpha}^2}
+								&= \Gamma \frac{m x_1 + 1}{4}
+\end{align}