331 lines
11 KiB
TeX
331 lines
11 KiB
TeX
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\chapter{Mathematische Sprache/Bühne: Hilbert-Raum (endl. Dim.)}
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\section{Definition}
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\begin{itemize}
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\item linerarer Vektorraum $\hilbert$ mit Vektoren $\set{\ket{\phi}, \ket{\psi}}$
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\begin{equation}
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c_1 \ket{\phi} + c_2 \ket{\psi} = \ket{c_1 \phi + c_2 \psi} \in \hilbert
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\end{equation}
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\item Inneres (hermitesches) Produkt: $\hilbert \times \hilbert \rightarrow \setC$
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\begin{equation}
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\ket{\phi}, \ket{\psi} \rightarrow \braket{\phi}{\psi} = \braket{\psi}{\phi}^*
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\end{equation}
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\begin{enumerate}
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\item linear im 2. Argument:
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$\braket{\phi}{c_1 \psi_1 + c_2 \psi_2} = c_1 \braket{\phi}{\psi_1} + c_2 \braket{\phi}{\psi_2}$
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\item $\underbrace{\braket{\phi}{\phi}}_{\in \setR} > 0$ falls $\ket{\phi} \neq \ket{0}$
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\item antilinear im 1. Argument:
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$\braket{c \phi}{\psi} = c^* \braket{\phi}{\psi}$
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\end{enumerate}
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\item \underline{ortho}normierte Basis $\set{\ket{n}} = \set{\ket{1}, \ket{2}, ... , \ket{N}}$
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\begin{equation}
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\braket{n}{m} = \delta_{n,m}
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\end{equation}
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Jeder Vektor kann entwickelt werden
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\begin{align}
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\ket{\psi} &= \sum^{N}_{n=1}c_n \ket{n} &\left| \bra{m} \right.\\
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\braket{m}{\psi} &= \sum^N_{n=1}c_n \braket{m}{n} = c_m\\
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\ket{\psi} &= \sum^N_{n=1} \braket{n}{\psi} \ket{n}\\
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\text{mit} \ket{\phi} &= \sum^{N}_{n=1}d_n \ket{n}\\
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\braket{\phi}{\psi} &= \sum^{N}_{n=1}d_n^* \sum^{N}_{m=1}c_m \braket{n}{m} = \sum^{N}_{n=1}d_n^* c_n
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\end{align}
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\item Norm
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\begin{equation}
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\norm{\ket{psi}} \equiv \norm{\phi} \equiv \left( \braket{\phi}{\phi} \right)^\frac{1}{2} = \left( \sum_n c_n^* c_n \right)^\frac{1}{2}
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\end{equation}
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\item Schwarz'sche Ungleichung
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\begin{equation}
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\abs{\braket{\chi}{\phi}}^2 \leq \braket{\chi}{\chi}\braket{\phi}{\phi} = \norm{\chi}^2\norm{\phi}^2
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\end{equation}
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Gleichheit falls $\ket{\chi} = c \ket{\phi}$
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\end{itemize}
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\section{Lineare Operatoren}
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\subsection*{linearer Operator A}
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\begin{align}
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A \ket{\phi} &= \ket{A \phi}\\
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\text{mit} A\left(c_1\ket{\phi_1}+c_2\ket{phi2}\right) &= c_1 \ket{A\phi_1} + c_2 \ket{A\phi_2}
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\end{align}
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\subsection*{``Darstellung'' in Basis}
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\begin{align}
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\ket{A\psi} &= \sum_n c_n \ket{An} &\left| \bra{m} \right.\\
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\braket{m}{A\psi} &= \sum_n c_m \braket{m}{An} \equiv \sum_n c_n A_{m,n}c_n
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\end{align}
