Liouville-von-Neumann-Gleichung

$\dot{\rho }=ℒ\rho =-\frac{i}{\hslash }\left[H,\rho \right]$ mit

$\rho$	Dichteoperator
H	Hamiltonoperator
$[\mathrm{\_},\mathrm{\_}]$	Kommutator

Herleitung

Schrödingergleichung

$𝔦{\partial }_t\Psi (t)=\hat{H}\Psi (t)$

Dirac Notation

Ket: $\begin{array}{cc}& |𝔦{\partial }_t\Psi \left(t\right)⟩=|\hat{H}\Psi \left(t\right)⟩\\ & 𝔦{\partial }_t|\Psi \left(t\right)⟩=\hat{H}|\Psi \left(t\right)⟩\Rightarrow {\partial }_t|\Psi \left(t\right)⟩=-𝔦\hat{H}|\Psi \left(t\right)⟩\\ \end{array}$

Bra:

$\begin{array}{cc}& ⟨𝔦{\partial }_t\Psi \left(t\right)|=⟨\hat{H}\Psi \left(t\right)|\\ & \text{-}𝔦{\partial }_t⟨\Psi \left(t\right)|=\hat{H}⟨\Psi \left(t\right)|,\hat{H}={\hat{H}}^{+}\Rightarrow {\partial }_t⟨\Psi \left(t\right)|=𝔦\hat{H}⟨\Psi \left(t\right)|\\ \end{array}$

Dichtematrix

$\hat{\rho }=|\Psi \left(t\right)⟩⟨\Psi \left(t\right)|$

Einzelnachweise

Wikipedia-Eintrag