65 lines
2.0 KiB
TeX
65 lines
2.0 KiB
TeX
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\chapter{Rotationsinvarianz in d=3}
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\section{Drehimpulsalgebra}
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Drehung mit dem Winkel $\phi$ um $\vec{n}$:
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\begin{equation}
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\ket{\tilde{\psi}} = D(\phi,\vec{n})\ket{\psi}
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\end{equation}
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mit
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\begin{equation}
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D(\phi,\vec{n}) = 1 - i\frac{\phi}{\hbar} J_{\vec{n}} + O(\phi^2)
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\end{equation}
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In () hatten wir die Relation
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\begin{equation}
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[J_x,J_y] = i\hbar J_z
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\end{equation}
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(etc. zyclisch). Diese Vertauschungsrelation bestimmt das Spektrum der $J$-Operatoren vollständig. Wir definieren ein $J^2$ zu:
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\begin{equation}
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J^2 = \vec{J}^2 = J_x^2 + J_y^2 + J_z^2
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\end{equation}
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und daraus folgt $\forall \alpha = x,y,z$
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\begin{equation}
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\left[ J^2, J_\alpha \right] = 0
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\end{equation}
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also haben $J^2$ und $J_z$ gemeinsame Eigenvektoren.
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\begin{align}
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J^2 \ket{\alpha,\beta} &= \alpha \ket{\alpha,\beta}\\
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J_z \ket{\alpha,\beta} &= \beta \ket{\alpha,\beta}
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\end{align}
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In Anlehnung an Erzeuger und Vernichter definieren wir:
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\begin{equation}
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J_\pm \equiv J_x \pm iJ_y
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\end{equation}
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mit dem Kommutator
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\begin{align}
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[J_z, J_\pm] &= [J_z,J_x + iJ_y]\\
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&= i\hbar J_y \pm i(-i\hbar) J_x
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&= \pm \hbar J_\pm
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\end{align}
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und
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\begin{equation}
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[J^2,J_\pm] = 0
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\end{equation}
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Vergleiche Harmonischen Oszillator:
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\begin{equation}
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[\nOp,\aDs] = -\aDs; ~ [\nOp,\aCr] = \aCr
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\end{equation}
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mit $J_\pm$ ist dann
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\begin{align}
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J_z J_+ \ket{\alpha,\beta} &= (J_+ J_z + i\hbar J_+) \ket{\alpha,\beta}\\
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&= (\beta + \hbar) J_+ \ket{\alpha,\beta}\\
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J_z J_- \ket{\alpha,\beta} &= (\beta - \hbar) J_- \ket{\alpha,\beta}
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\end{align}
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und
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\begin{equation}
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J^2 J_+ \ket{\alpha,\beta} = J_+ J^2 \ket{\alpha,\beta} = \alpha J_+ \ket{\alpha,\beta}
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\end{equation}
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also
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\begin{align}
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J_+ \ket{\alpha,\beta} &= c_+(\alpha,\beta) \ket{\alpha,\beta + 1}\\
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J_- \ket{\alpha,\beta} &= c_-(\alpha,\beta) \ket{\alpha,\beta - 1}
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\end{align}
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$\beta$-Spektrum ist eingeschränkt wegen:
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\begin{equation}
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0 \leq \dirac{\alpha,\beta}{J_x^2 + J_y^2}{\alpha,\beta} = \dirac{\alpha,\beta}{J^2-J_z^2}{\alpha,\beta} = (\alpha-\beta)^2 \underbrace{\braket{\alpha,\beta}{\alpha,\beta}}_{\geq 0}
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\end{equation}
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