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$A_{m,n}$ ... ``Matrixelemente''
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\subsection*{Adjungierter Operator $A^\dagger$}
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definiert durch
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\begin{align}
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\braket{\phi}{A^\dagger \psi} &\equiv \braket{A \phi}{\psi} = \braket{\psi}{A\phi}*\\
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A^\dagger &\equiv \braket{m}{A^\dagger n} = \braket{Am}{n} = \braket{n}{Am}^* = A_{n,m}^*\\
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\left(A^\dagger \right)^\dagger &= A
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\end{align}
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Produkt $(AB)^\dagger$
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\begin{align}
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\braket{\phi}{(AB)^\dagger \psi} = \braket{(AB)\phi}{\psi} &= \braket{B\phi}{A^\dagger \psi}\\
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&= \braket{\phi}{B^\dagger A^\dagger \psi}\\
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\Rightarrow (AB)^\dagger = B^\dagger A^\dagger
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\end{align}
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Definition:
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\begin{equation}
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\text{$A$ ist hermitesch} \gdw A^\dagger = A \gdw \underbrace{A_{n,m} = A_{m,n}^*}_\text{Diagonale reel!}
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\end{equation}
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\section{Dirac-Notation}
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\begin{align}
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\ket{A\phi} &= A \ket{\phi}\\
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\bra{A^\dagger \psi} &\stackrel{\text{DN}}{\equiv} \bra{\psi A}\\
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\braket{\psi}{A \phi} &= \braket{A^\dagger \psi}{\phi} \stackrel{\text{DN}}{=} \braket{\psi A}{\phi} \equiv \dirac{\psi}{A}{\phi}
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\end{align}
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Merke: Operatoren wirken entweder normal nach rechts oder adjungiert nach links.\\[10pt]
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Jedem Vektor $\ket{\phi}$ (``ket'') wird ein dualer Vektor $\bra{\phi}$ (``bra'') zugeordnet.\\
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\begin{tabular}[2]{c|c}
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ket & bra \\ \hline
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$\ket{\phi}$ & $\bra{\phi}$ \\
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$\ket{c_1 \phi}$ & $c_1^* \bra{\phi}$
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\end{tabular}
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\subsection*{Darstellung als Spalten- und Zeilenvektoren}
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\begin{align}
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\set{\ket{1}, ..., \ket{N}} &\cequiv \set{\inlinematrix{1 \\ 0 \\ : \\ 0}, \inlinematrix{0 \\ 1 \\ : \\ 0}, ..., \inlinematrix{0 \\ : \\ 0 \\ 1}}\\
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\ket{\phi} &\cequiv \inlinematrix{c_1 \\ : \\ c_N}\\
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\bra{\phi} = \sum_n d_n^* \bra{n} &\cequiv (d_1^*, ..., d_N^*)
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\end{align}
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\begin{equation}
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\braket{\phi}{\psi} = \sum_n d_n^* c_n
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\end{equation}
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\subsection*{Operator $A$}
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\begin{align}
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A \ket{\phi} &= \sum_n c_n A \ket{n}\\
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\braket{\phi}{A \psi} &= \sum_{n, m} d_m A_{m,n} c_n\\
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&= (d_1^*, ..., d_N^*) \inlinematrix{A_{1,1} & \cdots & A_{1,N} \\ \vdots & \ddots & \vdots \\ A_{N,1} & \cdots & A_{N,N}} \inlinematrix{c_1 \\ : \\ c_N}
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\end{align}
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Natürlich ist auch
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\begin{equation}
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AB \neq BA
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\end{equation}
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.
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\section{Projektionsoperatoren}
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Seien $\hilbert_1$ und $\hilbert_2$ orthogonale Unterräume von $\hilbert$.
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Jeder ket $\ket{\phi}$ kann ein geuteig zerlegt werden:
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\begin{equation}
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\ket{\psi} = \ket{\psi_1} + \ket{\psi_2}
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\end{equation}
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\subsection*{Definition}
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\begin{equation}
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P_1\ket{\psi} \equiv \ket{\psi_1}
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\end{equation}
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\begin{itemize}
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\item $P_1$ linear $\checkmark$
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\item $P_1$ hermitesch
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\begin{align}
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\braket{\phi}{P_1^\dagger \psi} &= \braket{P_1 \phi}{ \psi}\\
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&= \braket{\psi}{P_1 \phi}^* = \braket{\psi}{\phi_1}^*\\
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&= \braket{\phi_1}{\psi} = \braket{\phi}{\psi_1}\\
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&= \braket{\phi}{P_1 \psi} \checkmark
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\end{align}
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\end{itemize}
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\subsection*{Projektion auf einen Basisvektor $\ket{n}$}
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\begin{equation}
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P_n = \ket{n}\bra{n}
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\end{equation}
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denn
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\begin{align}
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P_n \ket{\psi} &= \ket{n}\braket{n}{\psi}
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&= c_n \ket{n}
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\end{align}
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\subsection*{Zerlegung der $\one$}
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\begin{equation}
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\one = \sum_n \ket{n}\bra{n}
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\end{equation}
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denn
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\begin{align}
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\one \ket{\psi} = \ket{\psi} &= \sum_n \ket{n} \braket{n}{\psi}\\
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&= \sum_n \ket{n} c_n
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\end{align}
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Als Matrix:
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\begin{align}
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P_n &= \ket{n}\bra{n} \cequiv \inlinematrix{0\\ :\\ 1\\ :\\ 0} (0, .., 1, .., 0)\\
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\one &= \sum_n P_n = \inlinematrix{1 & & 0 \\ & \ddots & \\ 0 & & 1}
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\end{align}
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\section{Unitäre Operatoren}
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\subsection*{Definition}
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\begin{equation}
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\text{$U$ unitär} \gdw UU^\dagger = U^\dagger U = \one \gdw U^\dagger = U^{-1}
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\end{equation}
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\subsection*{Satz}
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\begin{equation}
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\text{$U$ unitär} \gdw \norm{U \phi} = \norm{\phi}
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\end{equation}
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\paragraph{Beweis:}
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\begin{align}
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\norm{\phi + c \chi}^2 &= \braket{\phi + c \chi}{\phi + c \chi} = \braket{\phi}{\phi} + 2 \re{c \braket{\phi}{\chi}} + cc^* \braket{\chi}{\chi} \\
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\norm{U (\phi + c \chi)}^2 &= \braket{U \phi}{U \phi} + 2 \re{c \braket{U \phi}{U \chi}} + cc^* \braket{U \chi}{U \chi}
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\end{align}
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\paragraph{``$\Leftarrow$'':}
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\begin{align}
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\re{c \braket{\phi}{\chi}} &= \re{c \braket{U \phi}{U \chi}}\\
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c = 1 \re{\braket{\phi}{\chi}} &= \re{\braket{U \phi}{U \chi}}\\
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c = i \im{\braket{\phi}{\chi}} &= \im{\braket{U \phi}{U \chi}}\\
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\rightarrow \braket{\phi}{\chi} &= \braket{U \phi}{U \chi}\\
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&= \dirac{\phi}{U^\dagger U}{\chi}
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\end{align}
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mit $U^\dagger U = \one$
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\paragraph{``$\Rightarrow$'':}
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\begin{equation}
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\norm{U \phi}^2 = \braket{U \phi}{U \phi} = \dirac{\phi}{U^\dagger U}{\phi} = \braket{\phi}{\phi} = \norm{\phi}^2
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\end{equation}
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\paragraph{Bemerkung:}
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Unitäre Operatoren vermitteln Basiswechsel:\\
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gegeben $\set{\ket{n}}$, definiere
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\begin{equation}
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\set{\ket{n'}} \equiv \set{U \ket{n}}
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\end{equation}
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denn:
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\begin{align}
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\braket{m'}{n'} &= \braket{Um}{Un} = \dirac{m}{U^\dagger U}{n} = \braket{m}{n}\\
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&= \delta_{m,n}
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\end{align}
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\begin{enumerate}
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\item $\ket{\psi} = \sum_n c_n \ket{n} = \sum_{n'} c_{n'} \ket{n'}$\\
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mit $c_{n'} = \braket{n'}{\psi} = \braket{Un}{\psi}$
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\item Matrixelemente $A'_{m,n} \equiv \dirac{m'}{A}{n'} = \sum_{k,l} U_{m,k}^\dagger A_{k,l} U_{l,m}$
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\end{enumerate}
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\section{Sektralzerlegung von hermitschen Operatoren}
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\subsection*{Satz}
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\begin{align}
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\text{A hermitesch} \Rightarrow &\text{(1) Eigenwerte $a_n$ sind reell}\\
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&\text{(2) Eigenvektoren zu unterschiedlichen Eigenwerten sind orthognal}
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\end{align}
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\paragraph{Beweis}
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\begin{equation}
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A \ket{a_n} = a_n \ket{a_n}
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\end{equation}
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\subparagraph{zu (1)}
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\begin{align}
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A \ket{a_n} &= a_n \ket{a_n} &\left| \bra{a_n} \right.\\
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\braket{a_n A}{a_n} &= a_n \braket{a_n}{a_n}
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\end{align}
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\begin{align}
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\braket{a_n A}{a_n} &= \braket{a_n (a_n)}{a_n} = a_n^* \braket{a_n}{a_n}
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\end{align}
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\begin{math}
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\Rightarrow a_n = a_n^* \text{ \QED}
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\end{math}
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\subparagraph{zu (2)}
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\begin{align}
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A \ket{a_m} &= a_m \ket{a_n}\\
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A \ket{a_m} &= a_m \ket{a_m}
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\end{align}
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\begin{align}
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\dirac{a_n}{A}{a_m} &= a_m \braket{a_n}{a_m}\\
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&= a_n \braket{a_n}{a_m}
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\end{align}
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\begin{math}
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a_m \neq a_n \Rightarrow \braket{a_n}{a_m} = 0 \text{ \QED}
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\end{math}
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\subsection*{Zwei Fälle}
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\begin{enumerate}
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\item alle $a_n$ unterschiedlich
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$\rightarrow \set{\ket{a_n}}$ bildet Basis
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\item nicht alle $a_n$ unterschiedlich
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Dann gibt es immer eine untäre Transformation $U$ mit
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\begin{equation}
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U^{-1}AU = \inlinematrix{a_1 & & 0 \\ & \ddots & \\ 0 & & a_N}
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\end{equation}
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$\rightarrow$ orthogonle Basis konstruiert.
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\end{enumerate}
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Sei $g(n)$ die Entartung von Eigenwert $a_n$. Im Unterraum gibt es also $g(n)$ Eigenvektoren:
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$\ket{n,r}$ mit $r=1, ..., g(n)$
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\begin{equation}
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A \ket{n,r} = a_n \ket{n,r}
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\end{equation}
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\paragraph{Projektion auf diesen Unterraum}
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\begin{equation}
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P_n = \sum_{r=1}^{g(n)}\ket{n,r}\bra{n,r}
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\end{equation}
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\begin{equation}
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\sum_n P_n = \sum_n \sum_{r=1}^{g(n)}\ket{n,r}\bra{n,r} = \one
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\end{equation}
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\begin{align}
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A \ket{\psi} = A \one \ket{\psi} &= A \sum_n \sum_{r=1}^{g(n)}\ket{n,r}\braket{n,r}{\psi}\\
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&= \sum_n a_n \sum_{r=1}^{g(n)}\ket{n,r}\braket{n,r}{\psi}
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\end{align}
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\begin{align}
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A &= \sum_n a_n \sum_{r=1}^{g(n)}\ket{n,r}\bra{n,r}\\
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&= \sum_n a_n \ket{n}\bra{n} = \sum_n a_n \ket{a_n}\bra{a_n} & \text{(nicht entartet)}
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\end{align}
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\section{Vollständiger Satz kommutierender Operatoren}
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\subsection*{Definition}
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\begin{equation}
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\text{$A,B$ kommutieren} \gdw AB -BA = [A,B] = 0
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\end{equation}
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\subsection*{Satz}
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\begin{equation}
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\text{$A$, $B$ hermitesch und $[A,B] = 0$} \Rightarrow \text{es existiert eine gemeinsame Eigenbasis}
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\end{equation}
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\paragraph{Beweis}
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\begin{align}
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A \ket{n,r} &= a_n \ket{n,r} & B_\rightarrow\\
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BA \ket{n,r} &= a_n B \ket{n,r}\\
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A (B \ket{n,r}) &= a_n (B \ket{n,r})
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\end{align}
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$\rightarrow$ $B \ket{n,r}$ liegt im Untrraum zu $a_n$
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\begin{description}
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\item[Fall (1)] $a_n$ nicht entartet ($\ket{n,r} \equiv \ket{n}$)
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\begin{equation}
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B \ket{n} = b_n \ket{n}
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\end{equation}
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\item[Fall (2)] $a_n$ entartet
|
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|
|
\begin{equation}
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\bra{m,s} \cdot B \cdot \ket{n,r} = B_{s,r}^{(n)}
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\end{equation}
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|
|
\begin{equation}
|
||
|
|
B \cequiv \inlinematrix{\boxed{B^{(1)}} & & 0 \\ & \boxed{B^{(2)}} & & \\ & & \boxed{B} & \\ 0 & & & \boxed{B^{(4)}}} \rightarrow \text{kann diagonalisiert werden in jedem Kästchen}
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|
|
\end{equation}
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Falls $B^{(n)}$ entartet, gibt es einen dritten Opertor $C$ mit $[A,C] = [B,C] = 0$.
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|
\end{description}
|
||
|
|
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\subsection*{Definition}
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||
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|
Das Ensemble $set{A^1, ..., A^M}$ wechselseitig kommutierender Operatoren, deren Eigenwerte $(a_{n_1}^1, ..., a_{n_N}^M)$ mit zugehörigen Eigenvektoren $\set{\ket{a_{n_1}^1, ..., a_{n_N}^M}}$ mit
|
||
|
|
\begin{equation}
|
||
|
|
A^k \ket{a_{n_1}^1, ..., a_{n_N}^M} = a_{n_k}^k \ket{a_{n_1}^1, ..., a_{n_N}^M}
|
||
|
|
\end{equation}
|
||
|
|
eine eindeutige Basis definieren einen vollständigen Satz kommutierender Operatoren (VSKO, CSCO).
|
||
|
|
|
||
|
|
\section{Operatorfunktionen}
|
||
|
|
Sei $A$ Operator, definiere $f(A)$
|
||
|
|
\begin{enumerate}
|
||
|
|
\item über die Potenzreihe
|
||
|
|
\begin{equation}
|
||
|
|
f(A) = \sum_{n=0}^{\inf} \frac{1}{n!} f^{(n)}(n) A^n
|
||
|
|
\end{equation}
|
||
|
|
$\lboxed{\text{Bsp: } e^A = \sum_{n=0}^{\inf} \frac{1}{n!} A^n}$
|
||
|
|
\item für hermitesche
|
||
|
|
\begin{align}
|
||
|
|
f(A) &= \sum_{n=1}^N f(a_n) \ket{a_n}\bra{a_n} & \text{(nicht entartet)}\\
|
||
|
|
&= \sum_n \sum_{r=1}^{g(n)} f(a_n) \ket{n,r}\bra{n,r}
|
||
|
|
\end{align}
|
||
|
|
beachte: $e^A \cdot e^B \neq e^{A+B}$
|
||
|
|
\end{enumerate}
|